=- ( cos qt— nja 2 cos Sqt \ 



the Steady State of Forced Oscillation. 709 



Thus if %fjLa?>/jL-q>inu' 2 and 2k < V 1 - k* . o?q, they 

 motion is in general made up of an oscillation of increasing 

 amplitude and another which dies away. In the particular 

 case when y is zero at a time #/27r of its period after the 

 spring is minimum, the effect of the spring variation is to 

 diminish y asymptotically to zero ; but if the fractional 

 interval differs from /3/27T in the slightest, the disturbance 

 acting through the spring gradually widens the difference 

 and ultimately brings the motion into such phase that y is 

 zero at time/3/27r of the period before the spring is minimum 

 — the state in which the passage of energy to the y motion 

 is most rapid. If the initial disturbance is sufficiently small 

 this adjustment of phase will be brought about before y 

 becomes appreciable in comparison with a. 



It is of interest to note the physical significance of the 

 above range of q defined by the magnitude of a. By 

 substitution in the left-hand side of (i.) we find, neglecting 

 powers of a above the third, that the oscillation 



2« / .1 



\/3c 



is steady under the force 



a 3 a ( / i\ 



4Z/u, 2 -y= cos qt — 4:K/jL .— sin qt, where q = fjb\l — [l-\-~\ 

 V'oc *J'6c ( \ 2/ 



and as I is taken smaller the force decreases to a minimum, 

 while the amplitude of steady motion remains unaltered, this 

 occurring through the approach to isochronism, at 1 = 0, of 

 the forcing disturbance and the free frictionless motion at 

 the given amplitude. The range of q then for instability 

 lies immediately on the lower side of that value of q for 

 which the amplitude of steady motion is maximum. 



The instability being established the magnitude of the y 

 motion has to be determined from (iii.) : for simplicity we 

 shall neglect friction. Putting 



2* , .-• 



y = —j= (u cos qt + v sin qt) 



in (iii.) , neglecting the terms of argument 3qt *, and equating 



* If it were desired to carry the approximation further these terms 

 could be taken account of by the addition of the small term 



-r= (u 3 cos 3 qt -f v s sin 3<^) 

 V 3c 



to the expression for y. 



Phil. Mag. S. 6. Vol. 14. No. 81. Dec. 1907. 3 B 



