Following Example 9.12, we would like to examine the sensitivity of the output spectral density \(S_{X X}(\omega)\) and the mean-square response \(E\left\{X(t)^{2}\right\}\) to various combinations of parameter values, where \(S_{0}\) is the input spectrum. For the following cases, plot \(|G(i \omega)|^{2}\) and \(S_{X X}(\omega)\), and evaluate \(E\left\{X(t)^{2}\right\}\) for unit mass, \(0

(a) \(\omega_{n}=0.1\) and \(\zeta=0.1\)

(b) \(\omega_{n}=0.5\) and \(\zeta=0.1\)

(c) \(\omega_{n}=1.0\) and \(\zeta=0.1\)

(d) \(\omega_{n}=2.0\) and \(\zeta=0.1\)

(e) \(\omega_{n}=0.1\) and \(\zeta=0.5\)

(g) \(\omega_{n}=0.5\) and \(\zeta=0.5\)

(h) \(\omega_{n}=1.0\) and \(\zeta=0.5\)

(i) \(\omega_{n}=2.0\) and \(\zeta=0.5\).

Compare the results and draw general conclusions.

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Example 9.12 Oscillator Response to White Noise Consider an application of the above ideas to an oscil- lator. What is the response of a damped oscillator to a force with white-noise probability density?