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Linearization and first‐order expansion of the rocking motion of rigid blocks stepping on viscoelastic foundation
Author(s) -
Palmeri Alessandro,
Makris Nicos
Publication year - 2008
Publication title -
earthquake engineering and structural dynamics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.218
H-Index - 127
eISSN - 1096-9845
pISSN - 0098-8847
DOI - 10.1002/eqe.799
Subject(s) - linearization , foundation (evidence) , block (permutation group theory) , viscoelasticity , eigenvalues and eigenvectors , equations of motion , motion (physics) , mathematics , set (abstract data type) , nonlinear system , structural engineering , mathematical analysis , engineering , control theory (sociology) , classical mechanics , computer science , geometry , physics , control (management) , archaeology , quantum mechanics , artificial intelligence , history , thermodynamics , programming language
In structural mechanics there are several occasions where a linearized formulation of the original non‐linear 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. Copyright © 2008 John Wiley & Sons, Ltd.