Question 3.5.2: Calculate the mean-square value of the response of the syste......

Since the PSD of the forcing function is the constant $S_0$, equation (3.68) becomes

$E\left[x^2\right]=\int_{-\infty}^{\infty}|H(\omega)|^2 S_{f f}(\omega) d \omega$         (3.68)

$E\left[x^2\right]=S_0 \int_{-\infty}^{\infty}\left|\frac{1}{k-m \omega_n^2+j c \omega}\right|^2 d \omega$

Comparison with equation (3.70) yields $B_0=1, B_1=0, A_0=k, A_1=c$, and $A_2=m$. Thus,

$\int_{-\infty}^{\infty}\left|\frac{B_0+j \omega B_1}{A_0+j \omega A_1-\omega^2 A_2}\right|^2 d \omega=\frac{\pi\left(A_0 B_1^2+A_2 B_0^2\right)}{A_0 A_1 A_2}$         (3.70)

$E\left[x^2\right]=S_0 \frac{\pi m}{k c m}=\frac{\pi S_0}{k c}$

Hence, if a spring-mass-damper system is excited by a random force described by a constant PSD, $S_0$, it will have a random response, $x(t)$, with mean-square value $\pi S_0 / k c$.