Javier García de Jalón

A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems

Abstract A modified Lagrangian formulation is presented for the dynamic analysis of constraint mechanisms. The proposed method is based on a Hamiltonian description of the dynamics which leads to the Lagranges equations. However, the constraint conditions are not appended to the Lagranges equations in the form of algebraic or differential constraints, but instead inserted in them by means of a penalty formulation, and therefore the number of equations of the system does not increase. In addition, this approach directly leads to a system of ordinary differential equations, as opposed to the classical Lagranges formulation which results in differential algebraic equations. The resulting set of equations is of the form dot y =g(y,t) , which can be integrated by standard numerical algorithms. Finally, the proposed method is very systematic and general, and can model any type of constraint conditions, either holonomic or nonholonomic. A series of illustrative examples are analyzed. The results demonstrate the capabilities of the proposed method for simulation analysis.

Dynamic analysis of plane mechanisms with lower pairs in basic coordinates

Abstract In this article, we shall present the numerical solution to the dynamic problem of planar mechanisms with lower pairs. This method is based on the basic coordinates and link constraints. We shall begin by describing the matrix formulation of the inertial forces on a link and creating the dynamic equilibrium equations in three different ways: in Lagrangian coordinates, generalized coordinates and coordinates with Lagrange Multipliers. Finally, some examples, obtained by numerical integration of the equations of movement, will be given of problems of evolution in time of the configuration of a mechanical system.

Computer method for kinematic analysis of lower-pair mechanisms—I velocities and accelerations

Abstract This work presents a new method for the analysis of lower pair mechanisms with the help of a computer. This method make use of the coordinates of the pairs and of those of other points of interest as Lagrangian coordinates of the problem. It also makes use of link constraint equations. Perhaps, the most attractive features of this method are its conceptual simplicity and the ease with which it can be programmed in a digital mini-computer. This work is divided into two parts. The first part describes the coordinates and constraint equations made use of, with emphasis on the analysis of velocities and accelerations. The second part presents the resolution of three typically non-linear problems: initial position, finite displacements, and static equilibrium position of a mechanism with elastic connections. It presents an iterative method of rapid convergence and demonstrates that a good initial estimate is not required.

An efficient computational method for real time multibody dynamic simulation in fully cartesian coordinates

Abstract An algorithm is presented for the dynamic analysis of mechanisms that is based on the combination of fully cartesian coordinates for the definition of the mechanism, a penalty and augmented Lagrangian formulation for the satisfaction of the constraint equations and the trapezoidal rule for numerical integration with the positions — rather than accelerations — as primary variables. The new method is very systematic and general, and shows very good convergence characteristics even for large time steps. The facts that the Jacobian is linear, that the mass matrix is constant, and that neither Coriolis nor centrifugal terms are present in this formulation make the algorithm be very efficient computationally and therefore suitable for real time simulations. A series of numerical simulations are performed which demonstrate the capabilities of the proposed method.

Computer method for kinematic analysis of lower-pair mechanisms—II position problems

Abstract In the first part of this study, a new method for solving the problem of kinematic analysis was presented, based on the concepts of “basic coordinates” and “link constraint equations”. In this second part, these same concepts are used to solve the problems of initial position, finite displacements and static equilibrium position of a mechanism with springs between its links. The proposed algorithms are elementary in their formulation and of exceptional efficiency in their performance. The methods describes are based on the solution of a problem of mathematical programming. Several examples are presented, giving an idea of the potential of said algorithms.

Multibody Dynamics Optimization by Direct Differentiation Methods Using Object Oriented Programming

In this work, general purpose methodologies and formulations are developed for the optimization of the dynamic behaviour of 3D multibody systems. In order to achieve these goals, the latest advances in three fields are used: Dynamic formulations, object oriented programming languages and symbolic computation techniques.

Numerical Integration of the Equations of Motion

It was shown in Chapter 5 how the application of the laws of dynamics to constrained multibody systems leads to a set of differential algebraic equations (DAE). These can be transformed to second order ordinary differential equations (ODE) by proper differentiation of the kinematic constraint equations, by use of an independent set of coordinates, or by penalty formulations. A stable and accurate integration of both DAE and ODE is of great importance for the solution of the equations of motion. Although analytical solutions may be found for some simple cases, the number and complexity of the equations resulting from the majority of multibody systems requires numerical solutions. Because the theory of ordinary differential equations has been known for a long time, the stability, convergence, and accuracy of many methods have been studied in great detail. This has led to a wide use of these methods as compared to the differential algebraic equations, not so thoroughly known at this stage. As a consequence, many of the computer programs currently available for the computer-aided analysis and design of multibody systems rely on well-established methods for the solution of ODE.

Static Equilibrium Position and Inverse Dynamics

This chapter deals with two important multibody problems related to forces: the determination of the static equilibrium position and the solution of the inverse dynamics. In both cases, it is assumed that the motion, that is, velocities and accelerations, is known, and, in the former case also, that the motion does not exist. At least, there is not relative motion with respect to the reference frame on which the problem is to be solved.

Introduction and Basic Concepts

The kinematics and dynamics of multibody systems is an important part of what is referred to as CAD (Computer Aided Design) and MCAE (Mechanical Computer Aided Engineering). Figures 1.1 to 1.6 illustrate some practical examples of computer generated models for the simulation of real multibody systems. The mechanical systems included under the definition of multibodies comprise robots, heavy machinery, spacecraft, automobile suspensions and steering systems, graphic arts and textile machinery, packaging machinery, machine tools, and others. Normally, the mechanisms used in all these applications are subjected to large displacements, hence, their geometric configuration undergoes large variations under normal service conditions. Moreover, in recent years operating speeds have been increased, and consequently, there has been an increase in accelerations and inertial forces. These large forces inevitably lead to the appearance of dynamic problems that one must be able to predict and control.

Improved Formulations for Real-Time Dynamics

The general purpose dynamic formulations described in Chapter 5 are simple and efficient, but they are not suitable for real time dynamic simulation. Real time performance requires faster formulations. These can be developed by taking into account the system’s kinematic configuration or topology. In the last two decades, a big effort has been dedicated to developing very efficient dynamic formulations for serial robots or manipulators. These formulations have been extended later on to general open and closed chain configurations.