ENME361 Quiz 11 Solution

ENME361 Quiz 11 Solution - ENME 361 Fall 2008 Solution for...

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ENME 361 Fall 2008 Solution for Quiz #11 (December 2, 2008; Duration: 20 minutes) 1) The displacement response of a single degree of freedom system subject to an initial displacement X o is given by ( ) 2 2 () s in 1 1 n t o n Xe xt t ζω ω ζϕ ζ =− + Given that the logarithmic decrement is defined as the natural logarithm of the ratio of any two successive amplitudes of the response of a damped, linear system that occur one period apart, determine the expression for the logarithmic decrement. Solution : The logarithmic decrement is given by ln d xt T δ ⎛⎞ = ⎜⎟ + ⎝⎠ = ( ) ( ) 22 sin 1 / 1 ln sin ( ) 1 / 1 n nd t on tT d t t T ωζ ϕ −+ +− + Since 2 2 1 d n T π = , it follows that ( ) ( ) ( ) 2 sin 1 sin 1 2 sin 1 n n t t ϕω ϕπ + −+= + = As an account of this, we have that ln( ) T e = 2 2 1 πζ = 2) A single degree-of-freedom system is subjected to a harmonic excitation F o sin( τ ), where is the non-dimensional excitation frequency. Given that the steady-state response of the system is of the form ( ) s i n o F xH k
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ENME361 Quiz 11 Solution - ENME 361 Fall 2008 Solution for...

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