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.     

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.     

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.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Flexible body dynamics in a local frame with explicitly predicted motion

This paper deals with formulation of dynamics of a moving flexible body in a local frame of reference. In a conventional approach the local frame is normally fixed to the corresponding body and always represents the positions and angles of the body: the positions and angles are represented by Cartesian coordinates and Euler angles or Euler parameters, respectively. The elastic degrees of freedom are expressed by, e.g. nodal coordinates in a finite element analysis, modal coordinates, etc. However, the choice of these variables as the generalized coordinates makes the resulting equations of motion extremely complicated. This is because the representation of the rotation of a body is highly non‐linear and this non‐linearity makes the coefficient matrices dependent on the coordinates themselves. In this paper, we propose an alternative way of treating the issue by explicitly predicting the body motions and regularly updating the local frame. First, the motion of the local frame is assumed to explicitly follow the associated moving body. Then, the equations of motion are derived in a set of generalized coordinates that express both rigid‐body and elastic degrees of freedom in the local frame. These equations are solved by a time integration with a given time interval. The motion of the local frame in the interval is estimated from a prediction of the rigid‐body motions. Then, the gap between the predicted and the actual motions is evaluated. Finally, the predictions are iteratively corrected by the obtained responses in the rigid‐body motions so that the gap should remain within an imposed tolerance. Copyright © 2009 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.     

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.     

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.     

Efficient dynamic modeling for rigid multi-body systems with contact and impact

Efficient simulation of dynamical systems becomes more and more important in industry and research. Dynamic modeling of multi-body systems yields highly nonlinear equations of motion. Usually, the accelerations are computed by an explicit inversion of the mass matrix that has the dimension according to the degrees of freedom. This classical foregoing implies high computational effort. In the present contribution, an O(n) formulation is introduced for efficient (recursive) procedure. It is based on the Projection Equation in subsystem representation, structuring the problem into parts and yielding interpretable intermediate solutions. The hereby necessary inversion refers to a reduced mass matrix that has the order of the considered subsystem. Additional constraints like endpoint contact are included via corresponding constraint forces. Avoiding an inversion of the total mass matrix is again successfully applied by a recursive procedure. The impact that occurs in the transition phase between the free system and constrained system is also solved in this sense. Results for the simulation of a plane pendulum motion with changing contact scenarios are presented.     

刚柔耦合系统动力学建模新方法

作高速大范围运动的机械系统，由于运动和变形的耦合将产生动力刚化现象，传统动力学理论难以计及这种影响。通过Kane方程在保证变形广义坐标完全精确到一阶项的前提下建立了系统一阶完备动力学方程。通过与传统动力学方程的对比分析，揭示了传统建模方法不仅遗失了动力刚度项。同时遗失了某些刚柔耦合惯性项。本文提出了一种通过对传统非完备动力学方程的修正以获得一阶完备动力学方程的新方法。     

Graphical model of an elastic medium in a cartesian coordinate system

A discrete model of a deformable elastic body in the form of an oriented graph is examined. The graph can be used as a nontraditional means of deriving resolvent equations, involving the transformation of systems of generalized coordinates describing elements of the sectioned body to a coordinate system that describes the body as a whole. It is shown that graphing (vertex and loop) laws can be interpreted as equilibrium and strain compatibility conditions that in the limit become the corresponding differential equations. With the use of a unit cell having eight degrees of freedom, the strain field is approximated by linear polynomials (which corresponds to approximation of the displacement fields by quadratic polynomials). The standard finite-element method requires 16 degrees of freedom (elements with eight nodes) for the same purpose. The proposed graphical approach thus reduces the number of equations that describe the model and the time and amount of memory needed to obtain a solution.Translated from Problemy Prochnosti, No. 12, pp. 60–70, December, 1993.     

Dynamics of Uniaxial Tension of a Viscoelastic Strain-Hardening Body in a System with One Degree of Freedom. Part 4. Action of an External Force on a Compound Body

A rheological model and the dynamics of an open mechanical system, comprising an elastic machine element and a compound body consisting of two irreversibly deformable bodies connected in series with one degree of freedom under the action of an external force are considered. If the system is subjected to a load below the elastic limit of both bodies, the motion of the mechanical system is described by a classical second-order dynamic system. If the system is subjected to a load above the elastic limit of one of the bodies, the motion of the mechanical system is described by a third-order dynamic system, and when it is loaded above the elastic limit of both bodies, the motion is described by a dynamic system of two third-order differential equations. The solutions of this dynamic system are damped oscillations in character, with the damping factor increasing with the ratios of the elastic stiffness and strain hardening values to the value of viscous resistance of deformable bodies.     

The strain hardening rotating hollow shaft subject to a positive temperature gradient

Summary The modeling of deformation in flexible multibody systems is still under intensive investigation. While the floating frame of reference formulation has become a standard for the modeling of deformable moving bodies, formulations based on absolute coordinates are comparatively new. The recently developed absolute nodal coordinate formulation uses solely nodal position and slopes as degrees of freedom for structural elements. The numerical treatment is similar to the absolute coordinate formulation, which is investigated in the following. An efficient formulation based on absolute coordinates with a reduced strain tensor (similar to corotational formulations) has been derived recently, and the analogy to the floating frame of reference formulation has been shown. The efficiency of this formulation is based on a co-rotated stiffness matrix which is factorized only once, however, only linear constraints have been treated up to now. The present paper treats the extension of the advantages of the co-rotated formulation with respect to nonlinear constraints, damping and contact. Constraints are discussed for the special cases of linear and nonlinear dependence on the deformation degrees of freedom. The (implicit) Newmark scheme is used to illustrate the time stepping procedure within the present method. In the case of a small number of nonlinear constraints, a single time step can be performed by solving only a small system of nonlinear equations by means of Newton's method. A numerical example shows a method to reduce the number of constraints and illustrates the computational advantages of the proposed method.     

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.     

弹性动力学的多变量响应问题的解法

基于瞬时混合变分原理与乘积型二元三次B样条函数,以板壳为例,建立样条动力方程。引入样条参数及其对时间的导数作为状态变量,导出状态方程,对空间域采用混合样条元法,对时间域采用现代控制论中的状态空间法。文中还建立了一种状态变量的递推计算格式,可以直接计算出多变量动力响应值,数值计算结果表明,本文方法的计算精度与效率是令人满意的。本文方法对计算多输入与多输出,时不变与时变系统和线性与非线性系统等多变量动力响应问题,有广阔的应用与发展前景。     

A simple finite element‐based framework for the analysis of elastic pseudo‐rigid bodies

This article advocates a general procedure for the numerical investigation of pseudo‐rigid bodies. The equations of motion for pseudo‐rigid bodies are shown to be mathematically equivalent to those corresponding to certain constant‐strain finite element approximations for general deformable continua. A straightforward algorithmic implementation is achieved in a classical finite element framework. Also, a penalty formulation is suggested for modelling contact between pseudo‐rigid bodies. Representative planar simulations using a non‐linear elastic model demonstrate the predictive capacity of the pseudo‐rigid theory, as well as the robustness of the proposed computational procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

Formulation of equations of motion of finite element form for vehicle–track–bridge interaction system with two types of vehicle model

Vehicle, track and bridge are considered as an entire system in this paper. Two types of vertical vehicle model are described. One is a one foot mass–spring–damper system having two‐degree‐of‐freedom, and the other is four‐wheelset mass–spring–damper system with two‐layer suspension systems possessing 10‐degree‐of‐freedom. For the latter vehicle model, the eccentric load of car body is taken into account. The rails and the bridge deck are modelled as an elastic Bernoulli–Euler upper beam with finite length and a simply supported Bernoulli–Euler lower beam, respectively, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The dynamic contact forces between the moving vehicle and rails are considered as internal forces, so it is not necessary to calculate the internal forces for setting up the equations of motion of the vehicle–track–bridge interaction system. The two types of equations of motion of finite element form for the entire system are derived by means of the principle of a stationary value of total potential energy of dynamic system. The proposed method can set up directly the equations of motion for sophisticated system, and these equations can be solved by step‐by‐step integration method, to obtain simultaneously the dynamic responses of vehicle, of track and of bridge. Illustration examples are given. Copyright 2004 © John Wiley & Sons, Ltd.     

Eigenderivative analysis of asymmetric non‐conservative systems

In general, the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation. For this reason the eigenvalues and eigenvectors and consequently their derivatives become complex. Expressions for the first‐ and second‐order derivatives of the eigenvalues and eigenvectors of these linear, non‐conservative systems are given. Traditional restrictions of symmetry and positive definiteness have not been imposed on the mass, damping and stiffness matrices. The results are derived in terms of the eigenvalues and left and right eigenvectors of the second‐order system so that the undesirable use of the first‐order representation of the equations of motion can be avoided. The usefulness of the derived expressions is demonstrated by considering a non‐proportionally damped two degree‐of‐freedom symmetric system, and a damped rigid rotor on flexible supports. Copyright © 2001 John Wiley & Sons, Ltd.     

Determination of macroscopic electro-mechanical characteristics of 1-3 piezoceramic/polymer composites by a concentric tube model

An axisymmetric concentric tube model of a piezoelectric rod and a concentric elastic tube is used to characterize 1-3 piezoelectric/elastic composites macroscopically. With average displacements of and total forces on the surfaces as the mechanical degrees of freedom, and with charge and potential at the ends of the rod as the electrical degrees of freedom, the relation between the electromechanical degrees of freedom is given in a matrix formulation. A recursive numerical scheme for combining the matrix for the tube and the piezoelectric rod into one for the composite piezoelectric rod is used to directly identify the majority of the e-set of constitutive constants for 1-3 composites. The remaining constitutive constants are estimated from a cubes model to allow inversion of the d-g and h-sets and calculation of other characteristics.