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Direct integration, BEAM53D3

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 $W(t)=W_0\sin\omega t$.

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

Solution: In this example an appropriate time step for the Newmark method can be best derived from the first natural period T₁ (section IV.1) as

$$\mbox{{\it TSTEP\/}}=T_1/16 = 7.640428\times 10^{-3} ~\mbox{sec}$$

and the end-time must be chosen sufficiently large so that the transient response will diminish. We have shown with example IV.2 that the amplitude of the transient part would decay exponentially

$$\frac{w({\bf x},5T_1)}{w({\bf x},0)}= e^{-10\pi\xi_1} = 0.0432$$

thus we set $\mbox{{\it TEND\/}}=5 T_1 = 0.611$ sec and round it up to an integer multiple of the time step as

$$\mbox{{\it TEND\/}}=80 \times\mbox{{\it TSTEP\/}} = 0.61123424 ~\mbox{sec}$$

The most difficult task is to propose the damping matrix which must be formed explicitly. We can use the Rayleight damping matrix

$${\bf C}=\alpha {\bf K} + \beta {\bf M}$$

or if transformed to the modal basis

$$2\omega_k\xi_k=\alpha \omega_k^2 + \beta ~~~ \mbox{for}~k=1,2,\ldots$$

Obviously, the equations cannot be solved uniquely as we have only two free parameters. We know, however, that the response will presumably consist of bending modes, therefore, we select the first and third equation corresponding to bending in the x-y plane

$$\begin{eqnarray*}2\omega_1\xi_1 &=& \alpha \omega_1^2 + \beta\\ 2\omega_3\xi_3 &=& \alpha \omega_3^2 + \beta \end{eqnarray*}$$

from which

$$\alpha=5.3495239\times10^{-4}~\mbox{sec}~,~~~ \beta=8.8663179~\mbox{sec}^{-1}$$

Conversely, the actual values of the modal damping parameters introduced by Rayleigh's matrix are shown in table.

k	1	2	3	4	5	6	7	8	9	10
$\xi_k$	0.10	0.07	0.10	0.18	0.25	0.48	0.49	0.89	0.96	1.20

Higher modes are heavily damped, for $k\ge 10$ the damping being even overcritical. Participation of these modes, however, should not play an important role.

Execute from prompt:
>rmd3 beam53d3.I1
>rpd3 beam53d3.I2
>srh3 beam53d3.I3
>hmot beam53d3.IM
>hcre beam53d3.IC
>hfro beam53d3.IR
>hnew beam53d3.IW

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