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.

Simultaneous Resonances of Geometric Nonlinear Nonuniform Beams

Abstract: Simultaneous resonances, superharmonic and subharmonic, of two-term excitation nonlinear bending vibrations in the case of moderately large curvature of nonuniform cantilever beams are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order, result. Using factorization method, the linear modes of the zero-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude-phase evolution relationship and consequently the simultaneous resonances response.Copyright © 2009 by ASME

Microwave filter response of nonuniformly corrugated circular waveguide

Abstract: A corrugated circular waveguide is proposed as a microwave filter. The analysis is carried out using the perturbation method of multiple scales for the case of TM modes. The analysis concerns the interaction of two and four modes satisfying the resonant condition (Bragg condition) imposed by the periodicity of the waveguide wall. The coupled mode equations derived via the method of multiple scales are used to formulate the filtering problem as a two-point boundary-value problem which is solved numerically. Desirable filtering characteristics may be realised by introducing tapered as well as chirped corrugations to control the frequency response of such a wave filter. In case of two-mode interaction the side ripples can be eliminated by tapering the waveguide wall. In the four-mode interaction case, a multichannel filter may be realised by combining taper and chirp.

Internal Resonance of Rectangular Laminated Plates with Degenerate Modes

Abstract: Internal resonance of thin rectangular laminated plates with degenerate modes is investigated using the method of multiple scales. The plates are assumed to be simply supported along all edges, and both movable and immovable edge conditions are considered. In order to use the method of multiple scales properly, the definition of the detuning parameter σ, which expresses the relationship between the linear natural frequency ω and the driving frequency Ω, is changed from Ω=ω+eσ to Ω2=ω2+eσ. The influence of the lamination sequence and the edge conditions on two-mode responses of the plates is shown. The accuracy of the analytical results is confirmed by comparing them with results of numerical integration of the equation of motion. Furthermore, it is shown that the two-mode responses that lose stability via a Hopf bifurcation lead to chaotic motion.

On controlling the response of primary and parametric resonances of a nonlinear magnetic levitation system

Abstract: In this paper, a proportional-derivative controller is proposed to reduce the horizontal vibration of a magnetically levitated system having quadratic and cubic nonlinearities to primary and parametric excitations. A second order approximate solution is sought using the method of multiple scales perturbation technique to clarify the nonlinear behavior for both amplitude and phase of the system. The effect of feedback signal gain is studied to indicate the optimum values for best performance. Validation curves are included to compare the approximate solution and the numerical simulation. A comparison with previously published work is included.

Transient Coupling of Launch Vehicle Bending Responses with Aerodynamic Flow State Variations

Abstract: Current analytic formulations for the self-sustained coupling of launch vehicle bending responses with aerodynamic e ow statevariations are restricted to prediction of the limit cycle state and associated stiffness and damping values. The theory is extended to include evolution of the structural response during transonic e ight to the limit cycle state. The method of multiple scales is used to analyze the nonlinear equation of motion. The resulting analytic expressions agree well with histories from a rigorous numerical solution for the idealized force ‐response coupling relationship. Itis shown that the transientresponseforlaunchvehicleaeroelastic coupling depends on the initial conditions, the structural frequency and damping, and the time lag associated with e ow state changes. It is also shown that the maximum response during the transonic regime can be signie cantly smaller than that for the limit cycle state, depending on the combination of these variables. In particular, the short time period that launch vehicles typically spend in the transonic regime precludes convergence of the bending mode response to the limit cycle state when the structural frequency and damping are low. The structure of the solution space is described, and it is shown that a subcritical Hopf bifurcation exists when the e ow state changes at 0 deg.