Symbolic Equations Research Articles

Symbolic equations and their applications to reaction design

The application of symbolic equations to problems of reaction design is described. Problems in their analytical and constructive enumeration are formulated in terms of graph and permutation group theories. Some features of the combinatorial algorithm and of its computer implementations are described. Several hypothetical processes corresponding to unknown types of reaction are outlined.

Dynamically changing the logical behavior of the microcomputer interface

Industrial programmable controllers and hardware simulators currently use the software method of direct logic simulation. In this method, programs contain the logical functions of several Boolean variables that are encoded directly. Each combinational circuit expressed by a group of Boolean equations requires an independent program. An alternative solution for the software implementation of the Boolean equations, named the product terms method, is presented. This method features the use of a single program to implement any multiple-output combinational circuit. To make the distinction between different circuits, a block of data defines each circuit. The product method is extended to synchronous sequential circuits, for which two tables are used. One table determines the next state of the combinational circuit, and the second table determines the outputs. An expert system has been developed to generate the table used in the product terms by interpreting the symbolic Boolean equations supplied by the user. The implementation and testing of the method are described.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Application of NEWEUL in robot dynamics

We consider the simulation of the dynamics of a complex robot system modeled as multibody system with rigid links and flexible drives. The symbolic equations, linearized with respect to a nominal motion, are generated by the program system NEWEUL. From these equations the dynamic behaviour of the system can be determined, using numeric standard procedures. The features of the program system as well as some problems occuring with the use of a symbolic formalism for the determination of equations of motion are discussed. The application of NEWEUL during the simulation of a robot system with complex dynamics leads to efficient program codes in the simulation program and guarantees high accuracy and stability during the numeric evaluation.

Solving symbolic equations with PRESS

We describe a program, PRESS, (PRolog Equation Solving System) for solving symbolic, transcendental, non-differential equations in one or more variables. PRESS solves autonomously, i.e. without guidance from the user. The methods used for solving equations are described, together with the service facilities. The principal technique, metal-level inference, appears to be relevant to the broader field of symbolic and algebraic manipulation.

Computer Architecture and Software Design of a Distributed Digital Integrating Machine

Computer integrating environments, based on differential equation of Shannon. have been successfully tested as a processing mechanism for the implementation of distributed homogeneous integrating networks in several fields of control system applications . The paper presents the mathematical models. the architecture. the microprogram design and the symbolic programming system of a distributed digital integrating machine. based on fast 16 - bit bipolar microprogrammed microprocessors. which homogeneous parallel integrating structures. After a short introduction in the mathematical theory the working power of the basic computing elements (DDA, servo-adder, comparator), the internal structure of the basic integrating modules, their microprogram design and the internal and external programmable commutation mechanism are explained. An expert system for programming of the machine, implemented in Prolog on IBM-PC, which determines by symbolic equation transformation the connections in the structure and the initial integrating values, is discussed.

Dolls and Bears: From Symbolic Equation to Symbol The Significance of Different Play Material for Sexually Abused Children and Others

SUMMARY. Finding that a significant number of children were unable to touch or play with the standard doll families and animals available, the therapist brought into the therapy room large and ordinary sized dolls and teddy bears (black, white, baby, teenaged). To her surprise these were quickly used to reveal and explore incidents of sexual and physical abuse. This led her to the conclusion that children who have experienced concrete attacks need different material to help them regain their capacity to symbolise. In this paper she focusses on the meaning of the new toys for two children in particular. She now includes a range of different‐sized dolls as standard equipment for all children and adolescents and comments on their applicability.

Commentary on: solving symbolic equations with PRESS

The paper "Solving Symbolic Equations with PRESS", by Sterling, Bundy, Byrd, O'Keefe and Silver [Sterling et al 82], describes a program PRESS (PROLOG Equation Solving System) which solves some symbolic, transcendental, non-differential equations. The problem of solving such equations is divided into two levels, a meta-level which is intended as a research vehicle for exploring search strategies in mathematical reasoning and a set of equation-solving modules used initially for the MECHO [Bundy et al 79] system (a project which aims to solve high-school mechanics problems stated in English.)

Symbolic computation and equation on the center manifold: Application to the Couette-Taylor problem

A particular case of the transition from the Couette flow to the Taylor cells is mathematically studied using the Center Manifold theorem. In this degenerate case, the stability of the Taylor vortex flow is calculated using fifth order terms in amplitude. We describe a method for computing these high order terms. Moreover, the numerical values are obtained with a Fortran program generated by a Macsyma program.

A Prolog-based expert system for modeling with partial differential equations

Initial results are presented from a project to develop intelligent expert system tools for the construction and execution of com plex mathematical models which employ partial differential equations. This paper reports on that portion of the system, the expert system front end, which is concerned with the represen tation and execution of meta-level knowledge about the model building process itself. The current problem domain is restricted to models involving linear as well as a limited class of nonlinear partial differential equations of up to second order. The system is capable of examining the user's input model for inconsisten cies as well as making judgments about suitable algorithms for stable numerical solutions. The symbolic equations, along with boundary and initial conditions, are represented in the form of Prolog clauses. The input model is analyzed by the expert system for equation type as well as appropriate boundary and initial conditions. Discrete approximation of equation terms is perform ed symbolically and the results are then expanded and simplified into an efficiently computable form. Prolog clauses from the discrete approximation phase are then automatically combined with an appropriate solution stepping scheme to arrive at a fully executable algorithm for numerical solution. The algorithm may then be executed serially for testing or further processed into a form suitable for passing to a separate high-speed processor.

A systematic formulation of dynamic equations for robot manipulators with elastic links

AbstractAlthough a variety of formulation schemes for the dynamic equations of robot manipulators with rigid links can be found in the literature, an efficient method of the formulation for robot manipulators with elastic links is not well known. Accordingly, this work presents the derivation of the equations of motion for application to mechanical manipulators with elastic links. The formulation is conducted analytically using Hamilton's principle. The resultant equations consist of the terms of inertial, Coriolis, centrifugal, gravitational, and exerted forces. They are expressed in terms of a set of independent generalized coordinates. In contrast to conventional variational approaches, the present method provides an efficient and systematic way for obtaining the compact symbolic equations of flexible manipulator systems. Two examples are presented to illustrate the proposed methodology. Firstly, a three‐link flexible manipulator with three revolute joints is studied. A flexible manipulator consisting of a prismatic joint and a discrete mass is the second model.

Mathematical modeling of complex mechanical systems

Mathematical modeling of mechanical systems based on multibody system models is a well tested approach. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom is difficult with paper and pencil. For this reason methods for automatic equation generation have been developed. Most methods result in numerical equations of motion without explicit information about the parameters. In this paper a method is described resulting in symbolic equations of motion. The method allows also the determination of the constraint forces which are important for design purposes. The inverse problem of dynamics is also easily solved.

Automatic Symbolic Modelling of Robots with a Deformable Link

In this paper we present a tool for the automatic generation of symbolic dynamic models of spatial mechanical chains with one elastic element, for the analysis of formal and structural properties of these models. The code computes the energy terms and the method of Lagrange is applied in order to get the symbolic differential equation which represents the model. Therefore, a few approximated models are suggested and the models are shown in the appendices for a simple case.

Computer generation of robot dynamics equations and the related issues

AbstractTwo programs have been developed using the computer algebra system REDUCE to generate the dynamics equations of motion for robot manipulators. One of these programs is based on a Lagrange formulation and the other utilizes a recursive Newton‐Euler formulation. Both programs produce equivalent scalar symbolic expressions for the generalized actuator forces, but the program based on the recursive Newton‐Euler formulation is more efficient for the generation of equations. These programs have been used to generate the dynamics equations of manipulators with as many as six degrees of freedom.The efficiency of computing forces using the generated scalar symbolic expressions is compared with the efficiency of a numerical algorithm (implemented in FORTRAN '77) based on the recursive Newton‐Euler formulation. Force computation by the method of symbolic equations is shown to be more efficient than the numerical recursive Newton‐Euler algorithm. The technique of symbolic equations is also better adapted to multi‐CPU processing.

The complete stoichiometer

The Complete Stoichiometer, a computer program used in solving stoichiometric problems in chemical, biochemical or metabolic systems, is described. The program contains general facilities for handling the stoichiometric and real matrices used in chemical network problems. The program contains a number of utility and application subprograms. The utility programs provide for input of matrices in both standard form and symbolic equation form, editing and combination of matrices; storing and retrieving stoichiometric matrices from the disk; printing of reports; a BlackBoard subprogram which allows stoichiometric matrices to be manipulated on the screen with a command-driven language. The application programs are: standard processing of network stoichiometry to obtain reaction trees and reaction loops, the independent driving potentials, and the conserved moieties; the standard processing of an atomic composition matrix to obtain the set of independent reactions interconnecting the reactants, and a possible moiety structure; factoring of an atomic composition matrix to obtain the structure of reactants in terms of moieties and of moieties in terms of atoms; generation of reduced equivalent networks when some reactions are in rapid equilibrium. and some reactants are in steady-state; a robust mass action chemical equilibrium solver.

Computationally Efficient Kinematics for Manipulators with Spherical Wrists Based on the Homogeneous Transformation Representation

Most manipulators in use today are kinematically simple, and closed-form symbolic equations are the most efficient means of expressing their kinematics. The analysis presented in this paper is based on the use of homogeneous transforma tions to describe position and orientation. Using these methods, kinematic equations are obtained directly in a form suitable for computer implementation. The equations are numerically stable and are obtained almost automatically. The resulting equations involve the minimum number of mathematical operations.

Computer analytics for thermalhydraulics

The development of an interactive software tool, EQNWRITE, to help in the setup and manipulation of engineering mass, momentum and energy equations, is discussed. This software is a building tool for building mathematical models of process systems. For a given problem, EQNWRITE enables the user to interactively derive and simplify the governing equations in the desired reference frame and coordinate system. At the user's request, specified mathematical transformations are performed automatically and resulting equations are displayed. EQNWRITE allows the user to better perceive the physical significance of each term in the symbolic mathematical equations, which describe the dominant properties of a system, as they are being derived and simplified, while leaving the detailed mathematical transformations to the computer. EQNWRITE was programmed in the symbol manipulation language, MUSIMP and developed using some of the capabilities of the microcomputer based computer algebra system, MUMATH. Major programming features and capabilities of MUSIMP and MUMATH are briefly reviewed, followed by a discussion of the specific automatic transformations implemented in EQNWRITE. Finally, engineering examples are given showing the use of the transformation functions and operators.

Computerized generation of symbolic equations of motion for spacecraft

Computerized generation of symbolic equations of motion for spacecraft

Symbolic Notation and Kinematic Equations of Motion of the Twenty-Two Basic Spur Planetary Gear Trains

This paper introduces a symbolic notation for describing two degree-of-freedom spur planetary gear trains with the configuration of two central gears, one carrier or arm, and at least one planet gear. This simple notation is then used to write the kinematic equation of motion for this type of planetary gear train without the need to develop any intermediate equations. It is also shown that, there are exactly 22 basic spur planetary gear trains with this configuration and not 34 as Levai [1] previously found. These 22 planetary gear trains are illustrated along with their symbolic notation and kinematic equation of motion.

Automated Systems Used for Rapid and Flexible Generation of Vehicle Simulation Models Exemplified by a Verified Passenger Car and a Motorcycle Model

This study treats two systems based on the Newton-Euler formalism which has been developed for building numerical or symbolic equations of motion. To illustrate the different functions and the model range, the procedure to be adopted in model construction is shown for an automobile and a motorcycle model. The models have been verified within detailed test programs by use of standard BMW vehicles which have been equipped with special measuring devices.

A novel symbolic ordinary differential equation solver

This paper describes a symbolic ordinary differential equation solver written as a package for the muMATH computer algebra system. The novel features of this solver are:1. Whereas previous ODE solvers generally require the equation to be quasi-linear and generally yield only implicit results, this new solver is highly integrated with an algebraic equation solver before, during and after solution. In fact, the same function named SOLVE is used for both algebraic and differential equations.2. By not insisting upon quasi-linear input, it is also practical to incorporate in the same solver methods for quasi-linear equations and a generalized version of Clairaut's equation, including singular envelope solutions. To my knowledge, previous ODE solvers have not treated even the ordinary Clairaut's equation, even using a separate solver.3. Since integration constants introduced at intermediate stages can become deeply embedded in a final explicit solution, the new solver automatically replaces subexpressions containing only an integration constant and numerical constants with a new constant, in order to greatly simplify the appearance of results, with no loss of generality.4. Whereas previous ODE solvers are generally restricted to first or second order (or to linear equations with constant coefficients) the new solver attempts to solve equations of any order, even if they are nonlinear and/or have variable coefficients.5. When the new solver is unable to obtain a closed-form solution, Picard iteration or a Taylor-series method is automatically employed so as to obtain a symbolic truncated series solution.muMATH does not yet provide the algebraic operations necessary to support the recently-discovered ODE decision procedures. Consequently the methods used to solve quasi-linear first and second order equations are essentially the same as those employed by other solvers of the pre-decision-procedure era.