Mr Harrison, Notes on the Theory of Vibrations 83 



Notes on the Theory of Vibrations. (1) Vibrations of Finite 

 Amplitude. (2) A Theorem due to Routh. By W. J. Harrison, 

 M.A., Fellow of Clare College. 



[Read 3 May 1920.] 



p^ (I) Lord Rayleigh in his Theory of Sound, Vol. i, has considered 



the effect of introducing terms depending on x^ and x^ into the 



d'^x 

 simple equation of vibratory motion -j-^ + n^x = 0. He treats the 



added terms as small and employs the method of successive ap- 

 proximatioD. The object of this note is to point out that exact 

 integrals can be obtained in the form of the series of which 

 Kayleigh determined the first two or three terms. The solutions 

 now obtained are valid for any relative magnitude of the added 

 terms subject to the motion remaining vibratory. 



(a) The Symmetric System. 



The equation of motion is 



d 00 



-^-^ + n^x T '2Bx^ = 0, 



dt'^ 



where /S is positive, and the upper sign is taken in the first instance. 

 A first integral is 



where a is the amplitude of the vibration. 

 We have 



<dx\^ (^2 _. ^2) („2 _ ^^,2 _ ^^2)^ 



fdxV 

 \dt) 



or 



Jt 



where ax^^ =- x, k^ ^ ^a^Kn^ — ^a^). 



Hence* 



X = axi 



= a sn {{n^ - Ba^ft, h], [x =-0 when t = 0) 



_ 277a ^ g^+^ . (2m + I) tt (n^ - ^a^ft 



^ Kko 1 - q'""^' "''' 2E: 



* For the expansions of elliptic functions quoted in this paper see Whittaker 

 and Watson, Modern Analysis, 1915, p. 504, and Example (5), p. 513; or Hancock, 

 Theory of EUiptic Functions, Vol. i, 1910, pp. 486, 494, 495. 



6—2 



