Forced Vibrations Experimentally Illustrated. 171 



In equation (4) 



tan S = 



Urn 



q 2 =p 2 — k' 



■ ■ (5> 



and E and e are arbitrary constants to be cbosen to fit the 

 initial conditions. 



As indicated, the first term on the right side of (4) represents 

 the forced vibration with which we are here chiefly concerned. 

 The second term denotes the free vibration of the system, 

 aud this must be present to complete the solution. If the- 

 responding particles were at rest in the zero position when 

 the impressed force was started, then the values of E and e 

 would have to be sucli as to express a free vibration which 

 would annul both displacement and velocity as given by the 

 forced vibration, whose amplitude and phase have nothing 

 arbitrary. 



If the forced and free vibrations coexist of differing 

 periods and comparable amplitudes, beats will occur between 

 them. These are easily obtained but are usually best 

 avoided. 



When, in virtue of the damping factor involving /c, the 

 free vibration has practically disappeared, the forced vibration 

 is left in possession of the field. No beats are then possible. 

 While the free vibration is dying away, the resultant motion 

 which is under observation grows from nothing to the fixed 

 amplitude and phase of the forced vibration. 



Considering now the forced vibration itself, we may note, 

 from the first term on the right side of equation (4), the 

 following points. 



1. The period of the forced vibration is identical with that 

 of the impressed forces whatever the period natural to the 

 responding system. 



2. The best response occurs for the best tuning. This is a. 

 brief statement which may convey the right idea with sufficient 

 accuracy for our present purpose. To make the statement 

 precise we must define best as applied both to response and 

 to tuning. This has already been done by one of the present 

 writers in " Range and Sharpness of Resonance, &c. ; ' (Phil. 

 Mag. July 1913). 



3. The phase of the forced vibration varies continuously 

 between and it with the tuning. Thus the phase angle 8 

 is almost zero for p 2 much greater than n 2 , i. e., for a 

 responding system whose natural frequency is much greater 

 than that of the impressed force. On the other hand, S is 

 almost 7r for p 2 much less than n 2 , i. e., for a responding 

 system of natural frequency much less than that of the 



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