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.     

Decoupling joint and elastic accelerations in deformable mechanical systems using non‐linear recursive formulations

The aim of this paper is to develop non‐linear recursive formulations for decoupling joint and elastic accelerations, while maintaining the non‐linear inertia coupling between rigid body motion and elastic deformation in deformable mechanical systems. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large. Consequently, the use of these recursive formulations for solving the joint and elastic accelerations becomes less efficient. In this paper, the non‐linear recursive formulations have been used to decouple the elastic and joint accelerations in deformable mechanical systems. The relationships between the absolute, elastic and joint variables and generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. By using the inertia matrix structure of deformable mechanical systems and the fact that joint reaction forces associated with elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The use of the approaches developed in this investigation is illustrated using deformable open‐loop serial robot and closed‐loop four‐bar mechanical systems. Copyright © 2007 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.     

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 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.     

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.     

A matrix formulation for the dynamic analysis of spatial mechanisms using point coordinates and velocity transformation

Summary. This paper presents a matrix formulation for the dynamic analysis of spatial mechanisms with common types of kinematic joints. The formulation is derived in two steps. Initially an equivalent constrained system of particles that replaces the rigid bodies is constructed and used to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion in terms of the rectangular Cartesian coordinates of the particles without introducing any rotational coordinates. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. An example is used to demonstrate the generality and efficiency of the proposed method.     

Dynamics of an elastic multibody chain: part a—body motion equations

In this paper—the first of a series describing the dynamics of an arbitrary multibody system—motion equations governing a set of individual bodies in a chain configuration are discussed. A chain consisting entirely of rigid bodies is considered first. Motion equations for a typical body of arbitrary shape and arbitrary mass distribution are then briefly summarized. Finally, the geometrical constraints necessary to connect the individual bodies into a chain are derived.

Large translational and rotational motions are permitted at the joints connecting contiguous bodies. In other words, both prismatic and revolute joints are included, alone and in combination. As well, the interbody force constraints required to ensure that equal but opposite forces and torques exist at each joint are developed. The resulting expressions are amenable to the introduction of constraint and control forces at the chain joints. This permits the number of actively controlled degrees of freedom at any specific joint to be arbitrarily specified. The equations are fully nonlinear in the ‘rigid’ velocities for all individual chain bodies.

The analysis is then extended to the case of a chain consisting of an arbitrary number of elastic bodies. Each elastic body is assumed to possess an arbitrary stiffness distribution, although structural deformations are assumed to be small.     

An o(n) complexity recursive algorithm for multi‐flexible‐body dynamics based on absolute nodal coordinate formulation

A new algorithm called recursive absolute nodal coordinate formulation algorithm (REC‐ANCF) is presented for dynamic analysis of multi‐flexible‐body system including nonlinear large deformation. This method utilizes the absolute nodal coordinate formulation (ANCF) to describe flexible bodies, and establishes a kinematic and dynamic recursive relationship for the whole system based on the articulated‐body algorithm (ABA). In the ordinary differential equations (ODEs) obtained by the REC‐ANCF, a simple form of the system generalized Jacobian matrix and generalized mass matrix is obtained. Thus, a recursive forward dynamic solution is proposed to solve the ODEs one element by one element through an appropriate matrix manipulation. Utilizing the parent array to describe the topological structure, the REC‐ANCF is suitable for generalized tree multibody systems. Besides, the cutting joint method is used in simple closed‐loop systems to make sure the O(n) algorithm complexity of the REC‐ANCF. Compared with common ANCF algorithms, the REC‐ANCF has several advantages: the optimal algorithm complexity (O(n)) under limited processors, simple derivational process, no location or speed constraint violation problem, higher algorithm accuracy. The validity and efficiency of this method are verified by several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.     

A recursive method for the dynamic analysis of mechanical systems in spatial motion

Summary. In the present study, a recursive method for generating the equations of motion of mechanical systems that undergo spatial motion is presented. The method uses the force and moment equations to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotation matrices. For the open loop case, the equations of motion are generated recursively along the serial chains. Closed loop systems are transformed to open loop systems by cutting suitable kinematic joints and introducing cut-joint constraints. The method is simple and suitable for computer implementation. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

A simplified recursive formulation for the dynamic analysis of planar mechanisms

Summary In the present paper, a numerical method for generating the equations of motion of planar mechanisms with only revolute joints is presented. The method rests upon the idea of replacing the rigid body by a dynamically equivalent constrained system of particles. For the open loop case, the equations of motion are generated recursively along the open chains. Geometric constraints that fix the distance between the particles are introduced. For the closed loop case, the system is transformed to open loops by cutting suitable kinematic joints with the addition of kinematic constraints. The method is conceptually easy and suitable for computer implementation. It eliminates the necessity of distributing the external forces and moments over the particles and uses the concepts of linear and angular momentums to generate the rigid body equations of motion without introducing any rotational coordinates. An example with closed loops is chosen to demonstrate the generality and simplicity of the proposed method.     

A nonsmooth frictional contact formulation for multibody system dynamics

We present a new node-to-face frictional contact element for the simulation of the nonsmooth dynamics of systems composed of rigid and flexible bodies connected by kinematic joints. The equations of motion are integrated using a nonsmooth generalized-α time integration scheme and the frictional contact problem is formulated using a mixed approach, based on an augmented Lagrangian technique and a Coulomb friction law. The numerical results are independent of any user-defined penalty parameter for the normal or tangential component of the forces and, the bilateral and the unilateral constraints are exactly fulfilled both at position and velocity levels. Finally, the robustness and the performance of the proposed algorithm are demonstrated by solving several numerical examples of nonsmooth mechanical systems involving frictional contact.     

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.     

Modeling railroad track structures using the finite segment method

In the finite segment method, the dynamics of a deformable body is described using a set of rigid bodies that are connected by elastic force elements. This approach can be used, as demonstrated in this investigation, in the simulation of some rail movements. In order to ensure that the rail geometry is not distorted as the result of the finite segment displacements, a new track model that consistently integrates the absolute nodal coordinate formulation (ANCF) geometry and the finite segment method is developed. ANCF finite elements define the track geometry in the reference configuration as well as the change in the geometry due to the movement of the finite segments of the track. Using ANCF geometry and the finite segment kinematics, the location of the wheel/rail contact point is predicted online and used to update the creepage expressions due to the finite segment displacements and rotations. The location of the wheel/rail contact point and the updated creepage expressions are used to evaluate the creep forces. A three-dimensional elastic contact formulation (ECF-A) which allows for wheel/rail separation is used in this investigation. The rail displacement due to the applied loads is modeled by a set of rigid finite segments that are connected by a set of spring-damper elements. Each rail finite segment is assumed to have six rigid body degrees of freedom. The equations of motion of the finite segments are integrated with the railroad vehicle system equations of motion in a sparse matrix formulation. The resulting dynamic equations are solved using a predictor–corrector numerical integration method that has a variable order and variable step size. The finite segments may be used to model specific phenomena that occur in railroad vehicle applications, including rail rotations and gauge widening. The procedure used in this investigation to implement the finite segment method in a general purpose multibody system (MBS) computer program is described. Two simple models are presented in order to demonstrate the implementation of the finite segment method in MBS algorithms. The limitations of using the finite segments approach for modeling the track structure and rail flexibility are also discussed.     

A perturbation method used for static contact and low velocity impact

An approximate method for the solution of static and dynamic contact problems between bodies with non-linear material behaviour is described. The method is a perturbation technique based on the linear elastic quasi-static solution. Here the method is applied to the problem of a sphere in contact with a half-space which means that the Hertz solution is used. The governing equations are rewritten so that the problem for the perturbed variables is one with surface forces in the contact region and volume forces inside the bodies. The latter are due to accelerations and strain gradients calculated from the quasi-static solution and the equation of motion. The contact condition results in an integral equation for the surface forces. Results are compared with FEM calculations, which show very good agreement for the dynamic case, both with linear elastic and non-linear (plastic) material behaviour. For the static case with non-linear material behaviour the results are good approximately up to the point where the inelastic zone reaches the surface of the bodies.     

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.     

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     

Large reference displacement analysis of composite plates. Part II: Computer implementation

This investigation concerns itself with the computer implementation of the dynamic formulation of thin laminated composite plates consisting of layers of orthotropic laminae that undergo large arbitrary rigid body displacements and small elastic deformations. A finite element preprocessor computer program is developed to automatically generate the invariants of the laminae, which may have arbitrary orientations. The laminae invariants are then used to obtain the invariants of the elements and the composite laminated plate. The consistent and lumped mass formulations of the invariants of motion of composite plates are compared and it is concluded that the two methods are comparable, if a fine enough finite element mesh is used. The structure of the dynamic equations of motion, based on the formulation presented in Part I of this paper, is examined. Non-linear centrifugal and Coriolis forces arising as the result of the finite rotations of the laminae are defined, and the solution schemes of the resulting non-linear differential equations of motion are discussed. Numerical examples illustrating the differences between homogeneous isotropic and laminated composite plates are presented. An RSSR (Revolute-Spherical-Spherical-Revolute) mechanism is used in the numerical examples, with the coupler modelled as a laminated plate flexible body. It is found that the inertia of the plate contributed greatly to the transverse deformation. The effects of laminae orientation is also investigated.     

大变形复合材料薄板多体系统动力学建模

本文对大变形复合材料薄板的多体系统动力学建模方法进行研究。基于Kirchhoff假设,法线与中面保持垂直,从格林应变的表达式出发,建立了面内应变和曲率与绝对位置坐标和斜率的关系,在此基础上推导了广义弹性力阵和弹性力阵对广义坐标的导数阵,用绝对节点坐标方法建立了大变形复合材料薄板多体系统的动力学方程,用广义法和和牛顿迭代法求解微分-代数混合方程。对外载荷作用下的复合材料薄板进行数值仿真,通过与ANSYS的仿真结果进行对比,验证了本文建模方法的准确性和快速收敛性。最后,将建模方法应用于复合材料太阳帆板展开机构的数值仿真,分析了不同铺层情况下驱动力和约束力的振动特性。     

Contact forces in the non-linear dynamic analysis of tracked vehicles

It is known that when two springs are connected in series, the stiffness coefficient of an equivalent system that consists of one spring is less than the stiffness coefficients of the original springs. Experimental observations indicate that this fact can be very useful in determining the overall vibration characteristics of tracked vehicles. This simple fact is used in this investigation to develop a computer aided analysis procedure for the dynamic simulation of large-scale tracked vehicles. The track is considered as a closed kinematic chain that consists of rigid bodies connected by revolute joints. The contacts between the track links and the rollers, the sprocket, and the idler are represented by non-linear continuous force models. The stiffness and damping coefficients in these contact force models are determined by studying the viberation characteristics of the tracked vehicle. The tooth of the sprocket is defined using three surfaces. These are the left, the bottom, and the fight surfaces. Three successive transformations are used to define the contact kinematic relationships between the sprocket teeth and the pins of the track links. The equations of motions of the vehicle are formulated using the Lagrangian approach. Non-linear constraint equations that describe mechanical joints and specified motion trajectories in the system are adjoined to the differential equations of motion using the technique of Lagrange multipliers. The resulting mixed system of differential and algebraic equations is solved numerically using a direct numerical integration method. A Newton–Raphson algorithm is used to check on the violations in the kinematic constraints. The results presented in this paper are obtained using a 54 body planer tracked vehicle in which the track consists of 42 rigid links connected by revolute joints.