Closed Form Periodic Solutions for Piecewise-linear Vibrating Systems

Piecewise-linear oscillators are often considered as finite degrees-of-freedom models of cracked elastic structures [1-3] but may also represent specifics of original systems. In many cases, the corresponding periodic solutions combine different pieces of linear solutions valid for two different subspaces of the configuration space [4,5]. It is always preferable though to deal with closed form solutions especially when the solutions are involved in further transformations, for instance, perturbation procedures [6]. This work brings attention to the fact that non-smooth transformations of time [7] can automatically match both pieces of the solution by means of elementary functions. The reasonable approach however is to build the non-smooth transformation into the entire procedure rather then transform the solution after a perturbation tool has been applied. The technique is illustrated on multiple degrees-of-freedom piecewise-linear system Mẍ+Kx = εH(Sx)Bx (1) where x(t) ∈ R is a vector-function of the system coordinates, M is a mass matrix, H(•) is the Heaviside unit-step function, S is a normal vector to the plane splitting the configuration space into two parts with different elastic properties so that the stiffness matrix is K when Sx < 0 and K − εB whenever Sx > 0; a 2DOF case is shown in Fig. 1. It is assumed that the stiffness jump is small, 0 < ε ¿ 1. The perturbation term on the right-hand side of (1) is continuous but non-smooth. Therefore, only first-order asymptotic solution can be obtained within the classic theory of differential equations. However, at some stages, if being applied the idea of non-smooth time transformation improves the class of smoothness of the perturbation and therefore justifies higher order asymptotics. In the case represented by Fig. 1, matrix equation (1) gives