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Test case, dynamic beam

Purpose

The purpose of this test case is to verify that the simulation model is second order accurate for the general alpha time integration applied on a dynamic beam (balance between inertia and bending/normal forces).

Method

Standing flexural waves of a beam have been computed numerically. Comparisons are made with the analytically calculated exact frequencies and displacements of the oscillation. The numerical calculation was done for a beam fixed at one end and oscillating in the lower frequency modes.

Click on graph below to start animation.

The relative strain and kinetic energies for one period of an oscillation are presented below

The relative error norm versus no of spatial grid divisions is presented below verifying second order error decay.

Beam with material damping

With a “low” material damping, the total energy decay of an oscillating beam may be computed. The energies are computed by the simulation program and compared with the exact energy decay.

Conclusions

· The comparisons for the standing waves demonstrate that the equations are solved correctly and that the numerical scheme is second order accurate (error decreases with the spacing between discrete points squared).

· The comparison for the decaying oscillations of a beam with material damping shows that the decay of the amplitude with time is correctly computed.