410 



APPEI^IDIX A 



DiiRIVATION OF Tr'.u EQUATIONS OF MOTION OF THIN PLATES 

 SUBJECT TO AXIALLY SYMMETRIC DEFORMATIONS 



In Part I, Section 2 (Eqs. (1) and (2)) the equations of motion 

 of thin plates were stated without proof. Here we give an elementary der- 

 ivation of these equations employing a procedure similar to that used by 

 M. P. Whitei/ for the case of static equilibrium. 



We now consider the geometrical descrip- 

 tion of the deformation. Strictly speaking, the position of a particle in 

 the deformed plate is described by the Euler coordinates ( r , 4> , z ) v^ich 

 are related by a time-dependent transformation to the initial or Lagrange 

 coordinates ( •i , ^, , «o ) °^ ^^^ particle. Since we consider only 

 axially syjunetric loading, there is no dependence on ^ . Since the plate 

 is very thin, we shall employ the plane stress approximation, and neglect 

 bending. Vv'e now write for points on the middle section, 



Holding t constant, these expressions aire the parametric equation of the 

 surface of the plate. Holding t^ constant in the above expressions we ob- 

 tain the parametric equations of the trajectory of a particle whose radial 

 Lagrange coordinate is r^ . The radial and vertical components of the ac- 

 celeration of an element of the plate are consequently denoted by [_efr/$t^)^ 



and (5\/3tVr respectively. 

 o 



Before going to the derivation of the equation of motion, we make 



the assumption that the diaphragm is sufficiently thin for the bending stresses 



T/ M. P. \Vhite, Div. 2 report, A-167. 



91 



