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The Linearized Equations of Motion

The equation of motion of the pendulum is nonlinear because of the term ₀²sin. Driving the suspension point leads to a driving force which is also nonlinear in the angle . For small angles, the nonlinear terms can be linearized, i.e., sin = + O(³) and cos = 1 + O(²).

Thus the linearized equations of motion read

(horizontal motion)

d2dt2 + d/dt + 02 = a cos 2ft,

(vertical motion)

d2/dt2 + d/dt + (02 + a cos 2ft ) = 0,

and

(rotation)

d2/ddt2 + d/dt + (02 + a cos 2ft ) = a sin 2ft.

Additional comments:

• The linearized driving force of a horizontally driven pendulum is identical to the driving force of a pendulum which is driven by a periodic force. Thus, in the linear regime driving the pendulum by a periodic force is equivalent to moving the suspension point of the pendulum horizontally.
• The linearized equation of motion of the pendulum is called harmonic oscillator.
• The driving term in the linearized equation of motion of a vertically driven pendulum is not additive as for the horizontally driven pendulum, but multiplicative. It is a harmonic oscillator where the oscillator frequency is modulated periodically. The equation of motion is the damped Mathieu equation. The driving term leads to an instability called parametric resonance.

QUESTION worth to think about:

What are the equations of motion linearized around = 180°?

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© 1998 Franz-Josef Elmer,  Franz-Josef doht Elmer aht unibas doht ch, last modified Sunday, July 19, 1998.