Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate are studied by the extended Melnikov method.The rectangular thin plate is subject to transversal and in-plane excitation.A two-degree-of-freedom nonlinear nonautonomous system governing equations of motion for the rectangular thin plate is derived by the von Karman type equation and the Galerkin approach.A one-toone internal resonance is considered.An averaged equation is obtained with a multi-scale method.After transforming the averaged equation into a standard form,the extended Melnikov method is used to show the existence of multi-pulse chaotic dynamics,which can be used to explain the mechanism of modal interactions of thin plates.A method for calculating the Melnikov function is given without an explicit analytical expression of homoclinic orbits.Furthermore,restrictions on the damping,excitation,and detuning parameters are obtained,under which the multi-pulse chaotic dynamics is expected.The results of numerical simulations are also given to indicate the existence of small amplitude multi-pulse chaotic responses for the rectangular thin plate.     

Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance

The nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to the in-plane and transversal excitations are investigated in this paper. Based on the Reddy??s third-order plate theory and the von Karman type equations for the geometric nonlinearity, the nonlinear partial differential governing equations of motion for the composite laminated cantilever rectangular plate are established by using the Hamilton??s principle. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two degree-of-freedom nonlinear system under combined parametric and forcing excitations. The case of foundational parametric resonance and 1:1 internal resonance is taken into account. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. The numerical method is used to find the periodic and chaotic motions of the composite laminated cantilever rectangular plate. It is found that the chaotic responses are sensitive to the changing of the forcing excitations and the damping coefficient. The influence of the forcing excitation and the damping coefficient on the bifurcations and chaotic behaviors of the composite laminated cantilever rectangular plate is investigated numerically. The frequency-response curves of the first-order and the second-order modes show that there exists the soft-spring type characteristic for the first-order and the second-order modes.     

On experiments in harmonically excited cantilever plates with 1:2 internal resonance

Nonlinear Dynamics - This work presents some experimental results for resonant nonlinear response of hyperelastic plates for 1:2 internal resonance. Previously developed topology optimization...     

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus.     

Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes van der Pol damping, is very complex. In this paper, Melnikov'smethod is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system for various resonant cases.Finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.     

黏弹性传动带1:3内共振时的周期和混沌运动

研究了参数激励作用下黏弹性传动带在1:3内共振时的周期解分岔和混沌动力学. 同时考虑传动带的线性外阻尼因素和材料内阻尼因素. 首先建立了具有线性外阻尼情况下的黏弹性传动带平面运动时的非线性动力学方程, 黏弹性材料的本构关系用Kelvin模型描述. 然后考虑黏弹性传动带的横向振动问题, 利用多尺度法和Galerkin离散法得到黏弹性传动带系统在1:3内共振时的平均方程. 最后利用数值模拟方法研究了黏弹性传动带系统的周期振动和混沌动力学, 得到了系统在不同参数下的混沌运动. 数值模拟结果说明黏弹性传动带系统存在周期分岔, 概周期运动及混沌运动.     

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ₃2₂ and ₂2₁ to a harmonic excitation of the third mode, where the _m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.     

On beams membranes and plates vibration backbone curves in cases of internal resonance

The amplitude — frequency relations of beams, membranes and plates in free vibration with moderately large amplitude are of interest. A two harmonics solution of motion equation and the reciprocal interaction of two modes of vibration are taken into account. The resulting backbone curves with loops, additional branches and bifurcation points are determined and discussed. The physical meaning of the considered curves is also explained.

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Sommario Si studiano le vibrazioni libere di ampiezza moderatamente grande di travi, membrane e piastre. Si sviluppano soluzioni approssimate con due armoniche delle relative equazioni di moto in due diversi casi di risonanza interna, e si analizza l'effetto dell'interazione di due modi di vibrazione. Vengono determinate e discusse le curve frequenza-ampiezza con l'associata ricchezza di rami addizionali e punti di biforcazione. Il significato fisico di tali curve viene altresì evidenziato.

     

Global bifurcations and chaos in a harmonically excited and undamped circular plate

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.     

Primary resonance in forced vibrations of in-plane translating viscoelastic plates with 3:1 internal resonance

Nonlinear forced vibrations of in-plane translating viscoelastic plates subjected to plane stresses are analytically and numerically investigated on the steady-state responses in external and internal resonances. A nonlinear partial-differential equation with the associated boundary conditions governing the transverse motion is derived from the generalized Hamilton principle and the Kelvin relation. The method of multiple scales is directly applied to establish the solvability conditions in the primary resonance and the 3:1 internal resonance. The steady-state responses are predicted in two patterns: single-mode and two-mode solutions. The Routh?CHurvitz criterion is used to determine the stabilities of the steady-state responses. The effects of the in-plane translating speed, the viscosity coefficient, and the excitation amplitude on the steady-state responses are examined. The differential quadrature scheme is developed to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the single-mode solutions of the steady-state responses.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

梁主共振情形下索梁结构1∶1内共振分析

研究梁产生主共振情形下索梁组合结构的1∶1内共振问题。基于斜拉桥中的索梁组合结构模型,忽略索梁纵向惯性力的影响,考虑弯曲刚度、几何非线性及垂度等因素,利用索梁连接处的变形协调条件,采用Hamilton变分原理建立了索梁结构面内耦合非线性偏微分方程,运用Galerkin离散和多尺度法研究了梁主共振情形下索梁的1∶1相互作用问题,获得了内共振时的平均方程和分叉响应曲线方程。以某斜拉桥中索梁结构参数为例,研究了内共振时索梁结构之间的相互影响及时程曲线。结果表明,索容易出现共振情形,并呈现出较强的非线性特点;梁振动对索振动影响显著,索振动对梁振动影响较小;索梁内共振时能量相互交换,索梁振幅呈现此消彼长的现象。     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

Interaction between torsional and lateral vibrations in flexible rotor systems with discontinuous friction

In this paper, we analyze the interaction between friction-induced vibrations and self-sustained lateral vibrations caused by a mass-unbalance in an experimental rotor dynamic setup. This study is performed on the level of both numerical and experimental bifurcation analyses. Numerical analyses show that two types of torsional vibrations can appear: friction-induced torsional vibrations and torsional vibrations due to the coupling between torsional and lateral dynamics in the system. Moreover, both the numerical and experimental results show that a higher level of mass-unbalance, which generally increases the lateral vibrations, can have a stabilizing effect on the torsional dynamics, i.e. friction-induced limit cycling can disappear. Both types of analysis provide insight in the fundamental mechanisms causing self-sustained oscillations in rotor systems with flexibility, mass-unbalance and discontinuous friction which support the design of such flexible rotor systems.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. II: Combination resonance

In Part I of this work nonlinear coupling between torsional motion and both in-plane and out-of-plane flexural motion was examined for inextensional beams in the presence of a one-to-one internal resonance. Here the nonlinear response of the system considered in Part I is investigated for the case of an internal combination resonance involving modes associated with bending in two directions and torsion. The analysis presented is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities and account for torsional dynamics.     

Nonlinear resonant response of the cable-stayed beam with one-to-one internal resonance in veering and crossover regions

Nonlinear Dynamics - In this study, the large amplitude vibration of the cable-stayed beam subjected to external excitation is investigated. Emphasis is focused on the one-to-one resonant...     

The bending,stability and vibrations of rectangular plates with free edges on elastic foundations

This paper discusses the problems of the bending, stability and vibrations of the rectangular plates with free boundaries on elastic foundations. In the present paper we select a flexural function, which satisfies not only all the boundary conditions of free edges but also the conditions at free corner points, and consequently we obtain a better approximate solution. The energy method is used in this paper.     

Experimental analysis of the natural vibrations and stability of cylindrical shells reinforced with rectangular plates

The paper presents experimental results on the effect of reinforcing cylindrical shells with rectangular plates on the axial compressive critical loads and the natural frequencies and modes of two sets of shells __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 100–103, May 2008.     

轴向运动正交各向异性板的超谐波共振及稳定性

研究了轴向运动正交各向异性条形薄板在线载荷作用下的超谐波共振问题。通过哈密顿原理导出了几何非线性下正交各向异性条形板的非线性振动方程。运用伽辽金积分法,推得了关于时间变量的量纲归一化非线性振动微分方程组。应用多尺度法求解三阶超谐波共振问题,得到了稳态运动下一阶、二阶、三阶共振形式的共振幅值响应方程。利用Liapunov方法推得不同共振形式稳态解的稳定性判据,并据此分析不同参数对系统稳定性的影响。绘制了振幅特性变化曲线图和与之对应的激发共振多解临界点曲线图,分析系统参数对共振的影响,并预测系统进入非线性共振区域的临界条件。得出激励在特定位置区间时可激发系统的超谐波共振,随着激励幅值的增加,上稳定解支减小,下稳定解支增加,且一阶模态振幅大于二阶、三阶振幅。     

A theoretical and experimental investigation of a primary resonance of a three circular plates torsion vibration system

I.IntroductionRotor-shaftingiswidelyusedinengineering,androtor-shaftingtorsionvibrationisgainingattentionwidespreadly.Inthispaper,themodeloftherotor-shaftingtorsionvibrationofa200MWturbogenerationsetisdesignedandathreecircularplatestorsionvibrationsystemwithcubicnonlinearitiestoasimple-harmonicexcitationisstudied.TheobjectiveofthepaperistostudythedynamicsphenomenaofthesystemwhileQ=co2.Themethodofaveragingisusedandthesteadystateresponsebifurcationequationanditssingularityanalysisaregiven.Thet…