# MapleTech Volume 3 Number 1, 1996

## Maple Functionality

Simple Engineering Mathematics with Maple
R.M. Corless
In this article we investigate some simple examples of how Maple can help you to solve basic mathematical problems, allowing you to concentrate on the formulation and analysis aspects of engineering problems.
Maple V Release 4: New Features for Engineers and Scientists
T. Lee
This article contains descriptions on some of the new additions and improvements to the latest version of the Maple V system that will be of particular interest to users in applied technical areas such as engineering and the physical sciences. Topics included user interface improvements, new symbolic and numerical computation functions such as support for piecewise functions, and a new series of ordinary differential equation solvers, new plotting facilities such as root locus plots, and programming utilities such as the new interactive debugger.
MacroC and Macrofort: C and Fortran Code Generation within Maple.
P. Capolsini, C. 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. 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
J.B. Layton, J.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.

## Electrical Engineering

Syrup - A Symbolic Circuit Analyzer
J. 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
F. Heinle, R. Reng, G. Runze
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. 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. In this paper the authors describe the fundamental building blocks and their implementation as well as the basic operations for connecting multirate systems.

## Mechanical Engineering and Computer Aided Design

Modelling Flexible Robots with Maple
J.-C. 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. 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
J. McPhee, C. Wells
In this article the authors present a method that combines graph-theoretic techniques with the Maple computer algebra system to automatically generate the DAEs (differential-algebraic equations) of motion for planar mechanical systems. Thus, in addition to exploiting the well-known advantages of this 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.
Using Groebner Bases in Kinematic Analysis of Mechanisms
O.E. Ruiz S., P.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's routines to manipulate polynomial ideals, and calculate their Groebner Bases.
The Basic Curves and Surfaces of Computer Aided Geometric Design
C. 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. This article demonstrates 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.
Design of Cam Mechanisms Using Maple
E. Pennestri, V. 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 attention is confined to the synthesis of a disk cam with translating follower, in particular by describing how Maple can assist a cam designer during all phases required to define and manufacture a cam profile.
Weibull Probability Plot and Maximum Likelihood Estimation of its Parameters
G. Bohoris, 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.

## Chemical Engineering

The Shooting Technique for the Solution of Two-point Boundary Value Problems
D. Meade, B. Haran, R. White
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
A. Ogunye
This paper demonstrates the 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 (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.

HTML originally written by Reid M. Pinchback
Copyright 1996, Massachusetts Institute of Technology