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)
= 1 + O(2).
Thus the linearized equations of motion read
- 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
© 1998 Franz-Josef Elmer,
last modified Sunday, July 19, 1998.