stodola method

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METHOD OF CALCULATING THE NORMAL NORM AL MO ES A

D FREQ FREQUE UENC NCIE IE

OF A BRANCHED TIMOSHENKO BEAM

by Francis

Shaker

Lewis Research Center

Cleveland, Ohio N A T I O N A L A E R O N A U T I C S A N D S PA P A CE CE A D M I N I S T R A T I O N

WASHINGT ON, D.

1968

TECH LIBRARY KAFB, NM

1111111111111111lll11111111III 0333235

M E T H O D O F C A L C U L A T I N G T H E N O R M AL AL M O D E S A N D F R E Q U E N C I E S O F A B R A NC N C H E D T IM IM O SH S H E NK NK O B E A M

By Fran cis

Shaker

Lewis Research Center Clevelan d, Ohi

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse Clearinghouse for F ede ral Scientific and Techn ical Informati Information on Springfield, Virginia

22151

CFSTl price $3.00

TECH LIBRARY KAFB, NM

1111111111111111lll11111111III 0333235

M E T H O D O F C A L C U L A T I N G T H E N O R M AL AL M O D E S A N D F R E Q U E N C I E S O F A B R A NC N C H E D T IM IM O SH S H E NK NK O B E A M

By Fran cis

Shaker

Lewis Research Center Clevelan d, Ohi

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse Clearinghouse for F ede ral Scientific and Techn ical Informati Information on Springfield, Virginia

22151

CFSTl price $3.00

METHOD OF OF CALCULATING THE THE NO RM AL MODES AN FREQUENCIES OF A BRANCHED TIMOSHENKO BEAM F r a n c i s J. S h a k e r L e w i s R e s e a r ch ch C e n t e r SUMMARY A method is presented in this re po rt for calculat calculating ing the normal modes and frequenfrequenci es

branched Timoshenko beam.

The method method is essentially

modified Stodola

method method and re qu ir es an iteratio n proce dure to determine the nor mal modes and frequenfrequen-

cies f the system.

In thi s method, method, an ar bi tr ar y deflecti deflection, on, consistent with the boundary boundary

condition, is assumed. frequency is not

Also, bec ause

f the prese nce

f the spr ing -ma ss sy stem, the

constant factor in the governing equations and must also be assumed.

Know Knowin ing g the deflection deflection and frequency, the corresponding sh ea rs , moments, slopes, and new new deflection can be d eterm ined by integrating the governing differential equations o the system.

The new deflection is then adjusted to satis fy the boundary boundary condition conditions, s, and

new frequency is subsequently calculated. quency is used as t h e c r i t e r i o n fo

The process is then repeated, and the fre-

convergence.

Althou Although gh the method can be applied fo

any type of of boundary conditions, par tic ula r attention wa s given to those boundary condiconditio ns of of int ere st in launch-vehicle launch-vehicle dynam ics, namely, the fr ee -f re e beam beam and and the cantilevered-free beam. In general, the iteration routine will always always converge converge o th e lowest lowest or fundamental mode

vibration.

determine the higher modes, the lower mode components

re

moved from the assumed shear and moment distribution by utilizing the second orthogonality conditi condition on of of th e branched beam. lowest mode whose components

The iter ation routine will then converge converge to the

not removed.

INTRODUCTION Experimental and theo retica l st udies have shown shown that

be idealized as

nonuniform beam

When major components of the vehicle

launch launch vehicle vehicle a irf ra me can

purposes of studying its lateral bending dynamics. cantilevered cantilevered fro m the airfram e, the vehicle vehicle

can be be repres ented

beam with cantilevered branc h beam s.

With With such

representa-

tion, tion, the bending bending dynamics of of th e components and th ei r effect on gr os s vehicle dynamics can be studied. idealized as

Fo

example, the engine shroud

the engine engine thrust struc ture coul could d be

branched beam at the aft porti on of of the vehi cle, while the payload payload or

s

nose fairing could could be repres ented

branched beam at the forward portion. portion.

staged vehicles, the engine engine and and engine engine-- support s tru ctu re frequently interstage-ada pter well, well, and this componen componentt can be repres ente

or multi-

suspended in the

an intermediate

branch. ., L/D)

Because of of the low low aspe ct rat io (i. of airframes

f c ur re nt vehicles, the bendin bending g dynamics

accu ratel y described by beam theory, only only if the effec ts of of sh ea r flexiflexi-

bility and rotational inertia of the beam elements

included. included.

Th is beam theory, know know

as the Timoshenko beam theory, is given in reference 1. There

sev era l methods for determining the norm al modes and frequencies o

nonuniform Timoshenko beam.

Fo example, reference

presents

modified Myklestad

method method which which take s into account account both both s he ar flexibility and ro ta ry ine rti a of of th e beam e lements, while refe renc iner tia.

gives

mat rix iteration technique technique which which accounts for rotationa

Shear flexibility flexibility could al so be handled handled in the s econd method method y including including the

she ar deformation in the matri x of of influence influence coefficients.

Refe renc

gives

modified

Stodola Stodola method method which which a lso includes the effe cts of of sh ea r flexibility and rotational ine rtia beam eleme nts.

Th e effects of of flexible bran ches could be handled handled by by extending extending eithe r o

these methods methods

by using the component mode method presented in reference

This

to account account fo r the effect f th e br an ch es on the

re po rt extends the method method of of ref ere nc

normal mo des and frequencie s of of the system.

SYMBOLS AS

effective effective shea

area

distance fro m tip o length of

ith bra nch beam to ce nt er of of g ravi ty

it

rigid mass

ith branch beam

Young's modulus of elasticity modulus modulus of of e lastic ity in sh ea bending bending moment of of i ne rt ia Icgi

s moment of of ine rti a o ma ss momen

inertia of

it

rigid m

s about about cente r of of gravity

it

rigid m

s about about tip o

t o r s i o n s p r i n g rate attached to it

branch beam

it

branch beam

length f p ri ma ry beam moment distribution along

ith

moment transferred to end of

branch beam ith branc h beam due to motion of sprin g-

ma ss syste moment distribution along primary beam mass

rig id body attached to

ith branch beam

moment t ran sfe rre d t o prim ary beam due to motion o natura l frequenc

it

beam

spring-mass system

it

shear distribution along it shear tra nsf err ed to end o

branch beam bran ch beam due o motion of spr ing -ma ss

it

system sh ea r distribution along primary beam ver tic al she ar tr ans fer red t o primary beam due to motion

branch

it

beam longitudinal coordinate of pri mar y beam longitudinal dis tan ce from th e end f p ri ma ry bea m to att achm ent point o ith branch beam

ve rt ic al deflection function of pri mar y beam bending deflection of pr im ar y beam ass ume d fixed at ve rt ic al deflection f p rim ary beam at bending slo pe of p ri ma ry beam at total angle f rotation of s pri ng- mas s sys tem rela tiv e to horizontal bending slope at tip bending slop

it

branch beam

primary bea

ver tic al deflectio

at attachment of

it

branch beam

at tip of ith branch beam

ver tic al deflection of p rim ary beam at attachment station of

ith

branch

beam ve rt ic al deflection function of it bending deflection of it

branch beam

branch beam a ssume d fixed

attachment station

angle of rotatio n of sprin g-mas s syste m relativ e to stati c equilibrium position

generalized m as s corresponding to longitudinal coord inate

th

normal mode

branch beam

it

ma ss p er unit lengt bending slope

primary beam

sh ea r deflection of p rim ary beam bending slope of

it

branch beam

circ ula r frequenc

integers ref erri ng to branch beams i n t e g er s r e f e r r i n g t o n o rm a l m o d e

THEORETICAL ANALYSIS Consider

launch vehicle idealized

beam attached at station xi showing ho

nonuniform beam with

shown in figure 1.

make the analysis more general by

spring -mass component can be handled, ass um e that

attached o the end of t he branch beam by

rate is

cantilevered branch

pin connection with

rigid body mass is

torsio n spr ing whose

will

with one branch, and then the equations will be genera lized t o include (in principle) any

LBranch beam

Figure

I d e a l iz e d l a u n c h v e h i c l e c o o r d i n a t e s y s te m .

Primary Beam Equations An element taken along th e length of th e pri mar y beam is shown in figure concentrated moment and s hear tra nsfe rred t o the prim ary beam fr om the branch bea rep res ent ed on the differentia l element

y mi6(

xi)& and qi6(x

xi)&, respec-

xi) is the Dirac delta function. Some f the prope rtie s o the D irac delta function that will be used in the subsequent analy sis follows: tively.

Th e expression 6(

XXi

Xi)dX

JXJX

where h(x

xi)&

dx

xi) is the unit st ep function.

Th e equations of motion of th e pr im ar y beam

determ ined by summing fo rc es and moments a cting on the eleme nt and neglecting differentials of higher order.

Thus, for forces,

t,

t)

xi)

+-dx

x$dx

F i g u r e 2.

Forces and moments acting on typical prim ary beam element.

and for moments,

Note that the rotata ry inertia t er m in equation slope tion.

(1

depends only on the bending

sinc e the s hea r fo rce s produce only distortion of the c ro ss s ection and no rotaFr om the geom etry

the bending slop

f the sys tem (fig

is given by

.-

Y$t)

where Y$t)

is the bending slope at

cantilevered at

0.

0 and

%/ax

is the bend ing,slope of the beam

The total slope will be the bending s lope minus

T h e s h e a r s l o p is subtracted f rom the bending slope in this ca se because

From elementary beam theory, the angle f s hear

is the effective she ar area.

positive

shown in figu re 2; t h e r e f o r e ,

shea r for ce tends to decre ase the total slope

where

shea r slope

at the neutral axis is given by

Substituting equation (6 into equation (7 gives

The moment-curvature rela tions for the Bernoulli-Euler beam and the Timoshenko beam the same; therefore,

m(x, t)

t,

E1 a2y(x, t)

(9)

ax To eliminate the t ime dependency fr om equations (2) to (5), (8), an motions

e considered; that is

(9), only harmonic

it is assumed that the shear, moment, slope, and

deflection functions

of the form m(x, t)

q(x, t) mi(t)

(3

Fro m equation

M(x)e

Qie

qi(t)

Q(x)eiwt, Y(x, t)

y(x)eiwt)

@(x,t)

q(x)eiwg

Mie

(lo),

(6) an

the following ordin ary differ entia l equations

obtained: dQ(x) dx dM(x)

w2pAy(x)

Q(x)

MiG(x

M(x)

QiS(x

xi)

xi)

w21pq(x)

d2y (x

Equations ( ll a) repr ese nt the required differential equations f the prima ry beam. Stodola method, to obtain

defle ction function y(x is assumed and equations ( l l a )

new impro ved deflection.

the boundary conditions The va lues

satisfied.

The con sta nts f int egr ati on The proces

In the

integrated

adjusted such that

is then repeated until it converges.

Mi and Qi in th es e equations depend on the dynamics of the branch beam.

B o u n d a r y C o n d i t io n s F o r P r i m a r y B ea m The boundary conditions fo

two cases will be considered:

(1)

free-free beam and

(2)

Other typ es of boundary conditions can be handled in an

free-cantilevered beam.

analogous manner. beam free at

Free-free beam. -~

the boundaries must vanish.

first of equations

Thus,

( l l a ) an

Q(0)

Q(l)

M(0)

M(1)

(12) yield the relation pAy(x)dx

The secon

equations

he first f equations (

t h e s h e a r s a n d m o me n t at

an

Qi

) and (13) yield the rela tion

) yie lds

Substituting equation (16) into (15) results in

It is easi ly shown (ref. 6) that

Substituting equation (18) into (19) and rearranging yield

pAy(x)dx

QJ

Ipcp(x)dx

pAxy(x)dx

first te rm in brackets in equation (19) is simply the s hea

this case.

Thus, equation (19) beco mes

at

Qixi which is z e r o f o r

pAxy(x)dx

'Ipcp(x)dx

Qixi

Mi

Equations (14) and (20) rep rese nt the equations which must be satisfied to ensure

at the boundaries of the prim ary beam. beam free at Free-cantilever beam. an fixed at

van-

ishing moment and s hea

1, the boundary

conditions M(0)

Q(0) y'(1)

--Q(1)

y(l)

Branch Beam Equations Th e governing differential equations of the bran ch beam

be developed in

manner

The coordinate syst em of th e branch is as shown in figure 1, and an elem ent along the length is shown in figure 2, except that the con-

analogous to those of th e pr im ar

beam.

centra ted sh ea rs and mome nts qi and mi will not be presen t.

Th e governing equations

f the branch, corresponding o equations (l la ) f the prim ary beam, will theref ore be given by

where the subscripts bi refer to the it

branch (for one branch

1) an

is th

bending slope at the attachment station xi.

B r a n c h B ea m B o u n d a r y C o n d i ti o n s To obtain express ions fo the boundary conditions diagram

the branch beam shown in figure

the slopes

pi

and yi

the branch beam, the free-body

is consider ed.

6.

assumed to be sm all (i.

or sm al l deflection s, Newton's second la

In keeping with beam theory, and tan

y.).

gives the following equations

spring-mass system in harmonic motion: -mciw 2-la

ei

ei where

K.8.

i, the mass moment of inertia

c g i (e

[Ji(ei

+pi)

pi)

.a

m c1.a.

the spr ing- mass syste m about the tip of the

branch beam, may be expressed as

cgi

.a

ci

Free body diagram of ith b r a n c h

F i g u r e 3. - C o o r d i n a t e s y s t e m f o r it b r a n c h b e a m

(24)

Solving equation (24) for

ei

yields

,.=Piti+ w

,f

mciai

where pi is the natural frequency of the simple spri ng-m ass system defined by

Substituting equation (2 5) into equations (23) and (24) yields the following expressions for shea r and momen at the righ t s ide of the branch beam.

-w

Me

(Jipi

KiOi

.a.

c1

.)

The s he ar and momen at the left s id e of th e bran ch beam c an be obtained y integ rating first and se con d f equ ati ons (22).

and

Thu s,

Th

first integral in equation (29) can be writte n in th e following form

Using this relation in equation (29) yields

or the adopted sign convention, Mbi(0)

-Mi

Mbi(bi)

Qbi(bi)

&bi(O)

Then, fr m equations (26

(29) and (32), b. (33)

@A)biqi(