The general equation of the stigmatic lenses: its history and what we have learned from it

2021 ◽  
Author(s):  
Rafeal G. Gonzalez-Acuna ◽  
Simon Thibault
Keyword(s):  
General Equation

Modelling Electromigration and Induced Stresses In Aluminum Lines

1993 ◽  
Vol 309 ◽  
Author(s):  
R. Kirchheim
Keyword(s):  
Beneficial Effect ◽  
General Equation ◽  
Concentration Profiles ◽  
Simple Explanation ◽  
Vacancy Model ◽  
Induced Stresses

AbstractThe various equations used by different authors for describing electromigration are compiled. All of them can be derived from a general equation which is based on a vacancy model. During the derivation of simplified versions of the general equations assumptions have to be made and their effect on concentration profiles and/or electromigration stresses developing as a function of time are discussed. Consequences with respect to Blech's experiment or the current exponent in Black's equation are taken into consideration and a simple explanation is provided for the beneficial effect of a (111) texture on the reliability of Al-lines.

Line Contact Deformation: A Cylinder Between Two Flat Plates

1981 ◽  
Vol 103 (1) ◽  
pp. 21-25 ◽  
Author(s):  
M. R. Hoeprich ◽  
H. Zantopulos
Keyword(s):  
Plane Strain ◽  
General Equation ◽  
Roller Bearing ◽  
Experimental Results ◽  
Line Contact ◽  
Contact Deformation ◽  
Flat Plates ◽  
Special Cases ◽  
Analytical Work

Various line contact deformation equations used in roller bearing technology are analyzed. Many of these deformation equations, primarily involving plane strain, are shown to be special cases of a general equation derived in this paper. Experimental results are also presented to support the results of the analytical work.

scholarly journals The Probability That a Measurement Falls within a Range of Standard Deviations from an Estimate of the Mean

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Louis M. Houston
Keyword(s):  
Confidence Interval ◽  
Sample Size ◽  
General Equation ◽  
Sample Sizes ◽  
The Mean ◽  
Standard Deviations ◽  
Intermediate Value ◽  
Theoretical Results

We derive a general equation for the probability that a measurement falls within a range of n standard deviations from an estimate of the mean. So, we provide a format that is compatible with a confidence interval centered about the mean that is naturally independent of the sample size. The equation is derived by interpolating theoretical results for extreme sample sizes. The intermediate value of the equation is confirmed with a computational test.

Upper Ball Joint Force Variations due to Riser Tensioner and Vessel Motions—Part II: Analysis and Computer Simulation

1990 ◽  
Vol 112 (3) ◽  
pp. 200-207 ◽  
Author(s):  
T. J. Kozik ◽  
J. E. Lowell ◽  
A. Ertas
Keyword(s):  
Computer Simulation ◽  
General Equation ◽  
Ship Motion ◽  
Joint Motion ◽  
Ball Joint ◽  
Joint Force ◽  
Field Operation ◽  
Breakaway Torque ◽  
Cable Forces

An analysis of the variation of forces acting on the upper ball joint of a riser string due to the drill ship motion and riser tensioner dynamic has been conducted. The analysis includes the effect of breakaway torque on the tensioner sheaves while assuming vessel and upper ball joint motion to be independent. General equation for the tensioner cable forces and for the forces exerted on the riser upper ball joint by the ship joint-tensioner system derived in Part I are solved. The variation in the tensioner cable forces is compared to data generated in field operation.

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