A General Method for the Fatigue-Resistant Design of Mechanical Components—Part 2: Analytical

1975 ◽  
Vol 97 (3) ◽  
pp. 970-975
Author(s):  
D. T. Vaughan ◽  
L. D. Mitchell
Keyword(s):  
Analytical Solution ◽  
Fatigue Fracture ◽  
Fracture Criterion ◽  
Fatigue Loading ◽  
General Analytical Solution ◽  
Failure Theory ◽  
Straight Line ◽  
Mechanical Components ◽  
General Method

This paper develops the general analytical solution to the design of mechanical components under fatigue loading. Its only limitation is that the overloading lines must be a straight line on the σa−σm diagram. The designer is free to select his own failure theory for the material he intends to use as well as to select his own fatigue fracture criterion.

The Effect of Biot Number on a One-Dimensional General Conduction Solution

2021 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Biot Number ◽  
Analytical Solutions ◽  
Heat Fluxes ◽  
Graphical Form ◽  
Large Temperature ◽  
Convection Coefficient ◽  
General Analytical Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tablular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

scholarly journals On the theory of the elliptic transcendents

1837 ◽  
Vol 3 ◽  
pp. 60-60
Keyword(s):  
Elliptic Function ◽  
Elliptic Integral ◽  
Elliptic Functions ◽  
Algebraic Expression ◽  
Algebraic Equations ◽  
Similar Function ◽  
Circular Arcs ◽  
Straight Line ◽  
Regular Theory ◽  
General Method

Fagnani discovered that the two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and proved that any arc of a lemniscate, like that of a circle, may be multiplified any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. What he had accomplished with respect to the arcs of the lemniscates, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. Landen showed that the arcs of the hyperbola may be reduced, by a proper transformation, to those of an ellipse. Lagrange furnished us with a general method for changing an elliptic function into another having a different modulus; a process which greatly facilitates the numerical calculation of this class of integrals. Legendre distributed the elliptic functions into distinct classes, and reduced them to a regular theory, developing many of their properties which were before unknown, and introducing many important additions and improvements in the theory. Mr. Abel of Christiana happity conceived the idea of expressing the amplitude of an elliptic function in terms of the function itself, which led to the discovery of many new and useful properties. Mr. Jacobi proved, by a different method, that an elliptic function may be transformed in innumerable ways into another similar function, to which it bears constantly the same proportion. But his demonstrations require long and complicated calculations; and the train of deductions he pursues does not lead naturally to the truths which are proved, nor does it present in a connected view all the conclusions which the theory embraces. The author of the present paper gives a comprehensive view of the theory in its full extent, and deduces all the connected truths from the same principle. He finds that the sines or cosines of the amplitudes, used in the transformations, are analogous to the sines or cosines of two circular arcs, one of which is a multiple of the other; so that the former quantities are changed into the latter when the modulus is supposed to vanish in the algebraic expression. Hence he is enabled to transfer to the elliptic transcendents the same methods of investigation that succeed in the circle: a procedure which renders the demonstrations considerably shorter, and which removes most of the difficulties, in consequence of the close analogy that subsists between the two cases.

General analytical solution for photoacoustic effect with multilayers

2008 ◽  
Vol 92 (1) ◽  
pp. 014103 ◽  
Author(s):  
Hanping Hu ◽  
Wei Zhang ◽  
Jun Xu ◽  
Yi Dong
Keyword(s):  
Analytical Solution ◽  
Photoacoustic Effect ◽  
General Analytical Solution

A General Analytical Solution for the Concentration Profiles of Ion-Exchanged Planar Waveguides

1991 ◽  
Vol 244 ◽  
Author(s):  
Xiaoming Li ◽  
Paul F. Johnson
Keyword(s):  
Ion Exchange ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Surface Concentration ◽  
Concentration Profiles ◽  
Concentration Boundary ◽  
Glass Waveguides ◽  
General Analytical Solution ◽  
Exchange Processes ◽  
One Step

ABSTRACTDuring the recent years, a great variety of ion-exchange processes, including one-step or two-step electric field assisted ion-exchange processes, have been developed to fabricate different kinds of passive planar glass waveguides, e.g., surface waveguides, which correspond to surface maximum concentration, or buried waveguides, which correspond to inside maximum concentration [1,2,3]. Theoretical calculation of ionic concentration distribution has been of interest since refractive index is generally a linear function of concentration. A general analytical solution to calculate both surface and buried concentration distributions from different ion-exchange processes, however, has not yet been presented. In addition, traditional ion-exchange has been carried out only with constant surface concentration boundary conditions. Little attention has been paid, either experimentally or theoretically, to ion-exchange processes with variable boundary conditions. In fact, the time-dependent surface concentration is experimentally observed for the ion-exchange of GRIN glass in molten salt bath [4]. Very recently, a novel one-step technique [5,6] involving electric field assisted ion-exchange of Na+ in glass by Ag+ from molten AgNO3 bath with decaying silver concentration has been developed to produce buried Ag+ concentration profiles in glass. As the accurate and reproducible processes are very important for fabricating ion-exchanged glass waveguides, theoretical modeling and analysis on the new process are needed.In this paper, the one-dimensional field-assisted linear diffusion equation has been analytically solved by Laplace transformation to theoretically calculate concentration profiles produced by field enhanced ion-exchange process with exponentially decaying surface concentration boundary conditions. The applications of the solution to a variety of ion-exchange processes with different boundary or processing conditions for optical waveguide fabrication have been discussed. The theoretical results prove that the solution is a general analytical solution which can be used to calculate either surface concentration profiles or buried concentration profiles.

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