Asymptotic analysis of bilinear oscillators with preload

Bilinear oscillator is a spring-mass system with the spring having different stiffnesses in tension and compression. We analyze the behavior of the bilinear oscillator under a constant preload assuming that the spring has lower stiffness in tension than in compression. We formulate the conditions of the existence of the asymptotics of low stiffness ratios. Under these conditions, the bilinear oscillator can be approximated by an impact oscillator (an oscillator with zero stiffness ratio) under the same preload. In this asymptotics the damping of the stiff (compressive) part is modeled by the introduction of a restitution coefficient. We found that the use of the impact oscillator as an asymptotics of low stiffness ratios is only possible for sufficiently small preloads. We also determined the values of the restitution coefficient that simulate the given damping in the high stiffness region. The relation between the damping and the restitution coefficient is found to be independent of the magnitude of the preload.