TRANSIENTS IN MECHANICAL SYSTEMS 667 



This looks somewhat compUcated, but we can simpHfy by omitting the last 

 term, because we are only considering values of 8 during the pulse interval. 



,. 6 = 2r - ^5if^^ _ 2 ( (2r - 1) - ■^ ^■"'" ^ti ) u(2r - 1) (9) 



T<p 



_ 2 f (2r - 1) - «igTV^(2r-l)\ ^^^ _ j^ 



A plot of this equation for various values of <p is shown in Fig. 4. It is seen 

 that 5 becomes a maximum when <p is approx. .9 and r is then .75. The dis- 

 placement is approximately 1.5 times the peak displacement of the whip. 



After the whip has passed, or when r > 1, the transient has disappeared 

 and a steady-state condition exists. Since the system under consideration 

 is a simple harmonic system, the steady state is a harmonic motion of fre- 

 quency £0, with an amplitude to be obtained from equation (8). Indicating 

 the dimensionless values of the ampHtude by 8a when r > 1, equation 8 may 

 be written 



sin lipiTT 

 8a = 2t — 



. 2 f (2. - 1) - «Je^(?lii1)) 



\ Tip / 



+ (2(x - 1) - ^i'^ 2.^(r - 1) ^^ ^ ^ J 



TTip 



. . , .,. (10) 



Tr(p 

 After developing (10) and rearranging we obtain 



6. = ^^^-""^"^^ sin M2r - 1). (11) 



ircp 

 The maximum amplitude is 



2(1 — cos Tr<p) 



(12) 



TTip 



A plot of equation (12) is shown in Fig. 5. Before considering the action of 

 this whip in terms of what it does to the system, we shall take a brief look at 

 the analysis of the two other whips; viz., the sine whip and shifted cosine 

 whip (see Fig. 3). 



Sine Whip 



We have again equation (3). 



%-\- X = fit) = xiit) 



and equation (5) 



C-^') 



X(s) = F{s) 



