Find the harmonic transfer function for the linear system governed by the differential equation m x(t) + c x(t) = f (t) , for which the impulse response function was found in Example 5.2 as hx (t) = c^-1(l – e^-ct/m) U(t). Also consider the autospectral density and variance of the stochastic

Question

Find the harmonic transfer function for the linear system governed by the differential equation [latex] m\ddot{x}(t)+c \dot{x}(t)=f(t) [/latex] , for which the impulse response function was found in Example 5.2 as [latex] h_{x}(t)=c^{-1}(1-e^{-ct/m})U(t) [/latex]. Also consider the autospectral density and variance of the stochastic response of this system for a stochastic excitation with autospectral density [latex] S_{FF}(\omega ) [/latex] .

Accepted Answer

Strictly speaking, the Fourier transform of [latex] h_{x}(t) [/latex] does not exist for this problem because [latex] h_{x}(t) [/latex] is not integrable, which is the same condition as [latex] h_{x,static}=\infty [/latex] (as was pointed out in Example 5.2). Thus, one cannot directly apply Eq. 6.30

[latex]H_{x}(\omega )=2\pi h_{x}(\omega )=\int_{-\infty }^{\infty }{e^{-i\omega r}h_{x}(r)dr}[/latex]

There is no difficulty, though, in using Eq. 6.33

[latex]H_{x}(\omega )=\left(\sum\limits_{j=0}^{n}{a_{j}(i\omega )^{j}} ight) ^{-1}[/latex]

to write

[latex] H_{x}(\omega )=\frac{1}{i\omega c-\omega ^{2}m} [/latex]

Thus, the autospectral density of covariant stationary stochastic response will be

[latex] S_{XX}(\omega )=\frac{S_{FF}(\omega) }{\omega ^{2}c^{2}+\omega ^{4}m^{2}} [/latex]

Note that the denominator of this expression tends to zero as [latex] \omega [/latex] tends to zero. In particular, it behaves like [latex] \omega ^{2} [/latex] for small values of [latex] \omega [/latex] , and this ensures that the integral of the autospectral density will not exist if [latex] S_{FF}(0) eq 0 [/latex] . Thus, this system has infinite response covariance for most stationary stochastic excitations, which agrees with our observation in Example 5.4 that [latex] h_{x,var}(0)=\infty [/latex] . We do see, though, that if [latex] S_{FF}(\omega ) [/latex] is zero and behaves like [latex] \left|\omega ight| ^{b} [/latex] with [latex] b\gt 1 [/latex] at [latex] \omega\approx 0 [/latex] , then the response variance will be finite. This shows that, for some excitations, a system can have a finite response variance even though it has [latex] h_{x,var}(0)=\infty [/latex] .

Question 6.4: Find the harmonic transfer function for the linear system go...

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