588 



SEISMIC METHODS 



[Chap. 9 



ze> 



so 





Fig. 9-99. Phase shift of seismograph motion as a 

 function of damping and tuning factor. 



0.7 critical it is (/d),^.7 = 

 (coo' + oty^'\ Fig. 9-99 

 shows the variation of phase 

 shift with tuning factor n 

 and damping ratio r. For 

 an undamped seismograph 

 (r = 1:1) a phase shift of 

 180° occurs at the resonance 

 point. This is smoothed 

 out the more the damping 

 approaches critical (r = 

 CO :1). For small damping 

 ratios and ground frequen- 



cies less than instrument frequency, the seismograph moves in the same 

 direction as the ground. At resonance it is 90° out of phase. For ground 

 frequencies greater than the instrument frequency it moves in opposition 

 to the ground. Since in eq. (9-896) X sin {(jit — (p) = x = ground move- 

 ment, the dynamic magnification of a damped seismograph (for sustained 

 oscillations) is Wd = a^/x. The ratio Wd/V is represented in Fig. 9-100 

 as a function of tuning factors (0-1 up to resonance and inverse factors 

 past resonance) and relative damping r}. 



It is seen in Fig. 9-100 that, with an increase in damping, the resonance 

 peak moves toward increasing ground frequencies. The resonance fre- 

 quency for a damped seismograph is 



2 

 Wo 



Vcog - 2e2' 



(9-89c) 



If e = 0, cor = ojo ; the resonance frequency is equal to the natural fre- 

 quency. There is no resonance peak (cor = =o) if 2t = wo , that is, if 

 71 = \/| = 0.7 critical. To eliminate resonance, critical damping is 

 therefore not necessary but merely 0.7 critical damping. 



The complete solution of the original differential equation (9-89o) is then 



a = Ao-e *' sin (codf -|- ^) 4- 



YXo,' 



V('^o - "2)2 + 4e2o;2 



sin (oit - <p). (9-89d) 



This represents a superposition of a transient and a recurrent phe- 

 nomenon, the former being the natural oscillation of the instrument set 

 off by the ground impulse, the latter the forced oscillation. The stronger 

 the damping, the more rapidly the natural oscillation dies away. Hence, 

 eq. (9-896) and Fig. 9-100 describe the phenomenon fully for steady state 

 conditions. 



6. Forced oscillations , onset conditions. Stationary conditions are gen- 



