ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. US 



This is what we call resonance, and the explanation of the phenomenon is as follows : — 

 Let a material point vibrate with the constant frequency f, but with amplitudes which 

 also depend on the time, and let its velocity v be expressed by — 



V = /(;) sin 2 nft. 



We are here not concerned in the precise form of the function f(t), hut make the stip- 

 ulation that it is positive for any value of the time t. 



The frequency / is of course the frequency of the free vibration, for in the absence of 

 external forces nothing must change, except that the function f{t) becomes a constant. Let, 

 further, the above vibration be excited by an external force of the same frequency /, and ex- 

 pressed by 



P = ^ sin 2 nft. 



The rate of work done by the force P on the material point is Pt', for which we obtain — 



Pv = f{t) sin= 2 nft 



an expression of which the smallest value will be zero, but which never becomes negative. 

 But if the rate of work done on the material point never becomes negative, the energy 



J Pvdt absorbed by it will indefinitely increase with time, which is equivalent to saying 



o 



that the vibrations will occur with ever-increasing violence. 



Thus we see that feeble periodic forces in resonance with the free vibration will with 

 time — and only then — exert effects altogether out of proportion with their magnitude. 



And it follows that a machine may be run at a speed higher than the one which brings 

 about resonance, provided that it is passed through the critical speed fast enough. A cele- 

 brated example of this is the starting up of a De Laval steam turbine. 



In the foregoing, damping of the vibrations has been left out of consideration. With 

 damping the statement should be modified to say that at resonance the energy absorbed by a 

 vibrating system increases with time until a point is reached where the work done by the ex- 

 ternal force on the system is equal to the energy dissipated by it. 



Therefrom we see that vibrations are maintained by the expenditure of a given amount 

 of power which, in case of ships, must be supplied by the propelling machinery. 



From whatever point of view we look at it, one of the most important questions in naval 

 engineering is the predetermination of the periods of free vibration of the hull. 



As an example, it may be cited that in the two-cycle Diesel engine ship, the Monte 

 Penedo, designed initially for an engine speed of about 154 revolutions per minute, the 

 vibrations at that speed were such that the speed had to be reduced to about 135 revolutions 

 per minute. 



We are therefore greatly indebted to Mr. Akimofif for his paper, wherein he shows that 

 the necessary mathematical apparatus to deal with such problems already exists. 



With particular reference to Timoshenko's application of the Ritz principle to the cal- 

 culation of the periods of free vibration of a ship, it should be pointed out that the funda- 

 mental variational equation and the subsequent operations are true if the following condi- 

 tions are verified. 



