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R E T U R N TO
AFWL TECHNICAL LIBRAR
KIRTLAND
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
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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
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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
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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.
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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)
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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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- -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)
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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%)
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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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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)
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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
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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
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- -
- -
-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
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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
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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.
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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
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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
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--
--
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
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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 .
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--
-
---
----
-----
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
-
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-
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
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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
-
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--
-.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
-
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::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
-
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-_
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
-
8/20/2019 Vibration beams
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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
-
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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
-
8/20/2019 Vibration beams
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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
-
8/20/2019 Vibration beams
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( 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
-
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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
-
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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
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-
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.”
-NATIONAL AERONAUTICSN D SPACEACT OF 1958
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