stodola method

.. I , . . _ *

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METHOD OF CALCULATING THE NORMAL MODES AND FREQUENCIES OF A BRANCHED TIMOSHENKO BEAM

by Francis J. Shaker

Lewis Research Center Cleveland, Ohio

N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. M A Y 1968 I I

TECH LIBRARY KAFB, NM

I111111 11111 11111 lllll 11111 1111 IIIII Ill1 111 0333235

METHOD O F CALCULATING T H E NORMAL MODES AND

FREQUENCIES O F A BRANCHED TIMOSHENKO BEAM

By F r a n c i s J. Shake r

L e w i s R e s e a r c h C e n t e r Cleve land , Ohio

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION - ._ __ -

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTl price $3.00

L

METHOD OF CALCULATING THE NORMAL MODES AND

FREQUENCIES OF A BRANCHED TIMOSHENKO BEAM

by Francis J. Shaker

Lewis Research Center

SUMMARY

A method is presented in this report for calculating the normal modes and frequen- c ies of a branched Timoshenko beam. method and requires an iteration procedure to determine the normal modes and frequencies of the system. In this method, an arbitrary deflection, consistent with the boundary condition, is assumed. Also, because of the presence of the spring-mass system, the frequency is not a constant factor in the governing equations and must also be assumed. Knowing the deflection and frequency, the corresponding shears, moments, slopes, and new deflection can be determined by integrating the governing differential equations of the system. The new deflection is then adjusted to satisfy the boundary conditions, and a new frequency is subsequently calculated. The process is then repeated, and the frequency is used as the cri terion for convergence. Although the method can be applied for any type of boundary conditions, particular attention was given to those boundary conditions of interest in launch-vehicle dynamics, namely, the free-free beam and the cantilevered-free beam.

The method is essentially a modified Stodola

In general, the iteration routine will always converge t o the lowest or fundamental mode of vibration. To determine the higher modes, the lower mode components are removed from the assumed shear and moment distribution by utilizing the second orthogonality condition of the branched beam. The iteration routine will then converge to the lowest mode whose components are not removed.

INTRODUCTION

Experimental and theoretical studies have shown that a launch vehicle airframe can be idealized as a nonuniform beam for purposes of studying its lateral bending dynamics. When major components of the vehicle are cantilevered from the airframe, the vehicle

can be represented as a beam with cantilevered branch beams. With such a representa- tion, the bending dynamics of the components and their effect on gross vehicle dynamics can be studied. For example, the engine shroud or the engine thrust structure could be idealized as a branched beam at the aft portion of the vehicle, while the payload or the nose fairing could be represented as a branched beam at the forward portion. staged vehicles, the engine and engine- support structure frequently are suspended in the interstage-adapter well, and this component can be represented as an intermediate branch.

Because of the low aspect ratio (i. e . , L/D) of current vehicles, the bending dynamics of airframes are accurately described by beam theory, only if the effects of shear flexibility and rotational inertia of the beam elements are included. This beam theory, known as the Timoshenko beam theory, is given in reference 1.

There are several methods for determining the normal modes and frequencies of a nonuniform Timoshenko beam. For example, reference 2 presents a modified Myklestad method which takes into account both shear flexibility and rotary inertia of the beam elements, while reference 3 gives a matrix iteration technique which accounts for rotational inertia. Shear flexibility could also be handled in the second method by including the shear deformation in the matrix of influence coefficients. Reference 4 gives a modified Stodola method which also includes the effects of shear flexibility and rotational inertia of beam elements. The effects of flexible branches could be handled by extending either of these methods or by using the component mode method presented in reference 5. This report extends the method of reference 4 to account for the effect of the branches on the normal modes and frequencies of the system.

For multi-

7

'.

SYMBOLS

AS

"i

bi E

G

I

Icgi

Ji

5

effective shear area

distance from tip of ith branch beam to center of gravity of ith rigid mass

length of ith branch beam

Young's modulus of elasticity

modulus of elasticity in shear

bending moment of inertia

mass moment of inertia of ith rigid mass about center of gravity

mass moment of inertia of ith rigid mass about tip of ith branch beam

torsion spring rate attached to ith branch beam

2

X

xi

length of primary beam

moment distribution along ith branch beam

moment transferred to end of ith branch beam due to motion of spring- mass system

moment distribution along primary beam

mass of rigid body attached to ith branch beam

moment transferred to primary beam due to motion of ith beam

natural frequency of ith spring-mass system

shear distribution along ith branch beam

shear transferred to end of ith branch beam due to motion of spring-mass system

shear distribution along primary beam

vertical shear transferred to primary beam due to motion of ith branch beam

longitudinal coordinate of primary beam

longitudinal distance from the end of primary beam to attachment point of ith branch beam

vertical deflection function of primary beam

bending deflection of primary beam assumed fixed at x = 0

vertical deflection of primary beam at x = 0

bending slope of primary beam at x = 0

total angle of rotation of spring-mass system relative to horizontal

bending slope at tip of ith branch beam

bending slope of primary beam at attachment of ith branch beam

vertical deflection at tip of ith branch beam

vertical deflection of primary beam at attachment station of ith branch

vertical deflection function of ith branch beam

bending deflection of ith branch beam assumed fixed at attachment station

angle of rotation of spring-mass system relative to static equilibrium po-

beam

sition

3

generalized mass corresponding to jth normal mode

longitudinal coordinate of ith branch beam

mass per unit length

bending slope of primary beam

shear deflection of primary beam

bending slope of ith branch beam

circular frequency

integers referring to branch beams

integers referr ing to normal modes

THEORETICAL ANALYSIS

Consider a launch vehicle idealized as a nonuniform beam with a cantilevered branch beam attached at station xi as shown in figure 1. To make the analysis more general by showing how a spring-mass component can be handled, assume that a rigid body mass is attached to the end of the branch beam by a pin connection with a torsion spring whose spring rate is Ki. The governing equations will first be developed for a primary beam with one branch, and then the equations will be generalized to include (in principle) any number of branches.

LBranch beam

Figure 1. - Idealized l a u n c h vehic le coordinate system.

4

Primary Beam Equations

An element taken along the length of the primary beam is shown in figure 2. The concentrated moment and shear transferred to the primary beam from the branch beam are represented on the differential element by mi6(x - xi)& and qi6(x - xi)&, respectively. The expression 6(x - xi) is the Dirac delta function. Some of the properties of the Dirac delta function that will be used in the subsequent analysis are as follows:

i p X < X i

6(X - Xi)& = h(x - x.) = \ X > X i

JXJX 6(x - Xi)dX dx = Jx h(x - xi)& =

where h(x - xi) is the unit s tep function. The equations of motion of the primary beam are determined by summing forces and moments acting on the element and neglecting dif- ferentials of higher order. Thus, for forces,

t, + qi(x7 t)6(x - xi) = -pAY(x, t) ax

( m

i - x

x$dx

(3)

aM +-dx b X

Figure 2. - Forces and moments acting on typical primary beam element.

5

and for moments,

Note that the rotatary inertia t e rm in equations (1) to (4) depends only on the bending slope 5, since the shear forces produce only distortion of the cross section and no rotation. From the geometry of the system (fig. l),

.-

the bending slope is given by

\,

+ Y$t) (5)

where Y$t) is the bending slope at x = 0 and %/ax is the bending,slope of the beam cantilevered at x = 0. The total slope will be the bending slope minus the shear slope. The shear slope is subtracted from the bending slope in this case because a positive shear force tends to decrease the total slope as shown in figure 2; therefore,

From elementary beam theory, the angle of shear at the neutral axis is given by

where As is the effective shear area. Substituting equation (6) into equation (7) gives

The moment-curvature relations for the Bernoulli-Euler beam and the Timoshenko beam are the same; therefore,

m(x, t) = E1 t, = E1 a2y(x, t) ax ax2

(9)

To eliminate the time dependency from equations (2) to (5), (8), and (9), only harmonic motions a r e considered; that is, it is assumed that the shear, moment, slope, and

6

deflection functions are of the form

m(x, t) = M(x)e iwt , qi(t) = Qie iwt

q(x, t) = Q(x)eiwt, Y(x, t) = y(x)eiwt)

mi(t) = Mie i w t , - @(x,t) = q(x)eiwg

i

From equations (3) to (6) and (8) to (lo), the following ordinary differential equations are obtained:

-- dQ(x) - w2pAy(x) - QiS(x - xi) dx

~- dM(x) - Q(x) + MiG(x - xi) - w21pq(x) dx

d2y (x M(x) = E 1 2 dx2

Equations ( l la ) represent the required differential equations of the primary beam. In the Stodola method, a deflection function y(x) is assumed and equations (lla) are integrated to obtain a new improved deflection. The constants of integration are adjusted such that the boundary conditions are satisfied. The process is then repeated until it converges. The values of Mi and Qi in these equations depend on the dynamics of the branch beam.

Boundary Condi t ions For P r imary Beam

The boundary conditions for two cases will be considered: (1) a free-free beam and

7

(2) a free-cantilevered beam. Other types of boundary conditions can be handled in an analogous manner.

Free-free -~ beam. - For a beam free at x = 0 and x = 1 , the shears and moments at the boundaries must vanish. Thus,

Q(0) = Q(l) = 0

M(0) = M(1) = 0

The first of equations (lla) and (12) yield the relation

u2 J1 pAy(x)dx - Qi = 0

The second of equations ( l l a ) and (13) yield the relation

The first of equations ( l l a ) yields

Substituting equation (16) into (15) results in

It is easily shown (ref. 6) that

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

Z k 2 6' pAy(x)dx - QJ - U 2 6' Ipcp(x)dx - u2 6' pAxy(x)dx + Qixi + M i = 0

The first term in brackets in equation (19) is simply the shear at x = 1 , which is zero for this case. Thus, equation (19) becomes

8

w2 pAxy(x)dx + u2 f 'Ipcp(x)dx - Qixi - Mi = 0

Equations (14) and (20) represent the equations which must be satisfied to ensure a van- ishing moment and shear at the boundaries of the primary beam.

Free-cantilever beam. - For a beam free at x = 0 and fixed at x = 1, the boundary conditions are

? 7 Q(0) = 0 M(0) = 0

, e

Q(1) y(l) = 0 ASG 1 y'(1) = --

Branch Beam Equations

The governing differential equations of the branch beam can be developed in a manner analogous to those of the primary beam. The coordinate system of the branch is as shown in figure 1, and an element along the length is as shown in figure 2, except that the concentrated shears and moments qi and mi will not be present. The governing equations of the branch, corresponding to equations ( l la ) of the primary beam, will therefore be given by

9

where the subscripts bi refer to the ith branch (for one branch i = 1) and yi is the bending slope at the attachment station xi.

Branch Beam Boundary Condit ions

To obtain expressions for the boundary conditions of the branch beam, the free-body ?

diagram of the branch beam shown in figure 3 is considered. In keeping with beam theory, the slopes pi and yi are assumed to be small (i. e., tan 6. = p. and tan y. = y.).

spring-mass system in harmonic motion:

1 1 1 1 I

For small deflections, Newton's second law gives the following equations for the

Q~~ = -mciw 2- lai +

n I L M . = w I (e +pi) - Q .a. e i cgi i e i 1J

or

(24) 2 M ei . = K.8. 1 1 = w [Ji(ei + pi) + m c1 .a. 1 A . ] 1

where Ji, the mass moment of inertia of the spring-mass system about the tip of the branch beam, may be expressed as

2 Ji = I + m .a. cgi c i 1

Free body diagram of ith b ranch Mi

Figure 3. -Coordinate system for ith branch beam.

10

Solving equation (24) for ei yields

2 w

mciai 2 , .=Piti+ 1 w 2 Ji ) 1 - -

J ,f

where pi is the natural frequency of the simple spring-mass system defined by

Substituting equation (2 5) into equations (23) and (24) yields the following expressions for shear and moment at the right side of the branch beam.

2 -w m c i Q . = e i 3

wy 1 - -

2 w Mei - - KiOi = ~ (Jipi + m .a. A.) 2

1 - - c1 1 1

0 n

The shear and moment at the left side of the branch beam can be obtained by integrating the first and second of equations (22). Thus,

and

The first integral in equation (29) can be written in the following form:

Using this relation in equation (29) yields

For the adopted sign convention,

&bi(O) = Qi Qbi(bi) = Qei el Mbi(0) = -Mi Mbi(bi) = M

Then, f rom equations (26) to (29) and (32),

b. Qi = -a2 f 0 @A)biqi(<i)d<i + Qei

1 b. Mi = -Mei + biQei - 0 [Jbi dOA)bi<iqi(<i)d<i + f 0 (@)bi*i(<i)d<i

Eliminating Mei and Qei from these last two equations yields

(33)

(3 4)

J

12

These two equations represent the shear and moment transferred to the primary beam as a result of the dynamics of the branch-beam and spring-mass system.

Orthogonal i ty Condit ions

To determine the orthogonality conditions for the branched Timoshenko beam, eliminate m and q from equations (3) and (4) by using equations (8) and (9) to obtain

For a system vibrating in the jth normal mode,

i w j t 7 Y ( x , t) = y.(x)eiwjt, mi(t) = M..e J u

iw j t - +(x, t) = cp.(x)eiwt, qi(t) = Qije

J

13

I.

From equations (37) and (38),

(EIy;)' + [AsG(y; - qj)] - MijB(x - X$ =

3 J [A,G(y; - qj)]' - QijS(x - xi) = -O.pAy. 2

where

(39)

and the primes denote d/dx. Let yk and qk represent the deflection and slope of the beam vibrating in the kth normal mode. Then, multiplying the first of equations (39) by qk dx and the second by yk dx, integrating over the length, and adding the resulting equations yields

Integrating the first two integrals on the right side of equation (40) by parts yields

Now, for conventional boundary conditions (i. e. , free, pinned, o r fixed), the te rms in brackets in equation (41) vanish. Thus,

14

I I

Following the same procedure for the branch beam, the equation corresponding to equation (41) will be

I

Evaluating the te rms in brackets

then, substituting equations (44) into equation (43) and adding the resulting equation to equation (42) gives

Also, from equations (23) and (24)

15

2 MeijP, - .. Aik = - W . 0 . (JAY.. + m .a. A..) e11 J lk 1 1J C 1 1 1J

+ w j [ I c g i ~ i j ~ i k 2 + m .(A.- + a.(Y..)(A& + a (Y )] (46) C l 1J 1 1J i i k

From equation (24),

K.0.. J.o . . + m . a . ~ . . => 1 11 c1 1 1J 2

j w

c

Substituting equation (47) into equation (46) and substituting the resulting equation into equation (45) gives

All the te rms which appear in equation (48) are symmetric with respect to j and k except w 2 Therefore, if the system is considered to be vibrating in the kth mode and the same procedure is followed as for the jth mode, an equation identical to equation (48) is obtained except w2 will be replaced by ok; that is,

j '

2 j

16

Subtracting equation (49) from equation (48) and assuming w j 2 2 # Wk gives

+ ICgiaijaik + m c1 .(A.. 1J + a.a..)(A& 1 1J + a i i k a ) = 0 (50)

I

Equation (50) represents the first orthogonality condition of the branched beam. the second orthogonality condition, multiply equation (48) and (49) by wz and w2 re-

2 2 j , spectively, and subtract. Then, for w . # w J k'

To obtain

f

An alternate form for the second orthogonality condition is obtained by using the relations

M = EIq', Q = A,G(q - y')

Generalized mass. - Equations (50) to (52) are valid for w 2 2 # wk. To obtain the gen- j

era1 expression for these orthogonality conditions, let

17

where p is referred to as the generalized m a s s of the system vibrating in the jth normal mode. Then, in t e r m s of the generalized mass, the first and second orthogonality

.i

conditions become

+ IcgiCYijaik + m .(A.. + a.a..)(A& c1 1J 1 1J CY i i k ) = p 6 j j k (544

and

where 6 is the Kronecker delta function defined by

j = k

jk

j # k 6jk = f

Equations (54a) and (55a) satisfy equation (48) for j = k and equations (50) and (51) for j # k.

Natural frequency. - - An expression for the natural frequency of the system can be obtained by setting j = k in equation (48). Thus,

18

2 w, = L J

P r i m a r y Beam with Several Branches

The equations developed on the preceding pages are for a primary beam with one branch. The resul ts can be immediately generalized to include any number of branches by replacing Qi6(x - xi) and Mi6(x - xi) with

tively, in the equilibrium equations. The development will then proceed in the same manner, and the resulting equations will be similar to those previously developed, except that the branch beam parameters will be summed over all the branches. sume that there a r e n branches. Then, a summary of the pertinent equations will be as follows.

Governing differential equations of primary beam :

Qi6(x - xi) and i i

Mi6(x - xi), respec-

For example, as-

n -- dQ(x) = u2pAy(x) - E Qi6(x - xi) dx

i= 1

n 2 -- m(x) - Q(x) + Mi6(x - xi) - o Ipq(:

i= 1 dx

19

The last three equations are independent of the number of branches and are included herein only for later reference.

Free-free boundary conditions:

4

u2 L2 pAxy(x)dx + w

Generalize mass:

i= 1

Orthogonality conditions :

n

i= 1 + [ ~ ~ ~ ~ l y ~ ~ a ~ + mci(Aij + a.a..)(Aik 1 4 + ai A*)] = pj6jk (54b)

20

Frequency equation:

MODIFIED STODOLA METHOD

In the previous section, the governing differential equations of a branched Timoshenko beam have been developed. Also, the boundary conditions for the free-free and the free-cantilevered primary beam have been expressed in integral equation form. In this section, a modified Stodola method will be applied to solve the differential equations of motion subject to the free-free and free-cantilevered type boundary conditions.

shape of the beam which satisfies the boundary condition. Based on this assumed deflection, the corresponding shears, moments, slopes, and new deflections can be determined by integrating the governing differential equations. Then, by using the boundary condition equations, the new deflection is adjusted to satisfy the boundary conditions. At this point, the natural frequency of vibration can be determined. Using the new improved deflection, the process is repeated until the natural frequency converges. If the branched beam has an attached spring mass, as in the case considered in this report, it is also necessary to assume a frequency of vibration at the start of each iteration. The reason for this is that the frequency does not appear as a constant factor in the governing equations as developed. This point will be clarified as the Stodola method is applied in detail.

For the free-free case, the iteration routine is started by assuming that the deflected shape of the primary beam is a polynomial of the form

The Stodola method, as applied to beam vibration, starts by assuming a deflected

I

v=p

y(x) = Y(O)(X) = 1 - ( 2 ) j P + 2 P v= 1

(57)

21

where the superscript (0) re fers to the first iteration and the subscript p referes to the mode being determined. For a cantilevered beam, clamped at x = 1, the initial assumed primary beam deflection can be taken as

P

y(x) = Y (0) (x) = (x - 1) (A;) P v= 1

A deflection must a lso be assumed for each branch beam. flected shape can be taken as

For the ith branch, the de-

where xi") is the deflection of the primary beam at the attachment station and

is the slope at the attachment station. Before starting the iteration routine, it will be convenient to normalize the deflection

to unity at the tip of the primary beam. The branch beams should be multiplied by the same normalizing factor as the primary beam. Also for convenience, w2, which appears as a factor, is set equal to one, since the mode shapes can only be determined to within a constant factor. The w 2 t e rms which do not appear as a factor a r e set equal to one in the first iteration and then for subsequent iterations, the value determined by the previous iteration is used.

First, consider the iteration routine of the ith branch beam. For the assumed shape, the shear and moment at the t ip of the branch beam is given by equations (26) and (27) as

P;

(JiPi + m .a. A.) 1 c1 1 1

Mei = ~

2 1 - - w

2 Pi

22

Then, the shear and moment at the attachment station will be given by equations (33) and (34), respectively.

Since Qi or Pi is not known in the first iteration, it is set equal to zero, and for subsequent iterations the values from the previous iteration are used.

integrating the first and second of equations (22) and noting that, for the adopted sign con- vent ion,

The shear and moment distribution along the branch beam can now be calculated by

Therefor e,

When the shear and moment distribution is known, a new improved deflection of the branch beam can be determined by utilizing the third, fourth, and fifth of equations (22);

t thus,

and

23

I IIIII

To determine the values of si and yi in equations (67) and (68), the deflection and slope of the primary beam at the attachment point of the ith branch must be determined. This can be done by an integration procedure similar to that used for the branched beam.

tegrating the first and second of equations (llb). Thus, For the primary beam, the shear and moment distribution can be determined by in-

and

n n Qi LX h(x - xi)dx +

i= 1 i= 1 M(x) = Jx Jx pAy(x)dx dx - Jx Ipcp(x)dx - Mih(x - xi)

(7 0)

where h(x - xi) and Lx h(x - xi)& are given by equations (1) and (2), respectively.

Then, from the third, fourth, and fifth of equations ( l lb) ,

Y(X) = $ J X M O d x dx - J X Qo dx + ybx + yo E1 ASG

The constants of integration yu and yo must now be adjusted to satisfy the boundary conditions. First, for compactness, the following quantities are defined: 7

24

J

Using these substitutions, the deflection and bending slope of the beam can be written as

For the free-free beam, the boundary constants yh and yn must be adjusted to satisfy equations (14) and (20). The expressions for Qi and Mi, which should be used in these equations, are given by equations (35). Therefore, the following equations must be sat- isf ied

i= 1

i= 1

) ,> Pi

i= 1

[m .a.(x. c1 1 1 + bi) + Ji]pi =

25

If equations (73) are substituted into equations (74), two equations in two unknown constants a r e obtained, namely, yb and yo. Note that in equations (74) the expressions for hi and Pi are given by

A i = ~ibi) = {ibi) - Oibi) + [?'(xi) + Yb]bi + ?(xi) - @(xi) + Y& + YO } (75) Pi = +ibi) = {bi) +?'(Xi) + Yb

Substituting equations (73) and (75) into (74) yields

+ ,) ( - %){ti(bi) JiPi - Ci(bi)

1 - - 2

Pi

26

i= 1

Collecting t e rms in y' and yo and rearranging yields equations (78a) and (79a). 0

Shear boundary condition:

I , ' i= 1

c i= 1 J

27

Moment boundary condition:

/ , - P:

i= 1

1 - - 2

Pi

i= 1 1

28

To put equations (78a) and (79a) into a more convenient form, suitable for calcula- tions, we define the constants given in the appendix. with the constants K in this appendix signify that K depends on the particular mode being sought. The other constants are independent of the mode and depend only on the mass and geometric properties of the system. can be written as

The subscripts r which appear

Using the appendix, equations (78a) and (79a)

I

where

i= 1

n

i= 1

29

e- AQr = Mo + Po + 1

2 1 - -

2 w

Pi i= 1

n

n

i= 1 '21- = K2r - K3r + (K2ri - K3ri - K6ri + K7ri + K8ri - Kllri)

Solving equations (78b) and (79b) yields, for yb and yo,

Equations (80) give the boundary constants for the free-free case. For the free- cantilevered case, we obtain from equations (21) the following expressions for yb and yo:

After the integration constants have been determined from either equations (80) or (81), a new deflection can then be obtained from equations (73). The frequency of vibration can now be calculated from equation (56b). Before starting the next iteration, it will be necessary, in general, to recalculate yo and yb for the free-free case based on the new improved frequency. This is required because the assumed deflection must satisfy the boundary conditions. Then, after determining a new improved deflection, the deflection is normalized, and the process is repeated until two successive frequencies are equal. A flow chart of the iteration routine is given in figure 4.

30

Assumed pr imary beam deflection

Set ~ ( 0 ) = O

I ,

Nor ma I ize

I

I New improved free-free beam deflections, Y, 60, V i and Oi Sweeping

Q* - Q . M".- Mbi Q*- Q, M*- M bi b i bi

I I

Frequency, w Test if

Recalculate ?I 0) integration - 2 I constants

Frequency, w Test i f X

-1

Y =[ 4 d x (Enbi dci I

Branch beam bending slope Branch beam deflection 1

t f X si r=[ T a x 7i = / !id+

0 Primary beam bending slope Pr imary beam deflection p = Y + Y b Y = Y - 9 + YbX + Yo 1 1

t I

1L I , Integration X

constants, 9 = 4 $ d x Yb and Yo

Pr imary beam bending slope Pr imary beam deflection

X

r=[ T a x I t

Integration X

constants, 9 = 4 $ d x Yb and Yo

I I 1 t I

Figure 4. - Iteration rou t i ne for branched beam.

High Modes of Vibration

The iteration routine described in the previous section will always converge to the first or fundamental mode of vibration. To determine the higher modes of vibration, it is necessary to remove all lower mode components from the assumed shape. Let y,*(x) and * .(<) represent the assumed shape of the rth mode. Assume that the mode shapes of V r , 1

all the v = r - 1 modes a r e known and that it is required to determine the rth mode. Since any arbitrary shape can be represented by a ser ies involving the normal modes of the system,

where the Cvts a r e unknown constants that must be determined. The shear and moment distribution, corresponding to the assumed deflection, can be expressed by the ser ies

Q:(x) = C,Q1(X) + C ~ Q ~ ( X ) + - - + CVQv(x) + - . -

For < = bi, the last equation becomes

Next, multiply the first of equations (83) by [%(x)/A,G]dx, the second by [Mv(x)/EI]dx, the third by [Qvi(<)/(AsG)bild<, and the fourth by [Mvi(<)/(EI)bi]d<iy and integrate over their respective lengths. These operations yield the following set of equations:

Then multiply equation (84) by Mevi /Ki.

32

+ c v p + . E1 . .

2 . . 'vMevi + . + . . . Me liM evi

Ki Ki Ki M&.-iMevi = c1

Adding equations (85) and using the second orthogonality condition which is given by equation (55b) yield

where wv is the natural frequency of the vth mode and pv is the generalized mass of the vth mode given by equation (53b).

For n branches, equation (86) becomes

33

cv =

M:Mv QrQv E1 AsG

+-

i= 1 ~~

(% 9

Knowing av, the purified shear and moment distribution becomes

v= 1

- 1 t M*.M ri v i+ Qri *-Q vi I d E , +M;riMevi\

Mri = M*. ri - CvMvil v= 1 J

Equations (88) represent the shear and moment distribution which must be used in the iteration routine to determine the higher modes of vibration (i. e., 2, 3, . . .).

CONCLUDING REMARKS

A computer program, based on the analysis presented in this report, has been written for finding the normal modes and frequencies of a branched Timoshenko beam. Sev- eral test cases have been run and the results obtained show that the iteration routine converges in a reasonable length of time. A flow chart showing the iteration routine is given in figure 4. An examination of this flow chart shows that a number of integrals must be evaluated during each iteration. Any of the standard methods of numerical integration can be used to evaluate these integrals, and the best method will depend, in general, on

34

the mass and stiffness distribution along the beam. The summation method, which is thoroughly covered in reference 7, is particularly suitable for most practical cases.

Lewis Research Center, National Aeronautics and Space Administration,

Cleveland, Ohio, October 23, 1967, 180-06-06-03-22.

35

APPENDIX - DEFINITION OF CONSTANTS USED IN EQUATIONS (80)

mo = J6” p A dx

mbi = lbi bA)bi d6 i

Ndi = mci(Xi + bi)

No =Jz pAx dx

Nbi = dbi @A)bi(xi + 5,)dti

Nci = m c1 .(x. 1 + bi +ai)

Mo = J1 pAx2 dx

2 Mbi = lbi bA)bi(xi ti) d<i

2 Mci = m c1 .(x. 1 + bi + ai)

2 Mdi = m ci (x. 1 + bi)

36

I - w2 & ( N di -

Ki ci i

37

P;

I - - 2

Pi

38

I

REFERENCES

1. Timoshenko, S. : Vibration Problems in Engineering. Third ed., D. Van Nostrand Co., Inc., 1955.

2. Fowler, J. : Generalized Missile Dynamics Analysis. V. Calculation of Bending Modes. Rep. No. EM 8-16, GM-TR-0165-00445, Space Technology Labs., Apr. 7, 19 58.

3. Scanlan, R. H. : A Note on Transverse Bending of Beams Having Both Translating and Rotating Mass Elements. J. Aeron. Sci., vol. 15, no. 7, July 1948, pp. 425-426.

4. Ftheinfurth, M. : Modified Stodola Method Including Rotary Inertia and Shear Flexibil- ity. Rep. No. DA-TR-4-59. Army Ballistic Missile Agency, 1959.

5. Hurty, Walter C. : Dynamic Analysis of Structural Systems by Component Mode Syn- thesis. Tech. Rep. 32-530, Jet Propulsion Lab., California Inst. Tech., Jan. 15, 1964.

6. Hildebrand, Francis B. : Methods of Applied Mathematics. Second ed., Prentice- Hall, Inc., 1965.

7. Houbolt, John C. ; and Anderson, Roger A . : Calculation of Uncoupled Modes and Frequencies in Bending o r Torsion of Nonuniform Beams. NACA T N 1522, 1948.

NASA-Langley, 1968 - 2 E-4152 39

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