5.13 Nonlinear Normal Modes
Nonlinear normal modes (NNMs) offer yet another means of studying and illustrating important characteristics of vibrating systems, including their bifurcations.
Just like phase plane diagrams, frequency response functions, backbone curves, and
bifurcation diagrams, which are all data transforming and reducing tools that allows
for better communicating and understanding the complex motions of, in particular,
nonlinear systems. The science of NNMs is highly specialized, still in active
development, and important and difficult enough to warrant an entire book in itself
(like Kerschen 2014). Here we only devote it a short introduction and overview
(partly due to the authors lack of working expertise in the subfield), maybe enough
to stir further interest. For good overviews and introductions see, e.g., Vakakis
(1997), Kerschen (2014), Kerschen et al. (2009), Avramov and Mikhlin (2013),
Mikhlin and Avramov (2010), Noël and Kerschen (2017). For examples of application and more specialized subtopics see, e.g., Hill et al. (2017), Lacarbonara et al.
(2016), Renson, Kerschen and Cochelin (2016), Gendelman (2014), Peeters,
Kerschen and Golinval (2011), Lacarbonara, Rega and Nayfeh (2003), Shaw and
Pierre (1993).
The contemporary definition of NNMs is rather spacious and inclusive: A
nonlinear normal mode is any periodic (not necessarily synchronous) motion of a
system (Kerschen et al. 2009; Hill et al. 2017), though there is also a definition
based on invariant manifolds (Shaw and Pierre 1993). Here ‘system’ is mostly
understood to be a conservative system, or its conservative part, though the NNM
concept can also be extended to include damping and external forcing (e.g.,
Gendelman 2014). NNMs have a long history, back to the pioneering work of
Rosenberg (1960), but have gained increased interest since the 1990s, apparently
beginning with the works of, e.g. Shaw and Pierre (1993) and Vakakis (1997).
The various definitions of NNMs also covers the well-known (cf. Sect. 1.3.2)
linear normal modes (LNMs) – i.e. the linear mode shapes, if the system in
question is a physical structure where the concept of ‘shape’ makes sense. NNMs
can be seen as a generalization of linear normal modes. However, there are some
substantial differences:
First, LNMs are invariant, that is, motion started on any LNM remain in that
mode at all times. This generally breaks down with nonlinearity, so NNMs are not
necessarily invariant in that sense; energy imparted into on an NNM may go into
other NNMs, as we have already seen in Chap. 4 with nonlinear interaction.
Second, LNMs can be used in superposition: Owing to the orthogonality (i.e.
linear independency) of the LNMs, free and forced vibrations of a system can be
expressed as linear combinations of LNMs. NNMs cannot be used this way; generally they are not orthogonal, and with nonlinear systems the response to a sum of
influences does not necessarily equal the sum of responses to each separate
influence.
Third, with LNMs there is little debate on how to picture them: An LNM usually
comes in the form of a discrete set of numbers, or a continuous function, that
308
5 Bifurcation Analysis
Nonlinear normal modes (NNMs) offer yet another means of studying and illustrating important characteristics of vibrating systems, including their bifurcations.
Just like phase plane diagrams, frequency response functions, backbone curves, and
bifurcation diagrams, which are all data transforming and reducing tools that allows
for better communicating and understanding the complex motions of, in particular,
nonlinear systems. The science of NNMs is highly specialized, still in active
development, and important and difficult enough to warrant an entire book in itself
(like Kerschen 2014). Here we only devote it a short introduction and overview
(partly due to the authors lack of working expertise in the subfield), maybe enough
to stir further interest. For good overviews and introductions see, e.g., Vakakis
(1997), Kerschen (2014), Kerschen et al. (2009), Avramov and Mikhlin (2013),
Mikhlin and Avramov (2010), Noël and Kerschen (2017). For examples of application and more specialized subtopics see, e.g., Hill et al. (2017), Lacarbonara et al.
(2016), Renson, Kerschen and Cochelin (2016), Gendelman (2014), Peeters,
Kerschen and Golinval (2011), Lacarbonara, Rega and Nayfeh (2003), Shaw and
Pierre (1993).
The contemporary definition of NNMs is rather spacious and inclusive: A
nonlinear normal mode is any periodic (not necessarily synchronous) motion of a
system (Kerschen et al. 2009; Hill et al. 2017), though there is also a definition
based on invariant manifolds (Shaw and Pierre 1993). Here ‘system’ is mostly
understood to be a conservative system, or its conservative part, though the NNM
concept can also be extended to include damping and external forcing (e.g.,
Gendelman 2014). NNMs have a long history, back to the pioneering work of
Rosenberg (1960), but have gained increased interest since the 1990s, apparently
beginning with the works of, e.g. Shaw and Pierre (1993) and Vakakis (1997).
The various definitions of NNMs also covers the well-known (cf. Sect. 1.3.2)
linear normal modes (LNMs) – i.e. the linear mode shapes, if the system in
question is a physical structure where the concept of ‘shape’ makes sense. NNMs
can be seen as a generalization of linear normal modes. However, there are some
substantial differences:
First, LNMs are invariant, that is, motion started on any LNM remain in that
mode at all times. This generally breaks down with nonlinearity, so NNMs are not
necessarily invariant in that sense; energy imparted into on an NNM may go into
other NNMs, as we have already seen in Chap. 4 with nonlinear interaction.
Second, LNMs can be used in superposition: Owing to the orthogonality (i.e.
linear independency) of the LNMs, free and forced vibrations of a system can be
expressed as linear combinations of LNMs. NNMs cannot be used this way; generally they are not orthogonal, and with nonlinear systems the response to a sum of
influences does not necessarily equal the sum of responses to each separate
influence.
Third, with LNMs there is little debate on how to picture them: An LNM usually
comes in the form of a discrete set of numbers, or a continuous function, that
308
5 Bifurcation Analysis