Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.     

Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces

This paper addresses some important issues for multibody dynamics; issues that are basic and really not too difficult to solve, but rarely considered in the literature. The aim of this paper is to contribute to the resolution and clarification of these topics in multibody dynamics. There are many formulations for determining the equations of motion in constrained multibody systems. This paper will focus on three of the most important methods: the Lagrange equations of the first kind, the null space method and the Maggi equations. In all cases we consider singular inertia matrices and redundant constraint equations. We assume that the inertia matrix is positive-semidefinite (symmetric) and that the constraint equations may be redundant but always consistent. It is demonstrated that the aforementioned dynamic formulations lead to the same three mathematical conditions of existence and uniqueness of solutions, conditions that have at the same time a clear physical meaning. We conclude that the mathematical problem always has a solution if the physical problem is well conditioned. This paper also addresses the problem of determining the constraint forces in the case of redundant constraints. This problem is examined from a broad perspective. We will present several examples and a simple method to find practical solutions in cases where the forces of constraint are undetermined. The method is based on the weighted minimum norm condition. A physical interpretation of this minimum norm condition is provided in detail for all examples. In some cases a comparison with the results obtained by considering flexibility is included.     

A criterion on inclusion of stress stiffening effects in flexible multibody dynamic system simulation

This paper presents a criterion on inclusion of stress stiffening effects in dynamic simulation of flexible multibody systems. The proposed criterion examines numerically the eigenvalue variation of the total modal stiffness matrix that is a combination of the modal stress stiffness matrix and the conventional linear modal stiffness matrix prior to actual dynamic simulation. If the variation is sufficiently large for any flexible body in the multibody system, then stress stiffening effects must be included in dynamic simulation of flexible multibody systems for accurate prediction of dynamic behavior. Since the criterion uses the most general stress stiffness matrix contributed from applied and constraint reaction loads as well as from a system of 12 inertial loads, this criterion is applicable to any general flexible multibody dynamic system. Several numerical results are presented to show the effectiveness of the proposed criterion.     

The Absolute Coordinate Formulation with Elasto-Plastic Deformations

The present work contributes to the field of multibody systems with respect to the absolute coordinate formulation with a reduced expression of the strain energy and a non-linear constitutive model. Standard methods for multibody systems lead to highly non-linear terms either in the mass matrix or in the stiffness matrix and the most expensive part in the solution of the equations of motion is the assembling of these matrices, the computation of the Jacobian of the non-linear system and the solution of a linear system with the system matrices. In the present work, a consistent simplification of the equations of motion with respect to small deformations but large rigid-body motions is performed. The absolute coordinate formulation is used, therefore the total displacements are the unknowns. This formulation leads to a constant mass matrix while the non-linear stiffness matrix is composed of the constant small strain stiffness matrix rotated by the underlying rigid body rotation. Plastic strains are introduced by an additive split of the strain into an elastic and a plastic part, a yield condition and an associative flow rule. The decomposition of strain has to be performed carefully in order to obey the principle of objectivity for plasticity under large rigid body rotations. As an example, a two-dimensional plate which is hinged at one side and driven by a harmonic force at the opposite side is considered. Plastic deformation is assumed to occur due to extreme environmental influences or due to failure of some attached parts like a defect bearing.     

Twenty-five years of natural coordinates

In the early eighties, the author and co-workers created and further developed the natural coordinates to describe the motion of 2-D and 3-D multibody systems. Natural coordinates do not need angles or angular parameters to define orientation, leading to constant inertia matrices and to the simplest form of the constraint equations. Natural coordinates are composed by the Cartesian coordinates of some points and the Cartesian components of some unit vectors distributed on the different bodies of the system. The points and vectors can be located in the joints, being shared by contiguous bodies, decreasing or even eliminating the need to set joint constraints and reducing the total number of variables. However, other authors prefer not to share variables in order to get even simpler equations and to keep a bigger decoupling of equations, which is preferable in some cases. In this paper the history of natural coordinates is reviewed, as well as the main contributions coming from other research groups. In the second part of the paper some application areas in which natural coordinates can be particularly advantageous are examined. Commemorative Contribution.     

Elastic stability and buckling loads of multi-span nonuniform beams,shafts and frames on rigid or elastic supports by finite element method using planar uniform line elements

A numerical computer method using planar flexural finite line element for the determination of buckling loads of beams, shafts and frames supported by rigid or elastic bearings is presented. Buckling loads and the corresponding mode vectors are determined by the solution of a linear set of eigenvalue equations of elastic stability. The elastic stability matrix is determined as the product of the bifurcation sidesway flexibility matrix and the second order bifurcation sidesway stiffness matrix which is formed using the element bifurcation sidesway stiffness matrices. The bifurcation sidesway flexibility matrix is determined by partitioning the inverse of the global external stiffness matrix of the system which is formed from the element data using the element stiffness matrices. The method is directly applicable to the determination of the buckling loads of beams and frames partially or fully supported by elastic foundations where the foundation stiffness is approximated by a discrete set of springs. The method of the article provides means to consider complex boundary conditions in buckling problems with ease. Four numerical examples are included to illustrate the industrial applications of the contents of the article.     

Riccati discrete time transfer matrix method for elastic beam undergoing large overall motion

An efficient method for dynamics simulation for elastic beam with large overall spatial motion and nonlinear deformation, namely, the Riccati discrete time transfer matrix method (Riccati-DT-TMM), is proposed in this investigation. With finite segments, continuous deformation field of a beam can be decomposed into many rigid bodies connected by rotational springs. Discrete time transfer matrices of rigid bodies and rotational springs are used to analyze the dynamic characteristic of the beam, and the Riccati transform is used to improve the numerical stability of discrete time transfer matrix method of multibody system dynamics. A predictor-corrector method is used to improve the numerical accuracy of the Riccati-DT-TMM. Using the Riccati-DT-TMM in dynamics analysis, the global dynamics equations of the system are not needed and the computation time required increases linearly with the system’s number of degrees of freedom. Three numerical examples are given to validate the method for the dynamic simulation of a geometric nonlinear beam undergoing large overall motion.     

Rigid multibody system dynamics with uncertain rigid bodies

This paper is devoted to the construction of a probabilistic model of uncertain rigid bodies for multibody system dynamics. We first construct a stochastic model of an uncertain rigid body by replacing the mass, the center of mass, and the tensor of inertia by random variables. The prior probability distributions of the stochastic model are constructed using the maximum entropy principle under the constraints defined by the available information. The generators of independent realizations corresponding to the prior probability distribution of these random quantities are further developed. Then several uncertain rigid bodies can be linked to each other in order to calculate the random response of a multibody dynamical system. An application is proposed to illustrate the theoretical development.     

Flexible multibody dynamics using coordinate reduction improved by dynamic correction

This paper is concerned with the efficient dynamic analysis of flexible multibody systems using a robust coordinate reduction technique. Unlike conventional static correction, the formulation is derived by dynamic correction that considers the inertia effect. In this formulation, the constraint and fixed-interface normal modes, which are representative modes in the typical coordinate reduction, are corrected by considering the truncated modal effect with the residual flexibility. Therefore, the proposed method can offer a more precise reduced system without increasing the dimension, which consequently leads to a more accurate and efficient flexible multibody simulation. We implement here the proposed method under augmented formulations of the floating reference frame approach, and test its performance with numerical examples.     

Graph theoretic foundations of multibody dynamics

This is the first part of two papers that use concepts from graph theory to obtain a deeper understanding of the mathematical foundations of multibody dynamics. The key contribution is the development of a unifying framework that shows that key analytical results and computational algorithms in multibody dynamics are a direct consequence of structural properties and require minimal assumptions about the specific nature of the underlying multibody system. This first part focuses on identifying the abstract graph theoretic structural properties of spatial operator techniques in multibody dynamics. The second part paper exploits these structural properties to develop a broad spectrum of analytical results and computational algorithms.Towards this, we begin with the notion of graph adjacency matrices and generalize it to define block-weighted adjacency (BWA) matrices and their 1-resolvents. Previously developed spatial operators are shown to be special cases of such BWA matrices and their 1-resolvents. These properties are shown to hold broadly for serial and tree topology multibody systems. Specializations of the BWA and 1-resolvent matrices are referred to as spatial kernel operators (SKO) and spatial propagation operators (SPO). These operators and their special properties provide the foundation for the analytical and algorithmic techniques developed in the companion paper.We also use the graph theory concepts to study the topology induced sparsity structure of these operators and the system mass matrix. Similarity transformations of these operators are also studied. While the detailed development is done for the case of rigid-link multibody systems, the extension of these techniques to a broader class of systems (e.g. deformable links) are illustrated.     

Relative Finite Element Coordinates in Multibody System Simulation

In the paper a numerical approach for deriving the nonlinear explicitform dynamic equations of rigid and flexible multibody systems ispresented. The dynamic equations are obtained as Ordinary DifferentialEquations for generalized coordinates and without algebraic constraints.The Finite Element Theory is applied for discretization of flexiblebodies. The minimal set of the generalized coordinates includesindependent joint motions, as well as independent small flexibledeflections of finite element nodes. The node deflections and stiffnessmatrices are calculated with respect to the moving relative coordinatesystems of the flexible bodies. The positions and orientations ofelement and substructure coordinate systems are updated according to thenode deflections. A major step of the numerical process is the kinematicanalysis and calculation of matrices of partial derivatives of thequasi-coordinates (dependent joint motions and coordinates of points andnodes) with respect to the generalized coordinates. The inertia terms inthe dynamic equations are obtained multiplying the matrices of thepartial derivatives by the mass matrices of the rigid and flexiblebodies. Stiffness properties of flexible bodies are presented in thedynamic equations by stiff forces that depend on the generalizedrelative flexible deflections only. Several examples of large motion ofbeam structures show the effectiveness of the algorithm.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

Application of perturbation techniques to flexible multibody system dynamics

In this paper, a matrix perturbation technique is developed for flexible bodies (substructures) that undergo large reference translational and rotational displacements. Although the governing dynamic equations of motion of such systems are highly nonlinear because of the large angular rotations and the resulting nonlinear inertia coupling between the reference motion and the elastic deformation, a generalized linear eigenvalue problem that defines the deformation mode shapes of the body with respect to the selected body reference is identified. This eigenvalue problem is solved only once and the variations in the body stiffness and inertia properties due to a change in selected design parameters are evaluated by using perturbation analysis techniques. The main advantage of using the proposed technique is to avoid a new finite element discretization when some design parameters are changed. This, in turn, substantially reduces the computational time, especially when large scale flexible bodies with complex geometry are considered. A numerical example is presented in order to demonstrate the use of the perturbation techniques developed in this paper in the design of flexible multibody systems.     

Optimization of a hyper-elastic structure with multibody contact using continuum-based shape design sensitivity analysis

In this paper, a continuum-based shape design sensitivity formulation is presented for a hyper-elastic structure with multibody frictional contact. A nearly incompressible constraint is treated using the pressure projection method that projects a hydrostatic pressure into a lower order space to avoid a volumetric locking. The variational formulation for multibody frictional contact is developed using a penalty method that regularizes the solution of the variational inequality. The material derivative of continuum mechanics is utilized to develop the continuum-based shape design sensitivity analysis for the hyper-elastic constitutive relation and penalized contact formulation. The sensitivity equation is solved at each converged load step using the same tangent stiffness of response analysis due to the path dependency of the sensitivity of the frictional contact problem. A very accurate and efficient sensitivity results are shown through shape optimization of a windshield wiper. Received August 8, 1999     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.     

Maximum-entropy-type Lyapunov functions for robust stability and performance analysis

We present two Lyapunov functions that ensure the unconditional stability and robust performance of a modal system with uncertain damped natural frequency. Each Lyapunov function involves the sum of two matrices, the first being the solution to the so-called maximum-entropy equation and the second being a constant auxiliary portion. The significant feature of these Lyapunov functions is that the guaranteed robust stability region is independent of the weighting matrix, while the performance bounds are relatively tight compared to alternative approaches. Thus, these Lyapunov functions are less conservative than standard bounds that tend to be highly sensitive to the choice of state space basis.     

A unified approach for inverse and direct dynamics of constrained multibody systems based on linear projection operator: applications to control and simulation

This paper presents a unified approach for inverse and direct dynamics of constrained multibody systems that can serve as a basis for analysis, simulation, and control. The main advantage of the dynamics formulation is that it does not require the constraint equations to be linearly independent. Thus, a simulation may proceed even in the presence of redundant constraints or singular configurations, and a controller does not need to change its structure whenever the mechanical system changes its topology or number of degrees of freedom. A motion-control scheme is proposed based on a projected inverse-dynamics scheme which proves to be stable and minimizes the weighted Euclidean norm of the actuation force. The projection-based control scheme is further developed for constrained systems, e.g., parallel manipulators, which have some joints with no actuators (passive joints). This is complemented by the development of constraint force control. A condition on the inertia matrix resulting in a decoupled mechanical system is analytically derived that simplifies the implementation of the force control. Finally, numerical and experimental results obtained from dynamic simulation and control of constrained mechanical systems, based on the proposed inverse and direct dynamics formulations, are documented.     

多柔体系统动力学研究进展与挑战

首先回顾多体系统动力学的学科发展和学术交流情况,然后系统概述了多柔体系统动力学方程数值算法、多柔体系统接触/碰撞动力学与柔性空间结构展开动力学三个方面的研究进展及值得关注的若干问题,最后给出了开展多柔体系统动力学研究的若干建议.     

Transfer matrix method for linear multibody system

A new method for linear hybrid multibody system dynamics is proposed in this paper. This method, named as transfer matrix method of linear multibody system (MSTMM), expands the advantages of the traditional transfer matrix method (TMM). The concepts of augmented eigenvector and equation of motion of linear hybrid multibody system are presented at first to find the orthogonality and to analyze the responses of the hybrid multibody system using modal method. If using this method, the global dynamics equation is not needed in the study of linear hybrid multibody system dynamics. The MSTMM has a small size of matrix and higher computational speed, and can be applied to linear multi-rigid-body system dynamics, linear multi-flexible-body system dynamics and linear hybrid multibody system dynamics. This method is simple, straightforward, practical, and provides a powerful tool for the study on linear hybrid multibody system dynamics. This method can be used in the following: (1) Solve the eigenvalue problem of linear hybrid multibody systems. (2) Obtain the orthogonality of eigenvectors of linear hybrid multibody systems. (3) Realize the accurate analysis of the dynamics response of linear hybrid multibody systems. (4) Find the connected parameters between bodies used in the computation of linear hybrid multibody systems. A practical engineering system is taken as an example of linear multi-rigid-flexible-body system, the dynamics model, the transfer equations and transfer matrices of various bodies and hinges; the overall transfer equation and overall transfer matrix of the system are developed. Numerical example shows that the results of the vibration characteristics and the response of the hybrid multibody system received by MSTMM and by experiment have good agreements. These validate the proposed method.