86 



Mr Harrison, Notes on the Theory of Vibrations 



The calculations have been performed for illustrative purposes 

 only, and no special care has been taken to ensure the accuracy 

 of the digits in the final decimal places. 



(c) The solution of the equation 



dt^ 



+ n^x + f 



ax^ 



in the form of a Fourier Series requires rather more elaboration 

 of the algebra. 



The motion presents one novel feature which does not appear 

 in the previous solutions. If ^ be positive, however small, the 

 motion remains vibratory for any finite value of a, and if a and a/^ 

 be great, the amplitude of the motion on one side is approximately 

 a/jS times its amplitude on the other. 



(II) Kouth has shown (vide Advanced Rigid Dynamics, 1905, 

 p. 56) that an increase in the inertia of any part of a vibrating 

 system will increase all the periods in such a way that the modified 

 periods are separated by the periods of the original system. This 

 is true in general if the inertia of only one part of the system be 

 increased, the definition of a single part being that the effect of 

 increasing its inertia can be represented by a single term 



i (/^i^i +/2?2 + ■■■f 

 in the expression for the kinetic energy, where g'j, q2, ... are the 

 normal coordinates of the original system. For example, the 

 theorem is applicable to the case of an additional mass attached 

 at a single point of a stretched string, but not to the case of an 

 increase of mass spread over a portion of the string, or to the case 

 of two or more masses attached at different points. 



The theorem may be simply proved as follows. Let the modified 

 kinetic energy be 



i iii + ?2^ +...) + ! (/ii?i + Mi + •••)^ 

 and the potential energy be 



i(AiV + A2V +•••)• 

 The equations of motion are typified by 



'ir + W + Mr (M*i + Mi +...) = 0. 



