Table of Contents
Steady-state response
by Dr. Jiří Plešek
Problem description
The cantilever beam shown is subjected to kinematic excitation of its basis $W(t)=W_0\sin\omega t$ in the $z$ direction. Approximate solution for a slightly damped system can be written in the form
$$w(x,t)=w_h(x,t)+w_p(x)\sin(\omega t+\varphi)$$
where the transient part $w_h\to0$ as $t\to\infty.$ Establish the steady-state deflection $w_p(x)$ with the aid of the following methods:
(a) the mode superposition method,
(b) the direct integration method,
(c) the response spectrum method.
Compare the results.
Material properties
$E=2\times10^5\text{ MPa},$ $\nu=0.3,$ $\rho=7800\text{ kg/m$^3$}.$
Damping
Modal damping parameters: $\xi_k=0.1$ for $k=1,2,\ldots.$
Support
Clamped at $x=0.$
Loading
Kinematic excitation $W(t)=W_0\sin\omega t,$ $W_0=1\text{ mm},$ $\omega=25.7\text{ rad/s}$ (given as $\omega=\textstyle\frac{1}{2}\omega_1=\pi f_1=\pi\times8.18=25.7,$ see Natural frequencies).
Solution
The results of the computation are shown in the table below.
| method | $w_p\text{ [mm]}$ | ||||
|---|---|---|---|---|---|
| node 1 | node 2 | node 3 | node 4 | node 5 | |
| mode synthesis | $1.00$ | $1.05$ | $1.18$ | $1.34$ | $1.51$ |
| direct integration | $1.00$ | $1.05$ | $1.16$ | $1.30$ | $1.44$ |
| response spectrum | $1.00$ | $1.05$ | $1.18$ | $1.34$ | $1.52$ |
It is interesting to compare these solutions with the shape of the first eigenvector plotted below.
