scholarly journals He’s multiple scales method for nonlinear vibrations

2019 ◽  
Vol 38 (3-4) ◽  
pp. 1708-1712 ◽  
Author(s):  
Zhong-Fu Ren ◽  
Shao-Wen Yao ◽  
Ji-Huan He
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Multiple Scales Method

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.

The Multiple Scales Method

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

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

2006 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Nader Jalili
Keyword(s):  
Natural Frequency ◽  
Piezoelectric Layer ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Closed Form Solution ◽  
Cubic Form ◽  
Dynamic Representation ◽  
Microcantilever Beam

The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically. The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials. The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility). Galerkin approximation is utilized to discretize the equations of motion. The obtained equation is simulated to find the natural frequency of the system. In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system. The experimental results verify the theoretical findings very closely. It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear models.

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.

Nonlinear Vibrations of Non-Uniform Beams Including Inertia Effects

2005 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Ion Stroe
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Effects ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Inertia Effects

This papers deals with nonlinear vibrations of non-uniform beams with geometrical nonlinearities such as moderately large curvatures, and inertia nonlinearities such as longitudinal and rotary inertia forces. The nonlinear fourth-order partial-differential equation describing the above nonlinear effects is presented. Using the method of multiple scales, each effect is found by reducing the nonlinear partial-differential equation of motion to two simpler linear partial-differential equations, homogeneous and nonhomogeneous. These equations along with given boundary conditions are analytically solved obtaining so-called zero-and first-order approximations of the beam’s nonlinear frequencies. Since the effect of mid-plane stretching is ignored, any boundary conditions could be considered as long as the supports are not fixed a constant distance apart. Analytical expressions showing the influence of these three nonlinearities on beam’s frequencies are presented up to some constant coefficients. These coefficients depend on the geometry of the beam. This paper can be used to study these influences on frequencies of different classes of beams. However, numerical results are presented for uniform beams. These results show that as beam slenderness increases the effect of these nonlinearities decreases. Also, they show that the most important nonlinear effect is due to moderately large curvature for slender beams.

Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

2019 ◽  
Vol 24 (11) ◽  
pp. 3514-3536
Author(s):  
Mohsen Tajik ◽  
Ardeshir Karami Mohammadi
Keyword(s):  
Nonlinear Vibration ◽  
Bifurcation Analysis ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Vibration Stability ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

scholarly journals Energy harvesting in a delayed Rayleigh harvester device

2018 ◽  
Vol 241 ◽  
pp. 01026 ◽  
Author(s):  
I. Kirrou ◽  
A. Bichri ◽  
M. Belhaq
Keyword(s):  
Energy Harvesting ◽  
Output Power ◽  
Multiple Scales ◽  
Analytical Investigation ◽  
Multiple Scales Method ◽  
Amplitude Response ◽  
Power Coupling ◽  
Piezoelectric Harvester

In the present work, we report on energy harvesting in a delayed Rayleigh oscillator coupled to a piezoelectric harvester device. Analytical investigation using the multiple scales method is performed to obtain approximation of the periodic and QP amplitude response. The influence of different parameters of the harvesting system on the output power coupling to periodic and QP vibrations is examined. Results show that for appropriate values of the delay parameters, QP vibrations can also be exploited to harvest energy with good performence.

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