fS Ó Z O o. TECHNISCHE HOGESCHOOL VLIEGTUIGBCUV KUNDE REPORT No. 26
12 Juti1950
TEr.H»^?,^CH5 '.• !)ELFT K!üvvc»P'i/eT t - ?• J DELFTTHE COLLEGE OF AERONAUTICS
CRANFIELD
I
ON THE NATURAL FREQUENCIES OF A
REINFORCED CIRCULAR CYLINDER
by
W. S. HEMP, M.A.
of the Department of Aircraft Design.
I
VUEGTUIGBOUWKUNDE
REPORT No. 26 March. 19^9
T H ' E C O L L E G A E O F A E R O N A U T I C S . C R A N F I E L D
On the Natural Frequencies of a Reinforced Circular Cylinder
-*y-?if.S. Hemp, M.A.
of the Department of Aircraft Deslp^n. — o O o —
SUMMARY
This report is concerned with the
calculation of the natural frequencies of an ideal structure somewhat representative of an aircraft fuselage. The results are hased upon a simplified "shell" theory which permits proper allowance to be made for the shear stresses and the corresponding
displacements. No use is made of the so-called "shear deflection", but it is shown in the Appendix that, for the special case considered, this approach would yield the same answer. Numerical results are given in ^ 7 and comparison is made with both the usual beam theory results and the frequencies calculated on the assumption that flexibility in shear is of primary importance.
-1-CONTENTS
Statement of the Problem
Displacements and Strains ... Stresses and Equations of
Motion
The Differential Equation
for
V
The Boundary Conditions The Frequency Equation Numerical Results
Conclusions ...
It is well known that shear flexibility in a beam can affect the natural frequencies and modes of vibration. However the precise degree in which
this effect enters into the determination of the
frequencies of aircraft structures does not seem to be accurately known. This report attempts to throw
light upon this question, by the analysis of an ideal case somewhat representative of an aircraft fuselage,
Closely spaced ^ ^ \f igers stri Clo'sely spaced rigid rings - > Skin,"^thickness t. Fig. 1«
Consider a cantilever in the form of a reinforced circular cylindrical shell. (Pig.1.) The closely spaced rings, which for simplicity will be considered rigid in their planes, carry masses
amounting to a uniform distribution of M per unit length. The problem to be solved here consists in the determination of the natural frequencies for
flexural vibration parallel to a diametral plane which will be taken as xOy, (see Pig.1.) Comparison may
then be made with the more usual calculation which neglects the shear flexibility.
2. Displacements and Strains
A
-rdg
Since the rings are rigid their displacement parallel to Oy will be given by a function of x only,
say V(x). This displacement will be imposed upon the skin-stringer shell, whose remaining longitudinal freedom may be represented by a displacement u parallel to Ox.
"u" will be a function of both x and the polar angle ^ , The direct strain e,,,^ parallel to Ox will be ^^è X
The shear strain e ^ arises from both u and the tangential component V sin S of V. (see Pig.2). We
thus derive:-XX
i/^^i^ö C^téïJx)iU^
T
d X 11. XX V , _ -> ^he
he
^
' ^x6
^n - dx dV.n$
(1) iii,^ & Fig. 2.3.
-3"
Stresses and Equations of Motion
The stresses follow from the strains in the usual w a y . ^ Denoting the direct stress b y xx, the shear stress by x ^ and neglecting any hoop stresses in the skin we find
XX = E e XI
xx ' --^ "7®xÖ
where E,G are the direct and shear moduli respectivelyo (2)
xxt,rd@
I ^ + è£Edx)^tr^*8
^x
(xg +
^xm9
)tdx
Fig. 3.
The equation of motion for an element ( d x , r d ^ )
assumed to b e without m a s s , can be written down after inspection of Fig. 3. Denoting the equivalent thickness of the shell
for carrying direct stress x5 by t, (= stringer area per unit length of circumference plus skin thickness) we
find:-4^^
'*'» ^xx + t ^xé 0
(3)
i,<
The total shear force ^' acting across a section of our cylinder is given by,
2n-S '
oy = _ j"' / ^ sin. (9 . t. rd
|<dx >t
w
n
T ' -
dxMdx.dfv^
dt'^ o Fig, h'The equation of motion for an element of the cylinder dx follows from Pig.
k:-d^ = Md5[
= -
M p V
dx dfi- (5)
where t^ is the time and "p" the angular frequency in a
natural mode of vibration. All our variables vary with time. as exp (iot )
A * The ...
U. The Differential Equation for V.
Substituting from (1) into (2) and from thence into (3) we find,
Et, èfu + ^/ijfu - dV CosÖ ) = 0 (6)
We seek a solution of this equation in the form
u = U(x). cos /9 (7)
T h i s y i e l d s by (6)
E t , d^U. - Gtil - Gt dV = 0
dx*- TA. r dx (8) S u b s t i t u t i n g from (1) and (2) i n t o (U) and u s i n g (7)
we f i n d , y = ITGtr f (M + dV \ (9) S u b s t i t u t i n g from (9) i n t o (5) we f i n d :
-XiGtrf I ÖJX+ dfvN = - Mp^V
( r dx d x i y (10)
E l i m i n a t i n g L(. from (8) and (10) we o b t a i n , d V .^ M£^ dfv - M^fv = 0 dxt -^tv d x ^ - p E t , r ^ /^^N5. The Boundary Conditions
At the free end x = -£ we must have,
( ^ ) x = e ^ ° ' (/)x=.£=0 (12)
while at the root x = 0
:-(u)^^O, (V)^^O = ° (^^) Using (1),(2),(7) and (10) the first of (12) gives,
-5-Equations (9),(8) and (10) transform the second of (12) to
/ d \ \ = - Mpf /dv\ l^dx»;^ . TTGtr (dxj^
e
"^ '
V^Vx=^ (15)
Using (7),(8) and (10) the first of (13) becomes
/dvl = - Et, |r^ / ^ V \
W x = G Gt
j
^
l V
^^^\{ix,^^Q
(16)
For completeness the second of (13) is,
( V ) x = 0 = ° (^7) Since (11) is of the fourth order we can satisfy
(1i|), (1 5), (1 6) and (17) and obtain a frequency equation.
6, The Frequency Equation
Solving (11) and satisfying (1U),(15),(16) and (17) we find after considerable transformation a
frequency equation v/hich is best expressed in terms of the
parameters:-7rEt,r3 (18)
P = Etj_. tf
Gt £' (19)
(20)
The frequency equation is found to
be;-2 + (be;-2 + f ^ ^ coshCX-, cosO^^ -Qr^Q sinh Q(i sincC^ = (21)
In the case vi/here shear flexibility is neglected (i.e.p = 0 ) equation (21) reduces
to:-cosh ^ 4 cos \^^ + 1 = 0 (22)
I n t h e case where s h e a r f l e x i b i l i t y i s paramount
( i . e . p > 0 0 ) E q u a t i o n (21) y i e l d s :
-cos ( p ^ a ' ^ ) = 0
or i £ r ^ = (2n + 1) T T / . o i (2:)) 7. Numerical Results If we take E/^ = 2 . 5 t^ .^ = 1.6 ^/^r " ^' as fairly, representative of fuselage construction v/e find P = /25. Adopting th^s value the comparison o frequencies which vary as (Ü^2 (see(18)), as given by equations (21),(22) and (23) is asfollows:-No, of Mode (1) (2) (3) (U) (A) ^ ^ ( E q ( 2 1 ) ) ^ 5-226 11^.57 31.38 ^^8.27
(B) 0"^{J.'\{22))X 3.516 2 2 . 0 3 61.70 120.82 (C) {tT^(Eq(23)) 7.85U 23.56 39.27 5^1-93
Table of V a l u e s of (5b" " = TD £ / M V i T T E t j r ^ j
Line (A) includes shear flexibility, Line (B) disregards it, while
Line (C) treats it as paramount,
/8. Conclusion;
I am indepted to Dr. S. Kirkby and the Computing Section of the Aerodynamics Department for these results.
8 o Conclusions
Inspection of the table of ƒ 7. shows that,
while the usual theory (line(B)) gives a fair approximation to the fundamental frequency, it is already seriously
in error in estimating the first overtone. The assumption that shear flexibility is of primary
importance (line(c)) begins to yield fair approximations at the third overtone. The conclusion is therefore reached; that some method of calculation more accurate than the usual theory of beam vibration, is required for the estimation of fuselage frequencies, if anything more than a rough estimate of the fundamental is required.
APPENDIX
Note on the "Shear" Deflection
If we write
dx ^ °
E q u a t i o n (9) g i v e s ,
y = TTat{. dV^. = iöA dV^ , where A = 2i-rrt dx dx
Remembering t h a t t h e max. s h e a r s t r e s s i n a t h i n
c i r c u l a r tube i s e q u a l t o ty/ice t h e mean, we see t h a t V^ i s t h e s o - c a l l e d " s h e a r d e f l e c t i o n " .
E q u a t i o n s (8) and (9) y i e l d ,
y = jTEt, r^ 0^ = - E.(Trr^t,) d\-b
• ' dx- dx#
Since KT r t ] ^ is the moment of inertia of our section, v/e recognise a fundamental equation of beam theory and so can identify Vb with the "bending deflection". It follows that the results of this report may be obtained from the usual theory of beams combined with the notirj-^ of shear deflection. This is not really remarkable since equation (7) implies that
"plane sections remain plane". However this agreement is only valid for the thin circular tube. The
arguments of this Appendix and equation (7) upon which they are based are not valid for a tube of another section.
