vibrations of orthotropic rectangular plates with edges possessing different rotational flexibility...
TRANSCRIPT
VIBRATIONS OF ORTHOTROPIC RECTANGULAR PLATES WITH EDGES POSSESSING DIFFERENT
ROTATIONAL FLEXIBILITY AND SUBJECTED TO IN-PLANE FORCES
P. A. A. LAURA? and L. E. bJlSONI$
(Received 9 January 1978: received for publicorion 3 April 1978)
A-t-The title problem is solved using polynomial approximations and a weighted residuals approach. It is shown that a very simple, yet accurate. approximate fundamental frequency equation can be gknerated. A basic. forced vibrations problem is also analyzed.
The procedure is quite convenient for design purposes since all the resulting expressions for frequency coefficients. displacements and stress resultants can be easily implemented for numerical evaluation in a program- mable, pocket calculator.
IhTRoLWcTloN
Leissa’s classical monograph[l] and his two recent survey papers [2,31 contain a very thorough discussion of the subject of vibrating plates with elastic edge supports.
In general, previous investigations have dealt with plates with symmetrical slope restraints. Exception to this is a study previously published [4] where an isotropic plate is considered. No in-plane forces are considered in that investigation.
Soni and Amba Rao have studied the effect of in-plane forces on vibrations of orthotropic rectangular plates5[5]. Two opposite sides are taken to be simply supported whereas the other two edges are differently supported.
The title problem is solved in the present study ap- proximating the displacement function by a summation of polynomial coordinate functions which identically satisfy the boundary conditions. A frequency deter- minant is generated using Galerkin’s weighted residuals approach.
It is shown in the present paper that a very simple but accurate fundamental frequency equation is obtained if one approximates the fundamental mode shape by taking a one term expansion (formed as the product of two fourth-degree polynomials in x and y).
It is important to point out that the present approach can be used without any formal difficulties if additional complications are taken into account: concentrated masses, the presence of an elastic foundation, variable thickness, etc.11
Wesearch Scientist and Director, Institute of Applied Mechanics, Base Naval Puerto Belgrano, 811 I - Argentina.
*Teaching Assistant, U.T.N. (Campana Delegation, B. Aires) and Research Scientist. Comisi6n de lnvestigaciones Cientificas, Buenos Aires.
IThis problem is of great practical importance to the design engineer concerned with panels on rockets at take-off conditions [S].
This is also the case if one studies flexural vibrations. accord-
ing to classical orthotropic plate theory, and includes the effect of stiffener rotatory inertia. A thorough treatment of this problem is available in Troitky’s excellent textbook[6].
The notation is standard in the techniCal literature (see for inst.,[71.
GOVERNING DlnmmrlALsYmANn
lTsAP?mxlMATE~
Small amplitude vibrations of uniform orthqtropic
rectangular plates subject to in-plane. normal forces are governed by the partial differential equation7
4 , 4 4 , 2 2
D,~~ZH.,_d;iT+D,~-h:~-N,~ ax dy ay ay
=-ph$ (1)
where
D x ==h3. D/G.
12 ’ ‘- 12 ’ Hx, = D, + 20x,
D, =T; 4, =+t
It can be easily shown that [7]:
The boundary conditions are given by the functional relations (see Fig. 1):
w(-~;y;I)=w(~;Y;L)=w(x;-p;L) b =w _r;-’ ( ) 2’t =o
aw
ax I (2b)
X - 4012) =QI (a$$+D~$l~__,~~,
aw z I ._,,,=-+G+D1$>l x_(&)
(2c)
L?!! JY I ,--(b/2)
= 4s (4 $+ 0 2 )I
m )I__-(b,2,
aW
5 I =-,(D++D,$
>I (24
y-(b/2) y-<b/2)
Since an exact solution of this system seems out of the question, the displacement function w(x, y, t) will be
521
P. A. A. LAURA and L. E. LUIZONI
i I I i I
Fig. 1. Srructural system under study and some of its lower normal modes.
approximated by means of the expansion:
x (B1,y4+ PziY’+ B3,YZf bin + 1) X'Y'. (3)
It is assumed that each polynomial coordinate function satisfies identically the boundary conditions.
Substituting (3) in (2a) one obtains an error or residual function c(x. p, ~1. Several techniques are available to minimize the error function.
Galerkin’s method will be presently used in view of its simplicity and high accuracy. This method requires that 6(x, y, o) be orthogonal with respect to each coordinate function in the domain under consideration.
From the non-triviality condition one obtains a deter- minantal equation in the 0’s.
Judging from the good accuracy achieved in previous invest~ations in the case of isotropic ptates[4,8] it is reasonable to expect that the frequency equation generated using eqn (3) in the case of orthotropic plates will be accurate enough, at least from the point of view of a design engineer and for the lower modes of vibration of the structural system shown in Fig. 1.
The present study is restricted to the determination of the fundamental frequency of vibration, parameter of vital irn~~n~e from a st~ctur~ engineering viewpoint.
Fig. 2. Change of coordinate axes.
When performing this analysis it is considerably simpler to place the coordinate axes in the left, lower comer of the plate as shown in Fig. 2.
The fundamen~ mode shape is then approximate by the functional reIation:
W.(x.~)=Aoo(~ c&)(&,Y’). (4) r-1
Substituting (4) in the time independent boundary conditions one obtains:
al=2n2-=Uao!; k4
lQ=a;
a23kztSk,+l2tk,kz I a,= -2;
h&+6) =(I a;
a24kl +4k2+ 12+ ktkz i a4=;;L
k,(kz+ 6) =2fd
fit=t&$=b@i 3
82 = 8;
8, = _2 & 3k. + Sk, + 12 + kak, b k,tk, + 6)
B _3+4kd+4k,+ 12+ k,k, 4-
b kdk. f 61
where
iSa)
W
(5c)
(Sd)
@aI
(6b)
W)
b k,=-&: k2=-&; k,=osD,: Id=-&.
Replacing (4) in the governing, time independent diffe~nti~ equation and making use of Gale&in’s o~~on~~ation condition one obtains:
Vibrations of orthotropic rectanguiar plates 529
If the orthotropic plate under study is subjected to a uniformly distributed dynamic load p,cos or one
where
, , I t
p+!+!S+!z+~ obtains, using (4) and applying Galetkin’s technique:
P,=“;-+ 4 s d a$+ai+ai
wo(x,= Aoo --f. I%@) QXY)
+b’ $Li 0
(13)
%Y .=Y
@I where
P ’ &pi 8 -iJ- ai+&+ai
Q2,s;+E*E+& 2 3 4 5
*6,B;+Bi+B;+& 6 7 8 9
and
In a similar fashion one obtains dimensionless expres- sions for the amplitudes of the dynamic bending mo-
W8 ments:
Q24=81Qz+3p;Q,+ c&Q4 J-4 A&=_ Am Dx b2
Q~=BlQ3$BiQ4+BSQs+B;Qa. (lil pob [ 0 - - Pzl~x~ CKr~Y~
PO b ’ R, a -- 0 H,, a -,
The fundamental frequency coefficient will be defined as
+ $ P;4(x) Qi4ty 1 I
(15) XY
&t,=j/@+oo.bz. (12) XY
s=- Am z[f$P;Ax)Qh(~) &1?. XY
It has already shown that in the case of isotropic p&es 0 HX, a
a one term polynomial approximation yields very reasonable accuracy when c~cuIatin~ dvnamic dis-
+- i W 0
2 Pi*lx)Q;rtyi 1
(161
place~nts if the p&e is subjected to a-p0 cos or -type excitation [8,9].
where
The procedure yields also good accuracy when determining bending moments if the plate is ap- proximately square. If the parameter (b/a) differs greatly from unity it is necessary to consider additional terms in expression (4) (see f91).
PJ&)=2a;t6a; (17a)
Q&(y) = 2~~+6~~~(~) + 12~~4(~)*. (17W
Table 1. Comparison of fundamental frequency coelkients 0, (case of a sqnare. orthotropic plate rigidly clamped along four edges)
Tabl
e 2.
Fu
ndam
enta
l fr
eque
ncy
coef
ficie
nt&
of
an o
rtho
trop
ic
squa
re
plat
e
rigid
ly
clam
ped
alon
g th
ree
side
s as
a
func
tion
of
in-p
lane
no
rmal
st
ress
es.
orth
otro
py
para
met
ers
and
rigi
dity
of
th
e fo
urth
ed
ge
k,
Tabl
e 3.
Fu
ndam
enta
l fr
eque
ncy
coef
ficie
nt
&of
an
ort
hotr
opic
sq
uare
pl
ate
sim
ply
supp
orte
d al
ong
thre
e si
des
as a
fun
ctio
n of
in
-pla
ne
norm
al
stre
sses
, .
orth
otro
py
para
met
ers
and
rrgt
dtty
of
th
e th
ird
edge
ks
- L-
ia
1 - : - 2 - 1 - 2 - : - 2 - $ - 1 -
-20
-
r b3
-
0
1: m
-
0
1: _
-
0
11, m
-
0
1: m
-
! 10
m
-
0
1: m
-
0
1: _
-
0 1 10
_ -
0
1: ti
- 4
Nx.
b’
_ N
.b
’
&-I
S 0
t-
200
315
G
19.7
20
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21.9
23
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37.1
48
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37.3
---l
- 48
.8
38.6
50
.0
40.2
51
.8
65.8
79
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66.0
79
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67.2
80
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69.3
83
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17.1
17
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la.5
19
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65.1
65
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66.2
68
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13.7
15
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18.8
21
7
A
24.1
24
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27.5
29
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39.6
50
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40.1
50
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--I-
- 42
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52.7
44
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54.9
67.3
67
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69.3
71
.J
18.4
18
.9
20.8
22
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65.4
65
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66.9
68
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20.9
21
.J
24
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27.3
66.2
66
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68.2
70
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4.0
18.4
18
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19.8
21
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65.4
65
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66.6
68
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1T.i
lb:1
19
3
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6 7.
2
22.0
22
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25.7
28
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66.6
66
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68.6
71
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20.9
21
.l
22.1
23
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36.4
48
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36.6
48
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38.0
49
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39.6
51
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37.7
49
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38.2
49
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40.3
51
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42.5
53
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36.4
48
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36.5
48
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37.4
49
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38.7
50
.6
38.4
49
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38.8
49
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40.9
51
.a
43.1
54
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37.7
49
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37.8
49
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38.7
50
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40.0
51
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66.2
66
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67.3
69
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9.6
10.4
13
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15
2 A
22.0
22
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24.0
25
.6
38.4
49
.6
38.6
49
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-L
39.8
51
.o
41
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52.7
66.6
80
.0
66.7
80
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67.9
81
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70.0
03
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-I
Nx.
bL
b -c
- H X
Y
b‘
-
300
78.8
78
.9
80.0
82
2
L
80.6
80
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82.6
85
2
L 79
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19.2
80
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82
9 A
79.7
80
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81
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84.3
79.1
79
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80.3
82
5
A
80.0
80
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82.0
84
6
A 79.7
79
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80.9
83
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40
- a-
-6
-
28.5
36
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49.9
26
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34.0
48
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24.0
32
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46.9
23
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31 .a
46
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17.6
15
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13.8
13
2
A
42.7
39
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36.7
35
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28.1
44
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26.7
43
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25.6
42
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25.3
42
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48.0
45
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42.5
41
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23.7
32
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21 .
o 30
.1
18.0
28
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17.2
27
.6
59.1
56
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54.5
54
1
A 41
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45.4
44
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43.9
32.6
39
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28.8
35
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24.3
32
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23.1
31
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23.7
22
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20
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36.3
32
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29.0
28
0
A 32
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31 .J
30
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30
5 A
36.3
34
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32.8
32
4
A
32.3
31
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30.1
29
9
A 42
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39.3
36
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35.4
52.4
49
.5
47.1
46
5
A 47
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46.0
45
.4
45
4 A
54.1
52
.0
49.7
49
2
A
39.3
if.: 37’4
A
42.4
40
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39.3
.zu
52.4
51
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50.7
50
6
A 54
.7
53.1
52
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51
8 A
l-
/
Tim
78.0
76
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75.4
75
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92.1
JO
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89.5
89
.5
74.7
73
.1
72.8
72
8
A
59.4
87
.5
07.3
87
.3
84.8
81
.7
80.3
80
2
A 76.4
74
.4
73.7
73
.7
97.4
94
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93.7
93
.6
Ez
88:o
88
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79.6
93
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76.9
90
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75.5
89
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75.3
89
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76.4
74
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74.5
74
5
A al
.2
78.6
77
.1
76
? &
19.6
78
.1
77.8
77
8
A
:::
88.7
88
7
L 94.8
92
.1
90.9
90
.9
93.5
91
.7
91.5
91
.7
81.2
79
.3
78.6
78
b
A
94.8
92
.8
xz
d
8.3
t: 819
-
60.7
58
.9
58.0
57
.9
- 56.5
55
.0
54.5
54
.6
- 68.5
66
.0
64.3
63
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- 58.6
56
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55.7
55
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- 62.8
60
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58.1
57
7
1 58.6
57
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X
- 64.8
62
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60.2
59
9
&
62.8
61
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61.1
m
z*“o
62
:s
62.1
i I ; i i i i
) i
/ ; 1
/ i 9.
1
!4 .2
9.
1 1.
3 82
A
69::
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19.5
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I- P E 2
Vibrations of orthotropic rectangular pla!es 531
Table 4. Fuudameat~ frequency c~fficicnt & of a rectangular ptate rigidly clamped along three edges and elastically restrained against rotation at the fourth edge
Tabic 5. ~u~~n~ frequency co&Went &, of a rec~~~~ plate simply supine along three edges and elastically restrained against rotation at the third edge
Table 1 presents a comparison of results between lower than those offlO]. Since the Galerkin method
values previousIy calculated by Iyengar and generates upper bounds, it can be concluded that present vahzes are more accurate -than those available in the
~~ish~iO~, and those obtained using expression (7). literature in some instances. The agreement is excellent in all cases, in some in- stances the present approach yields results which are
Tables 2-5 contain values of fundamental frequency coefficient as a function of in-plane normal stresses, or
532 P. A. A. LAURA and L. E. LUMNI
“00 1
ssr ‘t-. 0.10 i h ---ii&--
k‘
Fig. 3. Fundamental frequency coefficient f& of a square plate with three edges rigidly clamped and the fourth elastically restrained against rotation as a function of k,, ff&/H,,) and
(~“~~~~) for (N, I b2fH,,) = (My . b?ff,,) = 100.
orthotropy parameters and (b/a) for rectangular plates with: (a) three rigidly clamped edges and (b) three simply supported edges, while the remaining edge is elastically restrained against rotation.
Figure 3 depicts graphically the variation of &O as a function of k4 and several combinations of values of (DA&.) and KVH,,) for
It appears at this point that the methodology presented in this paper is quite possibly, one of the simplest which can be developed to solve the title problem. For instance. use of the “beam coordinate functions method” would
require calculation of the characteristic values of the beam functions for each combination of orthotropic parameters and flexib~ity coefficients.
This is not required in the present approach.
Acknowledgements-The present investigation has been spon- sored by the Comisidn de Investigaciones Cienrificas (Buenos Aires Province).
I. 2.
3.
4.
5.
9.
IO.
Il.
A. W. Leissa. Vibrorion OJ Plafes. NASA, SP-140 { 19691, A. W. Leissa, Recent research in piate vibrations, 197~-1976: Classical theory. Tke shock and Vibration Digest !%I01 (1977). A. W. Leissa. Recent research in plate vibrations 19734976: Complicating effects. The Shock and Vibration LXgest 9fl I) (1977). P. A. A. Laura. L. E. Luisoni and C. Filipich. A note on the determination of the fundamental frequency of vibration of thin, rectangular plates with edges possessing different rota- tional &xibiiity coeflieients. 1. Sound Vibration 55f3) (19771. S. R. Soni and C. L. Amba Rao. Vi~tions of thin. ortho- tropic rectangular plates under in-plane forces. Compcrf. Strucrures 4(5), 110s1 I IS (19741. hi. S. Troitsky. Stiflened Plates: Bending, Stabiiity and Vibrafions. Elsevier, Amsterdam (1976). S. Timoshcnko and S. Woinowsky-Krieger, 77reory of Plates and Shells, 2nd Edn. McGraw-Hill. New York (1959). P. A. A. Laura and R L&an, A note on forced vibrations of a clamped rectangular plate. 1. Sound Vibration 42(l), 129- 135 (19751. E. A. Susemihl and P. A. A. Laura. Forced vibrations of thin, elastic, rectangular plates with edges elastically restrained against rotation. 1. Ship Research 2t(l), 2629 (1977t. K. T. Sundara Raja Iyengar and K. S. Jagadish. Vibration of rectangular orthotropic plates. Applied Scientific Research Section A, 13. 37-42 (1964). 8. L. Clarkson. Stresses in skin panels subjected to random acoustic loading. T!te Aeronautical J. 72(695). 10004010 t I968h