Calculation of Discharge Current Waveforms in High Voltage Spark Sources. II. Extensions and Limitations of Closed-Form Solution

Simulation of gap breakdown and dynamic impedance effects in high voltage spark sources is performed using an algebraically exact solution to an approximate model of source behavior. The importance of diode shunt capacitance in determining gap breakdown behavior is shown. Limitations in generality and implicit use of numerical methods in dynamic situations lead naturally to consideration of numerical integration methods. Comparisons to hardware sources are made.

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The Axially Loaded Circular Cylinder

Journal of Applied Mechanics ◽

10.1115/1.3640548 ◽

1962 ◽

Vol 29 (2) ◽

pp. 318-320

Author(s):

H. D. Conway

Keyword(s):

Exact Solution ◽

Fourier Transform ◽

Circular Cylinder ◽

Closed Form ◽

Cross Sections ◽

Closed Form Solution ◽

Form Solution ◽

Concentrated Force ◽

Transform Method ◽

Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

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Exact solution for time exponentially varying pulsed laser heating: Convective boundary condition case

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/0954406011520977 ◽

2001 ◽

Vol 215 (5) ◽

pp. 591-606 ◽

Cited By ~ 4

Author(s):

M Kalyon ◽

B S Yilbas

Keyword(s):

Boundary Condition ◽

Exact Solution ◽

Closed Form ◽

Temperature Rise ◽

Laser Heating ◽

Closed Form Solution ◽

Convective Boundary Condition ◽

Form Solution ◽

Heating Process ◽

Convective Boundary

Laser heating offers considerable advantages over conventional methods. The closed-form solution for the temperature rise in the substrate during the laser heating process gives insight into the physical phenomena involving during the heating process and the material response to a laser heating pulse. In the present study, the exact solution for the temperature rise due to a time exponentially varying pulse and convective boundary condition at the surface is obtained. The closed-form solution to the solutions available in the literature for a step input intensity pulse with a convective boundary condition at the surface as well as a time exponentially varying pulse with a non-convective boundary condition at the surface is deduced. A Laplace transformation method is used in the analysis. In order to account for a pulse resembling a typical laser pulse, an intensity function resulting in exponentially increasing and decaying intensity distribution is employed in the source term in the governing transport equation. The effects of the pulse parameters β′, β′/γ′ and Biot number Bi on the resulting temperature profiles are presented and the material response to a pulse profile resembling a typical actual laser pulse is discussed. It is found that the closed-form solution obtained in the present study becomes identical with those presented in the previous studies for different pulse and boundary conditions. Moreover, the coupling effect of pulse parameter β and Bi is significant for the temperature rise at the surface.

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Closed-Form Solution of Large Deflected Cantilever Beam a gainst Follower Loading Using Complex Analysis

Modern Applied Science ◽

10.5539/mas.v11n12p12 ◽

2017 ◽

Vol 11 (12) ◽

pp. 12 ◽

Cited By ~ 1

Author(s):

Ibrahim Mousa Abu-Alshaikh

Keyword(s):

Numerical Methods ◽

Closed Form ◽

Cantilever Beam ◽

Complex Analysis ◽

Numerical Solutions ◽

Closed Form Solution ◽

Form Solution ◽

Beam Deflection ◽

Integral Approach ◽

Loading Conditions

The literature reveals that the non-conservative deflection of an elastic cantilever beam caused by applying follower tip loading was investigated and solved by various numerical methods like: Runge Kutta, iterative shooting, finite element, finite difference, direct iterative and non-iterative numerical methods. This is due to the fact that the Euler–Bernoulli nonlinear differential equation governing the problem contains the “slope at the free end”, this slope however needs special numerical treatment. On the other hand, some of these methods fail to find numerical solutions for extremely large loading conditions. Hence, this paper is aimed to obtain a closed-form solution for solving the large deflection of a cantilever beam opposed to a concentrated point follower load at its free end. This closed-form solution when compared with other conventional numerical approaches is characterized by simplicity, stability and straightforwardness in getting the beam deflection and slopes even for extremely large loading conditions. The closed-form solution is obtained by applying complex analysis along with elliptic-integral approach. Very good results were obtained when the elastica of the beam compared with that of various numerical methods which are used in analyzing similar problem.

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Fractional Calculus Model of the Semilunar Heart Valve Vibrations

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2003/vib-48384 ◽

2003 ◽

Cited By ~ 5

Author(s):

Moustafa El-Shahed

Keyword(s):

Differential Equation ◽

Experimental Data ◽

Numerical Methods ◽

Fractional Calculus ◽

Closed Form ◽

Heart Valve ◽

Laplace Transformation ◽

Closed Form Solution ◽

Equation Of Motion ◽

Form Solution

The objective of this paper is to solve the equation of motion of semilunar heart valve vibrations. The vibrations of the closed semilunar valves were modeled with a Caputo hactional derivative of order α. With the help of Laplace transformation, closed-form solution is obtained for the equation of motion in terms of Mittag-Leffler function. An alternative Method for Semi-differential equation, when α = 0.5, is examined using MATHEMATICA. The simplicity of these solutions makes them ideal for testing the accuracy of numerical methods. This solution can be of some interest for a better fit of experimental data.

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Motion of a Bored Sphere in a Spinning Spherical Cavity

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3139681 ◽

1981 ◽

Vol 103 (4) ◽

pp. 389-394 ◽

Cited By ~ 2

Author(s):

R. H. Nunn ◽

J. W. Bloomer

Keyword(s):

Numerical Methods ◽

Closed Form ◽

Predictive Model ◽

Spherical Cavity ◽

Equations Of Motion ◽

Closed Form Solution ◽

Form Solution ◽

Analytical Expressions ◽

Theory And Experiment

Theory and experiment are combined to develop a predictive model for the motion of a bored sphere within a spinning spherical cavity. The motion is gyroscopic in nature with the sphere eventually aligning its hole with the axis of spin of the cavity. Analytical expressions are derived for the applied moments on the sphere due to its motion relative to that of the cavity, and the resulting equations of motion are solved by numerical methods. An approximate closed-form solution is also obtained. Experiments are described in which the measured nutation of the sphere substantiates the analytical predictions.

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