# 机械学、机器和振动Mechanics, Machines and Vibration FEEG2002W1-01

Another characteristic of a random variable is the power spectral density $S(\lambda)$, defined as the Fourier transform of the autocorrelation function
$$S(\lambda)=\int_{-\infty}^{\infty} \Psi(\tau) e^{i \lambda \tau} d \tau$$
The integral of function $S(\lambda)$ is the variance of function $y(t)$, i.e., if the average value is equal to zero, the square of its r.m.s. value
$$y_{r m s}=\sqrt{\int_{-\infty}^{\infty} S(\lambda) d \lambda} .$$

## FFEEG2002W1-01 COURSE NOTES ：

The resultant of the axial force $F_{z}$ exerted by the other parts of the bar is
$$F_{z}+\frac{1}{2} \frac{\partial F_{z}}{\partial z} d z-F_{z}+\frac{1}{2} \frac{\partial F_{z}}{\partial z} d z .$$
The dynamic equilibrium equation can then be written in the form
$$\rho A \ddot{u}{z}=\frac{\partial F{z}}{\partial z}+f_{z}(z, t) .$$
The axial force $F_{z}$ is easily linked with the displacement by the usual formula from the theory of elasticity
$$F_{z}=A \sigma_{z}=E A \epsilon_{z}=E A \partial u_{z} / \partial z .$$