27 



41 



z = Zc(t) f(x,y) 



[28] 



where z,, Is the deflection of a certain point on 

 the plate, which may be thought of as Its center, 

 and Is a function of the time ( alone, while 

 f{x,y) Is a shape factor represented by a fixed 

 function of the cartesian coordinates x,y specify- 

 ing position on the plate; see Figure ^1. The 

 natural small oscillations of a plate are actual 

 examples of proportional motion. 



After introducing this assumption into 

 Equation [l6], the equation can be reduced to an 

 ordinary integrodifferential equation in z and ( 

 by Integrating over the plate. The most useful 

 result is obtained if the equation is multiplied 

 through by f{x,y) before integrating, namely; 



Figure 17 - Illustration 



of Proportional Motion of 



a Plate or Diaphragm 



The deflection z is at every point 

 proportional to the displacement 

 jc of a chosen base point or center. 



M 





where 



^2F,^<P-^jfix,y)dSf{^l_J(x;y)^ 



M = fm[f(x,y)]^dS 

 F,= fpJix,y)dS, =j<t>f{x,y)dS 



[29] 



[30] 



[51a. b] 



In the first integral s is the distance between the elements of area dS and 

 dS', which could be replaced by dxdy and dx'dy', respectively. It must be 

 assumed that f{x,y) vanishes fast enough toward infinity to make the inte- 

 grals converge. 



The quantity M represents an effective mass of the plate, while F,- 

 and * represent effective forces; the last term In Equation [29] represents 

 an effective force due to release of pressure by the motion. The center of 

 the plate moves as would a mass M under a force equal to the right-hand mem- 

 ber of Equation [29]. Furthermore, the kinetic energy of the plate Is actu- 

 ally equal to M(dZc/dt)^/2; see Equation [115] In the Appendix. 



The proportional motion of the plate may be supposed to be guaran- 

 teed through the action of suitable Internal constraints which do no work on 

 the whole, so that the energy balance is not affected. These constraints 

 contribute nothing to 0, as is shown in the Appendix. 



Equation [29] is applicable either to an infinite plate or to a 

 plate mounted in an infinite fixed plane baffle; in the latter case the 



