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I N ~'(. 11; 2 : "l/6tt1 NO
File D 00.12/ 122/No. 3173
AIR CORPS INFO ATION CIRCULAR
PUBLISHED BY THE CHIEF OF THE AIR CORPS, WASHI NGTON, D. C.
Vol. VII March 1, 1930 No. 649
THE CALCULATION OF THE NATURAL FREQUENCY
OF A CANTILEVER MONOPLANE WING
(AIRPLA E BRANCH REPORT)
UNITED STATES
GOVERNMENT PRINTING OFFICE
WASHINGTON: 1930
Ralph Brown Draughon
l\BRARY
JUN 14 Z013
Non-Oepoitor~
Auburn Unh1ers1tv
THE CALCULATION OF THE NATIONAL FREQUENC
. A CANTILEVER MONOPLANE WING
Prepared by Stanley R. Carpenter, Materiel Division, Air Corps, Wright Field, Dayton, Ohio,
December 2, 1929
SUMMARY
The p urpose of this report is to present a practical
application of t he calculation of the natural frequency
of a stressed skin monoplane wing in bending and in
torsion. The t h eories of t he methods employed may
be found in the following papers:
Prelim inary Study of Fatigue F ailu res of Metal Prop
ellers Ca used by Engine Impulses and Vibration, Air
Corps I nformation Circular, Volume VII, No. 618, by
John E. Younger.
Simple Approximate Method of Determining the
Natural F requency of T orsional Vibration (With P art
icula r Reference to Monoplane V\Tings and P ropeller
Bl ades), Airplane Department Memorandum No. 1062,
Materiel Division, Air Corps, Wr ight Field, by J ohn
E . Younger.
The calculations a re applied to t he Fokker C-2A
monoplane wing, the final results of which a re compa
red to t he exp erimental r esults repor t ed in the fo llowing
paper :
Determination of t he Elast ic Axis and Natural P eriods
of Vibration of t he C- 2A Monoplane Wing, Airplane
Department Memorandum No. 1066, Matf riel
Division, Air Corps, Wright F ield, by Charles J . Spere.
The comparison of t he experimental and calcula ted
results is as fo llows :
I E xperi- Calcu· Per cent
mental lated or error
F requency in bending . .. . ..... ..1~ ~ ~ Frequency in torsion .······ · · · ··! 12. 00 9. 83 18. JO
In the Air Corps I nformation Circular No. 618 ment
ioned above, t he natural frequency of vibration in
bending is given as
in which-fb=
Natural freq uency of bending in complete cycles
p er second.
W n= Weight per running length of span at d iffe rent
inter vals.
Yn= Corresponding deflect ions d ue to Wn.
Loading = w __ _________ ________ _______ ___ _ (2)
Shear = J: w dx ___ __ __________________ (3)
Moment = J: (Shear) dx _________________ _ (4)
Slope = ~ J ~ (Mom;nt) dx _____ ___ ___ ___ (5)
Deflection= J: (Slope) dx ___ ___ __ _________ _ (6)
In all the following comp utations, the graphical
met hod of integration was used in finding t he results
of t he ::ibove fo rmulm. E ach curve was plotted against
semispan. The semispan was divided into a number
of small eq ual sections from which mean values were
found and summed in deriving one curve from another.
The loading curve w was com.puted from t he weight
summation of the various component p a rts of t he
wing (r ibs, spars, and ply\\·ood covering) plus a corr
ection factor for each section to make t he computed
weight of t he wing eq ual to t he a ctual weight . A correct
ion factor is usually necessary because of errors
in assumptions and neglect of weight of glue, p aint,
varnish, nails, fi ttings, etc.
The weight of t he semispan from t he computed
loading curve, AB, Figure 1, is equal to t he area
under the curve mul t iplied by t he scale to which it
is drawn and is a pproximately
2.8X 460 644 d
~= poun s. (See Figure 1.)
T h t 1 . ht f t h . 1730·25 e ac ua werg o e semi span= -- 865 2--=
pounds.
Difference of actual a nd computed weight = 865 -
644= 221 lb./semispan.
This weight d ifference was di vided equa lly into 10
pa rt s and assumed to be dist ri b uted over 10 equal
areas in the semispan.
A = a rea of semispan = 373.3 square feet.
373.3
= 10= 37.33 squa re feet = 5,380 squa re
inches.
220
!:::,. weight=w =22 pounds/ !:::,.A uniformly dis-t
ributed.
The computations for t he deflections Y are based
upon t he relat ionship between the loading, shear, moment,
slope, and defl ect ion curves expressed in the
following:
!:::,.A (equal to !:::,.x t imes mean chord)
scal ing from the 11·ing drawing, values of
(1)
was found by
!:::,.x, the in cre-
2
ment of span, and t he mean chord t o give 6 A equal
5,380 square inches.
6wl 22
6x = 6x
will give t he correction factors in pounds per uni t
length to be a pplied over in cremen t 6 x. The values
t hus found a re given in t he following table :
Area
----------------
?vl eau chord , inches ___ l49 Y.i 14 9 ~'! l45 140 134
6 x, inches __________ __ 36 36 37 38Vi 40
Coprerre cinticohn _, ___p__o_u_n_d__s_ 0. 61 0.6l 0. 60 0. 57 0. 55
Area 6 7 9 JO
, ___________ , ___ ---------
103 81Y.i
52 66
Mean chord , inches ___ 130 125 114
~x, inches ___________ 41 Y.i 43 47
Correction, pounds
per inch ____________ 0. 53 0. 51 0. 47 0. 43 0. 39
The corrections were plotted aga inst semi-spa n in
Figme 2 from whi ch values for every 20 in ches of span
were added to AB to find the loading curve.
This method of findin g the loading curve is somewhat
different from t hat u ed by the United States Army.
The Army method is to find-w=
weigbt per square foot
_ weight of wing 865 2.3 pounds per squa re
- a rea 375 foot,
so that t he weigh t per unit le11gth of \\'ing = w times
mean chord .
The secti ons we re di vided into 2-foot lengths. For
section 9, the mean ch ord is 10. feet, so t hat t he
weight per running foot of wing is-
10.8X 2.3= 24.9 pounds,
and t he weight per running inch of wing is-
2{29 =2.07 pounds.
The weight per running inch at t he va rious sect ions
is given in t he foll owing table:
Section __________ 3 6
-----------
Mean chord _____ l 2. 50 12. 50 l2. 50 12. 50 12. 25 JI. \JO
Wt.finch ________ 2. 40 2. 40 2. 40 2. 40 2. 35 2. 28
Section __ ________ ls 9 10 11 12
-----
~1ea n r hord _____ 11. i\O I I I. 12 10. 80 10. 50 IO. 12 9. i 5
Wt .finch ___ ___ __ 2. 21 2. l:J 2. 07 2. 02 I. 94 1. 87
Section __ _______ _ 13 14 15 16 17
-------- -----
Mean chord _____ 9. 37 9. 00 8. 65 . 25 8. 00 6. 75
page 7, and t he distance between each section is given
in column 2. Column 3 gives the mean weight of wing
per inch for each section ta ken from the upper curve of
Figure 1.
The mean results at any section (designated by two
station numbers) are indicated on the horizontal line
between t he two stations of t hat section. Results on
ihe same line as the stations represent t he values there
and a re computed from t he mean quantit ies for that
section .
Thus, column 4 gives the weight at each sta tion for
t he wing section to the right (if t he right half of the
span is considered) and,
Weight of section at station 19= mean w for section
19- 20 multiplied by dx= 0.94X 20 = 18.8 pounds.
Column 5 is t he evaluat ion of equation (3) representing
t he to ta l vertica l shear at a ny points, and is the
summation of t he shear at a ll t he st a tions above tha t
point in column 4, since t he ver t ical shear (zero at
t ip) increases to a maximum at the root. T h us, for
station 19,
Vertical hear= 18.8+ 16.6 + 9.2= 44.6 pounds, and
the mean shear between 19 nd 20=
44
·6 +
25
~I 2 ·8 35·?-
pounds.
These values are used to find t he results in column
7, which is t he rnean shear t imes dx .
The tota l moment gi ven by equatioo (4),
M = i L (Shear) dx
is represented in column 8, the summation of mean
shear t imes dx. For station 19,
total m o m e n t = ~ mean shear t imes dx of all the sect
ions between stat ion 19 and the tip .
= 704 + 350 + 78 + 0 = 1132 pound-inches.
From the moment the slope is found by equation (5) .
SI 1 f L (Moment) dx
ope=E Jo I
1 M
=E- ~ 7 dx
in which-
] = Mean moment of inertia in inches ' of both
spars and par t of t he skin for each secti on.
JYf = Conespondi ng mean moment in po11 nd-inches.
dx= Length of section in inches.
E= Mod ulus of elasticity in pounds per squa re
in ch.
In calcula ting J, t he plywood covering approxima tely
four t imes the wid th of spar on t he top a nd bottom of
both spars was co nsidered as pa r t of the spar . Figure
4 gives the values thus computed , which are listed in
column 10.
Column 11 is the mean values of
11
{ dx and for a ny
This table is plotted in Figu re 3 wi th t he loading station in column 12 the va lue is for the summation of
curve for th e p urpo ·e of compa ri on. The former all the sections below t hat station in column 11 . ~ Jllf
method is more accurate.
Wt .finch ___ _____ 1. 80 I. 73 I. 66 I. 58 L 53 J. 29
The semi-span was divided into small sections for dx for a ny t a t ion in column 12, when divided by E , will
t he graphical integration as designated in column 1, giv e t he slope at t hat point.
The slope a t sta tion 3, column 12,
_ l ,. MI
- E '"' Tex
= ~ (321+ 287+ 258)
l
E 66
The value E is not sub ti t uled un t il t he defl ec tion is
3
I
t he subs ti tutio n in form ula (1). His u ·ua ll.1· easier to
work in inch uni ts as in these compu tations. Jf inch
uni ts are employed , 11s
E in pounds per sq uare in ch.
J in inches '.
dx in i nches.
lV in pouuds per running in ch .
Jlt[ in pound-in ches.
found in column 16. Y " ·ill t hen be in inches, whi ch may be readi ly co n-
M verted lo feet for sub t it ut ion in t he formu la .
Column 14 is t he summation of t he average I d:r; in I
~olumn 13, starting at t he root, because the slope there
1s zer o.
Column 15 is the values of column 14 multiplied by
dx. In column 16, E (1,300,000 po unds por square
inch) is subst it uted to find t he deflection of the wing
in inches d ue to its own weight. The average defl ectio
n in feet for each sect ion is given in column 18.
The mea n values of WY a nd W Y 2 are listed in
columns 20 and 21, respect ively, t he summa t ion of
which is requirnd for formula (1),
f =-l-Y J~w ,. v ,.
b 21r 'V !; w 11 y n2
~W..Y ,. = 0 . 7622
~ W,.Y ,.2= 0.0324
fh= ~:·2 ~~:~~;!=0.905 X4.85
= 4.37 cycles per second.
The mean exper imental value of f = 3.95 cycles per
second.
P er cen t o f error 4.37 - 3.95 06 3.95 X 100= 1 . per cent.
The shea r, moment, slope, and deflection curves a re
pl otted in Figures 5, 6, 7, _and 8. The shear and
moment curves a re nega t ive; they a re zero a t the lip
of the wing a nd in crease l o a ma.ximurn a t the cente r.
The lope and defl ection cmves· a re also negative but
a re zero at the center with t heir maximum value at
the t ip.
Grea t care must be exer cised with t he uni ts. The
mea n deflections Y, in column 18, must be in feet fo:·
),
I
I
.....
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I I
F IGURE 2
- 1.oillill_n _i;:un1e I
I?- f1a1 tic C-ZA JnOl lane.11./lrlo
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IR",;;.
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2 '° 21!0 3 0 J c 0 ;00 '0
I "'" l -~!i!J.i lrl nr:i e .
I I "" ai- !n7~ ii r 1( ~•3
FIOURE 3 FIGURE 4
:. et rti -.SJ. 1a r1 l n n¢h1 ~s
0 <O fi/) J; 0 f~O ?~0 20 Z80 JO JM 410 .-ldll
v ..--
_ ,,~,., v
,,./
?,ll'l v
_,/
.c: "l ~n /
~ ./
CU A fin /
'10 /
~ ~rin v s e11r Cl Ir IVE = I' w r{
g / .I: ill n t( [ -2 A. !rlC ni>n a 1e win~
~ I nn v -
~ I/
r ·nn / •
/ ,,,,,, /
/
onn
F IGURE 5
--
1 · ~ , /
/ I
/ tlflanf c ""-~A mnnnr: 'arze 1'ti · n~
,/
&trvinn,
v
"" H /
J
unn111'.I
1<'1au 1rn 6
6
kS°E m ~ - oc n in irlc1 e~
(0 w fin f 0 11~0 2(~0 2f() 280 3 'O 360 40d' 0
"I'..
-1( 1n11 !'--
-..........._ ,, Inti ~!'-..
..............
.V 1n11 I'
"['...._
.!: ,, r. no ~..... ...... ' "
Sli '10 ...........
'-
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., \fa m "n ~ ... , = .,-l .,,.
• t;:;( oa • "< ~~ ...
)f1 'J.I ~ti I:'(' -2 4 JI m lfl1 JP a '1e w iri Q ' ' 7n rin - \
" - ~c m
I
FIGURE 7
')e m -s r2f n zn uic 1es
) 4 w f, 0 fl 0 2£ IQ 2•0 210 3 0 3 0 4XJ 440
-r--r--
. 1?111 ''" !"--..
-.... .......
ifnll ()N r-..~
['...._
"'" ?nt " " '"" llN '\
~ "' ~ ~1 hnt N1' 1 e i:-z~ ~c1 lO ri c 11t VE = s or e i:J.. •"\.
1:::-. ,, ~) I\. Ml Nii 14 t ant c r<_ 'A Irr: o·r 0 )/( n~ '\/! ~~ :\. ...,
\
I 'llll "'"' I\
I "'" hN
f!GURE 8
I 2 3 4 5 6 7 8 9
- - - -------__ I _ - - - - -----
Mean J\Coment M=
Station -dx w Wdx Shear= J\Ccan :i: Wdx shear shear =~mean l\lean dx shear dx moment
' --- --- -- -----1------------
I.AJs. per
lnche• in. Lbs. Lbs. Lbt. Lbs.in. Lbs. in. Lbs. in.
T ip 0 o· 0 0
17 0. 54 4. 6 39
21 9. 2 9. 2 7 78
20 .83 17. 5 253
20 16. 6 25.8 350 428
20 . 94 35. 2 780
10 18. 8 44. 6 704 I, 132
20 1. 02 54. 8 I. 680
18 20.4 65. 0 1,096 2, 22
20 1.09 75. 9 2, 9 7
17 21. 86. I. 518 3. 746
20 I. 18 98. 6 4, 732
16 23. 6 110. 4 1. 972 5, 718
20 I. 28 123. 2 6, 950
15 25. 6 136. 0 2, 464 , 182
20 I. 40 150.0 9, 682
14 28.0 164. 0 3,000 It, 182
20 I. 52 179. 2 12, 974
13 30. 4 194. 4 3, 584 14, 766
20 1.66 211. 0 16, 76
12 33. 2 227. 6 4, :no 18, 986
20 1. 81 246. 0 21, 446
' 11 36. 2 263. 8 4, 920 23, 906
20 I. 95 283. 5 26. 741
10 39. 0 302. 8 5,670 29, 576
20 2. 11 324. 0 32,816
9 42. 2 345. 0 6, 480 36, 056
20 2. 26 369. 0
43.' 436
39, 746
8 45. 2 390. 2 7, 380
20 2. 44 415. 0 47, - 6
7 48.8 439. 0 8, 300 51, 736
20 2. 61 465. 0 56, 386
6 52. 2 49 l. 2 9, 300 61,036
20 2. 77 519. 0 66, 226
5 55. 4
"' • 1
LO, 380 71,416
20 2. 92 576. 0 77, 173
4 58. 4 605. 0 ll, 520 82, 936
20 3. 06 636. 0 89, 296
3 61. 2 666. 2 12, 720 95, 656
20 3. 15 698.0 102, 636
2 63. 0 729. 2 13, 960 109,616
20 3. 21
i93. 4 I 76l. 0 11 i , 226
J 64. 2 15, 220 124. 836
20 3. 22 E26. 0 133. 096
Root 64. 4 858. 0 16, 520 141 , 356
I I
I
10 I II 12 13 14 15
--1- -
M Mean
r . ~rlx :i:y<I':i:¥ch
:i: Mean slope :!: ~[ean slope dx
l . E --E
-------------
ln :hn'
I: I
133 7,623 1, 450. 820
7,556 72, 54l
506 7, 490 1, 299, 700
7, 237 64, 985
624 6, 934 1. 154, 960
25 6, 672 57, 74
560 6. 360 1, 021,520
60 6,080 51, 076
597 5,800 99. 920
100 5,502 44, 996
473 5, 204 7 9. 0
200 I 4, 968 39, 494
397 <I, 731 690, 520
350 4, 532 34, 526
386 4. 334 599, 880
500 4, 141 29, 994
358 3, 94 517, 060
725 3, 769 25, 853
356 3. 590 441, 680
950 3. 412 22, 084
31-l 3. 234 373, 440
1, 250 3, 062 18, 672
334 2,890 312, 200
1,600 2, 723 15, 610
320 2, 556 257, 740
2, 050 2, 396 12, 87
289 2, 236 209, 820
2, i 50 2, 092 10, 491
23$ 1, 947 167, 980
4,000 I, 28 8, 399
198 1, 709 131, 420
5, iOO 1, GLO 6, 571
199 1, 5ll 99, 220
6. 650 1, 412 4, 961
213 I, 312 70, 980
7, 250 1, 205 3, 549
233 l, 099 46, 880
7, 650 9 3 2, 344
r 66 27, 220
7, 950 737 1, 361
287 608 12,480
8, 175 32~ I
464 624
321 3, 200
8.300 160 160
0 0
- I
I 16 17 18
Deflection= :i: Y=Meao
Mean slope cL• DcOec- de(lcc-
1,300,000 I~ Lion
----
Feet Feet
l. 11 0. 092'9
0. 0858
1.00 '07
. 0764
.885 . 0740
. 069
. 7 5 . 0655
.OUl5
. 691 '0576
'0541
'606 '0505
. 0473
'530 . 0442
. 0413
. 462 . 0384
. 03-
'398 . 0331
'0307
'339 . 0282
.0260
' 287 . 0239
.0220
. 240 .0200
. 01 25
'19 . 0165
'01495
. 161 ' 0134
.01208
. 129 . 01075
. 00959
. 101 . 00842
. 00739
'076 '00635
'00545
'056 . 00454
'003i7
.036 . 00300
' 00237
. 021 '00174
.00127
.001 . 0008
. 000502
'000205
0 0 1 · 0001025
19 20
Y' WY
0. 00735 0. 0464
'00582 . 0634
'00486 . 0656
'00379 '0626
. 00292 . 0590
. 00224 '0559
. 00171 . 0528
.00128 . 0501
'00094 . 0466
. 000675 . 0431
'000485 . 0402
'000333
1 · 0356
. 000224 . 0316
'000146 ' 0273
. 000092 .0234
. 0000546 .,0193
. 0000297 . 0151
. 0000142 ' 0110
. 0000056 '00725
'00000161 . 00400
. 000000252 '00161
' 00000001025 . 00033
21
0. 00396
. 00 483
00456
00386
0031
00264
.00 219
179
143
112
.00
.00
.00
I:
000878
00065
000474
00033
000224.
000143
0000824
0000115
OOOOL71
00000507
00000081
00000033
1: \V11Yn=.i622 ~ \\1 n Y nl = .03241
CALCULATION OF FREQUENCY l TORSION
The natural frequen cy of ,·ibralion in tors ion uf a
monoplane wi ng is showll i11 A. D . M. 1062 tu be-wher
J,=?L ~Y-''JJm O"2o _______________ (7)
~7r -' mu·o
f, = The uatural frequency of vibration in complete
cycles per seco nd .
J m = The mass polar moment of inertia per tmit
length for each wing section measured in
slng inches squared.
B0 = The total angle of ll\·ist of the shell in radians
between the root and the section unde r
consideration when subjected to a distributed
torsional moment.
= L':.01 + t.B2+ t.Ba+ - _____ t:.O •.
t.01 , t.02, t.Oa, etc., are increments of t\1·ist at the
,·arious sections.
The equat ion of 00 given in A. D. M. 1058 is
in whichQ=
U m.
0
QLdx
o= 4A 2tE , - - -- --- - -- ----- __ (8)
L= The mean length of periphery of the sect ion
in inches.
J m has t he same meaning as in cq uation (7) .
dx= The length of section in in ches subjected to
torque.
t= The mean t hickness of stressed cover in in ches.
A = The average area in square inches bounded
by periphery, L .
E ,= The modulus of elasticity of t he cover in
shear in pounds per square inch.
With reference lo A. D. M. 1059, !he total area of
lhe ai rfoil section, A, is-
A = 0.725 he lo 0.785 he __ _____ __ (9)
For these comp11lalio rrs the area of the airfoil is take11-
A = 0.731 he.
L = [ 2.7 (~ y + 2] c_ ____________ (lO)
l >=(0.119h+ 0.256c) h2t__ ____ _ ___ (J l )
/ ;;= 0.0435 (c+ 6h) c2/ __ ____ ___ ___ _ (12)
·p
J m = 32.2 J dx_ - - --- -- -- - - - - __ --- __ (13)
In " ·bich-l
= thickness of cover in inches .
h= maximum ordinate of sectiou in in ches.
c= chord of section in inches.
J = static polar moment of inertia of a irfoil shell
in inchcs.4
=l•+l•
J m = mass polar moment of inertia in s lug inches
squared.
p= density in pounds per cubic inch for shell.
L = lengtb of periphery of airfoil section in inches.
l > and JJ, are respectively the moments of inertia in
inches• of the shell about the X and the Y axis through
the center of gravity of the ection.
•
•
Tire same sections we re 11 .·ed as in lire case of bend ing.
The mean values of h, I, and c arc taken rcRpecLivcly
fronr Figmes 9, 10, and LL and arc Labulatcd in colirmns
24, 25, and 26. These results a rc used in the cval 11a l ion
of A , L , I x, and I ii from equations (9), (10), (11), and
(12), the results or which are listed in columns 27, 28 ,
29, and 30. l1 and L a re p lotted in Figures 12 and 13.
The mean value of J for any secti on is tabu lated in
column 31 , and is fouucl from columnti 29 and 30 at
that section. T.J for any section i given in column 32,
found by summing aU the J 's in column 31 between
!hat section and the wing t ip. T.J for eclion 19- 20
equals 322 + 1969+ 2877 = 5168 illches.4
T.J, however, must be multiplied by a factor 3.22
sirrce J is composed of not only the shell a lone, but also
the ribs and spar . It was assumed that t he proportion
of J contributed by the ribs, spars, and ·hell was the
same throughout the wing. By comparing the total
moment of inertia of a section with that of the wing
covering alone, the factor 3.22 was determined for the
section between stations 9 and 10.
Th11s, for ribs-l..,.;
0,= 0.0418 CJh _________ ___ (l4)
Im ;no,= 0.454 ch• __________ ___ (15)
(See Air Corps I nformation Circula r No. 597, Volume
v 1.)
T.., 0 ; 0 , and I m;., 0 , arc, respectively, the mon1ents of
in ertia of the rib about the major and minor axis
t hrough the center of gravity.
I majo ,= 0.0418 [132 -(2X 0.070)]3X [22.9- (2 X .070)].
= 0.0418 (131.86)3 22.76= 2, 170,000 inches .4
J 0 ,;., 0 , = 0.454X 131.86 (22.76) .3
= 704,000 jnches.4
J for rib = 2,170,000 + 704,000 = 2, 74,000 inches.'
For each section,
J for rib = 2, 74,000 )' ~~= L,825,000 inchcs,4 since
llrc rc arc J 4 ribs iu 22 secti ons. Frum eq uation ( 13)
J ,,, fur ribs= 3;.2 1,825.,000 rlx sl11g irr chcs,2 in whichp=
dcnsity of spruce in pounds per cubic iuch.
= 0.0156 lbs. per c ubic inch.
dx= width of rib = 0.062 inch.
Jm for ribs= 0 3°2~~6 x 1 , 25,000 X 0.062 = 55" slug
inches.2
In finding the mass polar moment of inertia of the
fron t and rear spa rs, it wa assumed t hat they were
concentrated masses, M1 and JI2, located on the principal
axis distant r 1 and r2 from t he center of gravity of
the airfoil ection Then-
J '" for spars = M1r 12 + M2r 22.
Weight of both spars at section 9- 10 = 0.7 lbs. per
inch span .
Weight of each spar per 20 in che = 0.35X 20 = 7 lbs.
r 1=,43.7 inches; M1=3~.2·
r2= 19.3 inches ; M2=3;.2.
heuce--
J m for spars=3~.2x (43.7)2+ 3;.2x (19.3)2
~ 415 + 81 = 496 slug inches squared .
9
This value of Jm is nol slriclly correct as lhc weigh t
of the spars is not cq 11ally divided according to the
assumption. The propurtiou uf frunL and rear spar
\\·eights is given in Appendix A by Mr. G. A. Zink.
J'" for shell = 3;.2 J dx
in which-
J = 15,642 in cl1cs 4 (Sec column 31, sec.
!}- 10.)
p= dcns ily 3-pl.v bin:h = 0.0257 pouud per
cubic inch .
dx= length of shell= 20 in ches.
.0255
J m for shell= 32.2 X l5642 X 20.
= 248 slug inch es2 aL sect ion !)- 10.
The total mass polar moment of inerLia at section
9-10 is the sum of-
Shcll= 248 slug inches'.
Ri bs= 55 slug inches2•
Spars = 496 slug inches2.
Total J,n for ection 9-10 = 799 slug inches2.
R t . f J '" Of total 799 = a JO O J m Of sh ell 248 3 .2 2
which, \\·hen multiplied by the static pola r momcnL of
inertia, J, gives the total tatic polar moment of
inertia at that section. Thus, equation (13) becomes
J m= J·3.22·
3
;.
2
dx ____________ (l6)
In the actual computations, it i not necessary to
change J to J'" for each section by equation (16)
since p , the density of pl.nvood covering, is a constant
in ord inary airpla ne consLruction. Equation (7) can
be modified to use J , whi ch simplifi es t he computatio ns.
From equations (8) and ( L6)-
-::.JLd:-c p
0 .. = 4/ 121E. · 3·22 .32.2 d:c
_ "::.JL I . . , p •
0" - 4A'l E , X3.22 / 32.2 d.c·
0 o =IJ.!E:. X3.2 2X P_ / ·2
32
.
2
rx ____________ ( l7)
where-
'J:.JL
0 o= 4A 2l - - - - - - - - - - - - - - - - - - - - - - - - - ( LS)
"1:,J is taken over the par t of the wing between the
Lip and the section considered since the formula for IJ 0
wa derived on the assumption that J is equal to the
clistriputecl torque. If J is LI eel instead Of J m, equation
(7) becomes
f 1
/ / T-Jo I 32.2
•= 2.,,. "' E . -y T-JO' -Y 3.22 p clx2 - - - -- - - _ (lO)
where p= density of ply \\·oocl covering in pounds per
cubic inch .
The average values of t:.IJ arc found from columns
25, 27, 28, and 32 which arc !isled in column 33.
For cxamplc-
.:ie at section 19-20=~J[, 5165X 1~= 1? 87
4A2t 4X6402X0.047 ~ .
The t:.IJ's are proportional to the increment of
twist in length dx of the wing. T-t:.O in column 34 is the
um of M's in column 33, tarting with zero at the
roo t, Kinrc Lhc 1\·ing thcrr is co nsidered Tigidl.r fixed.
~~O aL a n.v RLat ion ~ivcR Lhc t.utal an~lc of LwiRL, 00 in
rndia118 aL t hat pui11L when 1twltiplicd by 33;~;~~2 ,
in which case dx i 20 in ches, E . equals 685,500, and p
eq uals 0.0257 . (Sec cq11ations (17) and (18).) "'Lt:.0
plotted in Fig. 14 shows the shape of t he 00 cu rve.
"'LJ X T-ti.O and "2:-JX T- .:ilJ' are found from columns
32, 34, a nd 35, a nd arc tabulated in 36 and 37 the suru
of wh ich columns <UC used in Lhc subs titution of
cq u:.1.Lion ( 19) .
1 .JE- j T-JIJ ;-32.2
!t=2.,,. I.,, i J02-Y 1. 82pd.~
~J0 = 293 , 343 X l03 .
~JIJ2 = 51 2X 107•
p= 0.0255 lbs . per cubic inch.
dx= 20 inches. •
From tests by the Mater ia ls Branch on shear of birch
plywood, page 24,
E , = 685,500 pounds per square inch.
- 1 .J-- / 293,343X-103 I 32.2
f, - 2.,,. 685,5oo \ 5182X l0 7 -V 3.22X 0.025SX20'
--2.,!,_. X ?~ 9 X7150·23 ,x 0 .9 9 .
= 9. 83 cycles per scco11cl .
Experimental / 1= 12 cycles per second.
P e r cen t error= 21.:127 =1 . .1 per cen t .
The calculated fre quency of torsional vibration is
affected by the rigidity of t he spars and the modulus
of shear of birch p ly wood. T he r igid ity of the spars
has been neglected, which, if considered, ,,·oulcl give a
higher calculated frequ e nc.\r. Little testing has been
carried out to determine Lhe 1110d ulus of shear of birch
plywood, and u11Lil more exk1rn ivc Lcsts a rc made Lhe
modulus of shear 111111,;t be accei1tcd as given.
SHEAR TESTS ON BlRCH PLYWOOD
(Copy of Dcitn /1'urni i;hcd bu J\fal erfrlls Branch)
1. The values for the shear bst parallel to the face
grain on three equal ply, a ll birch p lywood of 1/16
inch t l1ickness for the beam o~ the spruce flanges are
given as follows :
I
Web No. Ultimate of elast ic·
shear ity in
I shear
Modulus
Remarks
----- -------- -
!_ ___ _____ __ _
2 _ _______ __ Lbs. per
sq. in.
2, 140
2, 606
.Lbs. per
sq. in.
715, 000
656, 000
2, 480 635, 500
Thi' web, located on upper face at
ea.'t end, failed first.
This web. upper face at west end,
failed second.
2. Tlie results we re obtained from tests on specimen
3 X 5}~ inches with a gap of } ~ in ch between the shear
tools.
3. The same average value,; for modulus of elasticity
and shear may be used for Lhe web of beams with
plywood flange . The values for the modulus of elasticity
in shear as given above are only approximate inasmuch
as the number of specimens tested is limited
and accu1·ate values are d iffic ult to obtain.
10
'
IO "fc ·:r 'nu mo d 'n 1t ~
r- -1
'I of I-I
... \. tru ~ ti < ·Z i'\1 nc fil>D ai1e w~
I\
2fi 'I
1- -- \.
74 i\.
--'QT( trot 21 e
I
I v
I
1\
~ "" 'I
'U \.
.:::1 M \ hDI1 · 'S. ~e !TI ·S " ~ rz .s
$:! 411. \
pf
A la nt c r. A mDr IOf le m ~ llit~Q
;::: i\.
i5 41< i\ " I
I
I i:: IA I \ I 1 >n I
i:::i ._. I\. I I
,t: n \
' t !
j_1. a. , ~ I I
r-l
I ~ lf/1 'I
i i I !\
: .. '
l I\
'" ,-W ~
_ J \
IA ' \
b~ ,....
I i -,....
1,. -,..._
Ql' I I
I , ~ .c:::' a n I I i \ '
SL I
,.5,.. 60 ,-1-L -- I . --,._ _J_ - I
I?
I - ·- . -t- I-- . ~~"
μ::; I
o ~,, I
,_ I
iv 1J <IO 110 f~O fpO 2Do z 0 2. 0 .:J 0 J , 0 400 1¥0
l ) ', 111 ! · Int?< i?. nh ~s
,,,
•0 !O J 0 fqQ ' <'IO "''° z 0 3 '0 J 0 uul .f. 0
Set ni s1 a, ~ i ri n hes
FIGURE 9 FIGURE 11
'""' I\ .r s ,,, i11n l f1 a o
I ' ti cu ti' ( 2 Df'I nr n/ "' •1L>i1\6
"'' I ? !~
'
~ ? ""
. .,, '" ~ ':li't11
(lJ I ~ '
ti I ~ >A \
:::: ':::: \
i::: I ""' ?~ I\
'ts I .::;
0
11"
J 'U ..11.00 \
~
:::: ,, rt'ir (TIO "' ril n "' - ({ "' ,,,, \
::,, ~t n "lite IC 12 l 0 "lflnl"' ew. n I t.. I\
"' <l> I .....!.'<3 "" ' I I I\
I ::; ii) >nt
!~ 1.0:' ' E::: I JMn I\.
I ' I I
I I I'\
' " 'M
" ~ ,, I\
I 1'-
h i ! AM I\
\
"h"
" IO 0 I 0 HO 210 2 0 2 0 HJ JPO 4 0 4•0 " ) 4P ~p { 0 f 0 2 0 2 0 no J 0 3, 0 .oo 41()
"111' l "" 111 i !'JC n. s Yem ·.•ni tf1.rte iitc ~- !'<
FIGURE 10 FIGURE 12
11
I
on
nn '
!'...
Lon r--.
" I Ir r '
!'...
I n"'<i '" ' 1"-
.c:; oon ' , t;;
,... "" , ·- r--.
1-+2 .,,
I \
~. ,...,., iV l".: 'ri ht>r 'J bf t:rh• ~ IPI ·t i1~n
·- I 1[
..!!..>. •n At lrn lztl -2 r. Ollrin/ ~ne 1\ li:i·u-l' I
<....[ "" ....:
~~Ltba. '
..C::1 I I
.~-. In
I ~
' ~()
i ' I I
lu1 I
nn I
. r
' 4() B tW f, 0 ZfJO zo 280 J 0 3~ 4 po · 4 ~0
se m -s bdn ir; i1 c!Kes I
I
FIGURE 13
4 hn
/ I--"""
-i"n ... v
/
~ Inn v
.... v
?i'>n /
v
L /
... I~ ~ nn .,...v
"' I ~ v
c i)
j 'j() ,,/ e-Ct r Je f c r
/"' H ~a t ti-:- ---L. A m Jn 'JD ~a 1e w ·nQ
v - nn
/ I
lli'n /
/
oV I
I () ~ o ~o 1/JO .t oO 200 I 2 ~o 2.~o J/O 3 0 4(0 4',10
~ e1 ini SJ Dah , n ~n "h r:>s
FIGURE 14
22 1 23 1 24 25 26 27 28 29 30
I
31 32 33 34 35 I 36 37
-------- -
Station I dx
h t=tbick- c A L l:JL
Max. ness o! Mean Airfoil Peripb- lx 1;; J = Ii+l >' 2:J t19 - 4A't
2:69 :l;Ci02 2:JX2:ti9 ux:i:t.0•
ord . plywood chord area ery
---,-- --- - -
Tip Inches Inches Inches Inches Sq. in. Inches Inches• Inches• Inches•
17 4. 3 0. 047 46 320 93 JI. 25 31J 322 322 1. 57 397 157XJO 3 I J27.5X LO 3 50.5Xl0'
21 I 20 7. 4 . 047 85 5\0 172 .6 l, 910 J, 969 2, 291 8. l 395 156 905 3-
20
I 20 9. 3 . 047 95 640 192 102 21 775 2, 877 5, 168 12. 9 387 150 2,000 775
19 I
I 20 10. 6 . 047 99 760 I 201 133 3, 260 3, 393 8, 561 15. 9 374 140 3, 200 1, 200
18
I 20 12. 0 . 047 103 880 2JO 188 3, 690 3, 87 12, 439 J .o 358 128 4, 460 I, 590
17
20 13. 4 . 053 106 1, 020 2J7 275 4, 560 4,835 17, 274 17. 4 340 J16 5, 870 2,000
16
I
20 14. 7 .050 110 l , 160 226 324 5, 220 5, 544 22, 1 19. 2 322 102 7, 350 2, 330
15
20 16. 1 . 047 114 1, 320 234 379 5, 590 5, 969 28, 7 7 20. 6 303 92 8, 730 2, 650
14
I
20 17. 5 . 047 117 1, 480 241 462 6, 220 6, 682 35, 469 20. 282. 4 79. 9 JO, 050 2, 35
13
20 18. . 0572 121 1,650 250 650 8, 500 9, 150 44,610 17. 9 261. 6 68. 1 11, 650 3, 040
12 I
I 20 20. 2 . 0675 124 l, 840 257 933 11, 100 12, 033 56, 652 16. 0 243. '/ 59. 1 13, 00 3, 350
11
20 21. 5 . 0675 128 2,020 266 1, 105 12, 400 13, 505 70, 157 16. 9 227. 7 51.8 16, 000 3, 640
10 I
20 22. 9 . 070 132 2, 200 275 l, 342 14, 300 15, 642 5, 799 17. 4 210. 8 44. 2 18, 050 3, 780
9
20 24. 2 . 081 135 • 2, 390 282 J, 780 18, 000 19, 7 0 105, 579 16. 1 193. 4 37. 4 20, 400 3, 940
8
20 25. 6 . 0 l 139 2, 590 291 2,050 20, 000 22, 050 127, 629 17.1 177. 3 31. 4 22, 600 4, 000
7
20 27. 0 . 081 142 2, 700 298 2, 340 21,600 23, 940 151, 560 17. 0 160. 2 25. 6 24, 200 3, 880
6
20 28. 3 . 081 146 3, 020 306 2, 645 23, 700 26, 345 177, 914 18. 4 142. 3 20. 3 25, 300 3, 610
5
20 29. 5 . 0 l 149 3, 210 314 2, 030 25, 500 28, 430 206, 344 19. 4 123. 9 15. 3 25, 600 3, 160
4
20 29. 8 . 081 150 3, 230 3J6 3, 020 26, 100 29, 120 235, 464 22. 0 104. 5 10. 9 24, 600 2, 560
3
20 29. .081 150 3, 230 316 3, 020 26, 100 29, 120 264, 584 24. 8 82. 5 6. 21 , 00 1, 800
2
20 29. 8 . 081 150 3, 230 316 3, 020 26, 100 29, 120 293, 704 27. 5 57. 7 3. 34 16, 900 980
l
20 29. 8 . 081 150 3, 230 316 3, 020 26, JOO 2<J, 120 322, 824 30. 2 30. 2 .91 9, 750 294
Root
I I
Suml:J X2:A9 =293, 343Xl0 a
Suml:J X:l:t.9'=51, 823XJO •
' APPENDIX A
The weights of the rear and front spars, as calculated
from the detail· of the spars between stations 9 and 10,
arc 5. 4 and 7.08 pounds respectively. These weight
were based on a density of 0.015tlpou nds per cubic inch
for t he spar material.
The dimensions referred to a re at a distance of 190.0
inches from the fuselage, and a re as follows:
c (chord)= 132.0 inches.
h (maximum ordinate) = 22. 9 inches.
t (thickness of plyll"oocl shell) = 0.070 in ches.
The formulas used in the following computations a re
the same as• t hose used in the discussion . A sample
computat ion for the dctcnnining of J m for the ll"ing cell
follows:
I z= [0.119 (22.9) + 0.256 (132.0)](22.9)2 (0.070)
= 1,342 inches•.
I .=(0.0435)fl32.0 + 6 (22.9) ](132)2 (0.070)
= 14,300 inches•.
J = l z+ f u
= 15,642 inches 4 •
J = (O.OZ55l (20) (15 642)
m 32.2 '
= 24 .0 pound -in ches 2 .
J ,,, for the ribs is found as folloll"s:
c1 = c-2 (0.070) ; c1= 131. 6 inches.
h1 = h-2 (0.070) ; h1 = 22.76 in ches.
Center distance of pars = 63.0 inches.
I u= 0.0-H 8 c13h1.
f z= 0.454 C1h13.
I .= O.OH (131. 6)3 (22.76)
= 2,170,000 inches•.
I z= 0.454 (131.86) (22. 76)3
= 704,000 inches•.
J = I z+ I , .
= 2,874,000 inche •.
Jm= (0 3o;~6) (0.062) (2,874,000) G~)
= 55.0 pounds-inches 2.
J m for the spars.
Distance of C. G. from leading edge of wing = 62.7
inches.
r1= 19.3 inches, r2 = 43.7 inches.
Jl!I 1 5. 4 "i\ll 7 .08 = 32.2' j 2= 32.2
J m=~z~~ (19.3)2+~z~2 (43.7)2
=487.5 pounds-inches 2.
Weight of rear spar/20 inches of length, stations 9- 10.
2.92 pounds weight of web.
2.92 pounds weight of beam.
Total weight of spa r, rear, 20 inches length = 5.84
pounds.
Weight of front spar/20 inches of length, stations
!}- 10.
4.21 pounds weight of web.
2.87 pounds weight of beam.
Total weight of spar, front, 20 inches length = 7.08
pounds.
C. G. along x axis=l32.0 (0.475) = 62.7 inches.
C. G. along y axis = 22.9 (0.37) = .47 inches.
Shell-
J = I z+ I •.
I z= (0.119h + 0.256c) h2t.
I z= [0.119 (22.9) + 0.25G (132.0)] (22.9)2 (0.070).
f z= l,342.
I .= 14,300.
J = l5,642.
J m =
0 3~~~5
(20) (15,642) = 248.0.
Rib-
J , = 0.0418 c13h1.
J, = 0.454 c1h13.
c1 = c-2 (0.070) = 131.86 inches.
h1 = h-2 (0.070) = 22. 76 inches.
C. L . rear spar to C. L . front = 63.0 inches.
The total J m for section considered is equal to the
sum of the J m's of the shell, rib, and spars, orJ
m= 487.5 + 55.0 + 248.0
= 790.5 pounds-inches 2 .
Ra t1. 0 0 fJJm oftota l 790.5= 319 m of shell 24 .0 · ·
(13)
0
..