84 



Mr Harrison, Notes on the Theory of Vibrations 



Let the units of length and time be chosen so that a= l,n= 1. 



doc 



It is necessary that ^ < |, otherwise -^ vanishes first f or < x < 1. 



The effect of the term 2^x^ on the vibrations can be exhibited 

 by the results of numerical calculation given in the following table: 



We proceed to consider the equation 

 d?x 



dt^ 



+ n^x + 2^x^ = 0. 



If a is the amplitude of the motion as before, we have 



f^f = (a2 _ ^2) (^2 + ^^2 + ^^2)^ 



or r^j = {n^ + ^a^) (1 - V) (1 + l^V), 



where ax^^ = x, and [x^ = ^a^/{n'^ + ^a^). 



Write 1 — a;j2 = z^, so that 



^^y = (n2 + 2^a2) (1 _ ^2) (1 _ ^^2)^ 



where k^ = ij?/{l + jjl^) = ^a^j{n^ + 2^a^). 



Thus 2 = sn {{n^ + 2^a2)^^, A'}, (^ = when t = 0). 

 Therefore 



dU CttJO-l 



= acn {{n^ + 2pa^)h, Jc} 



,m+i 12m + 1) 77 (%2 + 2i8a2)^^ 



cos r I 



^■na ~ q" 



v2m+l 



2Z 



X^ 1 + ?' 



In this case there is no limit to the value of ^, the motion 

 remains vibratory, but the period of the gravest mode decreases 



