Natural Frequencies of the Bias Tire

1976 ◽  
Vol 4 (2) ◽  
pp. 86-114 ◽  
Author(s):  
M. Hirano ◽  
T. Akasaka
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Composite Structure ◽  
Natural Frequencies ◽  
Variable Coefficients ◽  
System Of Differential Equations ◽  
Inflation Pressure ◽  
Point Collocation ◽  
Membrane Shell ◽  
Wide Range

Abstract The lowest natural frequencies of a bias tire under inflation pressure are deduced by assuming the bias tire as a composite structure of a bias-laminated, toroidal membrane shell and rigorously taking three displacement components into consideration. The point collocation method is used to solve a derived system of differential equations with variable coefficients. It is found that the lowest natural frequencies calculated for two kinds of bias tire agree well with the corresponding experimental results in a wide range of inflation pressure.

Free Vibration Analysis of an Inflated Toroidal Shell

2002 ◽  
Vol 124 (3) ◽  
pp. 387-396 ◽  
Author(s):  
Akhilesh K. Jha ◽  
Daniel J. Inman ◽  
Raymond H. Plaut
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Free Vibration Analysis ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Toroidal Shell

Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkin’s method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

scholarly journals A Study on the Complexity of a New Chaotic Financial System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Yi Liao ◽  
Yiran Zhou ◽  
Fei Xu ◽  
Xiao-Bao Shu
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Financial System ◽  
System Of Differential Equations ◽  
Bifurcation Diagrams ◽  
Complex Dynamic ◽  
System Parameters ◽  
Wide Range

The interaction of elements in a financial system can exhibit complex dynamic behaviours. In this article, we use a system of differential equations to model the evolution of a financial system and study its complexity. Numerical simulations show that the system exhibits a variety of rich dynamic behaviours, including chaos. Bifurcation diagrams show that the system behaves chaotically over a wide range of system parameters.

The successive differentiation computer-assisted method for solving well-known scientific and engineering models

2018 ◽  
Vol 28 (12) ◽  
pp. 2862-2873
Author(s):  
Suheil Khuri ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Problem Solving ◽  
Differential Equations ◽  
Undergraduate Students ◽  
Variable Coefficients ◽  
Abel Equation ◽  
Computer Assisted ◽  
Computational Tool ◽  
Content Type ◽  
Wide Range ◽  
Linear And Nonlinear

PurposeOrdinary differential equations (ODEs) are widely used in the engineering curriculum. They model a spectrum of interesting physical problems that arise in engineering disciplines. Studies of different types of ODEs are determined by engineering applications. Various techniques are used to solve practical differential equations problems. This paper aims to present a computational tool or a computer-assisted technique aimed at tackling ODEs. This method is usually not taught and/or not accessible to undergraduate students. The aim of this strategy is to help the readers to develop an effective and relatively novel problem-solving skill. Because of the drudgery of hand computations involved, the method requires the need to use computers packages. In this work, the successive differentiation method (SDM) for solving linear and nonlinear and homogeneous or non-homogeneous ODEs is presented. The algorithm uses the successive differentiation of any given ODE to determine the values of the function’s derivatives at a single point, mostly x = 0. The obtained values are used to construct the Taylor series of the solution of the examined ODE. The algorithm does not require any new assumption, hence handles the problem in a direct manner. The power of the method is emphasized by testing a variety of models with distinct orders, with constant and variable coefficients. Most of the symbolic and numerical computations can be carried out using computer algebra systems.Design/methodology/approachThis study presents a computational tool or a computer-assisted technique aimed at tackling ODEs. This method is usually not taught and/or not accessible to undergraduate students. The aim of this strategy is to help the readers to develop an effective and relatively novel problem-solving skill. Because of the drudgery of hand computations involved, the method requires the need to use computers packages.FindingsThis method is applied to a variety of well-known equations, such as the Bernoulli equation, the Riccati equation, the Abel equation and the second-order Euler equation, some with constant and variable coefficients. SDM handles linear and nonlinear and homogeneous or nonhomogeneous ODEs in a direct manner without any need to restrictive conditions. The method works effectively to the Volterra integral equations, as will be discussed in a coming work.Originality/valueThe method can be extended to a wide range of engineering problems that are modeled by differential equations. The method is simple and novel and highly accurate.

Solving systems of high-order linear ordinary differential equations with variable coefficients by means of exponential Chebyshev collocation method

2017 ◽  
Vol 8 (1-2) ◽  
pp. 40 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Variable Coefficients ◽  
High Order ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Linear Algebraic Equations

The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.

scholarly journals An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi ◽  
Eman S. Alaidarous
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Jacobi Polynomials ◽  
Nonlocal Conditions ◽  
Initial Boundary ◽  
Variable Coefficients ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Hyperbolic Pdes

One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

The course of consecutive reactions inside a nonisotropic catalyst particle, affected by internal difusion

1986 ◽  
Vol 51 (1) ◽  
pp. 54-65
Author(s):  
Jiří Hanyka ◽  
Alena Fialová
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Catalyst Surface ◽  
Spherical Geometry ◽  
Catalyst Particle ◽  
Diffusion Equations ◽  
System Of Differential Equations ◽  
Reaction Selectivity ◽  
Consecutive Reactions ◽  
Reaction Orders

A system of differential equations for consecutive reactions inside a nonisotropic catalyst particle under conditions of internal diffusion is solved. The system of diffusion equations for the spherical geometry of the catalyst grain is numerically solved by using the collocation method. The solution is sought for various radial activity profiles across the catalyst particle and for various values of Thiele's modulus for the two consecutive reactions. The effect of the reaction orders with respect to the reactants on the degree of utilization of the internal catalyst surface and on the reaction selectivity is examined.

A new exponential Chebyshev operational matrix of derivatives for solving high-order ordinary differential equations in unbounded domains

2016 ◽  
Vol 7 (1) ◽  
pp. 19 ◽  
Author(s):  
Mohamed Ramadan ◽  
Kamal Raslan ◽  
Talaat El Danaf ◽  
Mohamed A. Abd Elsalam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Operational Matrix ◽  
Variable Coefficients ◽  
High Order ◽  
Algebraic Equations ◽  
Linear Ordinary Differential Equations ◽  
The Given ◽  
Derivatives Of

The purpose of this paper is to investigate a new exponential Chebyshev (EC) operational matrix of derivatives. The new operational matrix of derivatives of the EC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. This method transforms the given differential equation and conditions to matrix equation with unknown EC coefficients. These matrices together with the collocation method are utilized to reduce the solution of high-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of EC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive with good accuracy.

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