Parametric Vibration of a Cardan Shaft and Sensitivity ...Hookes joint. Fly wheel. Gear box. Fig. 1 a Kinematic sketch of the cardan shaft system . Parametric Vibration of a Cardan

Abstract—In this paper a new model for analyzing self-

excited torsional vibrations of a heavy duty ground vehicle

cardan shaft is proposed. The model considers two heavy

inertias articulated by a circuit of torsional springs and a

damper. The inertias of the system at either ends are driven by

a gearbox through a Hooke’s joint. Traction resistance torques

at road wheels and engine are projected onto the shaft through

gearing transmission. The model has been used to study

sensitivity of the transmission to rigid body motion, elastic

deformation and stiffness perturbation of the Hooke’s joint. It

is shown that, cardan shaft vibration excitations are higher at

start-up and dampen away in after a short duration. Modelled

results reveal that the sensitivity of stiffness to velocity and

displacement inputs is higher at start-up than steady-state

motion of the shaft.

Index Terms—Cardan shaft, Coupling, Nonlinear, Torsional

Vibration

I. INTRODUCTION

A cardan shaft of a vehicle driveline is essential for

transmitting torque from the gearbox to the final drive [1-2].

Torsion is the most prevalent type of loading in ground

vehicle drivelines. Vibration of driveline systems is one of

the principal causes of noise in ground vehicle transmissions

[4-6]. Analysis of torsional vibration of a ground vehicle

driveline is a basic and an important step in safe design of

vehicle power transmission systems.

Identification of dynamic load spectra of a driveline

assembly is critical in evaluating the load carrying capacity,

structural integrity and maintenance of vehicle transmission

[2]. A theoretical study of driveline torsional vibration was

reported by Cathpole and Healy et al [7]. The research used

vibration histories taken from a number of stations along the

driveline, and passenger compartment noise levels recorded

in a series of road tests. Application of digital analysis to the

data revealed that, torsional resonances cause high noise

levels in a car. In [8], acceleration vibration response of a

rear driving axle caused by common excitation forces was

modeled by use of Finite Element software ABAQUS.

Manuscript received March 24, 2018; revised April 13, 2018. This work

was supported in part by Department of Mechanical Engineering, Vaal

University of Technology, Republic of South Africa.

AA Alugongo is with the Mechanical Engineering Department, Vaal

University of Technology, Vanderbijlpark 1900, Andries Potgieter BLVD,

South Africa,

phone: +27788018894; fax: +27169509797; e-mail: alfayoa@ vut.ac.za.

.

Rabeih, E. and El-Demerdash, S. [8], investigated the

effect of vehicle ride on the driveline vibration. Their study

established that, angularity of the driveshaft and its

universal joints cause torsional and bending vibrations in a

driveline. In [9], driveline torsional vibration in a vehicle

including a gearbox was investigated based on multibody

modeling of a car taking into account flexibility of major

components of the powertrain. In [10], transient

characteristics of a vehicle powertrain system were

investigated. The investigation compared free and forced

torsional vibration of the complex test rig with that of an

existing car.

This paper, presents a model for analyzing partial vibration of a driveline system. The system comprises, an elastic cardan shaft subsystem with a Hooke’s joint. The model is developed from vehicle dynamics and vibration theory. Parametric excitation due to Hooke’s joint is modelled as a perturbation of small order. The model has been used to determine fundamental torsional modes at low frequencies.

II. SYSTEM DESCRIPTION

The system model in figure 1b is developed by adopting basic dynamics of a Hooke’s coupling. The model considers a simple cardan shaft whose own inertia is represented by

four discrete rigid inertia disks, 1mJ , 2mJ , 3mJ ,

4mJ mounted on two massless torsional springs, 1k , 2k .

Inertia 1J comprises 1mJ , inertias of the, parts attached

to and rotating with the gearbox output shaft, inertias of the flywheel and of the parts of the engine rotating with the

gearbox output shaft. 4J comprises 4mJ , and inertias of the

parts attached to and rotating with the final drive shaft including the road wheels.

Shaft 1

[primary shaft] Shaft 2

[secondary shaft]

Bearing 1 Bearing 2

Bearing 3

Bearing 4

1T

4T

Hookes joint

Gear boxFly wheel

Pinion

Fig. 1 a Kinematic sketch of the cardan shaft system

Parametric Vibration of a Cardan Shaft and

Sensitivity Analysis

Alfayo A. Alugongo

Proceedings of the World Congress on Engineering and Computer Science 2018 Vol II WCECS 2018, October 23-25, 2018, San Francisco, USA

ISBN: 978-988-14049-0-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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1J

4J

1c

1k

2c

2k

2

3

4

2J

C

B

A

D

1b

2b

3b

4b

1mJ

2mJ

4mJ

3J

3mJ

Hookes joint

rT

T

Fig. 1b A dynamic model of cardan shaft system

The system’s d.o.f.s are lumped at the centres of the

inertias 1J , 2J and 4J . The approach assumes small

vibrations superimposed on the rigid-body motion of the gearbox output shaft. Individual inertias and their net displacements are as follows:

1J -Rigid-body rotation only

2J -Elastic deformation of spring 1k i.e., 1 superposed on

the rigid-body rotation of 1J

3J -Net displacement of 2J transmitted through the Hooke’s

joint

4J - Elastic deformation of spring 2k i.e., 2 superposed on

the rigid-body rotation of shaft 2 i.e. 2r

The model is developed by Lagrangian formalism.

III. DISPLACEMENT OF SYSTEM INERTIA

Let the net displacement of 3J be expressed as

3 2 2 (1)

Where, is a perturbation parameter that depends on

and 1 . Consistent with the assumptions and

considerations in section (2),

2 1 (2)

Substitution of equation (2) into equation (1) gives

3 1 1 (3)

In automotive design, the difference between input 2 and

output 3 motions are kept low to reduce vibration in the

coupling. This is achieved by keeping a low value of ,

usually below 6o . Consequently,

2 3 1 1 11 (4)

The kinematic relationship between 2 , 3 and is [1],

3 2tan tan , where cos (5)

Combining equations (2), (3), (4), (5), and allowing for

small angle approximation on 1 leads to

1

1 1

1

1 1

tantan lim

1 tan

(6)

Making the subject in equation (6) gives

1

2

1 1

tan

tan 1

a

(7)

Where 1a

Rigid-body rotation of4J ,

2r has been determined by

substituting, 1 0 in equations (3) and (7), whereupon,

2 2 1

r

an

n

(9)

Where, tann .

IV. SYSTEM KINETIC ENERGY

This comprises the kinetic energy of the parts of the driveline coupled to and rotating with the cardan shaft, and is expressed as

22

1 2 1

2

3 1 1

2

4 2 2

1 1

2 2

1

2

1

2r

G J J

dJ

dt

J

(10.1)

Differentiating equation (10.1), assigning and performing substitutions as follows:

2

2 2

1

11

r

d d an

dt dt cn

d ang

dt cn

(10.2)

Where,

1tanN ,

22

2 2

22

2 2

2 11

1 1

2 11

1 1

N NaN N

NN N

n nan ng

nn n

(11)

leads to equation (12) below

222

1 2 1 3 1

2

4 2

1 1 11

2 2 2

11

2

G J J J

J g

(12)

V. SYSTEM POTENTIAL ENERGY

This comprise the torsional strain energy between points

A and B and points C and D in figure 1b and is expressed

as,

22

1 1 2 1 2

1 11

2 2V k k (13)

VI. THE RAYLEIGH DISSIPATION FUNCTION

The Rayleigh’s dissipation function as follows:

2

2

1 1 2 1 2

1 11

2 2

dD c c

dt

(14)

Proceedings of the World Congress on Engineering and Computer Science 2018 Vol II WCECS 2018, October 23-25, 2018, San Francisco, USA

ISBN: 978-988-14049-0-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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VII. THE EQUATION OF MOTION

The Lagrangian equation in an i generalized coordinate

frame is

iq

i i i i

d G G D VT

dt q q q q

,

1 2 q , ,i (15)

Upon substitution of equations (13), (14) into equation (15),

performing requisite differentiation and manipulation, the

system dynamic equation reads,

1 2 1 2

1 1 1 1 2 1 1 1 1 2

2 2 1 2 2 2 2 1 2 2

1 1 1 2

2 1 2 2

1

2

1 1

2 2

0 0 0

0

0

m m m c c c

m m m c c c

m m m c c c

P

k k P

k k

1

2

1 1 1

2 1 4

gbs

s

s

P

TH R

H R T

H R T

(16)

Where, the elements of matrices in their final forms are,

2 2

1 1 1 1

2 1 1 2

2 2

2 2

1 2 3 4

4

2

2 3

4

1 1

1

1

0

m J J J J g

m m J g

m m m J J

m m

m J

1 1 1 2

2 1 1 2 2 2

2

1 2 2

2

1 , 1

,

k k k k k

k k k k

(17)

1

2

2

2 1

1

2

2 1 1

1 1

2 1

1

1

c c

c c

c c

(18)

1 1

1 1

1 2

2

1 2 1

1

2

2 1 1

1 1

2 1

1

1 1

1

1

c c

c c c

c

c c

(19)

2 2c c ,

2 1 1 2c c ,

2 2 2c c (20)

The vectors 1 1

T

s s sR R R ,

1 2

T

H H H ,

1 2

T

P P P correspondingly are, Hooke’s joint

quadratic-velocity excited torque, Hooke’s joint elastic

excitation torque (signifying elastic coupling elastic

deformations across the Hooke’s joint) and the i

T

q

term of

equation (15). The vectors are:

1

2

3 1 4 4 2

3 1

4

2 1 2 1

2 1

s

s

s

d dgJ J g J

dt dtRd

R Jdt

R dgJ

dt

(21)

1

2

2

2 1 1

2

2 2 1 1

1

1

1

0

H

H k

H

(22)

1

2

2

3 1 4 2

2

3 1

1 1

4 2

2 2

1 1

1

1

G gP J J g

GP J

G gP J g

(23)

The terms iq

,

i

g

q

in their final forms are,

2 2

2 3

1 1 2 1

2 ( 1)[(3 ) 3 1]

( 1)

0,

aN N N

N

g g

(24)

VIII. NUMERICAL SIMULATION AND RESULTS

The system was numerical solutions of equation (16) were

performed for: 1

1 2 200 N.m.s.radc c ,

1

0 20 rad.sec , 8 1

1 2 8 10 N.mk k ,

1 3o , 2 6o 2

1 4 rad.sa , 0.5 , 0.95 mr .

2 4 0 , 1

2 4 0.1 rad.s , 0.4 ,

Fig.3 Time-displacement history of gear box output shaft

Proceedings of the World Congress on Engineering and Computer Science 2018 Vol II WCECS 2018, October 23-25, 2018, San Francisco, USA

ISBN: 978-988-14049-0-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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Fig.4 : Elastic deformation of shaft 1 with gearbox displacement

Fig.5 Elastic deformation of shaft 2 with gearbox displacement

Fig.6 Rigid body displacement of shaft 2 with gear box output shaft

displacement

Fig.7 Variation of equivalent stiffness with displacement gearbox

output shaft

Fig.8 Variation of perturbation with displacement of gearbox output

shaft

Fig.9 Variation of non-dimensional equivalent stiffness with

perturbation

Fig.10 Sensitivity of nondimensional equivalent stiffness to gearbox

output shaft displacement

Fig.11 Elastic-torque excited by Hooke’s between shafts 1 and 2

Fig.12 Variation of elastic-torque excited by the Hooke’s joint and

rotation of the gearbox output shaft

Proceedings of the World Congress on Engineering and Computer Science 2018 Vol II WCECS 2018, October 23-25, 2018, San Francisco, USA

ISBN: 978-988-14049-0-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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Fig.13 Variation of quadratic-Velocity dependent torque excited by

Hooke’s joint with gearbox output shaft displacement

Fig.14Variation of quadratic-velocity dependent torque with gearbox

output shaft displacement

Fig.15 Variation of quadratic-velocity dependent torque with gearbox

output shaft displacement

Fig.16 Variation of quadratic-velocity dependent torque gearbox

output shaft displacement

Fig.17 Variation of quadratic-velocity dependent torque with the

gearbox output shaft displacement

Fig.18 Variation of dimensionless stiffness gearbox output shaft

displacement

Fig.19 Sensitivity of nondimensional stiffness to perturbation

parameter

Fig.20 FFT of the primary shaft perturbation

Proceedings of the World Congress on Engineering and Computer Science 2018 Vol II WCECS 2018, October 23-25, 2018, San Francisco, USA

ISBN: 978-988-14049-0-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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Fig.21 FFT of the secondary shaft perturbation

Fig.22 Sensitivity of nondimensional equivalent stiffness to

perturbation parameter

IX. CONCLUSION

A new model for analyzing self-excited torsional

vibrations of a heavy-duty ground vehicle cardan shaft

system has been developed. Quadratic-Velocity dependent

torques excited by the coupling of kinematic anisotropy of

the Hooke’s joint and the elastic motion of the primary shaft

have been analyzed. Elastic-torque excited by the coupling

of the primary and secondary shaft and due to change in

rigid motion across the Joint manifest a significant number

of excitation features in the power density and FFT

spectrum of the cardan shaft.

The cardan shaft vibration excitation is higher at start-up and

dampen in a short after start-up. Modelled results reveal that the

sensitivity of stiffness to velocity and displacement inputs are

higher at start-up than steady-state motion of the shaft.

ACKNOWLEDGMENT

The author acknowledges the Faculty of Engineering

and Technology, Vaal University of Technology, South

Africa, for facilitating this research.

REFERENCES

[1] A.A. Alugongo, A Nonlinear Torsional Vibration

Model of a propeller shaft with a Crack-induced

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61284-992-8, Livingstone, Zambia, September 13-

16 2011.

[2] E. Esmailzadeh and K.A. Farzaneh, Shimmy

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[3] M.R.F. Kidner and R.I. Wright, Vibration

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[7] A.P, Catchpole, S.P., Healy and D. Hotgetts,

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[8] Rabeih, E. and El-Demerdash, S., Investigation of

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[10] Vandenplas, B., Gotoh, K., and Dutre, S.,

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1726, 2003, doi:10.4271/2003-01-1726.

Proceedings of the World Congress on Engineering and Computer Science 2018 Vol II WCECS 2018, October 23-25, 2018, San Francisco, USA

ISBN: 978-988-14049-0-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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