Question 3.5.2: Calculate the mean-square value of the response of the syste......
Calculate the mean-square value of the response of the system described in Example 3.5.1 with equation of motion m \ddot{x}+c \dot{x}+k x=F(t), where the PSD of the applied force is the constant value S_0.
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.