TRANSIENTS IN MECHANICAL SYSTEMS 665 



Triangular whip (Fig. 3b). The Laplace transform of this pulse is 



The differential equation for a simple harmonic system is 



mx+ kx^ (1) 



or 



-,-{-x = 0. (2) 



If we let the whip operate on this system, then 



4 + ^ = /(/) = xt{t) (3) 



in which xi{t) represents the displacement of the whip as a function of time. 



Let 2lx{t)] = X(s) 

 and 



J?[x,(0] = £[/(/)] = F(s) = Xi(s) 

 then 



m = s'xis) - sm - f\o) 



By definition the initial conditions are zero, so that 



^[x] = s'X{s) (4) 



The Laplace transform of equation (3) is then 



4 X{s) + X{s) = £[/(/)] = X^(s) 

 or 



("-J^) XW = X.W. (5) 



Now 



Substituting and rearranging, 



2 /< -6«\2 



2 /< -6«\2 



«■>-.-Tl^.?(^^) 



(6) 



