MONTHLY WEATHER REVIEW Editor, EDGAR W. WOOLARD
CLOSED FEBRUARY 3, 1939 ISSUED MARCH 20, 1939 DECEMBER 1938 VOL. 66, No. 12
W. B. No. 1256
A SOLUTION TO THE PROBLEM OF ADJUSTING THE COUNTERBALANCE OF A SHIP-
BOARD THEODOLITE
By A. R. STICKLEY
[Weather Bureau, Washlngton; May l9%]
The usual shipboard theodolite is essentially a sextant mounted on gimbals and equipped with a horizontal circle.
In order to keep the sextant upright in the gimbal mount- ing, a shaft with a heavy weight or counterbalance on it is
rigidly attached to the outer spindle of the horizontal circle assembly. On some types of instruments, such as
that shown in figure 1 , this weight or counterbalance may be adjusted vertically. In the use of such an instrument, then, the problem naturally arises as to what is the optimum counterbalance adjustment for a given set of conditions aboard ship.
As far as can be learned, the only reference to any nt- tempt to solve this problem is that contained in an article written b F. Eredia on the “Esploration of the At-
Vessels” which wns published in volume IV the “Annali dell’UfXcio Presngi” appearing in 1932. A translation’
of his remarks regarding the problem of offsetting the effects of the motion of the ship is as follows:
This problem has been solved by supplying a Csrdsn suspension (gimbal mounting) with a pcndulum rigidly attached to it (the theodolite)-giving the pendiilum sufficient masR since the center of gravity should be rather lorn. The length of thc pendulum may be varied contiuuoudy in such a manner as to modify the period of oscillation so that it nvoids refionance with the period of om5llation of the ship. Moreover, three springs notably reduce the oscillation of the pendulum itaclf. Experience has shown that, having once regulated the period of the pendulum by means of trial in the best possible manner, pilot balloon soundings can be taken with the same ease as on land.
Since conditions, ns far ns the motion of a given slip is concerned, vary greatly from time to timeTde ending upon the type of sea encountered and the ship's [eading into it among other things-it is believed that I t will be of considerable advnntnge to be able to compute the optimum position of the counterbalance for a iven set of conditions investigations
have been pursued:
To begin with, the problem as to the proper counter- balance adjustment fof a free suspension (1. e., without springs) of the theodolite was considercd and an attempt
was made to apply the classical equation for forced vibra- tions with viscous dampipg to it. To do this the following
assum tions and limitations were imposed initially:
1. .e he motion of the theodolite was considered to be confined to n singleeplanethis plane being that per- pendicular to the &us of the most pronounced ringular
motion of the ship-its roll.
2. It was assumed that the theodolite had been so oriented that the axis about which the instrument oscillates within the gimbal ring was parallel to the ship’s axis of roll.
3. It wm assumed that it would be possiblo, finally, to so
mosphere % y Means of Pilot Balloons on Board Merchant
aboard ship. To this end, the folowing 7.
I Fwlsbed by Carl RUBSO of the Aarologkal Dlvlrlon.
126769-38-1
adjust the counterbalance that its amplitude of oscilla- tion with respect to the vertical would be less than 3O.
4. Damping proportional to the first pouer of the angular speed with respect to the vertical (viscous damp- ing) was assumed. The torques due to other types of damping were assumed to be negligible.
5. In accordance with the theov of the seismograph, the inertia effect on the part of the mstrument on gimbals was assumed to be concentrated at its center of gravity.
6. The error introduced by considering the direction of
the acceleration of the point of suspension due to the roll of the ship to be at all times normal to the counterbalance shaft was assumed to be negligible. 7. The effect of any torques which might arise when the theodolite moves through the air with the rolling of the ship or which might arise due to the blowing of the wind is assumed to be negligible. Under these assumptions, then, the equation of motion is:
(1) e+-+-=- Kde p3 -Mh e dt2 dt X K = where
e is the angle which the theodolite makes with the ver- tical,
K is the factor by which the angular speed must be multiplied in order to obtain the angular acceleration due to damping,
t is the time,
g is the acceleration of gravity,
h.1 is the mass of the part of the instrument on gimbals, h is the distance of the point of support from the center
of gravity of the part of the instrument on gimbals,
K is the moment of inertia of the part of the theodolite
on gimbals,
K X=m is the length of the equivalent simple pendulum,
y is the displacement (along the arc) of the point of sup-
If, now, Y bein the maximum displacement along the
and p being 2r divided by the period of roll of t e ship, the equation
is assumed to hold, then
and
port from its equilibrium position.
gosition
arc of the point o 4 support from its equilibrium
y=Y cos p t (2)
--- e- Yp4cospt d t 2 (3)
Letting a,, represent the amplitude of roll of the ship and H the distance of the point of support of the instru-
40 1
Monthly Weather Review. December 1938 Volume 66, No. I2
FIGURE 1.
402 MONTHLY WEATHER REVIEW DECEMBER 1938
ment from the axis of roll of the ship, equation (3) becomes
p’y= -@J€ p2 cos pt dt2
Substituting in equation (l), the required differentia
(4)
The expression for the amplitude of vibration obtained from this equation is found to contain, among others, the constant K. In order to evaluate this constant, the loga- rithmic decrement of the theodolite was measured. It was found, however, that instead of having a constant logarithmic decrement, as is demanded by the assumption of viscous damping, the decrements in the amplitudes of
free swing were approximately constant. This means that instead of the angular acceleration due to friction
being represented by t,he expression K- it must be repre-
sented by a constant. This is evident from the following
considera tions: If el represents the amplitude of any swing of the in- strument other than any one of the final three swings before the instrument comes to rest, and q and e, be the succeeding amplitudes of swing, the angular acceleration
equation becomes
E!+$+Se- @QHp? cos p t
dt2 dt - x
de
dt
IS’
where T represents the natural period of swing of the in- strument. Letting A0 represent the constant decrement
4 AB of the amplitude of swing, the expression - is finally T2 obtained for the frictional angular acceleration. Since T may, for the Purposes of this investigation, be regarded as
a constant, it follows then that the resulting angular
acceleration is constant. Lett,ing F represent the value of this constant, then, the equation of motion of the theodolite assumes the form:
the plus sign before F being used for one direction of swing and the minus sign being used for a swing in the opposite direction. A solution of this equation has been published by Den Hartog in Vol. 53 of the Transactions of
the American Society of Mechan ica.1 Engineers (Sect,ion
APM, page 107). Den Hartog, however, substitutes the ex ression cos Cpt+@) for cos pt so that the equation to be
so P ved finally becomes
(5 4
This change was made “for the purpose of subsequently writing the boundary conditions 111 a simple form.” Selectlng the negative sign before F the solution of this equation, which corresponds to Den Hartog’s (6)) is
+ *‘OHP2 [cos @(cos pt-cos pnt) (P:-p2>
+sin @ -sin p,t-sin pt
G. -
where p,=df. The values of cos I$ and sin 4 may be
obtained as is shown by Den Hartog (with the excep- tion that a positive instead of a negative sign should be used for the given values of sin @) and there results, then, as the equation corresponding to Den Hartog’s equation (9)
where L=KX F= the torque due to friction.
contained in the
last term on the right-hand side of this equation is ex- amined, it is found to become indeterminate whenever u assumes any of the odd integral multiples of T as a value. An application of L’Hospitnl’s rule for evaluating inde- terminate forms to this function shows that it becomes infinite a t these points. Since the coefficient of the square of this funct,ion does not vanish for finite distances of the counterbalance from the point of suspension and the first term on the right-hand side becomes infinite only when Xp2 approaches g as a limit (which, assuming that the period of roll varies between 6 sec. and 20 sec., can happen
only if takes on a value lying between 10 and 10,000 meters), it might appear that this equation would give negative values of 0; a t the points in question. As Den Hartog has pointed out, however, (L limiting condi- tion as to the application of (7) is furnished when it is considered that the negative sign before F was chosen in obtaining the desired solution. This, of course, means
tbat - must be considered negative for the interval for
which (7) applies. When equation (6) is differentiated with respect to the time and this fact is utilized, it is found
that the relation
sin u
1+cos u When the function of the form
de
dt
sin E a
l+cos p”T
F P
sin p,t + (cos pt-cos p,t)
(8) sin pt eo>- PPn
must hold for all values of t in the interval O<t<z P (assuming that t =O a t the beginning of the period for which the negative sign of F applies). If it can be shown now that the expression on the right-hand side of (8) is posit,ive for any one value of t in the given interval, Bo must always be greater than such a value in order for (7) to apply and hence will always be positive. The existence of the value required may be shown by applying L’Hospital’s Rule to the function for the case in which
t=O (which may be done since the function becomes indeterminate for this value of t ). Upon doing this it is
found that the limit - is approached by the function as
t approaches zero from above. This, then, means that eo
can only be less than - by an inhitely small amount
in the neighborhood of t =O .
If it is now assumed that equation (7) holds for all po- sitions of the counterbalance except those eliminated by (S), it is evident that the slope of the curve so obtained
will be positive on one side of the region for which (7)
does not hold and negative on the other side of such a
region. This indicates that a minimum value of eo lies
F
PPn
F
and the required value o pP the function is thus found to exist
DECEMBER 1938 MONTHLY WEATHER REVIEW 403
within these regions and that the position of the counter- balance for which a minimum occurs is approximately that for which the last term of equation (7) becomes infinite. On the basis of this conclusion and with the use
of the data for the theodolite previously mentioned the curves shown in figure 2 have been drawn to indicate the manner in which the counterbalance should be ad'usted.
ship varies more than a second from the mean period for which the counterbalnnce has presumably been adjusted, the position of the counterbalance will then be in the neighborhood of thnt for a maximum instead of a mini- mum. That such a variation is of an extremely frequent occurrence is shown by an inspect,ion of the record of the
I t is evident from this figure that if the period of rol 1 of the
The center of mass will still move toward the lower side of the ship in its rolling as before. A necessay condition that the center of mass of the theodolite shot d be at an extreme in this motion is that there should be equilibrium between the torques exerted by dl the forces involved-- i. e., between the torques exerted by the forces due to the spring, friction, gravity, and the inertia of the t,heoclolite. Assumin (1) that one leg of the tripod lies in a plane
opposite side of the tbeodolite from the lower side of the ship in t,he roll considered, and (3) that the springs are taut but not extended when the deck is level and the counterbalance shaft is vertical; the torque exerted by the spring under tension is given by the expression
perpendicu Y ar to the roll axes, (2) that this leg is on the
roll period of the U. S. S. Pensacola while she wns underway in a calm sea in which a moderate swell existed. The mean
period of roll for 155 observations was found to be 11.5
seconds and the average deviation from this mean was found to be 1.2 seconds. Hence, even with very nearly ideal conditions as far as the sea was concerned and with
a 10,000-ton ship, the variation in the period of roll was found to be quite su%icient to give just about as many maximum amplitudes of oscillation as minimum &rnpli- tudes. It is then evident that for a free suspension of the
theodolite, the position of the counterbalance is practically immaterial. If, however, the device of connecting the counterbalance shaft to the tripod legs by light springs is employed (which device was mentioned in the previously uoted article
States Navy since 1928, or before) and an absence of pitching together with a fixity of the axis of roll with respect to the ship is assumed, verticality of the theodolite
at the extreme of a given roll can be attained if springs of
the proper stiffness are used and the counterbalance is
properly adjusted. The stiffness of the springs to be used
and the way in which the counterbalance adjustment
should be made is derived from the following considera-
tions.
by Signor Eredia and has also been in use 1 y the United
kqw (I--t) sin (y+@,,+8) z
where, referring to figure 3:
r=AC= the distance between the points of attach- ment of the spring to the counterbalance shaft and to the tripod leg respectively when the spring is unextended. q=CO=the distance of the point of suspension of the theodolite from the point of attachment of the spring to the tripod leg. w=AO=the b t a n c e of the point of suspension of the theodolite from the point of attachment of the spring to the counterbalance shaft.
y= LAOC= the angle formed a t the point of suspension by straight lines through the pomts of attachment of the spring when the deck and tri od head are level and when the counter-
z=the distance between the points of attach- ment of the spring at the instant considered. k=the stsness of the spring.
For a roll in the opposite direction, the torque exerted
ba P ance shaft is vertical.
by the two springs combined is given by the expression
2kaw - ("+ sin (a+a,,+e) 9
MONTHLY WEATHER REVIEW D~DCZYBEB 1938 404
in which, referring to figures (3) and (4),
8=DO=the slant height of the pyramid formed by the point of suspension of the theodolite and the three points of attnchent of the springs to the tripod legs.
a= LBOD= the angle formed by drawing lines from the point of suspension t h o u h the extremities
oints of tho parallel sides of the trapezoid formed by the four points of attachment of the springs. concerned,
of the altitude drawn t H, rough the mid-
and u is given by the equatlon
v2 = az- 2a Jy2 - s2 + q2+ WP- 2208 cos (a + a,,+ e)
in which u=A'B=A''B is one-half the distance between
the two points at which the springs are attached to the coun terbalanco shaft. The torque excrtcd by friction is, as has been men- tioned previously, L=KXF=const., K being the moment of inertia of the theodolite and F being the angular accol- erntion duo to friction. The torque due to the inertia of the theodolite is:
M M H ~* COS (*+ao+e)
where *=the angle whoso tnngont is givon by tho ratio of the distance of tho point of suspension from the medial lane of the ship (the plane which divides tho ship kngitudinnlly into two symmetlicnl parts) to the height of the point of suspension above the water
I
I I I
I I
I
I I I
I I
I I I
\\
\
DICEMBEIB 1938 MONTHLY WEATHER REVIEW
C
haws 4. Sca/e /:5
405
406 MONTHLY WEATHER REVIEW DECEYBZB 1938
plane of the ship when the ship is in its equilibrium position. Now it is evidently possible to adjust the counterbal- ance so t,hat illh has such a value that when it is multi-
plied by H@,p2 cos Qo i t is equal to tfhe torque exert.4 by fric.t,ion plus the torque which the spring e.serts for a vortical position of the theodolite when the ship is in an estrenic position in its rolling. This adjustment having been made, the only value of 0 which will satisfy the necessary condition set forth abovo for having the cent.er of mass at an extreme position in its motion toward the IOMW side of the ship in its rolling is the value zero-i. e., assuming that ?Ir=O. For, considering 0 to increase as the center of ma.ss of the theodolite (lying below thepoint of suspension) moves towards the lower side of the ship in its rolling, a positive value of e a t the extreme of a roll will mean t,liat the inertial torque-acting. in positive
direc, tion--has decreased as compared with its value when 8=0, while, on the other hand, the spring torque- ac.t,ing nega t'ivelg-has increased as com ared with it,s
a resultant negative torque which when augmented by the negative t,orque due to gravity becomes even great,er. Similarly, t,he chaiiges in the inertial and spring torques c.aused by assuming a negative value. of 0, will procluce a
positive t,orquc? cind, when the positive t>orqiie cluc? to
gravity under these circumstances is d d e d in, the result- ant. torque has an even reater positive value. It then
required condit,ion is, as has been said, tallat of zero. Therefore,, a.ssuming that t,he lower side of the ship in its rolling is on the opposite side of the point of suspension from the tripod leg which lies in the plane athwart ship, the torque equilibrium equation is:
,llh&,Hp2 COS *o=L+kgW- sin (y+a0) (9)
where ho= the, dista.nce of the point of suspension from the cen- ter of mass of the theodolit,e corresponding t,o t,he, required c,ounterbitlance adjustment.
and
zo= the distance bet,wccn the points of attachment of the spring when tho theodolit,e is vertical at the ex-
treme of the roll.
raluc when e=o. These two chaiige,s wil P then produce
follows tlint, t,lm only va B ue of 0 which will satisfy the
ZO
NOW A&=L?!f~O+iIfiX1
where M,=the mass of the counterbalance, xo=the distance of the pomt of suspension from the cen- ter of mass of the. counterbalance w h n the re-
quired adjustment,is made, it41=the mass of the part of the theodolite on gimbals less the mass of the counterbalance,
and xl=the distance of the center of mas8 of MI from the point, of suspension. Therefore,
Similarly, for a roll in the opposite direction
do k' and a. being the quantities corresponding to ro, k, and zo, respectively, for the direction of roll first considered. In deriving the quantity H which appears in both (10)
nnd (l l ), i t is bcliewd that it will be found sufficiently iiccurate to assume that the mean position of the axes of roll lies a t the intersection of the medial plane of the ship and the ship's water plane. If the point of sus- pension lies in the medial plane of the ship, therefore,
the qunntity H will merely be the height of this point above the water plane of the ship when the ship is iii an equilibrium position. If, however, the point of sus- pension doas not lie the ship's medial plane, H must be tnken as Height of suspension point
cos JI
Also, the following facts are to be borne in mind in tipplying these equations:
(1) Since in the determination of the constant L. the piinbal bearings were perfectly dry, all traces of anything that tends to act as a lubricant should be removed from tl)ein. This should be done, if possible, by washing them
in solvent for such lubricants.
(2) The points at which the springs are attached to tlic counterbalance shnft are assumed to be fixed with respect to the shaft. This means that in the case of
the instrument shown in figure 1, the movable ring into nliich the springs are hooked must be fixed so that the ring cannot slide along the counterbalance shaft. One of
t1.c best wnps to do this would seem to be that of putting
LI set screw in the ring in such a manner that when it is tizhtened all sliding on the shaft is prevented.
To facilitat,e the application of equations (10) and (l l ),
1111 necessary measurements have been made on an instru- ment of the type "Aero 1928" manufactured for the
U. S. Navy. Since the instrument is equipped with an extension type tripod a.fixed spread of the legs was arbi- tr:iriIg adopted-it being assumed that the device of stretching a given length of chain between the tripod legs woiild be iised to give the chosen amount of spread uhenever the theodolite is set up. The spread adopted
was a little less than 25'. Assuming that the three lmgths of chain are connected by the closed ends of t h e e small hooks which are furnished by the instrument mnnufacturer and which have the other ends attached
to the tripod legs (as is shown in figures 3 and 5) the lcngth of chain required for the adopted spread is 25.27
inches. This corresponds to the length of 128 links of
t lie chain furnished by the manufacturer with the instru- ment. When this length of chain is used and the movable ring is clamped against the bottom.flange of the counter- balance shaft, the constants used in equations (10) and
(1 1) arc found to have the following values:
u= LBOD= 13.29O
y= LAOC=24.65'
r=AC=12.39 in.
w=AO==24.92 in. q=C0=29.28 in.
.s=DO=27.01 in. a=A'B=A''B=0.68 in. rl=-0.15 in. (the minus sign meanin that the center
of mass of Ml lies a % ove the point of
suspension)
-If*= 10.01 lb.
M2=8.93 lb. L=5.05 lb. ft. in. sec.'p
DECEMBER 1938 MONTHLY WEATHER REVIEW 407
1.696 1.170
0.206
Subst*ituting these values of the constants in equat,ions
(10) and (11) there results:
_____ _____ ___-_
zo- 13.39 sin (24.65°$@0) r,-0.17 in.=2.070k a, cos a,
1.744
1.440
O.1M
where
_____ _____ _____
~0=[1478.4349- 1459.3152 COS (24.65'+@0)]''~
1.771 1.679 0.070
a.nd
_____ _____ _____
sin (13.39°+~o)
a0 cos a0 xo' - 0.17 in. = 3.8 19k'
1.793 1.893 0.053
(1 W +1.432X10-2(g). cos 1 @o
__-__ --___ _____
where
uO= [1463.6445- 1346.1748 COS (13.39' +@o)]"2
I n order to apply these equations it, is further necessary to know where the center of mass of tliP counterbalance is
with reference to, say, the top of tjhe countr*rbalance, and
it is also necessary to know how far some convenient reference point on the theodolite proper is from the point of suspension. The distance of the counterbalmce's center of mass from the top of the ring which serves to
clamp it to the shaft was found to be 2.53 in. The distance of the counterbalance shaft flange which serves to limit the upward motion of the counterbalance from the point of suspension was found to be 2.22 in. If, therefore, it is
desired to use a value of x of, say, 10 inches, the counter- balance would be lowered (10.00-2.22-2.53) in. = 5.25
in. from the uppermost position which it is possible for it to have on the shaft. To further facilitate the application of these equations to this instrument the table given below has been worked out in the followin, 0 manner: An inspection of equations (loa) and (l l a ) shows that
1.777 1.734 0.065
M I X I Tz both x o +X a n d x;tA* are directly proportional to -. H
_____ _____ __-__
The factors of proportionality me seen to be composed of
two terms. Let the first term of the proportionalitmy factor in equation (loa) be represented by
- -12.39) sin (24.650+a0)
A= cz*070(~0 zo@, cos a, -1 and the first term of t,he proportionality factor in equabion
(1 la) be represented by
1.816 2.080 0.044
3.819(Vo- 12.39) sin (13.29°+@0)
voao cos a0 -I
_-___ _____ _____
The second terms of the two factors of proportionality are seen t.0 be identical and will be represented by
1.842 2.200 0.037
1.432X10-2 B= a, cos a0
_____ _____ ___._
Now the proportionality factor to be used in obtaining the required adjustment of the counterbalance is evidently
the mean of the two proportionality factors for z o +x A I l X ,
Mixi a n d z o ' +X - This mean can, of course, be obtained by
1.822 2.132 0.042
adding the mean of the values of A and A' to the value of B. This operation is provided for as follows in the table mentioned:
A set of six spaces has been provided for each whole degree of value of 4j0 from lo to 24' inclusive. These six spaces are numbered as is shomn in figure G and the data intended to be used in these spaces are those indicated in this figure. Hence in order to obtain the required value of the proportionality factor to be used in obtaining
X O f X g ) JlIXl +- for a given angle, the product of the datum 2 hf2
given in space 1 and the value of k used is entered in space 4, the product of the datum given in space 2 and the value of k' used is entered in space 5, the inem of the data in spaces 4 and 5 is added to the datum in space 3
and the result of the addition is entered in space 6. It is
then only necessary to multiply the datum in space 6 by the proper ~- value of T2/H and subtract (algebraically) the
_____ _____ _____
value of M1xl - from this product in order to obtain dl2
required setchg of the counterbalance.
10
bo
FIGUKX 6.
TABLE 1
20
I
I 1
I
7a
6 O
100 I 110
14' 1 l k
I I I
the
408 MONTHLY WEATHER REVIEW DECEMBPE 1938
I n computing the quantities given in spaces 1 ,2 , and 3, the British Absolute system of units has been used, i. e ., the unit of mass is the pound and the unit of force is the poundal. Hence, the stiffnesses k and k' are to be deter- mined in poundals per inch, i. e., the stiffness of a spring is found by observing the number of inches of elongation I produced when a mass of m pounds is suspended from it and then multiplying the quantity m/l by the acceleration
of gravity, 32.2 ft. sec.-2. In determining the value of P /H to be used, Tis to be measured in seconds and His to be measured in feet. Regarding the computation of the proper spring stiff- nesses, the guiding principle is to select springs of such stiffness that the counterbalance will be approximately at the mid-point of a convenient adjustment range for the average conditions aboard the ship to be used, and also to select them in such a way that zo and ro' will agree for these prevailing conditions. These computations have been carried out for three types of ships: destroyers, cut- ters, and battleships or heavy cruisers. For destroyers
P /H has been assumed to have a value of 1.5 seca/ft. and
8' has been chosen as the average value of a0. Selecting z=15.34 inches as the midpoint of a convenient counter- balance adjustment range, k should have a value of 5.21
poundals/in. and k' should be 6.30 poundals/ii. For cut- ters T2/H has been assumed to have 6.0 sec.'/ft. as a mean value and 8 O is assumed to be the average amplitude of roll encountered. This, then, gives 1.39 poundals/in. as the value for k and 1.68 poundals/in. as the value for k'. Aboard battleships and heavy cruisers P /H has been assumed to have-a mean value of 6.0 sec?/ft. while the mean value of a0 IS assumed to be 5O. The required stiil'- nesses under these circumstances turn out to be 1.38
poundals/in. for k and 1.90 poundals/in. for k'. An inspection of table 1 will show that for rolling in the neighborhood of 14O, k and k' should have the same value in order to produce good agreement between A and A'. Assuming the same values of T2/H as before, k and k' should both, therefore, have the value of 5.21 poundds/in. when 10°<@0<170 aboard destroyers. Aboard cutters, their common value will be 1.39 poimdals/in. when
1O0<@,,< 17O, and aboard battleships and heavy cruisers this common value will be 1.38 poundals/in. when 8 ' <4 ~~
<17O. Should the mean amplitude of roll equal or ex- ceed 17O, k' should be given the values 5.21, 1.39, and 1.38
poundals/in. for destroyers, cruisers, and batkleships, respectively, while the corresponding values of k will be
5.87, 1.57, and 1.55 poundals/m., respectively. Except when expressly stated otherwise, i t has been assumed in the whole of the foregoing calculations thzt,
P=O. When this is not the case, the factor cos a0 appear- in in the denominators of the terms on the right hand
by cos (ao-*) while this factor in the denominators of the terms on the right hand sides of (11) and (l l a ) should be replaced by cos @J~+P), i. e. assuming that the "athwart
ship leg" of the tripod has been placed outboard. To better the agreement between z,, and do under these cir-
cumstances, the value of k to be used should be taken from the formula
si % es of both equations (10) and (loa) should be replaced
15.17 COS (a0-q)
k=&[ (Tz/B) cos a0
and the value of k' should be taken from the formula
-.1
15.17 COS (+o+q)
In these formulae both A /k and A'lk' aro regarded as constants, whose value, along with that of B is t.0 be obtained from spaces 1, 2, nnd 3, respectively, in table 1.
Assuming that the springs having the stiffnesses given by the above formulae have been supplied, spaces (4) and (5) of Table 1 are then to be filled in using these values
of k and k'. Next 7-. "( x o+g) must be found by adding
B, the datum in spa.ce (3), to A, the datum in space (4), r;
cos @" and then multiplying the result by cos Similarly .- ,
must be found by adding A' to B and
The mean of the cos +o
cos (@o+P)- multiplying the result by
values $(~+x) A l ,X , and ;(%'+%)is then, of course,
~.
equal to F(-T+%), H ro+l0'. the datum to be put in space A&
(6), and the remainder of the procedure is the same as
that for the case when P=O. To illustrate the various ways in which table 1 is to be used, the following examples are given:
(1) Suppose that the point of suspension of a the- odolite set up on board a battleship lies in the medid plane of the ship and 40 feet above the water plane and that the mean period of roll is 16.2 seconds, the mean amplitude of roll is 6' and that, conse uently, the stiffnesses of the springs being used are?t=1.38
poundals/in. and k'= 1.90 poundalslin. The entry for space 4of the 6°sectionof table lis then1.726Xl. 38=
2.3s. The entry in space 5 will be the product of 1.311 and 1.90 or 2.49. The mean of these two products is 2.44. Adding this mean to the datum in space 3 we have 2.58 as the entry for space 6 and the completed section of the table for the roll a.mplitude of 6O will then read
Hence the required counterbalance adjustment will hn --
(16'2)aX2.58- (-0.17) = 17.1 in. 40.0
This means that the counterbdance will be lowered
from the topmost osition which it is possible for it
the counterbalance along the shaft furnished by the interference of the tripod legs is not, of course, to be considered in making this adjustment).
(2) Suppose, next, that all conditions are the same as those stated in the first example with the exception that the point of suspension has been moved 30 feet
from the medial plane of the ship (the value of @ being,
consequently, tan" 30=36.90 instead of zero as was
the case in the first example) and that the spring stiffnesses are, therefore, different from those used in
the first example. Assuming that the athwart ship
17.1- (2.2+2.5)= 17.1 -4 .7 ~ 12.4 in.
to have on the sha P t (any limitation to the motion of
40
DECEMBEE 1938 MONTHLY WEATHER REVIEW 409
leg of the tripod (to which the spring with the stiffness
k is attached) has been placed outboard and assum- ing, further, that tho same avera e values of a0 and
when the point of suspension lay in the ship's medial plane are to be used in computing the spring stiff- nesses in this case, we have for the valucs of these spring stithesses:
P /H 8s were used in computing t % e spring stiffnesses
k=-- 1 15.17Xc0~ (5-36.9)0-0.165
1.713 G.00 cos 5'
= 1.16 poundals/in. and
k'=- 1 15.17 cos (5+3fi.9)o-0.165
1.244 6.OOXO.996'2
= 1.39 poundds/in.
The datum entered inspace4 will then be 1.1 6 X 1.726 =
2.00. That tobeent~eredinspace5willbe1.39X1.311=
1.82. For the value of F~ "( za+x M1xl) we have:
and for the value of G+M, we have:
(
Hence for the datum to be entered in space 6 we take the mean of 2.48 and 2.66 which is 2.57. The com- pleted section for the 6O amplitude of roll then reads:
Mult-iplying the datum -- in space 6 by T2/H and
MIX~
M2 subtracting the value of -2 we have
2-57 (16.2)' COS 36.9'- (-0.17)=13.7 in.
40
as the required counterbalance ad'ustment. As
lowered 13.7-4.7=9.0 in. from its topmost posiLion
on the shaft.
In figure 3 it will be noticed that instead of having only
a spring stretching between A and C, a spring with a short piece of chain attached to it is stretched between these two points. If the spring is fastened to the connterbal- ance shaft, and the chain is fastened to the tripod leg, this arrangement will eliminate any torques that mimht arise
on account of the buclding of the springs. Furtgermore, the part of the distance covered by the length of the
s ring should be very nearly the same as that shown in
much greater than that shown will buckle for large swings of the theodolite, while the elastic limit of a spring which
is very much shorter than the one shown is more apt to be exceeded for large swings of the theodolite than is the cme for the longer spring. Hence if 10 links of the kind of chain furnished by the manufacturer for limiting the spread of the tripod legs is usod with a spring 9.22 in.
before, this means that the counterba 1 ance must be
t E e figure. This is true because a spring whose length is
long (9.22 in. is the distance between the inside edges of the spring couplin s when the spring is unstretched), the
ing the length of the two hooks which are linked with the eyelets nt A and C) will be equal to 12.39 in.-which is the length of AC required. In order to secure s rings of t.he proper length and stiff-
formula giving the stiffness of a closed coil helical spring.
This formula is
combined length o K the spring and chain assembly (includ-
ness it will, in gencrrr., 7 be necessary to make use of the
d4B
G4np k=-
where d is the diameter in inches of tho wire used, B is the coefficient of rigidity in poundals per square inch, n is the number of turns used and p is the mean radius of the coil in inches. If phosphor-bronze (the alloy usually used
in making such springs) is used, Ghas the valuo 1.996X lo7
poundals per square mch. Finally, the effects on the motion of the theodolite of variations of T and +o from the assumed mean values may be determined. In order to do this it is necessary to make us0 of the more general torque equations, i. e., those equs- tions which allow for other values of B than that of zero. These equations are:
kqw- sin(~+ao+e)+Mgh sin B+L
=ma,+ COS (*+ao+e)
2--f
2
(14) 4x2
where
and
z = [a'+ W B -~~W COS (7 +ao+ e)l1'*
2k'sw ('-') sin(a+aO+B) +Mgh sin B+L
v =[a ' -2 a J ~+q 2 +d -2 w s cos(a+a,,+e)
It is at once evident that it would be very difficult to solve these equations for Bo u s b T and 9, as the independ-
pendent variables and the corresponding values of T ascertained. Using a mean period of T= 15.58 sec. and a value of 5' for the mean value of a0, the values of T which d l give -3', -2O, -lo, O', lo, 2 O , and 3' as values of 8
have been computed for each of the cases in which a. takes on the values of 5 O , go, 13', 17O, and 19':this being done both for tho case where the single spring is being stretched and also for the case where the two s rings are being stretched, and the resulting vdues of T ! eing designated
a- T' and T" res ectively.% Usinp the same assumed
ao, t'he values of T which will cause 6 to vanish when +0y50, go, 13O, 17O, and 19' have also been computed.
This latter set of computations bas also been carried out for the case in which tbe coiinterbalance has been adjusted
for 17' as a mean value of a0. Also the values of T, which will give -3', O', and +3O as values of 8 when mean values of T and eo of 15.58 sec. and 19', respectively, aro assumed, have been computed for the cases in which Q0
takes on the values ti', go, 13', 17', and 19'. Finally, assuming a mean value of T=11.S sec. and 5' as a mean value of a0, the values of T which will give -3' and +3' as values of 8 have been computed for the 5', Be, 13', 17',
ent variables. B and a0 can, E owever, be used as inde-
period (15.58 sec.) f ut a value of 9' or the mean value of
8 'Y was arbitrarily piren the value zero In these computations, M well M In all the am- putations suresedlnu them.
410 MONTHLY WEATHER REVIEW DECEMBER 1938
0 0 50 go
T' 14.m 15.24
T' 17.37 16.18
_______-
tions are shown in the following tables:
13O 17' 19'
-----~
15.10 14.97 14.93
15.19 14.68 14.08 -
TABLE 2
Asswned mean values: Qo=5', T-15.53 sec.
I I
-Tn 5 O 9" 13O li0
-----~
T' 15.48 15.51 15.48 15.42
T' 18.94 17.66 16.57 16.72 ---____-
and 19O values of a,,. The results of all of these computa- 5 O and 19O were selected as mean values of @o in these computations since, with the use of the method of selecting spring stiffnesses previously outlined, r0 and x ' ~ a.gree for these mean values of a,,. Similarly 9' and 17' were also
1Q0
15.39
15.37
~
T' I 7.28 8.98 10.02 10.71 10.Y7
10.20 11.19 11.81 12.04
@-+30 _-____---
~
I g:: I
8.44
1
9.06
I
9.32
1
9.38
1 --
r
Assumed mean values: 0 0 =1 9 ~, T=15.58 see.
% so Be 13' 17O 19"
__----____
"- ,- [I T' I 8.32 I 9.58 1 10.03 1 10.16 1 10.17 I
#=+3"
{
11 T' 1 15.58 I 15.84 1 15.61 1 15.56 I 15.53 I
T' 6.41 7.68 8.39 8.54 9.00 -----____ T' 6.21 7.08 7.39 7.47 7.53
------- T' __.______ ______ ___ 31.61 19.84 18.20
T' .________ _________ 21.08 1 4 4 2 13.08
-- ---- -
.-- T' 16.58 14.90 13.51 12.74 12.51 ---__---
d :; I
________.
1
27.20
1
21.41
1
19.49
1
19.41
I
@=-lo ---___-
~
.________ 23.38 17.40 15.16 14.45
I1 T' I .-_____._ 1 .________I 59.51 1 3 .5 1 1 26.25 1
8-0
TABLE 5
TABLE 6
I Assumed mean values: %:So. T=11.5sec. -1
selected as mean values of Go due "to the fact that xo and x ' ~ differ most for these mean values of @o when the spring stiffnesses just mentioned are use.d. -3' and f3' were selected as limiting values of 8 because the theodolite for which these computations a.pply has a field of view of 6'
and any grea.te,r values of e (ta.ken absolutely) would, of course, carry the bnlloori out of t,his field. It will be noted that no value of T' or T" is give.n for a
good man cases where 0 was assumed to have a negative
value. $for instance, equat.ion (14) is solved for T2, t'he expression representing its value will be a fraction having
the coefficient of in equation (14) as the numerator
and the left-hand side of (14) d l constitute the denom- inator. Whenever the denominator of this fmction took
on a negative value t'he omissions just referred to were made. This was done bec.ause the va.liies of T in these
cases became imaginary unless c.os (\k+$+O) became negative in the ncimerator. Since, if for no other reason, the mechanical arrangement of t,he theodo1it.e mould pre- vent this,. it wns not thoiight worth while to finish the cornputations for these negative denominat,or values. An inspection of these tables shows that for O=Oo, T' in contrast to T", rema.ins practically constant throughout the whole of the range of the values of @,, chosen--this being true regardless of the assumed mean value of a0. It will further be seen that, as was to be expected, these values of T' difler more from the assumed values of T for
@o=90 and a0= 17' than is the case for the assumed values a0=5' and 30=190-tmhe error being due, of course, to the fact that the counterbalance setting corresponds to the mean of the values of x, and and to t,he fact that these two quantities differ most when the assumed mean values
of a0 are 9' and 17Orespectivel . In view of these two facts then, i t is apparent that the su % stitutinn of a second spring lying in the at,hwartship plane of the, ship for the two springs not lying in that plane would greatly improve the
results secured. *4 very simple way to construct a spring and chain assembl of this kind is to fasten the chain end to a ring or toroiJwhich can in turn be fastene,d to the tripod by resting it on the tripod legs and providing stops to prevent its sliding up the legs. Since considerable work is involved in prepanng tables similar to table 1, i t will, of course, be advantageous to design this ring and to locate the stops in such a way that the triangles formed
by the point of suspension and the pivoting points of the spring and chain assembly are congruent to the corre-
sponding triangle shown in figure 3. Should it be desired to use a device of this kind, the necessary trigonometrical calculations for its design have been carried out and may be had upon request. Furthermore if such a two-spring assembly is used instead of the usual three-spring assembly, tables 2 and 5
in conjunction with the roll record of the U. S. S. Pensacola previously referred to indicates tho possibility of a con- siderable reduction in the optical field of the theodolite together with the consequent possibility of an increase in the theodolite's msgmfication. To see this, reference
need only be made to figure 7. In this figure the curves marked 8=l0, 201 301 -lo and -2' indicate the ampli-
tudes and periods necessary to produce the designated values of 8 when (1) the two-spring assembly described above is used, (2) T is assumed to have a mean value of
11.5 sec., and (3) the mean value of a,, is assumed to be 3O.
The scattered points in the figure indicate the values of
the various amplitudes and penods of roll observed aboard
1
.
DECEMBER 1938 MONTHLY WEATHER REVIEW 41 1
the Pensacoh in the set of observations previously re- ferred to.a When it is now considered that the natural period of roll of the Pensacola is 11.5 sec., it mill be seen that a fact which is known to those familiar with the behavior of ships at sea is well illustrated here, i. e., that, excluding storm conditions, virtually all hea.vy rolling has a period which is closely coincident with the natural period of roll. If, therefore, the natural period of roll is used in obtaining the counterbalance adjustment from table 1
instead of the average period RS obtained from tiinin5 a comparatively small number of rolls of the ship (which may, of course, differ considerably from the natural penod), the large amplitudes of roll of the ship will have
an increase in magnification would only mean t,hat the field of view of the shipboard theodolite would have to be reduced from 6’ to 2.4’.
Summing up then, under the assumptions made it has first been shown that, due to the scattering of the periods of roll depicted in figure 7, the position of the counter- balance with the theodolite swinging free is immaterial. Secondly, under somewhat more general assumptions it has further been shown that it is possible to obtain a fairly satisfactory amount of stability for the theodolite by con- necting the end of the counterbalance shaft to the tripod legs with light s rings and coordinating the adjustment of the counterba I; ance with the stiffnesses of the springs
I4
13
12
1 1
10
$9
f 7
4
’t, $8
$6
Q
P 5
4
3
2
1
n
F
0 1 2 3 4 5 6 7 8 9 IO It 12 13 14 15 I6 17 18 19 20
Period, s e corrd.
v
FIGUBE 7.
very little effect onthe verticahtyof the theodolite.‘ When
this fact is considered along with the fact that of the 155
amplitudes and periods of roll observed, not a single roll
waa sufficient to produce a 1’ deviation of the theodolite from the vertical under the circumstances described above,
the possibilities as to the reduction of the field of view of the theodolite along with the consequent increase in mag- nification are evident. It seems quite possible, in fact,
that the 20 power magnification possessed by the ordi- nary “shore” theodolite may, under these circumstances, be available for use in the shipboard theodolite since such
a These data are published wlth the kind permisdon of the Chief of the Bureau of Construction and Repalr of the Navy Department.
4 It is evtdont that, under these ctrcumstances. tho obtaininu nf the proper counter- balance adjustment In greatly simplifled. For T being taken as a constant for a gircn ship and H being taken as a constant for any one position ofthe theodolitn on the ship the cmmterhnlance adjustments themselves may he entered In spa- 8 of the amplltudd of roll sections of table 1 and, hen-. a qimple reference to the table thus filled in will rep& tho computations previously Indlcated.
used. Finall , using the same set of more general assump- tions, i t has a 9 so been shown that a much more satisfactory amount of stability can be secured by using two springs instead of three to counteract the inertial torque on the theodolite and that it seems probable that this stability will be sufficiently great to warrant a considerable decrease in the theodolite’s fieId of view along with a corresponding increase in the magnification secured.
ACKNOWLEDGMENTS
I wish first to acknowledge my indebtedness to Mr. D. M. Little, Chief of the Aerological Division, for m a b p it possible to carry out the investigations necessary for
the attainment of the results set forth. My indebtedness to Dr. E. W. Woolard, Chief of the Meteorologicnl Re-
412 MONTHLY WEATHER REVIEW DECEMBER 1938
search Division, for his helpful advice on the mathe-'- the advice on the aspects of the principle of the seismom- matical aspects of the problem is also gratefully acknowl- eter applicable to the problem furnished by Dr. Frank edged. The aid and advice furnished in connection with Wenner of the National Bureau of Standards. Dr. H. L. the design and construct,ion of the apparatus for deter- Dryden, Chief of the Mechanics and Sound Division of mining the theodolite's moment of inertia by many of the tho Bureau of Standards, has also very kindly read over personnel of the Instrument Division and, in particular, the aper and has made many suygestions which have
is also to be acknowledged with thanks. Further indebt- are expressed to Mr. J. W. Clary. Principal Engineer of edness is expressed to Lt. Comdr. T. J. O'Brien and to the Bureau of Construction and Repair of the Navy, for Lt. Comdr. J. B. Anderson who, as officers in charge of his generous cooperation in the matter of supplying the the Aerology Section of the Bureau of Aeronautics of the necessary information in reqard to ship characteristics as Navy Department, coo eratod fully in the matter of well as in regard to questions on matters of a marine
lendmg equipment. AcLowledgment is also made of architectural nature.
that furnished by Mr. J. A. Balster and Mr. A. H. Mears resu P ted in its improvement. Finally, many thanks
PRELIMINARY REPORTS ON TORNADOES IN THE UNITED STATES DURING 1938
By J. P. KOHLEB
[Weather Burcau, Wsshlngton, February 2,19381
The present stud is based largely on the data contained
issues of the MONTHLY WEATHER REVIEW published dur- ing the year 1938. A final and more detaded study will be published in the United States Meteorological Yearbook,
1938. The figures given here are substantially correct; however, it must be remembered that all are subject to change after the final study mentioned above.
In contrast to the previous year, 1938 brought some highly destructive tornadoes; none of them, however, com- pared in violence or severity with the Georgia tornadoes of 1936, or the outstanding St. Louis storm of 1927 or
the tri-State tornado of 1925. Table 1 shows that 189
tornadoes occurred in 25 States; during the preceding year the total number reported was 123, a difference of 66
storms. The number of deaths caused by tornadoes in
1938 was reported as 1i8; this is a considerable excess over the comparative1 small figure of 29, which was
tornadoes in 1938, based on all available reports, were in excess of 2,189; for the preceding year the list of in'ured was only 192. Total property damn, e
$8,000,000.
Table 1, herewith, enumerates tornado frequency, deaths, injuries, and damage figures, by States durmg the year. It will be seen that the greatest number of tor- nadoes occurred in March with a total of 60 storms which is 44 greater than the 22-year average (1916-37) for that month. May was second in tornado frequency with 42
storms, an excess of 11 above normal; in April there were
26 or 3 more than normal. Thirteon tornadoes occurred in July, 8 each in August, September, and November, 2
in February, and 1 in December. October was the only
month in which no storm of even possible tornadic chmac-
ter was reported.
The greatest loss of life resulting from tornadoes during
the year occurred in March when i 4 deaths were reported.
There were 32 deaths in September, 21 in February, 17
in April, 15 in May, 14 in June, 3 in July, and 2 in Novem-
in table 3, severe 9 oca1 storms, appearing in the several
the tornado death to1 P in 1937. Injuries suffered from
resulting i rom the 1938 tornadoes is at'imated a t near K y
ber. The tornadic activity of August and December caused no loss of life. Practically all of the total property loss, approximately
$7,790,000, occurred during the months of March, June, and September. The greatest monthly loss was in September and was due primarily to the series of destruc- tive tornadoes which occurred in Charleston, S. C., and vicinity on the 29th. Some of the most outstanding tornadoes (from the point of view of loss of life and property damage) in the United States during the year are as follows: In Kansas on March
10 a single tornadic disturbance caused 10 deaths, 150
injuries, and damage of about $575,000. In Arkansas in the same month 1 tornado on the first, 4 on the 28th and
9 on the 30th resulted in 18 deaths, 287 injuries and damage
of $366,000. I n Missouri during March, 4 tornadoes on the 15th caused 11 deaths, 27 injuries and damage esti- mated at $257,000; and near the close of the month, on the 29-30th, 8 tornadoes were responsible for 5 deaths.
56 injuries and property damage exceeding $224,000.
On March 15 a series of 3 destructive tornadoes in Illinois caused 30 deaths, 77 injuries, and property damage to the extent of $215,000, and 3 storms on tho 30th resulted in 13
deaths, 89 injuries, and damage of more thnn $1,500,000.
On the same date a series of 6 minor tornadoes caused 4
injuries, but no deaths, and property damage estimated at $59,000.
The most outstanding instance of destructive tornadic action during the year occurred in Charleston, S. C., and vicinity on September 29 when a series of 5 tornadoes in Charleston County killed 32 persons, ilijured 160 or more, and damaged property to the estimated extent of $2,000,-
000. In this connection it may be stated that this is the greatest loss of life and property caused by tornadoes within a restricted area in that State during the period for which records are available.
In the event that the possible tornadoes enumerated in Table 2 are later adjudged to be true tornadoes, the 1038
figures will be 200 tornadoes, 181 deaths, 1,321 injuries,
nnd property losses exceeding $8,045,000. (See tables 1
and 2 on pp. 413414.)