Multibody Systems Handbook

In the design of mechanical systems, such as vehicles, machinery or robots, it is now customary to use computer simulations to observe the dynamic response of the system being designed. This form of computer-aided design drastically reduces the need to construct and test prototypes. However, the equations of motion of the system must first be obtained.

The overview summarizes the most important features of the codes or software systems, respectively, in writing and by tables. Altogether 20 codes are considered.

This chapter presents two mechanical systems which are appropriate as test examples for multibody systems. The plane motion of the mechanism with one degree of freedom was firstly used by Giles [1] and Manning [2] for testing. Later, Homolka [3] and Leister [4] analysed the mechanism using the programs NEWEUL and ADAMS, respectively. The topology of the mechanism is characterized by closed kinematic loops. The second example is the spatial motion of a robot with five degrees of freedom. The topology of such a mechanical system shows chain configuration.

In this paper the theoretical background as well as the application of the program NUBEMM is presented. Handling of the program will be demonstrated by using an example.

This contribution presents a new program package for the generation of efficient manipulator kinematic and dynamic equations in symbolic form. Since the computational efficiency of the generated symbolic models is extremely high, the real-time implementation of rather complicated mechanical equations become possible even on low-cost microcomputers.

The program package SYM is concerned with serial-link manipulators with stiff or elastic joints. The basic algorithm belongs to the class of customized algorithms that reduce the computational burden by taking into account the specific characteristics of the manipulator to be modeled. The high-level program code for computing various kinematic and dynamic variables (elements of homogeneous transformation matrices, Jacobian matrices, driving torques, elements of dynamic model matrices etc.) is generated. The application of recursive symbolic relations yields high, but not the minimal numerical complexity. Thus, we also apply the second-step optimization of the generated symbolic code. This step is based on the extraction of the expressions which appear more that once in the output code.

CAMS is an integrated graphical interactive software system for symbolical modelling, numerical evaluation and design of technical devices which can be considered as multibody systems. No constraints are imposed on the type of the hinges or on the topological structure of the system, i.e. the system may contain an arbitrary number of bodies interconnected by hinges in a kinematic chain with an arbitrary number of closed loops. A 3D geometric modelling system DESCARTES and a system MOVIES for Movement simulation and animation are incorporated in the CAMS software system. CAMS could be applied both — to the analysis of the already existing technical systems for full utilization of their resources in a specific technological process, and on the stage of the preliminary design — for synthesis of systems with optimal characteristics. CAMS is created in several versions for IBM PC compatible, COMPAQ, PDP 11-34, PERQ II etc. Here the theoretical foundations of CAMS software are described and a brief specification of input data, main functions, output data and post processing posibilities are presented.

AUTOLEV, an interactive symbolic dynamics program based on the method set forth in [1] for formulating equations of motion, differs fundamentally from other multibody dynamics programs in that it gives the user step-by-step control of the equation formulation process. Its unstructured format places virtually no restrictions on the types of dynamical systems that it accommodates, so that one can deal equally easily with one, two, and three dimensional holonomic and nonholonomic systems, closed loops, moving constraints, etc. Moreover, the process of formulating equations of motion is unencumbered by the computer memory limitations and slow run times associated with classical methods of mechanics, and the program thus can be used on a desktop computer.

DYNOCOMBS simulates the dynamics of constrained multibody systems [1]. The systems simulated may have general (three-dimensional) motion including translation between adjoining bodies and closed loops. Arbitrary force systems may be applied to the bodies and between the bodies.

DYNOCOMBS uses Kane’s equations to simulate the dynamics [2]. Partial velocity and partial angular velocity vectors form fundamental arrays in the development of the governing equations. DYNOCOMBS also uses Euler parameters to avoid geometrical singularities.

Constraint equations arising from the geometric and kinematic constraints (closed loops or specified motion) are appended to the dynamics equation. The resulting system of equations is reduced to a consistent set by using orthogonal complement arrays.

The governing differential equations are solved using a fourth-order Runge-Kutta procedure — although other integration techniques may be employed.

As input, the user supplies a body connection array, geometrical and physical properties of the bodies, constraint conditions, applied forces, and initial conditions. As output, DYNOCOMBS provides a description of the system configuration and movement at equal time intervals. Forces exerted on the bodies of the system are also provided.

The language of Dynocombs is Fortran.

The following paragraphs provide an outline of the theoretical basis of DYNOCOMBS. Example simulations are also discussed.

In this contribution the theoretical background of the computer program SPACAR is presented. The program is based on the finite element formulation for multi-degree of freedom mechanisms. A brief survey of the program and its major modules is given. Three examples, the mechanism and the robot in this handbook and a cantilever beam vibrating with large amplitude, are included to illustrate the capabilities of the program.

The general purpose multibody dynamics program NBOD and its follow-on companion DISCOS were developed with the expressed purpose of supporting spacecraft attitude control system design and analysis needs. During the past 2 decades, the interplay of NASA requirements with a desire to maximize software application generality has guided their development. The intent of this chapter is to chronologically outline the major technological advancements that enabled the development of these codes. Underlying theory is presented at the highest level with an emphasis on providing engineering insight needed for basic understanding and to judge program suitability for particular applications.

The Dynamic Analysis and Design System software is a set of general purpose computer programs that can be used to model and predict the motion of a variety of real world mechanical systems. Using a set of data that describes the machine to be modeled, DADS builds a mathematical model of the real system that calculates positions, velocities, and accelerations of the various parts of the machine, as well as resultant forces that act in the system. By using such a computer program to analyze a machine, the designer can simulate the behavior of a wide range of alternate designs prior to building and testing prototypes. The DADS program is often used for analysis of existing mechanical systems also.

In this contribution the theoretical background of the software NEWEUL is presented, the necessary input information is given and the output generated by the program is discussed. Two examples, a mechanism and a robot, are included for demonstration of the program. NEWEUL is written in FORTRAN 77 resulting in an excellent portability to all kinds of computers from personal computers to main frames.

Simulation and computer-aided analysis of complex mechanical systems has become a task of increasing importance in computational mechanics. It requires software tools combining modeling support, efficient generation of the equations of motion as well as modern numerical solution and system analysis techniques.

This contribution presents the main features of the AUTODYN programme. This programme based on d’Alembert Potential Power Principle, permits to derive the equations of motion of any mechanical system which can be represented by a set of interconnected rigid bodies. In particular, it has important applications in the fields of robotics and vehicle dynamics. The variables of the system are the generalized variables describing the relative motion of the various joints of the system. A joints’ library including surface rolling interconnections (rail-wheel joint) is available. Constraints can be considered and in particular those resulting from loops of bodies are automatically generated. The Lagrange multipliers technique permits to derive the complete set of equations of motion; a system reduction via the elimination of these multipliers and a coordinate partioning method is available. The obtained programme can be used as a sub-routine for any desired application such as numerical integration, stability analysis, control design, numerical linearization, eigenvalues determination.

This contribution presents the main features of the ROBOTRAN programme. This programme, based on d’Alembert Potential Power Principle, permits to derive, in symbolic form, the equations of motion of mechanical systems which can be represented by a set of rigid bodies, interconnected by one degree of freedom joints; it has important applications in the fields of robotics and vehicle dynamics. The variables of the system are the generalized variables of the various joints — linear displacement for prismatic joints and angular rotation for revolute joints. Constraints can be considered and in particular those resulting from loops of bodies can be generated by means of an auxiliary programme, CINEMA. The Lagrange multipliers technique permits to derive the complete set of equations of motion; a system reduction via the elimination of these multipliers and a coordinate partioning method is possible. The obtained programme can be used as a sub-routine for any desired application such as numerical integration, stability analysis, control design, numerical linearization, eigenvalues determination.

SIMPACK is a computer program for the simulation and calculation of motions and interacting forces of three-dimensional mechanical systems. The individual bodies can carry out geometrically large (non-linear) movements and may also exhibit elastic deformations.

The theoretical foundations of a 3-D multibody program called COMPAMM (COM-Puter Analysis of Machines and Mechanisms) are presented. Instead of using Euler angles or Euler parameters in order to define the spatial orientation of a rigid body, COMPAMM uses the cartesian coordinates of two or more points and the cartesian components of one or more unit vectors rigidly attached to the body. With this coordinates the constraint equations are quadratic and then the jacobian matrix is a linear function of them, needing for its evaluation far less arithmetic operations than with other methods. In addition to this, the mass matrix in the inertial reference frame is constant and Coriolis or centrifugal forces do not appear in the formulation. COMPAMM has also very advanced interactive and graphical capabilities that are very briefly described in this paper. Finally some examples are presented.

DYMAC (DYnamics of MAChinery) is an all FORTRAN computer program for the solution of problems in Dynamics of interconnected bodies which undergo planar motion. It is used to find displacements, velocities, accelerations, and joint reactions for multi-degree of freedom systems subjected to arbitrary, user supplied, forces. The mechanism may contain any number of closed or open loops, and be subjected to auxiliary constraints imposed by gears, cams, and user-supplied motion generators.

Descriptions of the program variables, input, output, modelling and preparation of input data, as well as solutions of sample problems, are given. A post-processing procedure is described which allows the user to provide for plotting, or otherwise processing any variables of interest to him, utilizing his local computer facilities.

DYSPAM is a computer program, written in FORTRAN, for the analysis and simulation of the Kinematics, Statics, and Dynamics of spatial interconnected systems of bodies. The system may include both open kinematic chains, as typified by robot manipulators, and closed kinematic chains, as typified by automotive suspensions and general linkage machinery. DYSPAM automatically generates the differential equations of motion, using Lagrange’s form of d’Alembert’s Principle (Virtual Work). The program finds displacements, velocities, accelerations, joint reactions, motor torques, etc. for systems with multiple degrees of freedom which are subjected to user-supplied forces, and geometric constraints. A brief description is given of modelling techniques and of input/output procedures. The capability for post-processing (e.g. plotting) data produced by DYSPAM is provided so that users may utilize their own graphics hardware/software facilities. Interactive pre- and post-processors for DYSPAM, using personal computers, are available to simplify data input and to provide for graphics and animation.

The purpose of this paper is to present the program package Mesa Verde for dynamics simulations of general multibody systems. The program name stands for “MEchanism, SAtellite, VEhicle and Robot Dynamics Equations”. Mesa Verde generates in symbolical form a minimal set of nonlinear differential equations for large motions as well as expressions for kinematical quantities and for constraint forces in joints. The program package is used by university institutions as well as by engineering consulting companies and by engineers in industry. It is commercially available at the third author’s address.

Multibody system analysis software forms an integral part of modern mechanical computer aided engineering (MCAE) practice in providing a means to simulate the operating performance of a product design prior to building a prototype. In this way, it reduces product development costs, allows evaluations of more alternative designs, and shortens the time it takes to bring a new product to the marketplace. This paper reviews the genesis, application, technical features, underlying analytical formulation, and future prospects of the most widely used multibody system analysis software package, Adams.

In most computer codes developed for the dynamic analysis of mechanisms, such as the serial manipulators used in Robotics, differential equations of motion are derived from the hypothesis that the links are rigid while flexibilities, if any, are located at the joints. This simplification is widely used since it correctly models the sturdy members found in present robots while limiting the number of unknown parameters and of the related kinematical and dynamical equations. Indeed, a rigid body shows six degrees of freedom, and any connection between two adjacent members removes up to five possibilities of relative motion: the total number of independent position parameters of any complex mechanism remains small, and elaborating the kinematical models for the end effector is then a straightforward procedure (the geometrical model relates its position and orientation to the joint parameters, while the differential model expresses its linear and angular velocities in terms of the time derivatives of joint parameters), [1].