Dynamic stability of rotating blades with transverse...

Shock and Vibration 10 (2003) 187–194 187IOS Press

Dynamic stability of rotating blades withtransverse cracks

T.Y. Nga,b,∗, K.Y. Lama and Hua LiaaInstitute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn Singapore Science Park II,Singapore 117528bSchool of Mechanical and Production Engineering, Nanyang Technological University 50 Nanyang Avenue,Singapore 639798

Received 4 October 2001

Revised 12 December 2002

Abstract. In this paper, the main objective is to examine the effects of transverse cracks on the dynamic instability regions of anaxially loaded rotating blade. The blade is modeled as an Euler-Bernoulli beam. To reduce the governing equations to a set ofordinary differential equations in matrix form, Hamilton’s principle is used in conjunction with the assumed-mode method. Thecrack is accounted for by considering the energy release rate and the parametric instability regions are obtained using Bolotin’sfirst approximation. Benchmark results are presented for cracked rotating blades at different rotating speeds, crack lengths andcrack positions.

Keywords: Rotating blade, transverse crack, dynamic stability, parametric resonance, Hamilton’s principle, assumed-modemethod, Bolotin’s first approximation

1. Introduction

The dynamic behaviour of a beam may change dueto the presence of cracks. The changes in the dynamiccharacteristics due to the changes in the crack lengthsare important with respect to the science of crack detec-tion and design of structures. Adams et al. [1] showedthat for the axial vibration of a uniform bar, the re-duction in stiffness of the bar due to the presence of acrack could be modelled by the introduction of a linearspring. However, the relationship between the magni-tude of the spring constant and the crack length wasnot determined. Ju et al. [2] theoretically related themagnitude of the equivalent linear spring constant tothe length of the crack in the beam based on fracturemechanics. Several researchers, Moshrefi et al. [3], Juand Mimovich [4] and Rizos et al. [5], have experimen-tally diagnosed the changes in natural frequency due

∗Corresponding author. E-mail: [email protected].

to existing cracks and have concluded the feasibilityof modelling a crack by an appropriate combination oflinear and torsional springs.

Using perturbation and transfer matrix methods, thechanges in the natural frequencies due to the presenceof cracks were computed by Gudmundson [6,7]. Pa-padopoulos and Dimarogonas [8] modelled a crack us-ing a 2 × 2 compliance matrix to describe the localflexibility in the analysis of the coupled longitudinaland bending vibrations of a cracked shaft. In analysingthe coupled vibration of a cracked rotor, Papadopoulosand Dimarogonas [9] represented the local flexibilityby a6×6 compliance matrix. A finite element methodfor the analysis of cracked beams was developed byHaisty and Springer [10] and the crack in this case wasmodelled as a linear spring for axial vibrations and atorsional spring for flexural vibrations. Qian et al. [11]also presented an element stiffness matrix for a crackedbeam based on the principles of fracture mechanics.The Rayleigh quotient was used by Chondros and Di-marogonas [12] for estimating the change in the nat-

ISSN 1070-9622/03/$8.00 2003 – IOS Press. All rights reserved

188 T.Y. Ng et al. / Dynamic stability of rotating blades with transverse cracks

ural frequencies and modes for the torsional vibrationof a cracked rotor and also for the in-plane vibrationof frames. The small change in the local stiffness atthe crack was introduced into the Rayleigh quotientfrom which the variation in the eigenvalue was com-puted. Shen and Pierre [13], using both an approximateGalerkin formulation and finite element method ob-tained the natural frequencies and modes for the trans-verse vibrations of simply supported Bernoulli-Eulerbeams with a pair of symmetric double-sided cracks atthe mid-span position. Dimarogonas and Paipetis [14]constructed a crack related 5× 5 local flexibility matrixfor a rectangular beam from principles of fracture me-chanics and analyzed the coupling effects between theaxial loading and lateral bending. Subsequently, Di-marogonas and Papadopoulos [15] expanded the workto a circular shaft with a transverse crack and calcu-lated the flexibility coefficients due to bending in twoperpendicular directions.

Examples of radially rotating beams and blades in-clude rotor blades, propellers, turbines, robotic armsand numerous other parts that are commonly found inrotating machineries. This field has been studied inthe last few decades in relation to the behaviour of ro-tating blades in turbomachineries. As such, an exten-sive literature on radially rotating uniform, pretwistedand tapered beams are readily available. Prior doc-umented works include Sutherland [16], Renard andRabowski [17], Likins et al. [18], Anderson [19],Swaminathan and Rao [20], Hodges and Ruthoski [21]and Subrahnanvam et al. [22]. These studies composeboth uniform and pretwisted beams rotating at constantangular velocities. Other studies of similar configu-rations include Kane et al. [23] who modeled a Tim-oshenko beam built into a rigid base and undergoinggeneral three-dimensional motions. Rotating beamssubjected to base excitations have commonly been usedto model propellers, helicopter rotor blades and mo-bile linkages of robots. Hammond [24] treated a non-isotropic hub and four-bladed rotor with the incorpora-tion of the base movements.

In the present study, the effects of transverse crackson the parametric resonance of an axially loaded rotat-ing blade modeled as an Euler-Bernoulli beam is in-vestigated. Hamilton’s energy principle is used in con-junction with the assumed-mode method to obtain a setof ordinary differential equations in matrix form thestability of which is analyzed using Bolotin’s [25] firstapproximation. The crack is accounted for by consid-ering the energy release rate. The analysis is carriedout for various rotating speeds, crack lengths and crackpositions.

P

X

Y

Fig. 1. Coordinate system of the rotating blade.

2. Theory and formulation

Consider a uniform slender beam system as shownin Fig. 1 with one end clamped to a rotating hub ofradiusr and rotating with constant angular velocityΩ.An axial harmonic loading in the longitudinal directionof the beam is considered

P = Po + Ps cosωt (1)

wherePo is the stationary component andPs is theamplitude of the harmonic component of the loading.The beam is considered as an Euler-Bernoulli beam ofconstant cross-sectional area and thus only the trans-verse deflectionv is considered. The potential energyarising from the above loading is

Up = P∫ r+L

r

(v,x )2dx (2)

where the subscripts after the commas indicate the par-tial differentiation with respect to, and the kinetic en-ergy of the beam is given as

T =ρA

2(3)

∫ r+L

r

(v,t )2 + 2xΩv,t +(x2 + v2)Ω2dx

whereρ denotes the mass density,E the elastic mod-ulus,A the cross-sectional area,I the area moment ofinertia, andL the length. The potential energy due tobending is

Uv =EI

2

∫ r+L

r

(v,xx )2dx (4)

and that due to the initial stress is

UΩ =ρA

4Ω2

∫ r+L

r

((r + L)2 − x2)(v,x )2dx (5)

From Dimarogonas and Paipetis [15], the energy re-lease of the crack is

T.Y. Ng et al. / Dynamic stability of rotating blades with transverse cracks 189

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5 3 3.5

Ps/

Po

ω

Fig. 2. First two instability regions for a rotating beam with a single crack at mid-span,Ω = 1 anda/h = 0.5. ‘-.-.-’ L = 0.8 m, ‘–’ L = 1.0m,‘- - -’ L = 1.2 m.

Uc = b∫ a

0

J(a)da (6)

whereb is the width of the beam and

J(a) =K2

c (1 − ν2)E

(7)

whereν is the Poisson’s ratio,a is the crack depth and

Kc =6Pc

bh2(πa)1/2Fc

(ah

)(8)

Fc

(ah

)=

0.92 + 0.2(1 − sin(πa/2h))4

cos(πa/2h)(9)

×(

2hπa

tan(πa/2h))1/2

Pc = EI(v,xx )2|x=xc (10)

whereh is the beam height. The released energy of thecrack can be simplified to the form

Uc = 3EIb(1 − ν2)C55(11)∫ r+L

r

δ(x− xc)(v,xx )2dx

C55 =∫ a/h

0

π(ah

)F 2

c

(ah

)d

(ah

)(12)

Using the assumed-mode method, see Wu andHuang [26], the beam deformation can be approximatedin the following form

v(x, t) =N∑

i=1

Vi(x)qi(t) (13)

whereqi(t) is a generalized coordinate andVi(x) is theith normalized mode of a non-rotating clamped-freebeam. The equations of motion in matrix form can thenbe written in matrix form as

[M]q + [[K] + cosωt[Q]]q = 0 (14)

Mij = ρA∫ r+L

r

Vi(x)Vj(x)dx (15)

[K] = [Kv + KΩ + Kc + KP ] (16)

Kvij = EI

∫ r+L

r

Vi(x),xx Vj(x),xx dx (17)

KΩij = ρAΩ2

∫ r+L

r

Vi(x)(18)

xVj(x),x −Vj(x) − Vj(x),xx

× 12 ((r + L)2 − x2)

dx

Kcij = 6EIb(1 − ν2)C55Vi(x),xx |x=xc

(19)Vj(x),xx |x=xc

KPij = Po

∫ r+L

r

Vi(x),x Vj(x),x dx (20)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Ps/

Po

ω

Fig. 3. First two instability regions for a rotating beam with a single crack at mid-span,L = 1.0 m anda/h = 0.5. ‘-.-.-’ Ω = 0.5, ‘–’ Ω = 1,‘- - -’ Ω = 2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Ps/

Po

ω

Fig. 4. First two instability regions for a rotating beam with a single crack at mid-span,Ω = 1 andL = 1.0 m. ‘-.-.-’ a/h = 0.25, ‘– ’a/h = 0.50, ‘- - -’ a/h = 0.75.

Qij = Ps

∫ r+L

r

Vi(x),x Vj(x),x dx (21)

To obtain the principal instability regions, Bolot-in’s [25] method is applied to the Mathieu-Hill form ofEq. (14). In this method, we seek the periodic solutionwith period 2T in the form

q =∞∑

k=1,3,5,...

fk sinkωt

2+ gk cos

kωt

2(22)

For principal instability, Bolotin’s [25] first approx-imation, associated with the2ω harmonic, is given bythe substitution of the following simplified expression

q = f sinωt

2+ g cos

ωt

2(23)

wheref andg are arbitrary vectors. It is well knownthat the secondary instability regions associated withtheω harmonic are significantly smaller than the prin-cipal instability regions associated with the 2ω har-monic. In most practical applications, these already


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Ps/

Po

ω

Fig. 5. First two instability regions for a rotating beam with a single crack,L = 1.0 m, Ω = 1 anda/h = 0.5. ‘-.-.-’ xc = 0.25, ‘–’ xc = 0.5,‘- - -’ xc = 0.75.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5 3

Ps/

Po

ω

Fig. 6. First two instability regions for a rotating beam with double cracks atxcmean = 0.5L, Ω = 1 anda/h = 0.5. ‘-.-.-’ L = 0.8 m, ‘–’L = 1.0 m, ‘- - -’ L = 1.2 m.

very narrow secondary regions are very often dampedaway. Substitution into governing equation and equat-ing coefficients ofsin(ωt/2) andcos(ωt/2) terms, weobtain a set of linear homogeneous algebraic equationsin terms off andg, and for non-trivial solutions∣∣∣∣∣

(− 14ω

2MIJ + KIJ − 12QIJ

)0

0(− 1

4ω2MIJ + KIJ + 1

2QIJ

)∣∣∣∣∣ = 0 (24)

Rearrangement leads to a generalized eigenvalue

problem∣∣∣∣∣∣∣∣∣∣

[KIJ − 1

2QIJ 00 KIJ + 1

2QIJ

]

−ω2

[14MIJ 0

0 14MIJ

]∣∣∣∣∣∣∣∣∣∣= 0 (25)

The eigenvalues define the boundaries between thestable and unstable regions.


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Ps/

Po

ω

Fig. 7. First two instability regions for a rotating beam with double cracks atxcmean = 0.5L, L = 1.0 m anda/h = 0.5. ‘-.-.-’ Ω = 0.5, ‘–’Ω = 1, ‘- - -’ Ω = 2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Ps/

Po

ω

Fig. 8. First two instability regions for a rotating beam with double cracks atxcmean = 0.5L, Ω = 1 andL = 1.0 m. ‘-.-.-’ a/h = 0.25, ‘– ’a/h = 0.50, ‘- - -’ a/h = 0.75.

3. Results and discussions

In the following discussion, the normalized crackposition is

xc =xc − rL

(26)

and the loading frequency, rotational speed and naturalfrequency are nondimensionalized as

ω = ω1r2

√EI

ρA(27)

Ω = Ω1L2

√EI

ρA(28)

ωn = ωn1L2

√EI

ρA(29)

Dynamic stability results for a rotating beam witha single crack are presented in Figs 2 to 5. Thesefigures show the unstable regions for the first two modesas various parameters are varied. Each set of lines


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

Ps/

Po

ω

Fig. 9. First two instability regions for a rotating beam with double cracks,L = 1.0 m, Ω = 1 anda/h = 0.5. ‘-.-.-’ xcmean = 0.25, ‘–’xcmean = 0.5, ‘- - -’ xcmean = 0.75.

originating from the abscissa represents the boundarybetween the stable and unstable regions. The regionbetween each set of lines constitutes the unstable regionwhile that outside the lines constitutes the stable region.The loading is overall tensile withP o = 1 where

P o =PoL

2

EI(30)

and the cross section of the beam is taken to be squarewith the length to beam height ratio beingL/h = 50with hub radiusr = 0.2 m and lengthL = 1 m unlessotherwise stated.

In Fig. 2, the length of the beam is varied, hub radiusbeing kept constant. It is observed that modest changesin the length can cause substantial changes in the po-sition as well as size of the unstable regions with thechanges more pronounced in the second mode. Shorterlength beams are observed to have unstable regions athigher values on the excitation frequency axis. This isintuitively expected due to their higher stiffnesses. Itis also noted that the shorter beams are associated withsubstantially larger regions of instability where the un-stable regions for a beam of lengthL = 0.8 are almostthree times that for a beam of lengthL = 1.2. In Fig. 3,the rotating speed is varied. As expected, the higher thespeed, the higher along the excitation frequency axiswill be the position of the unstable region. Further, theregion sizes are observed to increase with decreasingspeeds.

In Fig. 4, the crack depth is varied for a beam witha single crack at the mid-point. It is observed that the

variation of the results is minimal if the crack is shal-low, i.e. a/h < 0.5. However, when the crack becomesdeeper, i.e.a/h > 0.5, there is appreciable changein the position of the unstable regions. It is observedthat the increases in the crack depths shift the unstableregions to the left on the frequency axis, which can beattributed to the overall reducing stiffness due to theincreasing severity of the crack. Also, it is interestingto note that the sizes of the unstable regions increasemoderately with the crack severity, especially for thesecond mode. In Fig. 5, the position of the crack is var-ied. There is no clear trend that can be drawn from theresults with regards to the unstable region characteris-tics. However, it is observed that the unstable regionsizes are not sensitive to variation in the crack position.The results also show that effects of increasing crackdepth are slightly more influential on the higher modeswith minimal effects on the fundamental mode.

Figures 6 to 9 are the corresponding results to Figs 2to 5 for a beam with double cracks in relatively closeproximity. The cracks are a distance0.1L apart fromeach other and their position are defined by a meanposition parameter

xcmean =xc1 + xc2

2(31)

wherexc1 andxc2 are the distances of the two cracksfrom the centre of the hub. The results for the beamwith double cracks show similar trends to that of thesingle crack beam except that the unstable regions aregenerally lower on the excitation frequency axis withlarger sizes attributable to the lower stiffnesses.


4. Conclusions

The dynamic stability of a rotating blade containingtransverse cracks has been modeled based upon crackrelease energy approach in conjunction with Hamil-ton’s principle and the assumed-mode method. Bolot-in’s first approximation was used to obtain the principalinstability regions. Further, it was observed that therewere no qualitative changes in the instability regionsdue to the cracks. The quantitative changes due to vari-ations in blade length, rotational speed, crack depth andcrack location have been discussed in detail.

References

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