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APPeNDIX 



To illustrate the application of the proposed approximate method consider the motion under 

 uniformly distributed impulsive pressure of a rectangular plate »*iich is clamped in a rigid frame 

 mounted on elastic supports as depicted sectionally In Figure 1. The plate and frame are assumed 

 initially at rest and of uniform mass so that all deflections are symmetrical about the centre of 

 the plate. The deflection of the plate relative to the frame is assumed small enough for the 

 usual theory of elastic bending of thin plates to apply. 



The following notation will be used;- 



t = time 



2a = length of plate. 



2b = width of plate. 



ki = total mass of plate. 



= flexual rigidity of plate 



V - Poisson's ratio for plate. 



x,y, = rectangular co-ordinates in plane of plate referred to centre as origin. 



u = dcfl 'Action of frdme, 



u * z - deflection of point (x, y) of plate. 



u + w = deflection of centre of plate. 



M, = total mass of frame. 



k, = tot.il stiffness of elastic supports to frame. 



F(t) = total force at time t due to applied pressure, 



T = total kinetic energy of system. 



V = total potential energy of system including applied pressure 



L = T-v 



using dots to denote differentiation with respect to time the kinetic energy of the frame is 



i Mj u^ 



(6) 



while the kinetic energy of the plate is 



M 

 2 ab 



(u ♦ e) -dx dy. 



(7) 



The potential energy in the elastic supports to the frame is 

 i kj u^ 

 while the potential energy of bending in the plate is 



20 



Jc 





t4 * tl * ^ (^ - "^l 



3^z 3^z 



Bx ctyj bT^ ^ 



(6) 



dx dy (9) 



v*ere it may be noted that the second tern in the integral vanishes on integration in view of the 

 boundary conditions. 



Finally the potential energy of the applied pressure is 

 ,a J, 



a b 



(u ♦ z) dx dy 



Jc Jo 



(10) 



As the first step in the approximate solution w; now assume for the deflection z of the 

 plate relative to the frame, 



which Sdlisfies conditions that the plate is clamped round its edge to the frame. 



Subst i tut ing ,, , ,, 



