# Equation of Motion of Mass Damper and Rotating Bar

1. Sep 9, 2014

### ThLiOp

1. The problem statement, all variables and given/known data
Consider the inverted pendulum system, where a uniform rigid bar of mass m and length L is elastically hinged on top of a lumped mass M. The bar is constrained by a torsional spring of coefficient kτ and the mass is constrained by a damper of coefficient c. Derive the nonlinear equations of motion for the system by the Newtonian Method.

3. The attempt at a solution
I have drawn the FBD of each mass and got the separate equations of motion:

For mass M:
M$\ddot{x}$+c$\dot{x}$ = F(t)

For rotating bar:
J$\ddot{θ}$+k$_{τ}$$\dot{θ}$ = 0

where J = (1/3)mL$^{2}$, resulting in

(1/3)mL$^{2}$$\ddot{θ}$+k$_{τ}$$\dot{θ}$ = 0

I am not sure how to relate the 2 in order to derive the nonlinear EOM. Any hints or suggestions would be greatly appreciated!

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