3.8 Two More Classical Nonlinear Oscillators
Next we consider the van der Pol oscillator and the Rayleigh oscillator, which are
classical in the sense that they have a long history, and still serve to illustrate and
understand essential phenomena occurring with nonlinear oscillators. Both are
basically unforced and linearly damped single-DOF oscillators, with an additional
cubically nonlinear damping term. For both oscillators the composite (linear + nonlinear) damping term is such that energy is fed into the system at lower
oscillation amplitudes (i.e. the linearized damping is negative), but drained at higher
amplitudes, so that oscillations will grow or shrink in amplitude until some balance
of stationary oscillations is established. With no external forcing involved such
oscillations are self-excited or self-exciting (Nayfeh and Mook 1979; Tondl 1991).
They play a significant role in numerous technical and natural contexts, such as
with friction-generated noise and vibrations, flow-induced vibrations, biological
clocks (e.g. animal heart beats and brainwaves), electrical circuits, and economical/
sociological processes.
A self-excited oscillator is not driven primarily by an explicitly time-dependent
forcing term, but rather by some energy source internal to the system. An example
is friction-induced vibrations, where the typically negative slope of the velocity
versus friction force characteristic near zero velocity corresponds to negative
damping; thus energy is fed into the system, rather than being drained out (cf.
Sects. 3.9.5–3.9.6). This causes oscillations to grow, as is commonly experienced
with, e.g., squeaking car and bike brakes, door hinges and train tracks, and with
bowed musical instruments. However, with growing oscillation amplitudes other
phenomena will always start to outbalance the energy input, e.g. with sliding
friction surfaces the negative slope in the friction versus velocity curve will turn
positive for higher velocities, corresponding to ‘normal’ positive damping.
Another example of self-excitation is flow-induced vibrations, such as with
fluttering aircraft wings, singing air cables, wind musical instruments, and human or
animal voicing and whistling: Explicit time-dependent input forces are absent,
while instead some physical mechanism allows turning energy of the fluid/air/gas
flow into mechanical vibrations, effectively corresponding to negative damping
which will make oscillations grow. Again, with any real physical system this
growth is sooner or later be limited by other forces, whose significance increase
with oscillation amplitude, and the oscillations may stabilize at a certain level (or
turn chaotic, cf. Chap. 6, or the device may eventually break or disintegrate).
3.8.1 The Van Der Pol Oscillator
A nondimensional form of the van der Pol oscillator for the variable u = u(t) is:
€ u À eð1 À u
2
Þ _
u þ u ¼ 0;
ð3:243Þ
3.8 Two More Classical Nonlinear Oscillators
171
Next we consider the van der Pol oscillator and the Rayleigh oscillator, which are
classical in the sense that they have a long history, and still serve to illustrate and
understand essential phenomena occurring with nonlinear oscillators. Both are
basically unforced and linearly damped single-DOF oscillators, with an additional
cubically nonlinear damping term. For both oscillators the composite (linear + nonlinear) damping term is such that energy is fed into the system at lower
oscillation amplitudes (i.e. the linearized damping is negative), but drained at higher
amplitudes, so that oscillations will grow or shrink in amplitude until some balance
of stationary oscillations is established. With no external forcing involved such
oscillations are self-excited or self-exciting (Nayfeh and Mook 1979; Tondl 1991).
They play a significant role in numerous technical and natural contexts, such as
with friction-generated noise and vibrations, flow-induced vibrations, biological
clocks (e.g. animal heart beats and brainwaves), electrical circuits, and economical/
sociological processes.
A self-excited oscillator is not driven primarily by an explicitly time-dependent
forcing term, but rather by some energy source internal to the system. An example
is friction-induced vibrations, where the typically negative slope of the velocity
versus friction force characteristic near zero velocity corresponds to negative
damping; thus energy is fed into the system, rather than being drained out (cf.
Sects. 3.9.5–3.9.6). This causes oscillations to grow, as is commonly experienced
with, e.g., squeaking car and bike brakes, door hinges and train tracks, and with
bowed musical instruments. However, with growing oscillation amplitudes other
phenomena will always start to outbalance the energy input, e.g. with sliding
friction surfaces the negative slope in the friction versus velocity curve will turn
positive for higher velocities, corresponding to ‘normal’ positive damping.
Another example of self-excitation is flow-induced vibrations, such as with
fluttering aircraft wings, singing air cables, wind musical instruments, and human or
animal voicing and whistling: Explicit time-dependent input forces are absent,
while instead some physical mechanism allows turning energy of the fluid/air/gas
flow into mechanical vibrations, effectively corresponding to negative damping
which will make oscillations grow. Again, with any real physical system this
growth is sooner or later be limited by other forces, whose significance increase
with oscillation amplitude, and the oscillations may stabilize at a certain level (or
turn chaotic, cf. Chap. 6, or the device may eventually break or disintegrate).
3.8.1 The Van Der Pol Oscillator
A nondimensional form of the van der Pol oscillator for the variable u = u(t) is:
€ u À eð1 À u
2
Þ _
u þ u ¼ 0;
ð3:243Þ
3.8 Two More Classical Nonlinear Oscillators
171