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.

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.

Cracked Elastic Layer Under a Compressive Mechanical Load

2009 ◽  
Author(s):  
Vahagn Makaryan ◽  
Michael Sutton ◽  
Tatevik Yeghiazaryan ◽  
Davresh Hasanyan ◽  
Xiaomin Deng
Keyword(s):  
Elastic Layer ◽  
Integral Transformation ◽  
Mechanical Load ◽  
Physical Parameters ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
Penny Shaped Crack ◽  
Shaped Crack ◽  
Parameter Values

In the present work, the problem of an elastic layer weakened by a finite penny shaped crack parallel to a layer’s surface that is loaded in compression is considered. Assuming that the surfaces of the crack have frictional slipping contact, Henkel and Legendre integral transformation techniques are employed to formulate solutions in the form of an infinite system of linear algebraic equations. The regularity of the equations is established and closed-form solutions are obtained for stresses and strains. Assuming shear stress on the crack surfaces is linearly distributed, numerical results show both geometric and physical parameters have an essential influence on the stress distribution around the crack, with specific parameter values indicating the normal stress along the crack surface can change its sign from negative to positive. The implications of the work will be discussed.

Non-Linear Behavior of Spherical Shells under Static Ring Loads

2016 ◽  
Vol 835 ◽  
pp. 583-590
Author(s):  
E. Eylem Karataş ◽  
R. Faruk Yükseler
Keyword(s):  
Finite Differences ◽  
Circular Ring ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
Spherical Shells ◽  
Linear Behavior ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Newton Raphson ◽  
Raphson Method

The present study investigates the non-linear behavior of spherical shells under the influence of static circular ring loads. It is assumed that the material is isotropic and linearly elastic. The differential equations comprising the equilibrium equations, constitutive laws and kinematic equations are converted into non-linear algebraic equations by employing the method of finite differences. Respective non-linear algebraic equations are solved numerically by using the Newton–Raphson Method. The curves pertaining to the circular ring load versus the deflection at the application point of the ring load and the circular ring load versus the deflection at the apical point of the shell are plotted and compared for various shell radius/thickness ratios and parallel circle radii values.

scholarly journals Matrix Approach To The Direct Computation Method For The Solution of Fredholm Integro-Differential Equations of The Second Kind With Degenerate Kernels

CAUCHY ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 100-108
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Geoffrey Kumlengand ◽  
Joshua Sunday
Keyword(s):  
Differential Equations ◽  
Computation Method ◽  
Matrix Approach ◽  
Algebraic Equations ◽  
Direct Computation ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sum Of Products ◽  
Finite Sum

In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.

Simultaneous Wiener–Hopf equations

1980 ◽  
Vol 58 (3) ◽  
pp. 420-428 ◽  
Author(s):  
A. D. Rawlins
Keyword(s):  
Integral Equation ◽  
Half Plane ◽  
Fredholm Integral Equation ◽  
Coupled System ◽  
Algebraic Equations ◽  
Branch Points ◽  
Linear Algebraic Equations ◽  
Upper Half Plane ◽  
Infinite Set ◽  
Special Case

In Noble's book The Wiener Hopf Technique, Pergamon, 1958, he considers the coupled system of Wiener–Hopf equations (§4.4, pp. 153–154)[Formula: see text]He shows that provided the functions L(α), M(α), Q(α), and R(α) have only simple pole singularities the solution can be reduced to two sets of infinite simultaneous linear algebraic equations. In this article a different approach is used which gives the solution in the form of a Fredholm integral equation of the second kind. This Fredholm integral equation can be reduced to infinite sets of simultaneous linear algebraic equations under the less restrictive conditions that either (i) L(α)/M(α) has no branch points in the lower α-half plane: Im(α) < τ+; or (ii) Q(α)/R(α) has no branch points in the upper α-half plane: Im(α) > τ−. In the special case considered by Noble if L(α)/M(α) (or Q(α)/R(α)) only have simple poles in the lower (upper) half plane then the Fredholm integral equation reduces to one infinite set of simultaneous equations. This extends the Wiener–Hopf technique to yet a larger class of boundary value problems, and simplifies the numerical computations.

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.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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