Chaotic Vibration of a Quarter-Car Model Excited by the Road Surface   Profile
  The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos in the case of a quarter car model excited kinematically by the road surface profile. By analyzing the potential an analytic expression is found for the homoclinic orbit. By introducing an harmonic excitation term and damping as perturbations, the critical Melnikov amplitude of the road surface profile is found, above which the system can vibrate chaotically. 
Grzegorz Litak, Marek Borowiec, Michael I. Friswell, Kazimierz   Szabelski