Geometric non-linear substructuring for dynamics of flexible mechanical systems

A substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non-linear effects. Large relative displacement is permitted between points within bodies that undergo small strain elastic deformation. Components are divided into substructures, on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion. Normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure. Compatibility conditions are derived and imposed as constraint equations at boundary points between substructures. System equations of motion that include geometric non-linear effects of large rotation, in terms of generalized co-ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure, are assembled. The method is implemented in an industry standard flexible multibody dynamics code, with minimal modification. Use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation.     

A family of implicit partitioned time integration algorithms for parallel analysis of heterogeneous structural systems

An implicit time integration algorithm is presented for the solution of linear structural dynamics problems on parallel computers. The present algorithm is derived from the partitioned equations of motion for a structure which consist of the equilibrium equations of each substructure due to its deformation energy, the self-equilibrium condition for each substructure under rigid-body motions, the partition boundary displacement compatibility condition and Newton's 3rd law along the partition boundary forces. A novel feature of the present algorithm is a flexibility normalization along the partitioned boundary nodes by using a localized version of the method of Lagrange multipliers, thus making the algorithm insensitive to both material and kinematic heterogeneities among the partitioned substructures. Numerical performance of the present algorithm demonstrates both its simplicity and efficiency. Hence, we recommend it for production analysis of heterogeneous structures modeled. Received November 1998     

变截面梁多次碰撞响应的动态子结构方法

该文考虑多次碰撞过程和多次分离过程,考虑阻尼对变截面梁运动的影响,将动态子结构方法推广应用于变截面梁的多次碰撞响应问题,导出了模态坐标表示的变截面梁-弹簧碰撞系统动力学方程。采用Newmark隐式积分方法进行求解,研究了开启-关闭过程中簧片阀与升程限制器和阀座的多次碰撞现象,碰撞动力学响应,以及阀片厚度及中空部分尺寸对阀片开启和关闭动作的影响。数值收敛性结果和与三维动力有限元方法的对比分析结果表明,该动态子结构方法可合理、有效地分析变截面梁的多次碰撞响应问题,适用于研究时间跨度较长的变截面梁多次碰撞问题,以及阀门结构的优化设计计算。     

结构物理参数时域识别的子结构方法研究

研究了输入、输出信息皆不完备情况下的结构参数识别以及荷载反演问题。阐述了一种通用的子结构动力方程及其参数识别方程建立的基本原理和方法,并针对实际工程检测中子结构参数识别方程的输入特性,分别采用一种与之相适应的分解反演算法或统计平均算法。子结构技术与分解算法或统计平均算法的有效结合,为有限测点条件下的结构参数识别及荷载反演问题提供了一个较好的解决方案。大量的数值计算结果表明,本文提出的方法具有很好的参数识别精度及荷载反演效果。     

结构动力分析的静态子结构法

文中把静态子结构法与 Ritz 向量直接叠加法相结合,构成了结构动力分析的静态子结构方法,这一方法对于大型结构的动力分析问题是十分有效的。     

作大范围运动的矩形板动力分析

作高速大范围运动的弹性体,由于运动和变形的耦合将产生动力刚化现象,传统的动力学理论难以计及这种影响.本文在有限元方法中首次引入了单元耦合形函数(阵),以此将单元弹性位移表示成为单元结点位移的二阶小量形式.利用几何非线性的应变-位移关系式,在小变形假设条件下确定了单元耦合形函数.在此基础上,根据Kane方程.运用模态坐标压缩,并采用适当的线性化处理,得到了包含动力刚度项的线性动力学方程.针对矩形板编制了动力刚化有限元分析程序.仿真算例证明了理论和算法的正确性.     

基于子结构虚拟变形的损伤识别方法

针对大型土木结构损伤识别优化效率低的问题,提出了子结构虚拟变形方法。虚拟变形方法是一种结构快速重分析的方法,该方法利用单元的虚拟变形模拟结构的损伤,可以在不重新建立有限元模型的情况下,快速计算出结构参数改变后的结构响应。该文基于虚拟变形法的基本思想,对子结构的刚度矩阵进行分解和对损伤后结构运动方程进行整理,推导出利用子结构的虚拟变形刻画损伤的方法,扩展了虚拟变形方法的适用范围;并且给出了虚拟变形和结构响应的相关性计算公式,通过相关性分析提取主要的虚拟变形,减少参与计算的子结构虚拟变形的数目,提高计算效率;最后利用一个五十层框架的数值仿真验证方法的有效性。     

Applications of a reduction method for reanalysis to nonlinear dynamic analysis of framed structures

 This paper is concerned with the nonlinear dynamic analysis of framed structures using a reduction method recently proposed by the authors. The reduction method is originally devised for structural static reanalysis and has been applied in optimal design of structures to speed up the design process. For nonlinear dynamic analysis of framed structures, the incremental or iterative equations of motion can be transformed into an algebraic system of equations if appropriate integration methods such as Newmark's method are used to integrate the equations of motion. The resulting algebraic system, referred to as the effective system in this paper, changes during the simulation for a nonlinear dynamic problem. Therefore, from the point of view of solving systems of equations, a nonlinear dynamic problem is very similar to an optimal design problem in that the system of equations changes for both types of problems. Hence, any reanalysis technique can be readily applied to carry out a nonlinear dynamic analysis of structures. As demonstrated from the presented numerical examples, the response obtained by the adopted reduction method is as accurate as that obtained by the Cholesky method, and as estimated from the operation counts involved in the method, it is more efficient than the Cholesky method when the half-band width is greater than about 50. Received 23 March 2000     

Optimization of large structures subjected to dynamic loads with the multiplier method

The multiplier method for optimization of large-scale mechanical and structural systems subjected to dynamic loads is investigated. A large-scale dynamic response optimization problem is formulated and solved in several alternate ways using the first- and second-order forms of the equations of motion. Results are compared with those obtained with the sequential quadratic programming algorithm—a primal method. In all the cases investigated, the multiplier algorithm is more efficient than the primal method. Therefore, it is concluded that the multiplier method is more appropriate for dynamic response optimization of large-scale problems.     

A CFD/CSD Model for Transonic Flutter

In this paper, a rapid deforming technique is developed to generate dynamic, three-dimensional, multi-block, mesh. The second-order Runge-Kutta time-marching method is used to solve the structural equations of motion. A dual-time method and finite volume discretization are applied for the unsteady Euler/Navier-Stokes equations to calculate the aerodynamic forces, in which the physical time step is synchronous with the structural equations of motion. The Spalart-Allmaras turbulence model is adopted for a turbulent flow. Due to mass dissimilarity, exiting in flutter calculations for a compressible flow, methods of variable mass and variable stiffness are developed to calculate the dynamic pressure of flutter at the point of mass similarity, and the flutter characteristics are then obtained in accordance with similarity rule. For completeness, the calculated transonic flutter characteristic results are presented and discussed for a double-wing and an aircraft model.     

半柔性悬挂结构分析的子结构模态法

针对一种半柔性悬挂减振结构提出其简化计算方法。根据这种结构的特点,将柔性悬挂层作为连接子结构,应用子结构模态综合法分析结构动力特性,得到结构的简化运动方程;分析了柔性层与悬挂楼面层间刚度比和综合模态数对结构简化计算精度的影响;提出了确定参与综合模态数的选取方法;基于模态综合法进行了算例的时程分析对比。结果表明:采用合理的子结构模态法减少了参与结构分析的自由度数目;综合模态数对结构计算精度的影响更显著。子结构模态法具有应用方便、计算精度高的特点,完全可以满足工程分析的需要。     

An unconditionally stable explicit algorithm for structural dynamics

This paper presents an unconditionally stable explicit algorithm for the direct integration of the structural dynamic equations of motion. The algorithm is restricted to a diagonal mass matrix and positive definite symmetric stiffness and damping matrices. The algorithm is based on splitting the stiffness and damping matrices into strictly lower and upper trangular form. Unconditional stability is proven, but only for the undamped case and a completely symmetric splitting of the stiffness matrix. An alternate splitting method is also presented and numerical examples indicate superior performance over the symmetric splitting, but only a conditional stability. A spring-mass-dashpot model is used to illustrate the algorithm.     

Dynamic investigation of laminated composite beams with shear and rotary inertia effect subjected to the moving oscillators using FEM

An algorithm based on the finite element method (FEM) has been developed to study the dynamic response of composite laminated beams subjected to the moving oscillator. The first order shear deformation theory (FSDT) is assumed for the beam model. The algorithm accounts for the complete dynamic interaction between the components of system. The proposed method can also be applied to the general moving mass and the simplified moving force problems. After deriving the governing equations of motion of beam and oscillator, the corresponding equations of motion are integrated by applying the Newmark’s time integration procedures to obtain the system responses in each time step. The numerical results of free vibration and moving force problems analysis of isotropic and composite laminated beams are presented and, whenever possible, compared to the available analytical solution and other numerical results in order to demonstrate the accuracy of the present method. In addition, parametric analysis is carried out over a wide range of velocities and mass, frequency and damping ratios of system components.     

Nonlinear dynamics of offshore structures under sea wave and earthquake forces

A study of dynamic response of offshore structures in random seas to inputs of earthquake ground motions is presented. Emphasis is placed on the evaluation of nonlinear hydrodynamic damping effects due to sea waves for the earthquake response. The structure is discretized using the finite element method. Sea waves are represented by Bretschneider’s power spectrum and the Morison equation defines the wave forcing function. Tajimi-Kanai’s power spectrum is used for the horizontal ground acceleration due to earthquakes. The governing equations of motion are obtained by the substructure method. Response analysis is carried out using the frequency-domain random-vibration approach. It is found that the hydrodynamic damping forces are higher in random seas than in still water and sea waves generally reduce the seismic response of offshore structures. Studies on the first passage probabilities of response indicate that small sea waves enhance the reliability of offshore structures against earthquakes forces.     

饱和两相介质近场波动问题的一种时域全显式数值计算方法

基于 u -p形式的饱和两相介质弹性波动方程,开展了饱和两相介质近场波动问题时域显式数值计算方法的研究。通过对波动方程中的质量矩阵和孔隙流体压缩矩阵进行对角化处理,消除了方程中的动力耦联,实现了波动方程的解耦。分别应用中心差分法和Newmark常平均加速度法求解固相位移和速度,基于向后差分法求解孔隙流体压力,推导得到了饱和两相介质动力响应的时域显式逐步积分的计算列式,建立了饱和两相介质近场波动问题的一种新的时域全显式数值计算方法。进行了该文方法中矩阵对角化合理性的验证。将该方法的数值解与相应的解析解进行对比,二者符合良好,验证了该方法的正确性。将该文建立的时域数值计算方法与透射人工边界方法相结合,应用于饱和两相介质的近场波动问题,进行了饱和土场地地震响应的计算研究,计算结果符合弹性波动理论的基本规律,表明该方法对于饱和两相介质近场波动问题时域计算求解的适用性。基于该方法中时域递推计算格式的传递矩阵,进行了该方法稳定性特性的研究。该文建立的数值计算方法具有时域全显式算法的基本特征。方法中对动力响应的全部分量均采用递推和迭代的模式进行求解,避免了求解耦联的动力方程组。该方法具有较高的计算效率,是进行饱和两相介质近场波动问题时域计算求解的一种有效的算法。     

A simple explicit–implicit finite element tearing and interconnecting transient analysis algorithm

A simple explicit–implicit finite element tearing and interconnecting (FETI) algorithm (AFETI‐EI algorithm) is presented for partitioned transient analysis of linear structural systems. The present algorithm employs two decompositions. First, the total system is partitioned via spatial or domain decomposition to obtain the governing equations of motions for each partitioned domain. Second, for each partitioned subsystem, the governing equations are modally decomposed into the rigid‐body and deformational equations. The resulting rigid‐body equations are integrated by an explicit integrator, for its stability is not affected by step‐size restriction on account of zero‐frequency contents (ω = 0). The modally decomposed partitioned deformation equations of motion are integrated by an unconditionally stable implicit integration algorithm. It is shown that the present AFETI‐EI algorithm exhibits unconditional stability and that the resulting interface problem possesses the same solution matrix profile as the basic FETI static problems. The present simple dynamic algorithm, as expected, falls short of the performance of the FETI‐DP but offers a similar performance of implicit two‐level FETI‐D algorithm with a much cheaper coarse solver; hence, its simplicity may offer relatively easy means for conducting parallel analysis of both static and dynamic problems by employing the same basic scalable FETI solver, especially for research‐mode numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.     

弹性DEM方法在杆系结构中的应用研究

基于颗粒离散元方法（DEM）,结合杆系结构的特点,提出了一种适于杆系结构问题分析的DEM模型。对颗粒元相应的质量和转动惯量计算公式进行了修正;通过能量等效原理推导了杆系DEM模型分析时弹簧接触刚度系数表达式;将瑞利阻尼引入到DEM方法之中,给出了阻尼常数计算公式并用算例进行了验证。将该方法应用于杆系结构弹性分析,包括静动力与几何非线性大变形问题的空间框架结构和网壳结构等多个算例,计算结果与有限元方法结果吻合良好。DEM方法的特点是将动力分析和几何非线性分析自动包含在运动方程的计算之中,不用组集刚度矩阵,无需迭代求解。杆系DEM模型非常适宜处理杆系结构大变形及动力非线性问题,尤其是在结构进入强非线性之后的模拟分析。     

Structural intensity study of plates under low-velocity impact

The transmissions of transient energy flow and dynamic transient response of plate structures under low-velocity impact are presented. The structural intensity approach is used to study the transient dynamic characteristics of plate structures under low-velocity impact. In the dynamic impact response analysis, nine-node degenerated shell elements with assumed shear and membrane strain fields are adopted to model the target and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process. The novel structural intensity streamline representation is introduced to interpret energy flow paths for transient dynamic response of plates under low-velocity impact. The effects of plates with and without structural damping on the energy flow and energy path are discussed. Numerical results, including contact force, deflection histories and transient energy flow vectors as well as structural intensity streamlines, show that the present method and representation are an efficient approach for exploring dynamic response for plate structures subjected to low-velocity impact.     

梁撞击弹塑性波传播的动态子结构方法的研究

该文针对简支梁横向弹塑性撞击问题,建立动态子结构模型,推导了相应的动力学控制方程,并采用Newmark隐式积分法进行求解,将动态子结构方法应用于撞击激发弹塑性波传播问题的研究。考虑局部弹塑性接触变形,通过对撞击激发的弹塑性波传播,包括弯矩波、挠曲波、速度波和应力波传播过程的计算,研究动态子结构法分析弹塑性波的传播特征,弯曲波的弥散特征,以及塑性铰形成机理等的合理性。经过与三维动力有限元计算结果的比较表明,动态子结构方法可以合理地应用于柔性梁中弹塑性撞击瞬态波传播问题的研究。     

An efficient semi-analytic time integration method with application to non-linear rotordynamic systems

This paper presents a simple and efficient time-integration method for non-symmetric and non-linear equations of motion occurring in the analysis of rotating machines. The algorithm is based on a semi-analytic formulation combining powerful methods of linear structural dynamics applied to non-linear dynamic problems. To that purpose, the total solution is separated into a linear and a non-linear part, and a further partitioning into quasi-static and dynamic parts is performed. Modal analysis is applied to the undamped equations of the dynamic parts. The quasi-static parts contain all degrees of freedom, while a cost-saving modal reduction may be easily performed for the dynamic parts. Duhamel's integral is utilized for the modal equations. The time-evolution of the unknown modal excitations due to the dissipative, non-conservative, gyroscopic and non-linear effects entering Duhamel's integral is approximated during each time-step. The resulting time-stepping procedure is performed in an implicit manner, and the method is examined in some detail, in view of stability and accuracy characteristics. A rotordynamic system serves as a benchmark problem in order to demonstrate the computational advantages of the present method with respect to various other time-integration algorithms.