Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles
Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles
Xinxiu Fan 0 1
Paul D. Walker 0 1
Qi Wang 0 1
B X. Fan 0 1
P.D. Walker 0 1
0 Faculty of Engineering and IT, University of Technology , Sydney, PO Box 123, 15 Broadway, Ultimo, NSW 2007 , Australia
1 School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics , 100191 Beijing , P.R. China
During the engagement of the dry clutch in automotive transmissions, clutch judder may occur. Vehicle suspension and engine mounts couple the torsional and longitudinal models, leading to oscillations of the vehicle body that are perceived by the driver as poor driving quality. This paper presents an effective formulation for the modeling and simulation of longitudinal dynamics and powertrain torsional dynamics of the vehicle based on non-smooth dynamics of multibody systems. In doing so friction forces between wheels and the road surface are modeled along with friction torque in the clutch using Coulomb's friction law. First, bilateral constraint equations of the system are derived in Cartesian coordinates and the dynamical equations of the system are developed using the Lagrange multiplier technique. Complementary formulations are proposed to determine the state transitions from stick to slip between wheels and road surface and from the clutch. An event-driven scheme is used to represent state transition problem, which is solved as a linear complementarity problem (LCP), with Baumgarte's stabilization method applied to reduce constraint drift. Finally, the numerical results demonstrate that the modeling technique is effective in simulating the vehicle dynamics. Using this method stick-slip transitions between driving wheel and the road surface and from the clutch, as a form of clutch judder, are demonstrated to occur periodically for certain values of the parameters of input torque from engine, and static and dynamic friction characteristics of tire/ground contact patch and clutch discs.
Clutch judder; Non-smooth dynamics; Longitudinal dynamics of vehicle; Lagrange's equations; Linear complementarity problem; Stick-slip
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Nomenclature
x1, x˙1 Longitudinal displacement/velocity of the center of vehicle body
y1, y˙1 Longitudinal displacement/velocity of the center of vehicle body
θ1, θ˙1 Pitch angular displacement/velocity of vehicle body
xc1, x˙c1 Longitudinal displacement/velocity of the center of front wheel
yc1, y˙c1 Vertical displacement/velocity of the center of front wheel
θc1, θ˙c1 Angular displacement/velocity of front wheel
xc2, x˙c2 Longitudinal displacement/velocity of the center of rear wheel
yc2, y˙c2 Vertical displacement/velocity of the center of rear wheel
θc2, θ˙c2 Angular displacement/velocity of rear wheel
θc, θf , θe Angular displacements of clutch discs, flywheel and engine
h Height of body mass center above roll axis
a Longitudinal distance of body mass center from rear axle
b Longitudinal distance of body mass center from front axle
k, c Stiffness and damping coefficient of the front or rear suspension
k1, c1 Stiffness and damping coefficient of the input shaft from engine to flywheel
k2, c2 Equivalent stiffness and damping coefficient of elastic half-shaft
Je Engine inertia
Jf Inertia of flywheel
Jc Equivalent inertia of the clutch disc and the transmission
Jc1, Jc2 The inertia of front or rear wheels
m1, mc1, mc2 Body mass, front or rear wheel mass
J1 Body inertia according to the mass center
Te, Tf Engine torque, friction torque acting on the clutch
TS , TC The maximum static or slipping friction torque
θ˙s the Stribeck velocity
Ff 1, Ff 2 Friction force acting on the front or rear wheel
μ0i , μi Static or dynamic friction coefficient on the contact patch between wheels
and ground
FN1, FN2 Vertical tire force acting on front or rear wheel
Mf 1, Mf 2 Tire rolling resistance acting on the front or rear wheel
δi Rolling resistance coefficient
sgn(x) If x > 0, sgn(x) = 1; if x = 0, sgn(x) = 0; if x < 0, sgn(x) = −1
Sgn(x) If x > 0, Sgn(x) = 1; if x = 0, Sgn(x) ∈ [
−1, 1
]; if x < 0, Sgn(x) = −1
1 Introduction
Clutch judder between friction pairs of sliding contact during the clutch engagement
process is typically defined as a friction-induced vibration and even torsional self-excitation
of the transmission system. These oscillations introduce undesired dynamic loads, increase
clutch slip and wear of dry clutches and reduce driver comfort [
1–7
]. Multibody simulation
tools are frequently used to assess the performance of a vehicle to offer the driver better
driving comfort characterized by the longitudinal dynamics of vehicle. The acceleration of
the vehicle is calculated by integrating the powertrain model with the equations
governing the longitudinal behavior of the vehicle [
4, 6
]. However, the simulation of a complete
acceleration process during the clutch engagement is not straightforward, as the kinematic
constraints imposed on the sys (...truncated)