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

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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  Other Math Pages | math software | Laplace Transforms | Factorial | Euler's Equation | Bessel Function | | Gamma Function | Partial Fractions | Root of Complex Number