Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.     

轴向移动局部浸液单向板的1:3内共振分析

考虑单向板的轴向速度、轴向张力、流固耦合作用以及阻尼等因素, 基于由 von Kármán薄板大挠度方程得到的轴向移动局部浸液单向板的非线性振动方程, 研究了外激励作用下单向板在1:3内共振情况时的非线性振动特性. 首先利用Galerkin法对非线性振动方程离散化, 然后分别应用数值法和近似解析法对离散后模态方程组进行求解, 获得了系统内共振情况下复杂的幅频特性曲线, 并讨论了周期解的稳定性. 最后研究了1:3内共振系统平均方程组的运动分岔现象.     

静载荷对夹层圆板超谐波共振的影响

研究静载荷作用下夹层圆板的超谐波共振问题．基于Hoff型夹层板理论,给出了静载荷作用下夹层圆板的非线性动力学方程．应用Galerkin法推导了静载荷作用下夹层圆板的轴对称非线性振动方程．运用多尺度法分别对系统的三次超谐波问题和二次超谐波问题进行了求解,并依据Lyapunov稳定性理论得到了系统稳态运动的稳定性判据．通过算例,得到了周边简支约束下夹层圆板三次超谐波共振和二次超谐波共振的幅频响应曲线图、振幅-静载荷响应曲线图、振幅-激励力幅值响应曲线图;研究了不同参数对系统振幅的影响规律,并对解的稳定性进行了分析．     

热环境中功能梯度圆形薄板的混沌运动

针对陶瓷-金属功能梯度圆板,同时考虑几何非线性、材料物性参数随温度变化且材料组分沿厚度方向按幂律分布的情况,应用虚功原理给出了热载荷与横向简谐载荷共同作用下的非线性振动偏微分方程。在固支无滑动的边界条件下,通过引入位移函数,利用伽辽金方法得到了达芬型非线性动力学方程。利用Melnikov方法,给出了热环境中功能梯度圆板可能发生混沌运动的临界条件。通过数值算例,给出了不同体积分数指数和温度的同宿分岔曲线,平面相图和庞加莱映射图,讨论其对临界条件的影响,证实了系统混沌运动的存在。通过分岔图和与其相对应的最大李雅普诺夫指数图,分析了激励频率和激励幅值对倍周期分岔的影响及变化规律,发现系统可出现周期、倍周期和混沌等复杂动力学响应。     

Multi-frequency analysis of the double circular plate system non-linear dynamics

This paper presents a multi-frequency analysis of non-linear dynamics in a double circular plate system. The original series of the amplitude–frequency and phase–frequency graphs as well as eigen forced time functions–frequency graphs are obtained and analyzed for stationary resonant states. The series of the frequency characteristic of the forced time non-linear harmonics are presented first. The analyses identify the presence of singularities and triggers of coupled singularities, as well as resonant jumps.     

Structural analogies on systems of deformable bodies coupled with non-linear layers

The paper is addressed at phenomenological mapping and mathematical analogies of oscillatory regimes in systems of coupled deformable bodies. Systems consist of coupled deformable bodies like plates, beams, belts or membranes that are connected through visco-elastic non-linear layer, modeled by continuously distributed elements of Kelvin–Voigt type with nonlinearity of third order. Using the mathematical analogies the similarities of structural models in systems of plates, beams, belts or membranes are obvious. The structural models consist by a set of two coupled non-homogenous partial non-linear differential equations. The problems to solve are divided into space and time domains by the classical Bernoulli–Fourier method. In the time domains the systems of coupled ordinary non-linear differential equations are completely analog for different systems of deformable bodies and are solved by using the Krilov–Bogolyubov–Mitropolskiy asymptotic method. This paper presents the beauty of mathematical analytical calculus which could be the same even for physically different systems.The mathematical numerical calculus is a powerful and useful tool for making the final conclusions between many input and output values. The conclusions about nonlinear phenomena in multi-body systems dynamics have been revealed from the particular example of double plate׳s system stationary and non-stationary oscillatory regimes.     

气动载荷下旋转运动圆板磁弹性主共振分析

基于Kirchhoff薄板理论与哈密顿原理,建立旋转运动导电圆板的磁-气动弹性非线性动力学方程．根据电磁场基本原理得到旋转运动圆板所受电磁力表达式,同时采用一种简化的气动模型以描述作用于板上的气动载荷．基于贝塞尔函数形式振型函数的选取,应用伽辽金法得到旋转圆板的磁气动弹性轴对称非线性振动微分方程．应用多尺度法推导出主共振下系统的幅频响应方程,并依据Lyapunov方法得到系统稳态运动稳定性判据．通过算例,得到周边夹之约束下圆板主共振的幅频特性曲线图,以及振幅随磁感应强度和激励力幅值的变化曲线图;阐述了不同参数对系统共振幅值的影响规律,并对解的稳定性进行了分析．     

旋转运动导电圆板的磁弹性参数振动分析

针对磁场中旋转运动圆板,在动能、应变能表达式基础上,根据哈密顿原理导出圆板的磁弹性振动方程．应用伽辽金积分法,得到横向磁场中旋转变速运动圆板的轴对称参数振动微分方程．通过坐标变换得到包含两个变系数项的马蒂厄振动方程．应用弗洛凯理论和平均法对系统的参数振动问题进行求解．通过数值计算得到周期稳定图、对应的振动响应特性图和相轨迹图．结果表明：在稳定区域内,系统的幅频曲线呈现为周期或概周期变化形式;在不稳定区内,系统的幅频响应曲线呈现为发散变化形式．     

计算圆板大振幅非线性振动频率的平均刚度法

本文用平均刚度法研究圆板大振幅非线性振动的频率问题，导出了相应的非线性广义特征值方程，构造了一种避免发散并能加速收敛的加权平均迭代法，计算结果与Ｋａｎｔｏｒｏｖｉｃｈ时间平均法的解十分吻合。     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

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.     

Non-linear vibrations of transversely isotropic rectangular'' plates

The large amplitude free flexural vibration of transversely isotropic rectangular plate, incorporating the effects of transverse shear and rotatory inertia, is studied using the von Karman field equations. A mode shape, consisting of three generalised-coordinates together with the Galerkin technique, results in a system of three non-linear simultaneous ordinary differential equations which govern the motion of the plate. These equations are integrated using a fourth-order Runge-Kutta method to obtain the period for each amplitude of vibration. The non-linear period vs amplitude behaviour is of the hardening type and it is also found that transverse shear and rotary inertia effects increase the period and that this increase is quite significant even for thin transversely isotropic plates. The results are compared with earlier results which were based on a one-term or one generalised coordinate solution and using the Berger approximation or the von Karman field equations.     

Finite amplitude vibrations of an orthotropic circular plate with an isotropic core

The free and forced non-linear vibrations of a fixed orthotropic circular plate, with a concentric core of isotropic material, are studied. Existence of harmonic vibrations is assumed and thus the time variable is eliminated by a Ritz-Kantorovich method. Hence, the governing non-linear partial equations for the axisymmetric vibration of the composite circular plate are reduced to a set of ordinary differential equations which form a non-linear eigen-value problem. Solutions are obtained by utilizing the related initial-value problems in conjunction with Newton's integration method. The results reveal the effects of finite amplitude and anisotropy of materials upon the dynamic responses. Further, the method developed in this paper, which is used to solve the title problem, is one of some generality. It can be applied to many differential eigenvalue problems with piecewise continuous functions.     

The bifurcation analysis on the circular functionally graded plate with combination resonances

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.     

内共振条件下轴向运动梁磁弹性超谐共振研究

以载流导线激发的磁场中轴向运动梁为研究对象,同时考虑外激励力作用,推导出梁的磁弹性非线性振动方程．通过位移函数的设定和伽辽金积分法,将非线性振动方程离散为常微分方程组．采用多尺度法得到系统的近似解析解．应用Matlab 和Mathematica 软件求解幅频响应方程,并对稳态解进行稳定性判定．通过具体算例得到前两阶假设模态的响应幅值随不同参数的变化规律．结果发现：系统在内共振条件下发生超谐波共振时,二阶假设模态幅值明显小于一阶;随着外激励的增大,多值解区域范围明显缩小;随着电流强度增加,振动幅值减小,表明载流导线能够起到控制共振的作用．     

Saddle-Node Bifurcations of Cycles in a Relief Valve

We extend the asymptotic perturbation (AP) method to the studyof a linear partial differential equation with nonlinear boundaryconditions. A relief valve under the combined effects of static anddynamic loadings is considered with the following resonances between thenth linear mode and the external periodic excitation: primaryresonance, subharmonic resonance of order one-half or one-third,superharmonic resonance of order-two or order-three and combinationresonance. The AP method uses two different procedures for thesolutions: introducing an asymptotic temporal rescaling and balancing ofthe harmonic terms with a simple iteration. We obtain amplitude andphase modulation equations and determine external force-response andfrequency-response curves. Stability of steady-state motions is alsoinvestigated. Saddle-node bifurcations of cycles are observed and underappropriate conditions the performance of the relief valve may beunsatisfactory due to the presence of jumps and hysteresis effects inthe system response. Global analysis is used in order to exclude theexistence of modulated motion. The validity of the method is highlightedby comparing approximate solutions with results of the numericalintegration.     

The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity

Theoretical analysis of the nonlinear vibration effects on the sound absorption of a panel absorber and sound transmission loss of a panel backed by a rectangular cavity is herein presented. The harmonic balance method is employed to derive a structural acoustic formulation from two-coupled partial differential equations representing the nonlinear structural forced vibration and induced acoustic pressure; one is the well-known von Karman??s plate equation and the other is the homogeneous wave equation. This method has been used in a previous study of nonlinear structural vibration, in which its results agreed well with the elliptic solution. To date, very few classical solutions for this nonlinear structural-acoustic problem have been developed, although there are many for nonlinear plate or linear structural-acoustic problems. Thus, for verification purposes, an approach based on the numerical integration method is also developed to solve the nonlinear structural-acoustic problem. The solutions obtained with the two methods agree well with each other. In the parametric study, the panel displacement amplitude converges with increases in the number of harmonic terms and acoustic and structural modes. The effects of excitation level, cavity depth, boundary condition, and damping factor are also examined. The main findings include the following: (1)?the well-known ??jump phenomenon?? in nonlinear vibration is seen in the sound absorption and transmission loss curves; (2)?the absorption peak and transmission loss dip due to the nonlinear resonance are significantly wider than those in the linear case because of the wider resonant bandwidth; and (3)?nonlinear vibration has the positive effect of widening the absorption bandwidth, but it also degrades the transmission loss at the resonant frequency.     

Nonlinear dynamic response of a stiffened plate with four edges clamped under primary resonance excitation

An approach is presented to study the nonlinear forced vibration of a stiffened plate. The stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as geometrically nonlinear Euler beams. Assuming the displacement of the stiffened plate, Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. A stiffened plate with four clamped edges subjected to harmonic excitation is studied by means of the method of multiple scales; the first approximation solutions of the double-modal motion of the system are obtained. Numerical examples for different stiffened plates are presented to discuss the steady response of the primary resonance and the amplitude?Cfrequency relationship; and some nonlinear forced vibration characteristics of the stiffened plate are obtained, which are useful for engineering design.     

Chaotic vibration and resonance phenomena in a parametrically excited string-beam coupled system

The nonlinear behavior of a string-beam coupled system subjected to parametric excitation is investigated in this paper. Using the method of multiple scales, a set of first order nonlinear differential equations are obtained. A numerical simulation is carried out to verify analytic predictions and to study the steady-state response, stable solutions and chaotic motions. The numerical results show that the system behavior includes multiple solutions, and jump phenomenon in the resonant frequency response curves. It is also shown that chaotic motions occur and the system parameters have different effects on the nonlinear response of the string-beam coupled system. Results are compared to previously published work.