SolSec3.4 - 3 32 Problems and Solutions for Section...

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3- 32 Problems and Solutions for Section 3.4 (3.29 through 3.32) 3.29 Calculate the response of mx cx kx F t ˙˙ ˙ ( ) + + = 0 Φ where Φ ( t ) is the unit step function for the case with x 0 = v 0 = 0. Use the Laplace transform method and assume that the system is underdamped. Solution: Given: mx cx kx F t x x x F m t n n ˙˙ ˙ ( ) ˙˙ ˙ ( ) ( ) + + = + + = < 0 2 0 2 1 µ ζω ω µ ζ Take Laplace Transform: s X s sX s X s F m s X s F m s s s F m s s s n n n n n n n n 2 2 0 0 2 2 0 2 2 2 2 2 1 2 2 ( ) ( ) ( ) ( ) / + + = = + + ( ) = + + ( ) ζω ω ζω ω ω ω ζω ω Using inverse Laplace tables, x t F k F k e t n t n ( ) sin cos ( ) = + ( ) 0 0 2 2 1 1 1 ζ ω ζ ζ ζω
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