Forced Oscillators

Abstract

In Mathematica File MF09, the student has already seen some of the exciting possible solutions that can occur for a forced oscillator depending on the amplitude F chosen for the forcing term. The nonlinear system in that file is the Duffing oscillator

$$\ddot{x} + 2\gamma \dot{x} + \alpha x + \beta {{x}^{3}} = F \cos (\omega t)$$

(8.1)

with γ the damping coefficient and the driving frequency. In mechanical terms, the lhs of the Duffing equation can be thought of as a damped nonlinear spring. With the forcing term on the rhs included, the following special cases have been extensively studied in the literature:

• 1 Hard spring Duffing oscillator: α > 0, β> 0

• 2 Soft spring Duffing oscillator: α > 0, β < 0

• 3 Inverted Duffing oscillator: α < 0, β > 0

• 4 Nonharmonic Duffing oscillator: α = 0, β > 0

For he being dead, with him is beauty slain, And,beauty dead, balck chaos comes again.

William Shakespeare (1564–1616), Venus and Adonis