Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

On the dynamics of rings rotating with variable spin speed

Abstract: This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.

Internal resonances and their bifurcations of a rigid-flexible space antenna

Abstract: The nonlinear dynamics of a flexible space antenna supported by a serial of two rigid arms mounted on a heavy satellite is studied under the conditions of 2:1 or 3:1 internal resonances. The nonlinear dynamic equations of the in-plane vibration of the antenna system of three degrees of freedom are derived first via the method of assumed modes. Then, the resonance and non-resonance regions are determined in terms of the dimensionless parameters of a rigid arm and a torsional spring in the antenna system. Afterwards, the approximate analytic solutions of frequency-amplitude response of the antenna system under 2:1 or 3:1 internal resonances are derived by using the method of multiple scales. Furthermore, the topologic description for the bifurcation of each internal resonance with respect to unfolding parameters is studied by means of the theory of singularity. Finally, the accuracy of the approximate analytic solutions is validated via the numerical solutions. The study shows that the internal resonances exist over a wide range of the detuning parameter. In addition, the 3:1 internal resonance exhibits much more complicated bifurcation topology than the 2:1 internal resonance. The approximate analytic solutions of internal resonances and their bifurcation analysis results provide an essential reference for designing the antenna system of concern.

Steady-state responses and their stability of nonlinear vibration of an axially accelerating string

Abstract: The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.

Single-mode response and control of a hinged–hinged flexible beam

Abstract: The problem of suppressing the vibrations of a hinged–hinged flexible beam that is subjected to primary and principal parametric excitations is tackled. Different control laws are proposed, and saturation phenomenon is investigated to suppress the vibrations of the system. The dynamics of the beam are modeled with a second-order nonlinear ordinary-differential equation. The method of multiple scales is used to derive two-first ordinary differential equations that govern the time variation of the amplitude and phase of the response. These equations are used to determine the steady-state responses and their stability. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters on the steady-state amplitude and on saturation phenomena at resonance have been reported.