Research Article Vibration Analysis of Hollow Tapered...

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Research Article Vibration Analysis of Hollow Tapered Shaft Rotor P. M. G. Bashir Asdaque and R. K. Behera Department of Mechanical Engineering, National Institute of Technology Rourkela, Odisha 769008, India Correspondence should be addressed to P. M. G. Bashir Asdaque; [email protected] Received 27 December 2013; Accepted 6 March 2014; Published 28 April 2014 Academic Editor: Abdelkrim Khelif Copyright © 2014 P. M. G. B. Asdaque and R. K. Behera. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Shaſts or circular cross-section beams are important parts of rotating systems and their geometries play important role in rotor dynamics. Hollow tapered shaſt rotors with uniform thickness and uniform bore are considered. Critical speeds or whirling frequency conditions are computed using transfer matrix method and then the results were compared using finite element method. For particular shaſt lengths and rotating speeds, response of the hollow tapered shaſt-rotor system is determined for the establishment of dynamic characteristics. Nonrotating conditions are also considered and results obtained are plotted. 1. Introduction Shaſt is a major component of any rotating system, used to transmit torque and rotation. Hence the study shaſt-rotor systems has been the concern of researchers for more than a century and will continue to persist as an active area of research and analysis in near future. Geometry of shaſt is of the main concern during the study of any rotating system. Most papers related to shaſt-rotor systems consider cylindri- cal shaſt elements for study and analysis of rotating systems. e first idea of transfer matrix method (TMM) was com- piled by Holzer for finding natural frequencies of torsional systems and later adapted by Myklestad [1, 2] for computing natural frequencies of airplane wings, coupled in bending and torsion. Gyroscopic moments were first introduced by Prohl [3] for rotor-bearing system analysis. Lund [4] used complex variables as the next significant advancement in the method. An improved method for calculating critical speeds and rotor stability of turbo machinery was investigated by Murphy and Vance [5]. Whalley and Abdul-Ameer [6] used frequency response analysis for particular profiled shaſts to study dynamic response of distributed-lumped shaſt rotor system. ey studied the system behavior in terms of frequency response for the shaſts with diameters which are functions of their lengths. ey derived an analytical method which uses Euler-Bernoulli beam theory in combination with TMM. On the other hand, there are large numbers of numerical applications of finite element techniques for the calculation of whirling and the computation of maximum dynamic magnitude. In this regard, Ruhl and Booker [7] modeled the distributed parameter turbo rotor systems using finite element method (FEM). Nelson and McVaugh [8] reduced large number of eigenvalues and eigenvectors identified, following finite element analysis, and the erroneous modes of vibration predicted were eliminated. Nelson [9] again formulated the equations of motion for a uniform rotating shaſt element using deformation shape functions developed from Timoshenko beam theory including the effects of translational and rotational inertia, gyroscopic moments, bending and shear deformation, and axial load. Greenhill et al. [10] derived equation of motion for a conical beam finite element form Timoshenko beam theory and include effects of translational and rotational inertia, gyroscopic moments, bending and shear deformation, axial load, and internal damping. Genta and Gugliotta [11] also analyzed element with annular cross-section based on Timoshenko beam theory having two degrees of freedom at each node. Mohiuddin and Khulief [12] derived a finite element model of a tapered rotating cracked shaſt for modal analysis and dynamic modeling of a rotor-bearing system, based on Tim- oshenko beam theory, that is, included shear deformation and rotary inertia. Rouch and Kao [13] presented numerically integrated formulation of a tapered beam element for rotor dynamics. Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2014, Article ID 410851, 14 pages http://dx.doi.org/10.1155/2014/410851

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Research ArticleVibration Analysis of Hollow Tapered Shaft Rotor

P M G Bashir Asdaque and R K Behera

Department of Mechanical Engineering National Institute of Technology Rourkela Odisha 769008 India

Correspondence should be addressed to P M G Bashir Asdaque pmgbashir2001gmailcom

Received 27 December 2013 Accepted 6 March 2014 Published 28 April 2014

Academic Editor Abdelkrim Khelif

Copyright copy 2014 P M G B Asdaque and R K Behera This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Shafts or circular cross-section beams are important parts of rotating systems and their geometries play important role in rotordynamics Hollow tapered shaft rotors with uniform thickness and uniform bore are considered Critical speeds or whirlingfrequency conditions are computed using transfer matrix method and then the results were compared using finite elementmethod For particular shaft lengths and rotating speeds response of the hollow tapered shaft-rotor system is determined for theestablishment of dynamic characteristics Nonrotating conditions are also considered and results obtained are plotted

1 Introduction

Shaft is a major component of any rotating system used totransmit torque and rotation Hence the study shaft-rotorsystems has been the concern of researchers for more thana century and will continue to persist as an active area ofresearch and analysis in near future Geometry of shaft is ofthe main concern during the study of any rotating systemMost papers related to shaft-rotor systems consider cylindri-cal shaft elements for study and analysis of rotating systemsThe first idea of transfer matrix method (TMM) was com-piled by Holzer for finding natural frequencies of torsionalsystems and later adapted by Myklestad [1 2] for computingnatural frequencies of airplanewings coupled in bending andtorsion Gyroscopic moments were first introduced by Prohl[3] for rotor-bearing system analysis Lund [4] used complexvariables as the next significant advancement in the methodAn improvedmethod for calculating critical speeds and rotorstability of turbo machinery was investigated by Murphy andVance [5] Whalley and Abdul-Ameer [6] used frequencyresponse analysis for particular profiled shafts to studydynamic response of distributed-lumped shaft rotor systemThey studied the system behavior in terms of frequencyresponse for the shafts with diameters which are functions oftheir lengths They derived an analytical method which usesEuler-Bernoulli beam theory in combination with TMMOn the other hand there are large numbers of numerical

applications of finite element techniques for the calculationof whirling and the computation of maximum dynamicmagnitude In this regard Ruhl and Booker [7] modeledthe distributed parameter turbo rotor systems using finiteelement method (FEM) Nelson and McVaugh [8] reducedlarge number of eigenvalues and eigenvectors identifiedfollowing finite element analysis and the erroneous modesof vibration predicted were eliminated Nelson [9] againformulated the equations of motion for a uniform rotatingshaft element using deformation shape functions developedfrom Timoshenko beam theory including the effects oftranslational and rotational inertia gyroscopic momentsbending and shear deformation and axial load Greenhillet al [10] derived equation of motion for a conical beamfinite element form Timoshenko beam theory and includeeffects of translational and rotational inertia gyroscopicmoments bending and shear deformation axial load andinternal damping Genta and Gugliotta [11] also analyzedelement with annular cross-section based on Timoshenkobeam theory having two degrees of freedom at each nodeMohiuddin and Khulief [12] derived a finite element modelof a tapered rotating cracked shaft for modal analysis anddynamic modeling of a rotor-bearing system based on Tim-oshenko beam theory that is included shear deformationand rotary inertia Rouch andKao [13] presented numericallyintegrated formulation of a tapered beam element for rotordynamics

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2014 Article ID 410851 14 pageshttpdxdoiorg1011552014410851

2 Advances in Acoustics and Vibration

In this era of machines tapered shafts are widely used forrotating systems Using the approach of Whalley and Abdul-Ameer [6] the dynamic analysis of hollow tapered shaft-rotorhas been done Later the results obtainedwere comparedwiththat obtained fromfinite elementmethodThe effect of lengthand speed on the dynamic analysis of the hollow taperedshaft-rotor system is also clearly shown Frequency responseof the rotor system for an impulse of unit force at the freeend is determined in terms of critical speeds for various rotorspeeds and shaft lengths

2 Transfer Matrix Method

21 Shaft Model The shaft model is derived in matrix formWhalley and Abdul-Ameer [6] as

1198842

1205792

1198721199102

1198761199102

= 119865 (119904)

1198841

1205791

1198721199101

1198761199101

(1)

where

119865 (119904) =

[[[[[[[[[[[[[

[

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

sinradicΓ (119897)119897 + sinhradicΓ (119897)1198972radicΓ (119897)

minusradicΓ (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)2

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

Γ (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2119862 (0)

radicΓ (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)2119862 (0)

minusΓ (0)radicΓ (0) (sinradicΓ (119897)119897 + sinhradicΓ (119897)119897)2119862 (0)

Γ (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2119862 (0)

119862 (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2Γ (119897)

119862 (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)

2Γ (0)radicΓ (0)

minus119862 (0) (sinradicΓ (119897)119897 + sinhradicΓ (119897)119897)

2radicΓ (0)

119862 (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2Γ (0)

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

sinradicΓ (119897)119897 + sinhradicΓ (119897)1198972radicΓ (0)

minusradicΓ (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)2

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

]]]]]]]]]]]]]]]

]

(2)

The complete derivation is present in [6]

22 Rigid Disk The output vector from the shaft will becomethe input for the rigid disk model as shown in Figure 1 thatis for disk model we have

1199103(119904) = 119910

2(119904)

1205793(119904) = 120579

2(119904)

1198721199103(119904) = 119872

1199102(119904) minus 119869Ω119904120579

2(119904)

1198761199103(119904) = 1198981199042119910

2(119904) + 119876

1199102(119904)

(3)

where (1199103(119904) 1199101(119904)) (120579

3(119904) 1205791(119904)) (119872

1199103(119904)119872

1199101(119904)) and

(1198761199103(119904) 1198761199101(119904)) are the deflections slopes bending

moments and shear forces at the free and fixed endrespectively

Hence writing in matrix form we have

(1199103(119904) 1205793(119904) 119872

1199103(119904) 119876

1199103(119904))119879

= 119877 (119904) (1199102(119904) 1205792(119904) 119872

1199102(119904) 119876

1199102(119904))

(4)

where

119877 (119904) =[[[

[

1 0 0 00 1 0 00 minus119869Ω119904 1 01198981199042 0 0 1

]]]

]

(5)

Form transfer matrix method [6]

119867(119904) = 119877 (119904) 119865 (119904) (6)

where

119867(119904) = [11986711

11986712

11986721

11986722

] (7)

and input-output vectors relationship is given by

[[[

[

1199103

1205793

1198721199103

1198761199103

]]]

]

= 119867 (119904)[[[

[

1199101

1205791

1198721199101

1198761199101

]]]

]

(8)

After applying the boundary conditions for cantilever beamdeflection at the free end is obtained and hence leads totransfer function

Advances in Acoustics and Vibration 3

Shaft length

x + dx

y(t x)A(x

x

)

120579(t x)

qy(t x)

my(t x)

my(t x + dx)

qy(t x + dx)

y(t x + dx)

Figure 1 Vibrating shaft element

3 Finite Element Method

The vector of nodal displacements is given by

119902 =

119881119882120573Γ

(9)

So each element is having eight degrees of freedom

31 Rigid Disc Rigid disk is having two translations and tworotations in 119884 and 119885 direction respectively (considering 119883coordinate in axial direction) For constant spin conditionthe Lagrangian equation of motion is given by

([119872119879119889] + [119872119877

119889]) 11990210158401015840119889 minus Ω [119866

119889] 1199021015840119889 = 119876119889 (10)

where

[119872119879119889] =

[[[

[

119898119889

0 0 00 1198981198890 0

0 0 0 00 0 0 0

]]]

]

[119872119877119889] =

[[[

[

0 0 0 00 0 0 00 0 119868

1198630

0 0 0 119868119863

]]]

]

[119866119889] =

[[[

[

0 0 0 00 0 0 00 0 0 minus119868

119875

0 0 119868119875

0

]]]

]

(11)

The forcing term may include mass unbalance and otherexternal forces

32 Finite Shaft-Rotor Element The rotor-shaft element con-sidered here has eight degrees of freedom that is four degreesof freedom per node as in Nelson and McVaugh [8] For

constant spin condition the Lagrangian equation of motionis given by

([119872119879119890] + [119872119877

119890]) 11990210158401015840119890 minus Ω [119866

119890] 1199021015840119890 + ([119870

119890]) 119902119890 = 119876119890

(12)

where

[119872119879119890] = int

119897

0

120588119860 (119909) [119873]119879 [119873] 119889119909

[119872119877119890] = int

119897

0

119868119863(119909) [120601]119879 [120601] 119889119909

[119873119890] = int

119897

0

119868119875(119909) [120601

Γ]119879 [120601120573] 119889119909

[119870119890] = int

119897

0

119864119868 (119909) [11987310158401015840]119879

[11987310158401015840] 119889119909

[119866119890] = [119873

119890] minus [119873

119890]119879

(13)

Except skew-symmetric gyroscopic matrix [119866119890] others are

symmetric matrices Since the element is linearly taperedarea and inertias are the function of the shaft-rotor lengthThe translational shape function is given by

[119873] = [1198731

0 0 11987321198733

0 0 1198734

0 1198731minus1198732

0 0 1198733minus1198734

0] (14)

where

1198731= 1 minus 3

1199092

1198972+ 2

1199093

1198973

1198732= 119909 minus 2

1199092

119897+1199093

1198972

1198733= 3

1199092

1198972minus 2

1199093

1198973

1198734= minus

1199092

119897+1199093

1198972

(15)

The rotational shape function is given by

[Φ] = [

[

lfloorΦ120573rfloor

lfloorΦΓrfloor]

]

= [0 minus1198731015840

111987310158402

0 0 minus1198731015840311987310158404

011987310158401

0 0 1198731015840211987310158403

0 0 11987310158404

]

(16)

The element matrices are assembled together to get theequation of motion for the complete system

4 Numerical Results

41 Rotating Condition By using the transfer matrixapproach as in the paper of Whalley and Abdul-Ameer wewill ultimately get the transfer function which will be plottedThe FEM will be applied and then is compared with theTMM approach of Whalley and Abdul-Ameer [6]

4 Advances in Acoustics and Vibration

Example 1 Let us consider a cantilever tubular shaft withuniform thickness and a disc at the free end with downwardunit force 119875 on the disc as shown in Figure 2 The defaultvalues of various parameters are tabulated in Table 1

Transfer function for tubular shaft with constant thick-ness as shown in Figure 2 for default values is given by

1344119904 + 1016 times 104

1199043 + 75641199042 + 8401 times 105119904 + 1589 times 109 (17)

Bode plots for different lengths and rotating speed have beenplotted using MATLAB software as shown in Figures 3 and4

Applying FEM on the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement is shown in Figure 5

Table 1 Various parameters of the system shown in Figure 2 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Thickness of the hollow shaft 119905 (m) 0002Beginning radius 119877

0(m) 00050

End radius 1198771(m) 00037

The stiffness matrix for hollow tapered shaft element withuniform thickness is given by is given by

119870119899=

[[[[[[[[[[

[

11989611989911

11989611989921

11989611989922

11989611989931

11989611989932

11989611989933

sym11989611989941

11989611989942

11989611989943

11989611989944

11989611989951

11989611989952

11989611989953

11989611989954

11989611989955

11989611989961

11989611989962

11989611989963

11989611989964

11989611989965

11989611989966

11989611989971

11989611989972

11989611989973

11989611989974

11989611989975

11989611989976

11989611989977

11989611989981

11989611989982

11989611989983

11989611989984

11989611989985

11989611989986

11989611989987

11989611989988

]]]]]]]]]]

]

(18)

where elements of the stiffness matrices are

11989611989911

= 3120587119864119905 (711987730+ 3119877201198771minus 121198772

0119905 + 3119877

011987721minus 611987701198771119905

+ 1011987701199052 + 71198773

1minus 121198772

1119905 + 10119877

11199052 minus 51199053)

times (51198973)minus1

11989611989941

= 120587119864119905 (3011987730+ 121198772

01198771minus 511198772

0119905 + 6119877

011987721minus 18119877

01198771119905

+ 4011987701199052 + 121198773

1minus 211198772

1119905 + 20119877

11199052 minus 151199053)

times (101198972)minus1

11989611989981

= 120587119864119905 (1211987730+ 6119877201198771minus 211198772

0119905 + 12119877

011987721minus 18119877

01198771119905

+ 2011987701199052 + 301198773

1minus 511198772

1119905 + 40119877

11199052 minus 151199053)

times (101198972)minus1

11989611989933

= 120587119864119905 (1111987730+ 5119877201198771minus 191198772

0119905 + 2119877

011987721minus 711987701198771119905

+ 1511987701199052 + 21198773

1minus 411987721119905 + 5119877

11199052 minus 51199053)

times (5119897)minus1

11989611989973

= 120587119864119905 (811987730+ 2119877201198771minus 131198772

0119905 + 2119877

011987721minus 411987701198771119905

+ 1011987701199052 + 81198773

1minus 131198772

1119905 + 10119877

11199052 minus 51199053)

times (10119897)minus1

11989611989977

= 120587119864119905 (211987730+ 2119877201198771minus 411987720119905 + 5119877

011987721minus 711987701198771119905

+ 511987701199052 + 111198773

1minus 191198772

1119905 + 15119877

11199052 minus 51199053)

times (5119897)minus1

11989611989921

= 11989611989942

= 11989611989982

= 11989611989931

= 0

11989611989943

= 11989611989953

= 11989611989983

= 11989611989964

= 11989611989974

= 0

11989611989952

= 11989611989953

= 11989611989965

= 11989611989975

= 11989611989961

= 0

11989611989965

= 11989611989986

= 11989611989971

= 11989611989987

= 0

11989611989922

= 11989611989955

= 11989611989966

= 11989611989911 119896

11989951= 11989611989962

= minus11989611989911

11989611989963

= 11989611989941 119896

11989932= 11989611989954

= minus11989611989941

11989611989976

= 11989611989981 119896

11989972= 11989611989985

= minus11989611989981

11989611989944

= 11989611989933 119896

11989984= 11989611989973

(19)Translational mass matrix is given by

Advances in Acoustics and Vibration 5

119872119899119905=

[[[[[[[[[[

[

11989811989911990511

11989811989911990521

11989811989911990522

11989811989911990531

11989811989911990532

11989811989911990533

sym11989811989911990541

11989811989911990542

11989811989911990543

11989811989911990544

11989811989911990551

11989811989911990552

11989811989911990553

11989811989911990554

11989811989911990555

11989811989911990561

11989811989911990562

11989811989911990563

11989811989911990564

11989811989911990565

11989811989911990566

11989811989911990571

11989811989911990572

11989811989911990573

11989811989911990574

11989811989911990575

11989811989911990576

11989811989911990577

11989811989911990581

11989811989911990582

11989811989911990583

11989811989911990584

11989811989911990585

11989811989911990586

11989811989911990587

11989811989911990588

]]]]]]]]]]

]

(20)

where elements of translational mass matrix are given by

11989811989911990511

=120587120588119897119905 (20119877

0+ 61198771minus 13119905)

35

11989811989911990541

=1205871205881199051198972 (15119877

0+ 71198771minus 11119905)

210

11989811989911990551

=9120587120588119897119905 (119877

0+ 1198771minus 119905)

70

11989811989911990581

=minus (1205871205881199051198972 (14119877

0+ 12119877

1minus 13119905))

420

11989811989911990533

=1205871205881198973119905 (5119877

0+ 31198771minus 4119905)

420

11989811989911990563

=minus (1205871205881198972119905 (12119877

0+ 14119877

1minus 13119905))

420

11989811989911990573

=minus (1205871205881198973119905 (119877

0+ 1198771minus 119905))

140

11989811989911990576

=1205871205881198972119905 (7119877

0+ 15119877

1minus 11119905)

210

11989811989911990577

=1205871205881198973119905 (3119877

0+ 51198771minus 4119905)

420

11989811989911990521

= 11989811989911990542

= 11989811989911990582

= 11989811989911990531

= 0

11989811989911990543

= 11989811989911990553

= 11989811989911990583

= 11989811989911990564

= 11989811989911990574

= 0

11989811989911990552

= 11989811989911990553

= 11989811989911990575

= 11989811989911990561

= 0

11989811989911990565

= 11989811989911990586

= 11989811989911990571

= 11989811989911990587

= 0

11989811989911990522

= 11989811989911990555

= 11989811989911990566

= 11989811989911990511

11989811989911990551

= 11989811989911990562

11989811989911990532

= minus 11989811989911990541

11989811989911990563

= 11989811989911990554

11989811989911990572

= minus 11989811989911990581

11989811989911990544

= 11989811989911990533

11989811989911990584

= 11989811989911990573

11989811989911990585

= minus11989811989911990576

(21)

Rotational mass matrix is given by

119872119899119903=

[[[[[[[[[[

[

11989811989911990311

11989811989911990321

11989811989911990322

11989811989911990331

11989811989911990332

11989811989911990333

sym11989811989911990341

11989811989911990342

11989811989911990343

11989811989911990344

11989811989911990351

11989811989911990352

11989811989911990353

11989811989911990354

11989811989911990355

11989811989911990361

11989811989911990362

11989811989911990363

11989811989911990364

11989811989911990365

11989811989911990366

11989811989911990371

11989811989911990372

11989811989911990373

11989811989911990374

11989811989911990375

11989811989911990376

11989811989911990377

11989811989911990381

11989811989911990382

11989811989911990383

11989811989911990384

11989811989911990385

11989811989911990386

11989811989911990387

11989811989911990388

]]]]]]]]]]

]

(22)

where elements of rotational mass matrix are given by

11989811989911990311

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (70119897)minus1

11989811989911990341

= 1205871205881199052 (minus811987720+ 611987701198771+ 201198772

1

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990351

= minus (31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052))

times (70119897)minus1

11989811989911990333

= 1205871205881199052119897 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (210)minus1

11989811989911990363

= 1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

6 Advances in Acoustics and Vibration

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990373

= minus (1205871205881199052119897 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (840)minus1

11989811989911990376

= 1205871205881199052 (2011987720+ 16119877

01198771

minus 281198770119905 minus 81198772

1+ 71199052)

times (280)minus1

11989811989911990377

= 1205871205881199052119897 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (210)minus1

11989811989911990321

= 11989811989911990342

= 11989811989911990382

= 11989811989911990331

= 11989811989911990343

= 0

11989811989911990353

= 11989811989911990383

= 11989811989911990364

= 11989811989911990374

= 0

11989811989911990352

= 11989811989911990353

= 11989811989911990375

= 11989811989911990361

= 0

11989811989911990365

= 11989811989911990386

= 11989811989911990371

= 11989811989911990387

= 0

11989811989911990322

= 11989811989911990355

= 11989811989911990366

= 11989811989911990311

11989811989911990351

= 11989811989911990362

11989811989911990332

= minus11989811989911990341

11989811989911990363

= 11989811989911990354

11989811989911990381

= 11989811989911990376

11989811989911990372

= minus11989811989911990376

11989811989911990344

= 11989811989911990333

11989811989911990384

= 11989811989911990373

11989811989911990385

= minus11989811989911990376

(23)

Gyroscopic matrix is given by

119866119899119890=

[[[[[[[[[[

[

11989211989911989011

11989211989911989021

11989211989911989022

11989211989911989031

11989211989911989032

11989211989911989033

skewsym11989211989911989041

11989211989911989042

11989211989911989043

11989211989911989044

11989211989911989051

11989211989911989052

11989211989911989053

11989211989911989054

11989211989911989055

11989211989911989061

11989211989911989062

11989211989911989063

11989211989911989064

11989211989911989065

11989211989911989066

11989211989911989071

11989211989911989072

11989211989911989073

11989211989911989074

11989211989911989075

11989211989911989076

11989211989911989077

11989211989911989081

11989211989911989082

11989211989911989083

11989211989911989084

11989211989911989085

11989211989911989086

11989211989911989087

11989211989911989088

]]]]]]]]]]

]

(24)

where the elements of the gyroscopic matrix are given by

11989211989911989021

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (35l)minus1

11989211989911989031

= minus (1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

minus 281198771119905 + 71199052))

times (140)minus1

11989211989911989071

= minus (1205871205881199052 (2011987720+ 16119877

01198771minus 28119877

0119905

minus 811987721+ 71199052))

times (140)minus1

11989211989911989043

= 1205871205881198971199052 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (105)minus1

11989211989911989083

= minus (1205871205881198971199052 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (420)minus1

11989211989911989087

= 1205871205881198971199052 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (105)minus1

11989211989911989052

= 11989211989911989065

= 11989211989911989021

11989211989911989061

= minus11989211989911989021

11989211989911989042

= 11989211989911989053

= 11989211989911989064

= 11989211989911989031

11989211989911989082

= 11989211989911989071

11989211989911989086

= 11989211989911989075

= minus11989211989911989071

11989211989911989075

= minus11989211989911989083

11989211989911989011

= 11989211989911989041

= 11989211989911989051

= 11989211989911989081

= 11989211989911989022

= 11989211989911989032

= 0

11989211989911989062

= 11989211989911989072

= 11989211989911989033

= 11989211989911989063

= 0

11989211989911989073

= 11989211989911989044

= 11989211989911989054

= 11989211989911989084

= 11989211989911989055

= 11989211989911989085

= 0

11989211989911989066

= 11989211989911989076

= 11989211989911989077

= 11989211989911989088

= 0

(25)

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 2: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

2 Advances in Acoustics and Vibration

In this era of machines tapered shafts are widely used forrotating systems Using the approach of Whalley and Abdul-Ameer [6] the dynamic analysis of hollow tapered shaft-rotorhas been done Later the results obtainedwere comparedwiththat obtained fromfinite elementmethodThe effect of lengthand speed on the dynamic analysis of the hollow taperedshaft-rotor system is also clearly shown Frequency responseof the rotor system for an impulse of unit force at the freeend is determined in terms of critical speeds for various rotorspeeds and shaft lengths

2 Transfer Matrix Method

21 Shaft Model The shaft model is derived in matrix formWhalley and Abdul-Ameer [6] as

1198842

1205792

1198721199102

1198761199102

= 119865 (119904)

1198841

1205791

1198721199101

1198761199101

(1)

where

119865 (119904) =

[[[[[[[[[[[[[

[

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

sinradicΓ (119897)119897 + sinhradicΓ (119897)1198972radicΓ (119897)

minusradicΓ (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)2

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

Γ (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2119862 (0)

radicΓ (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)2119862 (0)

minusΓ (0)radicΓ (0) (sinradicΓ (119897)119897 + sinhradicΓ (119897)119897)2119862 (0)

Γ (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2119862 (0)

119862 (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2Γ (119897)

119862 (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)

2Γ (0)radicΓ (0)

minus119862 (0) (sinradicΓ (119897)119897 + sinhradicΓ (119897)119897)

2radicΓ (0)

119862 (0) (cosradicΓ (119897)119897 minus coshradicΓ (119897)119897)2Γ (0)

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

sinradicΓ (119897)119897 + sinhradicΓ (119897)1198972radicΓ (0)

minusradicΓ (0) (sinradicΓ (119897)119897 minus sinhradicΓ (119897)119897)2

cosradicΓ (119897)119897 + coshradicΓ (119897)1198972

]]]]]]]]]]]]]]]

]

(2)

The complete derivation is present in [6]

22 Rigid Disk The output vector from the shaft will becomethe input for the rigid disk model as shown in Figure 1 thatis for disk model we have

1199103(119904) = 119910

2(119904)

1205793(119904) = 120579

2(119904)

1198721199103(119904) = 119872

1199102(119904) minus 119869Ω119904120579

2(119904)

1198761199103(119904) = 1198981199042119910

2(119904) + 119876

1199102(119904)

(3)

where (1199103(119904) 1199101(119904)) (120579

3(119904) 1205791(119904)) (119872

1199103(119904)119872

1199101(119904)) and

(1198761199103(119904) 1198761199101(119904)) are the deflections slopes bending

moments and shear forces at the free and fixed endrespectively

Hence writing in matrix form we have

(1199103(119904) 1205793(119904) 119872

1199103(119904) 119876

1199103(119904))119879

= 119877 (119904) (1199102(119904) 1205792(119904) 119872

1199102(119904) 119876

1199102(119904))

(4)

where

119877 (119904) =[[[

[

1 0 0 00 1 0 00 minus119869Ω119904 1 01198981199042 0 0 1

]]]

]

(5)

Form transfer matrix method [6]

119867(119904) = 119877 (119904) 119865 (119904) (6)

where

119867(119904) = [11986711

11986712

11986721

11986722

] (7)

and input-output vectors relationship is given by

[[[

[

1199103

1205793

1198721199103

1198761199103

]]]

]

= 119867 (119904)[[[

[

1199101

1205791

1198721199101

1198761199101

]]]

]

(8)

After applying the boundary conditions for cantilever beamdeflection at the free end is obtained and hence leads totransfer function

Advances in Acoustics and Vibration 3

Shaft length

x + dx

y(t x)A(x

x

)

120579(t x)

qy(t x)

my(t x)

my(t x + dx)

qy(t x + dx)

y(t x + dx)

Figure 1 Vibrating shaft element

3 Finite Element Method

The vector of nodal displacements is given by

119902 =

119881119882120573Γ

(9)

So each element is having eight degrees of freedom

31 Rigid Disc Rigid disk is having two translations and tworotations in 119884 and 119885 direction respectively (considering 119883coordinate in axial direction) For constant spin conditionthe Lagrangian equation of motion is given by

([119872119879119889] + [119872119877

119889]) 11990210158401015840119889 minus Ω [119866

119889] 1199021015840119889 = 119876119889 (10)

where

[119872119879119889] =

[[[

[

119898119889

0 0 00 1198981198890 0

0 0 0 00 0 0 0

]]]

]

[119872119877119889] =

[[[

[

0 0 0 00 0 0 00 0 119868

1198630

0 0 0 119868119863

]]]

]

[119866119889] =

[[[

[

0 0 0 00 0 0 00 0 0 minus119868

119875

0 0 119868119875

0

]]]

]

(11)

The forcing term may include mass unbalance and otherexternal forces

32 Finite Shaft-Rotor Element The rotor-shaft element con-sidered here has eight degrees of freedom that is four degreesof freedom per node as in Nelson and McVaugh [8] For

constant spin condition the Lagrangian equation of motionis given by

([119872119879119890] + [119872119877

119890]) 11990210158401015840119890 minus Ω [119866

119890] 1199021015840119890 + ([119870

119890]) 119902119890 = 119876119890

(12)

where

[119872119879119890] = int

119897

0

120588119860 (119909) [119873]119879 [119873] 119889119909

[119872119877119890] = int

119897

0

119868119863(119909) [120601]119879 [120601] 119889119909

[119873119890] = int

119897

0

119868119875(119909) [120601

Γ]119879 [120601120573] 119889119909

[119870119890] = int

119897

0

119864119868 (119909) [11987310158401015840]119879

[11987310158401015840] 119889119909

[119866119890] = [119873

119890] minus [119873

119890]119879

(13)

Except skew-symmetric gyroscopic matrix [119866119890] others are

symmetric matrices Since the element is linearly taperedarea and inertias are the function of the shaft-rotor lengthThe translational shape function is given by

[119873] = [1198731

0 0 11987321198733

0 0 1198734

0 1198731minus1198732

0 0 1198733minus1198734

0] (14)

where

1198731= 1 minus 3

1199092

1198972+ 2

1199093

1198973

1198732= 119909 minus 2

1199092

119897+1199093

1198972

1198733= 3

1199092

1198972minus 2

1199093

1198973

1198734= minus

1199092

119897+1199093

1198972

(15)

The rotational shape function is given by

[Φ] = [

[

lfloorΦ120573rfloor

lfloorΦΓrfloor]

]

= [0 minus1198731015840

111987310158402

0 0 minus1198731015840311987310158404

011987310158401

0 0 1198731015840211987310158403

0 0 11987310158404

]

(16)

The element matrices are assembled together to get theequation of motion for the complete system

4 Numerical Results

41 Rotating Condition By using the transfer matrixapproach as in the paper of Whalley and Abdul-Ameer wewill ultimately get the transfer function which will be plottedThe FEM will be applied and then is compared with theTMM approach of Whalley and Abdul-Ameer [6]

4 Advances in Acoustics and Vibration

Example 1 Let us consider a cantilever tubular shaft withuniform thickness and a disc at the free end with downwardunit force 119875 on the disc as shown in Figure 2 The defaultvalues of various parameters are tabulated in Table 1

Transfer function for tubular shaft with constant thick-ness as shown in Figure 2 for default values is given by

1344119904 + 1016 times 104

1199043 + 75641199042 + 8401 times 105119904 + 1589 times 109 (17)

Bode plots for different lengths and rotating speed have beenplotted using MATLAB software as shown in Figures 3 and4

Applying FEM on the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement is shown in Figure 5

Table 1 Various parameters of the system shown in Figure 2 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Thickness of the hollow shaft 119905 (m) 0002Beginning radius 119877

0(m) 00050

End radius 1198771(m) 00037

The stiffness matrix for hollow tapered shaft element withuniform thickness is given by is given by

119870119899=

[[[[[[[[[[

[

11989611989911

11989611989921

11989611989922

11989611989931

11989611989932

11989611989933

sym11989611989941

11989611989942

11989611989943

11989611989944

11989611989951

11989611989952

11989611989953

11989611989954

11989611989955

11989611989961

11989611989962

11989611989963

11989611989964

11989611989965

11989611989966

11989611989971

11989611989972

11989611989973

11989611989974

11989611989975

11989611989976

11989611989977

11989611989981

11989611989982

11989611989983

11989611989984

11989611989985

11989611989986

11989611989987

11989611989988

]]]]]]]]]]

]

(18)

where elements of the stiffness matrices are

11989611989911

= 3120587119864119905 (711987730+ 3119877201198771minus 121198772

0119905 + 3119877

011987721minus 611987701198771119905

+ 1011987701199052 + 71198773

1minus 121198772

1119905 + 10119877

11199052 minus 51199053)

times (51198973)minus1

11989611989941

= 120587119864119905 (3011987730+ 121198772

01198771minus 511198772

0119905 + 6119877

011987721minus 18119877

01198771119905

+ 4011987701199052 + 121198773

1minus 211198772

1119905 + 20119877

11199052 minus 151199053)

times (101198972)minus1

11989611989981

= 120587119864119905 (1211987730+ 6119877201198771minus 211198772

0119905 + 12119877

011987721minus 18119877

01198771119905

+ 2011987701199052 + 301198773

1minus 511198772

1119905 + 40119877

11199052 minus 151199053)

times (101198972)minus1

11989611989933

= 120587119864119905 (1111987730+ 5119877201198771minus 191198772

0119905 + 2119877

011987721minus 711987701198771119905

+ 1511987701199052 + 21198773

1minus 411987721119905 + 5119877

11199052 minus 51199053)

times (5119897)minus1

11989611989973

= 120587119864119905 (811987730+ 2119877201198771minus 131198772

0119905 + 2119877

011987721minus 411987701198771119905

+ 1011987701199052 + 81198773

1minus 131198772

1119905 + 10119877

11199052 minus 51199053)

times (10119897)minus1

11989611989977

= 120587119864119905 (211987730+ 2119877201198771minus 411987720119905 + 5119877

011987721minus 711987701198771119905

+ 511987701199052 + 111198773

1minus 191198772

1119905 + 15119877

11199052 minus 51199053)

times (5119897)minus1

11989611989921

= 11989611989942

= 11989611989982

= 11989611989931

= 0

11989611989943

= 11989611989953

= 11989611989983

= 11989611989964

= 11989611989974

= 0

11989611989952

= 11989611989953

= 11989611989965

= 11989611989975

= 11989611989961

= 0

11989611989965

= 11989611989986

= 11989611989971

= 11989611989987

= 0

11989611989922

= 11989611989955

= 11989611989966

= 11989611989911 119896

11989951= 11989611989962

= minus11989611989911

11989611989963

= 11989611989941 119896

11989932= 11989611989954

= minus11989611989941

11989611989976

= 11989611989981 119896

11989972= 11989611989985

= minus11989611989981

11989611989944

= 11989611989933 119896

11989984= 11989611989973

(19)Translational mass matrix is given by

Advances in Acoustics and Vibration 5

119872119899119905=

[[[[[[[[[[

[

11989811989911990511

11989811989911990521

11989811989911990522

11989811989911990531

11989811989911990532

11989811989911990533

sym11989811989911990541

11989811989911990542

11989811989911990543

11989811989911990544

11989811989911990551

11989811989911990552

11989811989911990553

11989811989911990554

11989811989911990555

11989811989911990561

11989811989911990562

11989811989911990563

11989811989911990564

11989811989911990565

11989811989911990566

11989811989911990571

11989811989911990572

11989811989911990573

11989811989911990574

11989811989911990575

11989811989911990576

11989811989911990577

11989811989911990581

11989811989911990582

11989811989911990583

11989811989911990584

11989811989911990585

11989811989911990586

11989811989911990587

11989811989911990588

]]]]]]]]]]

]

(20)

where elements of translational mass matrix are given by

11989811989911990511

=120587120588119897119905 (20119877

0+ 61198771minus 13119905)

35

11989811989911990541

=1205871205881199051198972 (15119877

0+ 71198771minus 11119905)

210

11989811989911990551

=9120587120588119897119905 (119877

0+ 1198771minus 119905)

70

11989811989911990581

=minus (1205871205881199051198972 (14119877

0+ 12119877

1minus 13119905))

420

11989811989911990533

=1205871205881198973119905 (5119877

0+ 31198771minus 4119905)

420

11989811989911990563

=minus (1205871205881198972119905 (12119877

0+ 14119877

1minus 13119905))

420

11989811989911990573

=minus (1205871205881198973119905 (119877

0+ 1198771minus 119905))

140

11989811989911990576

=1205871205881198972119905 (7119877

0+ 15119877

1minus 11119905)

210

11989811989911990577

=1205871205881198973119905 (3119877

0+ 51198771minus 4119905)

420

11989811989911990521

= 11989811989911990542

= 11989811989911990582

= 11989811989911990531

= 0

11989811989911990543

= 11989811989911990553

= 11989811989911990583

= 11989811989911990564

= 11989811989911990574

= 0

11989811989911990552

= 11989811989911990553

= 11989811989911990575

= 11989811989911990561

= 0

11989811989911990565

= 11989811989911990586

= 11989811989911990571

= 11989811989911990587

= 0

11989811989911990522

= 11989811989911990555

= 11989811989911990566

= 11989811989911990511

11989811989911990551

= 11989811989911990562

11989811989911990532

= minus 11989811989911990541

11989811989911990563

= 11989811989911990554

11989811989911990572

= minus 11989811989911990581

11989811989911990544

= 11989811989911990533

11989811989911990584

= 11989811989911990573

11989811989911990585

= minus11989811989911990576

(21)

Rotational mass matrix is given by

119872119899119903=

[[[[[[[[[[

[

11989811989911990311

11989811989911990321

11989811989911990322

11989811989911990331

11989811989911990332

11989811989911990333

sym11989811989911990341

11989811989911990342

11989811989911990343

11989811989911990344

11989811989911990351

11989811989911990352

11989811989911990353

11989811989911990354

11989811989911990355

11989811989911990361

11989811989911990362

11989811989911990363

11989811989911990364

11989811989911990365

11989811989911990366

11989811989911990371

11989811989911990372

11989811989911990373

11989811989911990374

11989811989911990375

11989811989911990376

11989811989911990377

11989811989911990381

11989811989911990382

11989811989911990383

11989811989911990384

11989811989911990385

11989811989911990386

11989811989911990387

11989811989911990388

]]]]]]]]]]

]

(22)

where elements of rotational mass matrix are given by

11989811989911990311

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (70119897)minus1

11989811989911990341

= 1205871205881199052 (minus811987720+ 611987701198771+ 201198772

1

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990351

= minus (31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052))

times (70119897)minus1

11989811989911990333

= 1205871205881199052119897 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (210)minus1

11989811989911990363

= 1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

6 Advances in Acoustics and Vibration

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990373

= minus (1205871205881199052119897 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (840)minus1

11989811989911990376

= 1205871205881199052 (2011987720+ 16119877

01198771

minus 281198770119905 minus 81198772

1+ 71199052)

times (280)minus1

11989811989911990377

= 1205871205881199052119897 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (210)minus1

11989811989911990321

= 11989811989911990342

= 11989811989911990382

= 11989811989911990331

= 11989811989911990343

= 0

11989811989911990353

= 11989811989911990383

= 11989811989911990364

= 11989811989911990374

= 0

11989811989911990352

= 11989811989911990353

= 11989811989911990375

= 11989811989911990361

= 0

11989811989911990365

= 11989811989911990386

= 11989811989911990371

= 11989811989911990387

= 0

11989811989911990322

= 11989811989911990355

= 11989811989911990366

= 11989811989911990311

11989811989911990351

= 11989811989911990362

11989811989911990332

= minus11989811989911990341

11989811989911990363

= 11989811989911990354

11989811989911990381

= 11989811989911990376

11989811989911990372

= minus11989811989911990376

11989811989911990344

= 11989811989911990333

11989811989911990384

= 11989811989911990373

11989811989911990385

= minus11989811989911990376

(23)

Gyroscopic matrix is given by

119866119899119890=

[[[[[[[[[[

[

11989211989911989011

11989211989911989021

11989211989911989022

11989211989911989031

11989211989911989032

11989211989911989033

skewsym11989211989911989041

11989211989911989042

11989211989911989043

11989211989911989044

11989211989911989051

11989211989911989052

11989211989911989053

11989211989911989054

11989211989911989055

11989211989911989061

11989211989911989062

11989211989911989063

11989211989911989064

11989211989911989065

11989211989911989066

11989211989911989071

11989211989911989072

11989211989911989073

11989211989911989074

11989211989911989075

11989211989911989076

11989211989911989077

11989211989911989081

11989211989911989082

11989211989911989083

11989211989911989084

11989211989911989085

11989211989911989086

11989211989911989087

11989211989911989088

]]]]]]]]]]

]

(24)

where the elements of the gyroscopic matrix are given by

11989211989911989021

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (35l)minus1

11989211989911989031

= minus (1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

minus 281198771119905 + 71199052))

times (140)minus1

11989211989911989071

= minus (1205871205881199052 (2011987720+ 16119877

01198771minus 28119877

0119905

minus 811987721+ 71199052))

times (140)minus1

11989211989911989043

= 1205871205881198971199052 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (105)minus1

11989211989911989083

= minus (1205871205881198971199052 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (420)minus1

11989211989911989087

= 1205871205881198971199052 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (105)minus1

11989211989911989052

= 11989211989911989065

= 11989211989911989021

11989211989911989061

= minus11989211989911989021

11989211989911989042

= 11989211989911989053

= 11989211989911989064

= 11989211989911989031

11989211989911989082

= 11989211989911989071

11989211989911989086

= 11989211989911989075

= minus11989211989911989071

11989211989911989075

= minus11989211989911989083

11989211989911989011

= 11989211989911989041

= 11989211989911989051

= 11989211989911989081

= 11989211989911989022

= 11989211989911989032

= 0

11989211989911989062

= 11989211989911989072

= 11989211989911989033

= 11989211989911989063

= 0

11989211989911989073

= 11989211989911989044

= 11989211989911989054

= 11989211989911989084

= 11989211989911989055

= 11989211989911989085

= 0

11989211989911989066

= 11989211989911989076

= 11989211989911989077

= 11989211989911989088

= 0

(25)

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 3: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

Advances in Acoustics and Vibration 3

Shaft length

x + dx

y(t x)A(x

x

)

120579(t x)

qy(t x)

my(t x)

my(t x + dx)

qy(t x + dx)

y(t x + dx)

Figure 1 Vibrating shaft element

3 Finite Element Method

The vector of nodal displacements is given by

119902 =

119881119882120573Γ

(9)

So each element is having eight degrees of freedom

31 Rigid Disc Rigid disk is having two translations and tworotations in 119884 and 119885 direction respectively (considering 119883coordinate in axial direction) For constant spin conditionthe Lagrangian equation of motion is given by

([119872119879119889] + [119872119877

119889]) 11990210158401015840119889 minus Ω [119866

119889] 1199021015840119889 = 119876119889 (10)

where

[119872119879119889] =

[[[

[

119898119889

0 0 00 1198981198890 0

0 0 0 00 0 0 0

]]]

]

[119872119877119889] =

[[[

[

0 0 0 00 0 0 00 0 119868

1198630

0 0 0 119868119863

]]]

]

[119866119889] =

[[[

[

0 0 0 00 0 0 00 0 0 minus119868

119875

0 0 119868119875

0

]]]

]

(11)

The forcing term may include mass unbalance and otherexternal forces

32 Finite Shaft-Rotor Element The rotor-shaft element con-sidered here has eight degrees of freedom that is four degreesof freedom per node as in Nelson and McVaugh [8] For

constant spin condition the Lagrangian equation of motionis given by

([119872119879119890] + [119872119877

119890]) 11990210158401015840119890 minus Ω [119866

119890] 1199021015840119890 + ([119870

119890]) 119902119890 = 119876119890

(12)

where

[119872119879119890] = int

119897

0

120588119860 (119909) [119873]119879 [119873] 119889119909

[119872119877119890] = int

119897

0

119868119863(119909) [120601]119879 [120601] 119889119909

[119873119890] = int

119897

0

119868119875(119909) [120601

Γ]119879 [120601120573] 119889119909

[119870119890] = int

119897

0

119864119868 (119909) [11987310158401015840]119879

[11987310158401015840] 119889119909

[119866119890] = [119873

119890] minus [119873

119890]119879

(13)

Except skew-symmetric gyroscopic matrix [119866119890] others are

symmetric matrices Since the element is linearly taperedarea and inertias are the function of the shaft-rotor lengthThe translational shape function is given by

[119873] = [1198731

0 0 11987321198733

0 0 1198734

0 1198731minus1198732

0 0 1198733minus1198734

0] (14)

where

1198731= 1 minus 3

1199092

1198972+ 2

1199093

1198973

1198732= 119909 minus 2

1199092

119897+1199093

1198972

1198733= 3

1199092

1198972minus 2

1199093

1198973

1198734= minus

1199092

119897+1199093

1198972

(15)

The rotational shape function is given by

[Φ] = [

[

lfloorΦ120573rfloor

lfloorΦΓrfloor]

]

= [0 minus1198731015840

111987310158402

0 0 minus1198731015840311987310158404

011987310158401

0 0 1198731015840211987310158403

0 0 11987310158404

]

(16)

The element matrices are assembled together to get theequation of motion for the complete system

4 Numerical Results

41 Rotating Condition By using the transfer matrixapproach as in the paper of Whalley and Abdul-Ameer wewill ultimately get the transfer function which will be plottedThe FEM will be applied and then is compared with theTMM approach of Whalley and Abdul-Ameer [6]

4 Advances in Acoustics and Vibration

Example 1 Let us consider a cantilever tubular shaft withuniform thickness and a disc at the free end with downwardunit force 119875 on the disc as shown in Figure 2 The defaultvalues of various parameters are tabulated in Table 1

Transfer function for tubular shaft with constant thick-ness as shown in Figure 2 for default values is given by

1344119904 + 1016 times 104

1199043 + 75641199042 + 8401 times 105119904 + 1589 times 109 (17)

Bode plots for different lengths and rotating speed have beenplotted using MATLAB software as shown in Figures 3 and4

Applying FEM on the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement is shown in Figure 5

Table 1 Various parameters of the system shown in Figure 2 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Thickness of the hollow shaft 119905 (m) 0002Beginning radius 119877

0(m) 00050

End radius 1198771(m) 00037

The stiffness matrix for hollow tapered shaft element withuniform thickness is given by is given by

119870119899=

[[[[[[[[[[

[

11989611989911

11989611989921

11989611989922

11989611989931

11989611989932

11989611989933

sym11989611989941

11989611989942

11989611989943

11989611989944

11989611989951

11989611989952

11989611989953

11989611989954

11989611989955

11989611989961

11989611989962

11989611989963

11989611989964

11989611989965

11989611989966

11989611989971

11989611989972

11989611989973

11989611989974

11989611989975

11989611989976

11989611989977

11989611989981

11989611989982

11989611989983

11989611989984

11989611989985

11989611989986

11989611989987

11989611989988

]]]]]]]]]]

]

(18)

where elements of the stiffness matrices are

11989611989911

= 3120587119864119905 (711987730+ 3119877201198771minus 121198772

0119905 + 3119877

011987721minus 611987701198771119905

+ 1011987701199052 + 71198773

1minus 121198772

1119905 + 10119877

11199052 minus 51199053)

times (51198973)minus1

11989611989941

= 120587119864119905 (3011987730+ 121198772

01198771minus 511198772

0119905 + 6119877

011987721minus 18119877

01198771119905

+ 4011987701199052 + 121198773

1minus 211198772

1119905 + 20119877

11199052 minus 151199053)

times (101198972)minus1

11989611989981

= 120587119864119905 (1211987730+ 6119877201198771minus 211198772

0119905 + 12119877

011987721minus 18119877

01198771119905

+ 2011987701199052 + 301198773

1minus 511198772

1119905 + 40119877

11199052 minus 151199053)

times (101198972)minus1

11989611989933

= 120587119864119905 (1111987730+ 5119877201198771minus 191198772

0119905 + 2119877

011987721minus 711987701198771119905

+ 1511987701199052 + 21198773

1minus 411987721119905 + 5119877

11199052 minus 51199053)

times (5119897)minus1

11989611989973

= 120587119864119905 (811987730+ 2119877201198771minus 131198772

0119905 + 2119877

011987721minus 411987701198771119905

+ 1011987701199052 + 81198773

1minus 131198772

1119905 + 10119877

11199052 minus 51199053)

times (10119897)minus1

11989611989977

= 120587119864119905 (211987730+ 2119877201198771minus 411987720119905 + 5119877

011987721minus 711987701198771119905

+ 511987701199052 + 111198773

1minus 191198772

1119905 + 15119877

11199052 minus 51199053)

times (5119897)minus1

11989611989921

= 11989611989942

= 11989611989982

= 11989611989931

= 0

11989611989943

= 11989611989953

= 11989611989983

= 11989611989964

= 11989611989974

= 0

11989611989952

= 11989611989953

= 11989611989965

= 11989611989975

= 11989611989961

= 0

11989611989965

= 11989611989986

= 11989611989971

= 11989611989987

= 0

11989611989922

= 11989611989955

= 11989611989966

= 11989611989911 119896

11989951= 11989611989962

= minus11989611989911

11989611989963

= 11989611989941 119896

11989932= 11989611989954

= minus11989611989941

11989611989976

= 11989611989981 119896

11989972= 11989611989985

= minus11989611989981

11989611989944

= 11989611989933 119896

11989984= 11989611989973

(19)Translational mass matrix is given by

Advances in Acoustics and Vibration 5

119872119899119905=

[[[[[[[[[[

[

11989811989911990511

11989811989911990521

11989811989911990522

11989811989911990531

11989811989911990532

11989811989911990533

sym11989811989911990541

11989811989911990542

11989811989911990543

11989811989911990544

11989811989911990551

11989811989911990552

11989811989911990553

11989811989911990554

11989811989911990555

11989811989911990561

11989811989911990562

11989811989911990563

11989811989911990564

11989811989911990565

11989811989911990566

11989811989911990571

11989811989911990572

11989811989911990573

11989811989911990574

11989811989911990575

11989811989911990576

11989811989911990577

11989811989911990581

11989811989911990582

11989811989911990583

11989811989911990584

11989811989911990585

11989811989911990586

11989811989911990587

11989811989911990588

]]]]]]]]]]

]

(20)

where elements of translational mass matrix are given by

11989811989911990511

=120587120588119897119905 (20119877

0+ 61198771minus 13119905)

35

11989811989911990541

=1205871205881199051198972 (15119877

0+ 71198771minus 11119905)

210

11989811989911990551

=9120587120588119897119905 (119877

0+ 1198771minus 119905)

70

11989811989911990581

=minus (1205871205881199051198972 (14119877

0+ 12119877

1minus 13119905))

420

11989811989911990533

=1205871205881198973119905 (5119877

0+ 31198771minus 4119905)

420

11989811989911990563

=minus (1205871205881198972119905 (12119877

0+ 14119877

1minus 13119905))

420

11989811989911990573

=minus (1205871205881198973119905 (119877

0+ 1198771minus 119905))

140

11989811989911990576

=1205871205881198972119905 (7119877

0+ 15119877

1minus 11119905)

210

11989811989911990577

=1205871205881198973119905 (3119877

0+ 51198771minus 4119905)

420

11989811989911990521

= 11989811989911990542

= 11989811989911990582

= 11989811989911990531

= 0

11989811989911990543

= 11989811989911990553

= 11989811989911990583

= 11989811989911990564

= 11989811989911990574

= 0

11989811989911990552

= 11989811989911990553

= 11989811989911990575

= 11989811989911990561

= 0

11989811989911990565

= 11989811989911990586

= 11989811989911990571

= 11989811989911990587

= 0

11989811989911990522

= 11989811989911990555

= 11989811989911990566

= 11989811989911990511

11989811989911990551

= 11989811989911990562

11989811989911990532

= minus 11989811989911990541

11989811989911990563

= 11989811989911990554

11989811989911990572

= minus 11989811989911990581

11989811989911990544

= 11989811989911990533

11989811989911990584

= 11989811989911990573

11989811989911990585

= minus11989811989911990576

(21)

Rotational mass matrix is given by

119872119899119903=

[[[[[[[[[[

[

11989811989911990311

11989811989911990321

11989811989911990322

11989811989911990331

11989811989911990332

11989811989911990333

sym11989811989911990341

11989811989911990342

11989811989911990343

11989811989911990344

11989811989911990351

11989811989911990352

11989811989911990353

11989811989911990354

11989811989911990355

11989811989911990361

11989811989911990362

11989811989911990363

11989811989911990364

11989811989911990365

11989811989911990366

11989811989911990371

11989811989911990372

11989811989911990373

11989811989911990374

11989811989911990375

11989811989911990376

11989811989911990377

11989811989911990381

11989811989911990382

11989811989911990383

11989811989911990384

11989811989911990385

11989811989911990386

11989811989911990387

11989811989911990388

]]]]]]]]]]

]

(22)

where elements of rotational mass matrix are given by

11989811989911990311

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (70119897)minus1

11989811989911990341

= 1205871205881199052 (minus811987720+ 611987701198771+ 201198772

1

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990351

= minus (31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052))

times (70119897)minus1

11989811989911990333

= 1205871205881199052119897 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (210)minus1

11989811989911990363

= 1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

6 Advances in Acoustics and Vibration

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990373

= minus (1205871205881199052119897 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (840)minus1

11989811989911990376

= 1205871205881199052 (2011987720+ 16119877

01198771

minus 281198770119905 minus 81198772

1+ 71199052)

times (280)minus1

11989811989911990377

= 1205871205881199052119897 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (210)minus1

11989811989911990321

= 11989811989911990342

= 11989811989911990382

= 11989811989911990331

= 11989811989911990343

= 0

11989811989911990353

= 11989811989911990383

= 11989811989911990364

= 11989811989911990374

= 0

11989811989911990352

= 11989811989911990353

= 11989811989911990375

= 11989811989911990361

= 0

11989811989911990365

= 11989811989911990386

= 11989811989911990371

= 11989811989911990387

= 0

11989811989911990322

= 11989811989911990355

= 11989811989911990366

= 11989811989911990311

11989811989911990351

= 11989811989911990362

11989811989911990332

= minus11989811989911990341

11989811989911990363

= 11989811989911990354

11989811989911990381

= 11989811989911990376

11989811989911990372

= minus11989811989911990376

11989811989911990344

= 11989811989911990333

11989811989911990384

= 11989811989911990373

11989811989911990385

= minus11989811989911990376

(23)

Gyroscopic matrix is given by

119866119899119890=

[[[[[[[[[[

[

11989211989911989011

11989211989911989021

11989211989911989022

11989211989911989031

11989211989911989032

11989211989911989033

skewsym11989211989911989041

11989211989911989042

11989211989911989043

11989211989911989044

11989211989911989051

11989211989911989052

11989211989911989053

11989211989911989054

11989211989911989055

11989211989911989061

11989211989911989062

11989211989911989063

11989211989911989064

11989211989911989065

11989211989911989066

11989211989911989071

11989211989911989072

11989211989911989073

11989211989911989074

11989211989911989075

11989211989911989076

11989211989911989077

11989211989911989081

11989211989911989082

11989211989911989083

11989211989911989084

11989211989911989085

11989211989911989086

11989211989911989087

11989211989911989088

]]]]]]]]]]

]

(24)

where the elements of the gyroscopic matrix are given by

11989211989911989021

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (35l)minus1

11989211989911989031

= minus (1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

minus 281198771119905 + 71199052))

times (140)minus1

11989211989911989071

= minus (1205871205881199052 (2011987720+ 16119877

01198771minus 28119877

0119905

minus 811987721+ 71199052))

times (140)minus1

11989211989911989043

= 1205871205881198971199052 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (105)minus1

11989211989911989083

= minus (1205871205881198971199052 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (420)minus1

11989211989911989087

= 1205871205881198971199052 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (105)minus1

11989211989911989052

= 11989211989911989065

= 11989211989911989021

11989211989911989061

= minus11989211989911989021

11989211989911989042

= 11989211989911989053

= 11989211989911989064

= 11989211989911989031

11989211989911989082

= 11989211989911989071

11989211989911989086

= 11989211989911989075

= minus11989211989911989071

11989211989911989075

= minus11989211989911989083

11989211989911989011

= 11989211989911989041

= 11989211989911989051

= 11989211989911989081

= 11989211989911989022

= 11989211989911989032

= 0

11989211989911989062

= 11989211989911989072

= 11989211989911989033

= 11989211989911989063

= 0

11989211989911989073

= 11989211989911989044

= 11989211989911989054

= 11989211989911989084

= 11989211989911989055

= 11989211989911989085

= 0

11989211989911989066

= 11989211989911989076

= 11989211989911989077

= 11989211989911989088

= 0

(25)

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 4: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

4 Advances in Acoustics and Vibration

Example 1 Let us consider a cantilever tubular shaft withuniform thickness and a disc at the free end with downwardunit force 119875 on the disc as shown in Figure 2 The defaultvalues of various parameters are tabulated in Table 1

Transfer function for tubular shaft with constant thick-ness as shown in Figure 2 for default values is given by

1344119904 + 1016 times 104

1199043 + 75641199042 + 8401 times 105119904 + 1589 times 109 (17)

Bode plots for different lengths and rotating speed have beenplotted using MATLAB software as shown in Figures 3 and4

Applying FEM on the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement is shown in Figure 5

Table 1 Various parameters of the system shown in Figure 2 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Thickness of the hollow shaft 119905 (m) 0002Beginning radius 119877

0(m) 00050

End radius 1198771(m) 00037

The stiffness matrix for hollow tapered shaft element withuniform thickness is given by is given by

119870119899=

[[[[[[[[[[

[

11989611989911

11989611989921

11989611989922

11989611989931

11989611989932

11989611989933

sym11989611989941

11989611989942

11989611989943

11989611989944

11989611989951

11989611989952

11989611989953

11989611989954

11989611989955

11989611989961

11989611989962

11989611989963

11989611989964

11989611989965

11989611989966

11989611989971

11989611989972

11989611989973

11989611989974

11989611989975

11989611989976

11989611989977

11989611989981

11989611989982

11989611989983

11989611989984

11989611989985

11989611989986

11989611989987

11989611989988

]]]]]]]]]]

]

(18)

where elements of the stiffness matrices are

11989611989911

= 3120587119864119905 (711987730+ 3119877201198771minus 121198772

0119905 + 3119877

011987721minus 611987701198771119905

+ 1011987701199052 + 71198773

1minus 121198772

1119905 + 10119877

11199052 minus 51199053)

times (51198973)minus1

11989611989941

= 120587119864119905 (3011987730+ 121198772

01198771minus 511198772

0119905 + 6119877

011987721minus 18119877

01198771119905

+ 4011987701199052 + 121198773

1minus 211198772

1119905 + 20119877

11199052 minus 151199053)

times (101198972)minus1

11989611989981

= 120587119864119905 (1211987730+ 6119877201198771minus 211198772

0119905 + 12119877

011987721minus 18119877

01198771119905

+ 2011987701199052 + 301198773

1minus 511198772

1119905 + 40119877

11199052 minus 151199053)

times (101198972)minus1

11989611989933

= 120587119864119905 (1111987730+ 5119877201198771minus 191198772

0119905 + 2119877

011987721minus 711987701198771119905

+ 1511987701199052 + 21198773

1minus 411987721119905 + 5119877

11199052 minus 51199053)

times (5119897)minus1

11989611989973

= 120587119864119905 (811987730+ 2119877201198771minus 131198772

0119905 + 2119877

011987721minus 411987701198771119905

+ 1011987701199052 + 81198773

1minus 131198772

1119905 + 10119877

11199052 minus 51199053)

times (10119897)minus1

11989611989977

= 120587119864119905 (211987730+ 2119877201198771minus 411987720119905 + 5119877

011987721minus 711987701198771119905

+ 511987701199052 + 111198773

1minus 191198772

1119905 + 15119877

11199052 minus 51199053)

times (5119897)minus1

11989611989921

= 11989611989942

= 11989611989982

= 11989611989931

= 0

11989611989943

= 11989611989953

= 11989611989983

= 11989611989964

= 11989611989974

= 0

11989611989952

= 11989611989953

= 11989611989965

= 11989611989975

= 11989611989961

= 0

11989611989965

= 11989611989986

= 11989611989971

= 11989611989987

= 0

11989611989922

= 11989611989955

= 11989611989966

= 11989611989911 119896

11989951= 11989611989962

= minus11989611989911

11989611989963

= 11989611989941 119896

11989932= 11989611989954

= minus11989611989941

11989611989976

= 11989611989981 119896

11989972= 11989611989985

= minus11989611989981

11989611989944

= 11989611989933 119896

11989984= 11989611989973

(19)Translational mass matrix is given by

Advances in Acoustics and Vibration 5

119872119899119905=

[[[[[[[[[[

[

11989811989911990511

11989811989911990521

11989811989911990522

11989811989911990531

11989811989911990532

11989811989911990533

sym11989811989911990541

11989811989911990542

11989811989911990543

11989811989911990544

11989811989911990551

11989811989911990552

11989811989911990553

11989811989911990554

11989811989911990555

11989811989911990561

11989811989911990562

11989811989911990563

11989811989911990564

11989811989911990565

11989811989911990566

11989811989911990571

11989811989911990572

11989811989911990573

11989811989911990574

11989811989911990575

11989811989911990576

11989811989911990577

11989811989911990581

11989811989911990582

11989811989911990583

11989811989911990584

11989811989911990585

11989811989911990586

11989811989911990587

11989811989911990588

]]]]]]]]]]

]

(20)

where elements of translational mass matrix are given by

11989811989911990511

=120587120588119897119905 (20119877

0+ 61198771minus 13119905)

35

11989811989911990541

=1205871205881199051198972 (15119877

0+ 71198771minus 11119905)

210

11989811989911990551

=9120587120588119897119905 (119877

0+ 1198771minus 119905)

70

11989811989911990581

=minus (1205871205881199051198972 (14119877

0+ 12119877

1minus 13119905))

420

11989811989911990533

=1205871205881198973119905 (5119877

0+ 31198771minus 4119905)

420

11989811989911990563

=minus (1205871205881198972119905 (12119877

0+ 14119877

1minus 13119905))

420

11989811989911990573

=minus (1205871205881198973119905 (119877

0+ 1198771minus 119905))

140

11989811989911990576

=1205871205881198972119905 (7119877

0+ 15119877

1minus 11119905)

210

11989811989911990577

=1205871205881198973119905 (3119877

0+ 51198771minus 4119905)

420

11989811989911990521

= 11989811989911990542

= 11989811989911990582

= 11989811989911990531

= 0

11989811989911990543

= 11989811989911990553

= 11989811989911990583

= 11989811989911990564

= 11989811989911990574

= 0

11989811989911990552

= 11989811989911990553

= 11989811989911990575

= 11989811989911990561

= 0

11989811989911990565

= 11989811989911990586

= 11989811989911990571

= 11989811989911990587

= 0

11989811989911990522

= 11989811989911990555

= 11989811989911990566

= 11989811989911990511

11989811989911990551

= 11989811989911990562

11989811989911990532

= minus 11989811989911990541

11989811989911990563

= 11989811989911990554

11989811989911990572

= minus 11989811989911990581

11989811989911990544

= 11989811989911990533

11989811989911990584

= 11989811989911990573

11989811989911990585

= minus11989811989911990576

(21)

Rotational mass matrix is given by

119872119899119903=

[[[[[[[[[[

[

11989811989911990311

11989811989911990321

11989811989911990322

11989811989911990331

11989811989911990332

11989811989911990333

sym11989811989911990341

11989811989911990342

11989811989911990343

11989811989911990344

11989811989911990351

11989811989911990352

11989811989911990353

11989811989911990354

11989811989911990355

11989811989911990361

11989811989911990362

11989811989911990363

11989811989911990364

11989811989911990365

11989811989911990366

11989811989911990371

11989811989911990372

11989811989911990373

11989811989911990374

11989811989911990375

11989811989911990376

11989811989911990377

11989811989911990381

11989811989911990382

11989811989911990383

11989811989911990384

11989811989911990385

11989811989911990386

11989811989911990387

11989811989911990388

]]]]]]]]]]

]

(22)

where elements of rotational mass matrix are given by

11989811989911990311

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (70119897)minus1

11989811989911990341

= 1205871205881199052 (minus811987720+ 611987701198771+ 201198772

1

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990351

= minus (31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052))

times (70119897)minus1

11989811989911990333

= 1205871205881199052119897 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (210)minus1

11989811989911990363

= 1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

6 Advances in Acoustics and Vibration

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990373

= minus (1205871205881199052119897 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (840)minus1

11989811989911990376

= 1205871205881199052 (2011987720+ 16119877

01198771

minus 281198770119905 minus 81198772

1+ 71199052)

times (280)minus1

11989811989911990377

= 1205871205881199052119897 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (210)minus1

11989811989911990321

= 11989811989911990342

= 11989811989911990382

= 11989811989911990331

= 11989811989911990343

= 0

11989811989911990353

= 11989811989911990383

= 11989811989911990364

= 11989811989911990374

= 0

11989811989911990352

= 11989811989911990353

= 11989811989911990375

= 11989811989911990361

= 0

11989811989911990365

= 11989811989911990386

= 11989811989911990371

= 11989811989911990387

= 0

11989811989911990322

= 11989811989911990355

= 11989811989911990366

= 11989811989911990311

11989811989911990351

= 11989811989911990362

11989811989911990332

= minus11989811989911990341

11989811989911990363

= 11989811989911990354

11989811989911990381

= 11989811989911990376

11989811989911990372

= minus11989811989911990376

11989811989911990344

= 11989811989911990333

11989811989911990384

= 11989811989911990373

11989811989911990385

= minus11989811989911990376

(23)

Gyroscopic matrix is given by

119866119899119890=

[[[[[[[[[[

[

11989211989911989011

11989211989911989021

11989211989911989022

11989211989911989031

11989211989911989032

11989211989911989033

skewsym11989211989911989041

11989211989911989042

11989211989911989043

11989211989911989044

11989211989911989051

11989211989911989052

11989211989911989053

11989211989911989054

11989211989911989055

11989211989911989061

11989211989911989062

11989211989911989063

11989211989911989064

11989211989911989065

11989211989911989066

11989211989911989071

11989211989911989072

11989211989911989073

11989211989911989074

11989211989911989075

11989211989911989076

11989211989911989077

11989211989911989081

11989211989911989082

11989211989911989083

11989211989911989084

11989211989911989085

11989211989911989086

11989211989911989087

11989211989911989088

]]]]]]]]]]

]

(24)

where the elements of the gyroscopic matrix are given by

11989211989911989021

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (35l)minus1

11989211989911989031

= minus (1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

minus 281198771119905 + 71199052))

times (140)minus1

11989211989911989071

= minus (1205871205881199052 (2011987720+ 16119877

01198771minus 28119877

0119905

minus 811987721+ 71199052))

times (140)minus1

11989211989911989043

= 1205871205881198971199052 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (105)minus1

11989211989911989083

= minus (1205871205881198971199052 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (420)minus1

11989211989911989087

= 1205871205881198971199052 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (105)minus1

11989211989911989052

= 11989211989911989065

= 11989211989911989021

11989211989911989061

= minus11989211989911989021

11989211989911989042

= 11989211989911989053

= 11989211989911989064

= 11989211989911989031

11989211989911989082

= 11989211989911989071

11989211989911989086

= 11989211989911989075

= minus11989211989911989071

11989211989911989075

= minus11989211989911989083

11989211989911989011

= 11989211989911989041

= 11989211989911989051

= 11989211989911989081

= 11989211989911989022

= 11989211989911989032

= 0

11989211989911989062

= 11989211989911989072

= 11989211989911989033

= 11989211989911989063

= 0

11989211989911989073

= 11989211989911989044

= 11989211989911989054

= 11989211989911989084

= 11989211989911989055

= 11989211989911989085

= 0

11989211989911989066

= 11989211989911989076

= 11989211989911989077

= 11989211989911989088

= 0

(25)

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 5: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

Advances in Acoustics and Vibration 5

119872119899119905=

[[[[[[[[[[

[

11989811989911990511

11989811989911990521

11989811989911990522

11989811989911990531

11989811989911990532

11989811989911990533

sym11989811989911990541

11989811989911990542

11989811989911990543

11989811989911990544

11989811989911990551

11989811989911990552

11989811989911990553

11989811989911990554

11989811989911990555

11989811989911990561

11989811989911990562

11989811989911990563

11989811989911990564

11989811989911990565

11989811989911990566

11989811989911990571

11989811989911990572

11989811989911990573

11989811989911990574

11989811989911990575

11989811989911990576

11989811989911990577

11989811989911990581

11989811989911990582

11989811989911990583

11989811989911990584

11989811989911990585

11989811989911990586

11989811989911990587

11989811989911990588

]]]]]]]]]]

]

(20)

where elements of translational mass matrix are given by

11989811989911990511

=120587120588119897119905 (20119877

0+ 61198771minus 13119905)

35

11989811989911990541

=1205871205881199051198972 (15119877

0+ 71198771minus 11119905)

210

11989811989911990551

=9120587120588119897119905 (119877

0+ 1198771minus 119905)

70

11989811989911990581

=minus (1205871205881199051198972 (14119877

0+ 12119877

1minus 13119905))

420

11989811989911990533

=1205871205881198973119905 (5119877

0+ 31198771minus 4119905)

420

11989811989911990563

=minus (1205871205881198972119905 (12119877

0+ 14119877

1minus 13119905))

420

11989811989911990573

=minus (1205871205881198973119905 (119877

0+ 1198771minus 119905))

140

11989811989911990576

=1205871205881198972119905 (7119877

0+ 15119877

1minus 11119905)

210

11989811989911990577

=1205871205881198973119905 (3119877

0+ 51198771minus 4119905)

420

11989811989911990521

= 11989811989911990542

= 11989811989911990582

= 11989811989911990531

= 0

11989811989911990543

= 11989811989911990553

= 11989811989911990583

= 11989811989911990564

= 11989811989911990574

= 0

11989811989911990552

= 11989811989911990553

= 11989811989911990575

= 11989811989911990561

= 0

11989811989911990565

= 11989811989911990586

= 11989811989911990571

= 11989811989911990587

= 0

11989811989911990522

= 11989811989911990555

= 11989811989911990566

= 11989811989911990511

11989811989911990551

= 11989811989911990562

11989811989911990532

= minus 11989811989911990541

11989811989911990563

= 11989811989911990554

11989811989911990572

= minus 11989811989911990581

11989811989911990544

= 11989811989911990533

11989811989911990584

= 11989811989911990573

11989811989911990585

= minus11989811989911990576

(21)

Rotational mass matrix is given by

119872119899119903=

[[[[[[[[[[

[

11989811989911990311

11989811989911990321

11989811989911990322

11989811989911990331

11989811989911990332

11989811989911990333

sym11989811989911990341

11989811989911990342

11989811989911990343

11989811989911990344

11989811989911990351

11989811989911990352

11989811989911990353

11989811989911990354

11989811989911990355

11989811989911990361

11989811989911990362

11989811989911990363

11989811989911990364

11989811989911990365

11989811989911990366

11989811989911990371

11989811989911990372

11989811989911990373

11989811989911990374

11989811989911990375

11989811989911990376

11989811989911990377

11989811989911990381

11989811989911990382

11989811989911990383

11989811989911990384

11989811989911990385

11989811989911990386

11989811989911990387

11989811989911990388

]]]]]]]]]]

]

(22)

where elements of rotational mass matrix are given by

11989811989911990311

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (70119897)minus1

11989811989911990341

= 1205871205881199052 (minus811987720+ 611987701198771+ 201198772

1

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990351

= minus (31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052))

times (70119897)minus1

11989811989911990333

= 1205871205881199052119897 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (210)minus1

11989811989911990363

= 1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

6 Advances in Acoustics and Vibration

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990373

= minus (1205871205881199052119897 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (840)minus1

11989811989911990376

= 1205871205881199052 (2011987720+ 16119877

01198771

minus 281198770119905 minus 81198772

1+ 71199052)

times (280)minus1

11989811989911990377

= 1205871205881199052119897 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (210)minus1

11989811989911990321

= 11989811989911990342

= 11989811989911990382

= 11989811989911990331

= 11989811989911990343

= 0

11989811989911990353

= 11989811989911990383

= 11989811989911990364

= 11989811989911990374

= 0

11989811989911990352

= 11989811989911990353

= 11989811989911990375

= 11989811989911990361

= 0

11989811989911990365

= 11989811989911990386

= 11989811989911990371

= 11989811989911990387

= 0

11989811989911990322

= 11989811989911990355

= 11989811989911990366

= 11989811989911990311

11989811989911990351

= 11989811989911990362

11989811989911990332

= minus11989811989911990341

11989811989911990363

= 11989811989911990354

11989811989911990381

= 11989811989911990376

11989811989911990372

= minus11989811989911990376

11989811989911990344

= 11989811989911990333

11989811989911990384

= 11989811989911990373

11989811989911990385

= minus11989811989911990376

(23)

Gyroscopic matrix is given by

119866119899119890=

[[[[[[[[[[

[

11989211989911989011

11989211989911989021

11989211989911989022

11989211989911989031

11989211989911989032

11989211989911989033

skewsym11989211989911989041

11989211989911989042

11989211989911989043

11989211989911989044

11989211989911989051

11989211989911989052

11989211989911989053

11989211989911989054

11989211989911989055

11989211989911989061

11989211989911989062

11989211989911989063

11989211989911989064

11989211989911989065

11989211989911989066

11989211989911989071

11989211989911989072

11989211989911989073

11989211989911989074

11989211989911989075

11989211989911989076

11989211989911989077

11989211989911989081

11989211989911989082

11989211989911989083

11989211989911989084

11989211989911989085

11989211989911989086

11989211989911989087

11989211989911989088

]]]]]]]]]]

]

(24)

where the elements of the gyroscopic matrix are given by

11989211989911989021

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (35l)minus1

11989211989911989031

= minus (1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

minus 281198771119905 + 71199052))

times (140)minus1

11989211989911989071

= minus (1205871205881199052 (2011987720+ 16119877

01198771minus 28119877

0119905

minus 811987721+ 71199052))

times (140)minus1

11989211989911989043

= 1205871205881198971199052 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (105)minus1

11989211989911989083

= minus (1205871205881198971199052 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (420)minus1

11989211989911989087

= 1205871205881198971199052 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (105)minus1

11989211989911989052

= 11989211989911989065

= 11989211989911989021

11989211989911989061

= minus11989211989911989021

11989211989911989042

= 11989211989911989053

= 11989211989911989064

= 11989211989911989031

11989211989911989082

= 11989211989911989071

11989211989911989086

= 11989211989911989075

= minus11989211989911989071

11989211989911989075

= minus11989211989911989083

11989211989911989011

= 11989211989911989041

= 11989211989911989051

= 11989211989911989081

= 11989211989911989022

= 11989211989911989032

= 0

11989211989911989062

= 11989211989911989072

= 11989211989911989033

= 11989211989911989063

= 0

11989211989911989073

= 11989211989911989044

= 11989211989911989054

= 11989211989911989084

= 11989211989911989055

= 11989211989911989085

= 0

11989211989911989066

= 11989211989911989076

= 11989211989911989077

= 11989211989911989088

= 0

(25)

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 6: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

6 Advances in Acoustics and Vibration

minus 281198771119905 + 71199052)

times (280)minus1

11989811989911990373

= minus (1205871205881199052119897 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (840)minus1

11989811989911990376

= 1205871205881199052 (2011987720+ 16119877

01198771

minus 281198770119905 minus 81198772

1+ 71199052)

times (280)minus1

11989811989911990377

= 1205871205881199052119897 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (210)minus1

11989811989911990321

= 11989811989911990342

= 11989811989911990382

= 11989811989911990331

= 11989811989911990343

= 0

11989811989911990353

= 11989811989911990383

= 11989811989911990364

= 11989811989911990374

= 0

11989811989911990352

= 11989811989911990353

= 11989811989911990375

= 11989811989911990361

= 0

11989811989911990365

= 11989811989911990386

= 11989811989911990371

= 11989811989911990387

= 0

11989811989911990322

= 11989811989911990355

= 11989811989911990366

= 11989811989911990311

11989811989911990351

= 11989811989911990362

11989811989911990332

= minus11989811989911990341

11989811989911990363

= 11989811989911990354

11989811989911990381

= 11989811989911990376

11989811989911990372

= minus11989811989911990376

11989811989911990344

= 11989811989911990333

11989811989911990384

= 11989811989911990373

11989811989911990385

= minus11989811989911990376

(23)

Gyroscopic matrix is given by

119866119899119890=

[[[[[[[[[[

[

11989211989911989011

11989211989911989021

11989211989911989022

11989211989911989031

11989211989911989032

11989211989911989033

skewsym11989211989911989041

11989211989911989042

11989211989911989043

11989211989911989044

11989211989911989051

11989211989911989052

11989211989911989053

11989211989911989054

11989211989911989055

11989211989911989061

11989211989911989062

11989211989911989063

11989211989911989064

11989211989911989065

11989211989911989066

11989211989911989071

11989211989911989072

11989211989911989073

11989211989911989074

11989211989911989075

11989211989911989076

11989211989911989077

11989211989911989081

11989211989911989082

11989211989911989083

11989211989911989084

11989211989911989085

11989211989911989086

11989211989911989087

11989211989911989088

]]]]]]]]]]

]

(24)

where the elements of the gyroscopic matrix are given by

11989211989911989021

= 31205871205881199052 (811987720+ 12119877

01198771minus 14119877

0119905

+ 811987721minus 14119877

1119905 + 71199052)

times (35l)minus1

11989211989911989031

= minus (1205871205881199052 (minus811987720+ 16119877

01198771+ 201198772

1

minus 281198771119905 + 71199052))

times (140)minus1

11989211989911989071

= minus (1205871205881199052 (2011987720+ 16119877

01198771minus 28119877

0119905

minus 811987721+ 71199052))

times (140)minus1

11989211989911989043

= 1205871205881198971199052 (1811987720+ 611987701198771minus 21119877

0119905

+ 411987721minus 71198771119905 + 71199052)

times (105)minus1

11989211989911989083

= minus (1205871205881198971199052 (1211987720+ 411987701198771minus 14119877

0119905

+ 1211987721minus 14119877

1119905 + 71199052))

times (420)minus1

11989211989911989087

= 1205871205881198971199052 (411987720+ 611987701198771minus 71198770119905

+ 1811987721minus 21119877

1119905 + 71199052)

times (105)minus1

11989211989911989052

= 11989211989911989065

= 11989211989911989021

11989211989911989061

= minus11989211989911989021

11989211989911989042

= 11989211989911989053

= 11989211989911989064

= 11989211989911989031

11989211989911989082

= 11989211989911989071

11989211989911989086

= 11989211989911989075

= minus11989211989911989071

11989211989911989075

= minus11989211989911989083

11989211989911989011

= 11989211989911989041

= 11989211989911989051

= 11989211989911989081

= 11989211989911989022

= 11989211989911989032

= 0

11989211989911989062

= 11989211989911989072

= 11989211989911989033

= 11989211989911989063

= 0

11989211989911989073

= 11989211989911989044

= 11989211989911989054

= 11989211989911989084

= 11989211989911989055

= 11989211989911989085

= 0

11989211989911989066

= 11989211989911989076

= 11989211989911989077

= 11989211989911989088

= 0

(25)

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 7: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

Advances in Acoustics and Vibration 7

Table 2 Various parameters of the system shown in Figure 9 andtheir default values

Parameters ValuesLength of the shaft rotor 119897 (m) 01Mass of the disk119898 (Kg) 07443Diameter of the disk119863 (m) 009Youngrsquos modulus of elasticity 119864 (GPa) 209Density of the material 120588 (Kgm3) 7800Rotational speed119873 (rpm) 10000Inner radius of hollow shaft 119877

119894(m) 0001

Beginning radius 1198770(m) 00050

End radius 1198771(m) 00037

t

xl P

Disk

D

Bearings

Ro R1r(x)

Figure 2 Hollow tapered shaft disc with uniform thickness

Discretizing the tapered shaft into six elements as shown inFigure 6 and then assembling we get the assembled equationof motion

[119872119904119899] 119902119904 minus Ω [119866119904

119899] 119902119904 + [119870119904

119899] 119902119904 = 119876119904

119899 (26)

where [119872119904119899] is the assembledmassmatrix containing both the

translational and rotational mass matricesThe assembled equation of motion is arranged in the first

order state vector form

[[0] [119872119904

119899]

[119872119904119899] minusΩ [119866119904

119899]] + [

[minus119872119904119899] [0]

[0] [119870119904119899]] 119911 = 119885 (27)

where

119911 = 119902

119902 119885 =

0

119876119904119899 (28)

The shaft rotor has been discretized into six elements of equallength Hence the order of assembled matrices after applyingthe fixed-free boundary condition is 7 times 4 minus 4 = 24

MATLAB program is used to find the bode plot fordifferent values of shaft length and rotor speed as shown inFigures 7 and 8 and are found to be in good agreement withbode plots found using TMM as shown in Figures 3 and 4

Example 2 Let us consider a cantilever hollow shaft withuniform bore and a disc at the free end as shown in Figure 9The values of various parameters are tabulated in Table 2

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)10

110

210

3

101

102

103

minus60

minus80

minus100

minus120

minus140

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 3 Bode plot for different lengths with TMMM

agni

tude

(dB)

Bode diagram

Frequency (rads)10

2610

2510

27

1026

1025

1027

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

minus70

minus75

minus80

minus85

minus90

minus95

minus100

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 4 Bode plot for various speeds with TMM

xl

tR0

Ri

R1

Figure 5 Hollow tapered shaft finite element with uniform thick-ness

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 8: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

8 Advances in Acoustics and Vibration

1 2 3 4 5 6 7

t

1 2 3 4 5 6R0

Ri(x)R1

l1 l2 l3 l4 l5 l6

Figure 6 Discretized shaft element

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)10

210

3

102

103

minus60

minus80

minus100

minus120

minus140

Phas

e (de

g)

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 7 Bode plot for different shaft lengths (FEM)

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2410

2510

2610

27

1024

1025

1026

1027

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 8 Bode plot for various rotor speeds (FEM)

The transfer function for hollow shaft with constantthickness for default values is given by

1344119904 + 1217 times 104

1199043 + 90591199042 + 1097 times 106119904 + 2483 times 109 (29)

The bode plots for different lengths and rotating speeds havebeen plotted usingMATLAB software as shown in Figures 10and 11

Applying FEM in the same system we get mass gyro-scopic and stiffness matrices A finite hollow tapered shaftelement with uniform bore is shown in Figure 12

The stiffnessmatrix for hollow tapered shaftwith uniformbore is given by

119870119906=

[[[[[[[[[[

[

11989611990611

11989611990621

11989611990622

11989611990631

11989611990632

11989611990633

sym11989611990641

11989611990642

11989611990643

11989611990644

11989611990651

11989611990652

11989611990653

11989611990654

11989611990655

11989611990661

11989611990662

11989611990663

11989611990664

11989611990665

11989611990666

11989611990671

11989611990672

11989611990673

11989611990674

11989611990675

11989611990676

11989611990677

11989611990681

11989611990682

11989611990683

11989611990684

11989611990685

11989611990686

11989611990687

11989611990688

]]]]]]]]]]

]

(30)

where elements of the stiffness matrices are

11989611990611

= 3120587119864 (1111987740+ 5119877301198771+ 31198772011987721

+ 5119877011987731+ 111198774

1minus 351198774

119894)

times (351198973)minus1

11989611990641

= 120587119864 (4711987740+ 221198773

01198771+ 91198772011987721

+ 8119877011987731+ 191198774

1minus 105 lowast 1198774

119894)

times (701198972)minus1

11989611990681

= 120587119864 (1911987740+ 8119877301198771+ 91198772011987721

+ 22119877011987731+ 471198774

1minus 11051198774

119894)

times (701198972)minus1

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 9: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

Advances in Acoustics and Vibration 9

11989611990633

= 120587119864 (1711987740+ 9119877301198771+ 41198772011987721

+ 2119877011987731+ 311987741minus 351198774

119894)

times (35119897)minus1

11989611990673

= 120587119864 (1311987740+ 4119877301198771+ 1198772011987721

+ 4119877011987731+ 131198774

1minus 351198774

119894)

times (70119897)minus1

11989611990677

= 120587119864 (311987740+ 2119877301198771+ 41198772011987721

+ 9119877011987731+ 171198774

1minus 351198774

119894)

times (35119897)minus1

11989611990621

= 11989611990642

= 11989611990682

= 11989611990631

= 0

11989611990643

= 11989611990653

= 11989611990683

= 11989611990664

= 11989611990674

= 0

11989611990652

= 11989611990653

= 11989611990665

= 11989611990675

= 0

11989611990661

= 11989611990665

= 11989611990686

= 11989611990671

= 11989611990687

= 0

11989611990622

= 11989611990655

= 11989611990666

= 11989611990611

11989611990651

= 11989611990662

= minus11989611990611 119896

11990663= 11989611990641

11989611990632

= 11989611990654

= minus11989611990641 119896

11990676= 11989611990681

11989611990672

= 11989611990685

= 11989611990672

= minus11989611990681

11989611990644

= 11989611990633 119896

11990684= 11989611990673

(31)

Translational mass matrix is given by

119872119906119905=

[[[[[[[[[[

[

11989811990611990511

11989811990611990521

11989811990611990522

11989811990611990531

11989811990611990532

11989811990611990533

sym11989811990611990541

11989811990611990542

11989811990611990543

11989811990611990544

11989811990611990551

11989811990611990552

11989811990611990553

11989811990611990554

11989811990611990555

11989811990611990561

11989811990611990562

11989811990611990563

11989811990611990564

11989811990611990565

11989811990611990566

11989811990611990571

11989811990611990572

11989811990611990573

11989811990611990574

11989811990611990575

11989811990611990576

11989811990611990577

11989811990611990581

11989811990611990582

11989811990611990583

11989811990611990584

11989811990611990585

11989811990611990586

11989811990611990587

11989811990611990588

]]]]]]]]]]

]

(32)

where elements of translational mass matrix are given by

11989811990611990511

=120587120588119897 (1451198772

0+ 70119877

01198771+ 191198772

1minus 2341198772

119894)

630

11989811990611990541

=1205871205881198972 (651198772

0+ 50119877

01198771+ 171198772

1minus 1321198772

119894)

2520

11989811990611990551

=120587120588119897 (231198772

0+ 35119877

01198771+ 231198772

1minus 811198772

119894)

630

11989811990611990581

=minus (1205871205881198972 (251198772

0+ 34119877

01198771+ 191198772

1minus 781198772

119894))

2520

11989811990611990533

=1205871205881198973 (51198772

0+ 511987701198771+ 211987721minus 121198772

119894)

1260

11989811990611990563

=minus (1205871205881198972 (191198772

0+ 34119877

01198771+ 251198772

1minus 781198772

119894))

2520

11989811990611990573

=minus (1205871205881198973 (51198772

0+ 811987701198771+ 511987721minus 181198772

119894))

2520

11989811990611990576

=1205871205881198972 (171198772

0+ 50119877

01198771+ 651198772

1minus 1321198772

119894)

2520

11989811990611990577

=1205871205881198973 (21198772

0+ 511987701198771+ 511987721minus 121198772

119894)

1260

11989811990611990521

= 11989811990611990542

= 11989811990611990582

= 11989811990611990531

= 11989811990611990543

= 0

11989811990611990553

= 11989811990611990583

= 11989811990611990564

= 11989811990611990574

= 0

11989811990611990552

= 11989811990611990553

= 11989811990611990575

= 11989811990611990561

= 0

11989811990611990565

= 11989811990611990586

= 11989811990611990571

= 11989811990611990587

= 0

11989811990611990522

= 11989811990611990555

= 11989811990611990566

= 11989811990611990511

11989811990611990551

= 11989811990611990562

11989811990611990532

= minus11989811990611990541

11989811990611990563

= 11989811990611990554

11989811990611990572

= minus11989811990611990581

11989811990611990544

= 11989811990611990533

11989811990611990584

= 11989811990611990573

11989811990611990585

= minus11989811990611990576

(33)

Rotational mass matrix is given by

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

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Page 10: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

10 Advances in Acoustics and Vibration

BearingsDisk

Ro

t(x)

xl P

R1

Ri

D

Figure 9 Hollow tapered shaft disc with uniform bore and vertically downward force 119875 on the disc

119872119906119903=

[[[[[[[[[[

[

11989811990611990311

11989811990611990321

11989811990611990322

11989811990611990331

11989811990611990332

11989811990611990333

sym11989811990611990341

11989811990611990342

11989811990611990343

11989811990611990344

11989811990611990351

11989811990611990352

11989811990611990353

11989811990611990354

11989811990611990355

11989811990611990361

11989811990611990362

11989811990611990363

11989811990611990364

11989811990611990365

11989811990611990366

11989811990611990371

11989811990611990372

11989811990611990373

11989811990611990374

11989811990611990375

11989811990611990376

11989811990611990377

11989811990611990381

11989811990611990382

11989811990611990383

11989811990611990384

11989811990611990385

11989811990611990386

11989811990611990387

11989811990611990388

]]]]]]]]]]

]

(34)

where the elements of the rotational mass matrix are given by

11989811990611990311

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus 241198772

11198772119894+ 421198774

119894)

times (140119897)minus1

11989811990611990341

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894)

times (560)minus1

11989811990611990351

= minus (120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894

+ 10119877011987731minus 36119877

011987711198772119894+ 511987741

minus 24119877211198772119894+ 421198774

119894))

times (140119897)minus1

11989811990611990381

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990333

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894

+ 3119877011987731minus 12119877

011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (840)minus1

11989811990611990363

= 120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus20119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990373

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus12119877211198772119894+ 141198774

119894))

times (1680)minus1

11989811990611990376

= 120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+8119877211198772119894+ 141198774

119894)

times (560)minus1

11989811990611990377

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721

minus 8119877201198772119894+ 5119877011987731minus 12119877

011987711198772119894

+ 1511987741minus 361198772

11198772119894+ 281198774

119894)

times (840)minus1

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 11: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

Advances in Acoustics and Vibration 11

11989811990611990321

= 11989811990611990342

= 11989811990611990382

= 11989811990611990331

= 11989811990611990343

= 11989811990611990353

= 11989811990611990383

= 11989811990611990364

= 11989811990611990374

= 0

11989811990611990352

= 11989811990611990353

= 11989811990611990375

= 11989811990611990361

= 0

11989811990611990365

= 11989811990611990386

= 11989811990611990371

= 11989811990611990387

= 0

11989811990611990322

= 11989811990611990355

= 11989811990611990366

= 11989811990611990311

11989811990611990351

= 11989811990611990362

11989811990611990332

= minus11989811990611990341

11989811990611990363

= 11989811990611990354

11989811990611990372

= minus11989811990611990381

11989811990611990344

= 11989811990611990333

11989811990611990384

= 11989811990611990373

11989811990611990385

= minus11989811990611990376

(35)

The gyroscopic matrix is given by

119866119906119890=

[[[[[[[[[[

[

11989211990611989011

11989211990611989021

11989211990611989022

11989211990611989031

11989211990611989032

11989211990611989033

skewsym11989211990611989041

11989211990611989042

11989211990611989043

11989211990611989044

11989211990611989051

11989211990611989052

11989211990611989053

11989211990611989054

11989211990611989055

11989211990611989061

11989211990611989062

11989211990611989063

11989211990611989064

11989211990611989065

11989211990611989066

11989211990611989071

11989211990611989072

11989211990611989073

11989211990611989074

11989211990611989075

11989211990611989076

11989211990611989077

11989211990611989081

11989211990611989082

11989211990611989083

11989211990611989084

11989211990611989085

11989211990611989086

11989211990611989087

11989211990611989088

]]]]]]]]]]

]

(36)

where the elements of the gyroscopic matrix are

11989211990611989021

= 120587120588 (511987740+ 101198773

01198771+ 121198772

011987721minus 241198772

01198772119894+ 10119877

011987731

minus 36119877011987711198772119894+ 511987741minus24119877211198772119894+ 421198774

119894)

times (70119897)minus1

11989211990611989031

= minus (120587120588 (minus511987740+ 61198772011987721+ 8119877201198772119894+ 8119877011987731

minus 16119877011987711198772119894+ 511987741minus 201198772

11198772119894+ 141198774

119894))

times (280)minus1

11989211990611989071

= minus (120587120588 (511987740+ 8119877301198771+ 61198772011987721minus 201198772

01198772119894

minus 16119877011987711198772119894minus 511987741+ 8119877211198772119894+ 141198774

119894))

times (280)minus1

11989211990611989043

= 120587120588119897 (1511987740+ 5119877301198771+ 31198772011987721minus 361198772

01198772119894+ 3119877011987731

minus 12119877011987711198772119894+211987741minus 8119877211198772119894+ 281198774

119894)

times (420)minus1

11989211990611989083

= minus (120587120588119897 (511987740+ 2119877301198771minus 121198772

01198772119894+ 2119877011987731

minus 4119877011987711198772119894+ 511987741minus 121198772

11198772119894+ 141198774

119894))

times (840)minus1

11989211990611989087

= 120587120588119897 (211987740+ 3119877301198771+ 31198772011987721minus 8119877201198772119894+ 5119877011987731

minus 12119877011987711198772119894+ 151198774

1minus 361198772

11198772119894+ 281198774

119894)

times (420)minus1

11989211990611989052

= 11989211990611989065

= 11989211990611989021

11989211990611989061

= minus11989211990611989021

11989211990611989042

= 11989211990611989053

= 11989211990611989064

= 11989211990611989031

11989211990611989082

= 11989211990611989071

11989211990611989086

= 11989211990611989075

= minus11989211990611989071

11989211990611989075

= minus11989211990611989083

11989211990611989011

= 11989211990611989041

= 11989211990611989051

= 11989211990611989081

= 11989211990611989022

= 0

11989211990611989032

= 11989211990611989062

= 11989211990611989072

= 11989211990611989033

= 11989211990611989063

= 0

11989211990611989073

= 11989211990611989044

= 11989211990611989054

= 11989211990611989084

= 11989211990611989055

= 0

11989211990611989085

= 11989211990611989066

= 11989211990611989076

= 11989211990611989077

= 11989211990611989088

= 0

(37)

As proceeded in Example 1 bode plots are obtained forvarious shaft lengths and rotor speeds as shown in Figures13 and 14 and are found to be in good agreement with bodeplots found using TMM as shown in Figures 10 and 11

42 Nonrotating Conditions Bode plot for nonrotating (1 or2 rpm) tapered shaft-rotor system is slightly different fromrotating conditions in terms of amplitude Hollow shaft withuniform thickness is considered Bode plots are obtained forzero rpm as shown in Figure 15 with TMM

Applying FEM then for zero rpm we get the bode plot asshown in Figure 16

5 Conclusions

Shaft geometry plays one of the important roles in dynamiccharacteristics of rotating systemsVibration analysis with thehelp of bode plots has been done for hollow tapered shaft-rotor system Both TMM and FEM have been used for thepurpose The equation of motion for a tapered beam finiteelement has been developed using Euler-Bernoulli beam

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 12: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

12 Advances in Acoustics and VibrationM

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

210

110

310

4

102

101

103

104

minus60

minus80

minus100

minus120

minus140

minus160

minus90

minus45

0

minus135

minus180

L = 010mL = 015mL = 020m

Figure 10 Bode plot for changing length for hollow tapered shaftrotor with uniform bore

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

0

minus70

minus80

minus90

minus100

minus110

minus45

minus90

minus135

minus180

1025

1026

1027

1028

1025

1026

1027

1028

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 11 Bode plot for different speeds for hollow tapered shaftrotor with uniform bore

xl

R0

R1

Ri

t(x)

Figure 12 Hollow tapered shaft finite element with uniform bore

Mag

nitu

de (d

B)

0

Phas

e (de

g)

Bode diagram

Frequency (rads)

minus60

minus80

minus100

minus120

minus140

minus90

minus45

minus135

0

minus180

102

103

102

103

L = 010mL = 015mL = 020m

Figure 13 Bode plot for different shaft lengths (FEM)M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)10

2710

28

1027

1028

minus70

minus60

minus80

minus90

minus100

minus110

0

minus45

minus90

minus135

minus180

1000 rpm3000 rpm

7000 rpm10000 rpm

Figure 14 Bode plot for different rotating speeds (FEM)

theory Mass stiffness and gyroscopic matrices are foundand values of all these elements are stated in a systematicmanner for ease of understandingThe results obtained fromboth methods are compared and are found to be in goodagreement However the above procedures show that themethod of TMM is simpler in calculations Two types ofhollow tapered shafts have been analyzed that is one withuniform thickness and another with uniform bore

The length of the shaft is a vital parameter that affectsthe frequency response of the shaft-rotor system As shown

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 13: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

Advances in Acoustics and Vibration 13M

agni

tude

(dB)

Phas

e (de

g)

Bode diagram

Frequency (rads)

Zero rpm

1025

1026

1027

1028

1029

1025

1026

1027

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 15 Bode plot at zero rpm with TMM for uniform thicknesshollow tapered shaft-rotor system

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rads)

Zero rpm

1027

1026

1025

1028

1029

1027

1026

1025

1028

1029

50

0

minus50

minus100

minus150

minus90

minus45

0

minus135

minus180

Figure 16 Bode plot at zero rpm with FEM for uniform thicknesshollow tapered shaft-rotor system

in Figures 3 7 10 and 13 there is increase in amplitude ofvibration for increased value of shaft length while reducingthewhirling speed of shaftThe systemexhibits this behaviourdue to the bending effect of the shaft and stiffness change Forchanging lengths there are large differences in frequencieseven for small increase in lengths Bode plots obtained inFigures 4 8 11 and 14 show that rotating speeds havevery little effect on the critical frequencies however withincreasing speed the amplitude is lowered due to gyroscopiccouple The phase angle changes abruptly for lower valueof shaft speeds than higher speeds This is due to the factthat there is reduction in gyroscopic couple as the rotational

speed of the shaft decreases thereby giving larger amplitude ofvibration Nonrotating conditions are also shown in Figures15 and 16 as bode plots which show that any rotating systemat a very low speed vibrates with high amplitudes due to lackof gyroscopic couple

Effects of bearingmay be included in the problemGearedsystems and other rotary elements can be mounted instead ofdiscs and further calculations can be made Multidiscs andother complex problems can be solved using these methods

Notations

119862(119909) Compliance per unit of length (function)119871(119909) Inertia per unit of length (function)119871 Length of shaft119864 Modulus of elasticity119865(119904) System model matrix119872119910(119909 119904) Bending moment in 119909-119910 plane (function)

119876119910(119909 119904) Shear force (function)

119884(119909 119904) Vertical deflection of shaft (function)120579(119909 119904) Slope of the shaft (function)119868(119909) Mass moment of inertia (function)120588 Material density119869 Shaft polar moment of inertiaΩ Whirling frequencyΓ(119904) Wave propagation factor (function)Ω Shaft-rotor rotational speed119877(119904) Rigid rotor model matrix119869 Polar moment of inertia of disc1198770 Beginning radius of the shaft element

119877119894 Inner radius of the shaft element

1198771 End radius of the shaft element

119898 Mass of the disc attached at free end[119872119879119889] Translational mass matrix for disc

[119872119877119889] Rotational mass matrix for disc

[119866119889] Gyroscopic matrix for disc

[119872119879119890] Translational mass matrix

[119872119877119890] Rotational mass matrix

[119866119890] Gyroscopic matrix for element

[119870119890] Stiffness matrix for shaft element

119876119890 External force matrix119860(119909) Cross-sectional area (function)119868119863(119909) Diametral inertia (function)

119868119875(119909) Polar inertia (function)

Ω Rotational speed in rads[119873] Translational shape function matrix[120601] Rotational shape function matrix

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] N O Myklestad ldquoA new method for calculating natural modesof uncoupled bending vibration of airplane wingsrdquo Journal ofthe Aeronautical Sciences vol 11 no 2 pp 153ndash162 1944

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 14: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

14 Advances in Acoustics and Vibration

[2] N O Myklestad ldquoNew method of calculating natural modesof coupled bending-torsion vibration of beamsrdquo Transactions ofthe ASME vol 67 no 1 pp 61ndash67 1945

[3] M A Prohl ldquoA general method for calculating critical speedsof flexible rotorsrdquo Transactions of the ASME Journal of AppliedMechanics vol 66 pp A-142ndashA-148 1945

[4] J W Lund ldquoStability and damped critical speeds of a flexiblerotor in fluid-film bearingsrdquo Transactions of the ASME Journalof Engineering for Industry Series B vol 96 no 2 pp 509ndash5171974

[5] B T Murphy and J M Vance ldquoAn improved method forcalculating critical speeds and rotor dynamic stability of turbomachineryrdquo Transactions of the ASME Journal of Engineeringfor Power vol 105 no 3 pp 591ndash595 1983

[6] R Whalley and A Abdul-Ameer ldquoWhirling prediction withgeometrical shaft profilingrdquo Applied Mathematical Modellingvol 33 no 7 pp 3166ndash3177 2009

[7] R L Ruhl and J F Booker ldquoA finite element model for dis-tributed parameter turborotor systemsrdquo Journal of Engineeringfor Industry vol 94 no 1 pp 126ndash132 1972

[8] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Transactions of theASME Journal of Engineering vol 98 no 2 pp 593ndash600 1976

[9] H D Nelson ldquoA finite rotating shaft element using timoshenkobeam theoryrdquo Journal of Mechanical Design vol 102 no 4 pp793ndash803 1980

[10] L M Greenhill W B Bickford and H D Nelson ldquoA conicalbeam finite element for rotor dynamics analysisrdquo Journal ofVibration Acoustics Stress and Reliability in Design vol 107 no4 pp 421ndash430 1985

[11] G Genta and A Gugliotta ldquoA conical element for finite elementrotor dynamicsrdquo Journal of Sound and Vibration vol 120 no 1pp 175ndash182 1988

[12] M A Mohiuddin and Y A Khulief ldquoModal characteristics ofcracked rotors using a conical shaft finite elementrdquo ComputerMethods in Applied Mechanics and Engineering vol 162 no 1ndash4 pp 223ndash247 1998

[13] K E Rouch and J-S Kao ldquoA tapered beam finite element forrotor dynamics analysisrdquo Journal of Sound and Vibration vol66 no 1 pp 119ndash140 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 15: Research Article Vibration Analysis of Hollow Tapered ...downloads.hindawi.com/journals/aav/2014/410851.pdf · Research Article Vibration Analysis of Hollow Tapered ... Hollow tapered

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of


Vibration of hollow core floors

Vibration of hollow core floors

Tapered Haircut

Tapered Haircut

Roller Mtd. Tapered Bearings - Cross & Morse Tapered roller.pdf · I-2 Mounted Tapered Roller Modular bearing assembly consisting of sealed and lubricated inch or metric tapered bearing

Roller Mtd. Tapered Bearings - Cross & Morse Tapered roller.pdf · I-2 Mounted Tapered Roller Modular bearing assembly consisting of sealed and lubricated inch or metric tapered bearing

Large Amplitude Free Vibration Analysis of Tapered ...vibsys.put.poznan.pl/_journal/2017-28/articles/... · Many authors developed different methods to find the free vibration behaviour

Large Amplitude Free Vibration Analysis of Tapered ...vibsys.put.poznan.pl/_journal/2017-28/articles/... · Many authors developed different methods to find the free vibration behaviour

Tapered Roller Bearings, RBC Tapered Thrust Bearings

Tapered Roller Bearings, RBC Tapered Thrust Bearings

Vibration Analysis of Tapered Rotating Composite …ziaeirad.iut.ac.ir/sites/ziaeirad.iut.ac.ir/files/file_pubwdet/... · Applied and Computational Mechanics 4 (2010) 1–14 Vibration

Vibration Analysis of Tapered Rotating Composite …ziaeirad.iut.ac.ir/sites/ziaeirad.iut.ac.ir/files/file_pubwdet/... · Applied and Computational Mechanics 4 (2010) 1–14 Vibration

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Tapered OPC

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Wing Tapered Plates

Wing Tapered Plates

Free Vibration Analysis of Functionally Heterogeneous ... · M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL... Table 1.

Free Vibration Analysis of Functionally Heterogeneous ... · M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL... Table 1.

CylindriCal and Tapered roller Bearings - Bearings ......much noise, vibration, or both can be tolerated. The time when a bearing becomes unsuitable for further service is sometimes

CylindriCal and Tapered roller Bearings - Bearings ......much noise, vibration, or both can be tolerated. The time when a bearing becomes unsuitable for further service is sometimes

Tapered Rolled Bearings and Parts Thereof, and Certain Housings Incorporating Tapered Rollers

Tapered Rolled Bearings and Parts Thereof, and Certain Housings Incorporating Tapered Rollers

Vibration Analysis of Tapered Beam

Vibration Analysis of Tapered Beam

Measurement of Fly Rod Spines Graig Spolek. Modern fly rods Hollow, tubular, and tapered Manufactured of carbon fiber reinforced plastic Formed by layering.

Measurement of Fly Rod Spines Graig Spolek. Modern fly rods Hollow, tubular, and tapered Manufactured of carbon fiber reinforced plastic Formed by layering.

Tapered Steel Beams_1996

Tapered Steel Beams_1996

Tapered Navigator System For Guided Surgery - Biometbiomet3i.com/FMEA/Tapered Navigator System For... · The Tapered Navigator System For Guided Surgery includes the Tapered Navigator

Tapered Navigator System For Guided Surgery - Biometbiomet3i.com/FMEA/Tapered Navigator System For... · The Tapered Navigator System For Guided Surgery includes the Tapered Navigator