简谐载荷下嵌入式单壁碳纳米管动态响应的Magnus方法

基于两端固支的弹性梁模型,研究嵌入式单壁碳纳米管在横向简谐载荷作用下的非线性振动问题。利用Galerkin方法对运动微分方程进行近似处理,将原方程从非线性动力学系统转化到二阶动力学系统,对于二阶动力学方程采用Magnus级数方法进行求解。通过数值实验,分析了嵌入式单壁碳纳米管非线性振动幅频特性,根据非线性动力学理论分析了碳纳米管动态响应,结果表明倍周期分岔产生混沌。     

大转动曲边柱壳非线性3-D动力学建模及分析

对曲边柱壳受轴向非均匀内压作用下的大转动几何非线性3-D动力学行为进行了研究.基于Nayfeh and Pai[1]非线性壳体理论,给出了考虑几何非线性的3-D混合型（含内力与位移）动力学模型.为了克服该强非线性模型难以求解的问题,依据分析获得的结构静动态变形关系,采用Lagrange方程推导建立了基于结构静态解的曲边柱壳多自由度3-D动力学方程,并对其进行了线性化与降阶处理,结合差分法获得了一套高效的求解算法.与LS-DYNA有限元结果的吻合,验证了本文方法的正确性.最后分析了单元数和计算时间步分别对有限元模型和本文方法的影响,发现求解精度随着计算时间步的减小不断提高直至趋于稳定.同时对采用本文方法获得的曲边柱壳动态变形模式的分析表明:结构动态响应与其所受内压载荷沿轴向的分布形式关系紧密,可以通过改变或者设计内压轴向分布形式来影响以及控制结构的动态变形模式,从而应用于曲边柱壳结构设计及优化的工程实际中.     

简支梁大变形的优化算法

提出一种求解几何非线性问题的优化算法,并研究了简支梁的几何非线性大变形问题。首先取简支梁大变形后的平衡状态为研究对象,分别创建它的变分模型和微分模型;然后基于微分模型,通过动坐标的迭代关系式求得微段端点坐标,构建微段端点未知坐标的目标函数;最后确定简支梁几何非线性大变形的最优化问题,并编制相应优化程序进行求解。通过分析典型算例,并同有限元方法的计算结果相比较,表明提出的优化算法在求解强几何非线性大变形问题中的正确性,为处理几何非线性大变形问题提供了一种新方法。     

耦合变形对大范围运动柔性梁动力学建模的影响

柔性梁在作大范围空间运动时,产生弯曲和扭转变形,这些变形的相互耦合形成了梁在纵向以及横向位移的二次耦合变量。本文考虑了变形产生的几何非线性效应对运动柔性梁的影响,在其三个方向的变形中均考虑了二次耦合变量,利用弹性旋转矩阵建立了准确的几何非线性变形方程,通过Lagrange方程导出系统的动力学方程。仿真结果表明,在大范围运动情况下,仅在纵向变形中计及了变形二次耦合量的一次动力学模型,与考虑了完全几何非线性变形的模型具有一定的差异。     

求解具有弹性接头桩基动力学大变形的微分—代数方法

本文采用弧坐标首先建立了求解具有弹性接头的桩基大变形分析的非线性动力学微分方程,其中, 广义Winkler模型用来模拟土对桩基的抗力.其次,在空间域内应用微分求积单元法来离散非线性微分方程组,并给出了处理弹性接头处连接条件的微分求积单元公式,得到了时间域内的一组微分-代数方程,采用二阶向后差分来代替二阶时间导数离散微分-代数方程组,得到一组离散化的非线性代数方程,应用Newton-Raphson方法求解了该非线性代数方程组.最后给出了数值算例,得到了桩基在顶部处受到组合动载荷作用时的响应,考察了弹性接头的刚度、位置对桩基动力学行为的影响.     

几何非线性摩擦阻尼隔振系统动力学行为研究

非线性隔振系统由于具有较线性系统更优的隔振性能,因此在工程中应用广泛.本文通过配置与被隔振对象的运动方向相垂直的库伦摩擦阻尼器,构建了几何非线性摩擦阻尼模型.由于引入了几何非线性,因此其摩擦力与位移正相关,这与传统与位移无关摩擦力模型有显著不同.首先,建立了具有几何非线性摩擦阻尼的数学模型以及隔振系统的受迫振动方程;然后,使用谐波平衡法求解了动力学方程,并使用数值仿真方法验证了谐波平衡法求解的准确性;最后,研究了几何非线性摩擦阻尼隔振器的绝对位移传递率和相对位移传递率.研究结果表明,在库伦摩擦阻尼选择适当,非线性摩擦阻尼系统可以在保持高频振动衰减效果的前提下,显著降低系统共振峰,其性能优于传统的恒定摩擦阻尼隔振模型.同时,几何非线性摩擦阻尼系统能够避免传统摩擦阻尼系统中的“锁定”现象,从传递率角度来说,不利于共振峰控制;但从激励环境改变引发隔振系统失效的角度来看,几何非线性摩擦阻尼系统可以拓宽系统对激励幅值的适应范围,避免隔振系统失效.本文的研究结果对此类隔振系统的设计和摩擦阻尼参数的选择具有通用的指导意义.      

弹性连杆机构广义刚度可靠性分析的数值模拟法

首先对响应面法进行了改进并应用于弹性连杆机构刚度可靠性分析，以迭代的格式和选择最优插值点的响应面法确定弹性连杆机构刚度可靠性分析的极限状态函数，编制了相应的有限元程序。然后在考虑安全、失效状态模糊性的基础上建立了弹性连杆机构的广义可靠性分析模型；提出了以重要抽样法与描述性抽样法相结合来求解弹性连杆机构广义失效概率的方法，此方法可以大量减少抽样时间，并且可以大大提高抽样效率，从而加快结果的收敛速度。     

黏弹性环形板的临界载荷及动力稳定性

利用线性黏弹性力学的Boltzmann叠加原理,在考察位移单值性条件的基础上,给出黏弹性环形板非线性动力学分析的初边值问题。通过Galerkin方法和引进新的状态变量,将其化归为四维非线性非自治常微分方程组,从而得到黏弹性环形板的四种临界载荷,同时考察了几何缺陷对黏弹性薄板临界载荷的影响。根据Floquet理论,得出黏弹性形板在周期激励下的线性动力稳定性判据。综合使用非线性动力学中的数值分析方法,研究了参数对黏弹性环形板非线性动力稳定性的影响。     

几何非线性摩擦阻尼隔振系统动力学行为研究

非线性隔振系统由于具有较线性系统更优的隔振性能,因此在工程中应用广泛.本文通过配置与被隔振对象的运动方向相垂直的库伦摩擦阻尼器,构建了几何非线性摩擦阻尼模型.由于引入了几何非线性,因此其摩擦力与位移正相关,这与传统与位移无关摩擦力模型有显著不同.首先,建立了具有几何非线性摩擦阻尼的数学模型以及隔振系统的受迫振动方程;然后,使用谐波平衡法求解了动力学方程,并使用数值仿真方法验证了谐波平衡法求解的准确性;最后,研究了几何非线性摩擦阻尼隔振器的绝对位移传递率和相对位移传递率.研究结果表明,在库伦摩擦阻尼选择适当,非线性摩擦阻尼系统可以在保持高频振动衰减效果的前提下,显著降低系统共振峰,其性能优于传统的恒定摩擦阻尼隔振模型.同时,几何非线性摩擦阻尼系统能够避免传统摩擦阻尼系统中的"锁定"现象,从传递率角度来说,不利于共振峰控制;但从激励环境改变引发隔振系统失效的角度来看,几何非线性摩擦阻尼系统可以拓宽系统对激励幅值的适应范围,避免隔振系统失效.本文的研究结果对此类隔振系统的设计和摩擦阻尼参数的选择具有通用的指导意义.     

多跨不平衡轴系的非线性动力学建模

给出了多跨弹性转子系统非线性动力学建模的准则，建立了一个两跨四支撑的16DOF不平衡非线性弹性转子的动力学模型，从有限元法导出了转子系统各段横向刚度系数的解析表达式，求出转子系统各轴颈处的等效负载，模型中采用非稳态油膜力模型。本文工作为研究多跨转子系统的非线性动力特性提供了条件。     

An Alternative Method for Formulating Closed-Form Dynamics for Elastic Mechanical Systems Using Symbolic Process*

ABSTRACT

In this paper an alternative method of formulating the equations of motion and using a computer-aided symbolic method for modeling of elastic mechanical systems is presented. The formalism is derived from the principle of virtual work and cast in an expanded and closed form. Symbolic generation of nonlinear closed-form equations of motion and their linearization are discussed. Symbolic equations for two examples are generated using the above method. Numerical results and comparison between full nonlinear and linearized robot models are presented     

A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.     

Dynamics of Articulated Structures. Part I. Theory

ABSTRACT

A finite element based method is developed for geometrically nonlinear dynamic analysis of spatial articulated structures; i.e., structures in which kinematic connections permit large relative displacement between components that undergo small elastic deformation. Vibration and static correction modes are used to account for linear elastic deformation of components. Kinematic constraints between components are used to define boundary conditions for vibration analysis and loads for static correction mode analysis. Constraint equations between flexible bodies are derived in a systematic way and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A lumped mass finite element structural analysis formulation is used to generate deformation modes. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled effects of geometric nonlinearity and elastic deformation. Examples are presented and the effects of deformation mode selection on dynamic prediction are analyzed in Part II of the paper.     

刚柔耦合约束多体系统的动力分析

本文提出了描述多体系统的牵连坐标轴系统。该坐标轴系统由惯性参考系、牵连坐标系、物体坐标系及单元坐标系组成，从而实现了对刚体平动、刚体转动及弹性运动的分解，较好地消除了由于刚体大角度转动而产生的刚弹耦合非线性特性，以有限元方法为基础，应用拉格朗日方程建立了在该坐标系下的刚弹耦合约束多体系统的动力方程。该方程具有形式简洁、易于推导和编程等优点，本文并对方程的数值解法进行了研究，提出求解的广义坐标分块法，最后给出了一个典型多体系统的数值算例。     

Stability and bifurcation analysis of a nonlinear transversally loaded rotating shaft

The dynamic stability and self-excited posteritical whirling of rotating transversally loaded shaft made of a standard material with elastic and viscous nonlinearities are analyzed in this paper using the theory of bifurcations as a mathematical tool. Partial differential equations of motion are derived under assumption that von Karman's nonlinearity is absent but geometric curvature nonlinearity is included. Galerkin's first-mode discretization procedure is then applied and the equations of motion are transformed to two third-order nonlinear equations that are analyzed using the theory of bifurcation. Condition for nontrivial equilibrium stability is determined and a bifurcating periodic solution of the second-order approximation is derived. The effects of dimensionless stress relaxation time and cubic elastic and viscous nonlinearities as well as the role of the transverse load are studied in the exemplary numerical calculations. A strongly stabilizing influence of the relaxation time is found that may eliminate self-excited vibration at all. Transition from super- to subcritical bifurcation is observed as a result of interaction between system nonlinearities and the transverse load.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

Imperfection sensitivity in the nonlinear vibration of functionally graded plates

In this paper, the nonlinear partial differential equations of nonlinear vibration for an imperfect functionally graded plate (FGP) in a general state of arbitrary initial stresses are presented. The derived equations include the effects of initial stresses and initial imperfections size. The material properties of a functionally graded plate are graded continuously in the direction of thickness. The variation of the properties follows a simple power-law distribution in terms of the volume fractions of the constituents. Using these derived governing equations, the nonlinear vibration of initially stressed FGPs with geometric imperfection was studied. Present approach employed perturbation technique, Galerkin method and Runge–Kutta method. The perturbation technique was used to derive the nonlinear governing equations. The equations of motion of the imperfect FGPs was obtained using Galerkin method and then solved by Runge–Kutta method. Numerical solutions are presented for the performances of perfect and imperfect FGPs. The nonlinear vibration of a simply supported ceramic/metal FGP was solved. It is found that the initial stress, geometric imperfection and volume fraction index greatly affect the behaviors of nonlinear vibration.     

Three kinematic representations for modeling of highly flexible beams and their applications

Presented here are three kinematic representations of large rotations for accurate modeling of highly flexible beam-like structures undergoing arbitrarily large three-dimensional elastic deformation and/or rigid-body motion. Different methods of modeling torsional deformation result in different beam theories with different mathematical characteristics. Each of these three geometrically exact beam theories fully accounts for geometric nonlinearities and initial curvatures by using Jaumann strains, exact coordinate transformations, and orthogonal virtual rotations. The derivations are presented in detail, a finite element formulation is included, fully nonlinear governing equations and boundary conditions are presented, and the corresponding form for numerically exact analysis using multiple shooting methods is also derived. These theories are compared in terms of their appropriate application areas, possible singular problems, and easiness for use in modeling and analysis of multibody systems. Nonlinear finite element analysis of a rotating beam and nonlinear multiple shooting analysis of a torsional bar are performed to demonstrate the capability and accuracy of these beam theories.     

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.