ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 127 



This is sometimes also given in the form — 



^=i/ 



2 ^o EJ 



The kinetic energy of a vibrating beam will be found from the following con- 

 sideration: — The elementary conception, inv^/2, can be extended to a continuous 

 beam if we agree to consider it as an infinity of very thin slices, each possessing 

 a motion of its own. Let the axis x be horizontal and the deflections be denoted in 

 function of the ordinate y. Then the mass of a typical slice will be dm =yadx ; and 

 its velocity in its vibratory motion, up and down, will be the time-rate of deflection, 



that is, V = -^. 

 dt 



Therefore the total kinetic energy will be: — 



where y and a can be taken from under the integral sign only in case of a uniform, 

 continuous beam. 



Professor Morley in his book, "Strength of Materials" (section 162), proposes 

 a method whereby the frequency of a vibrating bar can immediatey be found from 

 equating the kinetic energy to the strain energy, which the bar would have in its static 

 deflected position under the same load. The method, although but approximative, is 

 very easy and instructive. 



3. Virtual Work. — By virtual displacement is understood an infinitely small 

 change of form of a system which could take place without being inconsistent with 

 the general characteristics of the system. Such a displacement may or may not be 

 due to the applied forces, but, at any rate, the applied forces may perform a certain 

 amount of work, owing to such an imaginary deflection of the system from its actual 

 position. This deflection can take place in an infinite variety of manners, and all 

 such quantities as may depend on their respective positions will be altered to the ex- 

 tent of certain special differentials, perfectly arbitrary, which we shall denote by 

 h and call "variations," reserving the notation d for changes actually taking place. 

 By virtual work, then, of a force P, it is proper to call the quantity P.hs. cos (P,5 j), 

 that is, the product of the force upon the virtual displacement, projected upon the 

 direction of the force. 



The virtual work of a resultant, R, of the forces Pi, P2, . . . applied to the same 

 point, is the algebraic sum of the virtual works of the components — 



R.hs. cos {R,hs) =^P.bs. cos {P,hs). 



According to these principles the problems of equilibrium as solved by equating to 

 zero the sum of virtual work of forces ; after which we reason as follows : — Since 

 the virtual displacement itself is perfectly arbitrary, it may or may not be ^ o; 

 therefore to make the whole expression of virtual work equal to zero under all con- 



