BARS 131 



The theory of forced vibrations again, is of little acoustical 

 interest, although it has some technical importance. A simple 

 example is furnished by the coupling rod which connects the 

 wheels of a locomotive. Attending only to the vertical com- 

 ponent of the motion, and treating the bar as uniform, we have 

 to solve the equation (13) of 45 subject to the conditions 



where n is the angular velocity of the wheels, and & is the 

 vertical amplitude. The forced oscillation is evidently of 

 symmetrical type, and we therefore assume 



rj = ( A cosh mx + C cos mx) cos (pt + a) (5) 



This satisfies the differential equation, provided 



m*=p*p/K*E', (6) 



whilst the terminal conditions give 



A cosh^ml + Ocosra = /8, ) ^ 



A cosh \ml-G cos \ml = 0, J 



the latter equation expressing the absence of terminal couples 

 (" = 0). Hence 



2 cosh ml' 



The oscillations would become dangerously large if c 

 were small, i.e. if the imposed frequency (>/2?r) were to ap- 

 proximate to that of one of the symmetrical free modes of the 

 bar when " supported " at the ends ( 47 (20). 



49. Applications. 



The use of transverse vibrations of bars in music is re- 

 stricted by the fact that the overtones are not harmonic to the 

 fundamental. If a flat bar, otherwise free, be supported at the 

 nodes of the fundamental (Fig. 45), and struck with a soft 

 hammer, the production of overtones is, however, in some 

 measure discouraged, and musical instruments of a kind (such 

 as the " glass harmonica ") have been constructed on this plan. 



The most important application is in the tuning fork. 



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