118 Mr. A. Stephenson on the Forcing of 



where 



+ ct 2 (A r - 2 + A r+2 ) + ...}-2k{c-rn)B r = Q . (rj 



Bri^-ic-mYj-f-^icc^Br.^B^) 



+ aj(B r _a + B r+ 2) +_...}■ + 2*(c-m)A r =0. . (r 2 ) 



Comparing the two solutions, we observe that c must be 

 complex. For the forced oscillations we replace c by q and 

 the right-hand side of (0)' by a : c being complex, it follows 

 by the argument of § 2, that there is a definite limit to the 

 amplitude of the forced motion due to a sin (qt + e), although 

 this limit may be large compared with a if the frictional 

 resistance is sufficiently small. 



It may be noted that for the numerical evaluation of a 

 forced oscillation under friction, it is expedient to replace the 

 trigonometric functions in the equation of motion by their 

 exponential equivalents and to solve in exponential series. 



5. When the variation in the spring is small, the solution 

 for the forced motion may be obtained readily by approxima- 

 tion. We shall investigate this in the frictionless case, apart 

 from the general theory, thus arriving at a more distinct 

 conception of how the magnifying effect is produced. 



If the equation of motion is 



Iv + /r (1 -f 2ot cos nt)x = a sin (qt + e), 

 for the forced oscillation 



# = 2A r sin .{(g— rri)t + e\, .... (i)< 



where A r =a ^ — approximately, 



o 

 provided fi 2 —(q—m) 2 is not small for any of the values of 1\ 

 In general., therefore, the motion approximates to the forced 

 motion under constant spring. In the exceptional case when 

 fju 2 —(q—rn) 2 is small for some value of r, m say, we consider 

 the solution as made up of two parts, y and z, where y is 

 that part of the series in (i) which lies on the same side of 

 the term in sin {{q— mn)t + e} as the term in sin(<^ + e). 

 Thus 

 y -f- fi 2 (l + 2a cos nt)y = a sin (qt + e) + b sin { (q—mii)t + e } , 



where o=a— — K ?_^__ _ 



Ml 



m 



in 



n {^-(q-rn) 2 } 



