Application Of Symbolic Computation Research Articles

Frequency dependent mass and stiffness matrices of bar and beam elements and their equivalency with the dynamic stiffness matrix

Starting from the solutions of the governing differential equations of motion in free vibration, the frequency dependent mass and stiffness matrices of bar and beam elements have been derived in this paper, but importantly, their equivalency with the corresponding dynamic stiffness matrix is established. In sharp contrast to series solutions, reported in the literature, explicit expressions for each term of the frequency dependent mass and stiffness matrices of bar and beam elements are generated in concise form through the application of symbolic computation and their relationship with the single dynamic stiffness matrix (which contains both the mass and stiffness properties) for each of the two element types is highlighted. The theory is demonstrated by numerical results. By splitting the dynamic stiffness matrix into frequency dependent mass and stiffness matrices and at the same time retaining the exactness of results, the investigation paves the way for future research to overcome the difficulty to include damping in the dynamic stiffness research which has not been possible earlier. Furthermore, the frequency dependent mass and stiffness matrices derived in this paper permit the application of the Wittrick-Williams algorithm to compute with certainty the exact natural frequencies of structures comprising bar and beam elements.

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

The contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects. Particularly, the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines. In this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance, a task that is seldom considered in literature. By the use of computer algebra system Mathematica, we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity. Applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations, we construct iterative methods that serve (i) for approximating a multiple zero of a given function f when the order of multiplicity is unknown and, simultaneously, (ii) for finding exact order of multiplicity. In particular, we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps. Such approach, combined with a rapidly convergent method for multiple zeros, provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy. The properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction.

数式処理技術の工学設計への応用

数式処理技術の工学設計への応用

Applications of Symbolic computation in MATLAB

This paper contains brief information about symbolic computation techniques. The location of symbolic computation within Computer Language classification is defined. MATLAB + MuPAD as combined method have been introduced. After some introductory information about MuPAD, some important applications of MATLAB + MuPAD are given. The method, transfer from MuPAD into MATLAB has been discussed as well. These two methods have been illustrated to engineering students on selected topics.

Application of Symbolic Computation in Nonlinear Differential-Difference Equations

A method is proposed to construct closed-form solutions of nonlinear differential-difference equations. For the variety of nonlinearities, this method only deals with such equations which are written in polynomials in function and its derivative. Some closed-form solutions of Hybrid lattice, Discrete mKdV lattice, and modified Volterra lattice are obtained by using the proposed method. The travelling wave solutions of nonlinear differential-difference equations in polynomial in function tanh are included in these solutions. This implies that the proposed method is more powerful than the one introduced by Baldwin et al. The results obtained in this paper show the validity of the proposal.

Modelling and Simulation of Heat and Mass Transfer, via Symbolic Computation

This basically tutorial/case study paper illustrates the application of symbolic computation for two classical cases of chemical engineering, namely the modelling and simulation of a heat transfer process intensified by fin-covered surface, and a convective mass transport with first order irreversible chemical reaction. It has been shown that computer algebra provides a comfortable and elegant way to solve such technical problems in fully analytical form, with standard mathematical knowledge.

Specifying and simulating rotor machines using Maple

The symbolic computation language, Maple, is used tospecify and simulate the performance of time-varying substitution cipher systems known as rotor machines, the most famous examples of which are the ENIGMA machines used by Germany during World War II. A mathematical model is presented, initially for a one-rotor machine. The direct implementation of this model in Maple provides a prototype system from which more efficient implementations are derived via a process of program transformation. The one-rotor system is then generalized to an N -rotor system. The paper provides a tutorial on the application of symbolic computation to a problem of interest to students of mathematics and of computer science.

Generalized Variable-Coefficient KP Equation

The variable-coefficient generalizations of thecelebrated KP equation (GvcKPs) are realistic models forvarious physical and engineering situations. In thisnote, the application of symbolic computation and the truncated Painleve expansion leads toan auto-Backlund transformation and soliton-typedsolutions to a type of the GvcKPs.

126 Recent applications of symbolic computation in control system design

126 Recent applications of symbolic computation in control system design

Use of symbolic computation in robotics education

An application of symbolic computation in robotics education is described. A software package is presented which combines generality, user interaction, and user-friendliness with the systematic usage of symbolic computation and artificial intelligence techniques. The software utilizes MACSYMA, a LISP-based symbolic algebra language, to automatically generate closed-form expressions representing forward and inverse kinematics solutions, the Jacobian transformation matrices, robot pose error-compensation models equations, and Lagrange dynamics formulation for N degree-of-freedom, open chain robotic manipulators. The goal of such a package is to aid faculty and students in the robotics course by removing burdensome tasks of mathematical manipulations. The software package has been successfully tested for its accuracy using commercially available robots.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Why symbolic computation in robotics?

This article is organized to demosttrate various applications of symbolic computation in robot manipulator design and analysis. Four major areas—kinematics, dynamics, trajectory planning and control, and pose error modeling—are extensively surveyed. A concept of an intermediate symbolic manipulator module is also described to automate and couple the processes of an interactive robot geometry design and numerical simulation.

Application of symbolic computation in robot pose error modelling

This paper presents an application of symbolic computation in geometrical error modelling and simulation of an industrial robotic manipulator. A program named SCRPE (Symbolic Computation of Robot Pose Errors) has been developed to automatically generate error model equations for the end-eflector of N degree-of-freedom robots. The SCRPE utilizes symbolic manipulation capability of MACSYMA (a Lisp-base artificial intelligence language). The prime reasons of this work are to provide a base for comparison, performance assessment and accuracy judgment of numerous error calibration and compensation models available in the literature; and to represent output results in a concise form eliminating completely the manual derivation process. When such computer generated outputs are fed to a simulation program, saving in computational time for error estimation is realized. As an example, the mathematical error model considered here is based on small perturbation in link parameters defined in accordance with a classical Denavit-Hartenberg notation. The time required to compute first and second order terms of the model are compared for numerous robots. The worst case scenario takes 290 seconds for a PUMA 600 series robot, which is insignificant compared to the time required for accurate manual derivation. The SCRPE program is user-friendly, interactively menu-driven and has been developed on VAX 750 Digital computer under the VMS operating system. Interested readers can obtain the program copy by contacting the first author.

Symbolic computation application for the design of linear multivariable control systems

The application of symbolic computation to the algebraic design of linear multivariable control systems is presented. A software package for manipulating polynomial and rational function matrices and for solving the linear matrix equations referred to as unilateral and bilateral equations is implemented on the basis of the symbolic manipulation system known as REDUCE. Several basic functions in the package are explained and their usage is demonstrated by using an example of the design of a discrete-time control system.

An example of application of symbolic computation in solid state physics

A system of symbolic computation is applied to the calculation of rather lengthy and cumbersome polynomial which is a sum of traces of different matrix products. The polynomial is the essential part of a formula for the magnetic susceptibility of electrons. The output file of a symbolic computation program is included as a Fortran routine into the main program for the calculation of the magnetic susceptibility. A numerical test of the routine in comparison with a routine made by straightforward programming of the formula for the polynomial was carried out.