THE ELLIPTIC MULTIPLE SCALES METHOD FOR A CLASS OF AUTONOMOUS STRONGLY NON-LINEAR OSCILLATORS

2000 ◽  
Vol 234 (3) ◽  
pp. 547-553 ◽  
Author(s):  
M. BELHAQ ◽  
F. LAKRAD
Keyword(s):  
Multiple Scales ◽  
Multiple Scales Method ◽  
Non Linear

scholarly journals Perturbations of nonlinear autonomous oscillators

1994 ◽  
Vol 35 (4) ◽  
pp. 445-468 ◽  
Author(s):  
P. B. Chapman
Keyword(s):  
General Theory ◽  
Fourier Coefficients ◽  
Multiple Scales ◽  
Second Order ◽  
Multiple Scales Method ◽  
Suitable Parameter ◽  
Non Linear

AbstractA general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

scholarly journals Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 85
Author(s):  
Yasser Salah Hamed ◽  
Ali Kandil
Keyword(s):  
Time Delay ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Control Process ◽  
Control Force ◽  
Specific Level ◽  
Loop Control ◽  
Positive Displacement ◽  
Non Linear ◽  
Linear Vibrations

Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones.

The Multiple Scales Method

2011 ◽  
pp. 359-360
Author(s):  
Michael Paidoussis ◽  
Stuart Price ◽  
Emmanuel de Langre
Keyword(s):  
Multiple Scales ◽  
Multiple Scales Method

Non-Linear Dynamic Analysis of a Cable-Stayed Beam

2005 ◽  
Author(s):  
Carlos E. N. Mazzilli ◽  
Franz Rena´n Villarroel Rojas
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Local Mode ◽  
Numerical Application ◽  
Global Mode ◽  
Lower Mode ◽  
Reduced Equations ◽  
External Resonance ◽  
Non Linear

The dynamic behaviour of a simple clamped beam suspended at the other end by an inclined cable stay is surveyed in this paper. The sag due to the cable weight, as well as the non-linear coupling between the cable and the beam motions are taken into account. The formulation for in-plane vibration follows closely that of Gattulli et al. [1] and confirms their findings for the overall features of the equations of motion and the system modal properties. A reduced non-linear mathematical model, with two degrees of freedom, is also developed, following again the steps of Gattulli and co-authors [2,3]. Hamilton’s Principle is evoked to allow for the projection of the displacement field of both the beam and the cable onto the space defined by the first two modes, namely a “global” mode (beam and cable) and a “local” mode (cable). The method of multiple scales is then applied to the analysis of the reduced equations of motion, when the system is subjected to the action of a harmonic loading. The steady-state solutions are characterised in the case of internal resonance between the local and the global modes, plus external resonance with respect to either one of the modes considered. A numerical application is presented, for which multiple-scale results are compared with those of numerical integration. A reasonable qualitative and quantitative agreement is seen to happen particularly in the case of external resonance with the higher mode. Discrepancies should obviously be expected due to strong non-linearities present in the reduced equations of motion. That is specially the case for external resonance with the lower mode.

Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

2021 ◽  
Vol 16 (3) ◽  
Author(s):  
Yuanbin Wang ◽  
Weidong Zhu
Keyword(s):  
Governing Equation ◽  
Transverse Vibration ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Frequency Responses ◽  
Runge Kutta ◽  
Nonlinear Transverse Vibration

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

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