186 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
E’ be the force of the compensating spring and let h’ be the length of the arm measured 
from the knife-edge. When the pendulum is displaced thru an angle @, the moment of 
the forces of restitution becomes 
rs ah 6 — E'h! 6 = (ple — E'h') 6 | (128) 
LOL 
and the period of free vibration is 
T, =24-V[1 (ple — Eh) (129) 
If the pendulum points in the positive direction of y, it only records relative deflections 
around the axis (1). To find the equation of motion of these instruments when subjected 
cere ay to a disturbance, we proceed as in the former cases. For the last- 
°:| ; 73) described instrument, using the same notation as heretofore, we 
ie or find the equation of linear displacement of the CG, 




(130) 
d*x d*y dz va ' 
Mo =F, MOE =F,+5,—(E— plib) 0, M& =F, +f.—Mg+(E-plé)—E 
The cosines of the angles between the fixed axes and axis (3) are (figure 61) 
cos (a, 3) = — a, cos (y, 3) = — (w, + 8) cos (2, 3) =1 (131) 
and the general equation of angular acceleration around axis (1) becomes 
ado 1 (/dé Pw - Pw a? dw Fi em es 
ae meet ee EL ee ee eee cae ele eee me he eh dee eee 
a? L es dt? ba dt? ) Be & y dt? dt? Se oat 
i 4 -: Tw, Pw, (pl~—-EN)O TL, Pu, | 132 
= (get O40 Ge XR) a ae Oe 
The weight Mg has been eliminated through equation (125). If we make the CG, 
the origin of codrdinates, omit the negligible terms, and add terms for viscous damping 
and solid friction, the equation becomes 
a6 dé , pl~— E'h' 
iat 
2, oY 
dt? co dt + 
ef ee! Pes) 1 
a O+> Eh (133) 

If we had used the Ewing form of attaching the spring to a point below the bar we should 
have obtained a similar equation with # and h substituted for H’ and h’. The indicator 
equation becomes, writing n? for (2 7/T,,)’, or (el? — E’h’)/{I], 
2 aT Ae 
—+2xn—+n%c—— — F p'=0 (134) 
where c is the recorded displacement of the marking point. These 2 equations are entirely 
similar to equations (25) and (26) for the horizontal pendulum, except that they do not 
contain a rotation. The physical explanation of this is that the position of equilibrium 
of the horizontal pendulum relative to the support is altered by the rotation ,, but that 
of the vertical motion pendulum is practically unaffected by a small rotation about 
any axis. If we place our origin at the CG,, the only term in the general equation (132), 
containing the angular acceleration about (1), is (/,/[/])d’w,/dé?, which corresponds to 
the term we considered on page 158 for the horizontal pendulum. The factor J,/(J] 
will in general be small; for a beam carrying a brass sphere 10 cm. in diameter, at a 
distance of 40 cm. from the axis of rotation, it would not be as much as 72¢5; and since 
d’w,/dt is, in general, much less than d*6/dé?, the motion of these instruments is only 
affected to an entirely negligible extent by a rotation around the CG). 
