vibrations of double-span uniform beams subject to an axial force
TRANSCRIPT
Applied Acoustics 16 (1983) 95-104
Vibrations of Double-span Uniform Beams Subject to an Axial Force
P. A. A. Laura,* G. Shnchez Sarmiento** & Alicia N. Bergmann**
* Institute of Applied Mechanics, 8111 Puerto Belgrano Naval Base (Argentina)
** ENACE, Av. Alem 712, (1001) Buenos Aires (Argentina)
(Received: 17 February, 1982)
S U M M A R Y
This study deals with the determination of natural frequencies of transverse vibrations of simply supported, clamped, and clamped-simply supported beams with an intermediate support subject to an axial force.
Results are presented in graphical fashion as a function of the governing geometric and mechanical parameter.
I N T R O D U C T I O N
Transverse vibrations of beams with intermediate supports have been studied by several authors (for instance references 1 and 2). Apparently, previous studies on this important, practical problem have not con- sidered the effect of axial forces.
This paper presents values of natural frequency coefficients of double- span uniform beams subject to axial forces for the following com- binations of boundary conditions (Fig. 1):
Simply supported Case (a) Clamped Case (b) Clamped-simply supported Case (c)
as a function of L1/L and SL2 /EI (Fig. 1). The eigenvalues are determined using classical beam theory.
95 Applied Acoustics 0003-682X/83/0016-0095/$03.00 © Applied Science Publishers Ltd, England, 1983. Printed in Great Britain
96 P. A. A. Laura, G. Sanchez Sarrniento, Alicia N. Bergmann
Fig. 1.
s
$
/77;"/7
s
x 2 q l - - -
L z
/
L ~,
Z~
(a)
(b)
(c)
Structural elements executing transverse vibrations. (a) Simply supported; (b) clamped; (c) clamped-simply supported.
DETERMINATION OF THE TRANSCENDENTAL FREQUENCY EQUATIONS
The behaviour of the mechanical system under study is described by the partial differential equation:
~4w m ~Zw m _ c32wm EI~x4 +pA s ~ - S ~x~ ( r e= l ,2 ) (1)
where E and p are Young's modulus and mass density of the beam material, respectively, and I and A s are the moment of inertia and cross- sectional area of the beam, respectively. It is assumed that the structural element is subjected to an axial force S (see Fig. 1).
Vibrations of double-span uniform beams 97
For x m = 0 the displacement function w,,(x, t) must satisfy the boun- dary conditions specified on Fig. 1.
Since, in the case of normal modes one can express
w.,(x, t) = W,.(x) eitot (m = 1,2) (2)
the compatibility conditions at x I = L 1 and x 2 = L 2 result:
6~ W1 -- ~ W2 ] (3a) ~Xl x l=Ll ~X2 ]xz=L2
(~2 W1 ] ~2 W2 (3b)
~x, ~ Ix,=~,- ~x~ ~=L2
and eqn. (1) becomes EI d#W,. s d2Wm dx~ + PAso°2 Wm= dx 2 (m = 1,2) (4)
The solution of eqn. (4) is available in standard textbooks. It is convenient to express the compatibility conditions, eqn. (3), in the
form
W1 x, ~2 W2 x2 (~ W2 ~2 WI × + ×T~x~lx,= = 0 (5)
Replacing W,,(x) in eqn. (5) results in:
C1 m l +~LCOS m l l V A l f r l ~ e r , - B ' ( r ' ~ e - r , - sinm 1 L D, \ L , / D 1 \L , } D, L, ,
,~ B ( r ) 2 2 2 - r 2 C2{m2~ 2 ( ~ ) 2 ] - - - - e - I - - - - - e -- cos m 2 -- sin m 2
× D2 L2 D2 L2 D 2 \ L 2 J
[() (m9 ] A2 r2 B 2 ( r 2 " ] e _ , 2 C2(m2)sinm2 + cosm 2 + D22 L2 er~ D2 \ L 2 J - D 2 \ L 2 f L 2
VA x i/ rl'~ 2 , B1 f rl"~ 2 x / - - / - - / e ~ ' + N ~ ) e - "
L D, \L, J
C, (m,~2cosm,_(rn,~2sinmll:O D, \ L , J \L , J
(6)
98 P. A. A. Laura, G. Sanchez Sarmiento, Alicia N. Bergmann
w h e r e
/ l SL2~ /1 I'SL~'~ 2 po,:,2L2 rl = 4 2 ~ - + X/ 4 ~-Ei-- ) + E ~
l l (SLy) :
r2 = 4 2 E 7 + \ E I ) + E1
O n t h e o t h e r h a n d , u s i n g t h e g o v e r n i n g b o u n d a r y
( x I = x 2 = 0) o n e o b t a i n s t h e f o l l o w i n g e q u a t i o n s
Case (a): Simply supported ends
A 1 s i n m l A 2 s i n m 2 B 1 s i n m 1
D 1 2 s i nh r I D 2 2 s i n h r 2 D 1 2 s i nh r 1
B 2 _ s i n m 2 C1 - 0 C 2 - 0
D 2 2 s i n h r 2 D 1 D 2
Case (b): Clamped ends
A 1 1 ml/rlsinhr 1 - s i n m I 1 m 1
D1 2 c o s h G - c o s m x 2 r~
B 1 1 ml/rlsinhr 1 - - s i n m 1 1 m 1
D 1 2 c o s h r 1 - c o s t a 1 2 r 1
C1 ml/rlsinhrl - s i n m 1
D~ - c o s h rx + c o s m ~
A 2 1 m2/r 2 s i n h r 2 - - s in m 2
0 2
B2
D 2
C 2
D 2
1 m 2 I
2 c o s h r 2 - c o s m 2 2 r 2
1 m2/r 2 s i n h r 2 - - s in m 2 1 m 2
-----2 c o s h r 2 - c O s t a E + 2 r E
= m2/r 2 s i n h r E - s in m E
- c o s h r 2 + c o s m 2
c o n d i t i o n s
(7)
(8)
Vibrations of double-span uniform beams 99
S~ 1' t I ] I 1 I I 1 I I _~
~ I~ : . ~ . . ~ l i
. / _ . , ., ?k, \ . . ., , ' - \ . , / L X . . . . . . , , ~ \ , /
. - 'Vx " , . " / -~ ' , , , '~ ' . " . ,,,,A.-... ~ . . . . , ._., , , .... \ , , . _ . , / \ , "..,
.-" / \ ' , . . . / - ~ .. ,, , . - - - \ - .
10
4 s = ~ A A - - ' s
SG/E~: o 2
" - - SLZ/EI= SO
. . . . . . . . . S LZ/EI = 100 L I / L
0 I I I I I I I I I 0.0 0.1 0.2 0.3 0.4 05 0.6 0,7 03 03 1.0
Variation of ~-~1/2 with respect to the parameter L1/L. (Simply supported case.) Fig. 2.
100 P. A. A. Laura, G. S(mchez Sarmiento, Alicia N. Bergmann
~')" I I I 1 I 1 I I 1
2
, , ' / \ ' , ,',,",', ,,;; \', ,,v~', , '~ ' , , ,
/ " , ,',,, ,'/ \ , ' ,
14 - . ' " / \ - . . - . / / " ~ ' - .
1o
.... z:..- .." ........
,4 £ 4 s -~---~ ~, ~ ~ _~=s
:t oo
Fig. 3.
. . . . . . . . . L I / L I I I I I I I
0,1 0.2 0.3 04 0.7 O.e o,g 1.0
S L 2 / E I = 0
S L 2 1 El : 50 S L z I El = I00
I I 0.5 0.6
Variation of ~'~/2 with respect to the parameter L1/L. (Clamped case.)
Vibrations of double-span uniform beams 101
t
18
Fig. 4.
8
0,0 0.1 0.2 B.3 0.4 0.5 0.6 0.7 0.8 o.g 1.0
Variation of ~ / 2 with respect to the parameter L1/L. (Simply supported- clamped case.)
2.' 20 ~
i
15
"~
/
\~.=
2
%=
1
I
Fig.
5.
I S
S L~
=O.S
L
Q A
l J
....
. I
i m
o lo
o 20
0 3o
o 40
0 50
0 S L
z
EI
Va
ria
tio
n o
f ~
,z w
ith
res
pect
to
SLZ
/EI.
(S
impl
y su
ppor
ted
case
; L~/
L =
0.50
.)
%
25. 20.
i.=3
15
.~
10 ~
~-
, ~ --
0
Fig.
6.
S i
L~ =~
i '1
100
200
3 0
4OO
50
0 S
L 2
El
Varia
tion o
f f2~ :2
with
respe
ct to S
L2/E
I. (C
lampe
d ca
se; L
~/L
= 0.50
.)
t~
f%
Vibrations of double-span uniform beams 103
Case (c)." Simply supported-clamped A t s inm,
D 1 2 s i n h r I D 2 2 c o s h r2 - c o s t a 2 2 r 2
B 1 sin m I B 2 1 m2/r 2 sinh r 2 - - sin m 2 1 m 2
D, 2sinhr, D 2 2 coshrE-cosm 2 2 r 2
CI =0 D,
A 2 1 m2/r 2 s i n h r 2 - s i n m 2 1 m 2
C 2 = m2/r 2 s i n h r 2 - s i n m 2
D 2 - c o s h r 2 + c o s t a 2
(9)
By placing the appropr ia te relations A , / D , , B , / D , . . . C 2 / D 2 in eqn. (6) the frequency equat ion can be generated for each situation described in Fig. 1.
N U M E R I C A L R E S U L T S
Figures 2, 3 and 4 depict the variation of the square root of the frequency
coefficient (f~]/2 =,~/(pAs)/(Ei)x/~L) for the first five modes of
.0.1/2 !,
25.
/ 2o /
J
15 ~ 2
i,=1 10.~/ I-'~ I-I=0'51" ~--
; -'L _ ,.1
I I I
0.0 100 200 300 I I
4.00 500
F i g . 7 .
I l L
s ~ EI
Variation of D]/2 with respect to SLZ/EI. (Simply suppor ted-c lamped case;
L1/L =0.50.)
104 P. A. A. Laura, G. Sanchez Sarmiento, Alicia N. Bergmann
vibration of simply supported, clamped and simply supported-clamped beams with an intermediate support as a function of L1/L and for SL2/EI = 0, 50 and 100. When no axial force is acting on the system the results obtained are in excellent agreement with those available in the literature. 1,2
Figures 5, 6 and 7 present the variation of~/2 for the first five modes of vibration as a function of the force parameter SL2/EI and when the intermediate support is placed at the middle of the beam (L 1 = L/2) for three types of boundary conditions considered.
No claim of originality is made by the authors, but it is hoped that design engineers will find the present results useful in many practical situations.
A C K N O W L E D G E M E N T
The present investigation has been partially sponsored by Comision de Investigaciones Cientificas (Buenos Aires Province).
REFERENCES
1. D.J. Gorman, Free lateral vibration analysis of double-span uniform beams. International Journal of Mechanical Sciences, 16 (1974) pp. 345-51.
2. R. D. Blevins, Formulas for natural frequency and mode shape, Van Nostrand Reinhold Co., New York (1979) pp. 137-41.