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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, see Theory, applied on a dynamic beam (balance between inertia and bending/normal forces). Standing flexural waves of a beam have been computed numerically. Comparisons are made with the analytically calculated exact frequencies and displacements of oscillation. The numerical calculation was done for a beam fixed at one end and oscillating in the lower frequency modes.

Results, undamped dynamic beam

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.

Results, dynamic 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.