TRANSIENTS IN MECHANICAL SYSTEMS 



661 



Figure 3C consists of one-half cycle of a sine wave. 



Figure 3D is a cosine pulse of one-cycle duration and is shifted along the Y 

 axis an amount equal to the amplitude. 



These are the pulses to be used in the problems under consideration. 



If they represent a force as it varies with time then it is said that F{t) 

 represents a particular pulse. The Laplace transform of F(t) is given as 

 Fo(s), Fq(s) being some function in the complex domain. It is outside the 

 scope of this paper to prove or [how the mathematical technique in ob- 

 taining the transforms which produce Fo{s). We will present them here for 

 future reference. 



TIME=t 



Fig. 3 — Four pulses. 



The Laplace transform for a very short pulse is 



F{s) = Ar 



(1.01) 



and is a pulse which has a finite area but the time interval of which is 

 approaching zero. 



For a square pulse with finite time interval and magnitude a (hereafter 

 referred to as pulse amplitude) it is 



-bs 



Fis) = a 



1 



(1.02) 



For a triangular pulse 

 For a sine pulse 



F(s) = ^^, (1 + r"'"») 



(1.03) 



(1.04) 



