Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

A dynamical model for nonlinear viscoelastic corrugated circular plates with shallow sinusoidal corrugations

An integrated mathematic model and an efficient algorithm on the dynamical behavior of homogeneous viscoelastic corrugated circular plates with shallow sinusoidal corrugations are suggested. Based on the nonlinear bending theory of thin shallow shells, a set of integro-partial differential equations governing the motion of the plates is established from extended Hamilton’s principle. The material behavior is given in terms of the Boltzmann superposition principle. The variational method is applied following an assumed spatial mode to simplify the governing equations to a nonlinear integro-differential variation of the Duffing equation in the temporal domain, which is further reduced to an autonomic system with four coupled first-order ordinary differential equation by introducing an auxiliary variable. These measurements make the numerical simulation performs easily. The classical tools of nonlinear dynamics, such as Poincaré map, phase portrait, Lyapunov exponent, and bifurcation diagrams, are illustrated. The influences of geometric and physical parameters of the plate on its dynamic characteristics are examined. The present mathematic model can easily be used to the similar problems related to other dynamical system for viscoelastic thin plates and shallow shells.     

Studies on Bifurcation and Chaos of a String-Beam Coupled System with Two Degrees-of-Freedom

In this paper, research on nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitations is presented. The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin's method is employed to simplify the governing equations to a set of ordinary differential equations with two degrees-of-freedom. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam, and primary resonance for the string is considered. The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system. Based on the averaged equation obtained here, the techniques of phase portrait, waveform, and Poincare map are applied to analyze the periodic and chaotic motions. It is found from numerical simulations that there are obvious jumping phenomena in the resonant response–frequency curves. It is indicated from the phase portrait and Poincare map that period-4, period-2, and periodic solutions and chaotic motions occur in the transverse nonlinear vibrations of the string-beam coupled system under certain conditions. An erratum to this article is available at .     

Nonlinear study of the dynamic behavior of a string-beam coupled system under combined excitations

In this paper,the nonlinear dynamic behavior of a string-beam coupled system subjected to external,parametric and tuned excitations is presented.The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system which are described by a set of ordinary differential equations with two degrees of freedom.The case of 1:1 internal resonance between the modes of the beam and string,and the primary and combined resonance for the beam is considered.The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system and obtain approximate solutions up to and including the second-order approximations.All resonance cases are extracted and investigated.Stability of the system is studied using frequency response equations and the phase-plane method.Numerical solutions are carried out and the results are presented graphically and discussed.The effects of the different parameters on both response and stability of the system are investigated.The reported results are compared to the available published work.     

Nonlinear dynamics of extensible fluid-conveying pipes,supported at both ends

In this paper, the post-divergence behaviour of extensible fluid-conveying pipes supported at both ends is studied using the weakly nonlinear equations of motion of Semler, Li and Païdoussis. The two coupled nonlinear partial differential equations are discretized via Galerkin's method and the resulting set of ordinary differential equations is solved either by Houbolt's finite difference method or via AUTO. Typically, the pipe is stable at its original static equilibrium position up to the flow velocity where it loses stability by static divergence via a supercritical pitchfork bifurcation. The amplitude of the resultant buckling increases with increasing flow, but no secondary instability occurs beyond the pitchfork bifurcation. The effects of the system parameters on pipe behaviour as well as the possibility of a subcritical pitchfork bifurcation have also been studied.     

Nonlinear dynamics of cantilevered pipes conveying fluid: Towards a further understanding of the effect of loose constraints

The nonlinear dynamics of a fluid-conveying cantilevered pipe with loose constraints placed somewhere along its length is investigated. The main objective of this study is to determine the effects of several geometrical and physical parameters of the loose constraints on the characteristics and behavior of pipes conveying fluid. Based on the full nonlinear equation of motion, the dynamical behavior of the pipe system is investigated. Phase portraits and bifurcation diagrams are constructed for a selected set of system parameters. Typical results are firstly compared to numerical ones reported previously and excellent agreement is obtained. Then, the threshold flow velocities for several key bifurcations including pitchfork, period doubling, chaos, and sticking behaviors are predicted, showing that in many cases, the gap size, stiffness, and asymmetry of the loose constraints have remarkable effects on the nonlinear responses of the cantilevered pipe conveying fluid. For a pipe system with small/large constraint gap sizes, small constraint stiffness, or large constraint offset, some of the complex dynamical behaviors including chaos and period-doubling bifurcations would disappear, at least in the flow velocity range of interest.     

VIBRO-IMPACT DYNAMICS OF PIPE CONVEYING FLUID SUBJECTED TO RIGID CLEARANCE CONSTRAINT 1)

管道与间隙约束间的碰撞振动是工程输流管结构的一个重要动力学现象. 迄今,人们通常采用光滑的非线性弹簧来模拟管道与间隙约束之间的碰撞力,但这种光滑的碰撞力无法真实反映碰撞前后管道状态向量的非光滑传递特征. 本文基于非光滑理论建立了具有刚性间隙约束简支输流管的非线性碰撞振动模型,利用 Galerkin 法离散了无穷维的管道模型, 并引入恢复系数构造了碰撞前后管道各处状态向量的传递矩阵,运用四阶龙格库塔法分析了脉动内流激励下管道与刚性间隙约束的非光滑碰撞振动现象,着重讨论了刚性间隙约束参数对管道动态响应随流速脉动频率变化的影响规律,特别是碰撞振动的周期性运动规律. 研究结果表明,刚性约束间隙值对管道碰撞振动行为的影响较大,在某些脉动频率下管道会出现多周期和非周期性的运动形态,还可出现非光滑系统特有的黏滑现象. 此外,碰撞恢复系数对管道振动的影响也比较显著,较小的恢复系数值更容易使管道在大范围脉动频率区间出现复杂的非周期碰撞振动.     

具有刚性间隙约束输流管的碰撞振动

管道与间隙约束间的碰撞振动是工程输流管结构的一个重要动力学现象. 迄今,人们通常采用光滑的非线性弹簧来模拟管道与间隙约束之间的碰撞力,但这种光滑的碰撞力无法真实反映碰撞前后管道状态向量的非光滑传递特征. 本文基于非光滑理论建立了具有刚性间隙约束简支输流管的非线性碰撞振动模型,利用 Galerkin 法离散了无穷维的管道模型, 并引入恢复系数构造了碰撞前后管道各处状态向量的传递矩阵,运用四阶龙格库塔法分析了脉动内流激励下管道与刚性间隙约束的非光滑碰撞振动现象,着重讨论了刚性间隙约束参数对管道动态响应随流速脉动频率变化的影响规律,特别是碰撞振动的周期性运动规律. 研究结果表明,刚性约束间隙值对管道碰撞振动行为的影响较大,在某些脉动频率下管道会出现多周期和非周期性的运动形态,还可出现非光滑系统特有的黏滑现象. 此外,碰撞恢复系数对管道振动的影响也比较显著,较小的恢复系数值更容易使管道在大范围脉动频率区间出现复杂的非周期碰撞振动.      

输液管的非线性振动、分叉与混沌——现状与展望

围绕输液管的非线性振动、分叉与混沌问题，对近几年来的主要研究工作加以综述并提出预测，其中包括运动方程中非线性项的归纳与讨论、非线性动力分析的一些现代计算方法、定常流速下输液管的分叉与混沌行为、振荡流速下输液管的参数共振以及今后值得进一步研究的某些问题．     

Nonlinear Wave-Wave Interaction and Stability Criterion for Parametrically Coupled Nonlinear Schrödinger Equations

A rigorous mathematical reduction of the procedure widely usedfor studying a class of the nonlinear problems with perturbations,namely the method of the multiple scales, is used. A profound analysis,which provides an approach for deriving a coupled nonlinearSchrödinger equations. The investigation has been achieved byperturbing the nonlinear dynamical system about the linear dynamicalproblem. Modulated wavetrains are described to all orders ofapproximation. Moreover, we extend our approach to deal with equationshaving periodic terms. Two types of simultaneous nonlinearSchrödinger equations are derived. One type is valid at thenon-parametric system and the second type represents a modification forthe first type which is governed the non-resonance case. Two parametriccoupled nonlinear Schrödeinger equations are derived to govern thesecond-sub-harmonic resonance. In addition other two coupled equationsare found for the third-sub-harmonic resonance case. These systems ofequations control the stability behavior at the parametric resonancecases. The stability criteria for the several types of coupled nonlinearSchrödinger equations are studied. These criteria are achieved by atemporal periodic perturbation.     

Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.     

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.     

Nonlinear Forced Vibration of Cantilevered Pipes Conveying Fluid

The nonlinear forced vibrations of a cantilevered pipe conveying fluid under base excitations are explored by means of the full nonlinear equation of motion, and the fourthorder Runge-Kutta integration algorithm is used as a numerical tool to solve the discretized equations. The self-excited vibration is briefly discussed first, focusing on the effect of flow velocity on the stability and post-flutter dynamical behavior of the pipe system with parameters close to those in previous experiments. Then, the nonlinear forced vibrations are examined using several concrete examples by means of frequency response diagrams and phase-plane plots. It shows that, at low flow velocity, the resonant amplitude near the first-mode natural frequency is larger than its counterpart near the second-mode natural frequency. The second-mode frequency response curve clearly displays a softening-type behavior with hysteresis phenomenon, while the first-mode frequency response curve almost maintains its neutrality. At moderate flow velocity,interestingly, the first-mode resonance response diminishes and the hysteresis phenomenon of the second-mode response disappears. At high flow velocity beyond the flutter threshold, the frequency response curve would exhibit a quenching-like behavior. When the excitation frequency is increased through the quenching point, the response of the pipe may shift from quasiperiodic to periodic. The results obtained in the present, work highlight the dramatic influence of internal fluid flow on the nonlinear forced vibrations of slender pipes.     

Dynamical behaviors of nonlinear viscoelastic thick plates with damage

Based on the deformation hypothesis of Timoshenko's plates and the Boltzmann's superposition principles for linear viscoelastic materials, the nonlinear equations governing the dynamical behavior of Timoshenko's viscoelastic thick plates with damage are presented. The Galerkin method is applied to simplify the set of equations. The numerical methods in nonlinear dynamics are used to solve the simplified systems. It could be seen that there are plenty of dynamical properties for dynamical systems formed by this kind of viscoelastic thick plate with damage under a transverse harmonic load. The influences of load, geometry and material parameters on the dynamical behavior of the nonlinear system are investigated in detail. At the same time, the effect of damage on the dynamical behavior of plate is also discussed.     

分布式运动约束下悬臂输液管的参数共振研究

输液管道结构在航空、航天、机械、海洋、水利和核电等工程领域都有广泛应用,其稳定性、振动与安全评估备受关注.针对具有分布式运动约束悬臂输液管的非线性动力学模型,分别采用立方非线性弹簧和修正三线性弹簧来模拟运动约束的作用力,研究了管道在脉动内流激励下的参数共振行为.首先,从输液管系统的非线性控制方程出发,利用Galerkin方法进行离散化;然后,由Floquet理论得出线性系统在失稳前两个不同平均流速下脉动幅值和脉动频率变化时的共振参数区域;最后,考虑系统的几何非线性项和分布式非线性运动约束力的影响,求解了管道的非线性动力学响应,讨论了非线性项及运动约束力对管道参数共振行为的影响.研究结果表明,系统非线性共振响应的参数区域与线性系统的共振参数区域是一致的,分布式运动约束力对发生参数共振时管道的位移响应有显著影响;立方非线性弹簧和修正三线性弹簧模型所预测的分岔路径存有较大差异,但都可诱发管道在一定的参数激励下出现混沌运动.      

Resonant dynamics in strongly nonlinear systems

This work considers the effect of resonances in systems in which the two resonant frequencies are allowed to slowly change, depending on the state of the system. A strongly nonlinear system is introduced that allows for exact solutions. This system is then coupled to a second component, and through the method of averaging, a reduced-order model is developed that approximates the dynamical behavior near a 1:1 resonance between the two components. The resulting reduced system is studied using bifurcation theory and Melnikov analysis to obtain predictions of the near-resonant dynamics. Finally, these predictions are compared to numerical simulations of the original equations. Two main points appear: (i) for nonlinear systems, the period–amplitude dependence plays an important role in the evolution of the system, and (ii) the coordinates identified within the reduced system allow for the qualitative structure of the original equations to appear.     

Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity

The nonlinear dynamical behavior of a micromechanical resonator acting as one of the mirrors in an optical resonance cavity is investigated. The mechanical motion is coupled to the optical power circulating inside the cavity both directly through the radiation pressure and indirectly through heating that gives rise to a frequency shift in the mechanical resonance and to thermal deformation. The energy stored in the optical cavity is assumed to follow the mirror displacement without any lag. In contrast, a finite thermal relaxation rate introduces retardation effects into the mechanical equation of motion through temperature dependent terms. Using a combined harmonic balance and averaging technique, slow envelope evolution equations are derived. In the limit of small mechanical vibrations, the micromechanical system can be described as a nonlinear Duffing-like oscillator. Coupling to the optical cavity is shown to introduce corrections to the linear dissipation, the nonlinear dissipation and the nonlinear elastic constants of the micromechanical mirror. The magnitude and the sign of these corrections depend on the exact position of the mirror and on the optical power incident on the cavity. In particular, the effective linear dissipation can become negative, causing self-excited mechanical oscillations to occur as a result of either a subcritical or supercritical Hopf bifurcation. The full slow envelope evolution equations are used to derive the amplitudes and the corresponding oscillation frequencies of different limit cycles, and the bifurcation behavior is analyzed in detail. Finally, the theoretical results are compared to numerical simulations using realistic values of various physical parameters, showing a very good correspondence.     

Dynamic modeling and extended bifurcation analysis of flexible-link manipulator

Abstract

In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the occurrence of chaos, the system dynamical behavior becomes unpredictable which in turn brings about uncertainty and irregularity in the system motion. The importance of this investigation is pronounced in similar systems such as double pendulum and single-link flexible manipulator. What makes this study distinct from previous ones is the increase in the number of links as well as the changing the bifurcation parameters from system mechanical parameters to force and torque inputs. To this aim, the motion equations of the N-link robot, which are derived with the aid of the recursive Gibbs-Appell formulation and the assumed modes method, are used. In the end, the equations of motion are developed for a two-link flexible manipulator, and its nonlinear dynamical behavior is analyzed via numerical integration of discrete equations. The results are presented in the form of bifurcation diagrams (for variation of torque amplitude), time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms. The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates. In addition, as the excitation frequency decreases from 2π to π, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinates is different, while this behavior has more similarity after this point. An experimental setup is also used to check the torques as the system input.     

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.