Oscillations by Disturbances of Different Frequencies. 117 



Thus a large oscillation is forced if q is nearly equal to 

 + c — 7'»,and the effect is continually cumulative ilq== ± c Q —rn ; 

 i. e. if the frequency of the forcing disturbance is equal to 

 the frequency of any periodic element in the free motion under 

 variable spring. 



If the variable spring itself forces an oscillation c is 

 complex, and therefore whatever the value of q the forcing- 

 disturbance cannot help in the continual intensification of the 

 oscillation, although it may force a motion of amplitude large 

 compared with a. 



3. In the above it is assumed that the variation in the 

 force of restitution is periodic. If, however, it is made up of 

 two elements of incommensurable frequencies, 



x -f y u, 2 {l + 2a 1 cos (wit+Cj) -f 2a 2 cos (n 2 t + e 2 )}x=a sin qt, 



and for the free motion 



3fc=22A rs g sin {et + rfot + e x ) -f s(n 2 t + e 2 ) }, 



where the summation includes the zero and all positive and 

 negative integral values of r and of s. On substituting for x 

 we find 



J^,[^ 2 -(c + ™, + ^o) 2 ]+/^ 



a system of equations determining a convergent series : c is 

 obtained as before by eliminating the A's, and the roots are 

 all included in the series +c + m- l + sn 2 . The examination of 

 the forced motion follows a similar course to that of the 

 preceding case, and we find that the forcing disturbance has 

 a continually cumulative effect if q= + c -\-rn l + sn 2 , where 

 r and s have the zero or any positive or negative integral 

 values. 



4. If the system is subject to kinetic friction, 



i? + 2/ji + /r{l + 2a i cos nt + 2a 2 cos 2nt-\- ...}#== a sin^H-e). 



To find the free motion we put d' = e~ kf y ; then 



y + /J? { 1 — k 2 l[JL 2 + 2«, cos nt + 2a 2 cos 2nt + . . . } y = ae kt sin {qt + e) . 



The complementary function is found as before : it is clear 

 that in all cases, whether the " c" of y is real or complex, 

 ,r must in part contain a harmonic series with an exponential 

 function of t as factor. Now the free motion is also given 

 directly by 



oc 



x= X [Arsm {(e — rn)t + e\ +B,-cos {(c— rn)t + e}], 



