A Note on the ‘Stability of Systems’ Some older texts on linear vibrations and
stability briefly touch upon the effects of nonlinearity. One may here find statements
like: ‘When the linearized system is stable, then so is the corresponding nonlinear
system’. By this one could be misled into thinking that linear analysis would be
fully sufficient for checking that a physical system will always return to rest,
however nonlinear, and however disturbed. This is not so, as illustrated above for
the pendulum example: Just below resonance, when the state of rest is stable, a
strong disturbance may cause the pendulum to escape to a coexistent stable state of
large amplitude oscillations.
Nevertheless, the cited statement is true, when one conceives ‘system’ as an
equilibrium solution of a system of differential equations (not the underlying
physical system), and ‘stable’ as stable to sufficiently small disturbances.
For systems that are known to possess at most a single equilibrium – e.g. all
linear systems – this dissection of terms is rather irrelevant. If the ‘system is stable’,
then the system (mathematical or physical) will be either at the state of equilibrium,
or on the way to it, irrespective of the magnitude and character of disturbances
applied. Since the question on stability can be decided on by linearization, then one
can just as well neglect nonlinearities from the very beginning.
The lesson to be learnt is that it makes little sense to characterize a system
(physical or mathematical) as being ‘stable’ or ‘unstable’. States can be stable or
unstable. Systems may possess zero, one, or any number of static and dynamic
states of equilibrium. Some states may be stable and others unstable, according to
the specific values of system parameters. A system will be either in a stable state, or
on its way to one. Upon disturbing the system in some particular state of equilibrium, it may return to this state or escape to another one, depending on the
magnitude and character of the disturbance. And finally: linear analysis is sufficient
for determining the stability of a given state of equilibrium, but nonlinear analysis
may be required for determining the states to linearize around.
3.7 Externally Excited Duffing Systems
In this section we consider the harmonically excited Duffing’s equation:
€ u þ 2bx 0 _
u þ x
2
0 u þ cu
3
¼ q cos Xt;
ð3:152Þ
where for mechanical systems x 0
2 is the linear stiffness parameter (if positive, then
x 0 is the linear natural frequency), b the damping ratio, c the coefficient of a cubic
nonlinearity, and q and X the amplitude and frequency, respectively, of an externally applied harmonic excitation. Duffing’s equation appears so often in applied
mechanics that a basic knowledge of its possible solutions is indispensable. The
equation looks simple – basically just a linear damped single-DOF oscillator with
harmonic excitation, and an additional cubic nonlinearity. Nevertheless, the solutions of this equation exhibit an extremely rich variation (e.g. see Kovacic and
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3 Nonlinear Vibrations: Classical Local Theory
stability briefly touch upon the effects of nonlinearity. One may here find statements
like: ‘When the linearized system is stable, then so is the corresponding nonlinear
system’. By this one could be misled into thinking that linear analysis would be
fully sufficient for checking that a physical system will always return to rest,
however nonlinear, and however disturbed. This is not so, as illustrated above for
the pendulum example: Just below resonance, when the state of rest is stable, a
strong disturbance may cause the pendulum to escape to a coexistent stable state of
large amplitude oscillations.
Nevertheless, the cited statement is true, when one conceives ‘system’ as an
equilibrium solution of a system of differential equations (not the underlying
physical system), and ‘stable’ as stable to sufficiently small disturbances.
For systems that are known to possess at most a single equilibrium – e.g. all
linear systems – this dissection of terms is rather irrelevant. If the ‘system is stable’,
then the system (mathematical or physical) will be either at the state of equilibrium,
or on the way to it, irrespective of the magnitude and character of disturbances
applied. Since the question on stability can be decided on by linearization, then one
can just as well neglect nonlinearities from the very beginning.
The lesson to be learnt is that it makes little sense to characterize a system
(physical or mathematical) as being ‘stable’ or ‘unstable’. States can be stable or
unstable. Systems may possess zero, one, or any number of static and dynamic
states of equilibrium. Some states may be stable and others unstable, according to
the specific values of system parameters. A system will be either in a stable state, or
on its way to one. Upon disturbing the system in some particular state of equilibrium, it may return to this state or escape to another one, depending on the
magnitude and character of the disturbance. And finally: linear analysis is sufficient
for determining the stability of a given state of equilibrium, but nonlinear analysis
may be required for determining the states to linearize around.
3.7 Externally Excited Duffing Systems
In this section we consider the harmonically excited Duffing’s equation:
€ u þ 2bx 0 _
u þ x
2
0 u þ cu
3
¼ q cos Xt;
ð3:152Þ
where for mechanical systems x 0
2 is the linear stiffness parameter (if positive, then
x 0 is the linear natural frequency), b the damping ratio, c the coefficient of a cubic
nonlinearity, and q and X the amplitude and frequency, respectively, of an externally applied harmonic excitation. Duffing’s equation appears so often in applied
mechanics that a basic knowledge of its possible solutions is indispensable. The
equation looks simple – basically just a linear damped single-DOF oscillator with
harmonic excitation, and an additional cubic nonlinearity. Nevertheless, the solutions of this equation exhibit an extremely rich variation (e.g. see Kovacic and
146
3 Nonlinear Vibrations: Classical Local Theory