Welcome to Vibration Data
Laplace Transform Table


Laplace transforms are used to solve differential equations. As an example, Laplace transforms are used to determine the response of a harmonic oscillator to an input signal.

 Operation Transforms


 N

 F(s)

  f ( t ) , t > 0
 1.1

 

 definition of a Laplace transform

y(t)
 1.2

 Y(s)

 inversion formula

 1.3

 

 first derivative

 1.4

 

 second derivative

 1.5

 

 nth derivative


 1.6

 

 integration


 1.7

 F(s)G(s)

convolution integral 

 1.8

 

 
 1.9

 

shifting in the s-plane

 
 1.10

 

 f(t) has period T, such that

f( t + T ) = f (t)
 1.11

 

  g(t) has period T, such that

g(t + T ) = - g(t)

 Function Transforms


 N

 F(s)

 f ( t ) , t > 0

 2.1

 1

 

unit impulse at t = 0

 2.2

 s

 

double impulse at t = 0

 2.3

 

 

 2.4

 

unit step 

u(t)

 2.5

 

 

 2.6

 

 t

 2.7a

 

 

 2.7b

  , n=1, 2, 3,….

 

 2.8

  , k is any real number > 0

 

the Gamma function is given in Appendix A

 2.9

 

 

 2.10

 

 


 2.11

  , n=1, 2, 3,….

 
 2.12

 

 
 2.13

 

 
 2.14

 

 
 2.15

 

 
 2.16a

 

 
 2.16b

 

 
 2.17

 

 
 2.18

 

 
 2.19

 

 
2.20 

 

 
 2.21

 

 
 2.22

 

 

 2.23

 

 
 2.24

 

 
 2.25

 

 
 2.26

 

 
 2.27

 

 
 2.28

 

 
 2.29

 

 
 2.30

 

 
 2.31

 

 
 2.32

 

 
 2.33

 

 

 2.34

 

 
 2.35

 

 

Bessel function given in Appendix A

 2.36

 

 
 2.37

 

 

Modified Bessel function given in Appendix A

 2.38

 

 
 2.39

 

 


Examples of the Laplace Transform as a Solution for Mechanical Shock and Vibration Problems: 

Free Vibration of a Single-Degree-of-Freedom System: free.pdf

Response of a Single-degree-of-freedom System Subjected to a Classical Pulse Base Excitation: sbase.pdf

 References
1. Jan Tuma, Engineering Mathematics Handbook, McGraw-Hill, New York, 1979.

2. F. Oberhettinger and L. Badii, Table of Laplace Transforms, Springer-Verlag, N.Y., 1972.

3. M. Abramowitz and I. Stegun, editors, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D.C., 1964.
4. Peter O'Neil, Advanced Engineering Mathematics, Wadsworth, Belmont, California, 1983.

 
APPENDIX A
Gamma Function
Bessel Function

_______________________________________________________________________________________________

 Laplace Transform Books

Schaum's Outline of Laplace Transforms

The Laplace Transform : Theory and Applications (Undergraduate Texts in Mathematics)

An Introduction to Laplace Transforms and Fourier Series

The Laplace Transform: Theory and Application

 Other Mathematics Books

3000 Solved Problems in Calculus (Schaum's Solved Problems Series)
Jan Tuma, Engineering Mathematics Handbook

William Press, et al, Numerical Recipes The Art of Scientific Computing

Frank Bowman, Introduction to Bessel Functions


Please send comments and questions to Tom Irvine at: tomirvine@aol.com 

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