On Axially Symmetric Bending of Nearly Cylindrical Shells of Revolution

1956 ◽  
Vol 23 (1) ◽  
pp. 59-67
Author(s):  
R. A. Clark ◽  
E. Reissner
Keyword(s):  
Elastic Shell ◽  
Approximate Solutions ◽  
Rate Of Change ◽  
Axially Symmetric ◽  
Shells Of Revolution ◽  
Small Parameters ◽  
Constant Coefficients ◽  
Radial Dimension ◽  
Axial Dimension ◽  
Linear Differential

Abstract The words “nearly cylindrical” are used in this paper to describe a thin elastic shell of revolution which is such that (a) the maximum variation of the radial dimension is small compared to the average radial dimension, and (b) the rate of change of the radial dimension with respect to the axial dimension is small compared to unity. For any particular type of loading a nearly cylindrical shell may or may not exhibit a behavior similar to that of a shell which is exactly cylindrical. The purpose of this paper is to demonstrate this fact and to present a method for obtaining approximate solutions for the stresses and deflections in either event. The method involves a perturbation procedure based on the assumption that all desired quantities can be represented as expansions in powers of two small parameters. The procedure leads to a set of linear differential equations with constant coefficients, which may be solved successively.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

Comparisons of Simulation Methods for Motions of a Moored Body in Waves

1985 ◽  
Vol 107 (1) ◽  
pp. 34-41
Author(s):  
M. Takagi ◽  
K. Saito ◽  
S. Nakamura
Keyword(s):  
Wave Theory ◽  
Simulation Methods ◽  
Convolution Integral ◽  
Linear Water Wave Theory ◽  
Initial Stage ◽  
Constant Coefficients ◽  
Exciting Forces ◽  
Linear Differential ◽  
Experimental Values ◽  
The Time Domain

Based on the linear water wave theory, numerical simulations are carried out for motions in waves of a body moored by a nonlinear-type mooring system. Numerical results obtained by using the equation of motion described in the time domain with a convolution integral (C.I. method) are compared with those of the second-order linear differential equation with constant coefficients (C. C. method). These results are also compared with experimental values measured from the initial stage when the action of exciting forces starts and the validity of C.I. method is discussed.

scholarly journals A graphical solution of a differential equation with application to hill’s treatment of nerve excitation

1937 ◽  
Vol 123 (832) ◽  
pp. 382-395 ◽  
Keyword(s):  
Viscous Medium ◽  
Graphical Solution ◽  
Initial Value ◽  
First Order ◽  
Rate Dependent ◽  
Monomolecular Reaction ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Physical And Chemical ◽  
Nerve Excitation

Linear differential equations with constant coefficients are very common in physical and chemical science, and of these, the simplest and most frequently met is the first-order equation a dy / dt + y = f(t) , (1) where a is a constant, and f(t) a single-valued function of t . The equation signifies that the quantity y is removed at a rate proportional to the amount present at each instant, and is simultaneously restored at a rate dependent only upon the instant in question. Familiar examples of this equation are the charging of a condenser, the course of a monomolecular reaction, the movement of a light body in a viscous medium, etc. The solution of this equation is easily shown to be y = e - t / a { y 0 = 1 / a ∫ t 0 e t /a f(t) dt , (2) where y 0 is the initial value of y . In the case where f(t) = 0, this reduces to the well-known exponential decay of y .

scholarly journals Transient Analysis of Thermal Treatment Processes Carburizing and Nitriding of Steel Components by Numerical Modeling and Simulation

2019 ◽  
Vol 9 (2) ◽  
pp. 2584-2593
Keyword(s):  
Differential Equation ◽  
Numerical Modeling ◽  
Ambient Temperature ◽  
Modeling And Simulation ◽  
Geometric Dimension ◽  
Rate Of Change ◽  
Critical Parameters ◽  
Numerical Modeling And Simulation ◽  
Linear Differential ◽  
Energy Balance Approach

The purpose of carburizing, nitriding and carbonitriding is to increase the strength of components. Elements such as carbon, nitrogen and carbon-nitride are diffused into the components at high temperature convective environment. The amount of diffusion is to be regulated by controlling the temperature and time of diffusion. Time and temperature of process govern diffusion rate and strength of the component. Numerical modeling is applied by energy balance approach i.e., equating rate of change of energy is equal to energy transferred by conduction, convection and radiation. By non dimensionalising relations for the mentioned critical parameters were obtained. The phenomenon of convection, radiation and conduction are taken together for the purpose of numerical modeling. Variation of temperature and depth of diffusion of component for the taken components i.e., sphere and cube was plotted in transient state. For both numerical analysis and simulation the boundary conditions i.e., for carburization the ambient temperature is 9500C with carbon monoxide as the carburizing agent and for nitriding the ambient temperature is 5300C with nitrogen as nitriding agent and the component taken is of steel which is initially at room temperature were taken. Results obtained from numerical modeling and simulation were compared with each other and observed that in both analyses the variation of temperature with time and depth of diffusion is almost linear. Final differential equation obtained in numerical modeling is a single order non linear differential equation which is solved in MATLAB using finite difference approach. Data obtained from MATLAB were plotted for variation of surface temperature and geometric dimension with respect to time.

scholarly journals A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 275-283
Author(s):  
Kubra Bicer ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Mean Value ◽  
Computational Method ◽  
Mean Value Theorem ◽  
Residual Functions ◽  
Non Linear ◽  
Linear Differential

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

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