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Title
Free vibration analysis of cross-ply symmetrically laminated right
cantilever triangular plates
Author(s)
Huang, Mei; Sakiyama, Takeshi; Matsuda, Hiroshi; Morita, Chihiro
Citation
長崎大学工学部研究報告 Vol.32(58) p.125-133, 2002
Issue Date
2002-01
URL
http://hdl.handle.net/10069/5194
Right
This document is downloaded at: 2017-06-17T00:14:58Z
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Reports of the Faculty of Engineering, Nagasaki University, Vol. 32, No. 58
125
Free vibration analysis of cross-ply symmetrically laminated
right cantilever triangular plates
by
Mei Huang *, Takeshi Sakiyama * *
Hiroshi Matsuda ** and Chihiro Morita * *
An approximate method is proposed for analyzing the free vibration problem of cross-ply symmetrically laminated cantilever triangular plates. By adding an extremely thin part, a right triangular plate is
translated into an equivalent rectangular plate with non-uniform thickness. Discrete solutions for the
deflection of the equivalent plate are obtained and used as Green function. By using the Green function, the
characteristic equation of the free vibration can be derived. Frequencies and mode shapes are shown for
some laminated cantilever right triangular plates with various aspect-ratios. The efficiency and accuracy of
the numerical solutions by the proposed method are investigated.
1 . INTRODUCTION
their mode shapes are shown for some laminated can-
The free vibration problems of isotropic triangular
tilever right triangular plates with various aspect-ratio.
plates have been studied for many years. Early studies
The efficiency and accuracy of the numerical solutions
were well compiled in Ref. [1]. Further studies have
by the proposed method are investigated
been done for the past two decades [2 -10]. With the
increasing use of composite materials in engineering,
2. DISCRETE GREEN FUNCTION
the free vibration problems of orthotropic triangular
An xyz coordinate system is used in the present
plates have been investigated by some researchers
study with its x-y plane contained in middle plane of a
[7 , 11 - 13]. But the study of the laminated triangular
symmetrically laminated rectangular plate and the z-
plates is rather limited.
axis perpendicular to the middle plane of the plate.
In this paper, an approximate method is proposed
The thickness, the width and the length of the or-
for analyzing the free vibration of symmetrically cross-
thotropic square plate are h, a and b, respectively. The
ply laminated right triangular plate with various
principle material axes of the plate in the direction of
aspect-ratio. By adding an extremely thin part, a can-
longitudinal, transverse and normal directions are
tilever triangular plate can be translated into an e-
designated as 1,2 and 3. The differential equations of
quivalent rectangular plate with non-uniform thick-
the plate with a concentrated load
ness. Therefore, the free-vibration characteristics of
are as follows:
P at point
(Xq , Yr)
the triangular plate can be obtained by analyzing the equivalent rectangular plate. The characteristic equation of the free vibration is obtained by using Green
function, which is the discrete-form solution [I4] for
the deflection of the plate with a concentrated load at
each discrete point. The lowest eight frequencies and
Received on Oct., 2001
* JSPS Postdoctoral Fellow
* * Department of Structural Engineering
aQx+ aQy= -Po (x-x) 0 (y-y)
ax
ay
q
r
CIa)
126
M
M
Mei Huang, Takeshi Sakiyama, Hiroshi Matsuda, Chihiro Morita
x
y
=
D aox + D aoy + D (aO X+ oOy)
(Id)
D aox + D oOy + D (oOX + oOy)
(Ie)
11
=
12
ax
12
oX
22
oy
oy
16
26
ay
ay
ax
ax
(If)
ow
Qy=kA 44 (ay+ Oy)
ow)
+ kA 45 (ax
+ Ox
(Ig)
where Do = E2 k5!(l2 (I- 1J12 1J21)) is the standard bending rigidity, k o is the standard thickness of the plate.
Qx = kA 45
ow
(ay +
(ax
ow)
Oy) + kA 55
+ Ox
(Ih)
By using the above expressions, the differential Eqs.
(Ia) '"'-' (Ih) can be rewritten as
where Qx and Qy are the transverse shear forces, M x
and My are the bending moments, Mxy is the twisting
moment, k = 5I 6 is the shear correction factor, 0 (xxq )
and 0 (y - Yr)
are Dirac's delta functions, A ij is
the extensional stiffness (i, j
stiffness Ci, j
=
=
4, 5) , D ij is the bending
where P= Pal (Do (I- 1J12 1J21)) , oij is Kronecker's
1, 2, 6) .
A ij, D jj can be obtained by the following expressions.
By dividing a rectangular plate vertically into m e-
N
A jj =
L,(Qij)k(Zk- Zk-1)
k~1
D jj =
1
delta, FIts, F 2ts and F 3ts are given in Appendix A.
N
qual-length parts and horizontally into n equal-length
parts as shown in Figure 1 , the plate can be consi-
-
3
3
3L,(Qij)k(Zk- Zk-l)
k=1
QIl = Ql1c4 + 2 (Q12 + 2Q66)C2S2+ Q22S4
Q12 = Q12 (c 4+ S4) + (QIl + Q22 - 4Q66)C2S2
Q16 = (Ql1 - Q12 - 2Q66) c3s - (Q22 - Q12 - 2Q66) cs 3
Q22 = Ql1S 4+ 2 (Q12 + 2Q66)c2s 2+ Q 22 c4
Q26 = (Ql1- Q12 - 2Q66) C5 3 - (Q22 - Q12 - 2Q66)c3s
Q66 = (Ql1 + Q22 - 2Q12 - 2Q66) C252+ Q66 (c 4+ 54)
Q44 = Q44c2 + Q55S2 , Q45 = (Q55 - Q44) CS,
Q55 = Q4452 + Q55C2,
C= 1, S = 0
dered as a group of discrete points which are the intersections of the (m + 1) -vertical and (n + 1) -horizontal
dividing lines. In this paper, the rectangular area, 0::;;
Tj ::;; Tj
j,
e ~ ej,
corresponding to the arbitrary
intersection (i, j) as shown in Figure I is denoted as
the area Ci,
n,
the intersection (i, j) denoted by
tions denoted by
0
are called the inner dependent
points of the area, and the intersections denoted by •
are called the boundary dependent points of the area.
n
y
b
..Ji
j
j)
'"
the axial modulus in the 2-direction, 1J12 is the Poisson's ratio associated with loading in the I-direction
and strain in the 2-direction, 1J21 is the Poisson's ratio
associated with loading in the 2-direction and strain in
the I-direction, G23, G31 and G12 are the shear moduli in
2-3, 3-1 and 1-2 planes.
The following non-dimensional expressions are
given
0
is called the main point of the area [i, jJ, the intersec-
(
where E 1 is the axial modulus in the I-direction, E 2 is
0~
3
2
x
a
2 3
m
T/
Figure 1 : Discrete points on a rectangular plate
Free vibration analysis of cross - ply symmetrically laminated right cantilever triangular plates
By integrating the equation (2) over the areaCi,
n,
points of the following larger areas. Whenever the
quantity X pij at the main point Ci, j) is obtained by us-
the following integral equation is obtained
.± {F
127
ing the equation (5) in the above mentioned order,
lts
fo'itX
s(
17, 'j) -
Xs(r;, 0) Jd r;
the quantities X tkj, Xti/ and X tkl at the inner dependent
points of the following larger areas can be eliminated
s~l
f 'iif'iX s( r;, ,
+ F 3ts
0
0
)dr;d'
by substituting the obtained results into the corresponding terms of the right side of equation (5).
}
+Pu( r; - r;q)u(' - C)Olt=O
(3)
By repeating this process, the equation X pij at the main
point is only related to the quantities X rkO , (r= 1,3,4,
6,7,8) and X s01 ,
(s =
2,3,5,6,7,8) which are six in-
where u ( r; - r;q) and U(, - 'r) are the unit step
dependent quantities at the each boundary dependent
functions.
point along the horizontal axis and the vertical axis in
Next, by applying the numerical integration method,
Figure 2, respectively. The result is
the simultaneous equation for the unknown quantities
X sij = X s (r;i, (j) at the main point Ci, j) of the area Ci,
n
is obtained as follows
+ F2ts~;9jl(Xsil- Xs()l)
j
k~()
I=()
}
the discrete Green function is chosen as X 8 ijl [Pal Do
+ PUjqUjr01t = 0
(4)
where ;9 ik = a ikl m, ;9 jl = a jll n, a ik = , 1- ( aok + 0 ik) 12,
ajl= 1- ( aol+ Ojl) 12, t= 1"'--'8, i= 1"'--'m, j= 1"'--'n.
i
X pij = L {L ;9ik A pt [Xtko - X tkj (1 - Oik) J
1112 1121)].
3. EQUIVALENT RECTANGULAR PLATE
PLATE
A cantilever right triangular plate is quite different
from uniform rectangular plates, but it can be translat-
k~()
ed into equivalent rectangular plates with non-uniform
j
thickness (showed in Figure 2) by adding an ex-
+ L;9jIBpt[Xtlll- XtitCI- Ojl)J
k~()
tremely thin part to the original part.
L t ;9jk ;9jICptkIXtkl(1- aik ajl) }
In this paper, the thickness of the actual part of
k~() t~()
(5)
- AplPuiqUjr
original right triangular plate is expressed as h o, and
the thickness of additional part of the equivalent rec-
where p = 1"'--'8, Apt, B pt and Cptkl are given in Appendix B.
In the equation (5), the quantity X pij at the main
point Ci, j) of the area [i,
n is related to the quanti-
ties X tk () and X tlll at the boundary dependent points of
the area and the quantities X tkj, Xti/ and X tk1 at the inner
dependent points of the area. With the spreading of
the area[i, naccording to the regular order as[l, IJ,
[1,2J , ... , [1,nJ, [2,1], [2,2J , ... , [2,nJ,
em, 2J
(1-
OF A CANTILEVER TRIANGULAR
The solution Xpij of the simultaneous equation (4) is
1] ,
(6)
ing problem of a plate under a concentrated load, and
i
+ F3ts~ ~;9ik ;9jlXskl
... , em,
g~O
the fundamental differential equation (2) of the bend-
I~()
+
1=0
The equation (6) gives the discrete solution of
j
t= I
d~l
j
~bpijgdXSOg}
where apij/dbpijgd and qpij are given in Appendix C.
k=()
8
i
+qpijP
±{FltsL;9ik(XSkj-X sk())
s~1
6
X pij = ~ {Lapij/dXrjO+
, ... ,
em, nJ,
a main point of a
smaller area becomes one of the inner dependent
r
b
l
-----,
\
'-
F
\
\
\
C
•
A\
F1
\
\
\
\
\
\
F
I
\
\
ho
\
C
I
h
\
\
\
I
\
I
I
I
h_____
o F \\1J
---_..-.~
~a----..l
Figure 2 : Cantilever right triangular Plate and
its equivalent rectangular plate
Mei Huang, Takeshi Sakiyama, Hiroshi Matsuda, Chihiro Morita
128
tangular plate is expressed as h. The fixed and free
where k=O, 1, "', m, /=0, 1"" n
The characteristic equation of the free vibration of a
edges are denoted by the symbols C, F, respectively
and showed by solid line -
and dotted line - - - .
The first symbol indicates the conditions at x= 0, the
rectangular plate with variable thickness is obtained
from the equation (9) as follows.
second at y = 0 and the third at the hypotenuse. The
plate CFF is analyzed in the present study.
In the free vibration analysis of the equivalent rectangular plate for the cantilever right triangular plate,
the mass of additional triangular part is treated as zero
and the load is only applied on the original part.
K oo
K Ol
K 02
K om
K IO
K ll
K I2
KIm
K zo
K 21
K 22
K 2m
K mo
K ml
K mz
K mm
=0
M
where
4. CHARACTERISTIC EQUATION OF FREE
VIBRATION OF RECTANGULAR PLATE
K ij = ~mj
WITH NON-UNIFORM THICKNESS
J3nolljoG iOjO - kOij
J3nnlIjn G iOjn
J3nolljOGiljO
J3nJljnGiljn
{3nolljOGi2jO
~nnlIjnGiZjn
{3nolljOGinjO
{3nnlIjnGinjn - kO,'j
By applying the Green function w (xo, Yo, x, Y) / P
which is the displacement at a point (xo, Yo) of a plate
with a concentrated load
P at a point
(x, y) and point
support at each discrete point (xc> Yd), the displace-
5. NUMERICAL RESULTS
ment amplitude til (xo, Yo) at a point (xo, Yo) of the
Although the objection of the present work is to study
rectangular plate during the free vibration is given as
til (xo, Yo) =
the free vibration of cross-ply symmetrically laminated cantilever triangular plate, the results for the
f:f:
p hwzw (x, y) [w (xo, Yo, x, y) I P]dxdy
isotropic and orthotropic cases are also given to show
(7)
the convergence and accuracy of the numerical solution obtained by the proposed method. The graphite
where p is the mass density of the plate material.
The non-dimensional expressions are used as,
I
epoxy material is used for orthotropic and laminated
plates. Its properties are given as Ed E z= 17.57, G12/
E z = GI3/E2= G23 1E2= O. 7,1112=0.28.
p(x, y) hex, y)
H( r; , C)
5.1
ho
po
til (x, y)
W( r; , C)
CONVERGENCE OF SOLUTION
In order to examine the convergence, numerical calculation is carried out by varying the number of divisions
a
m and n. The lowest eight natural frequency
r
G( r;o,
~o,
r;,
r)
~
w(xo, Yo, x, y) D o( 1 - Lil2
a
Pa
liZI)
parameters of orthotropic isosceles right triangular
plate are shown in Figure 3 . Figure 4 is used to determine the suitable thickness ratio hiho of the additional
here po is the standard mass density.
By using the numerical integration method, equa-
results can be obtained when m and n are not smaller
tion (7) is discretely expressed as
m
kWk1 =
than 10. It is sufficient to set the thickness ratio hih 0
n
L LJ3mi J3 njHijGk1ij W ij
and original parts. It can be noticed that convergent
(8)
i=Oj=O
~
1/5.
In the present analysis, the thickness ratio h/ h o= 1/5
where k= 11 (fJ).,4)
is used for all the plates and the convergent values of
From equation (8) homogeneous linear equations
frequency parameter are obtained by using Richard-
in (m + 1) x (n + 1) unknowns Woo, Wo I , "', WOn, W IO ,
son's extrapolation formula for two cases of divisional
W ll " ' , WIn, "', W mO ' W ml , "', Wmn,are obtained as
numbers m (=n) of 10 and 12.
m
n
L L (J3mi J3njHijGklij- kOikOjD Wij= 0
i=Oj=O
(9)
Free vibration analysis of cross - ply symmetrically laminated right cantilever
triangul~r
plates
129
cases, and compared with the results reported previConvergence of natural frequency parameter
of cantilever isosceles right triangular plate
made from graphite/epoxy. (h o'h=5.0)
20
)(
16
....-.:::
12
8
4
M
M
M
M
)C
)C
M
M
K8th
K 7th
K 6th
~
M
M
)( 5th
ously. As application, the lowest natural frequencies
and their mode shapes of cross-ply symmetrically laminated cantilever right triangular plates are given.
5.2.1
ISOTROPIC AND ORTHOTROPIC
RIGHT TRIANGULAR PLATES
~
..
..
)C
M
M
K2nd
Numerical solutions for the frequencies of isotropic
)C
M
M
K1st
CFF isosceles right triangular plate are given in Table
~4th
3rd
1 with the values obtained by Kim and Dickinson[ 7 J
OL..-_.L-
.L-
6
..L..-
8
and Lam, Liew and Chow [11]. The present method
..L..-~
10
12
yields slightly lower values of frequency parameter as
Number of division m(= n)
compared with the those of Kim and Dickinson [7 ]
Figure 3 : The natural frequency parameter A versus
the divisional number m ( = n )for orthotropic CFF isosceles right triangular plate
and Lam, Liew and Chow [llJ, which derived from a
Rayleigh-Ritz solution. The close agreement is
achieved. The nodal lines of 8 modes of CFF isosceles
right triangular plate for isotropic case are shown in
Convergence of natural frequency parameter
of cantilever isosceles right triangular plate
made from graphite/epoxy. (m=10,12)
~
16
>e--."
~
~E
M
M
M
M
M
~ 8th
7th
K6th
K 5th
IS
~ 4th
3rd
= = =
::---, .. ..
12
8
)C--
)(
4
M
M
M
M
K2nd
M
M
M
M
K 1st
Figure 5.
o
o
2
3
4
5
1st
2nd
3rd
4th
5th
6th
7th
8th
Figure 5 : Nodal patterns for isotropic CFF isosceles
right triangular plate
6
holh
In Table 2, the natural frequency parameters for
Figure 4 : The natural frequency parameter A versus
the thickness ration hi ho for orthotropic
CFF isosceles right triangular plate
isosceles right triangular plate made from graphite /
epoxy are shown together with the results obtained by
Kim and Dickinson [7 J. The present method yields
5.2 ACCURACY OF SOLUTION
lower frequencies as compared with those of Kim and
To investigate the accuracy of the results obtained by
Dickinson [7 ] and the biggest error is smaller than
the proposed method, the lowest 8 natural frequencies
and their mode shapes of cantilever right triangular
3 percent. The nodal lines of 8 modes of CFF isosceles
right triangular plate made from graphite / epoxy are
plates are presented for isotropic and orthotropic
shown in Figure 6 .
Table 1 : Natural frequency parameter A for isotropic CFF isosceles right triangular plate
References
Present
10 x 10
12 x 12
Extrapolation
Kim [7]
Lam [11]
Mode sequence number
4th
5th
6th
1st
2nd
3rd
2.498
2.507
2.527
2.542
2.540
4.860
4.875
4.908
4.959
4.957
5.850
5.833
5.796
5.852
5.858
7.466
7.502
7.585
7.672
7.873
9.059
8.997
8.856
8.951
9.980
9.992
10.019
10.203
7th
8th
10.991
10.910
10.726
11.861
11.804
11.672
130
Mei Huang, Takeshi Sakiyama, Hiroshi Matsuda, Chihiro Morita
Table 2 : Natural frequency parameter). for orthotropic CFF isosceles right triangular plate
References
Present
10 x 10
12 x 12
Extrapolation
Kim [7]
Mode sequence number
4th
5th
6th
1st
2nd
3rd
4.631
4.630
4.628
4.659
7.236
7.250
7.283
7.350
10.355
10.331
10.277
10.398
10.075
11.042
10.967
11.091
13.266
13.227
13.139
13.313
15.283
15.123
14.757
14.968
7th
8th
16.583
16.489
16.275
18.449
18.061
17.179
Table 3 : Natural frequency parameter ). for isotropic and orthotropic CFF right cantilever triangular plate (bla= 1.5)
References
Present
10 x 10
12 x 12
Extrapolation
Kim [7]
References
Present
10 x 10
12 x 12
Extrapolation
Kim [7]
1st
L""
2nd
1st
2nd
2.432
2.437
2.447
2.464
4.298
4.308
4.333
4.372
1st
2nd
4.486
4.483
4.476
4.501
6.459
6.457
6.454
6.506
Isotropic
Mode sequence number
4th
5th
3rd
6th
8.006
5.614
6.283
8.880
6.311
7.992
5.589
8.811
5.532
6.374
7.959
8.654
5.588
6.449
8.072
8.828
Orthotropic
Mode sequence number
4th
5th
3rd
6th
8.768
8.732
8.652
8.730
10.765
10.652
10.396
10.561
11.128
11.036
10.826
10.896
13.255
13.099
12.719
12.858
7th
8th
9.564
9.473
9.266
9.925
9.941
9.978
7th
8th
14.198
14.010
13.581
15.915
15.548
14.715
~2, t'~,
3rd
4th
~ t3~ ~ ~~,
5th
6th
7th
8th
Figure 6 : Nodal patterns for orthotropic CFF
isosceles right triangular plate
5th
Table 3 gives the natural frequency parameters of
right triangular plate with aspect ratio bI a = 1. 5 for
isotropic and orthotropic cases, respectively. The
6th
7th
8th
(a) Isotropic case
present results are compared with those obtained by
Kim and Dickinson [7] and it is found that close
agreement is achieved. The noda11ines of 8 modes of
CFF right triangular plate with aspect ratio bI a =
1. 5 for isotropic and orthotropic plates are shown in
Figure 7.
5.2.2 CROSS-PLY RIGHT TRIANGULAR
PLATES
The results presented in Tables 1 '"'-' 3 validate the
present method for calculating the natural frequencies
5th
6th
7th
8th
(b) Orthotropic case
Figure 7 : Nodal patterns for CFF right triangular plates
for isotropic and orthotropic cases (b/a= 1.5)
Free vibration analysis of cross - ply symmetrically laminated right cantilever triangular plates
131
Table 4 : Natural frequency parameter Afor cross-ply CFF right triangular plate (b/a= 1.0)
References
Present
10 x 10
12 x 12
Extrapolation
References
Present
10 x 10
12 x 12
Extrapolation
1st
2nd
4.599
4.597
4.597
7.322
7.341
7.341
1st
2nd
4.517
4.517
4.517
7.422
7.447
7.447
[0° /90° /0°]
Mode sequence number
3rd
4th
5th
6th
7th
8th
10.529 11.262 13.579 15.842
10.479 11.277 13.562 15.702
10.479 11.277 13.562 15.702
[0° /90° /90° /0°]
Mode sequence number
3rd
4th
5th
6th
17.031
16.964
16.964
18.316
18.015
18.015
7th
8th
10.480
10.414
10.414
17.379
17.246
17.246
18.224
18.087
18.087
11.703
11.760
11.760
13.879
13.872
13.872
16.789
16.700
16.700
Table 5 : Natural frequency parameter Afor cross-ply CFF right triangular plate (b/a= 1.5)
References
Present
10 x 10
12 x 12
Extrapolation
References
Present
10 x 10
12 x 12
Extrapolation
1st
2nd
4.457
4.454
4.454
6.515
6.520
6.520
1st
2nd
4.374
4.371
4.371
6.580
6.592
6.592
["" L"" ~L'
~ b~ th
1st
5th
[0° /90° /0°]
Mode sequence number
3rd
4th
5th
6th
7th
8th
9.060 10.680 11.567 13.528
9.037 10.594 11.487 13.360
9.037 10.594 11.487 13.360
[0° /90° /90° /0°]
Mode sequence number
4th
3rd
5th
6th
14.635
14.473
14.473
16.329
16.040
16.040
7th
8th
9.440
9.431
9.431
15.210
15.122
15.122
16.973
16.700
16.700
10.551
10.488
10.488
12.049
12.014
12.014
[,~,
2nd
3rd
4th
6th
7th
hs,
8th
5th
(a) [0°/90 0 /ooJ
t'", t'", t'
~
_1,
1st
2nd
3rd
t'~
4th
b~ ~, t~ ri~,
5th
6th
7th
13.871
13.721
13.721
8th
Figure 8 : Nodal patterns for cross-ply CFF right
triangular plate (b/a = 1.0)
7th
8th
t\~ l~ b t\~
r~ [~~ ts ~
1st
5th
(b) [00/90 0 /90 0/0 0J
6th
(a) [0°/90° /ooJ
2nd
3rd
4th
6th
7th
8th
(b) [00/90 0/90 0/0 0J
Figure 9 : Nodal patterns for cross-ply CFF right
triangular plate for (b/a = 1.5)
132
Mei Huang, Takeshi Sakiyama, Hiroshi Matsuda, Chihiro Morita
of right triangular plates and show that any cross-ply
symmetrically laminated triangular plate can be stu-
[3 J D. ]. GoRMAN 1986 Journal oj Sound and Vibration 106, 419-431. Free vibration analysis of
died. Here [0°/90° /ooJ and [0°/90°/90° /ooJ lami-
right triangular plates with combinations of clam-
nated triangular plates with aspect ratio b/ a = 1 and b/
ped-simply supported boundary conditions.
a = 1.5 are analyzed. No comparison can be given be-
cause as so far no suitable references can be found.
Table 4 and Table 5 give the natural frequency
parameters for the cross-ply symmetrically laminated
[4 J D.].
GoRMANO
1987 Journal oj Sound and Vibra-
tion 112, 173-176. A modified superposition
method for the free vibration analysis of right
triangular plates.
right triangular plates with aspect ratio b/ a = 1 and 1. 5.
[5 J H. T. SALIBA 1990 Journal ojSound and Vibration
The nodal lines of 8 modes of these plates are shown in
139, 289- 297. Transverse free vibration of simp-
Figure 8 and Figure 9. From these Tables and
ly supported right triangular thin plates: a highly
Figures, it can be seen that for the same aspect ratio,
accurate simplified solution.
the first frequency parameter of [0° /90° /ooJ plate is
higher than that of [0°/90°/90° /0°1 That means the
[ 6 J H. T. SALIBA 1995 Journal ojSound and Vibration
183, 765-778. Transverse free vibration of right
fibres orientated vertically to the clamped edge in-
triangular thin plates with combinations of
crease the first frequency more highly than those
clamped and simply supported boundary condi-
orientated horizontally to the clamped edge. Their
mode shapes of [00/90 0/0 0J and [00/90 0/90 0/0 0J
tions: a highly accurate simplified solution.
plates seem similar.
[ 7 J C. S. KIM and S. M. DICKINSON 1990 Journal oj
Sound and Vibration 141, 291- 311. The free
flexural vibration of right triangular isotropic and
6. CONCLUSIONS
An approximate method has been proposed for the
free vibration of cross-ply symmetrically laminated
cantilever triangular plate with various aspect ratios.
orthotropic plates.
[ 8 J S. MIRZA and M. BULANI 1985 Computers and
Structures 21, 1129-1135. Vibration oftriangular
plates of variable thickness.
triangular plate in the free vibration analysis. Based on
[9 J K. M. LIEW, C. W. LIM and M. K. LIM 1994Journal oj Sound and Vibration 177, 479-501. Trans-
the Green function of the equivalent rectangular plate,
verse vibration of trapezoidal plates of variable
the characteristic equation of free vibration is ob-
thickness: unsymmetric trapezoids.
An equivalent rectangular is used to replace the right
tained. The lowest 8 values of frequency parameter
[lOJ O. G. MCGEE and G. T. GIAIMO 1992 Journal oj
and their mode shapes are given. These results show
Sound and Vibration 159, 279-293. Three-
that the numerical solutions obtained by the proposed
dimensional vibrations of cantilevered right trian-
method have a good convergence and satisfactory ac-
gular plates.
[11] K. Y. LAM, K. M. LIEW and S. T. CHOW 1990 Int.
curacy.
I Mech. Sci. 32, 455-464. Free vibrations analyAcknowledgements
The present study has been sponsored by the Japan
Society for the Promotion of Science (JSPS).
sis of isotropic and orthotropic triangular plates.
[l2J S. K. MALHOTRA, N. GNESAN and M. A.
VELUSWAMI 1989 Composite Structures 12, 17-25.
Vibrations of orthotropic triangular plates.
REFERENCES
[13J D. V. BAMBILL, P. A. A. LAURA and R. E. ROSSI
[ 1 J LEISSA 1969 Vibration oj plates (NASA SP-
1998 it Journal oj Sound and Vibration 210, 286-
160) . Washington, D. C.:Office of Technology
290. Transverse vibrations of rectangular,
Utilization, NASA.
trapezoidal and triangular orthotropic, cantilever
[2 J D.
J.
GoRMAN
1983 Journal oj Sound and Vibra-
plates.
tion 89, 107-118. A highly accurate analytical
[14J T. SAKIYAMA and M. HUANG 1998 Journal oj
solution for free vibration analysis of simply sup-
Sound and Vibration 216, 379-397. Free vibra-
ported right triangular plates.
tion analysis of rectangular plates with variable
Free vibration analysis of cross - ply symmetrically laminated right cantilever triangular plates
thickness.
A 44
P 82
F
Appendix A
111 = F l23 = F 134 = 1, F 146 = 75 12 , F
= 7522 ,
F l56
F l57
=F
l66
= 7526 ,
F 167
147
F256
= F 225 = F 233 =
=
fl75 26 ,
F 278 = F30907
F 322
fl75T
otherFkts =
F 246
F257 = fl 75 12'
= F 3lO06 =
= F 331 = -
= -
fl'
fl'
= F 267 =
F 266
=
D 16 ,
f3 ij75ij, P S6 = fl kA45 f3 ij' P
k(A45 f3 ii+ flA 55 f3 jj), other P tp=
=
F 247
D ll
fl
aliOiOl
=
b30jj02
= b50jj03 = b60jj04 =
=
a3iOi02
=
IPl'
= F 354 = F 363 = -
fl kA 55 ,
fl 75F371
apijjd=
t~
= F 3S2
A p2
+ 75 661 P6)'
= 0,
= 0,
B P7
= fl
(A451p7+A55IPs) ,
A p3
=
Ip2'
A P4
A P7 =
=
Ip3'
B P5
=
fl Ip2'
(75 111 p4
Cplkl =
A p5
75 161p6 + 75
B p6
=
26
A p6
(A45IP7+A55IPS) ,
fl (75 16 / P4
f1 75Tkl/p7,
CpSkl=O[IPt] = [ppt]-l,
b pij/ d =
L:
f3 jIBpt[atO/fd-ati/fd(l- a lj)]
t t
t {t
t~
1
j
f3 jIBpt[btOlgd- btilgd(l- 0 lj)J
1=0
j
i
= f1 k
CP2kl=
f3 ikApt[btkOgd-btkjgd(l- 0 ki)]
L
+ 7526 / P5
Bps
L L
+
fl
13
qpij=
i
L: {L
t~
a
{3 ik {3 jl Cptkl btklgd (l - 0 ki Ij) }
1=0
k~O
Pll=f3ii'
f3 ik f3 jI CptkIf1tkl/d(l- 0 ki 0 Ij) }
1=0
k~O
+
B p3
f3 jj' P 22 = - fl f3 ij' P 23 = f3 ii' P 25 = fl f3 jj' P 31 =
- fl f3 ij' P 33 = - f1 f3 ij' P 34 = f3 ii' P 45 = - f1 f3 iPij, P 46 =
75 12 f3 ii + fl75 16 f3 jj' P 47 = 75 16 f3 ii + f1 75 11 f3 jj' P 54 = fl f3 ij75ij,
P 56 = 75 22 f3 ii + fl 75 26 f3 jj'
P 57 = 7526 f3 ii + f1
75 12 f3 jj, P 63 = - fl f3 ij75ij, P 666 = 7526 f3 ii + f1 7566 f3 jj'
P 67 = 7566 f3 ii + fl 75 l6 f3 jj'
P 71 = - f1 f3 ij75ij,
P 76 = fl k
-
b300002
j
1p07 + 7566
B p2 =flIPl,
+ 75 121 p5 + 75 161p6),
fllp3+
= 0,
+ fl 75 T kl / p8, Cp3kl = fl 75k11 p6, CP4kl = fl 75kl / p7' Cp5kl
fl 75kl/ p4' CP6kl = - f1 k (A 441 p7 + A 451 pS' CP7kl = - f1 k
P 12 =
b20jjOl =
f3 ikApt[atkOjd-atkjjd(l- a ki)]
k=O
1p2
=
b70jj05 = bSOjj06,
a8iOi06,
k=O
I
°
B P4
= a6iOi04 = a7iOi05 =
t=O
Aps=k(A44/P7+A45IPS)Bpl=0,
= fl IP3'
a4iOi03
L: { L:
+
7512/' p4 + 75 221p5 + 75 261 p6'
IPOS'
°
i
13
=
Appendix B
=
45
Appendix C
kA 44
+
ApI
f3 J)'
87 = fl kA55 f3 ij'
= k(A 44 f3 u + f1 A
fl
/17566
fl kA 45 , F 288 = F 387
F 345
P 7S
= 75 16
= D 66 , F 178 =
fl
= /1 kA 45 f3 ij'
P 77
= -
P 8S =
Fl88=kA45
F 212
f3 ij'
133
f3ik A pt[qtko-qtkj(l- Oki)]
k= 0
1
j
+
L
f3 jIBptCqtOl-qti/(l- 0 lj)J
t=O
fl
+
i
j
k~O
/=0
L L
f3ik f3jICptklqtkl(l-
0 ki 0 lj) }