Though NMMs are mostly only defined and used for the conservative part of a
system, they can be useful also for predicting or understanding the damped and
forced response. This parallels the usefulness of frequency response backbone
curves (Sect. 3.6.6), which are also derived solely for the unforced and undamped
system and thus conservative system, but nevertheless provide valuable cues to also
the behavior of the forced, damped system: The backbone curves tell at which
frequencies resonance may occur, in dependency of vibration amplitude, and
whether the forced resonance peak is left/right-slanted (softening/stiffening nonlinearity), whether isolas (disconnected resonance curves) can be expected, and
whether and for which frequencies sub/superharmonic, internal, or combination
resonances may occur.
NMMs can also be useful in the decision of whether nonlinearity needs to be
accounted for at all, in a given application: If a FEP shows (almost) straight horizontal lines in the frequency and energy range of relevance, then nonlinearity can
maybe just be ignored (at least stiffness type nonlinearity; damping nonlinearity
may not necessarily introduce frequency-energy dependency). And with attempts of
model reduction NMMs can be used to identify the critical vibration modes of
interest, and help including consideration to features defining these, while ignoring
less significant features.
To see specific examples of how to calculate and analyze NNMs the interested
reader is referred to literature cited in the beginning of this section.
5.14 Examples of Bifurcating Systems
We return in this section to some of the examples dealt with in Chaps. 3 and 4,
focusing now on the bifurcations occurring with these examples.
5.14.1 Midplane Stretching (Duffing’s Equation)
Reconsider Duffing’s equation:
€ u þ x
2
0 u ¼ e q cos Xt À 2bx 0 _
u À cu
3
À
Á ;
ð5:51Þ
describing the dynamics of numerous physical systems, e.g., the transverse modal
deformations u(t) of a beam in the presence of midplane stretching and timeharmonic loading (Fig. 5.13(b) and Sect. 3.7.1).
A multiple scales perturbation analysis was performed for this system in
Sect. 3.7.2. The near-resonant response (X % x 0 ) was obtained in the form:
310
5 Bifurcation Analysis
system, they can be useful also for predicting or understanding the damped and
forced response. This parallels the usefulness of frequency response backbone
curves (Sect. 3.6.6), which are also derived solely for the unforced and undamped
system and thus conservative system, but nevertheless provide valuable cues to also
the behavior of the forced, damped system: The backbone curves tell at which
frequencies resonance may occur, in dependency of vibration amplitude, and
whether the forced resonance peak is left/right-slanted (softening/stiffening nonlinearity), whether isolas (disconnected resonance curves) can be expected, and
whether and for which frequencies sub/superharmonic, internal, or combination
resonances may occur.
NMMs can also be useful in the decision of whether nonlinearity needs to be
accounted for at all, in a given application: If a FEP shows (almost) straight horizontal lines in the frequency and energy range of relevance, then nonlinearity can
maybe just be ignored (at least stiffness type nonlinearity; damping nonlinearity
may not necessarily introduce frequency-energy dependency). And with attempts of
model reduction NMMs can be used to identify the critical vibration modes of
interest, and help including consideration to features defining these, while ignoring
less significant features.
To see specific examples of how to calculate and analyze NNMs the interested
reader is referred to literature cited in the beginning of this section.
5.14 Examples of Bifurcating Systems
We return in this section to some of the examples dealt with in Chaps. 3 and 4,
focusing now on the bifurcations occurring with these examples.
5.14.1 Midplane Stretching (Duffing’s Equation)
Reconsider Duffing’s equation:
€ u þ x
2
0 u ¼ e q cos Xt À 2bx 0 _
u À cu
3
À
Á ;
ð5:51Þ
describing the dynamics of numerous physical systems, e.g., the transverse modal
deformations u(t) of a beam in the presence of midplane stretching and timeharmonic loading (Fig. 5.13(b) and Sect. 3.7.1).
A multiple scales perturbation analysis was performed for this system in
Sect. 3.7.2. The near-resonant response (X % x 0 ) was obtained in the form:
310
5 Bifurcation Analysis