160 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
7 d=(e)-Ca)e-Ceyte(@) © 
 N=T/V14 (x T/2r)? (38a) 
1 (xT 2 
m= Tl1—-3(s7) I (39) 
If, as in the majority of pendulums now in use, there is no especial device for damping, 
« is a very small quantity and we may, to a close degree of approximation, write T = T, 
and 27/7 =27/T,,. 
The solution, equation (36), represents a simple harmonic motion with decreasing 
amplitude given by a,e“; to determine the successive maximum swings in opposite 
sides of the central line, we put t=0, 7/2, 27/2, etc., in the expression. The ratio of 
these successive values of the amplitude is constant and equals “e’”; 7.e., if a), a4, a,, ete., 
’ are the successive maximum displacements we have 
If « is small, we can deduce 
a) cs ty KT /2 — 40 
G, Gy tls : . (40) 
this quantity is called the damping ratio. To determine the value of «, take the natural 
logarithms of both sides of equation (40); we get 
log,“ = 2.3026 log @ ="? =A (41) 
ay a 2 
where log stands for the logarithm to the base 10. A is called the logarithmic decrement 
of the amplitude. From this equation we can calculate «, but as it is difficult to get a 
good determination of the ratio of two successive amplitudes, we can determine « from 
the ratio of the zeroth to the nth amplitude, as follows: Multiply together the successive 
ratios of equation (40) and we get 
ey 
a 
nxT/2 __ Pu (42) 
n 
take logarithms of both sides of the equation, and we get 

og, ar ct 2.3026 log Ao = cd A (43) 
nr An n Ay 2 
This gives us more accurate values of « and A. The quantity needed to determine 
T,, in equation (39) is «7/27, and this becomes 
KT _ 2.3026 1, dy __ 0.733 154 4 _ A 
2a nar n a, 
(44) 
In determining « or «7/27, one naturally observes a, and a,,; but the logarithmic decre- 
ment, A, is a recognized constant, and is the quantity usually recorded to indicate the 
damping of the instrument. It is to be noticed that the logarithmic decrement is not 
a constant, but is proportional to the damped period. 
We also have from equation (38) 
KTy\? _ (xT\? kT : 
(22) Ge) / {t+ Ge) } ida 
and through equations (40) and (41) 
Mie ate eS Ok 44b 
eS) m+log2e« 1.862 + loge Sone, 

