Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance

The aim of the study described in this paper is to investigate the forced dynamics of an axially moving viscoelastic beam. The governing equation of motion is obtained via Newton's second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin–Voigt, a two-parameter rheological model, energy dissipation mechanism, in which material, not partial, time derivative is employed in the viscoelastic constitutive relation. The dimensionless partial differential equation of motion is discretized using Galerkin's scheme with hinged–hinged beam eigenfunctions as the basis functions. The resulting set of nonlinear ordinary differential equations is then solved using the pseudo-arclength continuation technique and a direct time integration. For the system with the axial speed in the sub-critical regime, the response of the system is examined when possessing an internal resonance and when not. By employing a direct time integration, it is shown how the bifurcation diagrams of the system are modified by the presence of the dissipation terms—i.e. by both the time-dependant and steady (due the simultaneous presence of the axial speed and the energy dissipation mechanism) energy dissipation terms. Moreover, the amplitude–frequency responses and bifurcation diagrams of Poincaré maps are presented for several values of the system parameters.     

Dynamic Behaviors of Axially Moving Viscoelastic Plate with Varying Thicknessn

Structural components of varying thickness draw increasing attention these days due to economy and light-weight considerations. In view of the absence of research in vibration analysis of viscoelastic plate with varying thickness, this study devotes to investigate the dynamic behaviors of axially moving viscoelastic plate with varying thickness. Based on the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of motion of the axially moving viscoelastic rectangular plate is derived, the plate constituted by Kelvin-Voigt model has linearly varying thickness in the y-direction. The dimensionless complex frequencies of axially moving viscoelastic plate with four edges simply supported are calculated by the differential quadrature method, curves of real parts and imaginary parts of the first three-order dimensionless complex frequencies versus dimensionless moving speed are obtained, the effects of the aspect ratio, thickness ratio, the dimensionless moving speed and delay time on the dynamic behaviors of the axially moving viscoelastic rectangular plate with varying thickness are analyzed. When other parameters keep constant, with the decrease of thickness ratio, the real parts of the first three-order natural frequencies decrease, and the critical divergence speeds of various modes decrease too, moreover, whether the delay time is large or small, the frequencies are all complex numbers.     

轴向运动矩形板的谐波共振与稳定性分析

针对轴向运动矩形薄板的非线性振动问题,在给出薄板运动的动能和应变能的基础上,应用哈密顿变分原理,推得几何非线性下轴向运动薄板的非线性振动方程。通过位移函数和应力函数的设定,并应用伽辽金积分法,得到四边简支边界约束条件下受横向激励载荷作用轴向运动薄板的达芬型振动方程。利用多尺度法对系统的非线性谐波共振问题进行求解,得到稳态运动下关于共振幅值的幅频响应方程。依据李雅普诺夫运动稳定性理论对定常解的稳定性进行分析,得到解的稳定性判别式。通过数值算例,得到不同横向载荷和轴向速度下共振幅值的变化规律曲线图以及对应的相图,讨论分岔点变化以及倍周期运动规律,分析横向激励载荷和轴向运动速度对系统非线性动力学行为的影响。     

Nonlinear transient response analysis for double walls with a porous material supported by nonlinear springs using FEM and MSKE method

In this paper, we newly propose a fast computation method for the nonlinear transient responses including coupling between nonlinear springs and sound proof structures having porous materials using FEM. In this method, we extend our numerical method named as Modal Strain and Kinetic Method (i.e. MSKE method proposed previously by Yamaguchi who is one of the authors) from linear damping analysis to nonlinear dynamic analysis. We assume that the restoring force of the spring has cubic nonlinearity and linear hysteresis damping. To calculate damping properties for soundproof structures including elastic body, viscoelastic body and porous body, displacement vectors as common unknown variable are solved under coupled condition. The damped sound fields in the porous materials are defined by complex effective density and complex bulk modulus. The discrete equations in physical coordinate for this system are transformed into nonlinear ordinary coupled differential equations using normal coordinates corresponding to linear natural modes. Further, using MSKE method, modal damping can be derived approximately under coupled conditions between hysteresis damping of viscoelastic materials, damping of the springs and damping due to flow resistance in porous materials. The modal damping is used for the nonlinear differential equation to compute nonlinear transient responses.Moreover, using the proposed method, we demonstrate new vibration phenomena including nonlinear coupling between nonlinear springs and soundproof structures by use of a simplified model. As a typical numerical example of the soundproof structure, we adopt double walls with a porous material. The double walls are supported by nonlinear concentrated springs. We clarify influences of amplitude of the impact force on nonlinear transient responses. We focused on the vibration modes, which magnify the amplitudes of the double walls. In these modes, the internal air of the porous material played a role of a pneumatic spring. Under a very large impact force as a severe condition, there exist the complicated nonlinear couplings between these modes and the super harmonic components of the rigid modes of the whole structure with large deformations in the nonlinear springs.     

点弹性支承粘弹性杆的动力稳定性

利用Lagrange方程推导了在轴向周期压载作用下点弹性支承粘弹性杆的运动微分方程,并由动力稳定理论给出了杆的临界频率方程和动力不稳定边界。计算结果表明,对Kelvin模型粘弹性杆来说,粘性系数与点弹性支承处的弹簧刚度对杆的动力稳定性有显著影响,但当粘性系数小于某个量级时,可将该粘弹性杆的动力稳定性问题按弹性杆处理。     

Nonlinear vibration and instability of fluid-conveying DWBNNT embedded in a visco-Pasternak medium using modified couple stress theory

Nonlinear free vibration and instability of fluid-conveying double-walled boron nitride nanotubes (DWBNNTs) embedded in viscoelastic medium are studied in this paper. The effects of the transverse shear deformation and rotary inertia are considered by utilizing the Timoshenko beam theory. The size effect is applied by the modified couple stress theory and considering a material length scale parameter for beam model. The nonlinear effect is considered by the Von Kármán type geometric nonlinearity. The electromechanical coupling and charge equation are employed to consider the piezoelectric effect. The surrounding viscoelastic medium is described as the linear visco-Pasternak foundation model characterized by the spring and damper. Hamilton’s principle is used to derive the governing equations and boundary conditions. The differential quadrature method (DQM) is employed to discretize the nonlinear higher-order governing equations, which are then solved by a direct iterative method to obtain the nonlinear vibration frequency and critical fluid velocity of fluid-conveying DWBNNTs with clamped-clamped (C-C) boundary conditions. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, spring and damping constants of surrounding viscoelastic medium and fluid velocity on the nonlinear free vibration, instability and electric potential distribution of DWBNNTs. This study might be useful for the design and smart control of nano devices.     

Chaotic dynamic analysis of viscoelastic plates

The dynamic buckling of viscoelastic plates with large deflection is investigated in this paper by using chaotic and fractal theory. The material behavior is given in terms of the Boltzmann superposition principle. In order to obtain accurate computation results, the nonlinear integro-differential dynamic equation is changed into an autonomic four-dimensional dynamical system. The numerical time integrations of equations are performed by using the fourth-order Runge–Kutta method. And the Lyapunov exponent spectrum, the fractal dimension of strange attractors and the time evolution of deflection are obtained. The influence of geometry nonlinearity and viscoelastic parameter on the dynamic buckling of viscoelastic plates is discussed.     

Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well

In this paper a nonlinear model was developed for a drillstring system in deviated well with axially moving motion and axial loading, using the perturbation techniques.The drillstring was considered as a simply supported axially moving rotor. Governing equations of motion were obtained based on Hamilton's principle and method of multiple scales was employed to solve the nonlinear motion equations in order to obtain the steady state response and stability region of the system.Then the effects of rotating speed, axial compression load, imbalance mass and nonlinear fluid force on the drillstring responses were investigated in detail and nonlinear natural frequencies and their corresponding mode shapes were presented.Analytical and numerical results reveal the rich and interesting nonlinear phenomena such as primary and parametric resonance that is being investigated for the first time in this study on the nonlinear vibration of the drillstring system.Finally in effort to validate the theoretical approach employed in this study, the numerical solutions obtained here were compared with a set of existing experimental data.     

Nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells

Based on the nonlinear large deflection theory of cylindrical shells as well as the Donnell assumptions, this paper presents nonlinear buckling and postbuckling analyses for axially compressed functionally graded cylindrical shells by using the Ritz energy method and the nonlinear strain-displacement relations of large deformation. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Meanwhile, by taking into account the temperature-dependent material properties, various effects of external thermal environment are also investigated. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on nonlinear buckling and postbuckling behaviors. There is a mode-jumping observed after buckling. The present theoretical results are verified by those in the literature.     

Spectral element analysis for an axially moving viscoelastic beam

In this paper, a spectral element model is derived for the axially moving viscoelastic beams subject to axial tension. The viscoelastic material is represented in a general form by using the one-dimensional constitutive equation of hereditary integral type. The high accuracy of the present spectral element model is verified first by comparing the eigenvalues obtained by the present spectral element model with those obtained by using the conventional finite element model as well as with the exact analytical solutions. The effects of viscoelasticity and moving speed on the dynamics of moving beams are then numerically investigated.     

TRANSVERSE VIBRATION CHARACTERISTICS OF VISCOELASTIC RECTANGULAR PLATE WITH CRACK

Based on the two-dimensional viscoelastic differential constitutive relation and the thin plate theory, the differential equations of motion of the viscoelastic plate with an all-over part-through crack are established and the expression of additional rotation induced by the crack is derived. The complex eigenvalue equations of the viscoelastic plate with crack are derived by the differential quadrature method, and the 8method is used at the crack continuity conditions. Dimensionless complex frequencies of a crack viscoelastic plate with four edges simply supported, two opposite edges simply supported and other two edges clamped are calculated. The effects of the crack parameter, the aspect ratio and dimensionless delay time of the material on the transverse vibration of the viscoelastic plate are analyzed.     

转矩分流式齿轮传动系统的非线性动力学特性

为研究转矩分流式齿轮传动系统的非线性动力学特性,建立系统的非线性动力学模型,考虑了齿侧间隙、时变刚度、综合传动误差、阻尼和外激励等参数。使用PNF(Poincaré-Newton-Floquet)方法对系统的动力学微分方程进行求解发现,在不同的参数条件下系统会出现4种动态响应：简谐响应、次谐波响应、拟周期响应及混沌响应。4种状态下的时间历程、相图、Poincaré映射图、快速傅里叶变换(Fast Fourier transform,FFT)频谱图表明该系统有很强的非线性特性,应使用非线性理论进行更深入的研究。     

弧面分度凸轮机构传动系统非线性动力学特性研究

对弧面分度凸轮机构传动系统,用集中参数法建立了较为精确的5自由度等效动力学模型,在该模型中,考虑了电动机动力特性、传动副间隙、阻尼、油膜挤压、构件弹性和负载特性的影响,并推导出了该传动系统的5自由度非线性微分方程组,使用有限差分法,用FORTRAN77语言编制了该传动系统动态模拟计算机应用软件RGCAMD（Roller gear cam dynamic）,并进行了大量的动态模拟计算,详细地分析了系统参数对动态响应的影响,总结出了各参数对动态响应的影响规律,得到了一些工程上很有应用价值的结论。     

轴向运动耦合热弹梁的稳定性分析

研究了轴向运动梁的耦合热弹稳定性.根据轴向运动梁的运动微分方程和考虑变形影响时的热传导方程,得出了温度场和变形场耦合情况下梁的耦合热弹运动微分方程.对两端简支轴向运动梁耦合热弹振动的复频率进行了数值计算,得出了无量纲复频率与无量纲运动速度之间的关系曲线.分析了无量纲热弹耦合因子、无量纲运动速度和梁的长高比对梁的临界速度和稳定性的影响.     

Modeling nonlinear behavior in a piezoelectric actuator

A piezoelectric tube actuator is employed as a sample positioning device in Nanocut, a cutting instrument conceived to study the mechanics of nanometric cutting. Extension of functionality of the instrument as a nanometric machine tool motivates the search for an accurate model of the actuator for implementing feedback control. A simple nonlinear model describing longitudinal expansion of the piezoelectric tube actuator is presented in this paper. The model derivation is based on a non-formal analogy with nonlinear viscoelastic materials under uniaxial extension, for which the responses to a step input are similar to the piezoelectric tube. Suitability of the model structure for arbitrary inputs is tested by cross-verification between time and frequency domains. Two parameter estimation procedures are examined and the results of the experimental work for characterization, estimation and validation are presented.     

二次非线性圆板的1/2亚谐解

计及材料的非线性弹性和粘性性质，研究圆板在简谐载荷作用下的1／2亚谐解，导出相应的非线性动力方程。提出一类强非线性动力系统的叠加－叠代谐波平衡法。将描述动力系统的二阶常微分方程化为在本解为未知函数的基本微分方程和分岔解为未知函数的增量微分方程。通过叠加－叠代谐波平衡法得出圆板的1／2亚谐解。同时，对叠加－叠代谐波平衡法和数值积分法的精度进行比较。并且讨论了1／2亚谐解的渐近稳定性。     

弹性地基上输送振荡流粘弹性管道的动力稳定性

基于线粘弹性理论,建立了弹性地基上输送振荡流粘弹性管道的运动微分方程,采用Galerkin法和解初值问题的Runge-Kutta法对含有周期系数的偏微分方程进行了求解。根据Floquet理论,研究了材料的量纲一延滞时间、量纲一流速以及量纲一刚度比对输送振荡流Kelvin-Voigt粘弹性管道动力不稳定区域的影响,给出了在这些参数变化时,频率比与激励参数平面上管道的动力稳定性区域和不稳定区域。     

Axially constrained beams on elastic foundation

The problem of bending in axially constrained beams on a Winkler-type elastic foundation is considered. This nonlinear problem is approached by a differential equation method as well as a finite element method. Numerical results are presented which show that for a low modulus of the foundation the linear solution for deflection (which neglects axial constraint effect) yields deflection values which may be several times higher than the exact solution. The two methods are shown to be in good agreement.     

The dynamic contact area of elastomers at different velocities

The friction in tribo-systems that contain viscoelastic materials, such as elastomers, is relevant for a large number of applications. Examples include tyres, hoses, transmission and conveyor belts. To quantify the friction in these applications, one must first understand the contact behaviour of such viscoelastic materials, both in static and in dynamic situations. This work discusses an experimental study into the change of the contact area with the sliding velocity and relates the change in contact area with the mechanical properties of the elastomer. The results show that for a tribo-system containing an elastomer, there is a threshold velocity, above which the size of the contact area significantly reduces.     

磁场中轴向运动导电板磁弹性主共振分析

给出轴向运动薄板动能、应变能以及电磁力虚功的表达形式。应用哈密顿变分原理,推得横向磁场中轴向运动条形导电薄板的非线性磁弹性振动方程。针对对边简支边界约束条件,通过位移函数的设定并应用伽辽金积分法,得到三阶位移展开形式下轴向运动板的非线性振动微分方程组。利用多尺度法对系统的主共振问题进行求解,分别得到三种频率关系条件下关于稳态解的幅频响应方程。依据李雅普诺夫稳定性理论对解的稳定性进行分析,得到相应的稳定性判别式。通过数值算例,得到轴向速度、磁感应强度、激励力幅值及板厚不同时的振幅变化规律曲线图,分析不同参量对共振幅值和非线性特征的影响,并对不同频率关系进行比较。