Vibrations of a swept box

REPORT No. 47

LU'''"-Bibliotheek TU Delft

FacuM der LucWvaart- en Ruimtevaarttechn'-Kluyverweg 1

2629 HS Delft

K )HS DELFT

THE COLLEGE OF AERONAUTICS

CRANFIELD

VIBRATIONS OF A SWEPT BOX

by

J X M . R A D O K , B.A. (Melbourne)

This Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

A p r i l ,

1931-18 Juni 1951

T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D V i h r a t i o n s of a Swept Box by -J.R.M. R a d o k ^ B . A . , (Melbourne). oOo S U M M A R Y

The eciuations of motion of a uniform swept box with stringers and ribs are deduced. For the

case of vibrations of a cantilever they are transformed into integral equations, an approximate method of

solution of which is indicated.

YSB

_

Mr. Radok is a member of staff of the Structures Section of the Aeronautical Research Laboratories, Department of Supply, Australia, and is at present studying at the College. Acknowledgement is paid to A.R.I, for their agreement to publish this as a College report.

Bibliotheek TU Delft

Faculteit L & R C2344255

-2-• TABLE OF CONTENTS

Page

Notation 3

Introduction k

Deduction of the Lagrangian Function I4.

H a m i l t o n ' s V a r i a t i o n a l Eq^uations o f M o t i o n 7

T r a n s f o r m a t i o n o f t h e Eq.uations o f M o t i o n 8

A Method of Solution of the Integral 10

Equations

References . 12

NOTATION

Oxyz Oblique axes of reference (Fig.1). Us Vj w Oblique components of displacement. p(x,t), q(x,t) Oblique components of rotation of box

sections about Ox, Oy respectively. a Angle between the axes Ox, Oy (complement

of angle of sweep).

W(x,t) Displacement of centre of box s e c t i o n s

in Oz d i r e c t i o n .

• •

»"' ..' ^ • Time derivatives of u, v, etc. u, V, etc.

M'(x,y,z) Mass distribution of swept box. T Kinetic energy of box.

f^ Length of swept axis of box (Pig. 1 ).

OX, OY Reference axes in Oxy plane, perpendicular to Oy and Ox axes respectively.

L^, M^ Oblique components of couple about OX, OY respectively,

is j Unit vectors in Ox, Oy directions. i^, j. Unit vectors in OX, OY directions, U Potential (strain) energy of box.

.C. . Elastic constants given by equation (100) ^ of Ref.1.

T*. . Constants defined by (2.10). L = T-U Lagrangian function of the box.

I (x), ^2,^^"^ Moments of inertia of box sections. m(x) Mass per unit length of box.

'n(x) Position of centre of gravity of box sections. t Time

P(x), Q(x),i'2(x) Amplitudes of normal vibrations.

)V2% Frequency of vibration.

k , k Radii of gyration corresponding to I^^, I^.

n ^ y z

2 2

\ = 1/mK Frequency parameter.

-k-1. Introduction

The use of swept wings in high speed aircraft has naturally produced considerable interest in the dynamic behaviour of such structures. As far as complexity is

concerned, the static and dynamic problems arising out of the sweeping of wings may well be compared with the problems encountered in high speed aerodynamics. Although the trend towards low thickness wings for high speed aircraft may lead to the treatment of wings as flat plates for the purposes of dynamic and aero-elastic investigations, it is of interest to consider the present problem as it should facilitate assessment of the effects of such a simplifying assumption

on frequencies and modPS of vibration, flutter characteristics, etc.

In this paper Hamilton's Principle is applied to deduce the equations of motion and relevant end conditions for a uniform swept box, reinforced by stringers and ribs in a manner typical of aircraft wings. The expression for the potential energy, forming part of the Lagrangian function, is obtained using generalised curvature-bending and

twist-torque relations deduced in ref.1. This reference and the present report use throughout oblique coordinates (Fig.1)

and the same notation, wherever possilole.

The equations of motion, obtained thus, are integro-differential equations in terras of the vertical displacement and the twists about two oblique axes of the box sections (Fig. 1). Using the boundary conditions,

they are transformed into integral equations in terms of certain derivatives of the above quantities. Finally, an approximate method of solution of these equations by

reduction to a finite number of linear equations is indicated,

2, Deduction of the Lagrangian Function

First the kinetic energy of a swept box will be obtained, assuming in conjunction with ref.1 that p, q and W are functions of x and t only. Referred to

the oblique axes Oxyz (Fig.1 ), the displacements of a point P(x,y,2) are

u = z (p cot a + q cosec a)

V = -z (p cosec a + q cot a) (2.1) w = W + py sin a

and thus the components of velocity are u = z (p cot a + q cosec a)

V = -z (p cosec a + 4 cot a) (2.2) w = W + py sin a.

The square of the modulus of the velocity vector is then • 2 . . '2 • P

u + 2uv c o s a + v + w

0 0 0 0 0 0 • * o

= z (p + 2pq cos a + q ) + y p sin a + 2yp W sin a + W (2.3)

and hence the kinetic energy of a point mass |-L(x,y,z)dx dy dz

H.(x,y,z) j ^ 2 ^ ^ 2 ^ 2p4 cos a-f q^) + y ^ p ^ s i n ^ a + 2ypW s i n a + W^fdx dy dz.

2 L J

Integration of the last expression over a cross-section x = const, gives the kinetic energy of a cross-sectional element of thickness dx of the swept box.

dT = i | l ( x ) ( p ^ + 2 p q c o s a + q^) + l 2 ( x ) p ^ s i n ^ a

+ 2m(x)ri (x)pW s i n a + m(x)W M x , (2.I4.)

from which f o l l o w s the t o t a l k i n e t i c energy

' -

J,

_'0 dT. (2.5)

' In order to deduce an expression for the potential energy, consider a uniform swept box acted on by a uniform moment (i.e. neglecting shear),

^1 ^1 +

_H

h'

The c o r r e s p o n d i n g r e l a t i v e d i s p l a c e m e n t f o r an element dx i s

dpi + dqj

and hence the strain energy

dU = i(L^i^ + M^ó^). (dpi + dqj) = - ^ ^ ( L ^ g + M^ g ) dx

since obviously (Pig.1)

( 2 . 6 )

i-i-l = d'd^ = s i n a , i . D>| = i ^ . j = 0. Thus t h e t o t a l p o t e n t i a l energy of a swept box i s

nl U = dU . ( 2 . 7 ) JO By equation (99) of ref.1 q = - cosec a dW dx (2.8) /where . . .

-6-where the C., are known constants given by equation (100) of the same reference, and the last equation has been

integrated taking into account the neglecting of shear deflections. Inversion of (2.8) gives

where M.

r.

₁₀

/ n

22

-r-r.

hi

_r=

'11 '12 (2.9) '12 '22 <2.10) Substitution from (2.9) in (2.6) to the total potential energy

and then in (2,7) leads

U = 1 _{binaK '22Mx^ '^'l2 dx dx}

0

r (M)2)

11 ^dx-* J

dx.

.(2. 11) Using (2.5) and (2.11) the Lagrangian Function

L = T - U (2. 12) can now be written down. However, before üroceeding to the application of Hamilton's Principle to (2.12), it will be of interest to discuss the character of some of the

quantities appearing in (2.I4.) in connection with the theoretical backgroiind of (2.8).

The equations (2,8) have been deduced in ref, 1 assuming a uniform rectangular box with the swept axis Ox passing through the centres of the cross-sections x = const. Nevertheless the quantity r](x), the position of the centre

of gravity of these cross-sections has been retained in (2.1;), in order to allow for the addition of masses which, while not affecting the elastic properties of the box, may change the value of T), and of course also of I„ and I„. It is obvious that in the absence of additional

Ti(x) 5--" 0.

ma s s e s

Finally it should b e noted that the equation

dW q = - cosec a g—

in (2.8) implies that only two of the three functions p , q, W are independent, a fact which has to be taken

into account in the application of Hamilton's Principle.

3. Hamilton's Variational Equations of Motion By Hamilton's Principle

r*2

Ö L d t = 0 •'t ^1

provided that for t = t., t = tp

(3.1)

Ö p = ö W = 0

where the latter can be seen by (2,8) to imply

Ö q = 0 .

( 3 . 2 a )

( 3 . 2 b ) I n a d d i t i o n ther© a r e t h e f o l l o w i n g clamped end c o n d i t i o n s

p ( 0 ) = 0 , q ( 0 ) = 0 , W(0) = 0 ( 3 . 3 ) S u b s t i t u t i n g i n ( 3 . 1 ) f o r L from ( 2 . 1 2 ) and u s i n g (3-2) I y ] p o p + ( q ö p + p ö q) cos a + q ö q l + ^g®^'^ a p ö p + (Wöp+ pöW)mTi s i n a + mWöW- s i n a J T ^ g P ' ^ P ' " "H 2 ^ ^ ' ^ ^ ' "^ ^ ' ^ ^ ' ^

+ T"^iq'öq'j

ntg n€ dt t^ u dt '^t.

I

0 . 2 ly-jp op +(q5 p + p Ö q) cos a + q ö ql

+ I^ sin a p o p + (Wop + pö W ) mr, sin a + m W o W

- sinalTgs P " ö P - " ^ 2 ^ ' ! " ^ ^ "" ^ " ^ ^^ "^"^1 'l"^ "^.ll dx

+ sina T^2 P' ^ P " " ^ 2 ^ ^ ' ^ ^ "^ P'ö q) + F^.,q*ö q 0

= 0. (3.1+)

Choosing op, öq as independent variations, one finds immediately that for Oéxé-C»

1^2?" "•'^20." - Wm-q - I sina p - I (p cosec a + q cot a ) c= 0 (3.5) and

r 2 2 P ' ( ' e ) - T : ; 2 q ' ( ^ ) = o.

Since the coefficient of op must disappear.

(3.6)

When dealing with the remaining terms of {3.k)>

öW will be transformed into öq using (2.8). By the help of the last condition of (3.3)s the following trans-formation of the terms involving öW is possible:

8 -(mW + s i n a mr] p)ö W d^ = 0

ot

P ( S » t ) ö W ( ? , t ) d ^ O

P ( g , t ) dg I öW'dx

*/0 = ÖW' dx F ( ê , t ) dg

Jo ^'x

(- s i n a ö q ) d x P ( C , t ) d 5 .

J o *'x

( 3 . 7 ) Hence, s u b s t i t u t i n g from (3-7) i n (3.U)»

7 ^ 2 ? " ~ T^ - i l " + I (cot a p + cosec a q ) - | m W + s i n amr|pid£ = 0

"" ( 3 . 8 )

1^2 P' ('^) - ' H i ^•('^)'= 0 (3.9)

The boundary conditions (3-5) and (3.9) have been chosen to satisfy Hamilton's Principle at the free

end of the cantilever. The equations (3.5) and (3.8) are the desired equations of motion of a swept box} they are supplemented by the relation holding between q and W, given in (2.8), so that there are actually three equations. Before giving an interpretation of the various terms appearing in these equations, they will be transformed into integral equations in the next

section.

\\. Transformation of the Equations of Motion

Integration of (3.5) and (3.8) with respect to X from t, to € using (3.6) and (3.9) gives

T'22P' (^>t)-T;2<l' (^»t) = - ƒ {m(x)r>(x)W(x,t)+ I^ (x)Bina p (x,t)

+ I (x)rp (x,t )cosec a + q(x,t)cot ajJdx (U. 1 )

-TjgP' ^ ^ ' * ^ ^ ' ^ 1 ^ ' ^^'*^'' "J {^y(^)(°°* ap(x,t)+ cosec a qJ

-j(m(^)W(e,t) + sinam(fjTi(e)p(£,,t))dddx (i+.2)

Comparing the equations (i4.. 1 ) and (I4.. 2) v/ith (2,9) leads immediately to the conclusion that the right hand sides of (U-1 ) and (i+. 2) are respectively equal to L^ and M^ of (2.9). A further reference to (2.1)

suggests that the terms under the integrals are rates

of change of moment of momentum and of momentum of a

cross sectional element of thickness dx.

L = JJ Ml (yw - zv) dydz

M = jJ |i zu* dydz

Z = J J iJ, w dydz

which on consideration of the equilibrixim of such an

element of the box will be found to be related to L-]

and M-| in the manner suggested by (1+. 1 ) and {k.2).

The equations (U. 1 ) and (i|. 2) are

integro-differential equations which are easily transformed

into integral equations using (3.3). However, before

doing so, in order to simplify the further treatment,

normal vibrations and absence of additional masses

(see end of section 2) will be assumed, i.e.

p(x,t) = P ( x ) s i n X t , q(x,t) = Q(x)sin><t, W = ^ ( x ) s i n K t ,

(U.3)

r|(x) = 0 ik'k)

substituting (U-3) and {k'k) in (U* 1 ) and (i^.. 2 ) , and

inverting the order of integration in (i^.. 2 ) ,

/N^l^pP'(S)->V^T:^2^' (^^ = j /k^sinaP(x)+ ky(p(x)coseca+Q(x)cot(^dx

^

{k>5)

-)\Vi2P' i^)^^'^'\'^' (^) = {k^(p(x)cota + Q(x)coseca - (x-£ji2(x)|d:

where

I„ = m k , I = m k , ) \ ' = ^ (i+. 7)

y y z z ^ ^ 2

ty (3.3)

P(0) = Q{0) = Q(0) = i2'{0) = 0,

hence

P(x) = f P' (C)dC,

^ 0

Q(x) = f Q' (e)d^

Q{x) = f dgf Xi"(r,)dTi = f (x-ri)^"(r,)dr, = -cosecaf (x-g)Q'(C) d^^,

OQ J Q '^O

"^0

where in the last step use has been made of (2.8),

-10-Introducing these last expressions in {k-5) and (I4.. 6 ) , and inverting the orders of integration

(U.8)

'0 -X2T^2P'

(^) + ^ ^ ^ 1 ^ ' (x)= J {P' (e)f3(e.,x)+ Q' (Hjf^(?,x)]dg

(U.9)

with 0 é: X 4 •£. f-|(g>x) 2 2 k^sin a + k cosec a z y ^ f (E,,x) f,(?,,x) k cota k cota f^(e.,x) =

V-•. (^

\k cosec a + cosec a (TJ-X) (ri-g)> drj

fTx

J

where C>x = max(2;,x).

The equations (ij..8) and (U-9) represent a system of simultaneous homogeneous Predholm equations of the

second kind, An approximate method of solution of these equations will be indicated in the next section.

5. A Method of Solution of the Integral Equations

The method of solution, to be suggested here, has been applied to similar problems in references 2, 3. As it has been presented there in great detail it will be sufficient to concentrate attention here on a

modifi-cation of this method which, it is hoped, will reduce the amount of computation involved as well as improve the accuracy of the results obtained. However, before going any further, it should be noted that the method of solution given in references 2, 3 is one of many which could probably equally well be applied.

The principle of the method of solution in the above mentioned references is to replace the integrals in the integral equations by finite sums. For this purpose the range of values of the independent variable, in the present case 0 •& x t -^ is subdivided into n

equal-intervals. The integrals over each of these subdivisions are then replaced by the product of the values of the

integrand at their mid-points and the length of the interval -6/n. By giving the independent variable of the converted integral equation successively the values corresponding to .the mid-points of the subdivisions, one obtains a set of n simultaneous linear equations in terras of the approxiraate values of the solutions of the integral equation at the mid-points.

It follows clearly from the last paragraph that the choice of the mid-points of subdivisions in the

reduction of the integrals is arbitrary, and that the

degree of accuracy of approximation to the solutions of the integral equation will depend entirely on the suita-bility of this choice. As a result of unpublished work,

it was found that in the case of vibrations of a uniform cantilever the use of the mid-points as "reference points" necessitated the introduction of a comparatively large number of degrees of freedom, i.e. subdivisions, to ensure

satisfactory accuracy for a few of the lower frequencies. Hence an attempt was made to develop a method by which more appropriate "reference points" could be found. In

this way it was hoped to reduce the number of simultaneous equations and at the same time to improve the accuracy of the solutions.

This method, which proved to be very satisfactory in the problem mentioned above, assumes the knowledge of at least an approximation to one solution of the integral

equation. In the case of the vibrating cantilever, a polynomial was used as an approximation to the fundamental mode. Using this approximate solution, the following equations determining values x.. of reference points in the i*^ subdivisions may be written down:

4^

f{l,\^)g(^)Sii = J f (Xjj,x^^)g{x.j), 1 = 1,2,...,n

Where f is the kernel and g the approximate solution of the integral equation. Once the x.. have been found, their suitability can be checked by substitution in the equations

n^

tU,x..) g(H)d^ = |f(Xi^,x,,) g(x..)

|(d-1)

ass\iming in the first place x. . = x... In the case of

in 11

the cantilever it was found that the X^A varied very little for j = const., i = 1,2,...,n.

The use of the modified method of solution in the problem treated earlier in this paper is obvious. It is intended to apply it to the case of a swept box, already built, and to compare the theoretical and ex-perimental results.

-12-REPERENCES

Author Title, etc,

W. S. Hemp On the application of

oblique coordinates to problems of plane elasticity and swept back wings.

College of Aeronautics, Cranfield. Report No.31. January, 1950.

E.R. Love and

J.P.O. Silberstein

Vibration of stationary and rotating propellers.

Australian Council for Aeronautics. A, C.A. 36. June, 19^+7.

J.R.M. Radok and L.P. Stiles

Motion and Deformation of aircraft in atmospheric

disturbances.

Australian Council for Aeronautics. A. C.A. k1. July, 19U8.