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. By Tom Irvine Email: tomirvine@aol.com

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	sY(s) - y(0)	first derivative y' (t)
1.4		second derivative y" (t)
1.5		nth derivative
1.6	(1/s) F(s)	integration
1.7	F(s)G(s)	convolution integral
1.8		f (at)
1.9	F(s - a)	shifting in the s-plane  exp(-at) f(t)
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	d(t) unit impulse at t = 0
2.2	s	double impulse at t = 0
2.3		d(t-a)
2.4a	1/s	unit step  u(t)
2.4b		0 t < a 1 a < t < b 0 t > b
2.5		u(t-a)
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		exp(-at)
2.10		t exp(-at)

2.11
2.12		1 - exp(-at)
2.13
2.14
2.15
2.16a		sin(at)
2.16b		sin(at + f)
2.17		cos(at)
2.18		t cos(at)
2.19
2.20
2.21
2.22

2.23
2.24
2.25		exp(-at)sin(bt)
2.26		exp(-at)cos(bt)
2.27
2.28
2.29
2.30		cosh(at)
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 Unit Step Displacement: unit_step.pdf Response of a Single-degree-of-freedom System Subjected to a Classical Pulse Base Excitation: sbase.pdf Partial Fractions in Shock and Vibration Analysis: partial.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 Please send comments and questions to Tom Irvine at: tomirvine@aol.com

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