Stability analysis of constraints in flexible multibody systems dynamics

Automated algorithms for the dynamic analysis and simulation of constrained multibody systems assume that the constraint equations are linearly independent. During the motion, when the system is at a singular configuration, the constraint Jacobian matrix possesses less than full rank and hence it results in singularities. This occurs when the direction of a constraint coincides with the direction of the lost degree of freedom. In this paper the constraint equations for deformable bodies are modified for use in the neighborhood of the singular configuration to yield the system inertia matrix which is nonsingular and also to take the actual generalized constraint forces into account. The procedures developed are applicable to both the augmented approach and the coordinate reduction methods. For the modeling of the constrained flexible multibody systems, a general recursive formulation is developed using Kane's equations, finite element method and modal analysis techniques. The system may contain revolute, prismatic, spherical or other types of joints, as well as geometrical nonlinearities; the rotary inertia is also automatically included. Simulation of a two-link flexible manipulator is presented at a singular configuration to demonstrate the utility of the method.     

A direct violation correction method in numerical simulation of constrained multibody systems

 A new direct violation correction method for constrained multibody systems is presented. It can correct the value of state variables of the systems directly so as to satisfy the constraint equations of motion. During the integration of the dynamic equations of constrained multibody systems, this method can efficiently control the violations of constraint equations within any given accuracy at each time-step. Compared to conventional indirect methods, especially Baumgarte's Constraint Violation Stabilization Method, this method has clear physical meaning, less calculation and obvious correction effect. Besides, this method has minor effect on the form of the dynamic equations of systems, so it is stable and highly accurate. A numerical example is provided to demonstrate the effectiveness of this method. Received: 17 December 1999     

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.     

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.     

An effective solver for absolute variable formulation of multibody dynamics

This paper presents an effective and general method for converting the equations of motion of multibody systems expressed in terms of absolute variables and Lagrange multipliers into a convenient set of equations in a canonical form (constraint reaction-free and minimal-order equations). The method is applicable to open-loop and closed-loop multibody systems, and to systems subject to holonomic and/or nonholonomic constraints. Being aware of the system configuration space is a metric space, the Gram-Schmidt ortogonalization process is adopted to generate a genuine orthonormal basis of the tangent (null, free) subspace with respect to the constrained subspace. The minimal-order equations of motion expressed in terms of the corresponding tangent speeds have the virtue of being obtained directly in a resolved form, i.e. the related mass matrix is the identity matrix. It is also proved that, in the case of absolute variable formulation, the orthonormal basis is constant, which leads to additional simplifications in the motion equations and fits them perfectly for numerical formulation and integration. Other useful peculiarities of the orthonormal basis method are shown, too. A simple example is provided to illustrate the convertion steps.The research leading to this paper was supported in part by the State Committee for Scientific Research, Grant No. 3 0955 91 01     

Direct differentiation method for sensitivity analysis based on transfer matrix method for multibody systems

On one hand, the new version of transfer matrix method for multibody systems (NV‐MSTMM), has been proposed by formulating transfer equations of elements in acceleration level instead of position level as in the original discrete time transfer matrix method of multibody systems to study multibody system dynamics. This new formulation avoids local linearization and allows using any integration algorithms. On the other hand, sensitivity analysis is an important way to improve the optimization efficiency of multibody system dynamics. In this paper, a totally novel direct differentiation method based on NV‐MSTMM for sensitivity analysis of multibody systems is developed. Based on direct differentiation method, sensitivity analysis matrix for each kind of element is established. By assembling transfer matrices and sensitivity analysis matrices based on differentiation law of multiplication, the sensitivity analysis equation of overall transfer equation is deduced. The computing procedure of the proposed method is also presented. All these improvements as well as three numerical examples show that the direct differentiation method based on NV‐MSTMM is suitable for optimizing the dynamic sensitivity in multi–rigid‐body systems.     

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.     

A natural partitioning scheme for parallel simulation of multibody systems

A parallel partitioning scheme based on physical-co-ordinate variables is presented to systematically eliminate system constraint forces and yield the equations of motion of multibody dynamics systems in terms of their independent co-ordinates. Key features of the present scheme include an explicit determination of the independent co-ordinates, a parallel construction of the null space matrix of the constraint Jacobian matrix, an easy incorporation of the previously developed two-stage staggered solution procedure and a Schur complement based parallel preconditioned conjugate gradient numerical algorithm.     

Object-oriented implementation of a graph-theoretic formulation for planar multibody dynamics

The objective of this paper is to describe the object-oriented implementation and computational efficiency of a multibody dynamics algorithm for planar mechanical systems. The underlying formulation uses a unique combination of orthogonal projection techniques and graph-theoretic methods to automatically generate the equations of motion in terms of ‘branch’ co-ordinates that are selected by a user. The direct analogy between the physical components in a multibody system and the ‘objects’ in Object-Oriented Programming (OOP) methods is exploited in a C++ implementation of the dynamic formulation. Issues associated with this OOP implementation are discussed, and the results of computer simulations are presented and examined for different sets of user-selected co-ordinates. © 1997 John Wiley & Sons, Ltd.     

A general formulation in rigid multibody dynamics

The dynamics of rigid multibodies is traditionally formulated by means of either minimal or redundant co-ordinates methods. An alternative approach is here proposed whereby a highly redundant set of coordinates is adopted. As a result, the equations of motion of the constrained bodies are decoupled. Several meaningful parameters are directly available and the constraint conditions are enforced in a very natural way. The first part of the paper presents the basic meanings and the theoretical developments of the formulation. The second develops a numerical approximation for the methodology proposed in the first part. The non-linear system of differential-algebraic equations governing the motion of the multibody is reduced to its weak form. It is linearized by applying a Newton–Raphson procedure and approximated through the method of finite elements in time. The details of the numerical application of this method are discussed and a solution procedure is presented. Finally, some numerical examples involving tree and closed loop topologies prove the capability of the present formulation in handling multibody dynamics.     

A hybrid direct‐automatic differentiation method for the computation of independent sensitivities in multibody systems

The usefulness of sensitivity analyses in mechanical engineering is very well‐known. Interesting examples of sensitivity analysis applications include the computation of gradients in gradient‐based optimization methods and the determination of the parameter relevance on a specific response or objective. In the field of multibody dynamics, analytical sensitivity methods tend to be very complex, and thus, numerical differentiation is often used instead, which degrades numerical accuracy. In this work, a simple and original method based on state‐space motion differential equations is presented. The number of second‐order motion differential equations equals the number of DOFs, that is, there is one differential equation per independent acceleration. The dynamic equations are then differentiated with respect to the parameters by using automatic differentiation and without manual intervention from the user. By adding the sensitivity equations to the dynamic equations, the forward dynamics and the independent sensitivities can be robustly computed using standard integrators. Efficiency and accuracy are assessed by analyzing three numerical examples (a double pendulum, a four‐bar linkage, and an 18‐DOF coach) and by comparing the results with those of the numerical differentiation approach. The results show that the integration of independent sensitivities using automatic differentiation is stable and accurate to machine precision. Copyright © 2014 John Wiley & Sons, Ltd.     

三维柔性多体梁系统非线性动力响应分析

研究了三维柔性多体梁系统的非线性动力响应问题。将空间柔性梁的变形分解为轴向变形以及在x-y平面的弯曲变形和在x-z平面的弯曲变形，引用各自的精确振动模态描述变形场，利用拉格朗日乘子法建立起柔性多体梁系统约束非线性动力学方程。结合Newmark直接积分法和Newton-Raphson迭代法，导出了求解该非线性代数一微分方程组的数值方法。仿真算例证明了该方法的正确性和有效性。     

Unconditionally energy stable implicit time integration: application to multibody system analysis and design

This paper focuses on the development of an unconditionally stable time‐integration algorithm for multibody dynamics that does not artificially dissipate energy. Unconditional stability is sought to alleviate any stability restrictions on the integration step size, while energy conservation is important for the accuracy of long‐term simulations. In multibody system analysis, the time‐integration scheme is complemented by a choice of co‐ordinates that define the kinematics of the system. As such, the current approach uses a non‐dissipative implicit Newmark method to integrate the equations of motion defined in terms of the independent joint co‐ordinates of the system. In order to extend the unconditional stability of the implicit Newmark method to non‐linear dynamic systems, a discrete energy balance is enforced. This constraint, however, yields spurious oscillations in the computed accelerations and therefore, a new acceleration corrector is developed to eliminate these instabilities and hence retain unconditional stability in an energy sense. An additional benefit of employing the non‐linearly implicit time‐integration method is that it allows for an efficient design sensitivity analysis. In this paper, design sensitivities computed via the direct differentiation method are used for mechanism performance optimization. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient kinematic joint descriptions for flexible multibody systems experiencing linear and non‐linear deformations

Different finite‐element‐based strategies used to represent the components' flexibility in multibody systems lead to various sets of co‐ordinates. For systems in which the bodies only experience small elastic deformations it is common to use mode component synthesis to reduce the number of generalized elastic co‐ordinates and, consequently, the equations of motion are written in terms of modal co‐ordinates. However, when the system components experience non‐linear deformations the use of reduction methods is not possible, in general, and the finite element nodal co‐ordinates are the generalized co‐ordinates used. Furthermore, depending on the type of finite elements used to represent each flexible body, the nodal co‐ordinates may include all node rotations and translations or only some of each. Regardless of the type of generalized co‐ordinates adopted it is required that kinematic joints are defined. The complete set of joints available in a general‐purpose multibody code must include, for each particular type of joint, restrictions involving only rigid bodies, or only flexible bodies, or flexible and rigid bodies. Therefore, the effort put into the development and implementation of any joint is at least three times as much as the initial work done in the implementation of joints with rigid bodies only. The concept of virtual bodies provides a general framework to develop general kinematic joints for flexible multibody systems with minimal effort, regardless of the flexible co‐ordinates used. Initially, only a rigid constraint between the flexible and a massless rigid body is developed. Then, any kinematic joint that involves a flexible body is set with the massless rigid body instead, using the regular joint library of the multibody code. The major drawback is that for each kinematic joint involving a flexible body it is required to use six more co‐ordinates per virtual body and six more kinematic constraints. It is shown in this work that for small elastic deformations, for which the mode component synthesis is applied, the use of sparse matrix solvers can compensate for the computational overhead of involving more co‐ordinates and kinematic constraints in the system, due to the use of virtual bodies. For non‐linear deformations, where the generalized co‐ordinates are the global positions of the finite‐element nodes, the use of the virtual body concept does not require an increase in the number of system co‐ordinates or kinematic constraints. By introducing the rigid joint between the flexible body nodal co‐ordinates and the virtual body, with the use of Lagrange multipliers, and then solving the equations explicitly for these multipliers the resulting equations of motion for the subsystem have the same degrees of freedom as the original flexible body alone. The difference is that degrees of freedom associated to the virtual body are used as co‐ordinates of the subsystem instead of the nodal co‐ordinates of the nodes of the flexible body attached to the virtual body. Copyright ©2003 John Wiley & Sons, Ltd.     

A hybrid variational method for multibody dynamics

This paper presents a hybrid variational method to minimize computational effort in forming and solving the equations of motion for broad classes of rigid multibody mechanical systems. The hybrid method combines the O(n) and O(n³) recursive variational methods for forming the equations of motion in terms of joint relative co-ordinates. While the O(n³) method is more efficient than the O(n) method for systems with short chains and decoupled loops, the converse is true when the number of bodies in chains is large. The computational complexity of the O(n³) and O(n) methods in forming and solving the equations of motion is analysed as a function of the numbers of bodies, decoupled loops, joints, cut joints, cut-joint constraint equations and force elements. Based on complexity estimates, the method presented in this paper uses either the O(n) or O(n³) variational method to formulate the equations of motion for each open chain and decoupled loop in the system, to minimize the computational effort.     

Equations of motion for general constrained systems in Lagrangian mechanics

This paper develops a new, simple, explicit equation of motion for general constrained mechanical systems that may have positive semi-definite mass matrices. This is done through the creation of an auxiliary mechanical system (derived from the actual system) that has a positive definite mass matrix and is subjected to the same set of constraints as the actual system. The acceleration of the actual system and the constraint force acting on it are then directly provided in closed form by the acceleration and the constraint force acting on the auxiliary system, which thus gives the equation of motion of the actual system. The results provide deeper insights into the fundamental character of constrained motion in general mechanical systems. The use of this new equation is illustrated through its application to the important and practical problem of finding the equation of motion for the rotational dynamics of a rigid body in terms of quaternions. This leads to a form for the equation describing rotational dynamics that has hereto been unavailable.     

A TORSIONAL SPRING-LIKE BEAM ELEMENT FOR THE DYNAMIC ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS

A new finite element beam formulation for modelling flexible multibody systems undergoing large rigid-body motion and large deflections is developed. In this formulation, the motion of the ‘nodes’ is referred to a global inertial reference frame. Only Cartesian position co-ordinates are used as degrees of freedom. The beam element is divided into two subelements. The first element is a truss element which gives the axial response. The second element is a torsional spring-like bending element which gives the transverse bending response. D'Alembert principle is directly used to derive the system's equations of motion by invoking the equilibrium, at the nodes, of inertia forces, structural (internal) forces and externally applied forces. Structural forces on a node are calculated from the state of deformation of the elements surrounding that node. Each element has a convected frame which translates and rotates with it. This frame is used to determine the flexible deformations of the element and to extract those deformations from the total element motion. The equations of motion are solved along with constraint equations using a direct iterative integration scheme. Two numerical examples which were presented in earlier literature are solved to demonstrate the features and accuracy of the new method.     

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

This study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Dynamics of an elastic multibody chain: Part C—Recursive dynamics

In multibody dynamics, and particularly in the robotics field, there are essentially two problems to consider: inverse dynamics and simulation dynamics. In the former problem, the desired trajectory is specmed and me necessary control forces and torques must be determined, the latter is the converse problem and the one of greater interest for elastic multibody systems

Just as the formulation of motion equations for multibody systems can take numerous paths, so too can their solution. The 'standard' approach to inverse and simulation dynamics is to write the equations of motion in toto for the entire system. However, the topological nature of chains can be exploited using the notion of recursion. Recursive methods allow the chain to be considered on a body-by-body basis rather than on a 'global$apos; level. The attractive feature of recursive algorithms is their computational cost, which varies linearly with the number of bodies in the chain. Global methods, on the other hand, are typically cubic in the number of coordinates

In this paper, which builds on the foundation of the inaugural paper in this series, we develop a recursive simulation algorithm for chains of general elastic bodies having arbitrary (rotational and/or translational) interbody constraints