Chap. 9] 



SEISMIC METHODS 



585 



03d = V COo 



(9-866) 



the damped natural frequency, is always less than the undamped frequency, 

 and approaches zero for critical damping. With (9-866), eq. (9-86a) may 

 be written 



a = A-e — sm cjd • t + -V cos cod t , 



which by substitution of B/A = cos yf/ and C/A = sin yp becomes 



a = A-e"'' &m{mt + ^) (9-86c) 



and is represented in time-amplitude and vectorial form in Fig. 9-98. 

 The peak ampUtudes decrease in geometric progression. A line connecting 

 them is an exponential, since the amplitude at the time f is At = Ao-e ' . 



— r 



T„ -I 



Fig. 9-98. Oscillation of damped seismograph. 



Thus, the ratio r of two equiphase peak amplitudes within one period 

 Ai Ao-e"'' 



TaiQT 



X..Q-^(i+ra) 



Hence, 



r = e-" = e"", (9-87a) 



where r is the damping ratio and fd = l/Td is the damped frequency. In 

 logarithmic form 



(9-876) 



logel = loge ~ = tTd = J = Ae, 



A3 Jd 



with Ae as the natural logarithmic decrement. 



In highly damped seismographs it is difficult to measure accurately the 

 amplitude ratios for one complete period. More suitable is the deter- 

 mination of the overshoot, which is the (reciprocal) damping ratio referred 

 to successive amplitudes within one-half of a period. It is expressed as 

 the overswing AA in terms of the original amplitude A so that 



AA = Ao-e"^-''. (9-87c) 



Another characteristic of damping is the relaxation time (Fig. 9-98), 

 that is, the time which elapses until the amplitude has declined to 1/e 

 of its initial amount: 



