494 



A. jf 



- 10 - 

 Equations of motion . 

 We now apply Lagrange's equations to the expressions (19) and (20) to obtain: 



/Oh Aj V + Aj T w = A^ o{t) (21) 



ph B^ U = (T - T') 8j (22) 



/3h Cj V = (T- T') C^ (23) 



The initial condition of rest gives: 



when t = 



W = W = 



U = LI = 



V = V = 



Kow from (22), (23) and (2H) it follows immediately that: 



_J U s -i V 



(21) 



(25) 



"t 1 



whence by virtue of (ll) U -ind V can Doth be expressed in terms of S by the relations: 

 U 



h\ 



^2*3 



= 1=1 



C A 



'^2 "3 



where A, = 



(26) 



From (22), (2n) and (26) it then follows that the equation of motion for S, is 



pu Sj = Aj (T _ T') (27) 



with initial conditions; 



S, = S = when t - 



(28) 



If we denote S as the total increase of area of dished plate due both to stretching and to 

 pull-in, then from (8) and (ll): 



From (29), (27) and (21) we note for future use that: 



^"S^ 



»(t) + ^h A, W^ + A,(T' - T) - _L T W 



(29) 



(30) 



We now have the three equations (2l), (27) and (29) for five unknowns w, S , S,, T and T' and we 

 therefore require two more equations which will now be derived from consideration of the physical type 

 of motion. 



4,5 Physical conditions re stri ctmg type of motio n . 



We shall restrict ourselves tc the case where the applied load is always of one sign and we 

 can then without loss of generality assume that: 



A, P(t) >. 



(31) 

 since 



