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Finite Element Analysis in Structural Mechanics

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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$}.$


Modal damping parameters: $\xi_k=0.1$ for $k=1,2,\ldots.$


Clamped at $x=0.$


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).


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.

en/example/dyn/3/start.txt · Last modified: 2022-02-11 14:25 by Petr Pařík