Consider a linear system governed by the first-order differential equation c x(t) + k x(t) = f(t) The accompanying sketch shows one physical system that is governed by this differential equation, with k being a spring stiffness, c a dashpot value, and f(t) an applied force. Find the impulse

Question

Consider a linear system governed by the first-order differential equation

[latex] c \dot{x} (t)+kx(t)=f(t) [/latex]

The accompanying sketch shows one physical system that is governed by this differential equation, with [latex] k [/latex] being a spring stiffness, [latex] c [/latex] a dashpot value, and [latex] f(t) [/latex] an applied force. Find the impulse response function [latex] h_{x}(t) [/latex] such that Eq. 5.2

[latex] x(t)=\int_{-\infty }^{\infty }{f(s)h_{x}(t-s)ds} \equiv \int_{-\infty }^{\infty }{f(t-r)h_{x}(r)dr} [/latex]

describes the solution of the problem.

Accepted Answer

Rewriting Eqs. 5.9

[latex]\sum\limits_{j=0}^{n}{a_{j}\frac{d^{j}h_{x}(t)}{dt^{j}} } =0[/latex]

and 5.11

[latex]\left(\frac{d^{n-1}h_{x}(t)}{dt^{n-1}} \right) _{t=0^{+}}=a_{n}^{-1}[/latex]

gives

[latex] c\frac{dh_{x}(t)}{dt} +kh_{x}(t)=0 [/latex] for [latex] t\gt 0 [/latex]

With [latex] h_{x}(0^{+})=c^{-1} [/latex] . Taking the usual approach of trying an exponential form for the solution of a linear differential equation, we try [latex] h_{x}(t)= ae^{bt} [/latex] . This has the proper initial condition if [latex] a=c^{-1} [/latex] and it satisfies the equation if [latex] cb+k = 0 [/latex] , or [latex] b=-k/c [/latex] , so the answer is

[latex] h_{x}(t)=c^{-1}e^{-kt/c}U(t) [/latex]

Taking the derivative of this function gives

[latex] h^{'}_{x}(t)=-kc^{-2}e^{-kt/c}U(t)+c^{-1}e^{-kt/c}\delta (t) =-kc^{-2}e^{-kt/c}U(t)+c^{-1}\delta (t) [/latex]

in which the last term has been simplified by taking advantage of the fact that [latex] e^{-kt/c}=1 [/latex] at the one value for which [latex] \delta (t) [/latex] is not zero. Substitution of [latex] h_{t} [/latex] and [latex] h^{'}_{x}(t) [/latex] confirms that we have, indeed, found the solution of the differential equation [latex] c\acute{h} _{x}(t)+kh_{x}(t)=\delta (t) [/latex].

Note also that

[latex] h_{x,staic}\equiv \int_{-\infty }^{\infty }{h_{x}(r)dr=k^{-1}\lt \infty } [/latex]

so the response for this system has a bounded static value. For the physical model shown, it is fairly obvious that the static response to force [latex] f_{0} [/latex] should be [latex] f_{0}/k [/latex] , so it should be no surprise that [latex] h_{x,static}= k^{-1} [/latex]

Question 5.1: Consider a linear system governed by the first-order differe...

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