THEORY OF VIBRATIONS 



17 



with the frequency H/^TT proper to the system. On this is 

 superposed a " forced vibration " represented by the last term. 

 This is of simple-harmonic type, with the frequency p/2?r of the 

 disturbing force, and the phase is the same as that of the force, 

 or the opposite, according as p $ n, i.e. according as the imposed 

 frequency is less or greater than the natural frequency. 



The above theory is easily illustrated by means of the 

 pendulum. If the upper end of the string, instead of being 

 fixed, is made to execute a horizontal motion in which the 

 displacement at time t is (Fig. 7), the equation of motion (1) 

 of 4 is replaced by 



.(5) 



or 



.(6) 



This is the same as if the upper end were fixed, and the bob 

 were subject to a horizontal force whose accelerative effect is 

 ?i 2 f . If as a particular case we take 



acospt, ........................ (7) 



The annexed Fig. 8 repre- 

 sents the forced oscillation in the two cases of p < n and p > n y 

 respectively. The pendulum oscillates as if C were a fixed 

 L. 2 



we get the form (3), with /= ?i 2 ct. 



