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Undamped and Undriven Pendulum

Damped and Driven Pendulum

Linearized Equations of Motion

The Linearized Equations of Motion

The equation of motion of the pendulum is nonlinear because of the term omega02sinphi. Driving the suspension point leads to a driving force which is also nonlinear in the angle phi. For small angles, the nonlinear terms can be linearized, i.e., sinphi = phi + O(phi3) and cosphi = 1 + O(phi2).

Thus the linearized equations of motion read

(horizontal motion)

d2phidt2 + gammadphi/dt + omega02phi = a cos 2pift,



(vertical motion)

d2phi/dt2 + gammadphi/dt + (omega02 + a cos 2pift ) phi = 0,


and

(rotation)

d2phi/ddt2 + gammadphi/dt + (omega02 + a cos 2pift ) phi = a sin 2pift.

Additional comments:

QUESTION worth to think about:
  1. What are the equations of motion linearized around phi = 180°?

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© 1998 Franz-Josef Elmer,  elmer@ubaclu.unibas.ch last modified Sunday, July 19, 1998.