SEISMIC METHODS 789 



Let yi be the function of time represented by the right-hand side of Equation 136, and 

 ya that represented by the right hand side of Equation 144. Then if 



3' = 3'i + J'2 (145) 



and substituting in Equation 139 



yi + 2nhyi + n^yi + ^2 + 2nhy2 + ^''^y^ = a cos wt 

 Since yi is a solution of Equation 133 



.Vi + 2nhyi + n^i = 

 and y2 is a solution of Equation 139 



y2 + 2nhy2 + n^y^ = a cos zvt 



therefore yi + 3»2 is a solution of Equation 139. Since the part yi of this solution con- 

 tains two arbitrary constants, the sum ji + 3)2 is the complete solution of the differen- 

 tial Equation 139 



yi z= Ae-'"" cos (wt -\- /3) for h<l (136) 



Note that yi contains a damping factor which becomes smaller and smaller with time. 

 For this reason the oscillations represented by yi are called the transient terms. 



If a long time elapses in comparison with since the beginning of the motion, 



nh 

 then the transients will have been damped out and the oscillations will consist only of 

 the forced vibrations represented by y2 and given explicitly by Equation 144. A further 

 study shows that these vibrations have a frequency equal to that of the impressed force 

 but lag behind it by the phase angle 0. If zv<in, then lies between 0° and 90°, whereas, 

 if zc;>n, <j> lies between 90° and 180°, corresponding to tan 0>O and tan 0<O respec- 

 tively. 



It is interesting to consider the amplitude of the steady state in this case of forced 

 vibrations. 



A= ° =:r (143) 



yj {w'-'iu'y+ (2nhzvy 



To find the frequency zva of the impressed force for which A is a. maximum we dif- 

 ferentiate and set equal to zero. 



dw' 



Thus, 



This gives 



-^ { (»== - w'Y + {2nhwy} = - 2(11" - zva") -f An-h" = 

 dzir 



zva" = n' (1 - 2h'') (146) 



This frequency zva is known as the frequency of resonance for a damped oscillator 

 under forced vibration. When /j^>l/2 there can be no frequency of resonance since zva 

 is imaginary. It can also be shown that the greater the damping factor (nh) the less 

 the amplitude of the motion from the forced vibrations becomes. 



