scholarly journals An Analytical Approximation Method for Strongly Nonlinear Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Wang Shimin ◽  
Yang Lechang
Keyword(s):  
Vibration Amplitude ◽  
Analytical Approximation ◽  
Frequency Characteristics ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Nonlinear Restoring Force ◽  
Forced Oscillators ◽  
Stationary Vibration ◽  
Derivatives Of ◽  
Strongly Nonlinear Oscillators

An analytical method is proposed to get the amplitude-frequency and the phase-frequency characteristics of free/forced oscillators with nonlinear restoring force. The nonlinear restoring force is expressed as a spring with varying stiffness that depends on the vibration amplitude. That is, for stationary vibration, the restoring force linearly depends on the displacement, but the stiffness of the spring varies with the vibration amplitude for nonstationary oscillations. The varied stiffness is constructed by means of the first and second averaged derivatives of the restoring force with respect to the displacement. Then, this stiffness gives the amplitude frequency and the phase frequency characteristics of the oscillator. Various examples show that this method can be applied extensively to oscillators with nonlinear restoring force, and that the solving process is extremely simple.

scholarly journals An Optimal Iteration Method for Strongly Nonlinear Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Convergence Of Approximate Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

We introduce a new method, namely, the Optimal Iteration Perturbation Method (OIPM), to solve nonlinear differential equations of oscillators with cubic and harmonic restoring force. We illustrate that OIPM is very effective and convenient and does not require linearization or small perturbation. Contrary to conventional methods, in OIPM, only one iteration leads to high accuracy of the solutions. The main advantage of this approach consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solutions, which prove that OIPM is very efficient and accurate.

A Novel Method to Improve the Multiple-Scales Solution of the Forced Nonlinear Oscillators

2019 ◽  
Vol 16 (04) ◽  
pp. 1843010
Author(s):  
Hai-En Du ◽  
Guo-Kang Er ◽  
Vai Pan Iu
Keyword(s):  
Multiple Scales ◽  
Continuation Method ◽  
Inertial Force ◽  
Multiple Scales Method ◽  
Numerical Continuation ◽  
Strong Nonlinearity ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Forced Oscillators ◽  
Novel Method

We propose a novel procedure to improve the solutions obtained by perturbation methods for analyzing the solutions of strongly nonlinear systems in this paper. The proposed procedure is presented and then combined with the multiple-scales method for the optimum solutions of a class of forced oscillators with strong nonlinearity. The solutions obtained from conventional multiple-scales method and the proposed method are examined by the results from numerical continuation method. The results show that the proposed method is effective for the oscillators with nonlinear restoring force as well as nonlinear inertial force even if the nonlinearities are strong. Numerical results and comparison show that the proposed method can improve the solution a lot in comparison to the solution obtained by conventional multiple-scales method.

Combination of Kharrat-Toma Transform and Homotopy Perturbation Method to Solve a Strongly Nonlinear Oscillators Equation.

2020 ◽  
pp. 143-146
Author(s):  
Bachir Nour Kharrat ◽  
George Albert Toma
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Integral Transform ◽  
Physical Sciences ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Modified Method ◽  
Force Equation ◽  
Strongly Nonlinear Oscillators

This article introduces a new hybridization between the Kharrat-Toma transform and the homotopy perturbation method for solving a strongly nonlinear oscillator with a cubic and harmonic restoring force equation that arising in the applications of physical sciences. The proposed method is based on applying our new integral transform "Kharrat-Toma Transform" and then using the homotopy perturbation method. The objective of this paper is to illustrate the efficiency of this hybrid method and suggestion modified it. The results showed that the modified method is effectiveness and more accurate.

Nonlinear Restoring Force Identification of Strongly Nonlinear Structures by Displacement Measurement

2021 ◽  
pp. 1-29
Author(s):  
Qinghua Liu ◽  
Zehao Hou ◽  
Ying Zhang ◽  
Xingjian Jing ◽  
Gaëtan Kerschen ◽  
...  
Keyword(s):  
Energy Harvesting ◽  
Vibration Isolation ◽  
Displacement Measurement ◽  
Identification Accuracy ◽  
Geometrical Parameters ◽  
Nonlinear Structures ◽  
Restoring Force ◽  
Experimental Conditions ◽  
Strongly Nonlinear ◽  
Nonlinear Restoring Force

Abstract Strongly nonlinear structures have attracted a great deal of attention in energy harvesting and vibration isolation recently. However, it is challenging to accurately characterize the nonlinear restoring force using analytical modeling or cyclic loading tests in many realistic conditions due to the uncertainty of installation parameters or other constraints, including space size and dynamic disturbance. Therefore, a displacement-measurement restoring force surface identification approach is presented for obtaining the nonlinear restoring force. Widely known quasi-zero stiffness, bistable and tristable structures are designed in a cantilever-beam system with coupled rotatable magnets to illustrate the strongly nonlinear properties in the application of energy harvesting and vibration isolation. Based on the derived physical model of the designed strongly nonlinear structures, the displacement-measurement restoring force surface identification with a least-squares parameter fitting is proposed to obtain the parameters of the nonlinear restoring force. The comparison between the acceleration integration and displacement differentiation methods for describing the restoring force surface of strongly nonlinear structures is discussed. Besides, the influence of the noise level on identification accuracy is investigated. In experimental conditions, quasi-zero stiffness, bistable, and tristable nonlinear structures with various geometrical parameters are utilized to analyze the identified nonlinear restoring force curve and measured force-displacement trajectory. Finally, experimental results verify the effectiveness of the displacement-measurement restoring force surface method to obtain the nonlinear restoring force.

Nonparametric nonlinear restoring force and excitation identification with Legendre polynomial model and data fusion

2021 ◽  
pp. 147592172199474
Author(s):  
Bin Xu ◽  
Ye Zhao ◽  
Baichuan Deng ◽  
Yibang Du ◽  
Chen Wang ◽  
...  
Keyword(s):  
Data Fusion ◽  
Legendre Polynomial ◽  
Structural Parameters ◽  
Polynomial Model ◽  
Magnetorheological Damper ◽  
Dynamic Responses ◽  
Identification Accuracy ◽  
Restoring Force ◽  
Nonlinear Restoring Force ◽  
Dynamic Loadings

Identification of nonlinear restoring force and dynamic loadings provides critical information for post-event damage diagnosis of structures. Due to high complexity and individuality of structural nonlinearities, it is difficult to provide an exact parametric mathematical model in advance to describe the nonlinear behavior of a structural member or a substructure under strong dynamic loadings in practice. Moreover, external dynamic loading applied to an engineering structure is usually unknown and only acceleration responses at limited degrees of freedom of the structure are available for identification. In this study, a nonparametric nonlinear restoring force and excitation identification approach combining the Legendre polynomial model and extended Kalman filter with unknown input is proposed using limited acceleration measurements fused with limited displacement measurements. Then, the performance of the proposed approach is first illustrated via numerical simulation with multi-degree-of-freedom frame structures equipped with magnetorheological dampers mimicking nonlinearity under direct dynamic excitation or base excitation using noise-polluted measurements. Finally, a dynamic experimental study on a four-story steel frame model equipped with a magnetorheological damper is carried out and dynamic response measurement is employed to validate the effectiveness of the proposed method by comparing the identified dynamic responses, nonlinear restoring force, and excitation force with the test measurements. The convergence and the effect of initial estimation errors of structural parameters on the final identification results are investigated. The effect of data fusion on improving the identification accuracy is also investigated.

Soft X-ray emission from an anharmonic collisional nanoplasma by a laser–nanocluster interaction

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
R. Nemati Siahmazgi ◽  
S. Jafari
Keyword(s):  
Electric Field ◽  
Laser Beam ◽  
Resonant Frequency ◽  
Restoring Force ◽  
Cluster Radius ◽  
X Ray ◽  
Nonlinear Restoring Force ◽  
Cluster Temperature ◽  
Argon Clusters ◽  
Helium Neon

The purpose of the present paper is to investigate the generation of soft X-ray emission from an anharmonic collisional nanoplasma by a laser–nanocluster interaction. The electric field of the laser beam interacts with the nanocluster and leads to ionization of the cluster atoms, which then produces a nanoplasma. Because of the nonlinear restoring force in an anharmonic nanoplasma, the fluctuations and heating rate of, as well as the power radiated by, the electrons in the nanocluster plasma will be notably different from those arising from a linear restoring force. By comparing the nonlinear restoring force state (which arises from an anharmonic cluster) with that of the linear restoring force (in harmonic clusters), the cluster temperature specifically changes at the resonant frequency relative to the linear restoring force, while the variation of the anharmonic cluster radius is almost identical to that of the harmonic cluster radius. In addition, it is revealed that a sharp peak of X-ray emission arises after some picoseconds in deuterium, helium, neon and argon clusters.

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