Speed Oscillations of a Vehicle Rolling on a Wavy Road

Walter V Wedig

doi:10.3390/app112110431

Abstract

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential-algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

Highlights

Introduction to the ProblemVertical vibrations of a vehicle driven by a constant force and rolling on a sinusoidal road surface are coupled with its horizontal travel motion, affecting the vehicle speed which fluctuates around a mean value
The coupling between both planar motions is caused by the permanent direction change of the contact force to ground along the contour of the road profile
First investigations of velocity jumps and turbulent speeds in nonlinear vehicle road dynamics are given by Wedig in [3,4,5,6,7] applying sinusoidal and random road models introduced by Robson et al [8,9,10]

Summary

Introduction to the Problem

Vertical vibrations of a vehicle driven by a constant force and rolling on a sinusoidal road surface are coupled with its horizontal travel motion, affecting the vehicle speed which fluctuates around a mean value. Equations (1)–(3) represent a DA equation system where the role of algebraic terms is taken by sinusoidal terms To eliminate these terms, the road level z and slope u in Equation (3) are differentiated with respect to the way coordinate s in order to obtain the increments dz = −z0Ω sin(Ωs)ds and du = −z0Ω cos(Ωs)ds that leads to the homogeneous nonlinear oscillator equations z. For analytical and numerical investigations, it is appropriate to introduce the dimensionless time τ = ω1t and the related speed v = vΩ/ω1, as well as the related coordinates (z, u) = (z, u)z0 and (y, x) = Ω(y, x) Their insertion into Equations (2), (4) and (5) leads to v + 2D(z0Ω)2u2v = z0Ω(uy + 2Dux) − (z0Ω)2zu + f Ω/c,. This is the reason why both quantities have opposite signs

Speed Driving Force Characteristic of Traveling Vehicles

Stable and Unstable Oscillation Amplitudes of Vehicle Speeds

Stability in Mean and Robustness with Regard to Disturbances

Summary and Remarks on Main New Results