Transient Response of Distributed Parameter Systems

1997 ◽  
Author(s):  
Jianping Zhou ◽  
Zhigang Feng
Keyword(s):  
Differential Equations ◽  
Transient Response ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
One Dimensional ◽  
Linear Algebraic Equations ◽  
Global Equilibrium ◽  
Distributed Transfer Function

Abstract A semi-analytic method is presented for the analysis of transient response of distributed parameter systems which are consist of one dimensional subsystems. The system is first divided into one dimensional sub-systems. Within each subsystem, replacing differentials on time t by finite difference, the governing partial differential equations are reduced to difference-differential equations. The solution of derived ordinary differential equations is obtained in an exact and closed form by distributed transfer function method and represented in nodal displacement parameters. Assemling global equilibrium equations at each nodes according to displacement continuity and force equilibrium requirements gives simutaneous linear algebraic equations. Numerical results are illustrated and compared with that of finite element method.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Differential Algebraic Equations ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Generalized Coordinates ◽  
Axial Forces ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

scholarly journals A new method for solving a class of heat conduction equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1205-1210
Author(s):  
Yi Tian ◽  
Zai-Zai Yan ◽  
Zhi-Min Hong
Keyword(s):  
Monte Carlo ◽  
Heat Conduction ◽  
Dimensional Space ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Governing Equations ◽  
One Dimensional ◽  
Sparse System ◽  
Linear Algebraic Equations ◽  
Crank Nicolson

A numerical method for solving a class of heat conduction equations with variable coefficients in one dimensional space is demonstrated. This method combines the Crank-Nicolson and Monte Carlo methods. Using Crank-Nicolson method, the governing equations are discretized into a large sparse system of linear algebraic equations, which are solved by Monte Carlo method. To illustrate the usefulness of this technique, we apply it to two problems. Numerical results show the performance of the present work.

scholarly journals On the continuous analogue of the Seidel method

2018 ◽  
Vol 20 (4) ◽  
pp. 364-377
Author(s):  
I. V. Boikov ◽  
A. I. Boikova
Keyword(s):  
Differential Equations ◽  
Systems Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Seidel Method ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Continuous Analogue ◽  
Solution Methods ◽  
Linear And Nonlinear ◽  
Equations With Delay

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

The Stress Response of Radially Polarized Rotating Piezoelectric Cylinders

2003 ◽  
Vol 70 (3) ◽  
pp. 426-435 ◽  
Author(s):  
D. Galic ◽  
C. O. Horgan
Keyword(s):  
Analytic Solution ◽  
Coupled System ◽  
Special Problem ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
Constant Potential ◽  
Radially Polarized ◽  
Fundamental Boundary

Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder rotating about its axis at constant angular velocity. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distribution in the tube is obtained numerically for a specific piezoceramic of technological interest, namely PZT-4. For the special problem of a uniformly rotating solid cylinder with traction-free surface and zero applied electric charge, explicit closed-form solutions are obtained. It is shown that for certain piezoelectric solids, stress singularities at the origin can occur analogous to those occurring in the purely mechanical problem for radially orthotropic elastic materials.

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

Sign in / Sign up

Export Citation Format

Share Document