Closure to “Discussion of ‘A New Matrix Formula for the Inverse Laplace Transformation’” (1967, ASME J. Basic Eng., 89, p. 272)

1967 ◽  
Vol 89 (2) ◽  
pp. 272-272
Author(s):  
C. F. Chen ◽  
R. E. Yates
Keyword(s):  
Laplace Transformation ◽  
Inverse Laplace Transformation ◽  
Matrix Formula

A New Matrix Formula for the Inverse Laplace Transformation

1967 ◽  
Vol 89 (2) ◽  
pp. 269-272 ◽  
Author(s):  
C. F. Chen ◽  
R. E. Yates
Keyword(s):  
Laplace Transformation ◽  
Transfer Functions ◽  
High Order ◽  
Partial Fraction ◽  
Inverse Laplace Transformation ◽  
Numerical Errors ◽  
Matrix Operations ◽  
Simple Matrix ◽  
Matrix Formula ◽  
Single Matrix

A new matrix formula for the inverse Laplace transformation is established. After substituting the eigenvalues and coefficients and performing some simple matrix operations, one can obtain the inverse Laplace transformation of the function in question. The regular Heaviside techniques involving partial fraction expansions, function differentiations, and so on, are avoided. Since the formula is general, it is particularly advantageous for use on high-order transfer functions; since the formula is exact, the results have no numerical errors. Hundreds of commonly used transform pairs can be replaced by this single matrix formula.

Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

2018 ◽  
Vol 25 (3) ◽  
pp. 593-611
Author(s):  
Xiayang Zhang ◽  
Haoquan Liang ◽  
Meijuan Zhao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Impact Load ◽  
Moving Load ◽  
Numerical Results ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Load Case ◽  
Inverse Laplace Transformation ◽  
Damping Effect

This paper, taking the clamped boundary condition as an example, develops Su and Ma's fundamental solutions of the dynamic responses of a Timoshenko beam subjected to impact load. Based on that, a further extension regarding the general moving load case is also established. Kelvin–Voigt damping, whether proportionally or nonproportionally damped, is incorporated into the model, making it more comprehensive than the model of Su and Ma. Numerical inverse Laplace transformation is introduced to obtain the time-domain solution, where Durbin's formula and the corresponding convergence criteria are utilized in numerical experiments. Further, the real modal superposition method is applied at an analytical level to validate the numerical results by applying a proportionally damped condition. Total comparisons are made between the methods by sufficient case studies. The dynamic responses with and without damping effect are computed with wider slenderness to verify the correctness and effectiveness of the numerical results. Furthermore, parametric studies regarding the damping coefficients are performed to explore the nonproportional damping effect. The results show that the structural damping has significant influences on the dynamic behaviors and is especially stronger at small slender ratios. As the damping decreases the inherent frequencies and excites the low-frequency modal components more actively, a resonant phenomenon appears in high slenderness case when the beam experiences a low-speed moving load. Additionally, the computations in the moving load case indicate that the algorithm convergence is preferable when the number of grids exceeds 1000.

Numerical inverse Laplace transformation using concentrated matrix exponential distributions

2020 ◽  
Vol 137 ◽  
pp. 102067 ◽  
Author(s):  
Gábor Horváth ◽  
Illés Horváth ◽  
Salah Al-Deen Almousa ◽  
Miklós Telek
Keyword(s):  
Laplace Transformation ◽  
Matrix Exponential ◽  
Exponential Distributions ◽  
Inverse Laplace Transformation
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