Vibrations analysis of flexible multibody systems

This paper is the continuation of a previous work (Pascal, 1988a) in which we consider the dynamical analysis of flexible space vehicles modelled by a chain of rigid and elastic bodies with tree structure. The multibody system consists of n + 1 bodies (S$iiei:) (i = 0, 1,…n) interconnected by n hinges l_a (a = 1,…n). The only external forces and torques are exerted on the first body which is assumed to be rigid. On each individual flexible appendage (Si), the only external forces and torques are those introduced by the hinges. Assuming that the multibody system undergoes small vibrations around an equilibrium position, we define in the frequency domain the linear transformation giving the resultant forces and torques on the boundaries of each flexible appendage (Si) in terms of the displacements of these boundaries. The motion of each flexible appendage is represented by a set of component modes, which are the modes obtained when the appendage vibrates independently with respect to the other parts of the whole system. Two different sets of vibration modes are used to give an expansion of the transfer function between forces and torques exerted on the boundaries of the body (Si) and displacements of these boundaries     

Stability analysis of constrained multibody systems

Automated algorithms for the dynamic analysis and simulation of constrained multibody systems usually assume the rows of the constraint Jacobian matrix to be linearly independent. But during the motion, at instantaneous configurations, the Jacobian matrix may become less than full rank resulting in singularities. This occurs when the closed-loop goes from 3D to 2D type of configuration. In this paper the linearly dependent rows are identified by an uptriangular decomposition process. The corresponding constraint equations are modified so that the singularities in the numerical procedure are avoided. The conditions for the validity of the modified equations are described. Furthermore, the constraint equations expressed in accelerations are modified by Baumgarte's approach to stabilize the accumulation of the numerical errors during integration. A computational procedure based on Kane's equations is presented. Two and three-link robotic manipulators will be simulated at singular configurations to illustrate the use of the modified constraints.     

Sensitivity analysis of flexible multibody systems using composite materials components

In this paper, a general formulation for the computation of the first‐order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented. The direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables, which are the ply orientations of the laminated. In order to appraise the benefits of the approach suggested and to highlight the risks of the procedure, the analytical sensitivities are compared with the numerical results obtained by using the finite difference method. For the beam composite material elements, the section properties and their sensitivities are found using an asymptotic procedure that involves a two‐dimensional (2‐D) finite element analysis of their cross section. The equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems. The equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations, velocities and the sensitivities of accelerations and velocities are integrated in time using a multi‐step multi‐order integration algorithm. Through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure, with respect to finite difference approaches or to the direct implementation of the analytic sensitivities, are discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Domain decomposition approach to flexible multibody dynamics simulation

Finite element based formulations for flexible multibody systems are becoming increasingly popular and as the complexity of the configurations to be treated increases, so does the computational cost. It seems natural to investigate the applicability of parallel processing to this type of problems; domain decomposition techniques have been used extensively for this purpose. In this approach, the computational domain is divided into non-overlapping sub-domains, and the continuity of the displacement field across sub-domain boundaries is enforced via the Lagrange multiplier technique. In the finite element literature, this approach is presented as a mathematical algorithm that enables parallel processing. In this paper, the divided system is viewed as a flexible multibody system, and the sub-domains are connected by kinematic constraints. Consequently, all the techniques applicable to the enforcement of constraints in multibody systems become applicable to the present problem. In particular, it is shown that a combination of the localized Lagrange multiplier technique with the augmented Lagrange formulation leads to interesting solution strategies. The proposed algorithm is compared with the well-known FETI approach with regards to convergence and efficiency characteristics. The present algorithm is relatively simple and leads to improved convergence and efficiency characteristics. Finally, implementation on a parallel computer was conducted for the proposed approach.     

Equivalence of the driving elastic forces in flexible multibody systems

When the driving joint forces, determined using the inverse dynamics procedure, are applied in the feedforward control of mechanical systems, discrepancies between the specified and the actual motion are observed. In some recent publications, these discrepancies were attributed to the wave phenomenon. It is shown in this investigation that the solution of the inverse dynamics of flexible mechanical systems defines two types of driving forces which can be classified as driving joint forces and driving elastic forces. The driving joint forces which depend on the deformation of the flexible bodies define the torque and the actuator forces which must be applied at the joints. The driving elastic forces are associated with the deformation degrees of freedom, and therefore, there is no gaurantee that an algorithm that ignores these driving elastic forces will converge and achieve the desired solution. It is the objective of this investigation to examine the nature of the driving elastic forces in the solution of the inverse dynamics problem, and demonstrate that the driving elastic forces associated with two different sets of vibration modes which produce the same physical displacements are basically the same and they differ only by a co-ordinate transformation. The effect of the selection of the deformable body co-ordinate system on these forces is also examined numerically using a slider crank mechanism with a flexible connecting rod.     

A generalized recursive formulation for constrained flexible multibody dynamics

This research develops a relative co‐ordinate formulation for the multibody flexible dynamics. The velocity transformation method is notationally compact, because the Cartesian generalized velocities are simultaneously transformed to the relative generalized velocities in a matrix form. However, inherent computational efficiency in the recursive kinematics between two adjacent bodies has not been exploited. This research presents a recursive formulation which is both notationally compact and computationally efficient. The velocity transformation method is used to derive the equations of motion and their derivatives. Matrix operations associated with the velocity transformation matrix in the resulting equations of motion and their derivatives are classified into several categories. A joint library of the generalized recursive formulas is developed for each category. When one category is encountered in implementing the equations of motion and their derivatives, the corresponding recursive formulas in the category are invoked. When a new force or joint module is added to a general purpose programme in the relative co‐ordinate formulation, the modules for the rigid body are not reusable for the flexible body. Since the flexible body dynamics handles additional generalized co‐ordinates associated with deformation, implementation of the flexible dynamics is generally complicated and prone to coding mistakes. A virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. This makes a flexible body subjected to only the kinematic admissibility condition for the virtual flexible body joint. As a result, the only extra work to handle the flexible bodies is to add the virtual flexible body joint modules in all recursive formulas. Since computation time in a relative co‐ordinate formulation is approximately proportional to the number of relative co‐ordinates, computational overhead due to the additional virtual bodies and joints are minor. Meanwhile, implementation convenience is dramatically improved. Copyright © 2001 John Wiley & Sons, Ltd.     

Partitioned analysis of flexible multibody systems using filtered linear finite element deformational modes

This work presents a partitioned finite element formulation for flexible multibody systems, based on the floating frame (FF) approach, under the assumption of small deformations but arbitrarily large rotations of the bodies. In classical FF of reference methods, deformational modes are normally computed by modal analysis. In this approach, free‐floating modes are eliminated from the linear model using projection techniques and substituted by a complete set of non‐linear finite rotations. In this way, all deformational modes are retained, and no modal selection is needed. The main difference between this work and a classical FF of reference formulation is an algebraic separation of pure deformational modes from rigid‐body motions. The proposed methodology presents the following advantages. First, the position and orientation of the FF has no restriction and can be freely located in the body with identical results. Second, the formulation uses only the linear finite element matrices of a classical vibration problem; hence, they can be easily obtained from linear FEM packages. Third, no selection of modes is needed, all deformational modes are retained through the filtering process. And finally, thanks to the use of localized Lagrangian multipliers (LLM), a partitioned system is obtained that can be solved iteratively and in a distributed manner by available scalable solvers. Copyright © 2014 John Wiley & Sons, Ltd.     

A Lagrangian approach to the non-causal inverse dynamics of flexible multibody systems: The three-dimensional case

This paper addresses the problem of end-point trajectory tracking in flexible multibody systems through the use of inverse dynamics. A global Lagrangian approach is employed in formulating the system equations of motion, and an iterative procedure is proposed to achieve end-point trajectory tracking in three-dimensional, flexible multibody systems. Each iteration involves firstly, a recursive inverse kinematics procedure wherein elastic displacements are determined in terms of the rigid body co-ordinates and Lagrange multipliers, secondly, an explicit computation of the inverse dynamic joint actuation, and thirdly, a non-recursive forward dynamic analysis wherein generalized co-ordinates and Lagrange multipliers are determined in terms of the joint actuation and desired end-point co-ordinates. In contrast with the recursive methods previously proposed, this new method is the most general since it is suitable for both open-chain and closed-chain configurations of three-dimensional multibody systems. The algorithm yields stable, non-casual actuating joint torques and associated Lagrange multipliers that account for the constraint forces between flexible multibody components.     

Projection methods in flexible multibody dynamics. Part I: Kinematics

In this paper a recursive projection method for the dynamic analysis of open-loop mechanical systems that consist of a set of interconnected deformable bodies is presented. The configuration of each body in the system is identified using a coupled set of reference and elastic co-ordinates. The absolute velocities and accelerations of leaf or child bodies in the open-loop system are expressed in terms of the absolute velocities and accelerations of the parent bodies and the time derivatives of the relative co-ordinates of the joints between the bodies. The dynamic differential equations of motion are developed for each link using the generalized Newton-Euler equations. The relationship between the actual joint reactions and the generalized forces combined with the kinematic relationships and the generalized Newton-Euler equations are used to develop a system of loosely coupled equations which has a sparse matrix structure. Using matrix partitioning and recursive projection techniques based on optimal block factorization an efficient solution for the system accelerations and joint reaction forces is obtained. This solution technique yields a much smaller operations count and can more effectively exploit vectorization and parallel processing. It also allows a systematic procedure for decoupling the joint and elastic accelerations.     

A non-recursive Lagrangian solution of the non-causal inverse dynamics of flexible multibody systems: The planar case

A technique is presented for solving the inverse dynamics of flexible planar multibody systems. This technique yields the non-causal joint efforts (inverse dynamics) as well as the internal states (inverse kinematics) that produce a prescribed nominal trajectory of the end effector. A non-recursive Lagrangian approach is used in formulating the equations of motion as well as in solving the inverse dynamics equations. Contrary to the recursive method previously presented, the proposed method solves the inverse problem in a systematic and direct manner for both open-chain as well as closed-chain configurations. Numerical simulation shows that the proposed procedure provides an excellent tracking of the desired end effector trajectory.     

Dynamics of flexible multibody systems using loaded-interface substructure synthesis approach

A simple numerical method for dynamic simulation of multibody systems consisting of rigid and flexible bodies is presented. This paper investigates the multibody systems with inertia properties of flexible components that undergo large angular rotations. The equation of motion is derived using the finite element/Lagrange formulation. A substructure synthesis method is employed to reduce the number of elastic coordinates of the multibody system. A modification to the traditional boundary conditions at the free interface has been incorporated. An example is given to demonstrate the accuracy of the computed results which obtained from this new free interface method. This example has been analyzed using the present free interface method and also the finite element method in order to compare the efficient and accuracy of both methods. It was shown that the new free interface substructure synthesis method provides accurate results even with lesser elements.     

多柔体系统动力学部件两级降阶响应分析

利用传统的模态综合法进行多柔体系统动力学建模时,没有考虑到控制系统设计技术的承受能力和激励的特点,保留的大量低频子结构模态对于整体控制目标没有贡献,反而加大了组合的整体方程的复杂程度．因此,为考虑部件级的降阶,可以在传统的部件组合之前,对部件模态坐标独立变换处理,去掉系统中弱能控和弱能观部分,而剩下的模态仍可使后继分析有必要的精度,从而建立降阶的动力学方程．     

Subsystem Global Modal Parameterization for efficient simulation of flexible multibody systems

This paper presents a new model order reduction strategy for flexible multibody simulation, namely the Subsystem Global Modal Parameterization. The proposed method is based on a system‐level reduction technique, named Global Modal Parameterization, but offers significant improvements for systems with many independent DOFs. The approach splits up the motion of a mechanism or part of a mechanism into a relative motion, in which the members move relatively with respect to each other, and a global motion of the system, in which the relative position of the members does not change. The relative motion is described by a local Global Modal Parameterization model expressed in a mechanism‐attached frame, and the global motion is described by the motion of the mechanism‐attached frame. In order to improve simulation efficiency, mass invariants are used, which are also introduced in this paper. Two numerical examples are presented, which show the good accuracy and the major simulation efficiency improvements this new approach offers. Copyright © 2011 John Wiley & Sons, Ltd.     

Model order reduction based on successively local linearizations for flexible multibody dynamics

An efficient method of model order reduction is proposed for the dynamic computation of a flexible multibody system undergoing both large overall motions and large deformations. The system is initially modeled by using the nonlinear finite elements of absolute nodal coordinate formulation and then locally linearized at a series of quasi-static equilibrium configurations according to the given accuracy in dynamic computation. By using the Craig-Bampton method, the reduced model is established by projecting the incremental displacements of the locally linearized system onto a set of local modal bases at the quasi-static equilibrium configuration accordingly. Afterwards, the initial conditions for the dynamic computation for the reduced model via the generalized-α integrator can be determined from the modal bases. The analysis of computation complexity is also performed. Hence, the proposed method gives time-varying and dimension-varying modal bases to elaborate the efficient model reduction. Finally, three examples are presented to validate the accuracy and efficiency of the proposed method.     

基于柔性多体动力学的瓦楞成型系统建模与仿真研究

针对某型号瓦楞机的瓦楞成型系统,基于自主研发的多体动力学求解程序,建立其刚柔耦合动力学模型。其中张力辊、瓦楞辊等主要支撑辊简化为刚体模型;传送带由36自由度绝对节点坐标四边形壳单元划分网格,并考虑其树脂材料的正交各向异性特征;此外,传送带与支撑辊之间的接触采用赫兹碰撞模型和点-面检测方法描述。利用该模型,计算了传送带的偏心位移,传送带表层应力场等动响应。仿真表明:基于绝对节点坐标法建立的瓦楞成型系统的多体动力学模型,可为瓦楞机传送带的动力学行为和控制研究提供一种新的分析方法。     

基于柔性多体动力学理论的舰船机械设备隔振系统建模

将柔性多体动力学理论应用于非线性隔振系统建模过程中,建立了线性弹性元件及非线性弹性元件的力学模型,利用该力学模型和单柔性体的动力学微分方程推导出了对舰船机械设备隔振系统动力学研究具有普适性的一般理论模型。     

An accelerated iterative method for the dynamics of constrained multibody systems

An accelerated iterative method is suggested for the dynamic analysis of multibody systems consisting of interconnected rigid bodies. The Lagrange multipliers associated with the kinematic constraints are iteratively computed by the monotone reduction of the constraint error vector, and the resulting equations of motion are easily time-integrated by a well established ODE technique. The velocity and acceleration constraints as well as the position constraints are made to be satisfied at the joints at each time step. Exact solution is obtained without the time demanding procedures such as selection of the independent coordinates, decomposition of the constraint Jacobian matrix, and Newton Raphson iterations. An acceleration technique is employed for the faster convergence of the iterative scheme and the convergence analysis of the proposed iterative method is presented. Numerical solutions for the verification problems are presented to demonstrate the efficiency and accuracy of the suggested technique.     

Projection methods in flexible multibody dynamics. Part II: Dynamics and recursive projection methods

In Part I of this paper the kinematic relationships between the absolute, elastic and joint accelerations are developed. In this paper, these kinematic equations are used with the generalized Newton-Euler equations and the relationship between the actual and generalized reaction forces to develop a recursive projection algorithm for the dynamic analysis of open-loop mechanical systems consisting of a set of interconnected rigid and deformable bodies. Optimal matrix permutation, partitioning and projection methods are used to eliminate the elastic accelerations while maintaining the inertia coupling between the rigid body motion and the elastic deformation. Recursive projection methods are then applied in order to project the inertia of the leaf bodies onto their parent bodies. This leads to an optimal symbolic factorization which recursively yields the absolute and joint accelerations, and the joint reaction forces. The method presented in this paper avoids the use of Newton-Raphson algorithms in the numerical solution of the constrained dynamic equations of open-loop kinematic chains since the joint accelerations are readily available from the solution of the resulting reduced system of equations. Furthermore, the method requires only the inversion or decomposition of relatively small matrices and the numerical integration of a minimum number of co-ordinates. Open-loop multibody robotic manipulator systems are used to compare the results and efficiency of the recursive methods with that of the augmented formulations that employ Newton-Raphson algorithms.