On higher order methods of multiple scales in non-linear oscillations-periodic steady state response

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.…”

“…In particular, one can identify the results of [31] and [33] with our amplitude equation (3.4) and renormalized expansion (3.7), respectively.…”

Two-Dimensional Vortex-Induced Vibration of Cable Suspensions

“…The method of multiple-scales (MMS) (Nayfeh & Mook 1979;Rahman & Burton 1989) is employed by introducing independent 234 time scales and uniform expansions of the unknowns up to O( ) in the new time scales:…”

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

“…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…”

“…This process, known as reconstitution procedure [9±11], yields a power series for the evolution of the complex amplitude A in the form [3] dA dt…”