SolSec3.4 - 3- 32 Problems and Solutions for Section 3.4...

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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 xx x F m t nn ˙ ˙ ( ) ++ = < 0 2 0 21 µ ζω ω ζ Take Laplace Transform: sXs Xs F ms Fm ss s F m s n n 22 0 0 0 2 2 2 1 / = = = Using inverse Laplace tables, xt F k F k et n t n ( ) sin cos ( ) =− −+ 00 2 1 1 ωζ
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3- 33 3.30 Using the Laplace transform method, calculate the response of the system of Example 3.4.4 for the overdamped case ( ζ > 1). Plot the response for m = 1 kg, k = 100 N/m, and = 1.5. Solution: From example 3.4.4, mx cx kx t xx x m t nn ˙˙ ˙ () ˙ ( ) ++= ++ = > δ ζω ω 2 1 1 2 Take Laplace Transform: sXs Xs m m ss 22 2 1 1 2 / = = Using inverse Laplace tables, xt e m ee n t n tt = [] −−− ωζ 21 2 11 Inserting the given values yields e t .
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SolSec3.4 - 3- 32 Problems and Solutions for Section 3.4...

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