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

摘要

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