Question 3.5.1: Consider the single-degree-of-freedom system of Window 3.1 s......
Consider the single-degree-of-freedom system of Window 3.1 subject to a random (white noise) force input F(t). Calculate the power spectral density of the response x(t) given that the PSD of the applied force is the constant value S_0.
The equation of motion is
m \ddot{x}+c \dot{x}+k x=F(t)
From equation (2.59) or equation (3.53), the frequency response function is
\frac{X(s)}{F(s)}=\frac{1}{\left(m s^2+c s+k\right)}=H(s) (2.59)
H(j \omega)=\frac{1}{k-\omega^2 m+c \omega j} (3.53)
H(\omega)=\frac{1}{k-m \omega^2+c \omega j}
Thus,
\begin{aligned} |H(\omega)|^2=\left|\frac{1}{k-m \omega^2+c \omega j}\right|^2 & =\frac{1}{\left(k-m \omega^2\right)+c \omega j} \cdot \frac{1}{\left(k-m \omega^2\right)-c \omega j} \\ & =\frac{1}{\left(k-m \omega^2\right)^2+c^2 \omega^2} \end{aligned}
From equation (3.62), the PSD of the response becomes
S_{x x}(\omega)=|H(\omega)|^2 S_{f f}(\omega) (3.62)
S_{x x}=|H(\omega)|^2 S_{f f}=\frac{S_0}{\left(k-m \omega^2\right)^2+c^2 \omega^2}
This states that if a single-degree-of-freedom system is excited by a stationary random force (of constant mean and rms value) that has a constant power spectral density of value S_0, the response of the system will also be random with nonconstant (i.e., frequency-dependent) PSD of S_{x x}(\omega)=S_0 /\left[\left(k-m \omega^2\right)^2+c^2 \omega^2\right].