501 



(5) The S curve must became coincident with the S curve prior tc cutting the time axis ana when 

 this latter occurs a period of rest will ensuo. There will be no further motion unless 

 P(t) subsequently increases sufficiently to violate (6l). 



Since the equation to the S curve is obtained from (68) and (29) with initial conditions, 



w = W = (69) 



the application of the preceding geometrical criteria becomes a matter of simple analytical geometry. 



When the S curve has thus been determined any of the remaining variables S^^, T, etc., can 

 be determined by using the relevant equations from paragraph A. 8 according to type of motion. Since 

 observable quantities in box model trials are the final doflecticn and pull-in, we shall confine 

 attention to the values cf: 



final mean deflection, 

 area under S curve 



fin;il area of pull-in = area under S, curve 



(W) 



in the two specific forms of P(t) which we now proceed to consider, 

 A. 12 Motion due to an in stantancous impulse . 



If we let P(t) become infinitely great 3ut of infinitely short duration, the impulse 

 remaining finite, then from our general equations (2l) and (29) and the initial conditions (24) and 

 (69), the conditions after the impulse will be; 



A 



^(t) dt = V 



3=0 



S = 



with the equation (68) for W becoming: 



Oh A, W t T„ A, W = 

 '^2 1 



The solution for W from (72) and (7l) is thus: 



w = w Sin a t 



m 



where 



t > 



/ Oh A 



ph 



w = V/a 



m 



Thence from (29), (73) and (75): 



A. V' - 



1 "1 



S = A, V W 



A, Wj Sin' a t 



2 1 m 



= i A. a w "^ Sin 2 a t 

 2 1 m 



(71) 



S = A^ V cos 2 a t 



(72) 



(73) 



(74) 

 (75) 



(76) 

 (77) 



(78) 



Thus 



