Linearization and first-order expansion of the rocking motion of rigid blocks stepping on viscoelastic foundation

Abstract

In structural mechanics there are several occasions where a linearized formulation of the original nonlinear problem reduces considerably the computational effort for the response analysis. In a broader sense, a linearized formulation can be viewed as a first-order expansion of the dynamic equilibrium of the system about a `static¿ configuration; yet caution should be exercised when identifying the `correct¿ static configuration. This paper uses as a case study the rocking response of a rigid block stepping on viscoelastic supports, whose non-linear dynamics is the subject of the companion paper, and elaborates on the challenge of identifying the most appropriate static configuration around which a first-order expansion will produce the most dependable results in each regime of motion. For the regime when the heel of the block separates, a revised set of linearized equations is presented, which is an improvement to the unconservative equations published previously in the literature. The associated eigenvalues demonstrate that the characteristics of the foundation do not affect the rocking motion of the block once the heel separates.