Transcript
Composite Structures Vol. 38, No. 1-4, pp. 99-110 0 1997 Elsevier Science Ltd. All rights resewed Printed in Great Britain O263-8223/97/$17.00 + 0.00 PII:SO263-8223(97)00046-9
Multiple eigenvalue optimization of composite structures using discrete third order displacement models C. M. Mota Soares*, C. A. Mota Soares IDMEC1I.S T-Institute de Engenharia Meckca-Instituto
Superior Tkcnico, Av. Rovisco Pais, 1096, Lisboa, Codex, Portugal &
V
M. Fkanco Correia
ENIDH-Escola Nbutica Infante D. Henrique, Av. Eng. Bonneville France, PaGode Arcos, 2780 Oeiras, Portugal
This paper deals with the implementation and test of a non-smooth eigenfrequency based criterion to evaluate the directional derivatives applied to multilaminated plate structures, when non-differentiable multiple eigenfrequencies occur during the structural optimization process. The algorithm is applied to a family of C? Lagrangian higher order shear deformation theory discrete models. Angle ply design variables and vectorial distances from the laminate middle surface to the upper surface of each ply are considered as design variables. The efficiency and accuracy of the algorithm developed is discussed with an illustrative case. The analytical single and/or directional derivatives are compared to forward finite difference derivatives for the developed discrete models. 0 1997 Elsevier Science Ltd.
INTRODUCTION
formance of the structures by manipulating certain design variables. Design sensitivity analysis is important to accurately know the effects of design variables changes on the performance of structures by calculating the search directions to find an optimum design. To evaluate these sensitivities efficiently and accurately it is important to have appropriate techniques structural associated to good models. It is well known that the analysis of laminated composite structures by using the classical Kirchhoff assumptions can lead to substantial errors for moderately thick plates or shells. This is mainly due to neglecting the transverse shear deformation effects which become very important in composite materials with low ratios of transverse shear modulus to in-plane modulus. This can be attenuated by Mindlin’s first order shear deformation theory [l, 21, but this theory yields a constant shear strain variation through the thickness and therefore requires the use of shear correction factors [3] in order to approximate the quadratic distribution in the elasticity
Laminated composite materials are being widely used in many industries mainly because they allow design engineers to achieve very important weight reductions when compared to traditional materials and also because more complex shapes can be easily obtained. The mechanical behavior of a laminate is strongly dependent on the fiber directions and because of this the laminate should be designed to meet the specific requirements of each particular application in order to obtain the maximum advantages of such materials. Accurate and effidesign sensitivity cient structural analysis, analysis and optimization procedures are very important to accomplish this task. Structural optimization with behavioral constraints, such as stress failure criterion, maximum deflection, natural frequencies and buckling load can be very useful in significantly improving the per*Address for correspondence: IDMEC/I.S.T.-Instituto Engenharia Mecbnica-Instituto Superior Tecnico, Rovisco Pais, 1096 Lisboa, Codex, Portugal.
de Av.
99
100
C.
M. Mota Soares et al.
theory. More accurate numerical models such as three-dimensional finite elements models can be used with adequate refined meshes in order to contemplate acceptable aspect ratios, but these models are computationally expensive. A compromising less expensive situation can be achieved by using single layer models, based on higher order displacement fields involving higher order expansions of the displacement field in powers of the thickness coordinate. These models can accurately account for the effects of transverse shear deformation yielding quadratic variation of out-of-plane strains and therefore do not require the use of artificial shear correction factors and are suitable for the analysis of highly anisotropic plates ranging from high to low length-to-thickness ratios. Pioneering work on the structural analysis formulation based in higher order displacement fields can be reviewed in Lo et al. [4,5], where a theory which accounts for the effects of transverse shear deformation, transverse strain and nonlinear distribution of the in-plane displacements with respect to thickness coordinate is developed. Third order theories have been proposed and/or reviewed by several researchers [6- 151. Recently, Abrate [16] gave a perspective of work carried out by different researchers in the field of the optimum design of composite laminated plates and shells subjected to constraints on strength, stiffness, buckling loads and fundamental natural frequencies. Most of the papers reviewed are based on variational approxima-
tion methods. The use of higher order models as well as the problem of non-differentiability of multiple eigenvalues that may occur during the structural optimization process is not mentioned. Multiple eigenfrequencies do appear frequently on complex structures. It is known [17-341 that multiple eigenvalues are not differentiable in the common sense and therefore analytical single design sensitivity analysis [35] cannot be used. When this problem occurs it can be overcome by evaluating the directional derivatives using the concept of generalized gradient [24]. An overview of recent developments in this area, mainly related to isotropic structures, can be found in Ollhoff et al. [30], Seyranian et al. [31] and Seyranian [34] among others. In the present paper 6’ nine node Lagrangian higher order models applied to eigenfrequency sensitivity analysis of multilaminated thin-to-thick plate structures [36] is extended to contemplate the non-differentiability of multiple eigenfrequencies. The design variables considered are the ply orientation angles of the fibers and the vectorial distances from the laminated middle surface to the upper surface of each layer. The sensitivities of eigenfrequency response with respect to the design variables are evaluated analytically. An illustrative numerical example is presented to validate the feasibility of the proposed design sensitivity approach to evaluate directional derivatives when multiple eigenvalues are involved.
HIGHER ORDER DISPLACEMENT FIELDS
In order to approximate the three-dimensional elasticity problem to a two-dimensional laminate problem, the displacement components u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) at any point in the laminate space (Fig. 1) are expanded in Taylor’s series powers of the thickness coordinate z. The following higher order displacement fields can be written [l l] HSDT 11 N&Y,ZJ)
= ~O(X,Y,t)+Z~.I(X,y,t)+z2~~(x,Y,t~+z~~~~~(x,Y,t)
W,Y,ZJ)
= V~~X,Y,t~+Z6)\.~X,Y,t~+zzv~~x,y,t~+z~~~~x,Y,t~
W,y,z,t)
= w,(x,y,t)+zcp,(x,y,t)+z2w~(x,y,t)
(1)
HSDT 9 u(x,y,z,t) = u,(x,y,t)+z~.,(x,y,t)+z2u;;(x,y,t)+z-~~~(x,~,t) v(x,y,z,t) = v,~x,Y,t~+z~,~,(x,y,t)+z*v2;~x,y,t)+z~cp~(x,Y,t~ w(x,y,z,t)
= w&,y,t)
(2)
where U, v, w are the displacement components of generic point in the X, y, z directions, uO, vo, W" are the displacements of a generic point on the reference surface, t is the time, Q,, 0, are the rotations
Multiple eigenvalue optimization
101
+
1
/T I
hlu
... hk
4 .. .
hl ho .
\
Fig. 1. Laminate
geometry
and coordinate
systems (x, y, z) and (1, 2, 3).
of normal to the reference surface about the y and x axes respectively and cpz,u,,*, vO*, wO*, cpx*, cpz* are the higher order terms in the Taylor’s series expansions, defined at the reference surface.
FINITE ELEMENT MODEL The finite element formulation for the displacement field HSDT 11 represented by eqn (1) applied to 9-node Lagrangian quadrilateral elements, will be briefly described. The strain components are given by I
%I
w
8X
ax
ae,
%I =
vo, --
ay
+z
aY au0
au;, ax av;
av,
ay+ ax
aY 0
-Z2
2w;
au;, a$ -ay + ax
ae, ae, -ay + ax \
aw;
I I-
aqlT_
+z3
ax a(p; ay
I-
(3)
O
a& N ay + ax
/
/
3&+ -
ay
aw; 3&+ ax
(4)
The strains can be written as \
(5)
C. M. Mota Soares
102
et al.
where Z,, and 2, are matrices containing powers of z coordinate (z” with n = 0, .... 3) defined in accordance with displacement fields and strain relations and I;,?, and a: are vectors containing the bending and membrane terms of the strain components and the transverse shear, respectively, given bY
ah
c; =
u.,.+i
av
aw,9 G+ sav
3 or+ -
ax
acp_
aw; aw; T , 3cp,+ 3&+ ax ax I av ’
3 2u,*,+L
Using C” Lagrangian shape functions 11,371 the displacements and generalized defined in the reference surface are obtained within each element, respectively, by
Ii
(7) displacements
l4 V
=Zm{u()
V() W()
0, 0,. cp; Ll;
v;w;q: cpy
(8)
W
where 2, is a matrix containing powers of z coordinate defined in accordance with the displacement field, N; are the Lagrange shape functions of node i and & is the displacement vector of node i which is related to the displacement vector of the element, @“,by qS’= { . ..&..JT.
i=
1, .... 9
The strains in eqns (6) and (7) are represented
(10) as
{$j=[~;j@ where B,, and B,s are the strain-displacement matrices, respectively, for bending and membrane transverse shear, relating the degrees-of-freedom of the element with the strain components.
Fig. 2. Square plate with central circular hole -
finite element
mesh.
and
103
Multiple eigenvalue optimization
Mode 1
Mode 2
Mode 3
Fig. 3. Vibration
modes for lamination
sequence
Mode 4 [45”/ - 457.
Applying Hamilton’s principle to the Lagrangian functional assuming small displacements and adding the contributions of all finite elements in the domain one obtains for free harmonic vibration the equilibrium equation of the finite element discretized structure (K-Q!!)q,=O,
L?= 1, ...) N
(12)
where K and M are respectively the structure stiffness matrix and mass matrix and N represents the total number of degrees of freedom. The solution of this eigenvalue problem consists of N eigenvalues A,,= CO:and corresponding eigenvectors qn, where CO,is the natural frequency of mode n. K and M matrices are obtained by assembling in the usual way the corresponding element matrices K,, A4,.
Mode I
Mode 3 Fig. 4. Vibration
Mode 2
Mode 4
modes for lamination
sequence
[30”/-60”].
C. M. Mota Soares et al.
104 Table 1. Eigenfrequency
sensitivities
in order to ply angles obtained with QB-HSDT 11 model, for simply supported z-ply [45”/ -45”] square plate with central circular hole
146.0411
dto,ldc
259.5086
d[min(w,,
259.5086
445.8111
Analytical
Direction (01, 0,)
Frequency (radh)
LO 0,l LO 0,l I -I,
tu,)]/dc
d[max(tuz, tu,)]/dc
2 --X
110” 031 I ,5’ 170
dwddc
Multiple
Single
-44.7128 -44.7128 - 19.9962
-0.163641 x 1O-5 0.163641 x lo- ’ 43.6214 - 43.6214 - 19.5081
44.7 128 44.7128 19.9962
.s 2
for forward finite difference
Table 2. Eigenfrequency
sensitivities
.J -
derivatives
-0.498112 x 10 -’ 0.498058 x lo-”
130.0205
dw,ldc
254.0572
d[min(to,,
254.0572
d[max(tu,, tu,)]/dc
426.0753
dwddc
sensitivities
259.3818
d[min(w,,
Global finite difference”
Multiple
Single
- 26.7139 - 26.7139 5.20718
47.1639 47.1639 56.7199 - 18.3624 8.94208
47.1590 47.1527 - 26.7067 -26.7130 5.31705
65.0715 65.0715 46.2549
- 18.3623 56.7199 42.5200
65.0594 65.0622 45.9627
63.6208 63.6208
63.6195 63.6212
6 = 0.000001”.
in order to ply angles obtained with QkHSDT 9 model, for simply supported 2-ply [45”/ - 45”] square plate with central circular hole Direction (01, w
dto,idc
“Increment
derivatives
sensitivities
-
I,0 031
for forward finite difference
145.9545
445.5345
Analytical
130 O,l 1,o 071 I 2 ,5 ’ ,s I,0 O,l
t+)]/dc
Frequency (radh)
259.3818
6 = 0.000001”.
Direction (01, 02)
Table 3. Eigenfrequency
0.114643 x 10 ’ -0.114643 x 10 ’
in order to ply angles obtained with QFHSDT 11 model, for simply supported 2-ply [30”/ - 60”] square plate with central circular hole
Frequency (radh)
“Increment
-0.802172 x lo--’ 0.229122 x 10 -’ -44.7136 -44.7102 - 19.7928 44.7073 44.7079 19.79225
-43.6214 43.6214 19.5081
071 “Increment
Global finite difference”
sensitivities
w,)]/dc
d[max(to,, tu,)]/dc
dw,ldc for forward finite difference
170 0,l 1,O O,l 1 .s’ I,0 071 I ,5’ 1,o 091
-2 ,3
Analytical
sensitivities
Multiple
Single
-
-0.162070 x 10 -’ 0.162070x lo-’ 14.7842 - 14.7842 -6.61170
-44.7366 - 44.7367 - 20.0068 44.7367 44.7366 20.0068
2
- 14.7842 14.7842 6.61170
Global finite difference” - 0.200543 x 10 ’ -0.160434 x 10-l - 44.7446 - 44.7360 - 19.6851 44.7280 44.7245 19.6737
-,5
derivatives
6 = 0.000001”.
-0.458874 x lo-’ 0.458800 x 10 --’
-0.114643 x lo- ’ 0.171964 x 10 *
105
Multiple eigenvalue optimization
The stiffness and mass matrices of the element are evaluated,
respectively,
as
(13) M,=1+_1, I”_‘,NT
kF, p,k_,
ZzZ,dz
(14)
N det J d5dv
where ok is the constitutive matrix in the (x, y, z) laminate axes for kth ply [6,7,38], N is the matrix of the Lagrange shape functions and (k the material density of kth layer. NL is the number of layers of the laminate, hk is the vector distance from the middle surface of the laminate to the upper side of kth ply (Fig. 1). Finally, det J is the determinant of the Jacobian matrix of the transformation from (r, q) natural coordinates to element (x, y, z) coordinates. The finite element model having the displacement field represented by eqn (l), will be referred to as Q9-HSDT 11. The element which displacement field is given by eqn (2) is developed easily from this parent element by deleting the appropriate degrees of freedom leading to the finite element discrete model referred to as Q9-HSDT 9. The C? 9-node Lagrangian first order discrete model, referred to as QPFSDT 5, can also be obtained by deleting all high order terms and introducing shear correction factors [l-3,38].
SENSITIVITY ANALYSIS For single eigenvalues, considering a vibration mode sp which corresponds the natural frequency oP normalized through the relation qz M qp= 1, the sensitivities of natural frequency with respect to changes in the design variable bi are [35]
these sensitivities can be efficiently obtained do, -=db,
1 20,
2 q;’ IEE
i3KL --O, abi
2 aM: ab;
at element level through e/ qP
where E represents the set of elements in which the design variable In the case of multiple eigenvalues, if S = s+* = ... = Aj+, are plicity and C+= qi,, = ... = q+m are orthonormal eigenvectors of s, the direction c = {cl, .... ci, .... c,,~“}, where ndv is the number of calculating the eigenvalues of the x matrix of dimension (m+ 1) x ated by [31,33]
(16) bj is defined. the eigenvalues with m+ 1 multithen the directional derivatives in design variables, are obtained by (m+l) whose elements are evalu-
(17) where the vector c has the norm ((c((= ~c:+...+c~+...+c~,, = 1. Let us define the functions fl =max&
Aj+l,
.h=min(ilj,
Aj+l9
df, dc
(19)
. . .. Aj+J
If p1 and pL:!are respectively the maximum and minimum eigenvalues of matrix directional derivatives of the functions fi and f2 in the direction c are given by -
(18)
.. . . Aj+m)
df2
=p, and -
dc
=P2
x then
the
(20)
C. M. Mota Soares et al.
106
For comparison purposes the eigenfrequency obtained by forward finite difference by df __dc-
directional
derivatives
in the direction
c can be
f’(a) - SW
(21)
6
The perturbed
design represented
by vector a is given by
a = b+&
C-W
where 6 is a small positive design perturbation variables.
and b = {b,, .... bi, .... bndv} is the vector of design
nodes is used (Fig. 2). The simply supported boundary conditions for HSDT 11 model were taken as u~~=u~=wo=w~=~~=cp~=(p_=O at y = +a/2 and vO=v~=wO=w~=O~=(p~=(p~=O at x = *a/2. The boundary conditions for HSDT 9 and FSDT 5 models can be obtained from HSDT 11 by deleting the appropriate terms.
NUMERICAL APPLICATION The free vibration problem of a 2-ply simply supported square plate with a central circular hole is considered. The side dimension of the plate is a = 2 m and the hole diameter is a/3. A finite element mesh with 288 elements and 64 Table 4. Eigenfrequency
sensitivities
in order to ply angles obtained with Q9-HSDT 9 model, for simply supported 2-ply [30”/ - 60”] square plate with central circular bole
Frequency
Direction
(radh)
(O,, b)
129.9607
dw,/dc
253.9611
d[min(cv,, w,)]/dc
253.9611
d[max(tuz, (+)]/dc
425.8654
dw,ldc
Analytical
1,o 0, I 1,o 0. I I 2 --- . 5 i ‘_ IlO 0, I I 7 5 IlO
for forward finite difference
Table 5. Eigenfrequency
sensitivities
derivatives
Direction
(0,. 02)
146.2792
1,O 0.1 1.o 0, I
259.8325
I --z.
446.528 1
dwddc
‘Increment
Single
- 26.8269 - 26.8269 5.06551
47.0852 47.085 1 63.9992 - 25.8454 5.50448
47.0931 47.0874 - 26.8150 - 26.8173 5.37664
64.9806 64.9806 46.1231
- 25.8454 63.9991 45.6841
64.9803 64.9769 45.7667
63.3884 63.3884
63.3926 63.3926
-
in order to ply angles obtained with QB-FSDT 5 model, for simply supported 2-ply [45”/ - 45”] square plate with central circular hole
(radh)
d[max(cu,, cl~,)]/dc
Multiple
6 = 0.000001”.
Frequency
259.8325
Global finite difference”
5
0, 1
“Increment
sensitivities
for forward finite difference
2
Analytical
sensitivities
Multiple
Single
-44.5991 -44.5991 - 19.9453
-0.165879 x lo-’ 0.165879 x 10 ’ - 27.6453 27.6453 12.3634
Global finite difference” 0.401086 x 10 ~’ 0.000000 - 44.5967 - 44.5985 - 19.6192
-Y
5 5 110 ‘_ 0. I I 2 --- . 5 5 1% I. 0, I derivatives
44.599 1 44.599 I 19.9453 ii = 0.00000 I”.
27.6453 - 27.6453 - 12.3634
44.6002 44.5956 19.6249
-56.2312 56.2316
-0.573212
0.000000
x 10 r
Multiple eigenvalue optimization
107
Table 6. Eigenfrequency sensitivities in order to ply angles obtained with Q9-FSDT 5 model, for simply supported 2-ply 130’7- 6O”Isquare plate with central circular hole Frequency (rad/s)
Direction (h 0,)
130.1740
do,/dc
254.2542
d[min(w,, w,)]/dc
254.2542
d[max(w,, o,)l/dc
426.5209
dwddc
Analytical sensitivities
LO 031 LO 071
Global finite difference”
Multiple
Single
-26.3020
47.4291 32.8684 47.4291
- 26.3020 1.71635
6.21572 20.2587
47.4421 47.4426 - 26.2925 - 26.2982 6.01434
65.3861 65.3861 46.7205
6.21573 32.8684 32.1782
65.3894 65.3980 46.3563
64.3679 64.3679
64.3707 64.3724
-
aIncrement for forward finite difference derivatives 6 = 0.000001”.
The
material
properties are: E, = 138 GPa, GPa, G12 = Cl3 = Gz3 = 7.1 GPa, ~12 = ~13 = ~23 = 0.3, p = 2000 kg/m3 (mass density). The thickness of each ply is 0.01 m. Two lamination sequences are considered: [45”/ - 451 and [30”/- 60“]. The 0” direction is aligned with the X axis. Both lamination sequences lead to repeated eigenfrequencies corresponding to the second and third vibration modes. The vibration modes for both lamination schemes are shown in Figs 3 and 4. Due to the high side-to-thickness ratio of 100 and low degree of anisotrophy (El/E2 ratio) the natural frequencies and corresponding sensitivity results obtained with higher order and first order models are all in a good agreement.
E,=E3=
The eigenfrequency sensitivities in order to ply angles 8, and 13~obtained with Q9-HSDT 11, QPHSDT 9 and Q9-FSDT 5 discrete models, for the above lamination sequences, are shown, respectively, in Tables l-6. The multiple derivatives (eqns analytical directional (17)-(20)) evaluated along chosen directions c and the single design derivatives evaluated using eqn (16) and corresponding directional derivatives are compared with alternative values obtained with global forward finite difference (eqns (21) and (22)) with a perturbation 6 = 0.000001” using the same discrete model. Tables 7-9 show the eigenfrequency sensitivities in order to vectorial distances from middle surface of the laminate to the upper surface of
8.96
Table 7. Eigenfrequency sensitivities in order to vectorial distances hk obtained with Q9-HSDT 11 model, for simply supported 2.~1~ [30”/ - 60’1 square plate with central circular hole
130.0205
do,/dc
254.0572
d[min(w,, w,)]/dc
254.0572
426.0753
Analytical sensitivities
Direction (h h,? h,)
Frequency (rad/s)
d[m=(w,,
doddc
w,)l/dc
LO,0 0, LO 0, 0, 1 1, 0, 0 0, 1, 0 0, 0, 1 1 1 XT%‘% LO,0 0, LO 0, 0, 1 1 1 lY$‘o%? 0: 1:o 0, 0, 1
Global finite difference”
Multiple
Single
2
- 0.148266 x - 0.446987 x 0.103567 x 0.422810 x
lo5 lo4 lo5 lo4
- 0.64815 x lo4 -0.259227 x 1O-4 0.648158 x lo4 -0.123981 x lo5 - 0.386985 x lo3 0.127851 x lo5 0.521950 x lo4
-0.648147 x lo4 0.448799 0.648169 x lo4 -0.148263 x lo5 -0.446871 x lo4 0.103570 x lo5 0.421704 x lo4
2
-0.103567 x 0.446987 x 0.148266 x 0.605292 x
lo5 lo4 lo5 lo4
-0.127851 x 0.386985 x 0.123981 x 0.506150 x
- 0.103564 x 0.447104 x 0.148268 x 0.604523 x
x
-
‘Increment for forward finite difference derivatives 6 = O.OOOOO1 m.
lo5 lo3 10’ lo4
-0.211640 x lo5 0.516952 x 1O-4 0.211640 x lo5
10” lo4 lo5 lo4
-0.211636 x lo5 1.67645 0.211645 x 10”
C. hf. Mota Soares et al.
108 Table 8. Eigenfrequency
sensitivities in order to vectorial distances h, obtained with Q9-HSDT 9 model, for simply supported 2-ply [30”/ - 60’1 square plate with central circular hole (ho, h I>4 -
129.9607
dto,/dc
253.9611
d[min(o,,
w,)]/dc
1, 0, 0 0, 1,o 0, 0, 1 1, 0, 0 0, 1, 0 0, 0, 1
I 253.9611
d[max(tu,, w,)]/dc
I 425.8654
-7
dto,ldc
I
,7;‘,6. I, 0, 0 O,l,O 0, 0. 1
Global finite difference”
Analytical sensitivities
Direction
Frequency (rad/s)
-
2
Multiple
Single
-. -0.148196 - 0.446896 0.103507 0.422565
-0.647778 x lo4 -0.258566 x 10V4 - 0.647778 x lo4 -0.10825 x lo5 - 0.352033 x lo4 0.143453 x 10” 0.585643 x lo4
-0.647767 0.449340 0.647789 -0.148194 -0.446780 0.103510 0.421460
x x x x x
lo4 10’ lo4 lo5 lo4
-0.143453 0.352033 0.108250 0.441930
-0.103504 0.447013 0.148199 0.604241
x x x x
lo’ lo4 10’ lo4
x x x x
lo” lo4 lo5 lo4
x lo4
,ii;
I --
--0.103507x 105 0.446896 x lo4 0.148196 x 10” 0.605009 x lo4
2
x x x x
10’ lo4 lo5 lo4
,6 ’ .6
1) 0, 0
-
0, 1, 0 0, 0, I
-
-0.211485 x 10” 0.516745 x 1O-4 0.211485 x lo5
-0.211481 x lo5 1.67717 0.211489 x 10’
Yncrement for forward finite difference derivatives b: = 0.000001 m.
agreement between the multiple analytical directional derivatives (eqns (17)-(20)) and global forward finite difference is obtained.
kth ply - hk, k = 0, .... 2 obtained with Q9HSDT 11, Q9-HSDT 9 and QB-FSDT 5 discrete models for the lamination sequence [30”/-60”]. Again in this case the directional derivatives are compared with single derivatives and global forward finite difference with a perturbation 6 = 0.000001 m. In both cases it can be observed a good agreement between the single analytical sensitivities and the global forward finite difference whenever the eigenvalue is unique. In the case of repeated eigenvalues the single analytical method gives wrong results. In this case a good Table 9. Eigenfrequency
254.2542
Direction (h,, h,> hz) d(o Jdc
1) 0, 0
d[min(co,, co,)]/dc
0, I, 0 0, 0, 1 1, 0, 0 0, 1, 0 0, 0, I
I 254.2542
426.5209
d[max(w,, co,)]/dc
dwddc
A non-smooth eigenfrequency based criterion has been applied to multilaminated plate structures in order to evaluate the directional derivatives when non-differentiable multiple eigenfrequencies occur. The algorithm has been applied to a family of Co Lagrangian higher
sensitivities in order to vectorial distances hL obtained with QkFSDT supported 2-ply [30”/ - 60”] square plate with central circular hole
Frequency (rad/s) 130.1740
CONCLUSIONS
5 model, for simply
Analytical sensitivities Multiple
Global finite difference”
Single
- 0.649469 x lo4 -
I
:6’ --,6 , 1, 0, 0 0, 1. 0 0, 0, 1 I I ,6’,6‘~
2
- 0.148339 -0.447825 0.103556 0.422765
x x x x
10’ lo4 10’ lo4
-0.258359 0.649469 -0.105351 -0.411929 0.146544 0.354504
-0.103556 0.447825 0.148339 0.605589
x x x x
lo’ lo4 lo5 lo4
- 0.146544 0.411929 0.105351 0.430093
x x x x x x
1Om-4 lo4 lo5 lo4 10’ lo4
- 0.649687 0.451860 0.649709 -0.148552 - 0.447709 0.103775 0.422541
x 1O6 x x x x x
lo4 lo” lo4 lo5 lo4
x x x x
lo5 lo4 10” lo4
-0.103770 0.447942 0.148558 0.605703
x x x x
lo” lo4 10’ lo4
,z
2
1, 0, 0 0, 1, 0
-
0, 0, 1
-
“Increment for forward finite difference derivatives (5= 0.000001 m.
-0.211879 x lo5 0.515666 x lop4 0.211879 x 10”
-0.212105 x lo’ 1.68202 0.212114 x 10’
Multiple eigenvalue optimization order shear deformation theory discrete models suitable for the analysis of thin-to-thick rnultilaminated structures. The feasibility of the described method was validated through the numerical solution of an illustrative example. From the results shown for natural frequencies one can observe a good agreement between global forward finite difference sensitivities and analytical directional derivatives when multiple eigenfrequencies are involved. When multiple eigenfrequencies occur single analytical sensitivities evaluated by eqn (16) should not be used. The frequent occurrence of multiple eigenfrequencies that can be observed from the simple test case carried out, emphasizes the need to introduce adequate algorithms into the structural optimization codes to overcome this non-differentiability problem.
ACKNOWLEDGEMENTS This work was partially supported by EU project ‘Diagnostic and Reliability of Composite Materials and Structures for Advanced Transportation Applications: HCM-CHRTXCT93-0222’.
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