Vibration beams

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AFB, N

M a

EFFECT

OF

AXIAL LOAD

ON

MODE SHAPES AND

FREQUENCIES OF BEAMS

Fm nc i s J. Sha k e r

Lewis Research Center

Cleveland, Ohio 44135

N A T I O N A L A E R O NA U T I CS A N D S PA C E 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 D E C E M B E R

1975

c


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TECH LIBRARY KAFB, NM

. .

.

1.

Report No.

2. Government Accession

No.

NASA TN D-8109

I

.

4. Title and Subtitle

EFFECT OF AXIAL LOAD ON MODE SHAPES

AND

FREQUENCIES OF BEAMS

7.

Author(s)

Francis J . Shaker

9.

Performing Organization Name and Address

Lewis Research Center

National Aeronautics and Space Administration

Cleveland, Ohio 44135

12. Sponsoring Agency Name and Address

National Ae rona utic s and, Space Ad ministration

Washington, D.

C.

20546

5. Supplementary Note3

_ _ .

16. Abstract

3.

Recipient's Catalog No.

5. Report Date

December 1975

6. Performing Organization Code

8. Performing Organization Report No.

E-8433

10.

Work Unit No.

506 -22

11.

Contract or Grant No.

13. Type of Report and Period Covered

Technical Note

14.

Sponsoring Agency Code

An investigation of the effect

of

axial load on the n atur al freq uenc ies and mode sh apes of uniform

beams and

of

a cantil evered beam with a concentrated ma ss a t the tip is presented. Character

ist ic equations which yield the frequ encies and mode shape functions for the var ious cases are

given. The solutions to these equations ar e presented by a series of graphs

so

that frequency as

a function of axial load can read ily be dete rmin ed. The effect

of

axial load on the mode shapes

are als o depicted by another series of graphs.

. -

17.

Key Words (Suggested by Author(~)

18.

Distribution Statement

Struc tura l dynamics Unclassified - unlimited

Beam vibration under prelo ad STAR cat egor y 39

-~ .~

19.

Security Classif. (of this repor t)

20.

Security Classif. (of this page)

Unclassified

Unclassified


3/33

EFFECT OF A X I A L L O A D O N M O D E S H A P E S A N D F R EQ U EN C IE S OF B E A M S

by

F r a n c i s

J.

S h a k e r

L ew i s R e s e a r c h C e n t e r

SUMMARY

An

investigation o the effect of ax ial load on the natural ,,,equencies and mode

shape s of uniform b eams with various types of boundary conditions and of

a

cantilevered

beam with

a

concentrated mass at the tip

is

presented.

Th is investigation yielded e x

pres sion s for the mode shapes and cha racte ristic equations fo r the various cas es con

sidered.

For

the uniform beams the char acte ristic equations were solved either numer

ically or in closed form , and the res ult s a r e presented by a se r ie s of graph s showing the

effec t of preload on the diffe rent types of beam s. The effect of axial load on the mode

shapes is also shown in graphical form for several different loading conditions. For the

cantilevered beam with a t ip mas s, two types of axial loads were considered: In the

first case the axial load vector remained constant, and in the second case the load was

directed through the root of the beam

at

all tim es. The re su lt s of thi s portion of the in

vestigation a re presented in graphs that show the effects of both tip -mas s variation and

axial load on the fundamental frequency of the system.

INTRODUCTION

In the design of c ert ain space craft st ruc tur al components it sometimes becomes

necessary to determine the normal modes and frequencies of beam type components

which are in a state of preload

or

pre str ess . This preload may result from inertial ef -

fects,

as

in the ca se of a spinning spacec raft, or it may be induced by mechanical

means.

For

example, curre nt designs for large, flexible solar ar ra ys (refs. 1 to 4 )

are

such that the boom that supports the a rr ay is in

a

state

of pr es tr es s due to the ten

sion that must be maintained in the s olar cell s ubstrate.

Also,

cer tain types of attach

ment and ejection mechanisms

can

cause preload

in

parts of the primary spacecraft

structure.

In

most of thes e cases

a n

exact solution for the normal modes and frequen

ci es of the syst em would be intractab le and rec ou rse would be made to one of the many

approximate solutions such as the Rayleigh-Ritz method, Galerk in's method, finite


4/33

elements, etc. (refs.

5

o

8).

There exists, however, a

class

of problems for which

an

exact solution can be obtained.

This class consists of uniform beams under constant

axial load with var ious types of sim ple boundary conditions. Th is type of struct ure may

represent some structural components reasonably well for preliminary design informa

tion. Even in those case s where

it

does not adequately model the structure, the modes

obtained for these ca se s can be used to obtain Rayleigh-Ritz

or

Galerkin type of

solutions.

For these reasons an investigation was made to dete rmine the effect of an axial load

on the modes and frequencies of

a

uniform beam with various boundary conditions and of

a

cantilevered beam with

a

tip mass. The boundary conditions considered for the uni

form beam include

all

possible s imple boundary conditions at both ends of the beam

(i.e. , free, hinged, clamped, and guided). For the cantilevered beam with

a

t ip mass,

two

types of

axial

loads were considered. The first type was a constant axial load ap

plied at the tip, and the second was an axial load whose direc tion is always through the

fixed s u p p r t . The second

is

the limiting case for some current large solar a rr ay de

signs if the mas s of the blanket is negligible compared with the m as s of the support

beam. When the mas s of the blanket is not negligible, which is usually the situation,

these modes could

still

be used in

a

Rayleigh-Ritz ana lys is for thi s type

of

array.

k

-

k

-

kcr

1

'cr

2

SYMBOLS

ar bi tr ar y constants of integration

constants of integration corresponding to nth mode of vibration

uniform beam bending stiffness

axial load parame ter, f z I

nondimensional axial load pa ram ete r, )/Pl/EI

nondimensional axial load parameter corresponding to critical load,

length of beam

moment dis tribution along length of beam

total mass of beam, p l

tip ma ss f or cantilevered beam

axial

load

critical or Euler 's buckling load for beam


5/33

X

-

X

VT

P

w

wn

Subscripts

:

B

n

T

Superscript:

?

shearing fo rc e distribution

time

beam displacement

nondimensional beam displacem ent , v/Z

mode shape of nth mode

lengthwise coordinate

nondimensional lengthwise coordinate, x/Z

chara cteris tic values, eq. (9)

chara cteris tic values for nth mode

frequency param eter,

a

nondimensional frequency parameter, P

1

nondimensional frequency pa ra met er corresponding to nth mode of

vibration, pwn

ra tio of tip m as s to total beam ma ss , +.,/AB

ma ss p er unit length of beam

circul ar frequency of vibration

ci rcular na tural frequency of nth mode

be am

integral number designating natural bending mode and frequency of beam

tip of cantilevered beam

differentiation with res pec t to

x

THEORETICAL ANALYSIS

Uniform Beam Under Axial Load

Equations of motion.

- The development of th e equations governing the bending

vibrations of uniform beams are presented in numerous texts on vibration theory

(refs.

5,

6,

o r

8). In the development tha t follows th e effect of an axia l load is included.

3


6/33

Consider

a

uniform beam vibrating freely under th e action of

a

constant axial load

P

as

shown in figu re

l(a). Let

V(x,

t )

be the displacement of any point

x

along

the

neutral axis of the beam. The internal shea r, moment, and inertial forces acting on an

element dx in length will be as

shown

in figure l(b). Summing forces and moments on

this element yields the following

two

equilibrium equations:

( la

From elementary beam theory the moment-curvature relationship for the sign convention

depicted

in

figures

1 is

M(x,t) = E1

a2v(x,t)

The shearing fo rce Q(x,

t,

can be expressed in t er m s of the displacement variz-lle by us

ing equations ( lb ) and (IC). Thus

Q(X,t)

= - [.' a

3

v(x,

t)

av(x,t)

I

(2 )

ax

ax

Finally, from equations (la )

and (2)

the equation of motion in te rm s of th e displacement

variable can be written

as

Equation 3) represents the governing partial differential equation describing the motion

of the beam, and equations (IC)and (2) represent the moment and shear distribution

along the beam.

Solution of equation and boundary conditions.

-

The time variable in equations

(1)

to

3) can be eliminated by assuming

a

solution of the form

V(x,t) = v(x) sin ut

(4 a)

4


7/33

M(x,t) = m(x)

sin Wt

Substituting equations (4) into equations (1)to (3) yields

4

2

+

k2

- p4v(x) =

0

dx

4

dx2

d2v(x)

m(x)

=

E1-

x 2

q(x) = -E1

d3v(x)

2

dv(x)

+

k

1x

where

k2 =

P/EI

and

p4

= pd2/EI.

It

will be convenient at

this

point to put equa

tion

(5)

into nondimensional form by letting

x=

x/l,

v = v/l,

k =

k l ,

and

p = Pi?

Us-

ing these express ions allows equations 5 ) and (6) to become

-2-tt

T (x) + k v

-

m(x)

- -4-

(x) - p

V(X)

= 0

(7)

EI-tt

=

-V

1

The solution to equation (7), which can be readily verified by dir ect substitution, is

given by

v(z)

=

A

cosh

a I z

+ B

sinh a,F +

D

cos

a,:

+ E

sin

a2x

9)

where

A, B,

C, and

D

ar e constants and

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8/33

- -l = 7+ / p ) l 2+ p

(10)

C Y 2 =

-+

{ - y

1

+

The frequency equation can now be dete rmined from equation (9) once the boundary con

ditions

are

stipulated. The boundary conditions being considered

at

either end of the

beam are as follows:

Pinned

v =

0 and m = 0 or v f f = O

Clamped

-

v = O and

v ' = O

Free

-2-1

m

=

0 or = 0 and q

=

0 or

7'

+ k v

=

0

Guided

-

The various combinations of th ese boundary conditions at both ends of the beam a r e

listed

in

table

I.

If

the boundary conditions given by equations (11)ar e used

at

X = 0,

two of th e unknown constan ts in equation (9) can be evaluated, and the resu lting solutions

are for the beam f re e at E

= 0

@2 sin

a2 )

(12a)

,?

+

1

for the beam pinned at

x

= 0

v(z)

=

A

sinh al

+ E

s in

a2x

(12b)

6


9/33

for the beam clamped at X = 0

v(2)= A

(cosh CY,^

-

cos cy

2

3

B

alx -- sin cy

2

x

cy2

and fo r the beam guided at ?i = 0

Note that each of equations (12) contains two unknowns. When the boundary conditions at

-

x =

1 are

specified, these equations will yield

two

homogeneous, algebraic equations

in

these two unknowns. The determ inant of the coeff icien ts of the unknowns must be equal

to zero for

a

nontrivial solution to exist. Equating thi s determinant to zero then yields

the characteristic equation from which the frequency parameter

p

can be determined as

a

function of axial load P or axial load parameter

E.

To illustrate, consider the case

where the beam is

free

at

x

=

1.

Fro m equation ( l l c ) the boundary conditions

are

k (1) = 0 ? '(l)

+

-2-1v

(1)

=

0

(13)

From equations

(l2c)

and (13) the following two homogeneous, algebraic equations in

A

and

B

ar e obtained after some manipulation:

Yos

a l )+

Bb; s inh cyl

-

(144

3

cosh a l

-

o2 cos a l ) = 0

(

14b

1

Equating the determinant of the coefficients of

A

and B in equations (14) to zero, ex

panding, and simplifying will then yield the following cha rac ter isti c equation for the f ree-

fr ee beam:

-2-4

2p6(1 - cosh

CY^

cos a2)+ k

(k

+ 3z4) sinh cy1 sin a2 = 0

Similarly, for the other boundary conditions at x =

1

the following characteristic equa

tions

are

obtained: for the free-guided beam

3

CY

2

3

sinh a1 cos CY

2

+ a1 cosh cy1 sin CY^ = 0

(1%)

7


10/33

for the free-pinned beam

3

3

-a2 cosh al sin a2 + a1sinh al cos a2 = 0

for the clamped-free beam

-4 -2-2

2P

+ (2p4

+

E4)

cosh a1 cos

a2 -

p

k

sinh

a1

sin ai = 0

for the clamped-pinned beam

a1 cosh cy1 sin

a2

- a2 sinh a1 cos a2 = 0

for the clamped-guided beam

a1 sinh

cy1

cos

cy2

+

a2 cosh

cy1 sin

a2 = 0

for the clamped-clamped beam

-2 -2

2p

(1

- cosh a1 os a2)

-

k sinh al

sin a2

=

for the guided-guided and pinned-pinned bea ms

sin a2

=

0

and for the guided-pinned beam

cos cu2 = 0 (151)

Equations (15g) and (15i) yield the following expres sions for the frequency param eter:

for the pinned-pinned or guided-guided beam

(16)

and for the pinned-guided beam

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11/33

Equations (16) and (17) exhibit the main characterist ics

of

all the frequency equations

given by equations (15). Namely, a s

P (or

E) Increase, the frequencies decrease

until

P equals the Euler buckling load Per of the beam. At this point the lowest elastic fr e

quency becomes zero. For P greater

t h n

Per,

the fundamental frequency becomes

complex,

and

the corresponding mode shape is unstable (ref.

6,

p.

4 5 1

and ref.

8,

p.

302).

Thus, the maximum value for P of practical interes t is Per. The values of

Per and

Ecr

for beams with the different boundary conditions ar e tabulated in table

E.

To

determine the frequency parameter for the beams with other boundary conditions,

equations (15) can be solved numerically for

for any value of

le ss than the cor

-

responding kcr

.

Cantilevered Beam Under Axial Load W i t h Tip Mass

For

the case

of

a cantilevered beam with a concentrated mass a t the fr ee end,

two

types of axial loads

will

be considered. In he first case (fig.

2(a))

the

axial

load

will

be

taken as a constant in both magnitude and direction so that this case corresponds to the

uniform clamped-free beam case except that a concentrated mass is added to the free

end. In the second case (fig.

3 a))

the axial load is assumed to be directed

through

the

root or fixed end of the beam at all times. For this case the axial load

is

not constant

since it changes direction during the motion. This type of loading w i l l be referr ed to as

a directed axial load. For both cases the deflection is given by (eq. (12c) since the

boundary condition at the fixed end is the sam e as for a

uniform

cantilevered beam.

Also, the moment boundary condition at the fre e end is

the

same

for

both cases

and

is

given by

?'(I) =

The difference between the two cases will be in the shear boundary condition at the f re e

end. For the case of constant

axiai

load, the shear boundary conditions can be found

by

satisfging the equilibrium condition at the f re e end which is depicted in figure 2@).

Thus

From equations

(6)

and

(9)

th is boundary condition becomes

2

F '(I) -i- k v

1)t

qT P

4-

~ ( 1 )

0

(20)

9


12/33

where qT is the ratio of the tip ma ss AT to total beam mass, AB

=

p Q . From equa

tions (12c), (18), and 20) the following two homogeneous equations in A and

B

are

obtained :

A(

cosh al

+

cos

a 2 +

B 4 sinh al +32sin

1

{ g sinh

a1

p2

sin a2

+

qTD2(cosh a1 - cos

+ B{g

cosh cy1 +

al

cos

a

+

q

-

2(a2

inh

a l - al

sin a2)}

=4

2 T

Equating the determinant of the coefficients of A and B o zero, expanding and simplify

ing yields the following char act eri stic equation for

a

cantilever beam with a tip mass

under constant axial load.

- zp4

+

-2-2

k

sinh

a1

sin

a2

-

2B4

+

E4)

cosh

a1

cos

a2

a1cosh a1 sin

a2

- a2 sinh a1 cos a2) =

0

(22)

For the case of

a

cantilevered beam with a tip mass and a directed

axial

load, the con

dition a t the free end is shown in figure 3 @ ) . From this fi gur e the equilibrium condition

can be written as

(23)

Comparing this equation with equation

(20)

shows that the te rm 77TF4 -

ii

replaces

7Tp4.

Thus multiplying equation

2 2 )

by p and replacing qTp4 with (qTp4- E 2 yields

the char acteri stic equation

for

the beam with a directed

axial

load.

- 2 p 6

+- 4 2

k

sinh a1 sin

a2

- 2p+k4)cosh a1cos a2

+ (qT? -x2) 4

ai)(al cosh al sin a2 - a2 sinh a1 cos a2)= 0

Mode Shapes For Beams Under Preload

The mode shapes for a beam under preload can be determined

from

equations

(12)

and one other boundary condition. Fo r example, ass ume that the nth root

of

the

fre

quency equation

is

known and is given by Fn. Corresponding to there w i l l be an

10


13/33

cyln and

a

2n

which ar e obtained by substituting

on

into equations (10).

For

the case

of

a

fre e-f ree beam , the displacement in the nth mode

is

given by equation (12a)

as

a

2n

2n

For the free-fr ee beam ca se ei ther of equations (14) can be used to determine

a

relation

between An and Bn. U s i n g equation (14a) yields

B

n

= -An

F

cash oln - COS

CY

)

)

(26)

X

@ln

Substituting equations (26)

into

(25) and ar bi tr ar il y setting An

=

1

will

then yield the

desir ed mode shape

or

eigenfunction for the free-free beam.

In a

simil ar manner the

mode s hapes for the other boundary conditions ca n be determined.

These functions

are

summa rized in the following equations: for free -free and free-pinned beams

2

- @ln

C *X +-

Y

2

2n

11


14/33

- -

- -

-sin

for

a

free-guided beam

for

a

clamped-free beam

-

vn(x) =

(cash C Y ~ XCOS C Y ~ ~ X )

for clamped-pin and clamped-clamped beams

-

@ln

v,(x) =

cosh

C Y ~ X

~ ~

cos

C Y -

CY2n

C Y

(27d)

for

a

clamped-guide beam

for a

guided-guided beam

1 2


15/33

for a guided-pinned beam

Vn@ =

and fo r a pinned-pinned beam

It

can readi ly be shown that the mode shapes for the cantilevered beam s with a tip ma ss

ar e the

same

as those given by equation (27~).

RESULTS AND DISCUSSION

Uniform Beam Under Axial Load

To determ ine the effect of

axial

load on the natura l frequencies of a uniform beam,

the cha ract er ist ic equations (given by eqs. (15a) to (15g) were solved numerically by the

method of bisection (ref. 9). In this investigation the axial load rat io P/Pcr was varie d

between -1 and

1,

and the first th re e lowest frequencies were determined. The re su lts

for the fundamental ela stic body frequencies a re presented in figure 4 . The pinned-

pinned, guided-guided, and pinned-guided curves

in this

figure were developed from

equations (16) and (17). In

this

figure the curves for the fre e-f ree and clamped-clamped

beam s appear a s a single curve. Although the frequencies for these two cas es were not

identically equal over the ent ire range of P/ Pc r, the differences could not be

shown

for

the s cal e used in the figure. The same is true for the free-guided and clamped-guided

cases . The variation in frequency a s

a

function of axia l load for the first few lower fre

quencies for each uniform beam case is given in figu re 5. It can be noted that the va r

iation in the second and third frequency for all cases shown is almost linear for the

ran ge of P/P cr considered. These higher mode frequencie s

w i l l

go to ze ro when the

second and third buckling loads ar e reached. However, these portions of the curv es a r e

of no pract ical i nt eres t unles s the beam is prevented from buckling in its fundamental

mode.

To show the effect of axial loading on the mode shapes of the beam, equations (27)

were used to determ ine the modes for

P/Pcr =

-1.0, 0, and 0.

8.

These calculations

were normalized, and the results ar e shown in figures 6 to 12 for the first thr ee modes.

These fig ure s indicate that the

effect

of axial load

is

greatest on the fundamental mode

and that the effect decr eas es rapidly as the mode number inc reases . The most

13


16/33

pronounced effect occurs for the free-pinned beam i n its fundamental mode (fig.

8(a)).

Note, however, that all the cases exhibit this effect as P/PCr approaches 1.

Cantilevered Beam With Tip M a s s

For

a

cantilevered beam with a tip m as s equation (22) was solved for the case of a

constant axial load and equation (24) for an axial load direc ted through the root of the

beam. The method of solution of th es e equations was the same as that used for the uni-

for m beams. The effects of both tip mas s rat io, qT = JT/.AB, and axial load ratio

P/PCr on the fundamental frequencies wer e determined, and the resu lts a r e shown in

figures 13 to 16. Figure 13 shows the variation in the frequency par am ete rs g;

i

as a function of ti p mas s rat io , qT, for values of P/PCr between -1 and 0. 8. Note that

8

s equals w1 d W E I , so that the frequency param eter for this case

is related to the total ma ss of the system and not just the beam mass. Figure 14 shows

the sa me data plotted as a function of axial load ratio with the tip m as s ratio as a param

eter. Figures 15 and 16 are

a

similar set of graphs for the cantilever with

a

directed

axial load.

Lewis Research Center,

National Aeronautics and Space Administration,

Cleveland, Ohio, August 5, 1975,

506 -22.

REFERENCES

1. Coyner, J.

V.

Jr . ;

and Ross, R. G.

Jr.

:

Pa rame tri c Study of the Performance

Characte rist ics and Weight Variations of Large-Area Roll- Up Solar Arrays.

(JPL

TR 32-1502, Jet Propulsion Lab. ; NAN-100. ), NASA CR-115821, 1970.

2. Greble, F. C.

:

Solar Arr ays for the

Next

Generation of Communication Satellites.

J. Brit. Interplanetary SOC. , vol. 26, no.

8,

Aug. 1973, pp. 449-465.

3. Wolff, George: Oriented Flexible Rolled-up Solar Ar ray .

AIAA Paper

70-738,

Apr. 1970.

4. Hughes,

P.

C. : Attitude Dynamics of a Three-Axis Stabilized Satellite with

a

Large

Solar Array . AIAA Pape r

72-857, Aug. 1972.

5. Hurty, Walter C.

;

and Rubenstein, M .

F.

:

Dynamics of Structures.

Prentice-Hall,

Inc., 1964.

14


17/33

6. Meirovitch, L eona rd: Analytical Methods in Vibrat ions. The MacMillan Co ., 1967.

7. Bisplinghoff, Raymond L.

;

Ashley, Holt; and Hoffman, Robert L.

:

Aeroelasticity.

Addison-Wesley Publishing Co.

,

1955.

8.

Tong, Kin N.

:

Theory of Mechanical Vibrations.

John

Wiley

& Sons.,

Inc., 1960.

9. Ralston, Anthony: A

First

Co urs e in Numerical Analysis. McGraw-Hill Book

Co.,

Inc., 1965.

15


18/33

TABLE

I.

- BOUNDARY CONDITIONS FOR UNIFORM BEAM UNDER CONSTANT AXIAL LOAD

~

.

. ~ .

Free-free

p rZ

P

. (l) = 0

-

0

1 T'l'(1)

+ - E 2 7

(1)

=

I

Fr e e-guided

p P

7' (0) = ? ( l ) = 0

-

v '(0) +k (O)=

0

v '(1) + k% (l ) = c

Free-pinned

v (0) =

0

v(1) = 0

-

v ' (0) +k 27 (0 ) = 0 v (1) = 0

Guided-guided

-

ki de d- pinned

P P

v'(0) =

0

G(1) =

0

;y yo)

+ i 2 7 ( 0 )

=

0 v (1)

= 0

Clamped-free

T(0) =

0

v (1) = 0

-

v'(0) = 0

v '(1) +25.,(1)

=

0

~

Pinned-pinned

P t-

J(0) = 0 T(1) = 0

V (0)

= 0

F' (1 ) = 0

__

Xamped- pinned

G 0)

=

0 v(1)

= 0

7 ( 0 )

=

0

v (1) = 0

~

.

_

.

.

.

-~

-

Xamped-guided J(0)

=

0 v'(1)

= 0

v (0) = 0 v ' (1 ) +k 27( 1 ) =

0

. -

~ ~

-

Xamped- clamped

J 0)= 0

v(1) = 0

v'(0)

=

0

v'(1) = 0

16


19/33

--

--

TABLE

11.

- BUCKLING CHARACTERISTICS OF SIM PLE BEAMS

Boundary condition

Characteristic

Eigenvalue

Critical load,

Buckling mode

-

equation

kn

'cr

-

2 E1

ree-free

sin kn

= 0

nn

n

l 2

-

2n - 1)

n 2

EI

ree-

guided

C O S

k

=

0

r

n

2

l 2

-

Free-pinned

sin k

= 0

n n

n

E1

(Same

as

free-free)

l 2

-

2

E1

ide d- guided

sin k

=

0

n n

n

-

l 2

v1

=

cos n Tz

-

-

hide d- pinned

COS

k

n

= 0

-n

2n -

1)

r 2

E1

_

2

L2

v1 = cos n(x/2)

.

-

(2n

- 1)

n 2 E1

amped-free

COS

k

= 0

-n

2

l 2

-

'inned- pinned

s i n k =

0

nn

n

E1

(Same

as free-free)

l 2

..

Xamped-pinned

tan

Ln = i

4 . 4 9 3

2

E1

(0. 6991)2

-

2

E1

lamped-guided

s i n k

= 0

nn

n

n

l 2

-

k

lamped- clamped

sin

=

o

2nn

4n

2

E1

Per

2

l 2

v1

=

cos 27rz

-

1

17


20/33

Ill

t v

...

.,

E l e m e n t

( a ) C o o r d i na t e s y s t e m f o r u n i f o r m v i b r a t i n g b e a m

( b ) S i g n c o n v e n t i o n f o r f o r ce s a n d m o m e n t

u n d e r c o n s t a n t a x i a l lo ad .

a c t i n g o n a r b i t r a r y e le m e n t.

F i g u r e 1 - V i b r a t i n g b ea m w i t h a r b i t r a r y b o u n d a r y c o nd i t io n s .

v

fli

( a ) C o o r d i na t e s y s t e m f o r c a n t i l e v e r e d b e am w i t h

( b ) F o rc e s a c t i n g a t f r e e e n d o f c a n t i l e v e r e d b e a m

t i p m a s s u n d e r c o n s t a n t a x i a l l o ad .

u n d e r c o n s t a n t a x i a l l o ad .

F i g u r e 2. - C a n t i l e v e r e d be a m w i t h t i p m a s s u n d e r c o n s t a n t a x i a l l oa d .

-

AV

-

-

X

( a) C o o r d in a t e s y s t e m f o r c a n t i l e v e r e d b e am w i t h

( b ) F o r ce s a c t i n g a t f r e e e n d

of

c a n t i l e v e r e d b e a m

t i p m a s s u n d e r d i r e c t e d a x ia l l o a d.

u n d e r d i r e c t e d a x i a l l oa d .

F i g u r e 3. - C a n t i l e v e r e d b ea m w i t h t i p m a s s u n d e r d i r e c t e d a x ia l l oa d .


21/33

--

-

---

----

-----

L

- - - - -

----

------- --------

---

----

32-

F r e e - f r e e o r c l a m p e d- c la m p e d

F r e e - p i n n e d

- e

C l a m p e d - p i n n e d

24

N

1 -

1

aJ

-

E

16

E

P

Y

-

-

4-

-

---

-

-_

--_

P i n n e d - p i n n e d o r g u id e d -g u id e d

F r e e - g u id e d o r c l a m p e d - g u i d e d

C l a m p e d - f r e e

G u i d e d - p i n n e d

-

-1.0 -.8 -.6 -.4

-.2 0 .2 . 4 . 6

.8

1.0

Ax ial lo ad rat io . PIP,

F i g u r e

4 -

F u n d a m e n t a l f r e q u e n c y a s f u n c t i o n o f a x i a l l o ad fo r u n i f o r m b e a m w i t h v a r i o u s b o u n d a r y c o n d it i o n s.

19

i


22/33

-

02

-

6:

2 2 5

90

i

P

40

20

10

(a)

~~r pinned-pinned and guided-guided b e a m s .

140

6;

120

100

80

6

-

60

20

0

15.0

12.5

Lo.ol

.5

h

2.5

1

I I

I I I

U

(b)

For guided- pinned b e a m .

-2

80 P3

0

10-

B

0

-1.0 - 8

-,6

-.4

-.2 0

.2 .4

.6 . 8 1.0

-1.0

-.8 -.6

- .4 - .2

Axial load

ratio,

PIP,,

(di For

free-guided

b e a m .

(c)For free-free beam.

Figure 5.

-

Frequency as function

of

axial load

20

-_____......

.

,

.

.,

.

,...

.-

I IU I I.

l l


23/33

L

m

...

.__. _..

1 2 o r

(el For f ree-pinned beam

80

160

70

140

60

-

120

B;;

50

c

'

100

m

40

80 -

B

a

30

2

60

I

20-

*

40

10

20

B

-

0

I 1 I I 1 1 7 - m

gl For clamped-gu ided beam.

( f l For c lampe d-pinned beam.

70

140-

6;

60

20

50

100

0

80

60

30

6;

40 20

F

20 10

0 1 1 I

0 I

I

I I

I

I

(i )

For clamped-free beam.

( h l For clamped-clamped beam.

F i g u r e

5

-

Concluded.

21


24/33

--

-.a

I I - L I _ L L - I _ L _ l

m

E

(a1 First mode.

-1. 0 - . 8 k - ‘ ” 1 1 1

I

1

I I

I

0 .1 . 2 . 3

. 4 . 5 . 6

. 7 .8

9 1.0

0 .1

2

. 3 .4 . 5 . 6 .7 . 8 . 9 1.0

Nond imens iona l beam coord ina te , x l l

( b l Second mode.

(c)

Thi rd mode.

F igure 6 - Effect of axial load on mode shape of f ree-free beam.

22


25/33

::h

xial loadatio,

PI Pc

r

-_

8

I

l l l l l l l l u

(at Fi rst mode.

-.

4

-. 6

- x

I I I I I I

1 . 8 1 I I

I I

0

. 1 . 2 . 3

. 4 .5 .6

.7

.8 . 9 1.0 0

. 1

. 2

. 3

. 4 5 . 6

7

Nond imens iona l beam coord ina te , x l l

bl Secon d mode.

( c l Th i rd mode

F i g u r e

7. -

Effect of ax ia l l oad on mode shape

o f

free-guided beam.

23


26/33

-_

2

-.

4

-.6

L - 1

-.8

(a) Fi rst mode.

- . a k l I I

I

L I

I

1-1

0 .1 . 2 . 3 . 4

.5 .6

.7 .8 9 1.0

Nondimensional beam coordinate, XU

b)

Second mode. (c )T hi rd mode

F i g u r e 8. - Effect of axial load on mode shape of f ree-pinned beam.

24


27/33

c

1.0

9-

.8

.7

. 6

.5

.4

. 3

. 2 -

-

Axial load

///

m

. t l

I I I

I I

I I

\

E

la) Fi rs t mode.

-.

8

-1.0

0

coordinate,

XU

(b l Second mode. (c ) Th i rd mode.

F i g u r e

9. -

Effect of axial load on mode shape of clamped -pinned beam.

25


28/33

Axial load

L k

m

m (a) First mode.

( b l Second mode.

( c l Third mode

Figure 10.

-

Effect of axial load on mode

shape

of clamped-guided b e a m .

26


29/33

beam

(b l Second mode .

(c)

Th i r d mo d e .

Figure

11. - Effect o f

a x i a l l o a d o n

mode

shape

of

c lamped-c lamped beam.

27


30/33

( b l

Second m o de. W T h i r d m o de .

F i g u r e 12

- E f f e c t

o f ax ia l load

on

mode shape of c lamped- f ree beam.

28


31/33

5.

c

I\

Rat io

o

b

of

t i p m a s s

~ . .

A x i a l l oa d r a t io , P I P c r

Figure 14. - F r e q u e n c y a s f u n c t i o n o f c a n t i l e v e r b e a m w i t h t i p m a s s .

4 .4 8L

Ax ia l l oad

ra t i o ,

P I P c

r

-1

-.6

-.4

-.2

0

. 2

.4

. 6

L \

I

0

1 2

3 4 5 6 7

8

R at io o f t i p mas s to beam mas s ,

q

Figure 13.

-

F u n d a m e n t a l f r e q u e n c y o f c a n t i l e v e r e d b e am w i t h t i p m a s s

u n d e r c o n s t a n t a x i a l l o ad .

4 . 5


32/33

N-

L

R a t i o o f t i p m a s s

4 . 0

4 .5

to

beam mass ,

VT

Ax i a l l oad

r a t i o ,

L

a

1 Q

E

E

m

LL

I .

t

. 8 ,

0

I

Rat i o

of

t i p m a s s t o b e a m m a s s ,

VT

F i gu re 15.

-

F u n d a m e n t a l f r e q u e n c y o f c a n t i l e v e r e d b e am w i t h t i p m a s s

a n d a x i a l l o a d d i r e c te d t h r o u g h r o o t.

M

I

03

p

w

w


33/33

-

L A E R O N A U T IC S 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.

2 0 5 4 6

P O S T A G E A N D F E E S P A I D

N A T I O N A L A E R ON A U T IC S A N D

O F F I C I A L B U S I N E SS S P A C E A D M I N I S T R A T I O N

4 5 1

PENALTY FOR PRIVATE US E 3 0 0 S P E C I A L F OU RT H- CL A S S R A T E

U S M A I L

B O O K

095 C U I

Cl

IJ

D

7 5 1 1 2 8

S009g3DS

DEPT OF

THE

AIR FORCE

B F

WEAPONS

L A 3 O R A T O R Y

RTTN: T E C H N I C A L LIBRARY

(SUL)

KIRTLAWD A P E N M 87117

.

If

Unde l ive r ab le Se c t ion

158

PO ST M AST E R

:

Postal

M annnl)

Do

Not Re tur n

.

f

‘“The aeronautical and space uctiu ities of t he Un it ed States shall

be

conducted

so

as to contribute

. . .

o the expansion of hum an Rnowl

edge of phenomena

in

the atmosphere and space. Th e Administration

shall provide for the wzdest practicable and appropriate disseminution

of information concerning

i t s

activities and the results thereof.”

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Vibration beams