PII: S 1359-8368(96)00003-0
ELSEVIER
Composites: Part B 27B (1996) 395-405 Copyright © 1996ElsevierScienceLimited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
Vibration and optimum design of rotating laminated blades
Ting Nung Shiaut, Yung Dann Yu¢ and Chun Pao Kuot tDepartment of Mechanical Engineering National Chung Cheng University, Chia Yi, Taiwan, ROC ~lnstitute of Aeornautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, ROC (Received 17 April 1995, accepted 18 December 1995) The vibration and optimum design of a rotating laminated blade subject to constraints on the dynamic behavior are investigated. Restrictions on multiple natural frequencies as well as the maximum dynamic deflections of rotating laminated blades are considered as constraints on the dynamic behavior of the system. Aerodynamic forces acting on the blade are simulated as harmonic excitations. The optimality criterion method and the modified method of feasible directions have been successfully developed for optimizing the weight of the rotating laminated blade. Effects of radius of the disk, aspect ratio, and rotating speed on the system dynamic behaviors and/or the optimum design are also studied. The vibration analysis shows that most of the bending modes can be significantly affected by the rotating speed and the radius of the disk. Results also show that the optimum weight with constraints on the dynamic response is higher than that with frequency constraints. Moreover, results show that the weight of the rotating laminated blade can be greatly reduced at the optimum design stage. Copyright © 1996 Elsevier Science Limited (Keywords: optimum; laminated blade; rotating; vibration)
INTRODUCTION The vibration of a rotating blade is a particularly important item in the design of modern high-speed engines. Much has been published on the vibration analysis of rotating blades. The simplest model of a beam simulates the blade with a high aspect ratio. Carnegie ~ first studied the natural frequencies of a rotating blade based on the beam model using a theoretical expression for potential energy in Rayleigh's method. Putter and M o n o r 2, Kaza and Kielb 3, and Ansari 4 have studied the natural frequencies of tapered blades, the effects of warping and pretwist angle on the torsional vibration, and the effects of shear deformation, rotatory inertia and Coriolis effects on the vibrations of a rotating blade. For blades with a low aspect ratio, the shell or plate models will be good approximations. Rawtani and Dokainish 5 investigated the influence of rotation on plane plates by using a finite element method. Petericone and Sisto 6 used the Rayleigh-Ritz method based on thin-shell theory to investigate the influence of pretwist and skew angle of nonrotating blades. Henry and Lalanne 7 determined the modes and frequencies using plane triangular elements. Sreenivasamurthy and Ramamurti 8 considered the Coriolis effect using triangular shell elements. They found that the increase
in Coriolis effect would decrease the bending and first torsional natural frequency. Assuming algebraic polynomials for displacement functions in the Ritz method in conjunction with a shallow-shell theory, Leissa e t a / . 9'1° studied the effects of setting angles, variable thickness, and curvature on the vibration characteristics of a rotating blade. For composite blades, Crawley II determined the natural frequencies and mode shapes of a number of graphite/epoxy and graphite/epoxy-aluminum stationary plates and shells both analytically using quadrilateral finite elements and experimentally. Crawley and Dugundji 12 used the partial Ritz method to determine the nondimensional natural frequencies of stationary composite laminated plates. Wang et a[. 13 studied the vibration of symmetrically stacked rotating composite laminated plates using complete sets of orthogonal polynomials in conjunction with the Galerkin method. The optimum structural design with constraints on dynamic behaviors has been studied by many authors 14-16. The optimum design of a rotating blade under constraints on structural frequencies and dynamic deflections has been a very interesting topic in the field of turbomachinery-blade design. The optimization techniques employed in this study are the optimality
395
Rotating laminated blades." Ting Nung Shiau et al. criterion method (OCM), the method of feasible directions (MFD) proposed by Zoutendijk 17, and the method of modified feasible directions (MMFD). MMFD, described by Vanderplaats 18, combines the best features of MFD and the generalized reduced gradient method of Gabriele and Ragsdel119. Kiusalaas. and Shaw 2° improved the numerical technique for the case with multiple frequency constraints. In this study, the OCM is first applied for the optimum design of a rotating laminated blade with multiple frequencies and/or dynamic deflection constraints, and the MFD and MMFD are then employed for the comparison of optimum results.
DYNAMIC ANALYSIS A symmetrically stacked composite laminated blade rotating with a constant angular velocity ~ about a fixed axis is shown in The blade is fixed with setting angle ~ on the rotating rigid disk of radius R. Damping and Coriolis effects are neglected, and the effects of pretwist as well as taper are ignored. In addition, the external forces are assumed to be harmonic and functions of time only. To analyze the dynamic behavior of the system, a finite element method using eight-node isoparametric plate elements based on the Mindlin plate theory is used. The displacement field p, which includes the previbrational deformation in each element, can be expressed in term of nodal translational and rotational displacements (ui, vi, wi) and respectively. It can be expressed as
Z
y
0
Y
,t
p_ = [Nl]q
0.1/¢¸
/
7
""
9or////////////
"k..dlsk
~ blade
Figure 1 Configuration composite laminate blade fixed on the rotating disk with rotating speed f~
where [a°] is the inplane initial stress matrix of the blade due to the effect of the centrifugal force of the rotating system (see Appendix B), and [Q] is the transformed reduced stiffness matrix (see Appendix C). To find the kinetic energy of the element, one first defines the position vector as follows:
OP= y + v z
w
(3)
=_+p The velocity is obtained by differentiating equation (3) with respect to time: -----, =-dtd (OP) + fi x Off
(4)
where ~ = {f~l, f~2, ~3} r. The kinetic energy T of the rotating blade becomes
(1)
where the displacement field vector p = {u, v, w}r, the nodal displacement vector _q= {uj, vl, wt, ~Pxl,~Pyl,..-, u8, v8, Ws, ~bx8,¢y8}, and [Nl] is shown in Appendix A. The strain/displacement relationships are
ir
,>' / 7 -
4 5"~ J / - / / / / ~ / / -45".,~-'-~~//
Figure1.
(~)xi,ff)yi),
I It
~JJVd
T2=~IIvlp{O}T[NI]r[N1]{O}dV
(6)
TI =IIvJP{il}T[NI]T[AI[NI]{q}dV
{c} = [Uz]{q }
+ I IV I p{O}T[NI]T[A][N4]{x-'n}dV
{eg} = [N3]{q } where the strains
{e}={ex, ey,')'xy, Tyz,')'zx}r ,
r0 {eg} = { - ~ , 0-~} r ,
and matrices [N2] and IN3] are shown in Appendix A. The total potential energy U of an element can be expressed as the summation of the strain energy due to plate bending (Ub) and that due to the inplane initial force (Up)21 which are of the following forms:
U = Ub + Up
Ub = ~IIIv{q}r[N2]r[O][N2]{q}dV Up=
396
(7)
~ I J Jv{q}T[N3]T[a°][N3]{q}dV
-~ ~ I IV J D{X--'n}T[N4]T[A]T[A][N4]{X--n}dV
+ I JV J p{q}r[Nllr[A]r[A][N4]{x--n}dV where [A] =
(2)
(8)
I 0 f~3 -f~2
-f~3 0 f~l
f~2 1 -ill 0
and [N4] is shown in Appendix A. It is assumed that the external force is harmonic and uniformly distributed on the surface of the blade and can
Rotating laminated blades." Ting Nung Shlau et al. be expressed as fi =
P2
ei~p"t
(9)
P3 n
([K]g - ~vZ[M]g){C} = 0
where Nf denotes the number of frequencies of the excitation forces. The corresponding equivalent nodal force becomes " ) dA {F}e = [a [JA[N1](nN__~l{ P1 P2 } ne~p.t P3
(10)
With the system potential energy, kinetic energy, and the excitations established, one can obtain the equations of motion for an element using the Lagrangian approach as follows: [M]e{q} + [Gle{q} + [K]e{q} + {fc}e = {F}e
where ~ denotes the natural frequency and {C} is the corresponding response amplitude. Substituting equation (13) into the homogeneous part of equation (12), one obtains
(11)
The natural frequencies and corresponding mode shapes can be obtained by solving equation (14) as a standard eigenvalue problem.
Steady-state response The steady-state response due to external periodic forces with frequency 9tp, is assumed to be of the following form: {6} = {O},e i~p.'
[M]e = J J IvPINI]T[NI]dV
(15)
Substituting equation (15) into equation (12) yields ([K]g -
where
(14)
~2[M]g){a} n =
{F}g,
The solution of equation (16) gives the dynamic response due to excitation with frequency f~p. If the system is excited by multiple frequencies, the total dynamic response can be obtained by superimposing all the responses due to forces with different frequencies, i.e.
[G]e=2 I I IvP[N1]T[A][NI]dV [K]e=- J J L P[N,]T[A]r[A][N1]dV
Nf
{6} = Z { Q } n e i~p°'
+I J [v[N3]T[ °][N3]dV
OPTIMUM DESIGN
=[Kr]e + [Ke]e-+-[gg]e
Consider the layer thickness (tg) of the laminated blade as the design variables and define the design variable vector d as follows:
I p[N1]T[A]T[A][N4]dV{x-~(t)}
The matrix [G]e is due to the Coriolis effect, and {fc}e is the centrifugal force. [Kr]~ is the rotary stiffness matrix, [K~]e is the elastic stiffness matrix, and [Kg]~ is the geometric stiffness matrix. The equations of motion for the system of the rotating blade in the global coordinate system can be obtained by assembling all the governing equations of each element as follows: [M]g{S} + [K]g{6} -- {F}g
(12)
d= {tl,t2,...,tNd} T where Nd is the number of design variables. The optimum design problem can be mathematically expressed as Minimize: F(d)
(18)
Subject to: &(d) < 0, = 0,
where {6} denotes the displacement vector including the total number of degrees of freedom of the system. It should be noted that the matrix [G]~ is ignored in this study.
Naturalfrequency The natural frequencies of the system can be solved from equation (12) by neglecting the forcing term and assuming the response of free vibration to be of the form {6} = {e}ei~'
(17)
n=l
+ J I Jv[N2]T[O][N2]dV
{fc}e = - J Iv
(16)
(13)
j = 1,J
(19)
k = l, K
(20)
t I <_ti <_tu, i=
1,N d
(21)
where F(d) is the so-called objective function, &(d) is an inequality behavior constraint, hk(d) is an equality behavior constraint, and equation (21) represents the side constraints. The objective of the optimum design problem is to minimize the weight of the rotating laminated blade subject to the constraints of multifrequencies and/or the dynamic behavior of the blade. The design variables are the thicknesses of all layers. Two optimization techniques are employed to study the optimum design of a rotating laminated blade. One
397
Rotating laminated blades: Ting Nung Shlau et al. is the optimality criterion method, and the other is the modified method of feasible directions of the mathematical programming method.
or
ti-f.ti
Optimality criterion method The optimum design problem of the present study can be stated as Nd Minimize: F(d) = ~ piAiti i=1 Subject to: gj(d) = w]2 - w} < 0 j = 1, J
g'~(d) = IOzlk -
The corresponding Kuhn-Tucker conditions which are the necessary conditions for minimizing F(d) 20'22 are:
o,~
_
~
_
Ogj( d)
~ ,, Ogk(a_4) ' x--"
j=l
k=l
L +
,,
f = 0
At ~// y~. /=1 .,, og, ( a)
if tl <
ti
I > 0 if t i =
ti
< 0
< tu
(24)
if ti = t u
(27)
where t'i is the new design variable. From equation (27), when an optimum design is reached, J} is equal to unity. Note that the speed of convergence depends not only on the initial design variables but also on the relaxation factor a.
Calculation of Lagrange multipliers. From equations (26) and (27), the Lagrange multipliers Aj, A~, A~' must be calculated before recursive design. It is assumed that the change in the natural frequency can be approximated by the linear approximation Nd ,~z 2 = S-~ o~wjt (w}l) 2 - wj2 ~ at i (t'i - t,) (28) If the Lagrange multipliers are positive, then the constraints &(d), j = 1, J must be zero. Replacing coal by w]l, and substituting equations (26) and (27) into equation (28), one obtains -- Z
)~-/"
ti Oti pTAi
0(0321) 0(092)
Oti
j=l
if t~ = t u
< 0
f fit i if t~ < fit i < t u 'ti = { t I if fiti <__tit i [, t~ if f,.ti >_ t p
Q~k < 0 k = 1, K
where w], Qzk, Qy,, and wj, IOzlk, IQy[t, are the specified and calculated natural frequencies, and the components of the dynamic response amplitude in the y- and zdirections, respectively. The side constraints are given as < t i <_ t u. The Lagrangian functional is defined as Nd J K L (b = Z p i A i t i + Z Ajgj(d) + Z A'kg'k(d) + Z A}'g;'(d) i=1 j=l k=l 1=1 (23)
if tl < ti < t u ifti=tl
where a is the relaxation factor that determines the magnitude of the redesign vector, 0 < a < 1. The optimum value of a is case dependent. The recursive design formula is taken from equation (26) as
(22)
g~'(d) = IOylt- Oy, <- o l = 1, t
= 0 >0
and kj
> 0
ifgj(_d) = 0
=0
if&(_d)<0
-}- ~k=l )~k ~/GX~aet Ogi
L
ifgT(d) = 0
=0
ifgy(d)<0
X-" O(co~,),
1 -
a
'
,----. O(wj )
' , - ~ + p-TyT,~ ~j 1 _ ~ - - , .,O([Qylz) AI ~ ti PiAi ~=l Oil
1
l_-_~ ~, o(IQ=l~) piAi ~ =0
Oil
>_0
ift~
<0
ifti=t u
u
(26)
398
1
O(w2t ) (tl - ti)
=-z-,--a77"+l--z-a ~ at, iEact t iEpassl
where Aj, A~,, A~' are the Lagrange multipliers. Substituting equation (22) into equation (24) and multiplying both sides by (1--ol)t i with rearrangement, one obtains
[
(g..,o(wZl)O(lQylt) ) Oti O~Ai
(25)
ifg~(d)<0
> 0
p~4;
"{- ~-"~ ' = "~111 1 ~i0E/~tact i
~j" > o ifg~(_a) = o
/ =0
Ot i
0 ( 4 ) (t u _ ti ) iEpass2
1
2
+~_~(wjl-wt),
,2
j l - - 1,g
(29) Similarly, the change in dynamic response can be assumed as
Q'zkl --IQzM = ~
3
ti)
(30)
, ~-~',O(IQzl,l) Qytl - I Q y l n = ~ Oti (t'i - ti)
(31)
Rotating laminated blades: Ting Nung Shiau et al. Sub.stituting equations (26) and (27) into equations (30) and (31) yields
Oti - ([K]g - f~p,[M]g)-' k,--~-/"- f~2°
+ ~ (V~O(lQz[kl)°(~) ti ) - z.~ -~;t oti P~i j = I J Iz..~ \,E:act ,
+
k=l
o,,
tz
Equations (34) and (35) express the sensitivities of system frequencies and dynamic response with respect to the design variables, respectively.
o,, pSA;
0(IQ~lk~)O(IQylt) ti
. /= 1
{Q}n
(35)
O(IQzlkL)O(IQ~Ik)
Z ,Eact
to the design variables gives
iEact
= _ ~ O(IQ~lkl)ot_____~ t, + ~ 1 iEact
~
0(IO~l~)0ti(tl
- ti)
iEpass 1
+-i---al
Z
O(IQzlkl)ot, (tu - t,)
icpass2
1 +l_-L-s(IOzlk~-a*k,), kl =
1,K (32)
Modified method of feasible directions The modified method of feasible directions 18 is a technique that combines the best features of the method of feasible directions 17 and the generalized reduced gradient method 19. It includes two procedures: (1) to determine the searching direction and (2) to decide the searching step size. Details of the algorithm have been given by Vanderplaats TM.
NUMERICAL RESULTS
,"act
.j=1
at,
Or, PiAil/
Natural frequency
K ' fV'°(IQylll)CO(IQzlk) t, ) + ~~)~k k-I ~kicZ~act Oti Ot, p~mi + Z ,k'/(S-" O(]QYIt')O(IQyl,) t, w-, O(tQylzl) icact ~
1
(33)
Ot------"~ti + 1---- ~
Nondimensional natural frequencies of rotating laminated plates with various ply orientations are studied for plates having various aspect ratios. Nondimensional frequencies wa2v/~/Dll for bending modes, wabx/ph/48D66 for torsional modes, and cob2v/~/Dz2 for chordwise modes are suggested by Crawley and Dugundji 12. The bending stiffness D~ is defined as 1 U
)< iEpassl Z
o(Iayfzl)
Oti
+-(------ ~
O(IQy[ll)oti (t u ti) _
,Epass2
1 +~(IOylz~ -
Q~*,~),
_
Z3
~ k~__l(Q~/)k( k - z3-*)
(t~ - ti)
ll =
1,L
The active (act) design variables are the ones in the range of their lower and upper bounds. The passive (pass) design variables are governed by their side constraints, i.e. t i t~ or ti = t~. The Lagrange multipliers are obtained by solving equations (29), (32), and (33) simultaneously. =
Sensitivity analysis. Multiplying both sides of equation (14) by {C} r and then taking partial derivatives with respect to the design variables, one obtains
when N is the number of layers. The effects of rotating speed and aspect ratio on the system natural frequencies of various laminates for the first two lengthwise modes are shown in Figures 3-5. The results indicate that increasing the rotating speed increases the system natural frequencies and the effect is considerably higher for bending modes. Furthermore, the effect of the aspect ratio of the blade on the natural frequencies is case dependent. However, it is insignificant for the cases studied. The effect of the nondimensional disk radius is shown in Figures 6-8 for various laminates. Numerical results show that the nondimensional natural frequencies increase as the disk radius increases. The effect of disk radius is more significant for the bending mode. It should be noted that predominantly lengthwise bending modes are shown in Figures 3-8.
Optimum results or,
-
(34)
Taking partial derivatives of equation (16) with respect
The optimum design of the rotating laminated blade [ 0 / ± 45/90]s with constraints on the natural frequencies and/or dynamic response is considered. The design
399
Rotating laminated blades: Ting Nung Shiau et al.
variables are the ply thicknesses of the laminates. The material properties are listed in Appendix D. The rotating blade with a/b = 3 and R/a = 1, fixed on a disk of R = 0.1143 m is considered and the rotating speed is fixed at fl = 600 7r rad/s with setting angle ~ = 30 °. The aerodynamic forces acting on the laminated blade for both lift and drag 16 are assumed to be of the following form:
]
--a/b =2 - a/b=3 .~....a/b--4
I
....,,/b=s
7 >,
6
°°
=,
5
r
4
(ll ""1
1
I
....
t° .
.9 irl
drag = 5.0 x 103(ei~lt + ei~p~t)N/m 2
................................ i .................................................
E 3
-~ o'Z
i I I r l - o....... r s i o n aT-'~.-,----,---'1"~_ l - - -.~....-...-----.-" - - - - ~ .....-
I
2
.......................... •......... ..-...-................ t.............. -i . . . . . . . . . . . . . . . . . . . . . . .........
--4. . . . . . . . . . . . . . . . . . . .
i
,
natural frequency constraints: ~1 ~ 1620 7rrad/s
1
i
I
The dynamic analysis in the optimization process is performed by using a finite element method (FEM) with four elements. Eight design variables (ply thicknesses) are considered in the problem. Four of them are associated with the four layers of elements 1 and 2 expressed as q, t2, t3 and t 4. The others are associated with 3 and 4 and are expressed as ts, t6, t7 and t8. The schematic plot of these elements is shown in Figure 2. The optimum weight results using MFD, M M F D , and OCM are shown in Table 1 for the. case under multiple frequency constraints. The constraints on the natural frequencies are chosen such that the fundamental natural frequency is higher than 1.5f]p~ and the second natural frequency is approximately
20
8
I
. . . .
0.6
,
,
i
t
0.8
--a/b=2 - a/b=3 ......a / b = 4
1
I
.................................. I..................... .-.alb=5
~
6
................................. i ............................... i- Ben6ing--~
o
4
.....
"o
3
o= c-
i
I
,.
I
!../J .............
~
j
i
I
r..................... ,............................ i-.... ~
:
N:N
. . . . . . . . . . ~ .......... ....
................. [..................... :,.~
i i
21
16,
t
7
0
15
0.4
i t
1
14
'
! . . . .
Figure 3 Effect of nondimensional rotating speed on natural frequency of [ 0 / + 45/90]s with R/a = l, ~ = 30 °
oc Z
19
I
Nondimensional Rotating speed (fl)
dynamic response constraints: IOyltip < 0.00001 m m
. . . . . .
0.2
0
~2 -> 4000 7rrad/s
18
i .........................................
................................................................... i ........................ i ................... ! .....................
side constraints: ti >_0.00013m, i = 1,4
17
. . . . . . . . . . . . .
r-
in which flp~ = 1080zrrad/s and ~"~P2 = 32407rrad/s are chosen. All the constraints are listed as follows:
[Qzltip --% 0 . 0 0 1
1_
°°n°'iO
U.
lift = 1.5 x 105(ei~plt + ei~p2t)N/m 2
i
I / J.. _.z/ ......
i ....
0.2
i
2"2
N:'~:N:N::~:2:
"r[orsional i t ....
0.4
I ~ ....
0.6
t ....
0.8
1
Nondimensional Rotating speed (fl) Figure 4 Effect of nondimensional rotating speed on natural frequency of [ 0 2 / + 30Is with R/a = 1, ~ = 30 °
9
I0
Figure 2
400
12
7q
6 I
II
2
3
13
8,
4
Eight-node element mesh of laminated blade
5
higher than 1.25 ~'~P2" The design history of the weight with frequency constraint for various algorithms is shown in Figure 9. For the case under a dynamic response constraint, the design history is shown in Table 2 and Figure 10. For the constraints on both multiple frequencies and dynamic response, the design history is shown in Table 3 and Figure 11. Since the choice of the relaxation factor (c0 for OCM influences the convergent speed, it is advisable to choose c~ carefully. A more efficient scheme suggested by Sadek 14 which has been
Rotating laminated blades. Ting Nung Shiau et al.
I-a/b=2[
- a/b=3] ......a / b = 4 [
30
I
1
!
] - a/b=3 [ ......a / b = 4 /
[
-a/b=Sl
......................... .......................
8
.................i. . . . . . .
..........................I......................~...... .. a / b
[
=5
i ~
]
-
-
_.f~_ -
~
-2
/
°o
O
E
7
i
........................................................ T ~" "':
.............................. '...................................
"~ 20
O"
P
.g
° r"
E
g 6
i2nd Bending *ode
LL
10
-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1st Be~ding M
.E
xJ
~
=0
5 ~ : ' _ L 2 _ . I
.................................
........ 0
0.4
3
- -.:.'-...T..-.-,--
!
I
I
;
i
......
_ ~=,=.
i
i
1
,
,
,
J
0
i
,
i
i
0.5
Nondimensional Rotating speed ~) speed
I
i
.......................[..........................t.......mOi~si°~ ........... -~I..........................
i
0.8
Figure 5 Effect o f n o n d i m e n s i o n a l r o t a t i n g f r e q u e n c y o f [ ± 4 5 / z g 45]s w i t h R/a = 1, ~ = 30 °
F i
...................................................................... '......................................... &. . . . . . . . . . . . ._
. . . . . . . .
0.6
t1
i
i
2
I .... i
0.2
4
Z
I
o
i
5
t"
.................................................. +................................ ~................................... i...........................
oC z
e._o
i................................................................ i i '
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
!
I
l
,
1
,
i
I
i
,
1.5
,
i
2
Nondimensional Disk Radius (R/a)
on
natural
Figure 7 Effect o f n o n d i m e n s i o n a l disk r a d i u s o n n a t u r a l f r e q u e n c y [ 0 2 / i 30]~ w i t h ft = 1, ~ = 30 °
--alb--2I 9
- a/b=3[ ......a / b = 4 l
i
~ a / b = 2
30
i
~,
c g
I ~ 6
t-
o
/ 5
i
i
P
................................................................................... ~ ..............................................................
i
"~
t
' I•
t-
.
O tO -I O"
Bending
.............................................................
"
4
!
!
..........................,.................................T..............................T................................ ........
...................... I ~................................... ..il. ~.;,..z_i.7:.._~7'"............ ~....................................... 2rid
Bending
Mode
I
20
.................................... ]........................................... i.................................................................
t'09 t~
._ c 0
U-
25
t
_
_ ; ~ ;,z ,;'''= - .,:.,~,z,~,-" '
.o
i i"
t,
j
- a/b=3
......a / b = 4 .... a / b = 5
Torsional
3
............................i........................... i ~
1................................ ......
..................
~ . . . - . . . . ~ . . . - . . . -
Z
i
0
i
i
i
i
0.5
~
i
t
I
1
,
,
,
,
~
~
1.5
E o~ "13 tO z
,
i
i
1st B e n d i n g M o d e
10
- - l z . ".--
" "--------i
0
0.5
................................
,
2
Nondimensional Disk Radius (R/a)
1
1.5
2
Nondimensional Disk Radius (R/a)
F i g u r e 6 Effect o f n o n d i m e n s i o n a l disk r a d i u s o n n a t u r a l f r e q u e n c y o f [ 0 / + 45/90]s w i t h ~ = 1, ~ = 30 °
Figure 8 Effect o f n o n d i m e n s i o n a l disk r a d i u s o n n a t r u a l f r e q u e n c y [ + 4 5 / : F 45Js w i t h ~ = 1, ~ = 30 °
incorporated into the computer program is to initially select a smaller a and check the result after each cycle. From Tables 1-3 and Figures 9-11, one finds that the optimum weight for the case with multiple frequency constraints is much smaller than that of the case with dynamic response constraints. This is because the multiple frequency constraint case allows more dynamic response. The optimum design weights are shown in Table 2 and Figure 10 for the dynamic response constraints, as well as in Table 3 and Figure 11 for both
the multiple frequency and dynamic response constraints. The results are very close to each other since the optimum design weight is dominated by the dynamic response constraint. In Tables 3 and 4, one can also find that the initial design significantly affects the optimum design as well as the convergent speed. It should be noted that the optimum thickness for each layer shown in the paper may not be an integer multiple of normal ply thickness 0.00013m, in which case one more layer for each ti is suggested for the practical design.
401
Rotating laminated blades." Ting Nung Shiau et
al.
Table 1 Initial and final design of various optimization algorithms with constraints on natural frequencies and a / b = 3, R / a = 1, ~ = 30°, f / = 6007r rad/s; mesh size: 2 x 2
140
~MFD ............
Optimum design Initial design
MFD
MMFD
t~ 2.40 2.07110 2.23130 t2 2.40 0.53868 0.53862 t3 2.40 0.72405 0.55186 t4 2.40 0.13000 0.13000 t5 2.40 0.44679 0.32837 t6 2.40 0.21525 0.30877 t7 2.40 0.40835 0.32411 ts 2.40 0.50333 0.53771 Weight? 125.43 32.906 32.339 Iteration number 10 9 Function evaluations 94 313 Maximum dynamic response of the tip (10 -3 m) y-direction 0.00761 0.00991 z-direction 3.9263 4.8195
' 2.11430 0.63124 0.71108 0.13000 0.45144 0.30012 0.38861 0.51443 34.239 14 15
" ._~
f
...............
I
L
J
...... i ................... j..................
......................... i ................ C
-
..... .......... ~ .........~................ l.................................... i..................
0
0.00687 3.663
i 1
60
20 i
i
0
2
I
4
J
6
]
i
8
i
i
10
12
14
Iteration Number 9 Comparison of design histories of various algorithms, for the case with natural frequency constraint Figure
140
Optimum design
t~ 2.40 t2 2.40 t3 2.40 t4 2.40 t5 2.40 t6 2.40 t7 2.40 t8 2.40 Weight~125.43 Iteration number Function evaluations Maximum dynamic response of y-direction z-direction
I -*-- OCM
L
4{)
Table 2 Initial and final design of various optimization algorithms with constraints on dynamic responses and a / b = 3, R / a = 1, ~ = 30°, f~ = 6007rrad/s; mesh size: 2 × 2
MFD
FO,
OCM
* Design variable Ii unit is 10 3 m t Weight unit is 10-3 kg
Initial design
[ i
,o
MMFD
3.00000 2.04000 0.84903 0.50880 0.70623 1.23720 0.17220 0.2500 1.67900 0.58781 0.73255 0.88125 0.45819 1.60810 0.74779 1.14000 54.512 58.974 20 7 191 238 the tip (10 -3 m) 0.00336 0.00534 0.9922 0.9935
'
i
i
I -'- MFD '- ................ ~........ r................... I~.MMFDII
120
i II
l .......... t............
OCM 2.81330 0.98212 0.83751 0.24387 1.79410 0.61557 0.52319 0.88266 56.785 16 17
-.
loo
.......
............. ............. i .......... i ..........
o)
%=
.................................................... f......................... [
8o
'°r [ ...... ,i............I........... ......................) ..............................I......... r..............I t t I , ! j
0.00435 0.9926
o 0
* Design variable ti unit is 10-3 m "~Weight unit is 10-3 kg
2
4
6
8
, l , l , , , l , t , I 10 12 14 16 18
20
Iteration Number 10 Comparison of design histories of various algorithms, for the case with dynamic response constraint
Figure
Table 3 Initial and final design of various optimization algorithms with constrains on both natural frequencies and responses and a / b = 3, R / a = l, ~ -- 30°, f~ = 6007r rad/s; mesh size: 2 × 2 CONCLUSIONS
Optimum design Initial design t~ 2.40 t2 2.40 t3 2.40 t4 2.40 t5 2.40 t6 2.40 t7 2.40 t8 2.40 Weight~" 125.43 Iteration number Function evaluations Maximum dynamic response of y-direction z-direction
MFD
2.9940 2.22100 1.01840 0.13896 0.97598 1.57290 0.13000 1.19710 1.06170 0.36771 0.78916 1.00510 0.68548 1.12320 0.99222 1.27410 56.519 58.137 12 13 100 497 the tip (10 -3 m) 0.00277 0.00431 0.9983 0.9927
* Design variable t~ unit is 10-3 m ~ Weight unit is 10-3 kg
402
MMFD
OCM 2.96510 1.12770 1.22350 0.37453 1.00410 0.92363 0.62144 0.89613 59.685 16 17 0.00392 0.9943
In this study, isoparametric elements are employed to model the system of a rotating laminated blade. The natural frequencies and dynamic response of the system are investigated. Three optimization algorithms are applied to study the minimum weight design with constraints on system and dynamic behaviors. The c o n c l u s i o n s c a n b e s u m m a r i z e d as f o l l o w s . (1) A n i n c r e a s e i n r o t a t i n g s p e e d a n d / o r d i s k r a d i u s i n c r e a s e s t h e s y s t e m n a t u r a l f r e q u e n c i e s f o r t h e first two modes. (2) T h e o p t i m u m w e i g h t w i t h d y n a m i c r e s p o n s e c o n s t r a i n t s is h i g h e r t h a n t h a t w i t h n a t u r a l f r e q u e n c y constraints. In addition, the optimum weight with
Rotating laminated blades: Ting Nung Shiau et al. Table. 4 Initial and final design of various optimization algorithms with constraints on both natural frequencies and response and a/b = 3, R/a = 1, ~ -- 30 °, f~ = 6007rrad/s; mesh size 2 × 2
4
Optimum design Initial design
MFD
t~ 2.70 t~ 2.70 t3 2.70 t4 2.70 t5 2.70 t6 2.70 t7 2.70 t8 2.70 Weightt 141.10 Iteration number Function evaluations Maximum dynamic response of y-direction z-direction
MMFD
5
OCM
2.73410 2.75570 2.05620 2.15660 2.07320 2.17660 1.90950 1.98850 1.04260 1.00310 0.89160 1.26050 0.91091 1.25530 0.88976 1.28730 81.704 90.961 15 4 137 115 the tip (10 -3 m) 0.00525 0.00548 0.9933 0.8455
2.71420 1.98720 2.18210 2.02440 1.10350 0.91221 0.90483 0.89112 93.087 18 19
6
0.00421 0.9521
10
I
i
,
J
~-MFD
I
13
I
1 2 0 ' ' ...............................i .............................................o - M M F D ...............'~...................
\
9
12
f
i
8
11
* Design variable t i unit is 10 3 m
f Weight unit is 10-3 kg 140
7
i
i
I
-*-OCM
,
100 " ~
r
I
14 15 16
60
40
r
i
F
'
I
l
j
r
18
............ ~............. , .............. 1............... ~.................. ! ........................................ I * ;
I
_
20
0
17
J
t
/
,
0
i
2
i
,
!
; ! t I ......... 1 ................ ................... i ................... ...................................... ' ~ i i
,
t
4
,
i
6
,
,
8
i
10
,
,
12
19 ,
14
20
16
Iteration Number Figure 11 Comparison of design histories of various algorithms, for the case with constraints on both natural frequency and dynamic response
constraints on both the natural frequency and the dynamic response is almost the same as for dynamic response constraints only. (3) The optimum design weight using OCM compared well with those using M F D and MMFD. Results also show that the choice of initial design significantly affects the optimum design as well as the convergent speed. (4) The weight of the rotating laminated blade can be greatly reduced when at the optimum design stage.
21 22
on torsional vibration of rotating beams. A S M E J. Appl. Mech. 1984, 51,913 Ansari, K. A. On the importance of shear deformation, rotatory, inertia, and Coriolis forces in turbine blade vibration. J. Engng for Gas Turbine and Power 1986, 10, 319 Rawtani, S. and Dokainish, M. A. Vibration analysis of rotating cantilever plates. Int. J. Num. Meth. Engng 1971, 3, 233 Petericone, R. and Sisto, F. Vibration characteristics of low aspect compressor blades. A S M E J. Engngfor Power 1971, 93, 103 Henry, R. and Lalanne, M. Vibration analysis of rotating compressor blades. A S M E J. of Engng for Industry 1974, August, 1028 Sreenivasamurthy, S. and Ramamurti, V. Coriolis effect on the vibration of flat rotating low aspect ratio cantilever plates. J, Strain Analysis 1981, 16, 97 Leissa, A. W., Lee, J.K. and Wang, A.J. Rotating blade vibration analysis using shell. A S M E J. Engngfor Power 1982, 104, 296 Leissa, A.W., Lee, J,K. and Wang. A.J. Vibration of blades with variable thickness and curvature by shell theory. A SMEJ. Engng for Gas Turbine and Power 1985, 1116, 11 Crawley, E.F. The natural mode of graphite/epoxy cantilever plates and shell. J. Compos. Mater. 1979, 13, 195 Crawley, E.F. and Dugundji, J. Frequency determination and non-dimensionalization for composite cantilever plates. J. Sound Vib. 1980, 72, 1 Wang, J.T.S., Shaw, D. and Mahrenholtz, O. Vibration of rotating rectangular plates. J. Sound Vib. 1987, 112, 455 Sadek, E.A. An optimality criteria method for dynamic optimization of structures. Int. J. Num. Meth Engng 1989, 28, 579 Lin, C.C. and Yu, A.J. Optimum weight design of composite laminated plates, Computers & Structures 1991, 38, 581 Shiau, T.N. and Chang, S.J. Optimization of rotating blades with dynamic-behavior constraints. J. Aerospace Engng 1991, 4, 127 Zoutendijk, M. 'Methods of Feasible Directions', Elsevier Publishing Co., Amsterdam, 1960 Vanderplaats, G.N. 'Numerical Optimization Techniques for Engineering Design: With Applications', McGraw-Hill Book Company, New York, 1984 Gabriele, G.A. and Ragsdell, K.M. The generalized reduced gradient method: a reliable tool for optimal design. ASME J. EngngJor Industry 1977, 90, 394 Kiusalaas, J. and Shaw, R.CJ. An algorithm for optimal structural design with frequency constraints. Int. J. Num. Meth. Engng 1978, 13, 283 Timoshenko, S. and Woinowsky-Krieger, S. 'Theory of Plates and Shells', McGraw-Hill Book Company, New York, 1959 Dobbs, M.W. and Nelson, R.B. Application of optimality criteria to automated structural design. AIAA J. 1976, 14, 1436
NOMENCLATURE a Ai
b
{c} {F}o {F}g
gj(_a) R E F E R E N C E S
Carnegie, W. Vibration of rotating cantilever blading: theoretical approach to the frequency problem based on energy method. J. Mech. Engng Sci. 1959, 1,235 Putter, S. and Monor, H. Natural frequencies of radial rotating beams. J. Sound Vib. 1978, 56, 175 Kaza, K.R.V. and Kielb, R.E. Effects of warping and pretwist
[M]g
Nd # {Q}
length of blade area of ith layer width of blade corresponding mode shape of blade equivalent nodal force vector force vector of global nodal D.O.F. jth inequality constraint kth equality constraint stiffness matrix of global nodal D.O.F. mass matrix of global nodal D.O.F. number of design variables external force vector dynamic response vector due to forces with different frequencies
403
Rotating laminated blades: Ting Nung Shiau et al. {Q}n
[0]
{q} R
ti
tl tr T U
Ub G
U~'0~ W Ub ~)i~ Wi
g x,y,z Xi ~Yi, Zi
dynamic response vector of nth external force transformed reduced stiffness matrix displacement vector of local nodal D.O.F. radius of rotating disk ith design variable lower bound of ith design variable upper bound of ith design variable kinetic energy total potential energy strain energy due to plate bending strain energy due to inplane initial force displacement components displacement components of ith node velocity vector rotating reference frame ith nodal coordinate
Greek letters relaxation factor search step displacement vector of nodal D.O.F strain vector of blade )~j , t Ak, A/t l Lagrange multipliers density of ith layer Pi inplane initial stress vector of blade ith shape function setting angle of blade rotation components of ith node O2 natural frequency of blade fundamental natural frequency of non030 rotating blade a0 nondimensional natural frequency f~ rotating speed of disk f~p, frequency of nth external force components of angular velocity of disk fl angular velocity of disk fi nondimensional rotating speed (fl/Wo)
[:o ,xoo oo [g3] =
[N4]=
0
051,y 0
0
i!0000
...
osxo:l
0
0
~S,y
0
051
0
0
0
...
0
058
00i]
0
051 0
0
...
0
0
058 0
0
0
where 051,052.... ,058 denote shape functions.
A P P E N D I X B: I N P L A N E INITIAL STRESS MATRIX The initial displacement vector {q} due to a centrifugal force is given by
Ot Ol*
[Ke]e{q} = {fc}e
(B1)
and the corresponding initial stress field {or°} for the previbration is described by o O"x Oy0 {crO} =
o O'xy
= [0][N2]{q }
(B2)
0
%z o
O'xz
where [Ke]e is the elastic stiffness matrix, {fc}e is the centrifugal force, and [~)] is the transformed reduced stiffness matrix. Solving equations (B1) and (B2), one can obtain the inplane initial stress field [a°], which is defined as
o]
(B3)
L~r~y A P P E N D I X A: [Nil, [N2], [N31, [N4] MATRICES [N|] =
A P P E N D I X C: T R A N S F O R M E D R E D U C E D STIFFNESS M A T R I X ~ 0 ... 0 01 ..-
0
~8
0
0
0
¢8
[N:] = ] O,O.X 0 Ol v [Ol,> Ot,x I
[00
0 0
404
0 zOs,x 0 ] z4h,x 0 ... 4'8,~ 0 0 zOl,y ... 0 OS.y 0 0 ZOs,y zOl,>, zOl,~ ... c)8~, OS,x 0 zO8,y z08,x (~)I,Y 0 (~| "'" 0 0 01,.,, ~l 0 ... 0 0 08,x O+ 0 0 0
o,6o!]
The transformed reduced stiffness matrix is a 5 x 5 symmetric matrix, and is defined as
[0] =
011 012 013 0 0
012 022 026 0 0
026
0
066
0
00440_-45 0
045
Q55 J
Rotating laminated blades: Ting Nung Shiau et al. with
where Qll =QI! cos4 0 + 2(Q12 + 2Q66) cos 20sin2 0
Q11 --
+ Q22 sin4 0 Q12
Q12 =(Qll + Q22 - 4Q66) cOs2 0sin2 0 + Ql2(COS4 0 + sin 4 0)
-
Q22 ---
El ul2u21
Q66 ~- G12
Elu21 1 - ulzU21
Q ~ = G2s
1 -
E2 1 - u12u21
Q55 = GI3
Ol6 =(Qll - Q12 - 2Q66) cOs3 0sin0 + (QI2 - Q22 + 2Q66)cos 0sin3 0 022 =-QlJ sin4 0 + 2(Q12 + 2Q66) cos 2 0sin 2 0
APPENDIX D: M A T E R I A L PROPERTIES OF GRAPHITE/EPOXY
+ Q22 cos4 0 Q26 =(QI1 - Ql2 - 2Q66) cos Osin3 0 + (Q12 - Q22 + 2Q66)cos3 0 sin 0 Q66 = ( Q l l 4- Q22 - 2Q12 - 2Q66) cos2 6- Q66(cos 4 0 6- sin 4 0)
044 =Q44 cos 2 0 + Q55 sin2 0
0sin2 0
Young's modulus: E1 128GPa 18.5 x 106psi E2 1 1 GPa 1.6 × l06 psi Shear modulus: G12 4.48 GPa 0.65 × 106 psi G]3 4.48 GPa 0.65 x 106 psi G23 1.53 GPa 0.222 × 106 psi Poisson's ratio: ul2 0.25 Density: p 1.5 x 103 kg/m 3 0.0551b/in 3 Thickness: t 0.00013 m 0.0052 in
Q45 =(Q55 - Q44) cos 0 sin 0 0_.55 =Q44 sin2 0 + Q55 cOS2 0
405