Oscillations hy Disturbances of Different Frequencies. 121 



leading cases given by q= e + n\. A body suspended by a 

 light spring forms a convenient system for this purpose. 

 When the body is set into vibration vertically and the spring 

 seized at the point of suspension and oscillated horizontally, 

 a comparatively large pendulum motion is generated if the 

 sum or difference of the vertical and free pendulum fre- 

 quencies is approximately equal to the suspension frequency. 



In a system with two. or more, degrees of freedom, if one 

 normal coordinate is subject to direct forced oscillation, a 

 variation of frequency n, say, is in general produced in the 

 spring of the other coordinate, which will therefore respond 

 to a direct force of frequency c + rn. 



8. If a system containing a cyclic coordinate is making 

 small oscillations about a state of steady motion under the 

 action of a periodic disturbance, the effect of the disturbance 

 at any instant depends partly upon the configuration of the 

 system in the small oscillations. If only two coordinates are 

 affected the equations for the small oscillations in frictionless 

 motion are of the form 



x + afju(l + 2«! cos nt + 2/9i sin nt-t- .. i)y + fi?(l + 2* 2 cos nt 

 + 2/3.2 sin nt ~r ...)#=E sin qt + F cos qt, 



y + a'fjJ(l + 2«j ' cos nt -f 2/5/ sin nt + . . .)i + H ,2 (l + 2ccJ cos nt 

 + 2/3./ sin nt+ ...)i/ = E' sin^-j-F' cos qt, 



where powers and products of x and y are neglected. The 

 complementary function is of form 



x = i [A r sin (c - rri)t + B r cos (c—rn)t] , 

 y = X [A/ sin (c—m)t + B/ cos (c— rn)t] . 



— X 



On substituting for x and y and eliminating the coefficients 

 from the conditional equations the frequencies c\ and c 2 are 

 obtained. Complex values of c indicate that the disturbance 

 acting through the spring alone produces a cumulative effect. 

 The forced oscillations are of similar form, but with g for c ; 

 and if c is real it is evident, by the argument previously 

 employed, that the oscillation is of continually increasing 

 amplitude if 



q= +Ci — m or +c 2 — rn. 



The approximation made in taking the equations of motion 

 as linear holds good for r=m if the amplitude of the forcing 

 disturbance is small compared with the mth power of that of 

 the spring ; otherwise higher powers of x and y must be 

 taken into account in the equation of motion. 



