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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 nonlinear instability problem of two superposed dielectric fluids is studied using the method of multiple scales, and the applied electric filed is taken into account under the influence of external modulations near a point of bifurcation.
Abstract: Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.

13 citations

Journal ArticleDOI
01 Mar 2008
TL;DR: In this paper, a two-degree-of-freedom non-linear system with quadratic and cubic nonlinearities and parametric excitation in the horizontal and vertical directions is investigated.
Abstract: The method of multiple scales is applied to investigate the non-linear oscillations and dynamic behaviour of a rotor-active magnetic bearings (AMBs) system, with time-varying stiffness. The rotor-AMB model is a two-degree-of-freedom non-linear system with quadratic and cubic non-linearities and parametric excitation in the horizontal and vertical directions. The case of principal parametric resonance is considered and examined. The steady-state response and the stability of the system at the principal parametric resonance case for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviours including multiple-valued solutions, jump phenomenon, hardening and softening non-linearity. Different effects of the system parameters on the non-linear response of the rotor are also reported. Results are compared with available published work.

13 citations

Journal ArticleDOI
TL;DR: In this paper, an analytical expression as a function of the system parameters describing the forced vibration of a spinning composite shaft in the neighborhood of the primary resonance is obtained, and the results are compared with both one and two modes, and it is shown that although the excitation is tuned in the neighbourhood of the first mode, one-mode discretization is not sufficient and it leads to inaccurate results.
Abstract: In this paper, nonlinear dynamics of an unbalanced composite spinning shaft are studied. Extensional–flexural–flexural–torsional equations of motion are derived via utilizing the three-dimensional constitutive relations of the material and Hamilton’s principle. The gyroscopic effects, rotary inertia and coupling due to material anisotropy are included, while the shear deformation is neglected. To analyze the rotor dynamic behavior, the full form of the equations without any simplification assumption (e.g., stretching or shortening assumption) is used. The method of multiple scales is applied to the discretized equations. An analytical expression as a function of the system parameters describing the forced vibration of a spinning composite shaft in the neighborhood of the primary resonance is obtained. The discretization is done with both one and two modes, and the results are compared. It is shown that although the excitation is tuned in the neighborhood of the first mode, one-mode discretization is not sufficient and it leads to inaccurate results. It shows the necessity of employing at least two modes in discretization due to the coupling in the equations. The effects of the external damping, eccentricity and the lamination angle on the vibration amplitude are investigated. In addition, the effect of the extensional–torsional coupling on the frequency response curves is investigated. To validate the perturbation results, numerical simulation is used.

13 citations

Journal ArticleDOI
TL;DR: In this article, the primary resonances of a pre-deformed rotating beam model including the quadratic and cubic nonlinearities are investigated in the presence of the 3:1 internal resonance.
Abstract: The primary resonances of a pre-deformed rotating beam model including the quadratic and cubic nonlinearities are investigated in the presence of the 3:1 internal resonance. The steady state responses of the beam are analyzed in two cases of the primary resonance with the method of multiple scales. The original dynamic equation is integrated numerically in two frequency sweep directions. The theoretical results are consistent with those obtained in the numerical simulation. The contributions of quadratic nonlinearities and cubic nonlinearities to the primary resonances behavior of the rotating beam are clarified. The frequency response curves are discussed by considering different model parameters such as the thermal gradient, the rotating speed, the damping coefficient and the gas pressure. The stability regions of coupled mode solutions are compared between the models with different nonlinearities in the case of the primary resonance of the second mode. A series of interesting findings are presented.

13 citations

Journal ArticleDOI
TL;DR: In this paper, an effective procedure using the component mode synthesis (CMS) and the method of multiple scales (MS) or the harmonic balance (HB) method for the nonlinear vibration analysis of rotor systems is proposed.
Abstract: In this paper, an effective procedure using the component mode synthesis (CMS) and the method of multiple scales (MS) or the harmonic balance (HB) method for the nonlinear vibration analysis of rotor systems is proposed. In the procedure, the system is divided into components and the differential equation for the component for each perturbation order or frequency is derived. The equation of motion for the overall system is then obtained using the CMS method. The dynamic analysis of a rotor system is carried out using the MS or the HB method. The distinguishing feature of the proposed procedure is that the nonlinear restoring force term is expressed using modal coordinates in a convenient form. The order of the modal equation of motion and the calculation time, therefore, can be reduced. In the numerical example, it is shown that the analytical methods proposed in this paper are effective for the nonlinear vibration analysis of rotor systems.

13 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202320
202237
202150
202042
201972
201851