3.7.5 Multiple Scales Analysis with Strong
Nonlinearity
Sometimes perturbation methods can be used even when nonlinearities are not
weak, as has been otherwise assumed in this chapter so far. The quality of a
perturbation-based prediction depend on the degree to which the underlying
assumptions are fulfilled. In the previous sections, when calculating resonant frequency responses, an essential assumption was on the nearness to resonance. This
was expressed as an assumed smallness of a detuning parameter r = X – x 0 , with X
being the excitation frequency and x 0 the resonance frequency in question, e.g. a
linear natural frequency. Even with the assumptions of weak damping and excitation fulfilled, an assumption based on small r will gradually worsen with the
increase of |r|, i.e. as the deviation of X from the constant frequency x 0 increases.
With a stronger (stiffness-)nonlinearity the resonance peak bends more over at
larger amplitudes, and even more so with smaller damping. This means that further
up the resonance peak r becomes too large for a prediction based on small r to
have acceptable accuracy. Thus, when comparing with numerical simulation, one
will typically note a marked drop in accuracy at the upper parts of a strongly
nonlinearly bent resonance peak.
But this can sometimes be cured by considering the detuning r instead as a
deviation from the nonlinear free oscillation frequency, i.e. the frequency on the
nonlinear response backbone, which depends on oscillation amplitude a (but not on
damping and excitation). Thus instead of r = X – x 0 , with a constant x 0 , one can
use r = X – x(a), where x is the nonlinear free and undamped oscillation frequency at oscillation amplitude a. So, if one can set up a reasonably simple
expression for x(a), a multiple scales analysis can be conducted that has good
accuracy as long as the excitation frequency is close to the backbone of the forced
frequency response. Considering a typical nonlinear frequency response, such as
Fig. 3.17, you will note that actually r will be smaller the further up the resonance
peak you come, to become eventually zero at the very top of the peak (which
intersects the backbone).
Burton and Rahman 1986 suggested and tested such an approach to derive
approximate analytical expressions for the stationary frequency response of a
Duffing oscillator with arbitrarily weak or strong nonlinearity, i.e. (3.152) with an
arbitrary ratio of c/x 0
2 (including zero and infinity). To find x(a) for this case
amounts to find the nonlinear oscillation frequency for (3.152) when b = q = 0.
This frequency can be expressed as a function of oscillation amplitude by a complete elliptic integral (Polyanin and Zaitsev 2003; Kovacic and Brennan 2011;
Kovacic 2020). But one can also for x(a) use a series solution in terms of elementary functions, as given by Burton and Hamdan (1983), and used in, e.g.
Thomsen (2008b). In any case, multiple scales approximate solutions for the corresponding forced, damped frequency response can be obtained that agree excellently with numerical simulation, even for very high ratios c/x 0
2 .
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3 Nonlinear Vibrations: Classical Local Theory
Nonlinearity
Sometimes perturbation methods can be used even when nonlinearities are not
weak, as has been otherwise assumed in this chapter so far. The quality of a
perturbation-based prediction depend on the degree to which the underlying
assumptions are fulfilled. In the previous sections, when calculating resonant frequency responses, an essential assumption was on the nearness to resonance. This
was expressed as an assumed smallness of a detuning parameter r = X – x 0 , with X
being the excitation frequency and x 0 the resonance frequency in question, e.g. a
linear natural frequency. Even with the assumptions of weak damping and excitation fulfilled, an assumption based on small r will gradually worsen with the
increase of |r|, i.e. as the deviation of X from the constant frequency x 0 increases.
With a stronger (stiffness-)nonlinearity the resonance peak bends more over at
larger amplitudes, and even more so with smaller damping. This means that further
up the resonance peak r becomes too large for a prediction based on small r to
have acceptable accuracy. Thus, when comparing with numerical simulation, one
will typically note a marked drop in accuracy at the upper parts of a strongly
nonlinearly bent resonance peak.
But this can sometimes be cured by considering the detuning r instead as a
deviation from the nonlinear free oscillation frequency, i.e. the frequency on the
nonlinear response backbone, which depends on oscillation amplitude a (but not on
damping and excitation). Thus instead of r = X – x 0 , with a constant x 0 , one can
use r = X – x(a), where x is the nonlinear free and undamped oscillation frequency at oscillation amplitude a. So, if one can set up a reasonably simple
expression for x(a), a multiple scales analysis can be conducted that has good
accuracy as long as the excitation frequency is close to the backbone of the forced
frequency response. Considering a typical nonlinear frequency response, such as
Fig. 3.17, you will note that actually r will be smaller the further up the resonance
peak you come, to become eventually zero at the very top of the peak (which
intersects the backbone).
Burton and Rahman 1986 suggested and tested such an approach to derive
approximate analytical expressions for the stationary frequency response of a
Duffing oscillator with arbitrarily weak or strong nonlinearity, i.e. (3.152) with an
arbitrary ratio of c/x 0
2 (including zero and infinity). To find x(a) for this case
amounts to find the nonlinear oscillation frequency for (3.152) when b = q = 0.
This frequency can be expressed as a function of oscillation amplitude by a complete elliptic integral (Polyanin and Zaitsev 2003; Kovacic and Brennan 2011;
Kovacic 2020). But one can also for x(a) use a series solution in terms of elementary functions, as given by Burton and Hamdan (1983), and used in, e.g.
Thomsen (2008b). In any case, multiple scales approximate solutions for the corresponding forced, damped frequency response can be obtained that agree excellently with numerical simulation, even for very high ratios c/x 0
2 .
170
3 Nonlinear Vibrations: Classical Local Theory