Linearization of EOM's: Influence of equilibrium state on response

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Consider the single-DOF problem of a thin bar pinned to ground with a constant horizontal force F acting at the free end of the bar.

Using Lagrange's equations, the nonlinear EOM for this system is:

This EOM, linearized about the equilibrium state of:

becomes:

where

This says that for a small perturbation about the equilibrium state, the response of the bar will oscillate about with this equilibrium state.

Shown below are animations of the free response of the bar for two values of F/mg. From these animations, can you see the influence of the force F on the equilibrium state and the frequency of oscillations about this equilibrium??

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