On the multi-scale analysis of strongly non-linear forced oscillators

The Renormalization Group: A Perturbation Method for the Graduate Curriculum

“…Nayfeh [31] derived higher order amplitude equations for the Duffing equation by the method of multiple scales in response to some uncertainty that existed at that time in the literature [32,33,34,35,36]. In particular, one can identify the results of [31] and [33] with our amplitude equation (3.4) and renormalized expansion (3.7), respectively.…”

On the Steady State Response of Oscillators With Static and Inertia Non-Linearities

“…These curves showed that the second order approximations, for this case where P, d and a are not relatively small, are in good (quantitative as well as qualitative) agreement with those obtained by numerically integrating equation (10); that is, these results indicate that the second order solutions obtained using the MMS with reconstitution in [9±11], represent merely a``slight'' additive correction to the ®rst order solutions up to relatively large response amplitude, i.e., for a response amplitude up to H0 Á 85. First and second order perturbation solutions for the primary resonance response of the softening oscillator in equation (10) were also obtained by Rahman and Burton [8] using a``modi®ed MMS'' procedure presented in reference [7] with reconstitution version II in conjuction with the O, instead of (O±z), expansion. In accordance with this MMS procedure, they used transformation of time T = Ot, and de®ned a new expansion parameter a = a(e) using the free oscillation frequency±amplitude relation (backbone of the steady state response curve) i.e., a = ea 2 /(4 À 3a 2 ), where a is the steady state amplitude of the response fundamental harmonic.…”

“…Furthermore, the essence of the MMS perturbation method is to seek asymptotically valid, usually low order, approximations to the steady state periodic response by using a number of time scales and power series expansions for the dependent variables and parameters of the assumed weakly non-linear system in terms of a small positive gauge parameter e. These series expansions are neither unique nor convergent, and several procedural steps have been devised by various authors in order to obtain consistently ordered (asymptotically valid) ®rst and higher order MMS results. This has led to, so called, different``versions'' of the MMS method which differ in, for example, whether or not a transformation of time, T = Ot, is used, the way the detuning parameters are introduced (i.e., the way the excitation frequency O is expanded in power series of the perturbation parameter e), and in the way the partial time derivatives for the amplitude and phase of the response main harmonic component are used to obtain the second and higher order steady state response [3±5, 7,8]. Consider, for example, the problem, discussed in detail in reference [3], of obtaining a uniformly valid second order MMS approximation to the steady state primary resonance response of the weakly non-linear oscillator…”

Study of a piecewise linear dynamic system with negative and positive stiffness

“…Burton put forward a modified Lindstedt-Poincaré method which can be applied to strongly nonlinear dynamic systems [21]. Burton and Rahman further adapted the method of multiple scale to address strongly nonlinear oscillators [22]. Cheung et al proposed a new parameter expansion technique which could keep the expansion parameter always small regardless of the magnitude of the original system parameters, and hence the standard Lindstedt-Poincaré method (L-P method) would be applicable thereafter [23].…”