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primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

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Applied Mathematics and Mechanics(English Edition) 2010年第10期31卷 1293-1304页

作者：原培新李永强 Mechanical Engineering and Automation School Northeastern University(Shenyang) College of Science Northeastern University(Shenyang)

A homotopy analysis method（HAM）is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic *** validity of the HAM is independent of the existence of small parameters in the considered *** HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary *** examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar＇e method and the incremental harmonic balance method.

关键词： homotopy analysis method primary resonance series solution strong nonlinearity multi-degree-of-freedom

A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF A primary resonance OF A THREE CIRCULAR PLATES TORSION VIBRATION SYSTEM

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Applied Mathematics and Mechanics(English Edition) 1998年第6期19卷 539-546页

作者：杨志安邱家俊李骊 Department of Mechanics and Engineering Measurement Tianjin University 300072 Tianjin PR China Basic Science Department Peking Polytechnic University 100022 Beijing PR China

The method of averaging is applied in this paper to deal with primary resonance of a three circular plates torsion vibration system having cubic nonlinearities which is excited by a simple-harmonic excitation. Bifurcation equation of the steady state response is obtained and its singularity analysis is given. The results of theoretical analysis are shown to be in good agreement with experimental ones.

关键词： three circular plates torsion vibration method of averaging primary resonance singularity theory jump

Bifurcation control of nonlinear oscillator in primary and secondary resonance

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Journal of Central South University of Technology 2007年第6期14卷 826-831页

作者：李克安萧寒崔荣繁 Department of Mechanical and Electrical Engineering Hunan Institute of Science and Technology College of Mechanics and Aerospace Hunan University Shenyang Liming Aero-Engine (Group) Corporation

A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.

关键词： nonlinear oscillator saddle-node bifurcation feedback controller primary resonance secondary resonance gas turbine

Study on primary resonance of Ceramic and Metal Functionally Graded Material Rectangular Plate Subjected to Mechanical Force

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Study on Primary Resonance of Ceramic and Metal Functionally...

Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation

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作者： YANG Zhi-An1,JIA Shang-Shuai21.Tangshan College and Tangshan Key Laboratory of Structure and Vibration Engineering,Tangshan 063000,P.R.China 2.College of Mechanical Engineering,Hebei Polytechnic University,Tangshan 06300,P.R.China

According to the nonlinear dynamics equation of the functionally graded material rectangular plate,the nonlinear vibration equation of ceramic and metal functionally graded material rectangular plate subjected to mechanical force excitation is obtained in order to study primary resonance of ceramic and metal functionally graded material rectangular plate subjected to mechanical force *** means of the method of multiple scales,the approximation solution of primary resonance of the system is acquired,and numerical analysis of the influence of damping excitation as well as geometric parameters on amplitude frequency response curve of the primary resonance system is carried out.

关键词： Functionally Graded Material Nonlinear Vibration The Method Of Multiple Scales primary resonance Amplitude Frequency Response Curves

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Applied Mathematics and Mechanics(English Edition) 2024年第8期45卷 1353-1370页

作者： Lele REN Wei ZHANG Yufei ZHANG State Key Laboratory of Mechanical System and Vibration Shanghai Jiao Tong UniversityShanghai 200240China Department of Mechanics Guangxi UniversityNanning 530004China Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures College of Mechanical EngineeringBeijing University of TechnologyBeijing 100124China

The chaotic dynamic snap-through and complex nonlinear vibrations are investigated in a rectangular asymmetric cross-ply bistable composite laminated cantilever shell,in cases of 1:2 inter-well internal resonance and primary *** transverse foundation excitation is applied to the fixed end of the structure,and the other end is in a free *** first-order approximate multiple scales method is employed to perform the perturbation analysis on the dimensionless two-degree-of-freedom ordinary differential motion control *** four-dimensional averaged equations are derived in both polar and rectangular coordinate *** from the obtained frequency-amplitude and force-amplitude response curves,a detailed analysis is conducted to examine the impacts of excitation amplitude,damping coefficient,and tuning parameter on the nonlinear internal resonance characteristics of the *** nonlinear softening characteristic is exhibited in the upper stable-state,while the lower stable-state demonstrates the softening and linearity *** simulation is carried out using the fourth-order Runge-Kutta method,and a series of nonlinear response curves are *** the excitation amplitude further elucidates the global bifurcation and chaotic dynamic snap-through characteristics of the bistable cantilever shell.

关键词： bistable composite laminated cantilever shell inter-well internal resonance primary resonance chaotic dynamic snap-through complex nonlinear vibration

预应力下具有初始几何缺陷的石墨烯增强多孔复合材料双曲率壳结构的非线性主共振分析

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Acta Mechanica Sinica 2023年第2期39卷 191-204页

作者：佘桂林丁浩轩 College of Mechanical and Vehicle Engineering Chongqing UniversityChongqing 400044China

本文研究了石墨烯增强多孔复合材料(GPLRMFs)双曲率壳结构的非线性主共振行为.考虑初始几何缺陷和预应力,并基于Reddy高阶剪切变形壳理论和冯卡门几何非线性,推导出GPLRMFs双曲率壳的非线性运动方程.接着,考虑简支边界条件,利用伽辽金... 详细信息

本文研究了石墨烯增强多孔复合材料(GPLRMFs)双曲率壳结构的非线性主共振行为.考虑初始几何缺陷和预应力,并基于Reddy高阶剪切变形壳理论和冯卡门几何非线性,推导出GPLRMFs双曲率壳的非线性运动方程.接着,考虑简支边界条件,利用伽辽金法进行离散得到了非线性常微分方程,随后,利用改进的Lindstedt-Poincare(MLP)法求解,便可以得到GPLRMFs双曲率壳结构的主共振幅频响应关系.在数值分析中,通过与现有文献进行比较,从而验证了本研究的正确性.最后,分析了各个参数(包括壳体类型、石墨烯片(GPLs)分布模式、孔隙率分布类型、初始几何缺陷、GPLs重量分数、孔隙率系数和预应力)对非线性主共振幅频响应曲线的影响.

关键词： primary resonance Initial geometrical imperfection Doubly curved shells Graphene platelet Metal foam

Forced response of quadratic nonlinear oscillator: comparison of various approaches

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Applied Mathematics and Mechanics(English Edition) 2015年第11期36卷 1403-1416页

作者： Wenan JIANG Guoce ZHANG Liqun CHEN Shanghai Institute of Applied Mathematics and Mechanics Shanghai University College of Sciences Shanghai University Shanghai Key Laboratory of Mechanics in Energy Engineering Shanghai University

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of dif- ferent analytical approximate approaches. The forced vibration of snap-through mecha- nism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincar method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are com- pared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method pre- dicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the Lindstedt- Poincar method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.

关键词： forced vibration quadratic nonlinearity primary resonance multiple-scalemethod Lindstedt-Poincar~ method averaging method harmonic balance method

OPTIMAL CONTROL OF NONLINEAR VIBRATION resonanceS OF SINGLE-WALLED NANOTUBE BEAMS

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Acta Mechanica Solida Sinica 2014年第6期27卷 648-656页

作者： Canchang Liu Chuanbo Ren Lu Liu Yingzi Xu School of Transportation and Vehicle Engineering Shandong University of Technology

An optimal time-delay feedback control method is provided to mitigate the primary resonance of a single-walled carbon nanotube （SWCNT） subjected to a Lorentz force excited by a longitudinal magnetic field. The nonlinear governing equations of motion for the SWCNT under longitudinal magnetic field are derived and the modulation equations are obtained by using the method of multiple scales. The regions of the stable feedback gain are worked out by using the stability conditions of eigenvalue equation. Taking the attenuation ratio as the objective function and the stable vibration regions as constrained conditions, the optimal control parameters are worked out by using minimum optimal method. The optimal controllers are designed to control the dynamic behaviors of tile nonlinear vibration systems. It is found that the optimal feedback gain obtained by the optimal method can enhance the control performance of the primary resonance of SWCNT devices.

关键词： nanotube nonlinear vibration optimal control time delay primary resonance

Vibration characteristics of multi-parametric excitations and multi-frequency external excitations of rolling mill under entry thickness fuctuation of strip

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Journal of Iron and Steel Research International 2020年第5期27卷 517-527页

作者： Teng Xu Dong-xiao Hou Zheng-nan Sun Da-wu Guo School of Control Engineering Northeastern University at QinhuangdaoQinhuangdao 066004HebeiChina

A dynamic rolling force model with multi-parametric excitations and multi-frequency external excitations caused by entry thickness fuctuation of strip was *** on the dynamic rolling force,a nonlinear vertical vibration model with multi-parametric excitations and multi-frequency external excitations was *** method of multiple scales was used to solve the amplitude-frequency characteristic equation of primary resonance of the nonlinear vibration system of a rolling *** transition set and the topology structure of systematic global bifurcation were obtained by using the singularity ***,primary resonance characteristics of the system under entry thickness fuctuation of strip were analyzed by using actual parameters of the rolling *** global bifurcation curves with the change of amplitude and frequency of entry thickness fuctuation of strip were obtained by using numerical simulation,and many dynamic behaviors were found such as single-cycle motion,multi-cycle motion and chaotic motion,which can provide a theoretical reference for further restraining the vibration of a rolling mill.

关键词： Entry thickness fluctuation Dynamic rolling force Nonlinear vertical vibration primary resonance Bifurcation