Nonlinear 3D vibrations of a heavy material point on a spring at resonance

We consider the problem of nonlinear 3D vibrations of a heavy material point suspended on a weightless spring at a 1:1:2 frequency resonance. To construct an asymptotic solution, we use the Hamiltonian normal form method. Just as in the plane problem, this asymptotic solution describes the periodic process in which the vertical vibration energy passes into the horizontal vibration energy. For an arbitrarily small nonzero angular momentum with respect to the vertical axis, an effect typical of 3D systems manifests itself. The projection of the trajectory of the point onto the horizontal plane (xy) is an ellipse of constant area with axes varying in time. For certain initial conditions, the ellipse almost degenerates into straight-line segments. The direction of the straight line does not vary on the time interval where the vibration energy is in the horizontal mode and then varies almost by a jump on the interval where the vibration energy is transferred into the vertical mode. The analytic results are in good agreement with numerical solutions of equations of motion of the system.     

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electro- magnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period- doubling motion and quasi-period motion are discussed.     

雨刮器非线性摩擦振动的稳定性与分岔特性

针对雨刮器建立2自由度非线性摩擦振动动力学模型,基于复模态理论计算复特征值并进行稳定性及其对刮刷速度的依赖性分析;通过数值计算分析摩擦振动对刮刷速度的分岔特性,并利用相轨迹、庞加莱映射、频谱特性分析不同刮刷速度下的非线性振动现象.研究发现:摩擦-速度特性的负斜率是导致系统不稳定的根本原因,增大刮刷速度有利于提高系统的稳定性;在高、低刮速区,随着刮刷速度的下降,系统振动形态遵循周期→准周期→混沌的演化规律,并会伴随显著的粘滑振动;仅高速区的周期振动和非振动条件下,刮刷时无附加的粘滑振动.     

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 .     

Application of reconstitution multiple scale asymptotics for a two-to-one internal resonance in Magnetic Resonance Force Microscopy

In this paper we formulate an initial-boundary-value-problem describing the three-dimensional motion of a cantilever in a Magnetic Resonance Force Microscopy setup. The equations of motion are then reduced to a modal dynamical system using a Galerkin ansatz and the respective nonlinear forces are expanded to cubic order. The direct application of the asymptotic multiple scales method to the truncated quadratic modal system near a 2:1 internal resonance revealed conditions for periodic and quasiperiodic energy transfer between the transverse in-plane and out-of-plane modes of the MRFM cantilever. However, several discrepancies are found when comparing the asymptotic results to numerical simulations of the full nonlinear system. Therefore, we employ the reconstitution multiple scales method to a modal system incorporating both quadratic and cubic terms and derive an internal resonance bifurcation structure that includes multiple coexisting in-plane and out-of-plane solutions. This structure is verified and reveals a strong dependency on initial conditions in which orbital instabilities and complex out-of-plane non-stationary motions are found. The latter are investigated via numerical integration of the corresponding slowly-varying evolution equations which reveal that breakdown of quasiperiodic tori is associated with symmetry-breaking and emergence of irregular solutions with a dense spectral content.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

多体系统指标2运动方程HHT方法违约校正1）

采用Cartesian绝对坐标建模方法,完整约束多体系统运动方程是指标3的微分--代数方程（differentialalgebraic equations,DAEs）,数值求解指标3的DAEs属于高指标问题,通过对位置约束方程求导,可使运动方程的指标降为2.位置约束方程求导得到的是速度约束方程.直接求解指标3的运动方程,速度约束方程得不到满足,而且高指标DAEs的数值求解存在一些问题.论文首先采用HHT（Hilber--Hughes--Taylor）直接积分方法求解降指标得到的指标2运动方程,此时速度约束方程参与离散计算,从机器精度上讲速度约束自然得到满足,而位置约束方程没有参与计算,存在“违约”.针对违约问题,采用基于Moore--Penrose广义逆理论的违约校正方法,消除位置约束方程的违约.指标2运动方程HHT方法违约校正,将HHT方法和违约校正方法很好地结合,在数值求解指标2运动方程的过程中,位置约束方程和速度约束方程都不存在违约问题,而且新方法没有引入新的未知数向量,离散得到的非线性方程组的方程数量与原指标2运动方程的方程数量相同,求解规模没有扩大.新方法的实用和有效性通过算例的数值实验得到验证,数值实验也说明新方法保持了HHT方法本身具有的数值阻尼可以控制和二阶精度的特性.最后从非线性方程组的求解规模和计算速度上与其他方法进行了比较分析,说明新方法的优势所在.     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Nonlinear vibration analysis of viscoelastic cable with small sag

Both the inplane and out-of-plane transverse vibrations of a viscoelastic cable subjected to an initial stress distributing uniform on the cross section are studied. The constitution of the cable material is assumed to be of the hereditary integral type. The partial differential-integral equations of motion are derived first. Then by applying Galerkin's method, the governing equations are reduced to a set of second-order nonlinear differential-integral equations which are solved by finite difference numerical integration procedures. Finally, the effects of the viscosity parameter and the elastic parameter on the transient amplitudes of the first mode are investigated by numerical simulation. Project supported by the National Natural Science Foundation of China (No. 59635140) and the National Postdoctoral Foundation of China.     

Piecewise constrained optimization harmonic balance method for predicting the limit cycle oscillations of an airfoil with various nonlinear structures

A method is proposed to calculate the periodic solutions of piecewise nonlinear systems. The method is based on analytical derivation of nonlinear multi-harmonic equations of motion. Since periodic variations of nonlinear forces are characterized by different states, the vibration cycle is broken into sequential transition intervals according to the instant sets of state transitions. Analytical formulations of the harmonic coefficients of the nonlinear forces and its derivatives with respect to the harmonic coefficients of displacements are developed. Sensitivities of the harmonic coefficients of periodic solutions are determined for constructing explicit expressions for vibration amplitude levels as a function of structural parameters. Numerical investigations of the limit cycle oscillations and its sensitivities of an airfoil with different piecewise nonlinearities have been performed. The results show that the developed method is capable of determining the periodic solutions and its sensitivities with respect to the structural parameters. In order to guarantee time continuity of the nonlinear force, for the hysteresis model it is not right to track the periodic solutions by using the preload or freeplay as the continuation parameters.     

非线性振动系统周期运动及其稳定性的数值研究

§1引言确定型非线性振动系统的运动可分类如下: 1.非定常运动;2.定常运动:(1)周期运动,(2)各态历经运动,(3)浑沌运动。其中非定常运动是一暂态过程,会随着时间的增长逐步衰减乃至实际上消失。定常运动中的各态历经运动,指系统至少有两个互不通约(即其比值为无理数)的振动频率,因此运动虽然局      

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

Numerical analysis of nonlinear modes of piecewise linear systems torsional vibrations

The nonlinear modes of essentially nonlinear piecewise-linear finite degrees of freedom systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the systems motions in configuration space. The numerical method for the nonlinear modes of essentially nonlinear piecewise-linear systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous system nonlinear modes. The nonlinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonlinear piecewise-linear system with fifteen degrees of freedom. The NNMs of this system forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the systems with dissipation are close to NNMs of the system without dissipation.     

多体动力学超定运动方程广义-α求解新算法

完整约束多体系统第一类Lagrange方程建模得到的运动方程是指标-3形式的微分-代数方程(differental-algebraic equations,DAEs).如果同时考虑速度约束,将得到超定运动方程,该方程是指标-2的超定微分-代数方程(over-determined differential-algebraic equations,ODAEs).基于结构动力学中常用的广义-α方法,将其拓展,求解包含速度约束的超定运动方程,相对于其他求解指标-2 ODAEs的算法,新的算法没有增加离散得到的非线性方程组方程的数目.通过数值实验验证算法,并说明其求解ODAEs不存在精度降阶的现象,仍然具有二阶精度,同时算法的数值耗散也是可以控制的.最后新方法与其他求解多体系统ODAEs形式运动方程算法的CPU时间进行了比较分析.     

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.     

NONLINEAR DYNAMICS OF AXIALLY ACCELERATING VISCOELASTIC BEAMS BASED ON DIFFERENTIAL QUADRATURE

This paper investigates nonlinear dynamical behaviors in transverse motion of an axially accelerating viscoelastic beam via the differential quadrature method. The governing equation, a nonlinear partial-differential equation, is derived from the viscoelastic constitution relation using the material derivative. The differential quadrature scheme is developed to solve numerically the governing equation. Based on the numerical solutions, the nonlinear dynamical behaviors are identified by use of the Poincare map and the phase portrait. The bifurcation diagrams are presented in the case that the mean axial speed and the amplitude of the speed fluctuation are respectively varied while other parameters are fixed. The Lyapunov exponent and the initial value sensitivity of the different points of the beam, calculated from the time series based on the numerical solutions, are used to indicate periodic motions or chaotic motions occurring in the transverse motion of the axially accelerating viscoelastic beam.     

Nonlinear vibrations of structures induced by dry friction

The chattering of machine tools, the squealing noise generated by tram wheels in narrow curves and the noise of band saws are examples of physical processes in which elastic structures exhibit self-sustained stick-slip vibrations. The nonlinear contact forces are often due to dry friction. Periodic, multiperiodic, and chaotic motions can occur, depending on the parameters.Because the governing equations of motion are non-integrable, solutions can only be determined by numerical integration methods. The numerical investigations of continuous structures requires themodal approach to reduce the number of degrees of freedom.As an example, a beam system has been investigated numerically and experimentally in this paper. The nonlinear motion of a point of the continuous structure has been measured by a specially developedlaser vibrometer.The friction characteristic has been measured directly and identified from a measured time series by means of amodal state observer. The correlation dimension, which represents a lower bound of thefractal dimension, has been calculated using thecorrelation integral method from a measured time series of the beam system.     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.