ION CIRCULAR
PUBLISHED BY THE CHIEF OF THE AIR CORPS, WASHINGTON, D. C.
Vol. VII August 15, 1928 No. 618
PRELIMINARY STUDY OF
FATIGUE FAILURES OF METAL PROPELLERS
CAUSED BY ENGINE IMPULSES AND VIBRATIONS
(AIRPLANE BRANCH REPORT)
Prepared by J. E. Younger
Materiel Division, Air Corps
Wright Field, Dayton, Ohio
September 12, 1927
UNITED STATES
GOVERNMENT PRINTING OFFICE
WASHINGTON
1928
Ralph Brown U ··~· .. ~:·~·'
UBRAfW
JUN 11 Z013
Non·Depoitor~
Auburn University
A
CER'I'IFICATE: By direction of t he Secretary of vVar t he matter contained
herein is p ublished as administrative inforrn aiion and is req uired for the proper
transaction or t he publi c b us iness.
(II)
PRELIMINARY STUDY OF FATIGUE FAILURES OF METAL
PROPELLERS CAUSED BY ENGINE IMPULSES AND
VIBRATIONS
SUMMARY
This is a report of the results of a preliminary study
of the cause and prevention of the breakage of certain
metal propellers in bending in a plane perpendicular
to the propeller axis. An examination of certain breaks
of this type has clearly shown them to be characteristic
fatigue failures. Two known causes which may
operate to develop this type of failure in the propeller
arc:
(a) The periodfo torque impulses and
(b) The transverse vibration of the engine.
Both of these causes a re investigated with particular
reference to the probability of the occurrence of the
phenomena of resonance. Formu las are developed
and methods suggested, which, it appears, ma); be
used effi ciently and practically in the routine design
and tests of metal propellers.
Since the problem of forced vibration, and hence the
phenomena of resonance is one always found in airplane
structu res, the basic theory is hereiu presented
in detail for ready reference. Care is taken to make
clear the processes involved in vibration problems in
order that these processes, as well as the final results,
may be available for general use.
The material concerning the special problem of propeller
vibration, being considered, is presented in the
body of the report, while the fundamental mathematical
processes are discussed in the appendices. That is,
instead of including long mathematical derivations in
the body of the report, which tend to obscure the mechanical
problems involved, references arc made to
appendices for derivations and mathematical discuss
ions.
INTRODUCTION
The immediate provocation of this analysis was the
breakage of propeller No. X- 49975 while in fl ight.
The observations made, which appearer! to have a
direct bearing upon the nature and cause of the failure,
are as follows (see figs. la and lb):
(a) The break was in the plane of the propeller disk;
that is, perpendicular to the direction of flight.
(b) The break occurred in the round portion of the
propeller blade at the tip of the steel hub.
(c) In the original design a sudden change of section
occurred at this point, but the blade was later faired
down to a gradual change of section.
(d) The engine on which the propeller operated was
a Liberty "12." The speed of operation was 1,400 to
1,7 0 revolutions per minute.
(e) It \Yas noted by pilots that the engine vibrated
excessively in a vertical plane, it being located in the
wing in such a manner that vibration in the horizontal
plane was very much more restrained than in the vertical
plane.
(j) The blades of the propeller were of duralumin, the
fatigue limit of which is 15,000 pounds per sq uare in ch.
Concentrated stress at the encl of the steel hub may
have been instrumental in causing the b reak, but the
direct strcs. cs, those clue to the centrifugal force and
th average torque, irrespective of the reversal of
stresses clue to periodic impulses, were shown by calculation
to be too small to be the probable cause of the
failure. We are thus lead to the study of periodic
impulses and v ibrations as the probable underlying
cau. es of the type of failure.
For the purpose of comparison, calculations arc carried
out for both propeller No. X- 49975, which is known
to have given trouble, and propeller No. 070717, which
is known to operate satisfacioriJy with the Libe;·ty " 12"
motor. The criterion developed to show the cau c of
the unsati. ·factory operation of the first propeller should
also show the satisfactory operation of the second
propeller.
CONCLUSIONS
1. Fatigue failures of metal propellers may, under
ordinary operating conditions, be cau ·ed by resonance
or near resonance of propeller vibrations with frequencies
of torque impulses and engine-mount vibrations.
2. The fa ilure of propeller No. X- 49975 in fligh t was
caused by resonance of propeller vibration with transvcr
e engine-mount vibrations.
3. An experimental study of the magnitudes and
frequenc ie of engine-mount vibration. should be made
in connection with further study of propeller vibrations.
4. A vibrating machine should be designed and constru
cted, following the principle ·hown in Figure 4, for
standard fatigue te ts on metal propellers, and an experimental
study of propeller vibration from this
standpoint should be made.
5. The idea of an "equivalent uniform beam" as
outlined in this report, for applying vibration calculation
to propellers, should be investigated experimentally
si nce the calculations would be greatly simplified.
(1)
MODES OF PROPELLER VIBRATION IN THE
TRANSVERSE PLANE
Figure 3 shows the modes in which vibrations of the
propeller, due to transverse oscillations and torque
impulses of the engine, may occur. If we assume the
propeller in the hori zontal position, as shown, and the
engine vibrating with a component in the vertical p lane
2
clearly illustrated by taking the wire ·ufficiently long
as the natural frequency of this mode of vibration, for
the same length of wire, is approximately 4.4 times the
natural frequency of the preceding modes.
If the torque impulses arc given by the hand at the
natural frequency of the wire, very large amplitudes
may be set up with a very slight effort. This, of course,
is a result of the phenomena of r eso na nce. Within
Fro. Ia
FIG. lb
of amplitude s., t hen one of the three modes of propeller
v ibrat ion, (b), (c), or (d) will occur 1Yi thin the
limit of the probable frequency of the engine vibration.
If this is not obvious from previous experience, a small
wire subjected to similar \·ibrations by the hand \Yill
clemoDstrate the type of curv<>s clearly.
The type of vibration du<> to torque impul.·e:;J (fig.
3 (e)), is not so obvious. Howc\·er, it ran be very
reasonable limit the magnitude of the amplitude of the
forcing vibration or the impulses, is immaterial at the
point of r e. onance, since excessive vibrations will be
built up in any case unless su ffi cient clamping forces a rc
provided.
A simple experiment illustrating the destructive
effect of resonance of forced and free vibration was performed
by the a uth or, as shown in Figure 4. " B " is
a small block through which a ;l-i-inch tool-steel rod
was inserted as shown. The block could be o ciliated
at any desired frequency and amplitude, within reasonable
limits, by the variable speed motor "A." ' Vith
very small amplitudes, less than 0.05 inch, of the
block "B," the rod could be bent out of shape by holding
the motor at a speed equal to t he natural frequency
of vibration of the rod in one of its modes of vibration.
TRANSVERSE VIBRATION IN ROTATION
Figure 2 shows a few of the possib ili ties of transverse
v1 bra t i on being induced in the propeller by a vertical
"omponent of vibration of the engine mou nt. The
width of the graph represents one revolu tion. The
One reiJolultO!l or ---~
propeller
B fa) -Le9end-
3
again subjected to t he maximum accelerating force
which tends to spring it into the posit ion as shown. It
is th us seen that two effective impulses tend to set up
transverse vib rations for this particular setting. In
(2) it is shown that four small effective impulses may
be obtained per revolu t ion. A ·tudy of these curves
and others which may be easily drawn will show a
number of combinations of vibration and rotation which
determine variou frequencies of impulses.
We should expect that t he frequency of these -mpulses
will always be some multiple or factor of the
revolutions per unit of t ime of the engine. vVith the
revolutions per unit of time and the natural frequency
of vibration of the propeller in the proper mode known,
one should be able to predict the possibility of
resonant vibration.
UESONANCE PHENOMENA
AB is lheneulralaxis
af"/IJe propeller
The problem thus becomes largely one of determining
the natural frequency of vibration of
the propeller in the proper mode of vibration
to produce resonance with the impulsive forces.
In the case of the torque impulses, these are
known to have a frequency six times the revolutions
per minute for a 12-cylinder engine. The
frequency of t ransverse vibration of the engine
may be determined by an oscillograph. However,
if the nat ual frequency of vibration of the
propeller in a mode similar to (b) (fig. 3), should
be a m ultiple of the revolutions per minute, the
opportunity for resonant vibration is so probable
that no furth er detail need be sought in regard
to the exact frequency, so far as the problem of
r esonance is concerned.
.D109rom shows Ille
relol1ile po.51/ion cf!l1e
neulrol axis of"l/Je propeller
wi/17 reterenceto
/IJe wbru/!{Jn displocemenl
(.bi}
7ivo effedive 1ir.pulses
per revo!ulion.
(b2)
rour small ef!ecl1JJe
_ _jc_.L-'~-l---',,..:::"'-',___,,<-1--~"'--=+--.,.1rhes
in fb) Ille tre1J11ency of"
\Jibroltro is //;e some as
//Je Rev. per Nin.
(cJ)
Two etrecl1ve impulses
per revolulton.
'B
(c-1)
rour erleclive impulses
P§r revolut1m
--1L--/-'"--'"""j..~,._---;,,f-~'--"'t<:::~-::7l--'-:/n fcJ /IJe trequenc!I or
v1brol!ro is lwice llJe
Rev.perNin
Ttme
ACCELEUATION FORCES
Another phase of the prnblem, which is not so
simple, involves the effect of the accelerations
irrespective of the vibrations which may be set
up. That is, even t hough resonance is not produced,
t he accelerations imposed upon the propeller
may be of such magnitude that the stresses
produced may be above t he fatigue limit, in
which case we may expect rupture with a sufficient
number of reversals of stresses.
If the center of gravity of a beam, with uni-
FIG. 2
vertical displacement of the curved line represents the
amplitude of vibration of the engine. It will be noted
t hat in (1) and (2) the frequency of vibration of the
engine and the revolutions per second are t he same.
In (3) and (4) the frequency of vibration of the engine
is twice that of the revolutions per second. I n (1)
the propeller is in a vertical position as the engine
mount passes its neutral point. When the engine
mount reaches its limit of vertical travel, the propeller
is in a horizontal position, L . In this po ition, the
propeller receives the effect of the maximum vertical
acceleration of the hub, tending to spring it into the
p osition as shown. When the engine again passes the
neutral position on its downward course, the propeller
is in the vertical position, 0 . At P , the p ropeller is
form or nonuniform cross section, be given an
acceleration in a direction perpendicular to the neutral
axis of the beam, an equivalent loading, that is, a loading
which will produce the same defl ection, is twice the
r eversed effective force per unit length. For example,
if the weight of a length of 1 inch at a distance from
the hub is 0.5 pound, the loading to be considered at
this point is-
Loading= 2 ~ a g
6A
= 2 -- a
g
= 2x 0·5x a
g
(1)
4
(o) Propeller, lrons1Jerse plone.
50 is /he omplilvde of enqine t.J1bro/1on
(b)Tre9uenc9 of' e f!ec// l)e impuls ~ due lo eaqine moulll 1)1brolion much
less thon Ille nolurol fre9uenct; of't.Ji.brolion Of !he propeller.
So
c)ffe9uenc9 of' ef'teclive impulses e lo en9/ne mounl vibration
s!t9hllS1 less !hon !he nalurol ffe9 '17CIJ of' vibral!on of' /he propeller. Approoch1i79
resononce.
(d )Tre9uenct/ of'impulses due lo e 9ine mounl v1brakon sliqhlltj 9realer
/hon !he 110/urol fre9uena; of' l-l ro/;on of' !he propeller. ---- --- -- - -- - -
(e) Mme of' vibrolion p roduced lJ lor<;ue impulses.
f laslic cunJes o/' !he steel rod
f"or vonous ratios of' f'ree lo
f'orced vibration.
F IG. 3
The vibralinq ~echom$m, l/Je
princf,ole of' which mo!f be used in
Ille conslruclion of' a melol propel/er
l'ali9ue leslin9 mac!J1i7e, and in f'indl/-J9
Ille nalural lre9(1enq of' iJibralion of'
the prO,tJeller.
FIG. 4
.l'iqhlh or an inch
tool steel rod
Slider crank
mechanism
t; = Not(lrol trequenc!f of'
vibration of'lhe rod.
complete vibrations per
minvle.
f'= J?evo/ut1011s per minvle
o/molor.
R~fo
Sa=Ampl!i'uae of' /'orced vibration.
In which w is the weight per unit length , 6. is the
weight per cubic inch of the material, A is the area
of the cross section i11 square in ches, "a" is the acceleration
in feet per second, and g is the gravity constant,
32.2.
5
If we take S as the fatigue limit of the material,
15,000 pounds per square in ch, and neglect all other
stresses, we have-
Formula (1) is proven in Appendix (A) . or
It should be part icula rly noted in this case that we
assume the acceleration to remain constant long enough
15,000 I = 35.7a
c
for the propeller to reach a maximum deflection.
Obviously, no such condition exists. However, it is
not intended that th is be a final analysis of the problem,
but rather a preliminary study of the forces with
which we shall deal in the more exact and complicated
analysis. Since the final analy is involves the solution
of certain differential equations and the evaluat ion of
several constants, it is not easy to follow the physical
significance of the mathematical processes, hence a
preliminary study of this nature should prove valuable
in fixing ideas.
Since the bending moment at any point x is-
M. = f f wdxdx (2)
Since w is used here as the general loading symbol,
from (1) we have-
M. = 2 ff [ ~A ad,1;dx]
Or by placing the constants outside the integral-
M.=2 ~a J J Adxdx (3)
If we let x=cL,
dx= Ldc
Thus-llf.=
2~ aV J J Adcdc
If we let -6. =p, the ma per cubic inch of the material, g
M. = 2paV ff Adcrlc.
The Yaluc of this in teg ral for propeller o. X- 49975
ltas been worked out for the vibration analysis of this
propeller and appears in Figure 5c. The integral
merely represents the moment derived from a loading
cur ve, and no difficulty should be experienced in understanding
same.
At the 7-inch station, where the propeller blade
enters the steel hub, the point where the rupture occu
rred, from Figure 5,
llf .=2paVX 1.3 (4)
Since, p= 0.1015 pounds per cubic inch for duralumin
and L= 66 inches,
M =2X 0.1015X aX 662X l.3
x g
= 0.203 X a X 5662.8=35.7a
g
We also have-
(5)
in which I is the moment of inertia of the cross-sectional
area, S is the allowable fiber stress and c is the distance
from the neutral axis to the outer fiber.
15,000X 11.07 in ches •
a= 35.7X l.93 inches
= 2,400 feet/second 2
= 28,800 inches/second 2
(5)
We will now see wl1at, approximately, the amplitude
and frequency of a vibration should be to give this acceleration
. Assuming simple harmonic mot ion we have-
(6)
in which 8 0 is the amplitude of vibration, and f is the
frequency per second. The acceleration, a, is a maximum
when sin 27rft is unity, so that--
a = S 0 (27rf) 2=39.4Sof2=2 ,800.
If we assume 60 vibrations per second, which is
reasonable, we bave-
28,800 2 8
S 0 -39.4X 60X 60 1,420=0.203 inch,
which is not unreasonable. This, together with other
calculations, show that it is not unreasonable to expect
that accelerations may occur in the vibration of the
engine mount which, aside from the vibrations set up
in the propeller may cause fatigue failure.
We will now investigate the probability of a fatigue
failure of the crank shaft under the assumption of the
fatigue limit stress given a bove.
The react ion on the shaft is twice the shear in one
blade of the propeller. We have the shear-
26 [I V=- aL Ade.
g • 0
(See fig. 5b.)
The value of the integral at the hub is 7.
Thus-
R = 2V= 2X 2X 6.XaX L X 7
g
= 4XO.lOl 5X 2,400 X 66 X 7 = 14,000 pounds.
(]
(7)
The bending moment on the crank shaft is 14,000X
6.6= 92,000 inch-pounds.
The fiber stress is-
Mc 92OOO X 1.312 . S=1=- '- 5
.18--=23,300 poun.ds per square mch.
This is less than half the fatigue limit stress of the
crank-shaft material. Vve need not, therefore, consider
the strength of the crank shaft further in our analysis.
NATURAL FREQUENCY OF VIBRATION OF A
PROPELLER BLADE, COEFFICIENT OF VIBRATION
If a periodic impulse or motion is applied to an elastic
structure at a frequency which is the same as the
natural frequency of the structure, experience teaches
us, without resorting to any complicated theory, that
the ampli tude of the vibration of t he structu re will
" build up " u ntil a point of equilibrium is reached
between the augmenting fo rce and the resisting forces.
Tl1is, which is know11 as reso na nt or synch ronous
vibration, is avoided wherever possible, since it represents
the worst possible co ndition. High stresses may
occ ur even if reso na nce be avoided , which depend on
the nearness to reso na nce. This last, the general case,
will be consid ered later in t his report, but it invol ves,
fu ndamentally, the natural frequency. vVe t h erefore
present a standard method of cl ctcrn;1ining the natural
frequency of a propeller blade wh ich appears to be
sufficiently accurate for practical purposes, and yet
involves only a small amount of labor . The method,
howc,·cr, is valid only fo r the fundamental mode of
vibration.
As in t he previous p roblem, we resort to t he energy
method of solut ion . I n t he posit ion of maximum deflection,
the kinetic energy is zero, and in the p osit ion
of no deflection, t he p otential energy is zero. Assuming
no dissipation of energy, we thus have by eq uating
the energies, and summing over t he entire length-
(8)
in wliich F is t he force per unit length required to hold
the beam in the maximum deflected posit ion, Y. A is
t he mass per unit length, and ~f is the velocity. To
solve equation (8), it is necessary to know y as a fun ction
of t he time, and F a.s a function of y. This requires
the solution of equation (4) in Appendix C, which is
not simple, and by far too tedious to be of any practical
value in this problem. I n this connection, we make
two a sumpt ions-
(a) That each clement of the beam moves \1·ith simple
harmonic motion, and
(b) T hat the maximum deflection curve of the beam
is t hat due to its own weight, so that F becomes pgA .
It should be noted here that the natural freq uency,
within reasonable limits, is independent of the amplit
ude of vibration . The fi rst ass umption is justified by
experiment, and t he second by the results obtained.
It is certain that this is not t he t rue elastic curve,
however. We have--
y= Y sin 2.,,.jt (9)
(10)
6
However, when y=O, cos 2.,,.ft= 1, thus-
(~t)2 =4Y2.,,.2p
Thus eq uation (8) becomcs-
.Lj(l /2pA) (4 Y2.,,.2p)= .LJ4pgAY (11)
or
2.,,.2pDAY2= ~ 9 DAY (12)
From 12, we obtain a formula for t he frequency, f-
(13)
Suppose A. and Y be taken every predetermined
interval on t he r adius, fo r instance every 5 inches, and
t he s tat ions numbered con ec ut ivcly by subscripts, we
have--
j - -Jg / ~A1 Y1+A2 Y2+ Aa Ya+ · · ·A . Y . (l 4)
- 2.,,. \f :!:A1Y 12+ A i Y 22+A3 Ya2+ · · · A0 Y0 2
In (14), it will be remembered, Y is the deflection of
t he propeller blade due to its own weight.
A sample calculation of Y based on propeller No.
X- 49975 is given in F igure 5 a, b, c, cl, e, f. The usual
successive grap hical integrat ion of t he load, based on
a unit length, is carr ied out. We fi nd in Figure 5 f
t ha t -
(15)
where d is a coefficient obta ined as indica ted in the
above-mentioned figures.
Substitu ting (15) in (14), we havc-
1 /E / 2::A1d1+A2d2+Aada+
f=2.,,.v-Y /,\! 2::A 1d12+A2d22+ Aada2+
Le tting-
We have--
K /E
f=21r.v-Y a-,;
(16)
(17)
(18)
Obviously, since d is obtainc!i from a beam of unit
length, K is con tant for geometr ically similar proK
pellers. Writ ing the constants, 2.,,.• a G, we have-r=
c. /~
L-\f P (19)
TABLE !.- Propeller No. X - 49975 /
Cross Sectiona l Are a and Moments of Ine rtia about the Principa l Axes
Area=0.738 bt. I ... ; • .,=0.0464 bt'. I ma; 0 ,=0.0446 b't. In which b=width, and t=thickness
Station _______________________ _
7" 10" 12'' 1 " 24" 30" 36" 42" 48" 54" 60" 66"
- ------------ --------- ------------ -----
)~id\h (inc~es)h _______________ ------- - 3. 875 7. 03 Ihw mess (me es) ____________ 7. 66 8. 45 8. 78 8. 80 . 70 8.12 6. 96 5. 41 3. 45 0 Area (square inches) __________ 3. 875 3. 17 2. 71 1. 768 l.185 . 943 . 774 . 621 . 4 7 .399 . 238 0 l minor (inches•) ____________ :_ 9.0 11. 79 16. 67 15. 3 11.06 7. 68 6. 12 4. 96 3. 73 2. 50 1. 595 .606 0 I major (inches•) ______________ 23. 6 11. 07 10. 38 7. 20 2. 16 . 678 . 327 . 186 . 09 . 0373 .0156 . 0026 0 I transverse• ________________ _ 23. 6 11. 07 49. l 54.4 47. 6 35. 28. 6 22. 7 14. 8 7. 36 2.83 . 045 0 23. 6 11. 07 32. 0 35. 23 34. 30. 9 24. 7 19. 7 13. 0 6.50 2. 70 . 40 0
•I transverse is in tbe plane of rotation.
7
- ,,, A.l" ,.j,r)--,,j,. ·~· ,;.,. - .J
;, .!- L ~ J. ,,
'" ~ '4J ; I I' ,I., N ;.,, <o
\ Ip.I- ' ' ·'-
v..- ,,.; U f -~L4r "" ""'- Cl 'A
\
IP ' M I\
I,,., .,,., ~/I \
,9
I"\.
- If; ' ' i.d
.......
I? ......
..... ,, -. a cw GIJ .Oi' C:5 J: 0 . 07 0'8 ~:;
ri; .,~M'
'7
"\ " ..· -Ir. L..r. ~. .... : '0. -'
'6
I \. ,,,,, .. ( ..,,1. ,,, " ·l. ,;, "' '",; " i..r.
;..- 0 ' . , •. .
\ " f ,?. .; I
IQ 1...- 1'\ I 0
'
1 \
'\
? '\
/ ['-., ,__
/1 ,...._~
' G G. Cj a..- ,<.> G ~ G 'ti 0,9 "' T; 1' "'• ;.b I
Fms. 5a and 5b
.• , .. , .
.,, I -- !'>Mr ·n - -- ..
\ '\
'" " ••• . J -·. ·- '" < U-"n ;-·/ ,: °"' - v4 /, r ~, .,q
iv -
'J \ " ,•.,/ - .., I. -- '\ I\, , \ - '7L "( I 1
IA 1"' = 0 .?)
~-
I'..
I'.. ,,, \ , "' I'
} \ i-...
10 I\ ,. r-..
' -,__
"~ \, AW - r-
!\.
n
r--.
A ,_
G GI'" ·~ ~ . '" GI" ·' GI" q;> !.IO
·~
-~ r--.,1
~ ..... I
r----
' "' C1 '" 'I" ' -0 7 'I<'" <9 'I"
IH - - - · - c -
,., -...... I ,.,nl ' ' ..... - ·In'
/A• r •I v ' 1::. ia
' ,_
' "' I - ., ~, ~ •. J .- ,-,M
l•r !
rtrJ' 7i I ' \
T- ~ 1.. ;?/ \,
_Ur '1~
, ,-,M ' I
I I ' 'd ' ,.,'.,.,, \,
I i I \, I
'
,.,,_, '\
I \.
I/ I
I
r ~ I
'\
I
0 \ - \,
..... ,.,_o
"'
,_
t/ C? tμ' 0# G:5 q t7 G6' q; '" \
,.,,,.,
"';J,,. ,,.,., ' ' "' a3 G.f' G:5 't5 G G,;' "' 1 0
I I / •a, r. ' jil'
Fras. 5c and 5d FIGS. 5e and 5f
1381-28--2
Sta·
ti on
--
0.1
.2
. 3
. 4
. 5
.6
. 7
.9
1.0
TABLE 2.-Propeller No. X -49975
A d d' I Ad Ad'
--
18. 0 o. 00018 0. 0000000324 0. 00324 0. 000000584
14. 3 . 00066 . 000000436 . 00945 . 00000624
9. 8 . 00139 . 00000193 . 01361 . 0000188
7. 0 . 00230 . 0000053 . 01620 . 0000371
5. 4. . 00334 . 00001115 . 01800 . 0000602
4. 0 .0044 . 0000201 . 0179 . 00008
2. 75 . 00570 .0000325 . 01570 . 0000895
1. 60 . 00699 . 0000487 . 0112 . 000078
. 75 . 00834 . 0000695 . 00625 . 0000524.
0. 0 --------- -------------- .11155 . 0004224.24.
K=
f
16.2
=2.,..X 662
32.2X 9,750,000
--0.102
=32.5 vibrations per second.
8
(20)
(21)
This result is probably within 20 per cent of the
true value. An independent calculation carried out
for the same propeller, using a much larger scale for
the graphical work gave the frequency as 28 vibrations
per second, or 1,6 0 per minute. It should be borne in
mind that the effect of the joints at the hub, the composite
structure at the hub, and the effect of the twist
in the blade is assumed or neglected. A 10 per cent
or larger error in the modulus of elasticity used is also
not improbable. It appears, however, from a study
of Figure 2, and the above results, that resonance
between the engine-mount vibration and the free
vibration of the propeller was quite probable, and
apparently was the cause of the failure .
In order to test further the value of the criterion,
calculations for the frequency of propeller No. 070717,
which was known to operate satisfactorily, on the
Liberty engine, were carried out. The frequency was
found to be 2,200 per minute, which was considered to
be out of the range of resonance.
NATURAL FREQUENCY EXPERIMENTALLY
Since there was no duplicate for propeller No.
X-49975, the above calculations could not be checked
experimentally. However, a test on propeller No.
070717 showed that the figure of 2,200 per minute was
400 too low. Assuming the same magnitude of error
for the first propeller, resonance may not have occurred,
but on account of the nearnes the stresses were probably
above the fatigue limit.
The natural frequency in this case was determined
by a reed tachometer. The propeller was clamped at
the hub on a heavy steel table for the experiment.
The tachometer was held against the hub after the
blades were set in the proper vibration. A very
definite point of resonance in the tachometer was noted.
EQUIVALENT UNIFORM BEAM
engine mount when resonance is not produced, we
introduce the idea of an "equivalent uniform beam."
We define the equivalent uniform beam as a beam which
has the sa~e section as the root of the propeller blade,
and the same natural frequency in the fundamental
mode. We assume that, since this equivalent beam
has the characteristics of the propeller blade in the
fundamental mode of vibration (fig. 3 (b) and (c)),
it will have at least approximately the same characteristics
for the following cases. The following appears
to be a simple method of dealing with the problems
being considered, but it should not be taken seriously
until proven by experiment.
The period of a uniform cantilever beam in free
vibration is-
T 0= l.8L02-fl;g= i.sv.J~~ (20)
(See Appendix C, formula 21.)
The period of a nonuniform cantilever beam is-
T =L2 /!:::.
n CVEg
(See equation 19.)
Equating '11
0 to T 0 , we have-
1.8L.2{#g=~{ig. or L.=.J1~20~f
(21)
(22)
L 0 is the length of the equivalent uniform beam, L
is the length of the propeller blade, j the ratio of the
moment of inertia at the root ection to the corresponding
area.
If the "equivalent beam" assumption proves satisfactory,
experimentally, the entire beam theory, as
applied to uniform beams, is open for use with the
nonuniform beam. vVe thus will have available for
use the equations of forced vibration of Appendb!: C.
ENGINE TORQUE IMPULSES
From Figure 3 (e) we note that the bending moment
at the center must be zero, since that is a point of inflection.
Thus, for the purpose of determing the frequency
of vibration in this mode, we can assume the propeller
pin jointed at the center.
We thus have Case III, Appendix C, equation (44)
available for use in considering the equivalent uniform
beam.
Since when-cos
nL sinh nL=sin nL cosh nL (23)
the shear, moment, and deflection become infinite, we
know that the value of nL given by equation (23) will
yield the natural frequency. We find nL to be very
5.,..
nearly 4 . Noting that-
(24)
In order to determine the natural frequency of
vibration of a propeller in a mode which would be produced
by the torque impulses (fig. 3 (e)) and to
determine the stresses due to forced vibrations of the
(25)
(26)
•
and comparing same to equation (20), we find that the
natural frequency of case (e) (fig. 3), is (2.45X 1.8=4.4)
(approximately) times the natural frequency of case
(b) or (c) of the same figure.
Since by measurement with a reed tachometer, the
natural frequency of propeller No. 070717 was found to
be 2,600 per minute for case (b), this would indicate
2,600X4.4 or 11,440 vibrations per minute for case (e).
And for propeller X- 49975, approximately 9,000 vibrations
per minute. The torque impulses of the Liberty
engine occur at the rate of 8,400 to 10,800 per minute.
It appears that there is ample opportunity for resonance
in the case of propeller X-49975, but not in the case of
9
propeller 070717. Several points should be borne in
mind in this connection however. They are--
a. The torque rmpulses of the Liberty engine do not
occur at equal intervals of time.
b. In the above calculations, we assume an equivalent
uniform beam.
c. The stresses at the hub due to this type of vibration
are small since the elastic curve changes slope at
this point.
It is dangerous to conclude, however, that the failure
did not result from this type of vibration. The case
should be proved, or disproved by experimental evidence.
APPENDIX A
CANTILEVER BEAM, DYNAMIC LOAD, ON A PROPELLER
BLADE DUE TO A CONST ANT ACCELERATION OF THE
HUB
I rrespective of the vibration set up in a propeller,
the acceleration to which the hub is subjected, due to
the motion of the engine mount, may cause stresse
above the fatigue limit. As an introduction to the
general problem of vibration, this special case will be
considered, since the problem embodies an excellent
illustration of how the energy method of analysis may
be used in considering the dynamics of elastic structures.
~--~---------- ------
Ace el= Cons/on!= a
VeloC1!51=V =O
(a )
We let the energy of the sy tern at rest be zero. The
energy in some position B, after an increment of time,
t, is the um of the kinetic and potential energy of the
sysl,em. The potential energy, that of bending, as
given b~· any text on the strcngl,h of materials is-
--,A- ------
1
I
I
I
I
1 ·Ir)
I ~
1><
I
I
I
I
Acee!= Cons/on!= o
V=ja12 l is'lhe lime
(h)
(1)
FIG. 6
Referring to Figure 6, we have a cantilever beam
subjected to a constant acceleration at the base. The
moment of inertia of the cross section need not be
constant. The beam is moved from rest to a sufficient
distance to allow the maximum deflection, D, to occur.
The maximum stress will occur simultaneously with
the maximum deflection, hence the problem we must
solve is that of finding this maximum deflection or the
condition under which it takes place.
in which :M is the moment at x, E is the modulus of
elasticity, and I is the moment of inertia of the crosssectional
area.
The kinetic energy is composed of two parts, (a) the
kinetic energy assuming no deflection of the beam and
(b) the"kinetic energy due to the elastic motion. The
first is-
E =!_(W)v2 l 2 g (2)
(10)
11
in which W is the total mass of the beam and V is the
g
velocity of the base.
The second is-
E2=l- fcL-w(d-y)2dx
2. 0 g dt
(3)
To integrate the left-hand side of the equation by parts,
WC let
Thus-du=
dM dx and v=c!:J!
in which w is the weight per unit length, and is a func- where
tion of x, and ~~ is the velocity of each element of
So that
dx ' dx
f udv=uv- f vdu
length relative to the base. Thus the total energy, U,
is-
U=!_(WL)v2+!. fLM dx+!. fL'!!!(dy)2dx
2 g 2 Jo El 2 Jo g dt
(4)
Since the first term does not involve the elasticity of
the beam, we may omit same and write equation (4)
relative to the moving base, thus-
_l fL M2 1 fL w (dy)2 u-2 Jo E I dx+z Jo g dt dx (5)
which is the internal energy of the beam.
Now imagine the beam removed in so far as its
restraining force is concerned. Or suppose it sawed
into 1 inch lengt h blocks, and the blocks all hinged
together with frictionless hinges. The force required
to hold the shape of the elastic curve is [~ a J pounds per
unit length, nor is this force diminished even if we hold
the beam straight vertically. It is clear, then, that the
work required to return each block clement of the beam
to the x-ax is is-u=
J0
L (~a) ydx (6)
Thus for any position of the beam-w
IJLMZ lJLw(dy)2 L - aydx= - - dx+- - - dx
g 2 o EI 2 o g dt (7)
Obviously, at the point of maximum deflection, the
vcloc1. ty, ddyt, 1. s zero, so tha t
lJLAf2. JLw - - dx = - aydx
2 o EI o g
d2y
Or si nce M = El -l ex2 • (8) becomes
JL (d2y)2 f Lw
0
E I dx2 dx=2
0
gaydx
(8)
(9)
JLE I (d- 2y)2dx = [ M-dy]L- JL- - dx dM dy
o clx2 dxo odxdx (10)
Now since. M = o, when x = L, and ddxy =O, when x= O,
[ Mdy] L =O
dx o
(ll)
We integrate the last term in (7) by parts.
dM dy du azM
Let ii=dX' dv= dx dx, then, dx = dxz, v=y.
Thus- -f LdM dy dx = [dMJL- J Ld2M1 dx
o dx dx clxy o o dx2 Y (12)
. dM
Now srnce "([;;i the shear, is zero when x=L, and y= O
when x=O,
then
[
dM JL - y =O
dx o
(13)
Thus-
JL (d2y)2--JLd2M E I axz - d 2 ydx 0 0 x
(14)
Finally we have--
JL d2 M JLw - -d 2 ydx=2 - aydx
0 x 0 g
(15)
Removing the integrals and canceling y, we have--
(16)
' . d2 l'hu , smce dxM . ti I d. b th d. 2 1s le oa rng on a earn, e con 1-
tion for maximum deflection is equivalent to the simple
cantilever beam loaded with twice the reversed effective
dynamic forces.
'
APPENDIX B
As an in d ication of t he probable mag ni t ude of t he
error in t he frequency of vibraiion of a beam clue to
t he assumpt ion t hat t he ca.lculations may be based on
an elast ic cu rve obtained by successive integration of
the weight of the beam itself, t he foJJowing example,
based on a uniform beam, is presented . 'Ve will take
For the calculaiion of K , formula (17), we have-
w [<L- x)' L3x £4] y--EI -----u-+ 6 - 24
If we let x= KL
...-1
I = IO, a nd A = lO, so t hat 7 = 1. The frequency, by
_ wL4 [ ( 1 - K) ~ K 1 J y- - EI ~+5 - 24
t he exact solut ion, is gh·en by t he formula- wL4 CL:-.AV CL:-. L4
=- EI C=~=-r
1 Wu (1 )
f i. 787 v -V w Since
(Sec Appendix C.)
Noting t hat w= 6 A, we have-
(
1 ) /ETfj 3.17 f E
!= 1.787 L2 y -A-'-= L2 -V 6 (2)
The calculation of C is given in the following table :
K 1-K (1-K) '
- - -----
0. 1 0.9 0.6561
. 2 .8 . 4006
.3 . 7 . 240
. 4 . 6 . 1295
. 5 .5 . 0625
. 6 . 4 . 0256
. 7 . 3 .0081
.2 . 0016
. 9 . l . 0001
LO 1.0 0. 0
Thus
T ABLE I. - Ap71endix B
I I
( l - K )' K (l- J{)• +!' - l - o 041 ' - 24- 6 24 624 ·"
---
o. 0273 0. 017 0. 0443 o. 0410
. 01 705 . 033 .01;00 ------------ . 0100 . 050 . 0600 ------------
. 0054 . 067
. 0724 1------------ . 00261 .0832 . 0858 ------------
. 00106 . 100 . 10106 ------------
.000338 . ll7 .117338 ------------
. 000066 . 133 . 133466 ------- -----
. 000041 .150 . 150041 ------------
0. 0 .16667 .16667 ------------
f = 0.902 /E I 0.566148
L 2 -YEV 0.0486068
=3229~~
Compa ring with (2), we find the error :
c
0. 0027
. 0084
. 0184
. 0308
. 0442
. 05946
. 075738
. 09286
. 10849
. 1251
-566148
E rror= 3.17 - 3.09 002 2 t 3.17 = . 5 = .5 per cen .
(12)
c•
o. 0000073
. 0000705
. 000339
• 00005
. 00195
. 00352
.00572
. 00860
. 01175
. 01 57
. 0486068
(3)
(4)
(5)
(6)
(7)
APPENDIX C
MATHEMATICAL ANALYSIS OF THE VIBRATION OF A
PROPELLER IN THE PLANE OF ROT A TION
EQUATION OF MOTION
From Figure 7, the absolute value of t he moment at
x due to any general type of loading, f (x) is-
M x = 3 J~ f(x') (x' - x) dx' (1)
Thus-and
El d2y J I, dx2= x f (x' ) (x' -x) dx'
ell (El ~~)
_Propeller,
li-onsverse plane
clx2
Tiie dflnom!C load is a
fl;nc/ton o/' l/Je eloslic
curve, ondl/Jemoss af'lhe
propeller .blade p er uni/
/enr;t/J.
f(x)
(3)
(4)
l'L£..LLLLLLLL..LLLLLLLLLLL.LL1..LL1.F..LL<:.t..LJ..Ll..l~I R lllis fiqure represents a
qenerol lflpe of' /oodinq
wllkll 1s o /'undion of' X.
Fro . 7
In which x is used to designate t he position of t he
moment, and x' to temporarily designate the position
of the loading. Obviously x and x' may designate the
same position but may also vary independently of each
other.
From standard beam theory-
M = E l cl2y • dx2 (2)
Since f (x) is a general type of loading, we may
consider it to be t he dynamic load due to vibration;
that is-cl2y
/(x) = -pA dtz (5)
in which p is t he mass per unit volume, A is the area of
the cross section, and ~% is t he acceleration. We note,
in this connection, that A is a function of x.
(13)
14
The equation of motion thus bccomcsd2y
El -dz -•i
d2 _ x + A '::Jl = 0
d.i;2 p dt2 (6)
Since 1 and A arc functions of x, a practical gene ral
solu t ion of th.i s equation is impossible. If, howeve r ,
I and A are constants; t hat is, t he beam is a uniform
cant ilever , a solution may be readily obta ined. Since
t his solu t ion gives an insigh t into the fundamen tals of
the general problem, it is t hought wo rthy of presenta tion
. The part dealing with. the free vibration of a
beam may be found in a ny text on vibrat ion . The
part dealing with forced vibratio n is taken from a
paper by J oseph N . LcConte, Professor of Ana lytic
Mecha nics, University of Califo rnia, and t he a ut hor,
Stre scs in a Ve r t ical E lastic Roel when Subjected to
Harmonic Mot ion at one End, pre pa red for the San
Francisco Earthquake Committee.
Equation (6) becomes-
Letting k'' = EwJg ' we ha ve'
Ve will now take up certain special cases.
I. 1' ertical rod with base fixed .
(7)
Since t he general case where the base is given a
simple harmonic motion involves t his as a cri t ical
condit ion, we will take it up first. Its solu t ion simply
involves an integrat ion of 7, 'li th t he prop er bounda ry
condit ions.
Let y= XZ where X is a function of x only, and Z
is a function of t only, then-and
7 becomes-
(8)
The condit ion reprc c ntcd by ( ) can exist only when
ea ch side is equal t o a constant. Let this co nstant be
n•. Then-a
nd,
d1X
---n'X = O
dx' (9)
(10)
Equat ion 10 can be in tegrated at once. If each side
which is the defining equat ion of harmonic motion, t he
period of which is,
'T= 21rk2
n2 (12)
and we may write down at once t he solution of 11 wl1ich
is-
. 2-,r
y= y0 Siil '.f' l (13)
I n eq uation 13, y0 is the maximum value of y or is the
amplit ude of motion of a ny point. '.L' is the period ,
which is t he same for eve ry p oint, b ut which st ill involves
t he unknown co nstant n4• I n o rder to obtain
n 4, we now use equation 9. Sinc:e y= X Z , i t is evident
that X = y0 and Z =sin
2
; t, h ence 9 can be written-d''
11
d~,0 - n4 y 0 = 0 (14)
aud this is tl1c different ial eq uation of the o utside curve.
The solu tion of cquat_ion (14) is-y0=
A cos nx+ B sin nx+C cosh nx+ D sinh nx (15)
For the freely vibrat ing rod we have the following
bounda ry condit ions:
1. vVhcn x= O, Yo=O.
2. When x= O, dly0= 0 (in it ial tangent vertical).
l X
3. Wl 1cn x= L , <~;'2°= 0 (bending moment at top.)
d3y
4. When x= L , dx1°= 0 (sh ear at top).
5. When x= L, y0= ye (or one point of given deflect
ion) .
The first four conditions e na ble us to det ermine the
co nstant n, for from equation (15)-
y0= A cos nx+ B sin nx+ C cosh n.-i;+ D inh nx (15)
!._ ~d?L"'=- A sin nx+ B cos nx+ C si nlt n.i;
n x
+ D cash ux
n12 cl2y · d:J= - A cos nx- B Sill nx+ c cash nx
+ D inh nx
1a d
1
3
Y3o= - A sin nx - B cos nx+ C sinh nx
n ex
H cnccO=
A + C
O= B+ D
+ D cosh nx
0= - A cos nL - B sin n L+ C cosh nL+ D sinh nL
O= A sin n L - B cosn L+ C sinh nL + D cosh nL
.110 = A cos n L+ B sin nL + C cash n L + D sinh nL
(1-6)
(17)
(18)
(15' )
(16')
(17' )
(18')
(19')
Substit uting equat ions (15') and (16') successively
in equations (17') a nd (18') we get in ea ch case a value
A
of "jj' These a re-
A sin nL +sinh nL
be mul t iplied by X , it becomes- and
B=-cosnL + cosh nL
(11)
~- + cos nL + cosh nL
B- sin nL - sinh nL
Equatin g t hese values 11-c get--
-sin2 n L +sinh2 nL = cos2 nL + cosh2 nL + 2 cos
nL · cash nL cos nL · cosh nL= - 1 (20)
15
This equation furnishes the required values of nL.
These arc found to bc-nL
1.875104 (fundamental).
(nL)' 4.6947
(nL)" 7.8547
(nL)'" = 10.9955
(nL)"" = 14.1371
all beyond the first being higher harmonics.
If the latter 4 of the boundary conditions in stead of
the first 4 be used , we can determine the 4 constant
A, B, C, and D and thus obtain a complete solution.
Since, however, we are primarily intere tee! in the
problem of forced vibration, where the base of the rod
is not fixed, but moves with simple harmonic motion
different, in general, from that of the freely vibrating
rod just discussed , we will not carry this first case any
further. It should be noted in passing that equation
(20) is the criterion for free vibration. Also that if
the first or fundamental value of n is substituted in the
expression for the period, this becomes-
II. Vertical rod with base clamped, and with base given
a si m]Jle harmonic motion of assumed period 11'0 , and
given amplitude S 0 •
In this ca e careful experiment as well as a complete
agreement between various equations shows that after
a considerable interval of time, during which the
starting irregularities smooth out, every point in the
rod moves with a simple harmonic motion of the same
period and phase as t he base, but of course, with
d iffe rent ampli tudes. If a node is developed, the
phases of points above and below the node will be
di rectly opposite.
We made tl1e usual assumptions fou nd in the theory
of vibrating strings, etc., namely, that any clement dx
of the rod moves in a 110rizont.al plane according to
the law-
. 271'
y=y0 Sill 'J'o t (22)
where Yo is the maximum value of y at any p oint, in
other words, Yo = f (x) is the eq uation of the outside
curve. Now equation (7) is-
~Y+k d2y=O
dzl { dt2 (7)
But by differentiating relation (22), we get--
d4y d•y0 • 271'
dx•= dx• sm T
0
t (23)
cl2y 411"2 • 211"
dt2 =-yo 'l'o2 Sill 7'o t (24)
If these arc substituted in (7), and the term, sin
[~,:J t, eliminated there results-
(25)
411"2
If now we represent by n• the constants k4 T----;} equation
(25) bocomcs-
(26)
This equation is identical in form with equation (14)
for the freely vibrating rod, and therefore has the ame
solution; namely, equation (15), but it should be part
icularly noted that n is here a known co nstant, for
/ 211" 4 ;-w 12 1.875 ; r
n = k-y T
0
= -y Eig'V'l'o. Or \YC can put k=-y;--y 211"
from equation (21), where '11 is the p eriod of free vibration
of the rod with fixed base. This gives for n-
(27)
This important equation shows that the value of nL
can be computed in terms of the ratio of the p eriods
of the free and forced vibrations, entirely independe
ntly of the dime nsions of the rod or structure.
As has been stated the integration of equation (26)
is identi cal in form with that of eq uation (14) for the
rod with fixed end, and is-y
0 = A cos nx+ B sin nx+C cosh nx+D sinh nx (15)
But the boundary conditions for the forced vibration
are-
2 - O dyo_O
· x- ' dx - ·
d2Yo
3. x= L, dx2=0.
4. x= L, ~;3°= 0.
If these conditions are substituted in equations (15),
(16), (17) , and (1 ) , there results-
A +C=So, or C=So- A
B+ D= O, or D=-B
(15')
(16')
- A cosnL - B sin nL + C cosh nL+D sinh nL= O (17')
A sin nL- B cos nL+ C sinh nL+D cosh nL=O (18')
Elim inating C and Din equations (17') and (1 ' ),
by means of the r elations given by (15') and (16'),
we obtain-
-A cos nL-B sin NL+So cosh nL (28)
-A cosh nL-B sinh nL=O
-A sin nL-B cos nL+S0 sinh nL (29)
-A sinh nL+B co h nL=O
Now we eliminate A between equations (28) an cl
(2!J), and solve for B
B=S0 ( sin nL cosh nL +cos nL sinh nL) (30)
2 l +co~ nL cosh nL
16
Hence
A=S0 (cos nL cosh nL -sin nL sinh nL+ l) (3l )
2 1 + cos 11L cosh nL
D=-So(sin nL cosh nL+cos nL sinh nL) (32)
2 l+cos nL cosh nL
C=S0 ( cos nL cosh nL+sin nL sinh nL+l) (33)
2 l +cos nL cosh nL
In regard to these four constants, the following
remarkable facts will be noted :
1. If the conditions are such that cos nL cosh
nL= -1 all four constants, and hence y 0 become
infinite unless S 0 =0. If, therefore, the forced period
of the base is the same as that of the freely vibrating
rod, all de fl~ction s and stresses become infinite.
2. The constants are expressed in terms of S 0 and
of nL only, and the portions with in the brackets are
functions of nL alone, and hence of the ratio ;
0
alone.
These constants can be computed once for all for
various values of this ratio, a nd for unit value of S 0 •
If the values of the four constants are substituted
in eq uation (15), we have t he equation of the outside
elastic curve.
The olut ion of t hese two ca es shows that the
a mplit ude of the freely vibrat ing rod is purely an
a rbit ra ry quantity, hence t he moments, s hear , and
slopes a re purely arbitrary depending on the amplitude
which has been chosen for a ny given point. But in the
case of the rod vibrating un lcr a given forced motion
of the base, these qua ntities depend entirely upon the
nature of the prescribed motion.
We can now compute the bending moment at any
point in the rod when in its outside position. This is
the position which iu tcrests us most since there the
moments will be greatest.
Differentiating equat ion (15) twice wit h respect to
x , and multiplying by EI, we havc-
EI ~;[2°= M0 = Eln2 (-A cos nx- B sin nx (34)
+c cosh nx+D sinh ?J,x)
If x=O, we get the moment at the base or-
M.=Eln2 (C-A)= EJn2 (S0 -2A) (35)
If we wish to get the shear at any point, we differen t iate
15 three times with resp ect to x and mult iply by ET.
- EI d3ly3°=V0=-Ein3 (A sin nx - B cos nx (36)
lX
+C sinh nx+D cosh nx)
The shear at the base will be-
V.=-Efn3 (- B + D)=-2 Efn3 B (37)
To get the nodes, if any, we put y 0 = 0 in 15 and solve
for x. This can be done by plotting the curve, or by
trail and error met hods only. The node will be a point
of maximum shear.
III. Vertical rod with base pivoted, and with base given
a simple harmonic motion of assumed 71eriod T 0
and am71litude S 0 •
This would correspond approximately to the case of
a tall structure under earthqua ke stress where the
foundations were not firm enough to maintain the first
tangent ver(,ical during motion.
\Ve have as before-y
0 = A cos nx+ B sin nx+ C cosh m+ D sinh nx (15)
The boundary condit ions a rc--
d2y 0 2. x=O, dx2 = 0.
3. x = L ' cdPx.2y o=O.
4. x= L, ~;:3°= 0.
These when substituted in equation (15), give-
(15')
O= C- A, C=A, A = 2s0 (17')
0=-A cosnL - B sin n L+CcoshnL + D sinh nL (18')
O= A. sin n L - B cos n L+C siDh nL+ D cosh nL (19')
Substi t ute (17') in (18') and (19').
0= -'~0 cos n L - B sin nL+ ~0 cosh n L+ D sinh nL (38)
0 = '~0 sin nL- B cos nL+ ~0 sin h nL + D cosh nL (39)
Multiply eq uation (3 ) by cos nL, and (39) by sin nL.
s s 0=- 2
°cos2nL-BcosnLsinnL+ 2°cosnL cosh nL
+ D cosnL sinhnL (40)
0 =~0 si ri2 nL - B cos nL sin nL+ ~0 sin nLsinh nL
+ D sin nL cosh nL (41)
Eliminate B between equations (40) and (41).
D= S 0 ( sin nL sinh nL -cos nL cosh nL+ 1)
2 cos nLsinh nL-sin nL cosh nL
B - _ S 0 ( sin nL sinh nL +cosnL cosh nL-1)
- 2 cos nL sinh nL -sin nL cosh nL
The bencling moment at any p oint can be
from 34, with the proper constants inserted.
bending moment at both ends will be zero.
The shear at any point can be taken from 36.
shear at t he base will be-
V. = EJna (B- D)
ET 3 ( cosnLcoshnL - 1 )
= n s cos nL sinh nL-sin nL cosh n l.,
(42)
(43)
t aken
The
The
(44)
,
APPENDIX D
FREQUENCIES OF SIMILAR PROPELLERS
Assume two geometri cally similar propellers, a model
and a full scale. Let Dm be the diameter of the model
and Dr the diameter of the full scale. Then
Dr= KDm, in which K i the . cale; that i , the
diameter of the full scale propeller is 2, 3, 4, etc., times
the diameter of the model. If Am is the area of the
cross section at any radius of the model and Ar the
area of the cross section at a co rresponding rad ius of
the full scale propeller, then-
Ar = K2Am (1)
And if the corre ponding moment of inertia of the
cross sections are Ir and I m-l
r= K'l m (2)
By substituting (1) and (2), in (4)-
We use these in equation (13) of this report. We
note first , howe ver, that-
Y= J J J J ~~dr·dr · dr·dr (3)
Substituting (3) in (13), and using the full scale
subsc ripts,
g I ~K2Amf f J J ~~~:~ ·K.drm·K.drm · K.drm ·K.drm
fr=
2
; V ~K2Am [J ff f ~~~::: ·K.drm· K.drm· K .drm· K .drmJ
(5)
Or-fr=
f{ ( )
That is to say, the frequencies are in versely proportional
to the scale.
This was checked experimentally on uniform rods,
then on similar propell ers. The frequency of a >-2 -inch
by 8-foot cold-rolled steel rod clamped in a vise as a
cantilever was compared to the frequency of a 7{-in.ch
by 4-foot r od of the same material, and clamped in the
same manner. The frequency of the first rod was
(6)
(7)
minute. The frequency .was measured by a vibrating
recd-type tachometer held against the hub of the
propeller clamped to a massive steel table.
The freq uency of the model, one-fourth the size, and
of the same material was found by the same method to
be 6,000 vibrations per minute. Multiplying the
1,300 vibrations by the scale of 4 we have 5,200 vibrations
per minute for the model.
The error is-
6,000-5,200 8
61000 60
- .13 or 133 .
found to be 46.5 vibrat ions in 30 seconds and that of This is considered a s uffi cient check ince the hub of
the second to be 93 vibrations per second, or twice as the model, was, in proportion, slightly stiffer than the
fast. full-scale propeller.
The frequency of vibration of propeller 070717, full The frequency of the model in the plane transverse
scale, 10 feet in diameter, in the pl a ne of the minimum to the above-mentioned plane was too high for the
moment of inertia was found to be 1,300 vibrations per iL1 truments at hand.
(17)
0