homework11.20100429.4bd997db273453.74775291

homework11.20100429.4bd997db273453.74775291 - TAM 412...

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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 [email protected] 1. Consider the differential equation m ¨ x + c ˙ x + kx = F cos ωt, 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 ω . (a) Introduce a rescaling ˜ x ( ˜ t ) = α − 1 x ( βt ) and find α and β such that ˜ x ′′ + 2 ζ ˜ x ′ + ˜ x = cos ˜ ω ˜ t, where ′ denotes differentiation with respect to ˜ t , for some non-dimensional quantities ˜ ω and ζ . Express ˜ ω and ζ in terms of the given physical quantities. (b) Show that for ˜ t ≫ 1 ˜ x ( ˜ t ) ≈ A cos ( ˜ ω ˜ t − φ ) where the amplitude and phase shift are given by A = 1 radicalBig (1 − ˜ ω 2 ) 2 + (2 ζ ˜ ω ) 2 and tan φ = 2 ζ ˜ ω 1 − ˜ ω 2 (c) Show that max ˜ ω ≥ A = braceleftBigg 1 for ζ > 1 / √ 2 1 2 ζ √ 1 − ζ 2 for 0 < ζ < 1 / √ 2 (d) Suppose that 0...
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.

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homework11.20100429.4bd997db273453.74775291 - TAM 412...

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