123 



NOTE ON AN APPROXIMATE METHOD FOR THE 

 SOLUTION OF DYNAMICAL PROBLEMS 



E. N. Fox 



Admiralty Undex Works 



September 1943 



Summary, 



4n approximate method, analogous to the energy method used in statical problems, is suggested 

 for the solution of dynamical probloms, in particular problems of impulsive loading on structures 

 such as a ship's hull composed of plates supported on frames. The method Is Illustrated by 

 application to the problsri of impulsive loading on a plate clamped In an elast ically-supported 

 rigid frfime. 



Proposed method . 



For the solution of problems relating to structures or structural units under static loading, 

 a widely used approximate m.;thod is that In which the shape of the deflections in the structure are 

 assumed and their magnitudes then detemincd by making the total energy a minimum. This method 

 derives from the fact thfit the true solution Is that which mal<es the total energy a minimum for ill 

 possible shapes and magnitudes of deflections. A corresponding theorem in dynamics is furnished 

 by Haniilton's principlt which for a conservative system may do written 



SL dt = u 



(1) 



where l = T - v 



T = Kirietic energy of system 

 V = Potential energy of system 



and S denotes any arbitrary small variation of the co-ordinates of particles forming the system 

 provided S does not affect tine or the initial and final co-ord>nat»s at t, and t . 



In general the problem is to determine dependent variables q , q ... in tenns of independent 

 variables, tine- t and space co-ordinates x, y, z. The integrand L will usually be expressible as 

 a function of the q"s, their partial derivatives with respect to t, x, y, z and of tine t. 



The proposed approximate method is to assume space di striout ions for q,, q, ... which 

 satisfy the boundary conditions and to determine the time variations of the q" s by use of equation 

 (l). In gener'il L will th'.:n become a function of time t, q,, q, ... and the velocities q , q ... 

 and the solution of equition (l) will be given as in particle dynamics by L'.grange's equations 



d L 



^ (3 1-) 



'ItR q) ^ 



(2) 



This system of ordinary differential equations will usually be more amenable to solution than 

 the corresponding system of partial differential equations for the exact solution, 



Th~ chief merit of the approximate method lies in its application to prcblcms where the uxact 

 equations '.re tither intr .ct,-.blc r.r nccessitat: very lengthy nuinoricdt computntion to .jive a solution 

 which is of prr.ctlc;l uee. 



Tr.e value of the approximate solution depends of course on the extent to which the assumed 

 space distributions approximate to those of the exact solution and in this particular connection one 



cannot 



