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Steady-state response

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 $w_h\to 0$ as $t\to \infty$. Establish the steady-state deflection w_p(x) with the aid of the following methods: i) the mode synthesis, ii) the direct integration method, and iii) the response spectrum method. Compare the results.

Mesh: Four beam elements--see appendix B.3.

Material properties: $E=2\times 10^5$ MPa, $\nu=0.3$, $\rho=7800$ kg/m³.

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

Support: All the degrees of freedom fixed at x=0.

$u=v=w=\varphi_x=\varphi_y=\varphi_z=0$ node: 1

Loading: Kinematic excitation $W(t)=W_0\sin\omega t$.

$W_0=1\,\mbox{mm},~~~\omega= 25.7~\mbox{rad/sec}$.
(Given as 

$\omega=\textstyle\frac{1}{2}\omega_1= \pi f_1 =\pi\times8.18 = 25.7$--see section IV.1.)

Solution: The results of computation detailed in next sections are shown in table.

node	1	2	3	4	5
wp(MS) [mm]	1.00	1.05	1.18	1.34	1.51
wp(DI) [mm]	1.00	1.05	1.16	1.30	1.44
wp(RSM)[mm]	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.