162 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
become ¢l'(v,,— 6’); this represents a moment proportional to the velocity of the lever, 
and a second proportional to the displacement. 
In the intermediate case where neither v’ nor v” is preponderatingly large, the fric- 
tional moment is a complex function of their ratio and of the angular displacement. 
In any large swing the recording point may pass thru its position of equilibrium with a 
velocity much larger than that of the paper, but as it reaches the limit of its swing its 
velocity gradually reduces to zero; hence the nature of the moment brought into play 
varies materially during the swing. As the lever passes its zero position the friction 
exercises a constant moment; and as it approaches the maximum displacement the 
friction exercises a damping moment, and a force of restitution. 
It sometimes happens, on account of a slight tilting of the pier, that the pendulum’s 
equilibrium position is not exactly in a line with the pivot of the indicator lever, so that 
~ 



Fie. 42. 
the lever stands at an angle with the pendulum. The frictional moment has the same 
expression as we have already found except that we must replace 6” by 0’+6,’, where 
6,’ is the angular displacement of the indicator when the pendulum is at rest, and 6” the 
displacement from this position during a disturbance. The limiting cases (as on p. 161) 
become $1’ and ¢l'(v,,' — 6’ — 6,') if 6’ and 0,’ are not large; that is, in the second ease, 
we must add to the moments already considered another moment which tends to bring 
the pendulum back to the proper position of equilibrium. 
Let us see what is the nature of the frictional moment in a special case; let us suppose 
we have a simple harmonic swing of the marking point of period, P= 15 sees., and ampli- 
tude 4cm.; let the velocity of the paper be 1.5 cm. per minute, or v’’ = 0.025 em. per sec- 
ond. We have supposed the swing simple harmonic, which it would not be under the 
action of the friction, but it would be approximately so, and we can get a fair idea of 
the variation of the frictional moment under this supposition. If y is the displacement, 
we shall have 
y=a sin 27 ¢; a a Ka oan 
ta dt D Mss 
then 2 7a/P = 25/15 = 1.67, and putting the successive values of the sine in the general 
equation for the frictional moment (45), we find that the force does not vary much for 
something over an eighth of the period on each side as the pointer crosses the zero posi- 
tion, and it changes very quickly near the ends of the swings; for movements therefore 
in which the maximum value of v'/v'’ is of the order of 1.67/0.025 = 67, the frictional 
moment does not vary much in value during a large part of the swing. It would pro- 
duce a much too complicated expression to introduce the actual value of the frictional 
moment into the equation of the pendulum; the best we can do is to look upon it as 
made up of a damping moment, which would enter the general damping term, a moment 
proportional to the displacement, which would combine with a similar term in the equa- 
tion, and of a constant moment opposed to the motion, which would be represented, to- 
gether with pivotal friction, by the constant term of the equation. The importance of 
reducing all this friction to a minimum is evident, for we can not take accurate account 
of it. Hence the adoption of very heavy pendulums, which reduce the effect of the 
frictional forces on their motion. That the friction at the recording point is, in general, 
very important, is shown by the rapid dying out of the vibrations of a Bosch-Omori 
