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
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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)
WCECS 2018
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)
WCECS 2018
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)
WCECS 2018
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)
WCECS 2018
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)
WCECS 2018
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
Parametric Excitation and a Hooke’s Joint Type of
Kinematic Constrain. IEEE Africon 2011, 978-1-
61284-992-8, Livingstone, Zambia, September 13-
16 2011.
[2] E. Esmailzadeh and K.A. Farzaneh, Shimmy
vibration analysis of aircraft landing Gears. Journal
of Vibration and Control Vol. 5: p.45-56, (1999).
[3] M.R.F. Kidner and R.I. Wright, Vibration
Absorbers: A Review of Applications in Interior
Noise Control of Propeller Aircraft. Journal of
Vibration and Control Vol.10: 1221-1237, (2005).
[4] T. Otake, Prediction of Torsional Vibration
Caused by Hook's Joint in Drive Train Total
Vehicle Dynanics, 1MechE XXX IV FISITA
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[5] H. B. Pacejka, Tyre Factors and Front Wheel
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No 2, p. 97-119,(1980).
[6] A.A. Alugongo, Parametric excitation and Wavelet
Transform Analysis of a Ground Vehicle Propeller
Shaft, Journal of Vibration and Control, DOI:
10.1177/1077546312463746145, 23 October
(2012).
[7] A.P, Catchpole, S.P., Healy and D. Hotgetts,
Torsional Vibrations of a Vehicle Drive Line,
Journal of Mechanical Design 100(4), 644-650,
October 01, 1978.
[8] Rabeih, E. and El-Demerdash, S., Investigation of
the Vehicle Ride Vibration Effect on the Driveline
Fluctuations, SAE Technical Paper 2002-01-3065,
2002, doi:10.4271/2002-01-3065.
[10] Vandenplas, B., Gotoh, K., and Dutre, S.,
Predictive Analysis for Engine/Driveline Torsional
Vibration in Vehicle Conditions using Large Scale
Multi Body Model, SAE Technical Paper 2003-01-
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)
WCECS 2018
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