Appendix 2: Kirchhoff's Integral Theorem

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

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On the Significance of the Inlet Pressure Build-Up in the Design of Tilting-Pad Bearings

Journal of Tribology ◽

10.1115/1.2920224 ◽

1990 ◽

Vol 112 (1) ◽

pp. 17-22 ◽

Cited By ~ 20

Author(s):

Cz. M. Rodkiewicz ◽

K. W. Kim ◽

J. S. Kennedy

Keyword(s):

Thrust Bearing ◽

Ambient Pressure ◽

Inlet Pressure ◽

Load Carrying Capacity ◽

Integral Theorem ◽

Lubricating Film ◽

Tilting Pad ◽

Load Carrying ◽

Momentum Integral ◽

Entrance Pressure

An operating tilting-pad thrust bearing generates a fore-region which is responsible for maintaining, at the bearing entrance, a pressure which is higher than the ambient pressure. This entrance pressure, in the presented analysis, is obtained by applying to the fore-region the momentum integral theorem. The solution of the lubricating film region is then obtained by using this modified inlet pressure. This solution yields the pressure distribution, the load carrying capacity, the film ratio and the frictional force for several values of the modified Reynolds number and various pivot positions. The analysis shows that there is a significant influence of the fore-region pressure on the bearing performance and that to properly design efficient tilting-pad bearing this effect should be taken into consideration.

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An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

International Journal of Engineering Mathematics ◽

10.1155/2015/523043 ◽

2015 ◽

Vol 2015 ◽

pp. 1-4 ◽

Cited By ~ 4

Author(s):

Rogelio Luck ◽

Gregory J. Zdaniuk ◽

Heejin Cho

Keyword(s):

Null Space ◽

Heat Conduction Problem ◽

Hankel Matrix ◽

Polynomial Equation ◽

Transient Heat Conduction ◽

Transcendental Equation ◽

Integral Theorem ◽

Transient Heat Conduction Problem ◽

Cauchy’S Integral Theorem ◽

Roots Of A Polynomial

This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

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