MacroC and MacroFort: C and Fortran Code Generation within Maple.

by Patrick Capolsini and Claude Gomez

Computer algebra has become a very powerful and useful tool for engineers. However, many applications require large-scale numerical in addition to symbolic computations.  Typical examples arise in the control of mechanical systems where computer algebra is used for deriving the equation of motion and numerical computations must be used for solving the simulation and control problem (see the example given later and the example of the control of a bike.  In Maple it is possible to perform floating-point numerical calculations with an arbitrary number of digits. This is very useful for ill-conditioned problems, but performance is poor. A number of Maple functions deal with standard numerical calculations but the running time of these functions is not as good as with compiled Fortran or C code. Moreover, the need to use existing C or Fortran programs arises often as well as the need to use them with code coming from computer algebra systems.  For these reasons tools are needed to link computer algebra and numerical computations. One way to do this is for the computer algebra system to generate C or Fortran code. The MacroC and Macrofort packages offer users new tools in Maple to generate complete C and Fortran programs. These packages are described in this article and their use illustrated through two engineering applications.

Integrating MathEdge with Multimedia for Instruction in Engineering

by Jeffrey B. Layton and James H. Kane

The unbundling of Maple to form MathEdge has provided a tremendous opportunity to take advantage of symbolic math in instruction. This paper will discuss one implementation of MathEdge in instruction, integrating MathEdge with Multimedia Toolbook in a multimedia lecture in a sophomore strength of materials course. It will discuss how Toolbook is integrated with MathEdge from a programming standpoint. It will then discuss some of the instructional design issues associated with using symbolic math in multimedia instruction focusing on possible roles for symbolic math. The paper will then finish with a brief discussion of a possible future for symbolic math in science and engineering instruction.

Syrup - A Symbolic Circuit Analyzer

by Joseph Riel

Computer algebra systems can solve the network equations of electric circuits. However, the task of generating the equations for any but the simplest of circuits is tedious and prone to error.  This paper describes a Maple package, Syrup, which uses the circuit descriptive language of Spice, a numerical circuit analysis program familiar to most electrical engineers, to simplify the task of analyzing an electrical circuit.

Symbolic Analysis of Multirate Systems

by Frank Heinle, Richard Reng, Gerhard Runze

Digital systems for the transmission and processing of signals have played a major role in modern telecommunications for a long time. In recent years a large number of new applications have been found that consistently use all opportunities offered by modern digital technologies. Most of these applications no longer have an equivalent in the analog world.  One class of digital systems that have become more important in the last few years is the class of multirate systems with different sampling rates. Some important applications of these systems include: subband and transform coding of still images, video, audio signals, and speech; efficient realization of very high speed digital systems, multiplexing and demultiplexing of digital signals in time or in frequency, e.g. for satellite communications; sampling rate conversions, e.g. between CD music and studio quality music,

Since multirate systems are widely used today there have been some efforts to develop methods for the analysis and description of such systems. The theoretical results are often not easy to handle. What is more, they are usually restricted to particular systems like M-channel filter banks with equal sampling rates at the input and at the output. However, a new consistent matrix description of arbitrary multirate systems was introduced that is well suited for an automated symbolic analysis of these systems.

Based on this description a Maple tool for the analysis of multirate systems has been developed. Since all linear multirate systems are composed of a few fundamental building blocks those blocks have been realized as Maple procedures. Furthermore functions for connecting multirate systems and for the analysis of the resulting composite systems have been implemented. Our analysis tool is far from complete, but even in this early stage it already permits the analysis of almost all multirate structures used in communications applications. In this paper we describe the fundamental building blocks and their implementation as well as the basic operations for connecting multirate systems.

Modelling Flexible Robots with Maple

by Jean-Claude Piedboeuf

Developing the equations of motion for flexible manipulators requires considerable effort, even for very simple systems. Obtaining a model free of error is even less obvious. This explains why programs generating the models of complex mechanical systems are so abundant and popular. In a 1989 survey, Schiehlen recorded twenty of those programs. More recently the German Research Council has established a nationwide research project devoted to dynamics of multibody systems. The goal was to develop a general purpose multibody system software package.

Traditionally, models were generated numerically.  In the last few years, there have been an increasing number of programs producing these models symbolically. These models are now more efficient from a computational point of view, which is important in real-time control and simulation. Moreover, the capability of having a symbolic model allows additional operations like linearization or optimization.

This paper discusses some practical aspects of the symbolic development of the model of a flexible robot. Since a rigid robot is a simplified version of a flexible one, the material presented here is applicable also to rigid manipulators. A brief description of SYMOFROS (Symbolic Modelling of Flexible Robots and Simulation), a symbolic modelling package developed by the author, is given and illustrated with a simple example

Automated Symbolic Analysis of Mechanical System Dynamics

by John McPhee and Cari Wells

Over the past 2-3 decades, the multibody dynamics approach has become a very popular method for simulating the kinematic or dynamic response of mechanical systems.  Due to its general nature, it has been successfully applied to the analyses of road and rail vehicles, mechanisms, satellites and space structures, and robotic manipulators. In this approach, a given system is modelled as a collection of discrete components, including rigid and flexible bodies, kinematic joints, springs, dampers, motors, and applied forces or torques.  Given only a description of the system as input, a computer implementation of a multibody dynamics formulation will automatically generate the differential-algebraic equations (DAEs) governing the time response of the system.  This allows an engineer to spend much more time on design since the hand derivation of these DAEs is a tedious and error-prone task, even for relatively simple mechanical systems.  Combined with numerical methods that generate an approximate solution to these nonlinear DAEs, and graphics routines for plotting and animating the simulated response of the system, these multibody codes represent powerful computer-aided design tools for a mechanical engineer.

In this article, the authors present a method that combines graph-theoretic techniques with the Maple computer algebra system to automatically generate the DAEs of motion for planar mechanical systems.  Thus, in addition to exploiting the well-known advantages of a implementation, the method allows the number of DAEs to be reduced by an intelligent selection of coordinates.  Heuristics governing this selection are presented via two examples:  an open-loop single pendulum, and a multi-loop quick-return mechanism.

A Maple Application in Kinematic Analysis of Mechanisms

by Oscar E. Ruiz S, Placid M. Ferreira

The Geometric Constraint Satisfaction or Scene Feasibility (GCS / SF) problem consists of a basic scenario containing geometric entities, whose context is used to propose constraining relations among still undefined entities. If the constraint specification is consistent, the answer of the problem is one of finitely or infinitely many solution scenarios satisfying the prescribed constraints. Otherwise, a diagnostic of inconsistency is expected.  The mathematical approach, previously presented in other publications, describes the problem using a set of polynomial equations, with the common roots to this set of polynomials characterizing the solution space for such a problem. That work presents the use of Groebner basis techniques for assessing consistency and redundancy of the constraints.  It also integrates subgroups of the Special Euclidean Group of Displacements in the problem formulation to exploit the structure implied by geometric relations.  In this article, the application of the discussed techniques to kinematic analysis of mechanisms is illustrated by an example.  It is implemented using MAPLE routines to manipulate polynomial ideals, and calculate their Groebner Bases.

GCS / SF underlies a number of problems in CAM / CAM / CAPP areas, for example, fixturing, assembly planning, constraint-based design, tolerancing and dimensioning, kinematic analysis, etc.  Therefore, it is evident that a strong theoretical and practical background in satisfaction of geometric constraints is crucial in  CAD / CAM / CAPP.

The Basic Curves and Surfaces of Computer Aided Geometric Design

by Colm Mulcahy

Computer Aided Geometric Design (CAGD) plays a major role in the design of cars, airplanes, and submarines, as well as in many modern manufacturing processes.  The mathematics behind CAGD is also indispensable in computer graphics.

We demonstrate the use of Maple V Release 3 as an educational tool in the construction, plotting and manipulation of the basic curves and surfaces of CAGD.  This can be done using a bare minimum of Maple, hence anybody who knows a little linear algebra and multivariate calculus can be introduced to this important material.  Maple's numerical and symbolic capabilities take the drudgery out of computing with the formulae, as well as providing immediate visual access to the resulting shapes.

Topics which can easily be explored in this way include: polynomial and parametric interpolation, least squares and Bezier curves, Hermite and natural cubic splines, tensor product surfaces, % (bilinear, biquadratic, bicubic), lofting and Coons surfaces, B-splines, B-spline curves and surfaces, interpolation with B-spline curves, least squares B-spline methods, and NURBS (non-uniform rational B-splines). We provide examples of many of these constructions, to give the general flavor of the subject. We concentrate on planar curves---the extension to space curves is routine.  The emphasis throughout is on (piecewise) polynomial methods, and their rational counterparts.

There are essential aspects of CAGD, which can also be investigated with the help of Maple, that we will not have time to touch on, such as recipes for drawing the curves and surfaces, important and illuminating connections with projective geometry, numerical analysis considerations, advanced spline algorithms, and differential geometry.

Design of Cam Mechanisms Using Maple

by Ettore Pennestri and Vanni Falasca

Cams are widely used by mechanical engineers. They are simple and inexpensive devices which are able to deliver a specified motion to another element called the follower.   The variety of cam topologies and their versatility are characteristic features of such mechanisms.  Although more complicated topologies can be considered, in this paper our attention will be confined to the synthesis of a disk cam with translating follower. In particular, we will describe how Maple can assist a cam designer during all phases required to define and manufacture a cam profile.

Using Maple for Modelling and Transient Analysis of Pneumatic Systems

by John Masse

Weibull Probability Plot and Maximum Likelihood Estimation of its Parameters

by George Bohoris and P.A. Kostagiolas

Density estimation is an important topic in the industrial applications of reliability and maintenance as it facilitates the description and study of survival data and the determination of the characteristics of the parent population. A possible approach towards density estimation is through a lifetime distribution. The way to proceed in that case is to postulate a distribution model, and then estimate from the data the parameters of that particular lifetime distribution. Density estimation is then carried out by the substitution of the calculated parameters in the relevant expressions of the various probability functions of a particular distribution model.

The Shooting Technique for the Solution of Two-point Boundary Value Problems

by Doug Meade, Bala Haran and Ralph White

One of the strengths of Maple is its ability to provide a wide variety of information about differential equations. Explicit, implicit, parametric, series, Laplace transform, numerical, and graphical solutions can all be obtained via the dsolve command. Numerical solutions are of particular interest due to the fact that exact solutions do not exist, in closed form, for most engineering and scientific applications. The numerical solution methods available within dsolve are applicable only to initial value problems. Thus, at first glance, Maple appears to be very limited in its ability to analyze the multitude of two-point boundary value problems that occur frequently in engineering analysis.

A commonly used numerical method for the solution of two-point boundary value problems is the shooting method.  This well-known technique is an iterative algorithm which attempts to identify appropriate initial conditions for a related initial value problem (IVP) that provides the solution to the original boundary value problem (BVP).

The first objective of this paper is to describe the shooting method and its Maple implementation, shoot. Then, shoot is used to analyze three common two-point BVPs from chemical engineering:  the Blasius solution for laminar boundary-layer flow past a flat plate, the reactivity behavior of porous catalyst particles subject to both internal mass concentration gradients and temperature gradients, and the steady-state flow near an infinite rotating disk.

Process Control and Symbolic Computation: An Overview with Maple V

by Ayowale Ogunye

This paper will demonstrate advantages obtained in the design and analysis of linear control systems using symbolic computation.  The representation of process control systems is by  linear state space models  or transfer functions models.  As a result, the design and analysis of control systems using transfer function models entails the manipulation of polynomials for the computation of system time responses, Laplace and inverse Laplace transformations, Z and inverse Z transformations, frequency domain responses, stability analysis, controller tuning, solution of Diophantine equations, etc.  In addition, the design of control systems represented by state space models involves the manipulation of matrices via the solution of matrix Lyapunov and Riccati equations, computation of eigenvalue-eigenvector decompositions, computation of system observability and controllability gramians, etc.  The aforementioned calculations, except for trivial cases, are complex, lengthy, laborious and error prone.  Furthermore, the solution of these problems in strict numerical computing environments, for example, MATLAB(R) (a matrix computing environment), results in a loss of the qualitative aspects of the design process. This happens because  the emphasis  is on the numerical manipulations being  performed.

Recently, there has been an interest in the application of symbolic computation to control design and analysis. Christensen (1994) describe the symbolic processor as a basic service in the computer aided control design process.  Munro and Tsapekis (1994) discuss the use of Mathematica for computing Smith-Mcmillan forms and minimal balanced realizations. Polyakov et al. (1994) and Ohtani et al.  (1994) describe the use of Mathematica packages for the design and analysis of nonlinear control systems. Ogunye and Penlidis (1995a, 1995b, 1996) discuss the use of MapleV to develop algorithms needed for the solution of state space design equations, computation of observability and controllability gramians,  computation of minimal balanced realizations and the solution of matrix Lyapunov equations.  Ogunye et al. (1995c) use MapleV to manipulate polynomial matrices in the rational functions field; algorithms for the solution of unilateral and bilateral Diophantine equations are developed.  Jager (1995) describe a MapleV nonlinear controller package designed to help along with the analysis and design of nonlinear control systems.