Approximate Solutions for Symmetrically Loaded Thick-Walled Cylinders

1947 ◽  
Vol 14 (4) ◽  
pp. A301-A311
Author(s):  
C. W. MacGregor ◽  
L. F. Coffin
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Approximate Solutions ◽  
Fourier Integral ◽  
Integral Methods ◽  
Pressure Distributions ◽  
Internal Strains ◽  
Walled Cylinder ◽  
Simple Approximate Solution ◽  
Good Agreement

Abstract Based upon an extension of the theory of a bar on an elastic foundation, a simple approximate solution is given in closed form for the analysis of the stresses and strains in a thick-walled cylinder loaded either internally or externally by an axially symmetrical system of forces. The analysis avoids the tedious computation of stresses inherent in exact solutions of this problem by the Fourier series or Fourier integral methods and is in a form which can easily be used by designers. The approximate solution for both semi-infinite pressure distributions and shorter bands of internal pressure are compared with the mathematically exact solutions and with experiment. Good agreement is found in all cases for external strains, while for internal strains the agreement is good except very close to the discontinuity in pressure. Since it is doubtful in practice that an abrupt discontinuity in pressure is often realized in such cases, the approximate solution may also be useful near this discontinuity. More important, however, is the fact that the effective stresses (based upon the distortion-energy theory of yielding), as determined both by the exact and approximate solutions, are in close agreement.

scholarly journals A solution method for integro-differential equations of conformable fractional derivative

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 7-14 ◽  
Author(s):  
Mustafa Bayram ◽  
Veysel Hatipoglu ◽  
Sertan Alkan ◽  
Sebahat Das
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Solution Method ◽  
Approximate Solutions ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Numerical Examples

The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.

Calculated Pressure Distributions and Shock Shapes on Thick Conical Wings at High Supersonic Speeds

1967 ◽  
Vol 18 (2) ◽  
pp. 185-206 ◽  
Author(s):  
L. C. Squire
Keyword(s):  
Integral Equation ◽  
Cross Sections ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Poor Agreement ◽  
Delta Wings ◽  
Newtonian Theory ◽  
Pressure Distributions ◽  
First Order Correction ◽  
Good Agreement

SummaryIn recent papers Messiter and Hida have proposed a first-order correction to simple Newtonian theory for the pressure distributions on the lower surfaces of lifting conical bodies with detached shocks. The method involves the solution of an integral equation which Messiter solved numerically for thin delta wings, while Hida gave an approximate solution for thick wings with diamond and bi-convex cross-sections. It is shown in the present paper that Hida’s approximate solutions give poor agreement with experiment, and a series of more precise numerical solutions of the equation are given for wings with diamond cross-sections. The pressures, and shock shapes, obtained from these solutions are in very good agreement with experiment at Mach numbers as low as 4·0.The method has also been extended to Nonweiler wings at off-design when the shock wave is detached from the leading edges. Again the agreement with experiment is good provided the integral equation is solved numerically.

Calculated Pressure Distributions and Shock Shapes on Conical Wings with Attached Shock Waves

1968 ◽  
Vol 19 (1) ◽  
pp. 31-50 ◽  
Author(s):  
L. C. Squire
Keyword(s):  
Shock Waves ◽  
Exact Solutions ◽  
Pressure Distribution ◽  
Newtonian Theory ◽  
First Order ◽  
Pressure Distributions ◽  
Present Solution ◽  
Design Behaviour ◽  
First Order Correction ◽  
Good Agreement

SummaryRecently Messiter has proposed a first-order correction to simple Newtonian theory for the pressure distribution on the lower (compression) surfaces of lifting conical bodies. Although the basic theory holds for bodies with and without attached shock waves, solutions have so far only been obtained for bodies with detached shocks. In the present paper an approximate method of applying the theory to bodies with attached shocks is given. In spite of the approximations involved the calculated shock shapes and pressure distributions are in good agreement with some exact solutions for flat wings, except near the incidence for shock detachment. Like the detached shock case, the present solution can be applied to Nonweiler wings in certain off-design conditions. The combined results for the detached shock and for the attached shock enable the off-design behaviour of Nonweiler wings to be discussed in a systematic manner.

Solution of a diffraction problem - Solution of a diffraction problem. II. The narrow double wedge

1959 ◽  
Vol 252 (1003) ◽  
pp. 31-51 ◽  
Keyword(s):  
Approximate Solution ◽  
Asymptotic Behaviour ◽  
Diffraction Problem ◽  
Approximate Solutions ◽  
Problem Solution ◽  
Evanescent Mode ◽  
Double Wedge ◽  
Resonance Effects ◽  
Reflexion Coefficient ◽  
Good Agreement

The problem of diffraction by a ‘narrow double wedge’ (width much smaller than wavelength) is investigated. Strong reflexion and quasi-static effects are the main features of this problem. The asymptotic behaviour of the solution is determined by the edge singularities. This leads to an approximate solution, which seems to be very accurate. This solution is found to be in good agreement with approximate solutions derived by different methods. The reflexion coefficient and the ‘end correction’ are evaluated. The results are compared with those obtained by other authors. It is shown that they contain a new effect, the 'evanescent mode correction’, which is very small in this region. Resonance effects in channels of finite length are analyzed.

scholarly journals On Fuzzy differential equation

2019 ◽  
Vol 11 (2) ◽  
pp. 1-9
Author(s):  
Eman Ali Hussain ◽  
Yahya Mourad Abdul – Abbass
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Exact Solutions ◽  
Turing Machine ◽  
Hybrid Method ◽  
Hybrid Methods ◽  
Fitness Function ◽  
Approximate Solutions ◽  
Fuzzy Differential Equation

In this paper, we introduce a hybrid method to use fuzzy differential equation, and Genetic Turing Machine developed for solving nth order fuzzy differential equation under Seikkala differentiability concept [14]. The Errors between the exact solutions and the approximate solutions were computed by fitness function and the Genetic Turing Machine results are obtained. After comparing the approximate solution obtained by the GTM method with approximate to the exact solution, the approximate results by Genetic Turing Machine demonstrate the efficiency of hybrid methods for solving fuzzy differential equations (FDE).

Computer Modeling of a Tire under Cornering Loads

1989 ◽  
Vol 17 (2) ◽  
pp. 86-99 ◽  
Author(s):  
I. Gardner ◽  
M. Theves
Keyword(s):  
Finite Element Analysis ◽  
Geometric Approach ◽  
Lateral Force ◽  
Slip Angle ◽  
Element Analysis ◽  
Pressure Distributions ◽  
Slip Effects ◽  
Improved Accuracy ◽  
Nonlinear Geometric ◽  
Good Agreement

Abstract During a cornering maneuver by a vehicle, high forces are exerted on the tire's footprint and in the contact zone between the tire and the rim. To optimize the design of these components, a method is presented whereby the forces at the tire-rim interface and between the tire and roadway may be predicted using finite element analysis. The cornering tire is modeled quasi-statically using a nonlinear geometric approach, with a lateral force and a slip angle applied to the spindle of the wheel to simulate the cornering loads. These values were obtained experimentally from a force and moment machine. This procedure avoids the need for a costly dynamic analysis. Good agreement was obtained with experimental results for self-aligning torque, giving confidence in the results obtained in the tire footprint and at the rim. The model allows prediction of the geometry and of the pressure distributions in the footprint, since friction and slip effects in this area were considered. The model lends itself to further refinement for improved accuracy and additional applications.

Fourier Integral Methods of Pattern Analysis

1946 ◽  
Author(s):  
MASSACHUSETTS INST OF TECH CAMBRIDGE
Keyword(s):  
Pattern Analysis ◽  
Fourier Integral ◽  
Integral Methods

A new algorithm for finding CRM-model coefficients

2019 ◽  
Vol 5 (3) ◽  
pp. 164-185 ◽  
Author(s):  
Alexander D. Bekman ◽  
Sergey V. Stepanov ◽  
Alexander A. Ruchkin ◽  
Dmitry V. Zelenin
Keyword(s):  
Approximate Solution ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Complex Form ◽  
Approximate Solutions ◽  
The Other ◽  
Programming Algorithm ◽  
Stochastic Algorithms ◽  
Large Set ◽  
Flow Rates

The quantitative evaluation of producer and injector well interference based on well operation data (profiles of flow rates/injectivities and bottomhole/reservoir pressures) with the help of CRM (Capacitance-Resistive Models) is an optimization problem with large set of variables and constraints. The analytical solution cannot be found because of the complex form of the objective function for this problem. Attempts to find the solution with stochastic algorithms take unacceptable time and the result may be far from the optimal solution. Besides, the use of universal (commercial) optimizers hides the details of step by step solution from the user, for example&nbsp;— the ambiguity of the solution as the result of data inaccuracy.<br> The present article concerns two variants of CRM problem. The authors present a new algorithm of solving the problems with the help of “General Quadratic Programming Algorithm”. The main advantage of the new algorithm is the greater performance in comparison with the other known algorithms. Its other advantage is the possibility of an ambiguity analysis. This article studies the conditions which guarantee that the first variant of problem has a unique solution, which can be found with the presented algorithm. Another algorithm for finding the approximate solution for the second variant of the problem is also considered. The method of visualization of approximate solutions set is presented. The results of experiments comparing the new algorithm with some previously known are given.

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

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