Bifurcation analysis of a rear axle tramp car model

Abstract

Axle tramp is a self-sustaining vibration in the driven axle of a vehicle with a beam axle layout, known to occur under heavy braking or acceleration. A 6DOF mathematical model of this phenomenon is used to identify how the key parameters of driveline stiffness, axle mass and fore/aft stiffness change the system’s dynamics. A bifurcation analysis is performed to study this nonlinear system’s dynamics. Four Hopf bifurcations in the underlying equilibria, along with a fold bifurcation in the outermost limit cycle branch, are shown to bound the parameter space where tramp occurs. The severity of tramp was found to be minimised by increasing drivetrain stiffness, reducing axle mass or increasing fore/aft stiffness: the trade-off for minimising tramp severity is that it may be easier to excite tramp when the drivetrain stiffness is increased, and the speed range over which tramp can occur is increased as fore/aft stiffness is increased. A key outcome from this work is that future electrified powertrains may experience more tramp, albeit at a reduced magnitude, than their combustion-powered counterparts.