A Study on the Modeling and Simulation Method

of Torsional Vibration Considering Dynamic

Properties of Rubber Parts for Engine Crankshaft

System with Rubber Damper Pulley

Tomoaki Kodama and Yasuhiro Honda Department of Science and Engineering, Kokushikan University, Tokyo, Japan

Email: kodama@kokushikan.ac.jp , honda@kokushikan.ac.jp

Abstract— In this paper, we first describe the dynamic

properties of rubber parts of rubber damper pulley that are necessary for the modeling and numerical simulation of

torsional vibration. Secondly, we describe an experiment in

which two crankshaft pulleys with a torsional rubber

damper pulley are fitted to a 6-cylinder, high-speed diesel

engine. Torsional waveforms of the rubber damper inertia ring and the pulley are measured by means of phase-shift

torsiograph equipment. The measured waveforms are

harmonically analyzed and the dynamic properties of the

torsional stiffness and the torsional damping are investigated from an experimental viewpoint. As a results of

comparisons with experimental data, certain dynamic

properties of damper pulleys with a torsional rubber

damper have been clarified. The model used for the

numerical simulation of the torsional vibration is a multi -degree-of-freedom equivalent torsional vibration system.

The tension acting on the damper pulley and the rotational

resistance of the alternator, the cooling water pump, the

valve train system, etc., as well as the frictional resistance of

other accessories, were considered. Moreover, as a numerical simulation method of the torsional vibration, a

transition matrix method is adopted. The validity of this

method has been confirmed from comparison and

examination of measured values of torsional vibration and

simulation values results.

Index Terms— damper pulley, rubber parts, torsional

vibration, modeling, simulation method, dynamic properties,

engine crankshaft system, experiment, numerical simulation

I. INTRODUCTION

The important requirements in the design stage are

that the vibration and noise (NVH) levels of automobile

engines be reduced and that the timbre be improved

further [1]-[4]. Torsional v ibration dampers of high

performance, and crankshaft pulleys with a torsional

rubber damper (hereafter called "damper pulleys"), which

can reduce both torsional and bending vibrations, and

flywheels with a torsional rubber damper have been

widely employed in automobile engines [5]-[8]. When

designers estimate the amplitude of angular displacement

of engine crankshafts with the abovementioned damper

Manuscript received July 1, 2019; revised May 21, 2020.

pulleys in the design process, the following issues are of

concern:

[1] the estimation of dynamic properties (torsional

stiffness and damping of the damper pulley).

These dynamic properties are indeterminable [9],[10].

Therefore, an experimental equation for determin ing the

values has not existed until now and the quantitative

values are inaccurate. Unless these values can be

accurately estimated to a certain degree, it is d ifficult to

achieve high-precision computation. We clarify some of

the complicated dynamic properties of the rubber

vibration isolator by measuring the torsional vib rations of

the engine crankshaft system with a damper pulley. The

model used for the numerical simulation of the torsional

vibration is a multi-degree-of-freedom equivalent

vibration system. The tension acting on the damper pulley

and the rotational resistance of the alternator, the cooling

water pump, the valve train system, etc., as well as the

frictional resistance of other accessories, were considered.

Moreover, as a numerical simulation method of the

torsional vibration, a t ransition matrix method is adopted.

The validity of this method has been confirmed from

comparison and examination of measured values of

torsional vibration and numerical simulation values

results.

II. EXPERIMENTAL EQUIPMENT AND EXPERIMENTAL

METHOD FOR MEASUREMENT OF TORSIONAL

VIBRATION WAVEFORMS

This section concerns the main specifications of the

test engine and damper pulley, as well as the method of

measuring the torsional vibration waveforms. In the

experiment, the engines are operated under fu ll load to

cause steady forced torsional vibration.

A. Main Specifications of Test Engine and Rubber

Damper Pulley

The test engine is a 6-cylinders, in-line, high-speed

diesel engine [5], [6]. Fig. 1 shows the dimensions of the

test damper pulleys. This Table I. also shows the static

stiffness and natural frequency obtained from the static

and free vibrat ion tests, respectively [5], [6]. Types A and

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International Journal of Mechanical Engineering and Robotics Research Vol. 9, No. 7, July 2020

© 2020 Int. J. Mech. Eng. Rob. Resdoi: 10.18178/ijmerr.9.7.1007-1011

B of the damper pulleys differ in Shore hardness but their

rubber parts have the same shape and dimensions. The

materials are similar-quality rubber (natural rubber and

nitrile rubber).

B. Method of Measuring Torsional Vibration Waveforms.

An eddy current dynamometer was connected via a

universal joint to the flywheel of the engine. The test

damper pu lley was fitted to the end of the crankshaft. The

gears for generating signal pulses were mounted on the

damper inert ia ring and pulley. The electric frequency

signals proportional to engine speed were obtained from

the electromagnetic pickup. The measured signals were

transmitted to the phase-shift torsiograph equipment via

the adapter which numerical calculated the average of

angular velocity (the center frequency). The torsional

vibration waveforms could be obtained from the torsional

angles, which were numerical calculated using the

relationship between the measured and center frequencies.

The measured torsional waveforms of the damper inertia

ring and pulley were harmonically analyzed using the

F.F.T. analyzer. The torsional angular d isplacement was

measured under full load from 800 [r/min] to 3200

[r/min]. The indicator diagrams, data from which were

necessary for the vibration analysis described later, were

measured using the piezotype indicator in the sixth

cylinder from the pulley side.

Figure 1. Dimensions and shape of test torsional vibration rubber damper pulley.

III. NUMERICAL CALCULATION METHOD FOR

DYNAMIC PROPERTIES OF RUBBER PARTS OF

DAMPER PULLEYS

A. Measured Amplitude Curves of Angular Displacement

Fig. 2 shows the measured amplitude curves of

angular displacements at the pulley end of the crankshaft

with the B-type damper pulley. These figures show

merely the main-order amplitude curves of angular

displacements. The large 6-th order resonant point occurs

at 2524 [r/min] and its resonant amplitude is 14.9x10-3

[rad] in the case of installing no damper pulley. Since the

tuning ratio of the B-type damper pulley is not optimum,

the 6-th order amplitude is not greatly reduced.

TABLE I. MAIN SPECIFICATIONS OF TORSIONAL VIBRATION RUBBER

DAMPER PULLEY.

Name of Rubber Damper Pulley Damper Pulley A Damper Pulley B

Measured Natural Frequency of

Rubber Damper Pulley [Hz]

148.0 228.0

Inertia Moment of Damper Inertia

Ring [kgm2]

0.0369 0.0369

Inertia Moment of Damper Pulley

[kgm2]

0.0462 0.0462

Static Torsional Spring Constant [Nm/rad]

2.548×104 3.332×10

4

Material of Rubber Natural Rubber and Nitrile Rubber

Figure 2. Measured Torsional Amplitude Curves of Angular Displacement [with Rubber Damper Pulley: B-Type, Amplitude of

Torsional Angular Displacement Curves at Pulley End].

B. Numerical Calculation of Dynamic Properties Values

from Measured Waveform.

The values of dynamic torsional stiffness and

damping coefficient of damper rubber parts can be

obtained from the harmonically analyzed results of

waveforms measured at the damper inertia ring and the

pulley. The equation of motion at the damper inert ia ring

is

0PULRDPRDPPULRDPRDPRDPRDP

KCI (1)

where tj

O,RDPRDPe and

tj

O,PULPULe

.

The following Eq. (2) can be obtained by rearranging the

expression of Eq. (1) by substituting the above-mentioned

relational expression into Eq. (1) and rearranging;

RDPRDPRDP

RDPRDPRDPRDPRDP

RDPcosMM

cosMMIK

212

2

,

RDPRDPRDP

RDPRDPRDPRDP

RDPcosMM

sinMIC

212

2

(2)

In Eq. (2), the values of amplitude ratio RDPM and

phase angle RDP can be obtained by analyzing

harmonically the waveforms measured at the damper

inertia ring and the pulley, where the values of RDPI ,

RDP are known. Therefore, the values of dynamic

torsional stiffness RDPK and damping coefficient RDP

C

can be determined from Eq. (2). The dynamic properties

of the vibration isolator rubber depend generally on [1]

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International Journal of Mechanical Engineering and Robotics Research Vol. 9, No. 7, July 2020

© 2020 Int. J. Mech. Eng. Rob. Res

temperature, [2] frequency, and [3] average strain and

strain amplitude effects on the basis of same shape and

material. To reveal the effect of one factor among the

above-mentioned factors on the dynamic properties, it is

necessary to keep the values of the other factors constant

in the experiment. In our experiments, the surface

temperature of the rubber part was kept at 313 [K], but

the other factors could not be controlled due to the

vibration properties of the crankshaft. In full

consideration of these conditions of the other factors,

strain rate RDP

, which is the product of strain

amplitude and angular frequency, is defined by the

following equation.

RDP

RDPRDPlRe

RDPb

(3)

3

1

3

2

4

1

4

2

4

3

,RDP,RDP

,RDP,RDP

RDPrr

rr

(4)

Eq. (4) is obtained for the unifo rm, hollow, circular

cross section under the condition that the torsional

moment (in such a case that the shearing stress in the

representative radius RDP

is uniformly distributed over

the entire cross section) is equal to that (in the case that

shearing stress distribution x,RDPRDPRDP

CGRDP

caused by the simple shearing deformat ion in the

circumferential direction.

IV. NUMERICAL SIMULATION METHOD OF RESULTANT

TORSIONAL VIBRATION WAVEFORM

A. Derivation of General Expressions.

From what we have just discussed, vibration

calculations will be made by replacing the crankshaft

system with such equivalent vibration systems as shown

in Fig. 3. The equation of motion for the m th mass

can be expressed as follows.

tFKK

CCCJ

mmmmmmm

mm

'

mmm

'

mmmmm

111

111

(5)

Considering equations of motion for all masses ( n :

number of masses) from Eq . (5), they can be expressed in

matrix as follows.

FKCJ (6)

In this method, numerical calcu lations are made,

taking into consideration the constant coefficients in Eq.

(6). Therefore, it has the advantage that the time required

for making numerical calcu lations can be so much

reduced. Eq. (6) can be rewritten

FJKJCJ 111 (7)

If Eq. (7) is successively differentiated with respect to

time, taking the constant coefficient into consideration,

the n th derived function 2n of can be

written as follows.

21

2111

n

nnn

FJ

KJCJ (8)

Now if the angular displacements at the time t and

t are replaced by k

and 1k

respectively and the

angular velocit ies by k

and 1k

, 1k

and 1k

are

expand into a Taylor series, considering up to the i th

derived function, they can be expressed by Eq. (9).

i

k

i

i

k

i

kkkkk

!i!i

!!!

1

1

3

32

1

1

321

ik

i

kkkkk

!i

!!!

1

3211

4

3

3

2

1

(9)

Where in kt is the step size. If Eq. (8) is

repeatedly applied to Eq. (9), the second and higher

derivatives on the right hand side of these equations may

be written in terms of the angular displacements k

and

velocity k

.

k,i

i

k,i

i

k.k,

k,k,ki

i

i

i

ki

i

i

i

k

W!i

W!i

W!

W!

W!

WY!i

Y!i

Y!

Y!

Y!

YX!i

X

!iX

!X

!X

!X

1

1

3

3

2

2

101

1

3

3

2

2

101

1

3

3

2

2

101

132

11

321

1321

k,i

i

k,k,k,

ki

i

ki

i

k

W!i

W!

W!

W

Y!i

Y!

Y!

Y

X!i

X!

X!

X

121

121

121

1

3

2

21

1

3

2

21

1

3

2

211

(10)

Where,

KJE,CJD

FJWEWDW

FJWEWDW

FJWEWDW

W

XEYDYXEXDX

YEYDYXEXDX

YEYDYXEXDX

WIYX

YIX

i

kk,ik,ik,i

kk,k,k,

kk,k,k,

k,

iiiiii

k,

11

21

21

1

123

1

012

0

2121

123123

012012

111

00

0

00

0

(11)

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Therefore, in this section, the equations will be

derived, considering up to the fourth derived function. If

the angular displacement 1k

and velocity 1k

at the

time t are expanded into the Taylor series,

considering up to the fourth derived function, they can be

expressed by Eq. (12) instead of Eqs. (10) and (11).

kkkkkkFDFCFBAA

11112111

kkkkkkFDFCFBAA

22222211

(12)

Eq. (13) can be rearranged.

k

kk

kk

FD

D

FC

CF

B

B

AA

AA

2

1

2

1

2

1

2221

1211

1

(13)

Where ij

A , i

B , i

C , and i

D 2121 ,j,,i

which constitute the transition matrix. are part ial matrices

of nn . The part ial matrices ij

A , i

B , i

C , and i

D

can be defined by the step size which satisfies the

condition for not diverging during the process of repeated

numerical simulat ions, that is the stabilizing condition,

and also satisfies the condition for obtaining the correct

solution and by the various factors constituting the

equivalent vibration systems shown in Fig. 3. The

excitation torque k

F and the time differentials k

F and

kF in Eq. (13). The angular d isplacement

1k and

velocity 1k

at the time t can be numerical

calculated by giv ing in itial values of k

and k

. This

numerical calcu lation can be repeated continuously. Since

no steady waveform was obtained for the first two or

three periods, the subsequent periods were adopted as the

final results. Since the values of the excit ing torques in

Eq. (13) are equals to zero in the case of free v ibrations,

only the first term need to be considered.

V. ANALYTICAL INVESTIGATION OF TORSIONAL

VIBRATION PROPERTIES OF ENGINE

CRANKSHAFT WITH DAMPER PULLEY

A. Input Data Necessary for Numerical Simulation of

Torsional Vibration Waveforms

Fig. 3 shows the equivalent torsional v ibration system

of a crankshaft with a rubber damper pulley rep laced

according to the analytical method described in section

(Measured Amplitude Curves of Angular Displacement).

The rubber part of the damper pulley is replaced with a

Voight model. Since the simulation program can yield the

torsional vibration waveform for a given engine speed,

the value of the strain rate is numerical simulation using

the measured dominant-order amplitude of angular

displacement at a given engine speed. Then, the values of

the dynamic torsional stiffness RDP

K and damping

coefficient RDP

C of the rubber part can be obtained by

referring to Eqs. (2) to (4), respectively. These obtained

values of RDP

K and RDP

C are used as the input data for

the numerical simulation. Since the curve-fitted curves in

Eqs. (3) and (4) are representative of all the experimental

data, the obtained values of RDP

K and RDP

C can be

regarded as the mean values.

Figure 3. Multi-degree of freedom of equivalent torsional vibration system of an engine crankshaft system with a rubber damper pulley [numerical simulation model].

PUL

RDP

IPUL

IRDP

KRDP

CRDP

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International Journal of Mechanical Engineering and Robotics Research Vol. 9, No. 7, July 2020

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B. Numerical Simulation Results and Considerations.

Fig. 4 shows the main-order amplitude curves of the

angular displacements at the damper pulley end obtained

by analyzing harmonically the numerical simulat ion

torsional vibration waveforms of the crankshaft with the

rubber damper pulley. The numerical simulat ion

amplitude curves in th is figure correspond to the

measured amplitude curves of the angular displacements

in Fig. 2. As compared with the experimental results,

these numerical simulation results contain slight error,

but this degree of error is allowable in practice.

VI. SUMMARY

Accurate estimation of the dynamic p roperties of the

rubber part of the damper pulley are proble ms in the

numerical simulation of torsional vibration of a

crankshaft with a rubber damper pulley. The results of

investigation of these properties from experimental and

analytical viewpoints are as follows:

[1] We expressed the absolute torsional stiffness and

damping of the rubber part of the damper pulley as a

function of the strain rate under the conditions of the

same kind rubber material, constant temperature and

invariable shape factor.

[2] The valid ity of this method has been confirmed

from comparison and examination of measured values of

torsional vibration and simulation results.

CONFLICT OF INTEREST

The authors declare no conflict of interest

AUTHOR CONTRIBUTIONS

Tomoaki Kodama analyzed the experimental and

measured data, concuted the research, wrote the paper.

Yasuhiro Honda wrote the paper. All authors had

approved the final version.

REFERENCES

[1] T. Kodama, Y. Honda, K. Wakabayashi, S. Iwamoto, “A

calculation method for torsional vibration of a crankshafting system with a conventional rubber damper by considering rubber form, SAE 1996 International Congress and Exposition , SAE Technical Paper Series No. 960060, 1996, pp. 103-121.

[2] Y. Honda, K. Wakabayashi, T. Kodama, S. Hama, S. Iwamoto, “An experimental study on dynamic properties of rubber test specimens for design of shear-type torsional vibration dampers,“ in Proc. the Ninth International Pacific Conference on

Automotive Engineering, vol. 2, no. 971400 (Abstract Code:

00088), 1997, p. 217-222.

[3] Y. Honda, K. Wakabayashi, T. Kodama, H. Okamura, “An experimental study of dynamic properties of rubber specimens for a crankshaft torsional vibration of automobiles,” in Proc.

ASME 1999 International Design Engineering Technical

Conference and The Computers and Information in Engineering Conference, 17th Biennial Conference on Mechanical Vibration and Noise Symposium on Dynamics and Vibration of Machine Systems, Session VIB-00037: Dynamics and Vibration of

Machine System 02, DETC99/VIB-08125, 1999, pp. 1-14. [4] Y. Honda, T. Kodama, K. Wakabayashi, “Relationships between

rubber shapes and dynamic properties of some torsional vibration rubber dampers for diesel engines,“ in Proc. the 15th Pacific

Automotive Engineering Conference – APAC15, No.0320, 2009, pp. 1-8.

[5] T. Kodama, Y. Honda, “A study on the modeling consideration dynamic properties of vibration damper rubber parts,” in Proc.

2018_The 7th International Conference on Engineering and Innovative Materials ICEMI 2018, EM028, 2018, pp. 01-07.

[6] T. Kodama, Y. Honda, “A study on the modeling consideration dynamic properties of vibration damper rubber parts,” 2nd

International Conference on Robotics and Mechatronics, IOP Conference Series: Materials Science and Engineering 517, 012003, 2019, pp. 1-7.

[7] A. S. Mendes, P. S. Meirelles, “Application of the hardware-in-the-loop technique to an elastomeric torsional vibration damper,” SAE International Journal of Engines-V122-3, 2013, pp. 1-11.

[8] K. Yoon, I. Oh, J. I. Ahn, S. Y. Kim, Y. C. Chung, “A study for

fuel economy improvement on applying new technology for torsional vibration reduction of crank pulley,” SAE/KSAE 2013 International Powertrains, Fuels & Lubricants Meeting, SAE Technical Paper 2013-01-2514, 2013, pp. 1-7.

[9] A. K. Yadav, M. Birari, V. Bijwe, D. Billade, “Critique of Torsional Vibration Damper (TVD) design for powertrain NVH,” Symposium on International Automotive Technology 2017 , SAE Technical Paper 2017-26-0217, 2017, pp. 1-5.

[10] G. Nerubenko, “Engine noise reduction using self-tuning torsional vibration damper,” in Proc. SAE 2016 World Congress and Exhibition, SAE Technical Paper 2016-01-1063, 2016, pp. 1-9.

Copyright © 2020 by the authors. This is an open access article distributed under the Creative Commons Attribution License (CC BY-NC-ND 4.0), which permits use, distribution and reproduction in any

medium, provided that the article is properly cited, the use is non-commercial and no modifications or adaptations are made.

Tomoaki Kodama

h as a Ph.D. degree in

engineering, with the Course of Mechanical En gineer in g, Dep artment of Scien ce an d En gin eer in g, Sch o o l o f Sc ien ce an d Engineering, Kokushikan University, T okyo,

Japan.

His research interests are NVH of int ern al com bust ion en gines ( in cludin g automotive and

marine engines), engineering

education, experiments in mechanical engineering, and mechanical

design.

He can be reached by e-mail at

kodama@kokushikan.ac.jp.

Yasuhiro Honda

has a Ph.D. degree, with the

Course of Mechanical Engineering, Department of Science and Engineering, School of Science and Engineering, Kokushikan University,

Tokyo,

Japan.

His research interests are NVH of internal combustion engines, engineering education, vehicle kinematics,

mechanics, and automotive design.

He can be reached by e-mail

at

honda@kokushikan.ac.jp.

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© 2020 Int. J. Mech. Eng. Rob. Res