Topic
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.
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TL;DR: In this article, the authors presented both linear and nonlinear stability theories for characterization of viscoelastic film flows down on the outer surface of a rotating infinite vertical cylinder and derived a generalized nonlinear kinematic model to represent the physical system.
16 citations
TL;DR: In this article, the Routh-Hurwitz criterion is used to determine the instability boundaries of axially accelerating beams with 1:3 internal resonance and the effects of viscoelastic coefficient and the viscous damping coefficient are examined on the instability boundary.
16 citations
TL;DR: In this paper, weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical microbubbles is theoretically investigated, especially focusing on an initial small polydispersity of both the bubble radius and the number density of bubbles (i.e., void fraction), which appears in a field far from the sound source.
16 citations
TL;DR: In this article, the response of two-degree-of-freedom systems with quadratic and quartic nonlinearities to a principal parametric resonance in the presence of a two-to-one internal resonance is investigated.
Abstract: The response of two-degree-of-freedom systems with quadratic and quartic nonlinearities to a principal parametric resonance in the presence of two-to-one internal resonance is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order nonlinear ordinary differential (averaging) equations governing the modulation of the amplitudes and phases of the two modes. These equations are used to determine the steady state (fixed point) solutions and their stability. Bifurcations of the fixed points are investigated. Numerical solutions are carried out and graphical representations of the results are presented. The effect of the different parameters on the system response is studied. It is found that each mode has a single-valued curve and there exist zones of multivalued with the increasing and decreasing of some parameters. Both modes lose stability with the varying of some parameters. The region of definition decreases with the increasing and decreasing of some parameters.
16 citations
TL;DR: In this article, the principal parametric resonance of rotating Euler-Bernoulli beams with varying rotating speed is investigated, and a closed-form relation is derived to determine the stability region boundary under the condition of a small harmonic variation.
Abstract: In this paper, principal parametric resonance of rotating Euler–Bernoulli beams with varying rotating speed is investigated. The model contains the geometric nonlinearity due to the von Karman strain–displacement relationship and the centrifugal forces due to the rotation. The rotating speed of the beam is considered as a mean value which is perturbed by a small harmonic variation. In this case, when the frequency of the periodically perturbed value is twice the one of the axial mode frequencies, the principal parametric resonance occurs. The direct method of multiple scales is implemented to study on the dynamic instability produced by the principal parametric resonance phenomenon. A closed-form relation which determines the stability region boundary under the condition of the principal parametric resonance is derived. Numerical simulation based on the fourth-order Runge–Kutta method is established to validate the results obtained by the method of multiple scales. The numerical analysis is applied on the discretized equations of motion obtained by the Galerkin approach. After validation of the results, a comprehensive study is adjusted for demonstrating the damping coefficient and the mode number influences on the critical parametric excitation amplitude and the parametric stability region boundary. A discussion is also provided to illustrate the advantages and disadvantages of the fourth-order Runge–Kutta method in comparison with the method of multiple scales.
16 citations