Design and Development of a Pendulum Type Dynamic Vibration Absorber for a Sdof Vibrating System...

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) IAEME

214

DESIGN AND DEVELOPMENT OF A PENDULUM TYPE

DYNAMIC VIBRATION ABSORBER FOR A SDOF VIBRATING SYSTEM

SUBJECTED TO BASE EXCITATION

Irshad M. Momin1, Dr. Ranjit G. Todkar

2

1Assistant Professor, Department of Textile Engineering, DKTEs Textile & EngineeringInstitute, Ichalkaranji, Maharashtra, India. E mail: [email protected]

2Professor, Department of Mechanical Engineering, Annasaheb Dange College of Engineering

and Technology, Ashta, Maharashtra, India.E mail: [email protected]

ABSTRACT

The use of centrifugal pendulum for dynamic vibration absorber design (CPVAs) is a

proven method for reducing undesired torsional vibrations in rotating systems. These devices are

in use for many years, most commonly in light aircraft engines, helicopter blade rotors etc. These

devices have also been reported for reduction of rectilinear vibrations. Pendulum type dynamicvibration absorbers use impact forces for effective reduction of rectilinear vibrations describing

modeling method and transient state analysis in view of spring impact absorbers, floating impact

absorbers and pendulum impact absorbers. Bond graph technique for modeling of SDOFvibrating system excited by rotation unbalance at the sprung mass using a pendulum type

dynamic vibration absorber is reported in the recent literature. This paper deals with the

modeling and design procedure for a centrifugal pendulum type dynamic vibration absorber(CPVA) subjected to base excitation. This paper presents a detailed analysis and experimental

investigations of the effect of parameters affecting the motion transmissibility of the sprung mass

such as size of the pendulum mass, eccentricity of the pendulum pivot with respect to axis ofrotation of the pendulum assembly, mass ratio (ratio of the pendulum mass to the sprung mass),

gear ratio (the ratio of the pendulum rotational frequency to the excitation frequency) and thefrequency ratio (the ratio of the excitation frequency to the natural frequency of the SDOFsystem). It has been proved that the CPVA is effective in reduction of motion transmissibility of

the sprung mass of the SDOF system with the proper selection of the affecting parameters.

Keywords: Centrifugal Pendulum Vibration Absorber (CPVA), Motion transmissibility, SDOF,

Mass ratio, Gear ratio, Frequency ratio.

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

AND TECHNOLOGY (IJMET)

ISSN 0976 6340 (Print)

ISSN 0976 6359 (Online)

Volume 3, Issue 3, Septmebr - December (2012), pp. 214-228

IAEME: www.iaeme.com/ijmet.htmlJournal Impact Factor (2012): 3.8071 (Calculated by GISI)

www.jifactor.com

IJMET

I A E M E


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1. INTRODUCTION

The use of centrifugal pendulum for dynamic vibration absorber design (CPVAs) is aproven method for reducing undesired torsional vibrations in rotating systems. These devices are

in use for many years most commonly in light aircraft engines, helicopter blade rotors [1].

Vibration suppression of helicopter blade assembly is extremely important as it is principal causeof the entire system vibration leading to harmful effects to the machinery and the passengers. In

this case the blade vibrating system can be modeled as a 2DOF as well as 3DOF vibrating

system consisting of a rigid blade and a pendulum type dynamic vibration absorber system basedon non-linear dynamics [2]. Pendulum type dynamic vibration absorbers have also been reported

using impact forces for effective reduction of rectilinear vibrations describing modeling method

and transient state analysis in view of spring impact absorbers, floating impact absorbers and

pendulum impact absorbers [3]. These absorbers continue to find new applications due toincreasingly stringent vibration suppression requirements in variety of technical fields like

automotive racing car engine crank shafts etc. A complete modeling and analysis method

correlating the theory with experimental investigation has been reported recently. The method ofmodeling using bond graph technique and design of a dynamic vibration absorber for a SDOF

vibrating system subjected to excitation induced by rotational unbalance at the vibrating mass

using a rotating pendulum system has been presented. It takes into account the effect ofpendulum as well as system parameters and the presentation is said to be a fundamental design

related contribution [4].

This paper deals with the modeling and design procedure for a centrifugal pendulum type

dynamic vibration absorber (CPVA) for SDOF system subjected to base excitation.

2. THEORETICAL ANALYSIS

Figure 1 shows a schematic representation of a SDOF system associated with a pendulumtype dynamic vibration absorber. It shows a SDOF vibrating system which may be a simple

model of any real physical system subjected to base excitation u(t) resulting sprung mass

displacement x(t). Sprung mass M (the total mass i.e. sprung mass in addition to the mass ofpendulum assembly rotating drive system) is supported by suspension spring having spring rate k

and system damping having coefficient of viscous dampingc.

The pendulum assembly consists of a pair of absorbing masses (mp/2 each) suspendedfrom the top of the vertical rotor free to rotate about Z- axis. A speed controlled dc motor drives

the vertical shaft carrying pendulum assembly through at a controlled angular speed & such that

&= where is the gear ratio. The system has a mass ratio i.e. ratio of absorber mass mp to

the main massM. The pivots supporting the pendulums have an eccentricity e with respect to the

Z-axis. Pendulum masses (mp/2 each) are fixed diametrically opposite at a distance l from the

centre of the pivot in each case. The masses (mp/2 each), eccentricity e and length l are variables.

There are two possible states i) &=0 (the vibration absorber is ineffective) and

ii) &0 (the vibration absorber is effective). The system is analyzed for tuning the angular

rotational speed of the pendulum assembly so that motion transmissibility of the sprung mass is

controlled in the neighborhood of natural frequency of the system.


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2.1 Equations of motion

Figure 1: Schematic representation of the CPVA.

Equations of motion have been developed as follows,

1) At massM, the equation for the force balance is as under,

+ 2 2 ( + ) = ( ) ( )Let = , = & = 2 where is the angle made by the pendulum with respect to the horizontal during pendulum

rotation. For small values of, sin~, cos~1.

(1 + )

+ 2 +

+

= (2 +

) (1)

2) At the pivot O, the equation for torque balance is as under,

2 2 + 2 = 2 2 ( + ) 2 2 + 2 2 (2)where is frictional torque developed during pendulum oscillation process. Assuming Tfr=0.

= + + ( + ) (3)

O

c k

Primary Mass M

2

l

e

x t

u t

Z

YX

Absorber mass

Pendulum drive system


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Substituting equation (3) in equation (1) and simplifying we obtain,

+ + + + = + + + (4) where

= (1 + ) + , = 2 + , = + + (1 + ) ( + ) , = 2 ( + ) , = ( + ) and = 2 + , = + , = 2 ( + ) , = ( + )

Assuming harmonic solution of the type ( ) = & ( ) = and following theregular procedure for solution,

= = + + + + + + The motion transmissibilityMtis expressed as,

| | = + + + + + + (5) Setting the numerator of equation (5) equal to zero we obtain,

= () (6) Equation (6) provides a tuning rule for nullifying the motion transmissibility Mt. Equation (7)

represents the simplified form of equation (5).

= ( )(1 + 4 ) + ( ) + 2 ( )(7)

where,= , F = ( + ) , A = + , = (1 + )From equation (7) it can be concluded that in the parametric analysis, the parameters

affecting the motion transmissibility Mt are system damping ratio , frequency ratio

gear ratio , eccentricity e, length l and mass ratio.

3. MOTION TRANSMISSIBILITY ANALYSIS

The motion transmissibility analysis has been developed for lightly damped system(=0.05) and the frequency ratio in the neighborhood of resonance i.e. ( =0.9)

3.1 Effect of Radius of Pendulum Mass (r)

Figure 2shows the effect of the variation of the radius r (8, 14 and 20 mm), eccentricitye = 10 mm, mass ratio = 0.06 and gear ratio =1 when the pendulum length l is varied in the


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International Journal of Mech6340(Print), ISSN 0976 6359(O

range 30 to 100 mm. The corres

in Table 1

Table 1: Val

Pen

It is seen that larger spherical ba

30 to 60 mm. The pendulum len

3.2 Effect of Eccentricity (e)Figure 3shows the effect

r= 14 mm, mass ratio = 0.06 arange 30 to 100 mm. The corrtabulated in Table 2.

anical Engineering and Technology (IJMETline) Volume 3, Issue 3, Sep- Dec (2012) IAE

218

onding values of motion transmissibilityMtha

Figure 2:Mtvs l(Effect ofr).

ues ofMtfor e=10 mm, = and = 0.06ulum length

l(mm)

Mt

r(mm)8 14 20

50 6.886 5.831 4.292

70 4.364 3.893 3.210

90 3.122 2.873 2.505

ll radius ris preferred for smaller pendulum len

th is more effective in the range l = 60 mm on

of the variation in the eccentricity e (0, 10, an

nd gear ratio =1, when the pendulum lengthsponding values of motion transmissibility ra

), ISSN 0976 E

e been tabulated

gth l in the range

ards.

20 mm), radius

l is varied in theio Mt have been


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Figure 3: Mtvs l(Effect ofe).

Table 2: Values ofMtforr =14 mm, = and = 0.06Pendulum length

l

(mm)

Mt

e (mm)

0 10 20

50 0.5929 5.831 11.090

70 0.3157 3.893 10.24090 0.1945 2.873 8.010

It is seen that the eccentricity (e) plays a very important role in reducing the motion

transmissibilityMt. Vibration suppression is more effective for pendulum length l in the range 70

to 100 mm when e=0.

3.3 Effect of Mass Ratio ()Figure 4 shows the effect of the variation in the mass ratio (0.04, 0.06 and 0.08),

radius r= 14 mm, eccentricity e = 10 mm and gear ratio = 0.94, when the pendulum length l isvaried in the range 30 to 100 mm.


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Table 3: Value

Pendu

It is observed that= 0.08 is

range 30 mm to 100 mm andMt

3.4 Effect of Gear Ratio ()Figure 5 shows the effec

mm, eccentricity e = 10 mm and

range 30 to 100 mm. The corrtabulated in Table 4.


220

igure 4: Mtvs l(Effect of).

s ofMt forr =14 mm, =0.94 and e =10 mm.

lum lengthl

mm)

Mt

0.04 0.06 0.08

50 1.040 0.6522 0.4749

70 0.1856 0.1252 0.094490 0.7595 0.5325 0.4099

ore effective for increasing values of pendulu= 0 at l= 65 mm.

of the variation in gear ratio (0.9, 0.95 and

mass ratio = 0.06 when the pendulum length

sponding values of motion transmissibility ra

), ISSN 0976 E

length l in the

1), radius r= 14l is varied in the

io Mt have been


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Table 4: Valu

Pend

It is seen that gear ratio = 0.9and gear ratio = 0.95 is more e

4. EXPERIMENTAL SET-UPFigure 6 shows the expe

investigation of effectiveness of

harmonic base excitation. Thependulum drive D.C. motor (7)

The suspension springs (2) are s

excited with harmonic excitatioCPVA consisting of a pair of tw

pendulum drive D.C. motor (7).

(8). The measurement of exper

with the help of FFT analyzerfrequency ratio () in the neighb


221

Figure 5: Mtvs l(effect of).

es ofMtforr=14 mm, e=10 mm and = 0.06

ulum lengthl

(mm)

Mt

0.9 0.95 1

50 1.041 1.248 5.831

70 1.464 0.3336 3.89390 1.698 1.398 2.873

s more effective for pendulum length l in the raffective in the range 53 to 100 mm.

rimental set-up designed, developed and used

the developed CPVA for a SDOF vibrating sy

sprung mass M (1) consisting of the mass oand the sliding base (3) is supported on two h

pported on a sliding base assembly (4) and th

at its base (4) by an electro-magnetic vibratiopendulums with masses mp/2 (6) each is attac

The entire system is mounted on a massive co

mental values of motion transmissibility (Mt)

(Multi-purpose data acquisition system) forrhood of natural frequency of the sprung mass

), ISSN 0976 E

nge 30 to 52 mm

for experimental

tem subjected to

f the pendulum,lical springs (2).

entire system is

exciter (5). Theed to the shaft of

crete foundation

was carried out

arious values ofsystem.


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Figure 6: Experimental Set-up.

5. EXPERIMENTAL INVESTIGATIONSTable 5 and Table 6 show the details of the SDOF vibrating system subjected to

harmonic excitation at the base and of pendulum system respectively.

Table 5: Details of SDOF system under investigation.

Sr. No. Item. Magnitude

1 Sprung mass (M) in kg. 2

2 Damping ratio () of the primary system (determined byexperimental half power point method) 0.068

3 Effective spring rate of suspension springs kused in the primary

system in N/m

2855.38

4 Undamped frequency of the SDOF system Mkn= in Hz. 6

Table 6: Details of Pendulum mass under investigation for eccentricity e=0 mm.

Set No. Radius (r) of each spherical ball

(mm)

Mass of each spherical

ball i.e. (mp/2) in Kg

Mass ratio

()

I 10 0.035 0.035

II 13 0.07 0.07


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5.1 Verification of Tuning relation for Set No. ITable 7 shows the experimental values ofMt for observing the effectiveness of the

developed CPVA for variation in excitation frequency ratio in the neighborhood of resonance

frequency (0.833, 1.00, 1.166 Hz) and the corresponding pendulum assembly rotational speed in rpm for theoretical and experimental comparison purpose when eccentricity e and radius rare

0 mm and 10 mm respectively.

Table 7: CPVA withr=10 mm & e=0 mm.

Figure 7, Figure 8 and Figure 9 show the experimental observations in graphical form

respectively.

Figure 7: Mt vs for e=0 mm,r=10 mm & l=50 mm.

Frequencyratio

()

Pendulumlength

(mm)

Mt

CPVA

Off

MtCPVA

On

Theoreticalspeed

(rpm)

Experimentalspeed

(rpm)

0.833

50 1.253 0.780 302 285

70 1.000 0.776 301 295

90 1.230 0.771 300 275

1.000

50 5.080 0.641 363 368

70 4.910 0.551 362 335

90 4.390 0.615 361 350

1.166

50 1.339 0.698 423 435

70 1.461 0.490 422 390

90 1.479 0.476 421 408


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It is seen that the motion transmissibility Mt is substantially reduced in the neighborhood ofresonance frequency for the pendulum length l = 70 mm for Set No. I.


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5.2Verification of Tuning relation for Set No. IITable 8 shows the experimental values ofMt for observing the effectiveness of the

developed CPVA for variation in excitation frequency ratio in the neighborhood of resonance

frequency (0.833, 1.00, 1.166 Hz) and the corresponding pendulum assembly rotational speed in rpm for theoretical and experimental comparison purpose when eccentricity e and radius rare

0 mm and 13 mm respectively.

Table 8: CPVA withr=13 mm & e=0 mm.

Figure 10, Figure 11 and Figure 12 show the experimental observations in graphical form

respectively.


Frequency

ratio

()

Pendulum

length

(mm)

Mt

CPVA

Off

Mt

CPVA

On

Theoretical

speed

(rpm)

Experimental

speed

(rpm)

0.833

50 1.130 0.809 304 270

70 1.166 0.804 302 280

90 1.250 0.850 301 298

1.000

50 5.391 0.882 365 361

70 5.535 0.600 362 32090 5.636 0.648 361 330

1.166

50 1.460 0.706 425 412

70 1.356 0.684 422 430

90 1.372 0.534 421 380


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It is seen thatthe motion transmissibilityMt is substantially reduced in the neighborhood

of resonance frequency for the pendulum length l = 70 mm for Set No. II


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6. CONCLUSION

The objective of this paper is to present modeling, design and development of aCentrifugal Pendulum type Dynamic Vibration Absorber (CPVA) to suppress rectilinear

vibrations of a SDOF vibrating system subjected to base excitation. The expression for motion

transmissibility (Mt)has been derived in terms of the pendulum parameters such as size of the

pendulum mass, eccentricity of the pendulum pivot with respect to the axis of rotation of thependulum assembly, mass ratio, gear ratio and the frequency ratio. The expression for the

rotational speed of the pendula (i.e. tuning speed of the pendulum) has also been derived. This

tuning relation matched with the relation found by the research scholars earlier by different

methods.The expression for (Mt) has been analyzed to study the effect of pendulum parameters on

motion transmissibility (Mt). This analysis is quite useful in the component selection and in the

design of CPVA system. Beside this it also proves that for a given excitation frequency, thependulum parameters have great effect in suppression of vibrations. Experimental investigations

are further carried out in the neighborhood of the natural frequency of the sprung mass for the

proof of theoretical analysis. Some assumptions are made while deriving the expression for (Mt)

such as the friction in pendulum pivot is assumed to be zero which may not be exactly valid inactual practical conditions. It is seen that the vibration suppression of the sprung mass system is

strongly dependent on the tuning speed of the pendula and even little variation in its speed maynot give the desired results.

While performing the experimentation work the speed of the dc motor is controlled

manually which is bit complicated task therefore the speed of the motor is adjusted in such a

manner that the values for motion transmissibility (Mt) of sprung mass system are reduced in arange of 60 to 80% as compared to CPVA off conditions. The actual rotational speed of the

pendula is found to be in a close agreement range of theoretical values and a substantial

reduction of motion transmissibility (Mt) of the sprung mass system is found to be quiteimpressive especially in the neighborhood of resonance conditions with the proper selection of

the affecting parameters.

8. REFERENCES

[1] T. M. Nester, P. M. Schmitz, A. G. Haddow and S. W. Shaw 2004 Experimental

Observations of Centrifugal Pendulum Vibration Absorbers The 10th

International Symposiumon Transport Phenomena and Dynamics of Rotating Machinery Honolulu, Hawaii, March 07-11,

2004.

[2] Imao Nagasaka, Yukio Ishida and Takayuki Koyama 2008 Vibration Suppression of

Helicopter Blades by Pendulum Absorbers (First Elastic Mode of the Blade) ENOC-2008, Saint

Petersburg, Russia, June, 30- July 4 2008.

[3] S. Polukoshko, V. Jevstignejev and S. Sokolova 2011 Impact Vibration Absorbers of

Pendulum Type ENOC-2011, Rome, Italy, 24-29 July 2011.

[4] Raul G. Longoria and Vinoo A. Narayanan 1997 Modeling and Design of an Inertial

Vibration Reflector Journal of Mechanical Design, Vol. 119, No. 1, pp. 20-27, March 1997.


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[5] B. Diveyev, V. Hrycaj, T. Koval, V. Teslyuk 2009 Pendulum Type Dynamic Vibration

Absorber Applications Lviv National Polytechnic University (Ukraine), Computer AidedDesign Department.

[6] Cheng-Tang Lee & Steven W. Shaw Torsional Vibration Reduction in Internal Combustion

Engines using Centrifugal Pendulums. Department of Mechanical Engineering, Michigan StateUniversity, East Lansing, M148824.

[7] Singiresu S. Rao, 2005, Mechanical Vibrations, Pearson Education Pte. Ltd. Singapore.

[8] Korenev B. G. and L. M. Reznikov, 1993, Dynamic Vibration Absorbers: Theory andTechnical Applications, John Wiley and Sons, Chichester.

Design and Development of a Pendulum Type Dynamic Vibration Absorber for a Sdof Vibrating System...

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