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Free vibration of statically compressed clamped beams on nonlinear elastic foundation
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Vo1.12(5),303-308, 1985. Printed in the USA. Copyright (e) 1985 Pergamon Press Ltd.
FREE VIBRATION OF STATICALLY COMPRESSED CLAMPED BEAMS ON NONLINEAR
ELASTIC FOUNDATION V. Birman School o f Naval A r c h i t e c t u r e and Marine Engineering U n i v e r s i t y o f New Orleans P. O. Box 1098 New Orleans, La. 70148
(Rec~ved 8 July 1985; accepted for print 11 October 1985)
Introduction
The problem of free n o n l i n e a r v i b r a t i o n of simply supported beams was considered by Woinowsky-Krieger [ I ] and Burgreen [ 2 ] . They determined the frequencies of the beams w i t h a p r e s c r i b e d s i n u s o i d a l v i b r a t i o n a l mode shape e x a c t l y in terms of e l l i p t i c f u n c t i o n s . These r e s u l t s were confirmed e x p e r i m e n t a l l y by Ray and Bert [3] who studied v i b r a t i o n s of a slender beam w i t h large v i b r a t i o n h a l f - t h r o w s (up to 16 beam t h i c k n e s s ) . The s o l u t i o n was extended to the case of a beam w i t h i n i t i a l imperfect i o n s in [ 4 ] . Free v i b r a t i o n s of simply supported beams on n o n l i n e a r cubic e l a s t i c f o u n d a t i o n were studied by Massalas using a p e r t u r b a t i o n method [ 5 ] . Evenson was probably the f i r s t to analyze free n o n l i n e a r v i b r a t i o n of clamped beams [ 6 ] . He used a p e r t u r b a t i o n method to o b t a i n an approximate s o l u t i o n by assumption t h a t the motion can be represented by the s i n g l e mode shape defined v i a l i n e a r t h e o r y . Notably, some of the authors claim good agreement o f t h e o r e t i c a l r e s u l t s obtained by employment o f the fundamental mode shape based upon l i n e a r theory [3,7]. However, there is an i n d i c a t i o n t h a t such s i n g l e mode approximation may be i n c o r r e c t i f the amplitude of motion is large [8,9].
303
304
V. BIRMAN
In this paper we consider free nonlinear vibration of a clampud beam undergoing axial loading and resting on the nonlinear (cubic) elastic foundation. The amplitudes of vibration are assumed to remain in the range where the single mode analysis is valid. The effect of an axial compressive force on the frequencies and the influence of the elastic foundation are shown in numerical examples.
Analysis
Consider the beam clamped at the ends x = 0 and x = ~ and supported by a nonlinear cubic elastic foundation. The additional nonlinear effect of this problem is due to the nonlinear stretching of the elastic curve. The equation of motion of such beam subjected to the action of an axial compressive force P is: o
m~ + El ytV +
o-
2~
[
(Y')
Y
+ KIY - K3Y3 = 0
o !
(...)
•
: a ( .~. x. )
where y(x,t)
(.
'
represented
by
y(x,t)
(t)
which satisfies The substitution
3
m
and t is time.
sin 2
4
f3
El is the bending stifness,
A is
K
and K 3 are the I The lateral displacement is
~x
(2)
the boundary conditions at the both ends. of (2) into (I) and Galerkin procedure yield:
.. 4 ~ f (t) + 2 El ( ~ )
+ ~ EA C ¥~)
(I)
area, m is the mass per unit length,
foundation constants,
f
a ( .~. t. )
is the lateral displacement,
the cross-sectional
=
"" ) _
1 f (t) - -~
Ct) + _~ KI f (t)
2 (~ )
Po f(t) +
105 K3 f3 (t) = 0
(3)
CLAMPED
BEAMS ON NONLINEAR
T h i s equation can be c o n v e n i e n t l y
d2F dT 2
+
~2 F -
represented
FOUNDATIONS
305
in the nondimensional
form:
~F 3 = 0
(4)
where F = f (~)I~
(5)
~t = T
(6)
i.e.
T
is a nondimensional
2
1 + n! -~ - 2
=
=
time parameter. P =
2__
~I~o
~o
(~ 4 ~ )
16
-3
p
P
cr
(8)
m
=4 (Tt,
cr
AZ2 "-T'-
"~ )
m~
2
free nonlinear vibrations system w i t h o u t e l a s t i c (4)
K3~2 2 m~
n3 -
functions. yield:
(lo)
o
frequency ~ is the r a t i o
to the n a t u r a l
foundation'and
is a D u f f i n g
transformation
(9)
EI
KI nl -
Note t h a t the nondimensional
of elliptic
(7)
El m
o
Equation
n3
2
o
I M = " 16
~ - 2
P
P-
M+
o f the frequency o f
frequency o f the l i n e a r i z e d
compressive f o r c e .
type e~quation which can be i n t e g r a t e d
Multiplication
of
(4) by dF, i n t e g r a t i o n
in terms and some
306
V. BIRMAN
d--~- =
{
I
- 2
where F
\/
~F2max
( 1 _~2) (I - k2~ 2)
is the nondimensional
(11)
vibration half-throw and
max
F2
I
P
k2=
max
~2 _ ! 2
= F/F
uF 2 max
The period of the solution of (11) can be determined consideration
in
4 2
where K(k)
-
following
the
[10] as:
T =
V
(12)
max
K(k)
(13)
I ~ ~F2ma x
is a complete elliptic
integral of the first kind whose values
are tabulated. The nondimensional
period of vibrations of the linearized beam without
elastic foundation
and compressive
T
=
force is
(14)
2 ~
0
Therefore 0
-
_
T
11"
2"
1+n
1
I - ~ - 2
(- M +
.1~
n3)
F2
max
1
(15)
It can be noted that the condition of the existence of free nonlinear vibration
is
i 35 I + n I - P - ~ ( - M + ~-~
2 0 n3) F m a x ~
(16)
CLAMPED BEAMS ON NONLINEAR FOUNDATIONS
307
Therefore the beam vibrating with large amplitudes can be unstable even in the absence of the compressive force. Results and Discussion The nondimensional frequency squared versus the load r a t i o ~ is shown in Fig. 1 f o r the clamped beam unsupportedllbY e l a s t i c w i t h M = 50.
For the h a l f throws
IFmaxl ~__ O.O1 the frequency p r a c t i c a l l y Q
I
c o i n c i d e s w i t h t h a t given by the l i n e a r t h e o r y , The i n f l u e n c e of n o n l i n e a r i t y It
f o u n d a t i o n (n 1 = n 3 = O)
i.e.,
~
~/1 - ~.
becomes e s s e n t i a l when the throws increase.
is c l e a r from Fig. 1 t h a t the frequency amplitude r e l a t i o n s h i p of the
beam is hardening, i . e . , vibration
half-throws.
the l a r g e r f r e q u e n c i e s correspond to the l a r g e r The hardening c h a r a c t e r i s t i c s
have been a l s o observed
from hinged beams. The i n f l u e n c e o f the l i n e a r e l a s t i c
f o u n d a t i o n on the f r e q u e n c i e s is shown in
Fig. 2 for the beam with M = 50, ~ = O, n3 = O,
IFmaxl
= O.I.
As i t
could be expected the foundation increasing the s t i f f n e s s of the system increases the frequencies.
Fig. 3 shows the effect of nonlinear cubic I
I
e l a s t i c foundation on the frequencies (M = 50, ~ = O, nI = 0.5, I Fmaxl
= 0.1)
The softening n o n l i n e a r i t y of the e l a s t i c foundations is shown to decrease the frequencies of free nonlinear v i b r a t i o n s . References I.
S. Woinowsky-Krieger, The Effect of an Axial Force on the Vibration of Hinged Bars. J. AppI. Mech. (ASME), 17, p. 35 (1950).
2.
D. Burgreen, Free Vibrations of Pin-Ended Column with Constant Distance Between Pin-Ends. J. App1. Mech. (ASME), 18, p. 135 (1951).
.
J. D. Ray and C. W. Bert, Nonlinear Vibrations of a Beam with Pinned Ends. J. Eng. Industry, (ASME), 91, p. 997 ( 1 9 6 9 ) .
.
I. E i i s h a k o f f , V. Birman and J. Singer, I n f l u e n c e o f I n i t i a l Imperfect i o n s on N o n l i n e a r Free V i b r a t i o n of E l a s t i c Bars. To appear in Acta Mechanica.
.
C. Massalas, Fundamental Frequency of Vibration of a Beam on a Non-Linear E]astic Foundation. J. Sound & Vib., 54, p. 613 ( 1 9 7 7 ) .
.
D. A. Evenson, N o n l i n e a r V i b r a t i o n s of Beams w i t h Various Boundary C o n d i t i o n s . AIAA J., 6, p. 370 (1968).
.
W. Y. Tseng and J. Dugundji, Nonlinear Vibrations of a Beam under Harmonic Excitation. J. Appl. Mech. (ASME), 37, p. 292 ( 1 9 7 0 ) .
308
V. BIRMAN J. A. Bennett and J. G. Eisley, A Multiple Degree-of-Freedom Approach to Nonlinear Beam Vibrations. AIAA J., 8, p. 734 (1970).
.
M. M. Bennouna and R. G. White, The Effects of Large Vibration Amplitudes on the Fundamental Mode Shape of a Clamped-CHamped Uniform Beam. J. Sound & Vib., 96, p. 309 (1984).
.
10.
H. Jeffreys and B. Swirles, Methods of Mathematical Physics. 3rd Ed., Cambridge University Press, Cambridge, Oh. 25 (1980).
2
co I 1.0
I: 2:
0 FIG. I.
0.5
[Fmaxl
= O.1
Linear solution
1.0
Nondimensional natural frequency squared versus the nondimensional load
-,-,2
CO 3.0
~
°Uo's 21t 15
2
0
50
100 150 Na
nI FIG. 2. The influence of the linear foundation on the frequencies
': lFmaxl =0~ Z"
Lineer 5olution
FIG. 3. The influence of the nonlinear elastic foundation on the frequencies
Vo1.12(5),303-308, 1985. Printed in the USA. Copyright (e) 1985 Pergamon Press Ltd.
FREE VIBRATION OF STATICALLY COMPRESSED CLAMPED BEAMS ON NONLINEAR
ELASTIC FOUNDATION V. Birman School o f Naval A r c h i t e c t u r e and Marine Engineering U n i v e r s i t y o f New Orleans P. O. Box 1098 New Orleans, La. 70148
(Rec~ved 8 July 1985; accepted for print 11 October 1985)
Introduction
The problem of free n o n l i n e a r v i b r a t i o n of simply supported beams was considered by Woinowsky-Krieger [ I ] and Burgreen [ 2 ] . They determined the frequencies of the beams w i t h a p r e s c r i b e d s i n u s o i d a l v i b r a t i o n a l mode shape e x a c t l y in terms of e l l i p t i c f u n c t i o n s . These r e s u l t s were confirmed e x p e r i m e n t a l l y by Ray and Bert [3] who studied v i b r a t i o n s of a slender beam w i t h large v i b r a t i o n h a l f - t h r o w s (up to 16 beam t h i c k n e s s ) . The s o l u t i o n was extended to the case of a beam w i t h i n i t i a l imperfect i o n s in [ 4 ] . Free v i b r a t i o n s of simply supported beams on n o n l i n e a r cubic e l a s t i c f o u n d a t i o n were studied by Massalas using a p e r t u r b a t i o n method [ 5 ] . Evenson was probably the f i r s t to analyze free n o n l i n e a r v i b r a t i o n of clamped beams [ 6 ] . He used a p e r t u r b a t i o n method to o b t a i n an approximate s o l u t i o n by assumption t h a t the motion can be represented by the s i n g l e mode shape defined v i a l i n e a r t h e o r y . Notably, some of the authors claim good agreement o f t h e o r e t i c a l r e s u l t s obtained by employment o f the fundamental mode shape based upon l i n e a r theory [3,7]. However, there is an i n d i c a t i o n t h a t such s i n g l e mode approximation may be i n c o r r e c t i f the amplitude of motion is large [8,9].
303
304
V. BIRMAN
In this paper we consider free nonlinear vibration of a clampud beam undergoing axial loading and resting on the nonlinear (cubic) elastic foundation. The amplitudes of vibration are assumed to remain in the range where the single mode analysis is valid. The effect of an axial compressive force on the frequencies and the influence of the elastic foundation are shown in numerical examples.
Analysis
Consider the beam clamped at the ends x = 0 and x = ~ and supported by a nonlinear cubic elastic foundation. The additional nonlinear effect of this problem is due to the nonlinear stretching of the elastic curve. The equation of motion of such beam subjected to the action of an axial compressive force P is: o
m~ + El ytV +
o-
2~
[
(Y')
Y
+ KIY - K3Y3 = 0
o !
(...)
•
: a ( .~. x. )
where y(x,t)
(.
'
represented
by
y(x,t)
(t)
which satisfies The substitution
3
m
and t is time.
sin 2
4
f3
El is the bending stifness,
A is
K
and K 3 are the I The lateral displacement is
~x
(2)
the boundary conditions at the both ends. of (2) into (I) and Galerkin procedure yield:
.. 4 ~ f (t) + 2 El ( ~ )
+ ~ EA C ¥~)
(I)
area, m is the mass per unit length,
foundation constants,
f
a ( .~. t. )
is the lateral displacement,
the cross-sectional
=
"" ) _
1 f (t) - -~
Ct) + _~ KI f (t)
2 (~ )
Po f(t) +
105 K3 f3 (t) = 0
(3)
CLAMPED
BEAMS ON NONLINEAR
T h i s equation can be c o n v e n i e n t l y
d2F dT 2
+
~2 F -
represented
FOUNDATIONS
305
in the nondimensional
form:
~F 3 = 0
(4)
where F = f (~)I~
(5)
~t = T
(6)
i.e.
T
is a nondimensional
2
1 + n! -~ - 2
=
=
time parameter. P =
2__
~I~o
~o
(~ 4 ~ )
16
-3
p
P
cr
(8)
m
=4 (Tt,
cr
AZ2 "-T'-
"~ )
m~
2
free nonlinear vibrations system w i t h o u t e l a s t i c (4)
K3~2 2 m~
n3 -
functions. yield:
(lo)
o
frequency ~ is the r a t i o
to the n a t u r a l
foundation'and
is a D u f f i n g
transformation
(9)
EI
KI nl -
Note t h a t the nondimensional
of elliptic
(7)
El m
o
Equation
n3
2
o
I M = " 16
~ - 2
P
P-
M+
o f the frequency o f
frequency o f the l i n e a r i z e d
compressive f o r c e .
type e~quation which can be i n t e g r a t e d
Multiplication
of
(4) by dF, i n t e g r a t i o n
in terms and some
306
V. BIRMAN
d--~- =
{
I
- 2
where F
\/
~F2max
( 1 _~2) (I - k2~ 2)
is the nondimensional
(11)
vibration half-throw and
max
F2
I
P
k2=
max
~2 _ ! 2
= F/F
uF 2 max
The period of the solution of (11) can be determined consideration
in
4 2
where K(k)
-
following
the
[10] as:
T =
V
(12)
max
K(k)
(13)
I ~ ~F2ma x
is a complete elliptic
integral of the first kind whose values
are tabulated. The nondimensional
period of vibrations of the linearized beam without
elastic foundation
and compressive
T
=
force is
(14)
2 ~
0
Therefore 0
-
_
T
11"
2"
1+n
1
I - ~ - 2
(- M +
.1~
n3)
F2
max
1
(15)
It can be noted that the condition of the existence of free nonlinear vibration
is
i 35 I + n I - P - ~ ( - M + ~-~
2 0 n3) F m a x ~
(16)
CLAMPED BEAMS ON NONLINEAR FOUNDATIONS
307
Therefore the beam vibrating with large amplitudes can be unstable even in the absence of the compressive force. Results and Discussion The nondimensional frequency squared versus the load r a t i o ~ is shown in Fig. 1 f o r the clamped beam unsupportedllbY e l a s t i c w i t h M = 50.
For the h a l f throws
IFmaxl ~__ O.O1 the frequency p r a c t i c a l l y Q
I
c o i n c i d e s w i t h t h a t given by the l i n e a r t h e o r y , The i n f l u e n c e of n o n l i n e a r i t y It
f o u n d a t i o n (n 1 = n 3 = O)
i.e.,
~
~/1 - ~.
becomes e s s e n t i a l when the throws increase.
is c l e a r from Fig. 1 t h a t the frequency amplitude r e l a t i o n s h i p of the
beam is hardening, i . e . , vibration
half-throws.
the l a r g e r f r e q u e n c i e s correspond to the l a r g e r The hardening c h a r a c t e r i s t i c s
have been a l s o observed
from hinged beams. The i n f l u e n c e o f the l i n e a r e l a s t i c
f o u n d a t i o n on the f r e q u e n c i e s is shown in
Fig. 2 for the beam with M = 50, ~ = O, n3 = O,
IFmaxl
= O.I.
As i t
could be expected the foundation increasing the s t i f f n e s s of the system increases the frequencies.
Fig. 3 shows the effect of nonlinear cubic I
I
e l a s t i c foundation on the frequencies (M = 50, ~ = O, nI = 0.5, I Fmaxl
= 0.1)
The softening n o n l i n e a r i t y of the e l a s t i c foundations is shown to decrease the frequencies of free nonlinear v i b r a t i o n s . References I.
S. Woinowsky-Krieger, The Effect of an Axial Force on the Vibration of Hinged Bars. J. AppI. Mech. (ASME), 17, p. 35 (1950).
2.
D. Burgreen, Free Vibrations of Pin-Ended Column with Constant Distance Between Pin-Ends. J. App1. Mech. (ASME), 18, p. 135 (1951).
.
J. D. Ray and C. W. Bert, Nonlinear Vibrations of a Beam with Pinned Ends. J. Eng. Industry, (ASME), 91, p. 997 ( 1 9 6 9 ) .
.
I. E i i s h a k o f f , V. Birman and J. Singer, I n f l u e n c e o f I n i t i a l Imperfect i o n s on N o n l i n e a r Free V i b r a t i o n of E l a s t i c Bars. To appear in Acta Mechanica.
.
C. Massalas, Fundamental Frequency of Vibration of a Beam on a Non-Linear E]astic Foundation. J. Sound & Vib., 54, p. 613 ( 1 9 7 7 ) .
.
D. A. Evenson, N o n l i n e a r V i b r a t i o n s of Beams w i t h Various Boundary C o n d i t i o n s . AIAA J., 6, p. 370 (1968).
.
W. Y. Tseng and J. Dugundji, Nonlinear Vibrations of a Beam under Harmonic Excitation. J. Appl. Mech. (ASME), 37, p. 292 ( 1 9 7 0 ) .
308
V. BIRMAN J. A. Bennett and J. G. Eisley, A Multiple Degree-of-Freedom Approach to Nonlinear Beam Vibrations. AIAA J., 8, p. 734 (1970).
.
M. M. Bennouna and R. G. White, The Effects of Large Vibration Amplitudes on the Fundamental Mode Shape of a Clamped-CHamped Uniform Beam. J. Sound & Vib., 96, p. 309 (1984).
.
10.
H. Jeffreys and B. Swirles, Methods of Mathematical Physics. 3rd Ed., Cambridge University Press, Cambridge, Oh. 25 (1980).
2
co I 1.0
I: 2:
0 FIG. I.
0.5
[Fmaxl
= O.1
Linear solution
1.0
Nondimensional natural frequency squared versus the nondimensional load
-,-,2
CO 3.0
~
°Uo's 21t 15
2
0
50
100 150 Na
nI FIG. 2. The influence of the linear foundation on the frequencies
': lFmaxl =0~ Z"
Lineer 5olution
FIG. 3. The influence of the nonlinear elastic foundation on the frequencies