FORCED VIBRATIONS 369 



body goes slowly back to its equilibrium position although it 

 never reaches it. 



184. Forced Vibrations. A study of the curves obtained 

 from the examples given in the previous paragraph indi- 



cif^jc doc 



cates that the equation -^ + 2a -r + b 2 x = represents a motion 



which is damped that is, it gradually dies away. In some 

 cases this motion is oscillator}', while in other cases it is 

 not. If the whole system has a vibration of known frequency 

 forced upon it and it is necessary to make a study of this 

 forced vibration, it is as well to remember that when the 

 natural or damped vibration dies away, this forced vibration 



d 2 x dx 



will remain. Let the equation -nr + 2a -7- + b z x = c sin pt 



at 2 " at 



represent the motion of a body which has a forced vibration of 



27C 



periodic time impressed upon it. When the natural vibration 



dies away, the motion of the body will be one solely due to the 

 forced vibration, and as the periodic time remains unchanged this 

 motion will be given by x = A sin pt + B cos pt, and this solution 

 must satisfy the complete equation of motion and the values 

 of the constants A and B must be determined so that this 

 will be so. 



dx 



-j- = pA cos pt po sin pt 



d?x 



-;- = p 2 A. sin pt p 2 B cos pt 



at 



d^tK dx 



But c sin pt = -j-r + 2a -j- + b 2 x 

 di 2 - at 



= p 2 A sin pt p 2 B cos pt + 2a(pA cos pt pB sin pt) 



+ b 2 (A. sin pt + B cos pt) 

 = ( - p 2 A - 2apB + b 2 A) sin pt + ( - p 2 E + 2apA 



+b 2 E) cospt 

 Equating coefficients of sin pt, 



(b* - p*)A - 2apB = c (1) 



Equating coefficients of cos pt, 



2apA + (b* - p 2 )B = (2) 



and relations (1) and (2) can be solved as a pair of simultaneous 

 equations for A and B. 



A(6 2 - p 2 ) 2 - 2ap(6 2 - p 2 )B = c(b- - p*) 

 4aVA + 2ap(6 2 - p z )E = 



and {(b 2 - p 2 ) 2 + 4a z p 2 }A = c(b* - p 2 ) 



2 A 



