To obtain the absolute velocity of an airship by knowing the speed at which two routes are covered, we have only to determine the geographical direction of the routes which we locate from a map, and the angles of routes as given by the compass, after correcting for the variation (the algebraical sum of the local magnetic declination and the deviation)

Volterra, Vito

English

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I?ATIOHALADvIElti
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OBSEKVATIONS ON
.oa5-mE
HETHOD OF DETERMININGTHE VELOCITY
OF AIRSHIPS:
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OBSERVATIONS ON THE.METHOD OF’DETERMINING THE VELOC~TY
,. OF’JIHMMIPS.*
... .== ,Ifv i~~e, velocity of the airships, v the velocity of
.. . ... . ,
.&&!&..
the wind, the resultant VI will ‘~”ethe effective velocity ‘at
wh~oh the maohine covers the distance.
‘“i The angle Wi
!
= Y will be the.angle of drift, and the an-
\f:j gle”Viv = a the angle of the wind with the direction of route.
{,+ We then have:
.’
V1 = v Cosy + v Cos u
k =x .QzUL
v sin a
From these fomulas we obtain:
sin y = k sin a
-------. .--------
Qos ‘y k2 sins a=J3-
------ ~-.--- -------
Cos a sm2 a=J/l-~
Suppose that the distance da is covered in the time dt, we ‘
shall.”have:
dt=~= ds = (v Cos Y -vcosa)ds=
v (20sy +V cos a V2 COS2 Y - V2 COS2 a
(v Cos 1’ - vcos~)cis =( VCOSY-VCOSOCLS=
=
V2 (C082 y - k= COS2 U) V2 (1 - k~)
~>... ,,,+..__.. . . ...
= Cos v -kcos ad~
V(I - k2)
* Translated from “Rasseggna?JarittimaAeronautic Illustrate”
—-.-
No. 12j-Decemberj 1920..
..;,r.
.!,!,.,,,
#“’ .,-, ,::.
Y’ 2-
and therefore, in the time t the distanoe covered, s, will be
-. ,,-.. ,, .. t -= J C*6 v -kcosad~,. ,.-...-
V (l-_ kz)y.. ,,...,.,..
Supposing”thewind to be constant in magnitude and direc-
tion the foregoing formula becomes:::“’!,,h
1=— 1~:jCOS ‘/ ds - j COS ffds’
v(l-k2) - A
If we take the axis x parallel to the direction of the
wind and proje
we shall have:
ot the line of flight AB along x in A’Bf = X,
x= f cos a ds
therefore
t- ... a.
<
J C*S ‘y (3.s- kx-j
V(l-k2)
If the route covezed is in a closed circuit, that isj A
coincides with B and A1 with B1j :/eshall have x = O and the
result will be:
-----.......---
1 ““ jcosyds=- 1 /t = JJl - k2 COS2 a M
V (1 - k=) V (1 - k2)
.
The av;?age speed”at which ’’”’themaohinecovers the distance
will be:
assuming that
----------------
0 =Joosyd8=~~l-k2~s2 ad~
- . ----- :-----, s --’-” - “------”””-...=.4.- ..—.-.
.,
It follows that the ratio between the mean velocity at
We will now show
covered and the ve,locityproper of the m?.”
that this ratio is always less than 1 ‘Then
the velocity of the mind is less than the velpcity proper of
the machine, that is, k e 1. In point of fact;
--------
u fdl - ?& COS2 a ds
-------- ---
------ -
=JL.-k2 .f J1-k2di3
----- ..-...-- -------
J J1’-k’cos2’u ds
But:
----- ---- ------ -------
. . . . . .
~l-k’<~l- k’cos’ a-=1
~therefore
------------ -
---------- ------ -
f.jl - k2 COS2 U ds
and
c1
------
/
‘,&&.,- ‘2
-------
~
=s2<1
v
‘,
.,
.: . ..”.,
L .... -- :,:.@y“,
.-4~
IF, THEN, ti MEASURE THE MEAN VELOCITY ALCN?G.4NY0L05ED
==-~~~CIR~ XT mA-m~R, -WE--SHALL ALWAYS..OBTAIN.A .vEiJc)C!ITYLEsa THAN
THE VELOCITYPROPER Ol?THE AIRSH~PALTHOUGHTHE WIND MAY SOME-
TIMES’BE FAVORABLE,SOMETIMESOPPOSEIJOR SOMETIMESBE BLOWING.
SIDEWAYSTO THE DIRECTIONOF MOVEMENT. :
Let us examine some particular cases:
Suppose first that the circuit covezed,isa polygon
A13c DE=s,82s3s4s5.
Denote the values of CL along the sides by al, a2j a3~
U4 , U5, and the values of the Angle of Drift by Y1 , T2,
fy3 > Y4 > Y6 l We shall have:
k?)s
?= ‘l- W)S=. (1 --------- --,------ --
zoosY1q ~Sx J1-k2coa2~1
Suppoee the polygon to two superposed sides AB and BA, we shall
have:
w = f<v
that is, assume AB = L and denote by t’ and t“ the times in
whioh the distanoesAB ~d BA are covered.
2L L
~m— —<v
t!+tll=. l+ll
(tat)
,
Thus’,-’”’zwedivide’’thedistan&’’L’bythe mean v~ue” of the
times in which L is covezed either in the di%eot sense or in the
oppositedirection,we shall always find a velocity less than
the Velocitypxoper of the airship unless the wind is zero~
.,
,,
.’,
. ..
‘:*
-“5-
THE ANGLES OF’DRIFT FOR ROTH 7,1NESOF FLIGHT, DIRECT J?Nd
.=-,,
IfiTHE CON~E~R~””DIfitiTIQfi-;”*ILL 61 tiQUAL.
Denoting”them by ?’, We s~als ~ave:
~’
= v 00s y +Trcos cf- “L =“;11 v Cos y
t’
. v cos a
which means that the mean velocity of flight along both lines,
direct and inverse, is equal to the veloeity proper of the air-
ship multiplied by the cosine of the angle of drift.
IN MEASURING THE VELOCITY PROPER OF AN AIl?SHIPWHEN A CROSS
WIND IS BLOWING,WHICH IS USUALLY THE CASE, IT X3 ADVISABLE TO
MEASURE THE ANGLE OF DRIFT BOTH FOR THE DIRECT AND INVERSE LINE
OF FLIGHT. THERE SHOULD NOT BE MUCH DIFFEREITCEBETWEEN THESE
TWO ANGLES, WE TAKE THE AVEM&Z OF THESE ANGLES AND IT 1S AD-
VISABLE TO INTRODUCE A CORRECTION DUE TO DRIFT BY DIVIUING THE
MEAN OF THE TWO VELOCITIESOF THE TWO LINES OF FLIGHT BY THE
COSINE OF THE ANGLE OF’DRIFT.
Fox this correction to be of value, the distance covered
mu6t be pr~cti”%lly rectilinear and the angle of drift must ,“
be oonstant, thus showing th,atthe wind is constant.
,,,. ., ,.,, ...4 .,.,.,., ,..,?.,, ,-. ,..
We can also obtain the velocity and direction of the mind
from the velocity of the two lines of flight, direct and inverse,
and f~om the angle of drift.
.
In fact, from the equations:
$, =Vcosy+vco6a $, =Vcos?
it follows that
v=
*($+:!) ‘=.03:..(3-$3
therefore
t’ - tin
from which we deduce
--.__+_____-
and
CM Y
= JT:--aE?x--
k2
r .LL
----- __
J
——-----
therefore
----------
---1r& -~ 2t’ tnIL+&t’ tn!- -.
--.-.--e ______ -+ ____ _____ ___
/ [3XJ2v =V J sinzy + cosay~’,-~ ~,. =L*Lt’ *tt
:. 1=
2 Cos Y
whioh means
-.
[
&
t
that
v .*
-7. / ------------- -------- ------- ----
~ [.
~ ~a
+.IJ
]~
s*n2 y + Cosa y t ‘ 1- ;11~ II
& .++,,I
_l
\
We %hen have
sin a s siriy
k
--------------------- ------------- ---
J’p-q ‘ ~, +q’ ‘=2 ‘
sin .+ .&-.
=
-
/=-----”-----” --6------+7--
r -t
= “_--k!.Lid““”
Suppose that the distance covered forms a
R. I,asuoh a case we shall have:
Y
circle of radius
------ -----
------ ------- ----
_----- .
(j%,~ &k2=o~2(y &pR/2nJ/ l-k2cos2u da=
o
= R J2m 1 -k?cos% da=
------------------
0
71 - k2 COS2 a
4- -1
Now:
d
. -..---,-- ------- ___
J
/=*”f2n” da. ‘=4~(e~..
-------------+--0 /“ 1iptica;!Jl - k2 8i112U integra:
of 1st
grade)
f
‘2T1 co6 ef=j d a Tr/2
“1”
sin2 a da
/
------ ------------0 0 /
=---------------------
= 4 E (ellip-
tiCal inte-
J1 - k2 COf32 e Jl - k2 sin2 a gral of 2nd
grade)
therefore
e =4 R(K-k2E)
and
t
=4 R(K-k2E)
V (1 - k2)
whence
,V= 4R, (K- k2E)
t (3 - k’”)
Suppose that the airship flies suaeessively along the tmo
routes MlN3, Mall= and that the velocity of the wind is such
that it is represented by v,. The axis of the airship is repre- -
sente’dby V? when sailing ovez the fizst route and by Vn when
sailing over the secondroute.
1..- =. —— —
**?
The two vectors v will .beequivalent to eaoh othef, while
airs-nipwIII ham different directions~ but will have the same
modulus V if the absolute veZocity of the airship V is maintain-
ed at the same magnitude during the fZight over the two routes.
.
We will take the geographioal direction $ N (Sout~-North)
and will represent the’angles as follows:
(MiN,) , (Sx) = (3L
V1 . (SN) = :Xl
v, (SN) = p
(i!d&J l (m) = e=
V2 l “(SIN)= *2
V.(sll)=p
Lastly, we denote by V..and Vz the effective speed.at mhich
the airship covers the two routes; by yk and y~ ttle~~Q angles
of drift; by al and’Ua the angles formed by the wind with each
of the two zoute~, and by 0 the angle formed by the t’woroutes.
We have:
(1) .
7’2= e2 - x (2)
2
al = p - el .“ (a)
up = p - e (a’)
.
9 e2-e =al’,-u’= (!3)
L 2
VL = v Cos yl + v Cos a%J (!5)
V2 = ).v cos Ye + v cos a2 “
,,-~
. . . — ——
.,,.
.
., --.=- . . -—.,--
,..
..
-1o-
Frorn (4) we obtain:
there~ore (5) becomes:
i,
:
VI = V(cos YI + sin Vl ootan
V2 = V(cos. ‘Y= +- sin Y2 Cotan
Wt frOm (4) we have also:
(b)
(5’)
from which it follows that:
(c)
cotan al =
L
Cos sin WY
1
e-_——____—._ 1 (c’)
sin VI sin e
-“ 11 -
and substituting in (5’)
M, . . . ---- >.7. - ,,,
VI =V
[
Cos
. .
Yl + sin 7’2
... . . . .
1
2V sin
[
O+ Y1-Y
‘1 [ 1
fj+’Y+Y2 -
Cos
2 2
=
V2=V
[
C08 ‘Ya+ sin Y2 (sin Y%sin ‘Y2
V(cos T 6in 0 - cos e sin ?’2+ sin ~1 ) =
=
sin 9
.4.
“V(sin[e ,- ‘Y21+ sin yz) =
=-’-.
sin 0
sin 0
From formulas (6} and (7) we obtain:
-i
L-- .- —.— -.
1;
— .—. —.- —— -- ——-—
,~
.
VI sin 9.
VG =
q.+..”.7,- Ya e..+v + y ., ..:,,. ......r. ...
2sliz Gos 2
Now, from (l), (2), ad (3)”it follows that: ;
e -?- Y’= e-e-e
1 2 72 1 1
+
(8)
xl -
xl + $2-X =2
e -x -X2
2 1
x- 62+ X“2 =1
-i- X2 -201
therefore (8) becomes:
-. ..,.,,, ,.– .#- .,..’-
,,
W.p--+-q.0s p, - y]
IL ..__._ .—
I- “’13-
(A)
Thus, wishing to obtain the absolute velocity of the airship
by knowing the speed at which the two routes are covered, we have
only to determine the geographical direction of the routes whid
we locate from the map, and the angles of the routes as given by
the
sum
compass, after correcting tor the variation (the algebraical
of the local magnetic declination and the deviation).
The determination of the constmcy of the wind along the two
mutes is given by equalizing tbe two values of V given by the
second and third terms of (A).
We can alm obtain the directi,m and magnitude of the wind,
since from formulas (c) ~d (ct) we can have the angles ~ and
as and therefore from (a) and (at) we shall have p in two ways.
Then by applying fmnula (b) we obtain the veloaity v of the
w:ind
also
in two ways also.
These various ways of
serve for determining
mutes.
determining the values of v and P maY
the constanoy of the wind on the two
I
PRJv XI,GALEXAMPLE.
DETERMINATION OF THE VELOCITY OF THE AIRSHIP M,6.
We give here two examples of the determination of the speed
of the airship M,6 made on board along a straight I.ineof known
length, traversed in both directions, along the railway line
Signs-l?lorencebetween the
Paolil~(length 6.925 d~)
The measurements made
two localities ltSarGignalfand ltCastel-
and the results obtained are summarized
in the four following tables which contain:
The first, all the elements concerning the measurements and
the resultin~ values of the velocity observed in the firs% deter-
mination.
The second, all the data required for computing the angle of
drift presented by the route of the airship with respect to the
orientation of the track while the measurements were being taken.
The third and fourth tables give similar data concerning the
second determination.
,,
These latter data’~erve for passing from the observed valw$
of the velocity to ‘thetfie v&lue~ the effect of the wind being
,. ,, .,. .,..
eliminated in “eachcase. The data referring to the computation
of the angle of drift were deduced from the compass readings
,.. while the measure”ments”werebeiti$t“aken2corrections being made
fox deviations due to local magnetic conditions.
From the results obtained in the two ascensions, we may mn-
chxie that the airship M.6 had a mean velocity of 69.2 with the
regime of the engines maintained constant during the experiments
here recorded.
——
-u5-
FIRST DETERMINATIIOI?
Alti-:l?res-: Temper-: Route :pi6~.,:@gle’”of:““Time :Speed:Mean
tude :sure :ature. :,,,,.,... : :routeby: : per :speed
. _.,w-.,..---- . . :CXnngass: :houz :...- . l .
=-.
—.
m. :m/xn: *. : km. : l. : km. : km.
210 : 740 : 27.2° :C. Paoli-:6.925: 100° : 4m146496: 86*8:
l
.
l
. :Sard@na : .l ..l * ll
“68.60
~:22Q : 741 : 27.5° :Sard@na-: ‘t :. 2750 : 8m16*4& : 50+3:
l * :C. Paoli : . . .“., ll l l .
ANGLE OF DRIFT
Ro@J followed,-:Angle of: Actual :Orientation:Angleof:Mean sn#-e
:route by:angle of: of route : drift : of drift.
:GOmp8SS : routeA : j ll
C. Paoli-Stuxiigna: 1000 : 84° : 790 : 50 :
l l l l . 70 30’
Sardigna-O. Paoli ~ 275° ; 269° ~ 259° ; 10° ;
v’= 68l 60 = 69.19 km.
Cos 70301
SECOND DETERMINATION
Alti- :Pres-:Ternper-:Route “:Dist.:~g~.e Qf: Time :3peed:Mean
tude ;sure :ature : ; :routeby: : per :Speed.
. ., . .’
. l . . :oompass : : hour:
“El :
320 :
l
.
885 :
m/m l+
.
.
.
.
l
.
.
.
15.2°
.
.
; km.
: C.Piio13=:6.925
: Sardigna:
:Sardigna-: [’
:C. Paoli :
l. .,
* ,
*
. 950 : 6m3.3s
. l
. .
: 265° ,,;,5m$8.ls
. .
. .
: km. : km.
:68.68:
l
.
‘69.15
:69.62:’’’’”
.
l
.,,
~,-. w-s? -- -z—— —
-16”
ANGLE OF DRIFT.
Route followed :Angle of: A@ud. :Or:&&tit.#: An$:@-t:f: Me= =g~{~
:routeby:angleof: .. of drifi
:oompass: ruute”: . .l .
-----..
C. Paoli- : 95° : 80° : 790 : 10 :
$.ardigna: , l , .l . l s
2° 1/2
Sardigna~ : 265° : 263° : 2590 : 4° :
Gg Pa03i : l l .: l b .
v= 69,16
C06’20 30~ = 6g’a2
.,
, .
,,/
ii.,
i


Observations on the method of determining the velocity of airships

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