Chap. 9] SEISMIC METHODS 587 



whose solution 



a = -5 -„ sin o)t = — X. (9-88c) 



Substituting the dj'^namic magnification W for the ratio a/x, and the 

 "tuning factor" n for the ratio co/coo, 



W = V.-^?— ,. (9-88d) 



1 — n^ 



If w = Wo , resonance occurs; W =*= oo ; and, theoretically, infinite ampli- 

 tudes are obtained. In practice they do not occur because of a certain 

 amount of friction or damping which is always present. If co ^ uq , 

 W = —V; the dynamic magnification is equal to the negative static 

 magnification ; and very rapid ground motions are recorded with the indi- 

 cator magnification. If co <$C wo , W = V w /a>o ; the dynamic magnification 

 is proportional to the square of the ground frequencj'^; and the seismograph 

 acts as an accelerometer. 



Undamped seismographs are used in vibration engineering for deter- 

 mining, by resonance, the natural frequency of oscillating shafts, engines, 

 and the like. As a rule, however, seismographs are applied away from 

 the point of resonance. Theoretically, this is possible by a very low or 

 very high natural frequency. Some refraction receivers of the acceler- 

 ometer type are built in this manner (carbon button and piezoelectric 

 receivers). For the recording of first breaks only, the use of damping 

 offers no particular advantage. Conversely, in station seismometers and 

 reflection seismographs, resonance and continued oscillation after cessation 

 of ground motion are highly undesirable. Therefore, damping is applied. 



Forced oscillations of a damped seismograph may be treated by inserting- 

 the damping term 2 ed in eq. (9-886), so that 



a + 2 e d -H woa = VX<a sin tat, (9-89a) 



whose first particular solution oi = Ao-e"'* sin {udt + ^) is a transient and 

 vanishes with time the more c approaches wo . The second particular 

 solution is 



a, = V ^^'^i^M-^) _ , (9_896) 



in which (p is the phase shift between ground motion and record and is 

 given by tan ip = 2 cw/(o)o — (a), Avith (ojo — u)~^ as a frequency factor 

 (/o) characterizing the reaction of an undamped seismograph and 

 [(wo — u)^)^ + 4€V]"^'^ = (/d) as a damped frequency factor. For critical 

 damping, the damped frequency factor is (fd)f^i = (wo + w )~ and for 



