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Numerical Analysis of Oscillation Death in Coupled Self-Excited Elastic Beams  [PDF]
Miguel A. Barron,Isaias Hilerio,Gabriel Plascencia
Advances in Mechanical Engineering , 2012, DOI: 10.1155/2012/746537
Abstract: The emergence of the oscillation death phenomenon in a ring of four coupled self-excited elastic beams is numerically explored in this work. The beams are mathematically represented through partial differential equations which are solved by means of the finite differences method. A coupling scheme based on shared boundary conditions at the roots of the beams is assumed, and as initial conditions, zero velocity of the first beam and three normal vibration modes of a linear elastic beam are employed. The influence of the self-exciting constant on the ring dynamics is analyzed. It is observed that oscillation death arises as result of the singularity of the coupling matrix. 1. Introduction In the past years the collective behavior of coupled nonlinear oscillators has been widely studied in many disciplines, for example, physics [1], biology [2], ecology [3], chemistry [4], and mechanics [5]. A wide diversity of nonlinear dynamic phenomena such as locking [1], partial synchronization [6], full synchronization [7], antiphase synchronization [8], and clustering [9] have been reported in coupled oscillators. Many coupling schemes have also been tested: local [10], nearest [11], global [12], diffusive [13], adaptive [14], delayed [15], hierarchical [16], and so on. An interesting behavior of coupled oscillators is amplitude death and oscillation death, which are steady states where the coupled oscillators stop their oscillation in a permanent way and become frozen in time [17–19]. Sometimes this cessation of oscillations in time is named quenching [20]. Amplitude death arises through a Hopf bifurcation mechanism in coupled oscillators with an important parameter mismatch or in identical oscillators with time delays [21]. An already existing unstable steady state with zero amplitude is transformed by the coupling into a stable one allowing its observation; that is, the coupling induces stability at the origin of the phase space. On the other hand, oscillation death occurs through a saddle-node bifurcation mechanism allowing the emergence of new fixed points: a new stable steady state with nonzero amplitude is created by the coupling [19, 21]. Frequently, in the literature amplitude death is confused with oscillation death [22–27]. Even the famous finding of Lord Rayleigh [28] related to the quenching of two organ pipes standing side by side is indistinctly considered as amplitude death or oscillation death [29]. To date, in spite of the significant conceptual and technical differences between amplitude death and oscillation death, there is not yet a clear
Vibration Analysis Of a Self-Excited Elastic Beam
Barrón-Meza, M. A.;
Journal of applied research and technology , 2010,
Abstract: the vibration behavior and the energy exchange among the normal modes of a clamped-free self-excited elastic beam are analyzed in this work. to model this kind of beam, the damping term of a van der pol oscillator is directly added to the equation of a linear elastic beam, yielding a single nonlinear partial differential equation. to solve this equation, a spectral method is employed. three vibration modes are considered in the analysis, and the values of the self-exciting constant are varied in order to cover from linear to nonlinear vibration behavior. multiple frequencies of the nonlinear beam are determined through the power spectral density of the beam free-end time series. given that this relatively simple model mimics at least in a qualitative way some key issues of the fluid-structure problem, it could be potentially useful for fatigue studies and vibration analysis of rotating blades in turbomachinery.
Vibration Analysis Of a Self-Excited Elastic Beam
M. A. Barrón-M
Journal of applied research and technology , 2010,
Abstract: The vibration behavior and the energy exchange among the normal modes of a clamped-free self-excited elasticbeam are analyzed in this work. To model this kind of beam, the damping term of a van der Pol oscillator is directlyadded to the equation of a linear elastic beam, yielding a single nonlinear partial differential equation. To solve thisequation, a spectral method is employed. Three vibration modes are considered in the analysis, and the values of theself-exciting constant are varied in order to cover from linear to nonlinear vibration behavior. Multiple frequencies ofthe nonlinear beam are determined through the power spectral density of the beam free-end time series. Given thatthis relatively simple model mimics at least in a qualitative way some key issues of the fluid-structure problem, it couldbe potentially useful for fatigue studies and vibration analysis of rotating blades in turbomachinery.
THE INHERENT VIBRATION CHARACTERISTICS OF COMPOSITE LAMINATED BEAMS AND ELASTIC BEAMS 1)
复合材料叠层梁和金属梁的固有振动特性

Xing Yufeng,Zhu Dechao,Qiao Yuansong,
邢誉峰
,诸德超,乔元松

力学学报 , 1998,
Abstract: In this Paper,according to Euler beam theory,Timoshenko beam theory and higher order beam theory,the inherent vibration characteristics of elastic beams are compared with that of composite laminated beams.The patterns of mode shapes are divided into two types,in one of the two types,the bending gradients are relatively more than the shear,in the other one,the shear gradients are more,and the latter type consists of several pure shear mode shapes.It is founded that many order frequencies and mode shapes related much to shear would not be obtained with the use of Timoshenko beam theory.On the basis of the above results,the applications of Euler beam theory,Timoshenko beam theory and beam theory based on warping corrections of shear deformation of cross sections to the analysis of elastic impact problems of beam structures are investigated and compared.Some conclusions are obtained that Euler beam theory can meet the requirements of impact analysis of practical beam structures when contact deformation is considered,and Timoshenko beam theory can meet the requirements of theoretical analysis of impact problems of beam structures,and beam theory based on warping corrections of shear deformation of cross sections by Zhu Dechao is essential to the static and dynamic analysis of composite laminated structures.However,according to the results obtained here,some causes why the delamination occurs and the intensity becomes lower greatly when the composite laminates are subjected to impact with lower velocity can be found out.
Winkler-Pasternak弹性地基梁自由振动的二维弹性分析 Two-dimensional elastic analysis for free vibration of beams set on winkler-pasternak elastic foundations
 [PDF]

蒲育,滕兆春
- , 2016, DOI: 10.7511/jslx201602007
Abstract: 基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。
Based on the two-dimension linear elasticity theory and applied the Hamilton's principle,the governing differential equations of free vibration for beam set on Winkler-Pasternak elastic foundation are derived.Using differential quadrature method (DQM),the free vibration dimensionless frequencies of beams are investigated numerically.With application of DQM in this paper,it illustrated that the method was validated and accurate for Bernoulli-Euler beams as well as for Timoshenko beams by comparison of previously reported results.Finally,the effect of the geometrical parameter on the non-dimensional frequency parameter of the beams,the influence and the convergence for dimensionless frequency owing to elastic coefficients of foundation under different boundary conditions are considered.
弹性地基上转动FGM梁自由振动的DTM分析 DTM analysis for free vibration of rotating FGM beams resting on elastic foundations
DTM analysis for free vibration of rotating FGM beams resting on elastic foundations
 [PDF]

滕兆春,衡亚洲,张会凯,马永斌
- , 2017, DOI: 10.7511/jslx201706006
Abstract: 基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行量纲化,采用微分变换法(DTM)对量纲控制微分方程及其边界条件进行变换,计算了弹性地基上转动FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的量纲固有频率,再将控制微分方程退化到转动和地基时的FGM梁,计算其不同梯度指数时第一阶量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的量纲固有频率随量纲转速和量纲弹性地基模量的增大而增大;在一定量纲转速和量纲弹性地基模量下,FGM梁的量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶量纲固有频率几乎不变。
Based on Euler-Bernoulli beam theory,the governing differential equation of motion of the lateral free vibration a rotating functionally graded material (FGM) beam on elastic foundation is derived by using generalized Hamilton principle,and differential transform method (DTM) is used to transform the dimensionless governing differential equation and the boundary conditions.At the same time,the dimensionless natural frequencies of transverse free vibration of rotating FGM beam on elastic foundation at the clamped-clamped,clamped-simply supported and clamped-free three boundary conditions are determined,then the governing differential equation is degenerated to the FGM without rotation and elastic foundation.The values of first non-dimensional natural frequency with different FGM gradient index are calculated and they are completely consistent with the results by either the FEM or the Lagrange multipliers method in the literature.The results show:at the above three kinds of the boundary conditions,the dimensionless natural frequencies increase with the growth of the dimensionless rotating speed and the dimensionless elastic foundation modulus.Under a certain dimensionless rotating speed and dimensionless elastic foundation modulus,the dimensionless natural frequencies decrease along with the growth of the FGM gradient index.However,when at clamped-simply supported and clamped-free boundary conditions,the first dimensionless natural frequency is almost constant.
An Efficient Analytical Method for Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends  [PDF]
Mustafa ?zgür Yayli,Murat Aras,Süleyman Aksoy
Shock and Vibration , 2014, DOI: 10.1155/2014/159213
Abstract: An efficient analytical method for vibration analysis of a Euler-Bernoulli beam on elastic foundation with elastically restrained ends has been reported. A Fourier sine series with Stoke’s transformation is used to obtain the vibration response. The general frequency determinant is developed on the basis of the analytical solution of the governing differential equation for all potential solution cases with rigid or restrained boundary conditions. Numerical analyses are performed to investigate the effects of various parameters, such as the springs at the boundaries to examine how the elastic foundation parameters affect the vibration frequencies. 1. Introduction Beams resting on elastic foundations have wide application in engineering practice. The vibration analysis of beams is investigated using various elastic foundation models, such as, Vlasov, Pasternak, and Winkler models. A number of studies have been performed to predict the dynamic response of beams on elastic foundations with different boundary conditions. Numerous works have been performed to explore the static deflection and vibration response of the beams resting on various elastic foundations. Chun [1] has investigated free vibration of hinged beam. Maurizi et al. [2] have considered the vibration frequencies for a beam with different boundary conditions. Vibration of beams on partial elastic foundations has been studied by Doyle and Pavlovic [3]. Laura et al. [4] have investigated beams which carry concentrated masses subject to an axial force. Abbas [5] has investigated vibration of Timoshenko beams with elastically restrained ends. Free vibration and stability behavior of uniform beams and columns with nonlinear elastic end rotational restraints has been considered by Rao and Naidu [6]. Free vibration behaviour of an Euler-Bernoulli beam resting on a variable Winkler foundation has been considered by Kacar et al. [7]. Civalek [8] has implemented differential quadrature and harmonic differential quadrature methods for buckling analysis of thin isotropic plates and elastic columns. H. K. Kim and M. S. Kim [9] have considered vibration of beams with generally restrained boundary conditions. A number of studies have been reported investigating the free vibration of beams on elastic foundation [10–25]. Although vibration analysis of beams on elastic foundation is a widely studied topic, there are only few papers that exist in the literature pertaining to the analysis of beams with elastically restrained ends. In this study, an efficient method is introduced for the analysis of the free
Effects of Structural Parameters on the Dynamics of a Beam Structure with a Beam-Type Vibration Absorber  [PDF]
Mothanna Y. Abd,Azma Putra,Nawal A. A. Jalil,Jamaludin Noorzaei
Advances in Acoustics and Vibration , 2012, DOI: 10.1155/2012/268964
Abstract: A beam-type absorber has been known as one of the dynamic vibration absorbers used to suppress excessive vibration of an engineering structure. This paper studies an absorbing beam which is attached through a visco-elastic layer on a primary beam structure. Solutions of the dynamic response are presented at the midspan of the primary and absorbing beams in simply supported edges subjected to a stationary harmonic load. The effect of structural parameters, namely, rigidity ratio, mass ratio, and damping of the layer and the structure as well as the layer stiffness on the response is investigated to reduce the vibration amplitude at the fundamental frequency of the original single primary beam. It is found that this can considerably reduce the amplitude at the corresponding troublesome frequency, but compromised situation should be noted by controlling the structural parameters. The model is also validated with measured data with reasonable agreement. 1. Introduction A beam-type absorber is one of the techniques to reduce undesirable vibration of many vibrating systems, such as a synchronous machine, mounting structure for a sensitive instrument, and other continuous structure in engineering. The absorber system usually consists of a beam attached to the host structure using an elastic element. The natural frequency of the absorber is then tuned to be the same as the troublesome operating frequency of the host structure to create counter force, which in return reduces the vibration of the structure. As beams are important structures in civil or mechanical engineering, several works have also been established to investigate the performance of the absorbing beam which is attached also to a beam structure. Among the earliest studies of the double-beam system is one proposed by Yamaguchi [1], which investigated the effectiveness of the dynamic vibration absorber consisting of double-cantilever visco-elastic beam connected by spring and viscous damper. The auxiliary beam is attached to the center of the main beam excited at its end by a sinusoidal force. It is found that the amplitude at resonances of the main beam is sensitive to the stiffness and mass of the absorbing beam. The damping ratio was formulated as a function of mass and layer stiffness of the absorber. Vu et al. [2] studied the distributed vibration absorber under stationary distributed force. A closed form was developed by utilizing change of variables and modal analysis to decouple and solve differential equations. Oniszczuk [3] studied the free vibrations of two identical parallel simply
Synchronization of a class of time-varied nonlinear vibration system
一类时变非线性振动系统的同步控制方法

Qin Wei-Yang,Wang Hong-Jin,Zhang Jin-Fu,
秦卫阳
,王红瑾,张劲夫

物理学报 , 2007,
Abstract: For a class of nonautonomous nonlinear vibration system,a corresponding derived system is constructed.A novel control method to realize the synchronization between the derived and the original system is presented.The method is proved by Lyapunov stability theory.The principle for choosing coupling parameters is given.For a Mathieu system excited by parameters and external force,the simulation is carried out.The result shows that the derived system can achieve fast synchronization with the original system under the action of chaos.The simulation for Duffing system excited by quasi-periodic force shows that the derived system can reach synchronization with the original system under the quasi-periodic motion.
Synchronization of two self-excited double pendula  [PDF]
Piotr Koluda,Przemyslaw Perlikowski,Krzysztof Czolczynski,Tomasz Kapitaniak
Physics , 2014, DOI: 10.1140/epjst/e2014-02129-7
Abstract: We consider the synchronization of two self-excited double pendula. We show that such pendula hanging on the same beam can have four different synchronous configurations. Our approximate analytical analysis allows us to derive the synchronization conditions and explain the observed types of synchronization. We consider an energy balance in the system and describe how the energy is transferred between the pendula via the oscillating beam, allowing thus the pendula synchronization. Changes and stability ranges of the obtained solutions with increasing and decreasing masses of the pendula are shown using path-following.
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