,, w’-”” - NASA charts, &Qthough derived for a vertical diye starting from rest, may be used...
Transcript of ,, w’-”” - NASA charts, &Qthough derived for a vertical diye starting from rest, may be used...
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TECHNICAL MOTES
“’’”““”’’ylm!m@—._.....
No. 599.—. ——— —
w’-””— -— _-
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CHARTS EXPRESSING THE TIME, V“ELOCITY, AND ALTITUDE ._ _ “:
RELATIONS FOR AN &(RPLANE DIVING
IN A STANDARD ATMOSPHERE
33y H. A. PearsonLangley Memorial Aeronautical Laboratory
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Washington —
April 1937.-
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https://ntrs.nasa.gov/search.jsp?R=19930081344 2018-08-19T18:15:23+00:00Z
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
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TECHNICAL NOTE NO. 5S9
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CHART’S EXPRZiSSING THE TIME, VELOCITY, AND ALTITUDE
RXLATIONS FOR AN AIRPLAITE DIVING
IN A STANDARD ATMOSPHERE > -
By H. A. Pearson
SUMMARY
In this report charts are given showing the_relationbetween time, velocity, end altitude for airplanes havingvarious terminal velocities diving in a standard atmos-phere. The range of starting altitudes is from 8,000 to32,000 feet, and the terminal velocities vary from 150 to ‘“-550 mi3.es per hour. A comparison is made between an ex-perimental case and the results obtained from the charts;Examples pointing out the use of the charts are included.
Il~TRODUCTION
The velocity-altitude relations for airplanes in adive have been treated, in the past by several writers.Diehl (reference 1) assumed a constant density atmosphere.~ilson (reference 2) and Becker (reference 3), who havetaken the variation of density into account, though attack-ing the pro%lem differently, have given no expression forthe time to dive. Regardle~s of the manner In which thedensity is taken into account, ‘the velocity-altitude equa-tions become too lengthy and complicated for general use.
Because of this fact, and also because the time varia-%le has previously been omitted, it is the purpose of thispaper to present in a readily usable fo”rm charts express-ing all three relations with density variation taken intoaccount. #
N.A.C.A. Technical Note No; 599
CHARTS
The charts shown in figures 1 to 9 cover a range ofterminal velocities from 150 to 550 miles per hour by 50-mile-per-hour increments. The starting altitudes varyfrom 8,000 to 16,000 feet by 2,000-Fcot intervals, andfrom 16,000 to 32,000 feet by 4,000-foo.t increments. Thislatitude in both speed and altitude will, it is believed,cover all cases. of interest from the extremely slow air-plane UT to the fastest airplane available. ..
The terminal velocity U by which each chart is des-ignated, is that which would be obtained fn an atmosphereof constant sea--level density. The abscissas of the curvesare the true, not” the indicated, velocities.
In the establishment of the velocity-altitude curves,the equation develhped in reference 2 was used with slightmodifications to make it agree with the now recognizedIrstandard atmosphere!’ of reference 4~ The modified equa-tion was similar in form to that given in reference 2 ex-cept that-the factors 3 and 1,200 occtirring in the origi-nal were replaced by the factors 2.7 and 1,254, respec-tively. The new equation was:
f)125A a 1254 2———~
(
u H ()---w-Va = 2g 1 + –2*7h
; (1 + 6i;;;)dh (1)
64000
where 7 is the true air speed in f~p.s.
.-.
●
I
g, gravity constant, 32.2 .m
h, altitude i.n feet above sea level
H, start.in~ altitude in feet above sea level
U, terminal velocity of airplane in air &t stand-ard sea-level density, f.p.s.
..
The placing of the time network on the charts was accom-plished by using three methods: (1) by the use of” the vac-uum i?ormula, (2) by use of an equation developed in the .’appendix which takes variation of density into account ~and (3) by use of a step-by-step process of integration. &Xach of the foregoing methods had its particular rOgiOn in
9
I
N.A.C.A. Techhical Note No. 599 3
which it was more easil~ apylied than the others f-or thesame degree of accuracy.
The acceleration of an airplane in a vertical dive,at any instant, is given by
a= 3’*2 (’ -$3
where a is the acceleration in ft./sec.a
~! dynamic pressure
qu, dynamic pressure at terminal velocity19.h.iu
It is obvious from this equation that during the early ‘:”.::.jJ””[J
. .. ..-
part of the dive the acceleration differs only slig-ht-ly “( 1from g; hence the vacuum formula is applicable, wi:h_iy_ ._ - __the plotting accu~acy, for this range of the charts. Thisrange increases with increase in starting altitude and_.._terminal velocity. All timing lines below 6 seconds” ar”ecomputed by using the vacuum formula and in some cases the -range has been extended to 8 seconds. Above. these time-j .
● values, use was made of a step-by-step integration of thevelocity-altitude curves when the terminal velocities were
. 300 miles pe_r hour, or more. Below a terminal velocity of● 300 miles mer hour, the time-altitude formula developed in
the appegd~x was used, Application of this formula showed .that it was necessary to carry a large nu”m%er of termswhen the terminal velocity was greater.than 300 mile~ per”hour, and also when the altitude loss was ‘small, regard-less of the value of the terminal velocity. consequently,its practical use iS limited to altitude losses of morethan 2,000 feet and limiting speeds of less than 250 mile_sper hour. The foregoing limits are, of c_oyrs~, perfectlyflexible depending both upon the combination of U andd-h and upon the degree of accuracy desir-ed.
The charts, &Qthough derived for a vertical diyestarting from rest, may be used .to include various divingangle~ snd sts.rting velocities. If U is the terminalvelocity of an airplane in a vertical dive, its terminalvelocity in a dive where the flight path makes an ~gle/.Y with the ground is
.. —.— -.——. UJsinY
-.
-,
.-
4 N,A.C,A. Technical Note No. 599
The vertical component ofi this velocity, which is the on+used in choosing the appropriate chart, is
u? = u (sin Y)3 /2
These factors will be found tabulated on each of thecharts, except figure 1. The effe.ct..oflan “initial divingspeed is taken into accdmst in the char% simply by con-sidering that the airplane started .to dim from a higheraltitude.
PRll!CISION●
As far as the charts are concerned, the only errorsare those due to plotting! to neglecting terms in the timeequation, and to the discrepancies that will occur in any
-—
step-by-step integration. The error in the time lines du-eto the last two sources is believed to be within “2 ~ercent;the plotting error in the velocity-altitude curves is neg-ligible.
In the application of—the charts to diving airplanes,however, several. uncontrollable sources of error will ex- .ist; namely , those due to:
1. Vaiiation of.‘~~rn=~jher-e”fr”om-“s-t~~.dqrd.”—
2. Scale. and compressibility effect on the air-pl-e drag, .- -.
I
3. Manner of entry into the dive. 1
4. Dive-angle variation from that assumed.
5, Propeller effect.
The error due to the first source is negligible and neednot be considered. Present knowledge indicates that theerror due to source 2 is small up to 500 m~l-es per hourwith small angles of atitack (reference 5). The manner of––entry into the, dive will be relatively unimportant in thelonger dives but may be important in the ehorter ones;however, in practice, its effect may be obviatti if de-si~able. The effect Of errors in the dive angle may becomeconsiderable in the shallower dives. .
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N.A.C.A. Technical Note No. 599 5
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Probably the largest individual source of error isthat due to the propeller effect. In the computation ofthe values of U with”which to enter th-e c“h-~rts$liK 6f-fect should, if possible, be taken into account. .-
—.
In order to determine the cumulative effect of someof the a’hove-listed errors a comparison is made in-figure10 between an experimental case and the results given bythe charts. The initial air speed is taketi as 120 milesper hour in order to eliminate the entry into-the divefrom level flight. It can be seen that a satisfactory”- ”---agreement exists %etween experiment and the results ‘given- ‘
.—
by the charts. If the dive.h.ad been ,longer and the start-ing altitude greater, the agreement would have been muchbetter.
EXAMPLES
1. Given :
a. Airplane with U of 496 m.p.h.
-b. Altitude at start H, 16,000 ft.
c. Velocity at start, 100 m.p.h.
d. Dive angle 60° to the ground.
Required:
a! Velocity at 6,000 ft. in m.p.h.
l), Time to dive, sec.
Solution: \
Ur = U (sin p)3’a = 496 (0.866)3’a = 400 m.p.h. -
Using figure 6, point A is located at 100 miles per hourand 16,000 feet. An interpolated curve is drawn %etwe-enthe existing curves down to B at 6,000 feet. ~“roj8ct”ingfrom 3 to c, the velocity is found to be 406 miles perhour (an ind..icated velocity of 3?’2 m.p.h.). The time is28.3 - 4.6 = 23.7 seconds, found by subtracting the” t“imecorresponding to A from that at B.
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6 l?.A.C,A. Technical Note No. !399
2. Given :
a, Airplane with U of 500 m.p.h.
b, Altitude at start H, 14,000 ft.
c. Velocity at start, O m.fi.h.
d. Starts gull-out at 3,000 ft.
Required:
a, The velocity at pull-out.
1
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Solution:
Using figure 8 and f-ollowing 14=000-foot curvedown to 3,000-foot altitude, the true velocity is
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449 miles per hour or the indicated velocity is 430miles, per hour.
Langley Memorial Aeronautical Laboratory,National Advisory Committee for Aeronautics,
Langley Field, Vs., May 2, 1934...,
7.
. ..APPWJDIX ‘
Derivation of Time-Altitude Pormula
As stated in the introduction, ~revious investigatorshave not included an equation connecting time and alti-tude taking density variation into account. Consequently,the following derivation is made and is based Upori Wilson~svelocity-altitude equation (reference 2), modified to con-form to the standard atmosphere of reference.4.
.,
In equation (1) of the text, let && = a, and
(%) =C*dh
Then dt = .——___
f
————-—____
c’ H2g (1 + ah] J (1 + ah)-c’ dh
hfrom whence
t =J-—dh -
—— —————. + cl. ————————— ———— -
I~a
2g (1 + ah] ~H (1 + ah)-c2.’dh
If t = O when h = H, Cl disappears. After integratingthat portion under the. radical, the foregoing expression ,is -.=.. ___-—
t dh.JH —-.—— — —
/,
—.— ____ _Ca ~’.”l
2g (l+ah)[(l+aH)l-c” - (l+ah)l-c9
a T:P 1
NOT,2g
letting —--—._. = -ka and (l+aH)l-c2 = Ka (l-c=) ,.,
(Note:2g
––—-- is negative for U < 1254)a (l-ca)
N.A.C.A. Technical Note No. 5998
then t 1 f l__________q._______—=—‘h
J( l+ ah). K(l + ah)ca
Letting “=~ ‘he” ‘h= :~du
Subst-ituting, there results
This expression cannct be integrat~ directly but may beput- into a f~rm that can, lIy expanding bin.omiall.y the ex-pression
[
2(W)]-*1-Ku.
J!The resulting series converges when K u
2(C2-1)~ 1,
which is the case” here.
When the foregoing expansion is made the expressionfor t becomes
/li-aH
t’=— f[1 + ~ K~2(c2-’) + ~ ~~ U4(C2-1)
a’ m-8
1
& ~3”u6(Ca-1) + .,.: au+ 16
which becemes upon integrating . .
[
~2c2-1t2=— u+*K _ + 3 2 ~4c= -3
ak~K~~
2C2 - 1 -3
5 ~ ~6c2-5
1
~l+aHK+-— ––~— +
16 6c-5 ““” Gl-ca
Remembering that u = J l+ah and K = (1 + “&E) ,the upper limit is
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,.-
i—.:
,
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.H.A. C.A, Technical Note No. 59”9 9
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.
+ . . ● l“~”s”””(an-1) 1
2.4.6 ...2n 2nca - (2n - 1) 1
In ‘the same manner the lower limit 3ecomes -
*. , .1.3.5 ...2n - 1..——2,~,~Q..2n
Combination of the two
1-——211C2 - (2n - 1) (Hs(’’’a)]
.
-.
limits gives an equation for the ,time in terms of the altitude,
t[
:k x===np-~ 1— ““--” “=— = [211] 2nca - .(2n - 1)
1.—-—(;T:)n(& -l)]2nc2 - (2n - 1)
.
.
10 N.A.C.A. Technical Note No. 599.
REFERENCES ‘..
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1. Diehl , Walter S.: Engineering Aerodynamics. TheRonald Press Company, New York; 192B, p. 222. .—.
2. Wilson, .~.dwin Bid’ivell: The Limiting Velociby in,
I?alling from a Great Height. T.R. ~0. 78, N.A.C,A. ,1919. -.. .-
3. Becker, Fritz: Der Sturzflug in yer&nderlicher. Luft- . ._’_ _.dicht-e. Z.I?OM., November 28, 1932; pp. 659-663,
4. Diehl, Walter S.: Standard Atmosphere - Tables andData. T.R. NC). 218, N.A.C.A. , 1925.
. .-:’
!5. Stack, John: ‘~~ N.~.C,A. High-Speed Wind Tun~Q and ‘——
Tests of Six Propeller Sections. T.R. NO. 463,N.A.C.A. , ,19Z3.
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N.A.C.A. T.N. 599 Fig. 1
32,000
3fJ*ooo
28.000
u{ I I I I I I II I I I I I I 1 I II I I t II I IIIN 1A r v I I I I I JIIIII
IIII II I I I I26,
24,
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w l!!! !!u’!!! !!!! !/!!!!/!!!!Y!A W-Iy&y1nl 11
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8,000 —
12,000ft.,0 I 40 I 8D I 120 I 160 I200 I240 I ,. 8816,000ft.,O I 40 I 80 I 120 I 160 I200 I240 .-
20,000ft.,O I 40 \ 80 ] 120 I 160 I2oon!A~ofi.: ,, “ ** .,..
24.000ft.,o [ 40 80 [120 I 160 f28.000ft.,O \ 40 I 80 I 120! l~u IKIU IZ4U I II II
._—-
.-..—..-.:
!_v!_J-!w- 1- I “ “ - .
“32,000ft.,6 40 80 120 160 200 240 . 1!
Figure 1.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. u= 150 m.p.h.
N.A.C.A. T.N. 599 Fig. 2
32,000 0 w v
30,000
28,000
26,000
24,000
22,000+QJ
~ 20,000d:QJ18,000
:
m I6,000g
% 14,00C
$5 -----.++4
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8.000
6.000
1 1 1 1 1 , I {—
, ,,
I I I I I I I I I I I I I I 1%1/ I
VII /! IIAll/Ill i77 0,354j - j
4,000I 1 I I Iw
.-
.670~
1’
—
.8051161
.91OI182
8,01
32,000ft.,0 40 80 120 160 200 240 ““,, .
—
..—
..
.—
-.
—
Figure 2.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. U = 200 m.p.h.
.
.
N.A.C.A. T.N. 599 Fig. 3
““ I I i I ) r I I \ I
JI
~ 20,000I I
-1L—‘
8,0012
1 r 1 i 1 n t ! ,
IIt Y,000 I I I I 1/1 I I1A
I I I I I 1 I ! I w rbLll I I ? I I I I IN Ill II I A v A 1/1/111 1/ I/ 11 I1 /1 1 r J’ JIi I
.2,0001II ! Ill I I I
.0,
8,
6,000
4.0001-+ + + + + + f I 1 1 + +
! 1 1 I I 1 1 t t ! 1 , , r
2.(300I rl I I I 11 I I I I I I I I I.-,
“l-+ + + + + + 1 1 1 + -t
Figure 3.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. u= 250 m.p.h.
.-
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.
. ---
.
.
1“
N.A.C.A. T. N. 599 Fig. 4
32,000
30,000
28,000
—,—_
.0,000
8,000
1 .
6,000 \vI
I,I I
4.000
2,
8,00012,0
1 —
.
.-
—
.-
—..- —
.-
Figure 4.- Time–altitude-velocity relations for airplanes diving in astandard atmosphere. u= 300 m.p.h.
N.A .C.A. T.N. 599 Fig. 6
32,000
30,000
28,000
26,000
24.000
22,000+wQ1
I I I I I
VI Y ‘(,G~ 70 .910 364-80 .s77 391-
.90 1.000400-)d
I I I I II I I I I I I I I
I I I I ! ! I !
8,00C12,
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1A I t I I I I I I I II t I I } I I I
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8,000
6,000
4,000
10
47! I 1 I I I I I I 4
1 I 1 1 1 I I 1 I 1 1
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Ift.,”o1IIIIII I I I I I I I I I I u n r rll I All/l Ill Ul,rllxi II I I II I \ I
lbO 240 3 0 400 4d0 Velor!it$,k.p.h.Oooft.,o 8rElo IlJO I210
II
pdmh p -1 p16,000ft.,O I 80 j ]60 I2 0 I320 \4r’i3I48020.000ft..O I 80 \160 [240 [320 ~ 400 !480 “1,!, ‘1 M
:UU14WJI 1“ ,.‘-’-24,000ft.,L1 1’8b I160 ~ 240 I 320 I 4’:- : ::- : :28,000ft.,O ] 80 ]160~jo240[320 ]400 I480 I ,,
32,000ft.,O 80 240 320 400 480 ,, - .
Figure 6.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere. U = 400 m.p.h.
-. .
*- ,-.- .,
N.A .C.A. T.N. 599 Fig. 7b
b
32,000
30,000
28,000
26,0005U .5”(U3UL60 .80536270 .910 410
24,00080 .977 4409a ,1;000 ,45? !
?2. flnfl. --,+-QIabil“20,000.
d
8,
6,000 I I -k
t I M II I I I 1 t t 1-
4,
---
000
,---
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,000
,000
,000
,000
ft.,”
I I I I2,000 I I I I
8.00012,000ft.,6 I 8’0I160 \240 I320 I400 I480 I 560 \ “- ‘“
16,000ft.,fJ I 80 I 160 I240 I320 I400 ]480 560$20,000ft.,O \ 80 I 160 [240 \320 I400 I4 0 ]560 “ ,, “ M
24,000ft.,O I 80 \160J24QI,20I4001 480I 5p ~ ‘* 128,000ft.,O I 80 I 1 0 [240 I320 I400 I480 ,, .,
32,000ft.,O 80 160 240 320 400 4 0 560 M
Figure 7.- Time-altitude-velocity relations for airplanes diving in astandard atmosphere, U = 450 m.p.h.
?
,
N.A..C.A.T. N. 599 g. a
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...-M28,00CJ
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26,000 - ‘“ - ‘-.—.
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.910!4551
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.977\489j -1.000~soo;—>-
._=_ .—-.. ..
-+
.=..
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24,000
H22,000 ,. .
? 18.000R$JW
~ -&$7-w+
ElEli8,000
6,000
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--.._.:. —.12,000ft.,0 1 8“0[160 [240 1320 ~460 [480 I 560; I . ; ,,’“
16,000ft.,O I 8.0I 160 I240 \320 [400‘1480 I560 i ,, -,, . ..—..20,000ft.,o I 80 \lGO I240 I320 ~400 [480 \560 ; :!, - !’-~_&,, ::-..,
24,000ft.,O \ 80 I160 [240 ~320 ]400 I480 I5q0 ~ ,, ~~128,000ft.,O I 80 I 160 [240 320 400 \480
_—560 ; r, .,
>
32,000ft.,O 80 160 2 0 320 400 4i0 560 ;,””.— -:.:— .
Figure 8.– Time-altitude-velocity relatlons for airplanes diving in astandard atmosphere. u= 500 m.p.h. —-
●
N.A.C.A. T.N. 595 Fig. 9—
E!.-=. —.—.—
u’4195:
,.-—--
.-
.-.
—..—
___
28.000ft.,O ! 8’0 !160 ! 240 I 320 I4t!0I“460!560 !640 !. tI -1 ‘“32,000ft.,0 8’0
Figure 9.- Time-altitude-velocityQ standard atmosphere.,C
160 240 320 460 4d0 560 640 ~
relations for airplanes diving in aU = 550 m.p.h. -.—— ,-
.
N.A.C.A. T.N. !599 Pig. 10 —
40
0
Experimental, for vertical dive on 1?6C-4.Initial velocity 120 m.p.h., initial altitude12,000 ft.Calculated, assuming a standard atmosphere.
4
Figure 10.-
10,000
9,000— .—
8,000
3,000
2,000
1,000
.—
..—
.-.
-——
-—
..
.
_—--
— -—
8 12 16 20 24 28 0Time, seconds
Comparison between experimentaland calculated results.