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Unformatted text preview: TAM 412, Homework 11, due May 5, 2010 Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign firstname.lastname@example.org 1. Consider the differential equation m x + c x + kx = F cos t, where the dot denotes differentiation with respect to t , governing the motion of particle of mass m suspended from a spring with stiffness k and a damping element with damping coefficient c and excited by a harmonic excitation with amplitude F and angular frequency . (a) Introduce a rescaling x ( t ) = 1 x ( t ) and find and such that x + 2 x + x = cos t, where denotes differentiation with respect to t , for some non-dimensional quantities and . Express and in terms of the given physical quantities. (b) Show that for t 1 x ( t ) A cos ( t ) where the amplitude and phase shift are given by A = 1 radicalBig (1 2 ) 2 + (2 ) 2 and tan = 2 1 2 (c) Show that max A = braceleftBigg 1 for > 1 / 2 1 2 1 2 for 0 < < 1 / 2 (d) Suppose that 0...
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