Response spectrum method, BEAM53D4

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Response spectrum method, BEAM53D4

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.




 node: 1

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

Solution: The only input for the response spectrum method is the set of spectral accelerations corresponding to the loading and damping given. Denote the base acceleration as

$\begin{displaymath}a(t)=\ddot W(t)=-\omega^2 W_0 \sin\omega t \end{displaymath}$

A characteristic equation of the linear harmonic oscillator $(\omega_k,\xi_k)$ exited kinematically by W(t) has the form

$\begin{displaymath}\ddot x_k(t) + 2\xi_k\omega_k \dot x_k(t) + \omega_k^2x_k(t) = -a(t) \end{displaymath}$

with the solution being

$\begin{displaymath}x_k(t) = A_ke^{-\xi_k\omega_k t}\sin(\bar\omega_k t + \varphi_k) + x_{pk}(t) \end{displaymath}$

where A_k, $\varphi_k$ are constants to be determined from the initial conditions

$\begin{displaymath}x_k(0)=\dot x_k(0) = 0 \end{displaymath}$

and $\bar\omega_k$ is the damped angular frequency

$\begin{displaymath}\bar\omega_k=\omega_k\sqrt{1-\xi_k^2} \end{displaymath}$

Global acceleration can be computed as

$\begin{displaymath}g_k(t)=a(t)+\ddot x_k(t) \end{displaymath}$

and its maximum

$\begin{displaymath}G_k=\max_t \vert g_k(t)\vert=G_k(\omega_k,\xi_k) \end{displaymath}$

which is also called the spectral acceleration.

Thus, it is sufficient for us to calculate G_k's as functions of $(\omega_k,\xi_k)$ for the base acceleration a(t) defined above. In this particular example we make use of the steady-state part of the solution only, which yields

$\begin{eqnarray*}G_k &\simeq& \max_t\vert a(t) + x_{pk}(t)\vert \le \max_t\vert... ...ega_k^2-\omega^2)^2 + 4(\xi_k\omega_k\omega)^2] ^{-\frac{1}{2}} \end{eqnarray*}$

Substituting for $\omega_k=2\pi f_k$ with f_k read from the table shown in section IV.1 for the ITE=53 element type we finally obtain

k	1	2	3	4	5
G_k [m/s²]	0.8787	0.7045	0.6647	0.6615	0.6610
	6	7	8	9	10
	0.6606	0.6606	0.6605	0.6605	0.6605

Note that the FEM results are related to the reference frame that moves with the basis. For example, the total z-displacement must be calculated as

$\begin{displaymath}w_p=W_0+w(\mbox{FEM}) \end{displaymath}$

Execute from prompt:
>rmd3 beam53d4.I1
>rpd3 beam53d4.I2
>srh3 beam53d4.I3
>hmot beam53d4.IM
>fefs beam53d4.I4
>heig beam53d4.IE
>hmod beam53d4.ID

Next: INPUT Up: Steady-state response Previous: OUTPUT