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Test case, hanging chain

Purpose

The purpose of this test case is to verify that the simulation model gives second order accuracy for the general alpha time integration applied on a hanging chain (balance between inertia, gravity and axial line force).

Method

Free oscillations of a hanging chain 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 chain fixed in the upper end and oscillating in the lower frequency modes.

Click on graph below to start animation.

The relative potential 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.

Conclusion

· The comparisons 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).