# SolSec3.2 - 3 12 Problems and Solutions for Section...

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3- 12 Problems and Solutions for Section 3.2 (3.12 through 3.21) 3.12 Calculate the response of an overdamped single-degree-of-freedom system to an arbitrary non-periodic excitation. Solution: From Equation (3.12): xt F ht d L () = () − () ττ τ 0 For an undamped SDOF system (see Problem 3.4) ht m ee e d F m e d n t tt t n t n nn n = () = () −− ωζ ζω 1 21 1 2 11 0 2 22 ⇒ () = () − e m Fe e e d n n n t ζω τ 2 0

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3- 13 3.13 Calculate the response of an underdamped system to the excitation given in Figure P3.13. Plot of a pulse input of the form f ( t ) = F 0 sin t . 0 f ( t ) t F 0 2 Figure P3.13 Solution: xt m eF e t d Ft F t t t F m eet d d t d t d t d t nn () = () − () [] () = () < =− 1 0 0 0 0 ω τω τ π ζω ζω τ sin sin , sin sin From Figure 3.18 For F m e et t t t t d t d n t d n dd n d d n t d n d n n n ++ −− {} + +− 0 2 2 1 21 2 11 1 1 ωω ωζ ζ sin cos sin cos sin cos 1 sin cos d n d tt For , > F m d F m e d t d d t n 0 0 0 sin sin 1 1 1 1 1 2 2 + + d n t n d n d d n t t n n sin cos sin cos sin τζ cos sin cos d n d t 1
3- 14 3.14 Calculate and plot the response of an undamped system to a step function with a finite rise time of t 1 for the case m = 1 kg, k = 1 N/m, t 1 = 4 s and F 0 = 20 N. This function is described by Ft t tt t () = ≤≤ > 0 1 1 01 0 Solution: Working in Mathcad to perform the integrals the solution is:

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3- 15 3.15 A wave consisting of the wake from a passing boat impacts a seawall. It is desired to calculate the resulting vibration. Figure P3.15 illustrates the situation and suggests a model. This force in Figure P3.15 can be expressed as Ft F t t tt () = ≤≤ > 0 0 0 0 10 0 Calculate the response of the seal wall-dike system to such a load. Model Physical setting Input model Concrete seawall Water Wake k F F m l m K EA F ( t ) t 0 Dike Solution: From Eq. (3.12): xt F ht d t () = () − () 0 ττ τ From Problem 3.8, ht m t n n =− ω ωτ 1 sin for an undamped system For : 0 t t < m F t td F m t n t n n
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SolSec3.2 - 3 12 Problems and Solutions for Section...

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