Oscillations, Waves, and Interactions - GWDG

Figure 1. Resonance curves of the
damped and driven harmonic oscillator
Eq. (1) for high (d = 0.5) and
low (d = 0.2) damping with f = 1
and ω0 = 1. The maximal amplitude
amax occurs at the resonance
frequency ωr = p ω 2 0 − d2 /2.
Complex dynamics of nonlinear systems 407
a
6
5
4
3
2
1
d = 0.5
0
0 0.5 1 1.5 2
ω
d = 0.2
These relations completely describe the response of a (damped) harmonic oscillator
to a sinusoidal excitation and can be extended to the case of a general periodic force
using the superposition principle. Such a comprehensive treatment is possible only,
because this system is linear. Many natural oscillators, however, are nonlinear and
the question arises: how do nonlinearities change the dynamical behaviour? With
the harmonic oscillator nonlinearity can enter in terms of a nonlinear restoring force
and/or due to a nonlinear damping mechanism where the latter may render the
oscillator a self-sustained system oscillating without any external driving.
A prototypical example of an oscillator with a nonlinear restoring force is the
Duffing oscillator
¨x + d ˙x + ω 2 0x + αx 3 = f cos(ωt) , (6)
where the restoring force ω 2 0x+αx 3 may be interpreted as a nonlinear approximation
of a more general nonlinear force (Taylor expansion). By rescaling time t and x both
coefficients ω 2 0 and α can be set to one and the resulting normalised Duffing equation
reads
¨x + d ˙x + x + x 3 = f cos(ωt) . (7)
The class of nonlinear oscillators with cubic restoring force is named after Georg
Duffing engineer in Berlin in the beginning of the 20th century and published in 1918
a detailed study on ‘Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und
ihre technische Bedeutung’ 3 [4].
As pointed out in the title of Duffing’s book the “eigenfrequency” of nonlinear
oscillators depends in general very much on the amplitude of the oscillation and
actually provides a useful notion for relatively small amplitudes only, as we shall
show in the following. This phenomenon is illustrated in Fig. 2 where amplitude
resonance curves of the Duffing oscillator Eq. (7) are shown for d = 0.2 and three
different driving amplitudes. For weak forcing with f = 0.15 the resulting (black)
resonance curve still resembles the linear resonance curve (Fig. 1) but it is shifted
towards higher frequencies and it bends to the right. If the driving amplitude is
increased to f = 0.3 the resonance moves further to the right and the curve overhangs
3 In English: Driven oscillations with variable eigenfrequencies and their technical im-
portance.