Edge states in nonlinear model of valve spring

We address the dynamics of helical compression valve springs of an internal combustion engine. To this end, the spring is mathematically modeled as a finite non-homogenous one-dimensional mass-spring-damper discrete chain. Regarding the boundary conditions, the upper end of the chain is forced with periodic displacement, which mimics the actual camshaft profile, while the other end is fixed. In order to model the interaction between the valve and valve seat, the displacement of the upper mass is constrained to be nonnegative by adding an obstacle such that when it approaches the obstacle, it experiences an impact that satisfies the Newton impact law with restitution coefficient less than unity. Another source of damping in this model arising from the internal damping of the spring material. The nonlinearity of the model originates from the periodic impact interactions. This interplay between nonlinearity and discreteness supports time-periodic and spatially localized solutions characterized by a strong localization at the edge of the chain (i.e. edge states) such that the periodicity of the impact allows derivation of exact analytical solutions for the forced-damped edge state. Then, the governing equations are solved numerically in order to illustrate the exact solution. The results are compared to experimental findings from analysis of actual automotive valve spring.