represents the physical configuration of the system in a straightforward manner; we
perceive an LNM simply as the system configuration ‘frozen’ in time, and then
visualize (mentally or by computer animation) the system vibrating in this shape,
with all material points moving time-harmonically and synchronously in either
phase or anti-phase. For NNMs the definition is much more general – an NNM is
just ‘a periodic motion’, and there are many ways we can picture periodic motions.
So, while we think of LNMs as simply the vibrating system frozen at an instance of
time during trivial time-harmonic oscillation, with NNMs this is not possible – the
motion can rarely be represented in such a simple fashion. NNMs can therefore be
thought of rather as just ‘the periodic motion’ of the system, and then we have to
decide how to picture that motion. That means choosing which motion-describing
variables to work with, and which projection of these to graph.
A typical way of depicting NNMs is by configuration plots, where (at least) two
state variables are graphed versus each other. That could be two displacement
coordinates, or two modal coefficients for a (linearly) mode shape expanded nonlinear system. With linear systems such graphs appear trivial: Since LNMs can be
multiplied by any constant, and still be the same LNM, the graphs will be straight
lines in configuration space, with generally positive/negative slope for in/anti-phase
motion. For NMM’s the graphs in configuration plots will generally be curved
lines.
Frequency–energy plots (FEPs) is another, and very common and convenient,
way to picture NMMs, typically in combination with configuration plots. In a FEP,
NNM motions are represented by the oscillation frequency corresponding to the
minimal period of the periodic motion, versus the total mechanical energy associated with that motion. Several NNMs can be represented in the same plot. Again,
with LNMs such graphs are trivial, just straight horizontal lines, since with linear
system the free oscillation frequency is independent on amplitude and thus on
energy level. However, as we have seen repeatedly in Chaps. 3 and 4, with nonlinear systems the free oscillation frequencies typically depend on amplitude, so
NNM typically display as curved lines in FEPs – at least when the energy becomes
large enough for nonlinearity to be of significance. So typical FEPs of NNMs start
out for low energies with a number of straight horizontal lines, one for each NNM,
and then as energy increases the lines deform into curves. At certain critical levels
of energy bifurcations may occur, and thus FEP curves can appear rather complicated, maybe splitting out and even self-intersecting. FEPs are typically accompanied by miniature inserts of configuration plots along the FEP-curves, giving
hints of how the system vibrates at a given energy level and frequency.
Of what use are NMMs? As already mentioned they are not useful for modal
expansion the way LNMs are. But they can aid in understanding and communicating how the behavior of a nonlinear system changes with, e.g., the level of
energy, in particular when it comes to identify qualitative changes such as bifurcations and amplitude-frequency dependencies. To some extent NNM plots can be
seen as extension or supplements to the well-known nonlinear frequency responses,
and (in particular) frequency response backbones (cf. Sect. 3.6.6).
5.13 Nonlinear Normal Modes
309
perceive an LNM simply as the system configuration ‘frozen’ in time, and then
visualize (mentally or by computer animation) the system vibrating in this shape,
with all material points moving time-harmonically and synchronously in either
phase or anti-phase. For NNMs the definition is much more general – an NNM is
just ‘a periodic motion’, and there are many ways we can picture periodic motions.
So, while we think of LNMs as simply the vibrating system frozen at an instance of
time during trivial time-harmonic oscillation, with NNMs this is not possible – the
motion can rarely be represented in such a simple fashion. NNMs can therefore be
thought of rather as just ‘the periodic motion’ of the system, and then we have to
decide how to picture that motion. That means choosing which motion-describing
variables to work with, and which projection of these to graph.
A typical way of depicting NNMs is by configuration plots, where (at least) two
state variables are graphed versus each other. That could be two displacement
coordinates, or two modal coefficients for a (linearly) mode shape expanded nonlinear system. With linear systems such graphs appear trivial: Since LNMs can be
multiplied by any constant, and still be the same LNM, the graphs will be straight
lines in configuration space, with generally positive/negative slope for in/anti-phase
motion. For NMM’s the graphs in configuration plots will generally be curved
lines.
Frequency–energy plots (FEPs) is another, and very common and convenient,
way to picture NMMs, typically in combination with configuration plots. In a FEP,
NNM motions are represented by the oscillation frequency corresponding to the
minimal period of the periodic motion, versus the total mechanical energy associated with that motion. Several NNMs can be represented in the same plot. Again,
with LNMs such graphs are trivial, just straight horizontal lines, since with linear
system the free oscillation frequency is independent on amplitude and thus on
energy level. However, as we have seen repeatedly in Chaps. 3 and 4, with nonlinear systems the free oscillation frequencies typically depend on amplitude, so
NNM typically display as curved lines in FEPs – at least when the energy becomes
large enough for nonlinearity to be of significance. So typical FEPs of NNMs start
out for low energies with a number of straight horizontal lines, one for each NNM,
and then as energy increases the lines deform into curves. At certain critical levels
of energy bifurcations may occur, and thus FEP curves can appear rather complicated, maybe splitting out and even self-intersecting. FEPs are typically accompanied by miniature inserts of configuration plots along the FEP-curves, giving
hints of how the system vibrates at a given energy level and frequency.
Of what use are NMMs? As already mentioned they are not useful for modal
expansion the way LNMs are. But they can aid in understanding and communicating how the behavior of a nonlinear system changes with, e.g., the level of
energy, in particular when it comes to identify qualitative changes such as bifurcations and amplitude-frequency dependencies. To some extent NNM plots can be
seen as extension or supplements to the well-known nonlinear frequency responses,
and (in particular) frequency response backbones (cf. Sect. 3.6.6).
5.13 Nonlinear Normal Modes
309