Speed Oscillations of a Vehicle Rolling on a Wavy Road

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.