The Linearized Equations of Motion
The equation of motion of the pendulum is nonlinear because of the term _{0}^{2}sin. 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
and
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°?

© 1998 FranzJosef Elmer,
elmer@ubaclu.unibas.ch
last modified Sunday, July 19, 1998.