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A methodology for synthesizing symmetric laminated beams with optimal elastodynamic response characteristics
Int. d. Mech. Sci. Vol. 29, No. 12, pp. 821-830, 1987 Printed in Great Britain.
0020-7403/87 $3.00+ .00 Pergamon Journals Ltd.
A M E T H O D O L O G Y FOR S Y N T H E S I Z I N G SYMMETRIC L A M I N A T E D BEAMS WITH O P T I M A L E L A S T O D Y N A M I C R E S P O N S E CHARACTERISTICS C. K. SUNG,* B. S. THOmPSOn,* M. V. GANDHI* a n d C. Y. LEEr * Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan and * Department of Mechanical Engineering, and Composite Materials and Structures Center, Michigan State University, East Lansing, MI 48824-1226, U.S.A. (Received 30 June 1987) Abstraet--A methodology is presented herein for synthesizing symmetric laminated beams with optimal elastodynamic response characteristics by generalizing a previous publication of the first two authors [Liao et al., J. Composite Mater. 21,485-501 (1986)] in order to include the percentage of fiber and matrix in each ply, and to also accommodate fibers with transversely isotropic properties. Constitutive models are developed for the stiffness and damping properties of laminated beams as a function of the ply fiber volume fraction, the fiber orientation in each ply, the ply thickness and the stacking sequence prior to incorporating these models into an optimal design formulation. This formulation is then employed in illustrative transient response studies of cantilever beams in order to demonstrate the significance of incorporating these micromechanical parameters into design algorithms for optimally tailoring the properties of these laminated beams. A generalized reduced gradient algorithm is employed in this optimal design endeavor.
NOTATION
co-ordinate system oxyz: x-axis along the fiber co-ordinate system 0123: l-axis along the principal longitudinal axis of the beam E x longitudinal Young's modulus of a unidirectional ply Ef elastic Young's modulus of the fiber Em elastic Young's modulus of the matrix t/f volume fraction of the fiber Vm volume fraction of the matrix Er transverse Young's modulus of the ply G~, transverse shear modulus of the ply ky transverse plane strain bulk modulus of the ply ~x longitudinal Poisson's ratio Vf longitudinal Poisson's ratio of fiber ~m longitudinal Poisson's ratio of matrix qk stress partitioning parameter kb transverse plane strain bulk modulus of the fiber Ely transverse Young's modulus of the fiber km bulk modulus of the matrix Gm shear modulus of the matrix qG stress partitioning parameter in transverse shear Gs shear modulus of the ply r/s stress partitioning parameter in shear Gf shear modulus of the fiber P number of plies Z.i distance of the jth ply from the beam mid-plane 0 fiber orientation angle [c3 global damping matrix [K] global stiffness matrix ugi ith undamped natural frequency {tJ,} ith modal vector [M] mass matrix {Xo } initial displacement vector initial velocity vector Pf density of the fiber Pm density of the matrix 821
822
C.K. SUNG et aL
AW W ~, AW~ ~b~ ~ l, b, h m,n Qo U~ D~j d~j
strain energy dissipation maximum strain energy specific damping capacity i = x, y, s; strain energy dissipation components associated with ax, ay and % i = x, y, s; specific damping capacity components associated with a~, e~; ay, q ; as, r s i = x, y, s; damping capacity component coefficients associated with ~,, i = x, y, s length, width and height (total thickness) of the beam cos0, sin0 components of moduli; i,j = x, y, s, 1, 2, 6 i = 1, 2, 3, 4, 5; linear combinations of moduli i, j = 1, 2, 6; flexural moduli of a multidirectional symmetric laminate i, j = 1, 2, 6; flexural compliance of a multidirectional symmetric laminate; the [dlj] are the inverse of the [Dii]
tj MI El I kj
thickness of t h e j t h ply flexural moment per unit width o f the beam longitudinal Young's modulus relative to the 01-axis second moment of area curvature component associated with the strain component ~1
superscripts
j
jth ply
INTRODUCTION
Fibrous composite laminates are increasingly being employed in the fabrication of a very wide variety of products which are subjected to dynamical excitations. These products range from sporting goods and medical equipment to aerospace components and industrial machinery. In many of these applications components must be fabricated with high stiffness and also high values of structural damping in order to tailor their dynamical response. However, there is a distinct absence of viable design methodologies for synthesizing laminates in order to achieve these goals, and the work presented herein, is directed towards partially filling this void in the literature. The objective herein, is, therefore, to develop a design methodology for this important task. This objective is accomplished by generalizing a recent publication on this subject [1], by including the fiber volume fraction of each ply within the formulation and incorporating fibers with transversely isotropic properties. The resulting formulation enables symmetrical laminated beams to be designed with high stiffness and also high damping by optimally selecting the fiber volume fractions of each ply, the ply thicknesses, the stacking sequences and also the ply thicknesses. An illustrative example of the transient response of a simple cantilever beam serves to demonstrate the advantages of implementing the proposed methodology. The methodology developed herein is generic in nature and hence may be utilized in different applications where the design objectives may be significantly different. In order to develop this design methodology, a micromcchanical model for the constitutive properties of a laminate as a function of various constitutive properties is presented in the next section. This is then followed by the development of models for the flexural moduli of symmetrical laminated beams and also their structural damping capacity. These models are then incorporated into a optimal-design formulation which seeks to maximize the damping capacity subject to constraints imposed on the flexural rigidity of the beams. A finite-element approximation and also the generalized reduced gradient algorithm are employed as tools to determine an optimal solution for this class of design problems.
MICROMECHANICAL
MODEL
The methodology proposed herein for the optimal design of laminated beams considering damping and stiffness is based on the propositions that the global properties of the material are elastic and the damping properties are a function of the elastic strain energy dissipated during each stress cycle I-2, 3]. In order to establish the basis for an optimal design methodology for this class of problems, the elastic moduli must first be expressed as a
Synthesizing laminated beams with optimal response
823
function of the properties of the constituents. These micromechanical design variables, which represent the constitutive characteristics of the laminate, are the fiber and matrix properties and the fiber volume fraction. Subsequently, these variables are incorporated in macromechanical models for the global mechanical properties of the laminate which include such design variables as ply orientations, ply thicknesses and the stacking sequence. These variables serve as manufacturing specifications for the fabrication of the laminate and also govern the dynamical response of the fabricated part. The first objective of developing a viable formulation for the elastic moduli of each ply is accomplished by utilizing the modified rule-of-mixtures approach for fibers which are transversely isotropic [4]. Graphite and aramid fibers, which account for a substantial proportion of the industrial market, are typical fibers in this category. The longitudinal Young's modulus, Ex, of a unidirectional ply is defined by Ex = V f E f + V m Em,
(I)
where Ef and E m are the elastic moduli of the fiber and matrix, respectively, and Vr and Vm are the fiber and matrix volume fractions. Assuming that the material is free of voids, these volume fractions are related by Vm = 1 - Vf.
(2)
The transverse Young's modulus, Ey, of a unidirectional ply is defined by [4], p. 397, 4 kr Gy Er - k r + nGy'
(3)
where
• 4krv~ n = 1 -t. Ex
(4)
In equation (4), vx, denotes the longitudinal Poisson's ratio which is related to the ply volume fraction by Vx = Vf vf --[- VmVm,
(5)
where vf and vm are the Poisson's ratios of the fiber and matrix, respectively. In equation (3), k r denotes the transverse plane strain bulk modulus of the ply, which is defined by
kr 1 __ (Vf+r]kVm)~kfy 1 ( V f + ~mVm "k ),
(6)
where r/k is the stress partitioning parameter which measures the relative magnitude of the average stress in the matrix as compared with the average stress in the fiber for conditions of plane strain hydrostatic uniaxial loading. The stress partitioning parameter t/k is defined by I 1+ qk = 2(1 -- vm)
(7)
and kfy, which denotes the transverse plane strain bulk modulus of the fiber, is defined by
Efy kfy = 2(1 - vf)'
(8)
where Efy is the transverse Young's modulus of the fiber. In equation (6), km represents the bulk modulus of the matrix which is defined by Gm km - 1 - 2vm'
(9)
where Gm, which denotes the shear modulus of the matrix, is defined by Em G m - 2(1 + Vm)"
(10)
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C.K. SUNGet al.
/ h e term Gy, in equation (3) denotes the transverse shear modulus of the ply, and is defined by
l Gy -
( 1 )
[ Vf +
Vm'
Vf+/']G Vm ~Gfyfy ~]GG--mJ'
[|l}
where Gfy is the transverse shear modulus of the fiber and qG, which is the stress partitioning parameter in transverse shear, is defined by
qG=I4(llVm)](3--4Vm+~).
(12)
The shear modulus of the ply, Es, may be defined as a function of the previous expressions. Thus 1 1 Vf + qs Es Vf+qsV m where q~, which denotes the stress partitioning parameter in shear, is defined by
1( r/s=~ 1
+Gm~ Gf/"
(13)
(14)
The shear modulus of fiber, Gf is denoted by Ef Gf - 2(1 + vf)'
(15)
where vf is the transverse Poisson's ratio of the fiber. The above equations permit useful relationships to be developed for the following components of the elastic moduli of each ply by utilizing the expressions,
Qxx = mE~
(16)
Qxy = n vyE~
(17)
Q.. =
(18)
m Ey
Qrx = m vxEy
(19)
Qss = Es,
(20)
m = E1 - vxvr]- l
where
(21)
FLEXURAL MODULI The stiffness and damping characteristics of a structure are governed by the geometrical configuration and also the material properties. Having developed the necessary mathematical apparatus for representing the material properties, attention now focuses on establishing a viable model for the flexural moduli of a symmetrical laminated beam. The flexural moduli, Dab , a, b-l, 2, 6 governing the response of a symmetrical laminated beam may be expressed as a function of the micromechanical and macromechanical design variables using the foregoing expressions for each ply. These functions are 2 e/2
Dlx = ~ ~ 1 [ (z3 - z3-1 ) (U[JI + U~j' cos 20 OJ+ Uy ~cos 400~]
2 P/2
O,2 = ~Y~.=E(z~ ~ -
)(v?
v~'
z)3_ l
--
P/2
(U(~j)
2 e/2
(u~j~
2
DI6 = 3j--~t E(z3 -- z3-1 )
022 = ~j_E E(z3 - 4 - , )
(22)
cos 40 O~)-I
½sin2#J' +
U3 j)
- u ? cos20'~' +
(23)
sin4#~')] u~ j~
cos40%]
(24)
(25)
Synthesizingl a m i n a t e d
2 P/2 3s~.=, [(z 3 _ Z j3- 1 ) ~ ~,u,J, 2 ½sin20~J,
D26
066
b e a m s with o p t i m a l response
.~-
--
U3~J)sin40,J))]
2 P/2 3jE.=I [(z3 - - Z j _a 1 )(us~J,_ U3~J,cos40,J,)],
825
(26)
(27)
where the beam comprises a total of P plies, zi is the distance ofthejth ply from the mid-plane of the beam and 0 t ~ is the angle ofthejth ply relative to the longitudinal axis of the beam. The compliance [dab] may be determined by inverting matrix [Dab]. The terms U~J~, i = 1, 2, 3, 4, 5 in equations (22)-(27) are defined by ulJl
l rar~lJJ . ~otJ~ ~_ 2r~¢J~ . 4,~J~ ~xr ~Vss _1
= ~ l _ - , ~ x x -1- - ~ r r
U~2s~=
r,~'~
I "~ L ~ x x
--
~ L~,xx U(4J) U~J)
1 rs-~tJ)
,
W~ 1 ~YyA
I rr~(J)
= ~ L~
(29)
~r~yy - - - - Y ~ x y - -
~SS
()(J) d'- "--(J)
" ~¢J} ~ 4¢dss J
= g L ~ x x -I- ~ y y
(28)
O~xy
-
d
9 / ' ) (j1-4- " -(J} ± ,~yr - - - ~ x r - - 4 Q s ~ ] . , o~Jl
(31) (32)
This set of equations completes the development of the necessary expressions governing the stiffness characteristics of a symmetric laminated beam, as functions of both the micromechanical design variables and also the macromechanical design variables. Attention now focuses on utilizing these expressions in the development of relationships for the damping properties of a beam. STRUCTURAL DAMPING The model developed herein for the structural damping of a symmetrical laminated beam as a function of the fiber and matrix properties, and their volume fractions, the stacking sequences, the ply thicknesses and also the ply orientations, builds upon the theoretical arguments presented in Ref. [1] by including within the formulation the fiber volume fraction and also fibers which are transversely isotropic. This formulation generalizes the work presented in [2, 3]. The developments presented in the latter suite of publications are employed herein because of the favorable correlation between the analytical and experimental results presented in [2], thus validating the predictive capabilities of the analytical models for the structural damping. Consider the time-dependent flexural motion of an arbitrary symmetric laminated beam comprising P plies with different properties. The specific damping capacity, ~b, of the beam may be defined as AW = W'
(33)
where AW is the strain energy dissipated during each stress cycle and W is the total strain energy. The energy dissipated per cycle may be considered to comprise the sum of three components, which may be written AW = AWx + AWr + AWs,
(34)
where the subscript s denotes the shear component in the oxy plane. These three components may be written in terms of the strain energy dissipated during a single stress cycle. Thus for a beam of width b, total length I and total thickness h, AWx = 2b
AWy = 2b
do
~4)xaxexdz d/
I hI2 1 -~ qbytrrer dz dl dO
f•
(35)
(36)
C . K . SUNG et al.
826
f: (h'2 1 AWs = 2b
d,,
} 4~sasesdzd/.
(37)
These expressions contain terms defined relative to the principal fiber directions oxyz, rather than the principal geometrical axes of the beam 0123. A more convenient formulation may be established by employing orthogonal transformation expressions [4, 5] to yield
b P/2
Am
x
=
~jEI[49:, ,n~ {Q'/'ld,1 +Q':~2dl2+Q'/~d,6} (J).
2
.=
{mZda, +m~njdl6}
Mfdl
x 1- J, 3 - z 33 1)]
(38)
b P/2
AW, = 5 El.=t_.rrl-tfitJ'n2j {QTldl,+ Q':)zd12+ Q°~6d16} {n~d,l x (z~-z~_l)]
f'
-mjnjd16
M~ dl
} (39)
0
b e/2
=
.
Lq), mrnj~:z,,d,l +Q'a~2d12+Q':~d16}{2mjnjdll
E=1 '-'"'
- ( m ~ - n})d16 } (z3-z3-1)]
fo
M 2 dl,
(40)
where the integration from the beam midplane to the surface of the beam is written as a summation statement rather than an integral for the P/2 plies. In equations (38)-(40), mj = cos0 °~, n~ = sin0 I~, M1 is the bending moment per unit width of the beam, and the elastic moduli for the jth ply are written 4.- 2mj2 nj2 ()tj~ + 4 m j2n j2QsstJ)
/)i j}
~
~1/2)
~2.2,¢'~(j) i O I j p 4mj2 nj2 Qss (j~ = mj2 nj2 c.).(.j )._ I '"i "J ~rr + (m~ + n. j, *,~x~
QOl 16
DO) - - m j n j 3Q rtj~ = m3n ""j "'jr..,xx r +
--.4t'~tD
n 4 ( ' ~ {j)
(41) (42)
(mini3 _m3n~)Q~jrl
+ 2(mjn~ - "-Jm3n-J,~OIJ~ss"
(43)
Assuming that the flexural deformation of the beam is governed by the Bernoulli-Euler hypothesis, the total strain energy of the beam may be written as
W=
fo I b: M ~ dl 2 E~ I '
(44)
where Ea is the Young's modulus associated with the longitudinal axis 01 of the beam and I is the associated second moment of area. For this class of beams
bMl kl - E , I '
(45)
where kl is the curvature component associated with M~. Since the moment-curvature relationship for symmetric laminates is kl = dll M1,
(46)
b --. EI = -dll I
(47)
the following expression results
The specific damping capacity qJ, is assumed to be the sum of three components
qJ,, = AW/W, ~br = AWr/W and qJs = AWs/W. Utilizing equations (38)-(44) these three
Synthesizing laminated beamswith optimal response
827
components can be written as
=
2[~_~1 ~2 ,t,(j)__2 I~(J).s (J) (J) 2 "v~ "'j ~ t t u l l +Qt2dt2 +Qt6d16)(mjdll +mjnjd16)
× ev =32
3 ]
- z~_ 1 )
1
(48)
th,j)n2 .-j (Q]~dll +Q,l~d12 1=1
(49)
×
~Os=3 d-~xxj=l ~p°s)mjnj(Q',~d,t±t~°"tT y.g12~12 "~/"~(J)d, kgl6 16/' x (2mjnjdt 1 -- (mj2 --nj2 )dl6)(z~-zj_ 3 1) J
(50)
These three equations complete the development of a formulation for predicting the damping capacity of symmetrical laminated beams as a function of both micromechanical and also macromechanical design variables. Having developed models for the micromechanical material properties, stiffness and damping characteristics, these formulations provide the essential ingredients for predicting the elastodynamic response of symmetrically laminated beams by employing finite-element techniques. OPTIMAL DESIGN STRATEGY The mass, stiffness and damping matrices contained in the finite-element model govern the elastodynamic response of symmetrical laminated beams. The objectives of an optimal design formulation must first be specified before utilizing an optimization algorithm in order to develop a capability to synthesize optimal manufacturing specifications. Herein, the objective of the optimal design formulation is posed as one which seeks to maximize the damping capacity of a laminated beam subject to constraints imposed on the fiber volume fraction, ply thickness, ply angle, stacking sequence, number of plies and the flexural rigidity. The formulation may be written as maximize subject to 0max >~ 0j i> 0min tmax ~ tj ~ tmin
Vfm
Vo Vfmi°
El I = constant P = constant. This class of optimization problems is solved herein by employing the generalized reduced gradient algorithm. This algorithm has been shown to exhibit superior speed and robustness under a wide variety of problem conditions [6, 7]. The essential features of this non-linear programming algorithm employed are as follows. (1) Input the initial parameters. (2) Partition the design variables as basic and non-basic variables., (3) Search for a feasible starting point by using the constraint function as a temporary objective function and minimize the error. (4) Calculate the reduced gradient. (5) Calculate the search direction. (6) Check the convergence criterion: if this is satisfied the results are stored and the iterations stopped, otherwise the iterations continue. (7) Restricted line search: use the Newton-Raphson approach to determine the step size. ns
29:12-D
828
C . K . SUN~ et al.
(8) Develop a better partition if necessary: change non-basic and basic variables. (9) Update the Hessian matrix. (10) Check for the limit of the maximum number of iterations; if the total iterations are less than a user-prescribed limit, then the control returns to step (4) above and the cycle is repeated; otherwise the iterations stop. This class of problems in the optimally-tailored design of composite laminates is a non-convex programming problem, hence this optimization method will only converge to a local minimum. It is therefore evident that the several different initial points must be chosen in order that an approximate global solution be determined. ILLUSTRATIVE
EXAMPLE
The task of designing a fibrous composite cantilever beam with optimal damping which is to be fabricated in an E-glass fiber and DX-210 epoxy resin, is considered herein as being a viable illustrative example for demonstrating the capabilities of the proposed methodology. The elastic moduli and density for the fiber and matrix are written Ef = 72.0 GPa E m = 3.45 GPa
vf = 0.2 vm = 0.35 pf ---= 2.6 g cm- 3 Pm ----- 1.2 g cm- 3. The specific damping capacities q~x, ~y and ~bs are abstracted from results presented in Ref. [3]. The stacking sequence for the different plies in each laminate is defined using the notation [ + 03 / - 0a Is where the symmetry of the composite laminate with respect to the mid-plane is denoted by the subscript s. A solution was sought for the above optimal design problem using the following data set: tmin -----0 . 1 8 0 m m ,
re/2,
tma x
Omi n =
--
Omax =
Eli=
l x 1 0 6 N m m 2.
= 0.230
mm
1r/2
The manufacturing specification for an initial arbitrary design is presented in the upper portion of Table 1, which is a unidirectional laminate. Finite-element transient dynamic TABLE 1
VI
Initial design
Optimal design
Design variables zj (mm)
0j (°)
VI VI~ V~ V~ V~ V~
= = = = = =
0.55 0.55 0.55 0.55 0.55 0.55
zl z2 z3 z4 z5 z6
= = = = = =
0.185 0.370 0.550 0.740 0.925 1.110
01 02 03 04 05 06
= = = = = --
V[ V[ V~ vf, V~ V~
= = = = = =
0.6339 0.5586 0.5086 0.5242 0.5243 0.5211
zl z2 z3 z4 zs z6
= = = = = =
0.230 0.460 0.690 0.920 1.150 1.380
01 02 03 04 05 06
= -29.24 = -29.24 = - 29.24 --- 29.24 = 29.24 = 29.24
E1 = 1.0296 x 106 N m m 2
0 0 0 0 0 0
Damping capacity
Young's modulus (GPa)
0.009
44.46
0.036
17.9
Synthesizing laminated beams with optimal response
829
analyses of cantilever beams of length 800 mm and width 25.4 mm, fabricated with the initial specification and then the optimal laminate specification were undertaken in order to compare the two response histories and to evaluate the superior elastodynamic characteristics of the optimally designed beam. These simulations modelled the free vibrational response of the cantilever beams which were initially subjected to a deflection of 6.7 mm. In these simulations, Euler-Bernoulli beam elements were employed 1,8], which readily permit the flexural moduli expressions developed herein to be incorporated within the formulation. The model for the damping matrix is based on the concept of equivalent viscous damping and also the hypothesis that structural damping is a function of the strain energy dissipated during each stress cycle [9]. The damping matrix [C] is therefore written as
~=1\~i/)
[C]-2nA2
where subscript i denotes the ith mode, n denotes the number of active modes in the response of the beam, A~is the amplitude of the ith mode which responds with a natural frequency o9i. In equation (51) the term [K] represents the stiffness matrix, A represents the total energy content in the n modes and A~ = oJ? ( { C , } r i , M ] { X o } ) 2 + ( { V , } r [ M ] {.~'o }) 2,
(52)
where {Ui } is the ith modal vector, [M] is the mass matrix, and {Xo } and {)~o} are the initial displacement vector and initial velocity, respectively. It may be noted that the mass, stiffness and damping matrices are all functions of the micromechanical and macromechanical design variables of the laminate. Simulations based on the above finite-element formulation are presented in Figs 1 and 2, from which it is evident that the elastodynamic response of the beam with the optimal properties is considerably superior to the response of the beam fabricated in a unidirectional composite. The optimal laminate specification is presented in the lower portion of Table 1. This optimal manufacturing specification for a laminated beam yields a damping capacity four times larger than the initial design, while maintaining the same flexural rigidity as the initial design. CONCLUSION
A methodology for synthesizing the manufacturing specification of symmetrical laminated beams has been proposed which permits structural members with optimal damping properties to be fabricated. The design parameters employed in this methodology are the
•"~
4
E
0 "o ,o. I---4-
-8
o
i
5
i
~o
,'~
2'0
2'5
Time (s)
Original design FIG. I. A simulation of the transient response of the tip of a cantilever beam fabricated with a unidirectional laminate.
830
C.K. SUNGet al. 8-
!4 g
'i! a_
()
d 5
Io
15
21©
L
Time (s) OptimoL design
FIG. 2. A simulation of the transient response of the tip of a cantilever beam fabricated with the optimal material specification.
fiber a n d m a t r i x properties, their v o l u m e fractions, ply thicknesses, ply o r i e n t a t i o n s a n d the stacking sequence. An illustrative e x a m p l e d e m o n s t r a t e s the significance o f the p r o p o s e d m e t h o d o l o g y in laminate design, by s h o w i n g the critical d e p e n d e n c e o f d a m p i n g u p o n the various m i c r o m e c h a n i c a l a n d m a c r o m e c h a n i c a l design variables. The m e t h o d o l o g y presented here is generic in n a t u r e a n d can be clearly extended to various multi-objective multic o n s t r a i n e d o p t i m i z a t i o n p r o b l e m s in p o l y m e r i c c o m p o s i t e laminate design. Acknowledgements--This work was funded partially by the National Science Foundation, under Grant Number
MSM-8514087, the Defence Advanced Research Projects Agency under Grant Number DAAL03-87K-0018, the State of Michigan, Department of Commerce, Research Excellenceand Economic Development Fund, and also the Composite Materials and Structures Center at Michigan State University. The support is gratelully acknowledged. REFERENCES 1. D. X. LIAO,C. K. SUNGand B. S. THOMPSON,The optimal design of symmetric laminated beams considering damping. J. Composite Mater. 21, 485-501 (1986). 2. R. G. NJ and R. D. ADAMS,The damping and dynamic moduli of symmetric laminated composite beams-theoretical and experimental results. J. Composite Mater. 18, 104-121 (1984). 3. R.G. NI and R. D. ADAMS,A rational method for obtaining the dynamic mechanical properties of laminae for predicting the stiffness and damping of laminated plates and beams. Composites 15, 193 199 (1984). 4. S. W. TSA[ and H. T. HAHN, Introduction to Composite Materials. Technomic Publishing Co., Westport, Connecticut (1980). 5. R. M. JONES, Mechanics of Composite Materials. McGraw-Hill, New York (1975). 6. E. SANDGRENand K. M. RAGSDELL,The utility of nonlinear programming algorithms: a comparative study-parts 1 and 2. ASME J. Mech. Des. 102, 540-551 (1980). 7. K. SCHITTKOWSKI,Nonlinear Programming Codes: Information, Tests, Perjormance, Lecture Notes in Economics and Mathemmtical Systems, Vol. 183. Springer-Vertag, New York (1980). 8. L. MEIROVITCH,Elements of Vibration Analysis. McGraw-Hill, New York (1975). 9. L. MEIROVITCH,Analytical Methods in Vibrations. MacMillan, New York (1967).
0020-7403/87 $3.00+ .00 Pergamon Journals Ltd.
A M E T H O D O L O G Y FOR S Y N T H E S I Z I N G SYMMETRIC L A M I N A T E D BEAMS WITH O P T I M A L E L A S T O D Y N A M I C R E S P O N S E CHARACTERISTICS C. K. SUNG,* B. S. THOmPSOn,* M. V. GANDHI* a n d C. Y. LEEr * Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan and * Department of Mechanical Engineering, and Composite Materials and Structures Center, Michigan State University, East Lansing, MI 48824-1226, U.S.A. (Received 30 June 1987) Abstraet--A methodology is presented herein for synthesizing symmetric laminated beams with optimal elastodynamic response characteristics by generalizing a previous publication of the first two authors [Liao et al., J. Composite Mater. 21,485-501 (1986)] in order to include the percentage of fiber and matrix in each ply, and to also accommodate fibers with transversely isotropic properties. Constitutive models are developed for the stiffness and damping properties of laminated beams as a function of the ply fiber volume fraction, the fiber orientation in each ply, the ply thickness and the stacking sequence prior to incorporating these models into an optimal design formulation. This formulation is then employed in illustrative transient response studies of cantilever beams in order to demonstrate the significance of incorporating these micromechanical parameters into design algorithms for optimally tailoring the properties of these laminated beams. A generalized reduced gradient algorithm is employed in this optimal design endeavor.
NOTATION
co-ordinate system oxyz: x-axis along the fiber co-ordinate system 0123: l-axis along the principal longitudinal axis of the beam E x longitudinal Young's modulus of a unidirectional ply Ef elastic Young's modulus of the fiber Em elastic Young's modulus of the matrix t/f volume fraction of the fiber Vm volume fraction of the matrix Er transverse Young's modulus of the ply G~, transverse shear modulus of the ply ky transverse plane strain bulk modulus of the ply ~x longitudinal Poisson's ratio Vf longitudinal Poisson's ratio of fiber ~m longitudinal Poisson's ratio of matrix qk stress partitioning parameter kb transverse plane strain bulk modulus of the fiber Ely transverse Young's modulus of the fiber km bulk modulus of the matrix Gm shear modulus of the matrix qG stress partitioning parameter in transverse shear Gs shear modulus of the ply r/s stress partitioning parameter in shear Gf shear modulus of the fiber P number of plies Z.i distance of the jth ply from the beam mid-plane 0 fiber orientation angle [c3 global damping matrix [K] global stiffness matrix ugi ith undamped natural frequency {tJ,} ith modal vector [M] mass matrix {Xo } initial displacement vector initial velocity vector Pf density of the fiber Pm density of the matrix 821
822
C.K. SUNG et aL
AW W ~, AW~ ~b~ ~ l, b, h m,n Qo U~ D~j d~j
strain energy dissipation maximum strain energy specific damping capacity i = x, y, s; strain energy dissipation components associated with ax, ay and % i = x, y, s; specific damping capacity components associated with a~, e~; ay, q ; as, r s i = x, y, s; damping capacity component coefficients associated with ~,, i = x, y, s length, width and height (total thickness) of the beam cos0, sin0 components of moduli; i,j = x, y, s, 1, 2, 6 i = 1, 2, 3, 4, 5; linear combinations of moduli i, j = 1, 2, 6; flexural moduli of a multidirectional symmetric laminate i, j = 1, 2, 6; flexural compliance of a multidirectional symmetric laminate; the [dlj] are the inverse of the [Dii]
tj MI El I kj
thickness of t h e j t h ply flexural moment per unit width o f the beam longitudinal Young's modulus relative to the 01-axis second moment of area curvature component associated with the strain component ~1
superscripts
j
jth ply
INTRODUCTION
Fibrous composite laminates are increasingly being employed in the fabrication of a very wide variety of products which are subjected to dynamical excitations. These products range from sporting goods and medical equipment to aerospace components and industrial machinery. In many of these applications components must be fabricated with high stiffness and also high values of structural damping in order to tailor their dynamical response. However, there is a distinct absence of viable design methodologies for synthesizing laminates in order to achieve these goals, and the work presented herein, is directed towards partially filling this void in the literature. The objective herein, is, therefore, to develop a design methodology for this important task. This objective is accomplished by generalizing a recent publication on this subject [1], by including the fiber volume fraction of each ply within the formulation and incorporating fibers with transversely isotropic properties. The resulting formulation enables symmetrical laminated beams to be designed with high stiffness and also high damping by optimally selecting the fiber volume fractions of each ply, the ply thicknesses, the stacking sequences and also the ply thicknesses. An illustrative example of the transient response of a simple cantilever beam serves to demonstrate the advantages of implementing the proposed methodology. The methodology developed herein is generic in nature and hence may be utilized in different applications where the design objectives may be significantly different. In order to develop this design methodology, a micromcchanical model for the constitutive properties of a laminate as a function of various constitutive properties is presented in the next section. This is then followed by the development of models for the flexural moduli of symmetrical laminated beams and also their structural damping capacity. These models are then incorporated into a optimal-design formulation which seeks to maximize the damping capacity subject to constraints imposed on the flexural rigidity of the beams. A finite-element approximation and also the generalized reduced gradient algorithm are employed as tools to determine an optimal solution for this class of design problems.
MICROMECHANICAL
MODEL
The methodology proposed herein for the optimal design of laminated beams considering damping and stiffness is based on the propositions that the global properties of the material are elastic and the damping properties are a function of the elastic strain energy dissipated during each stress cycle I-2, 3]. In order to establish the basis for an optimal design methodology for this class of problems, the elastic moduli must first be expressed as a
Synthesizing laminated beams with optimal response
823
function of the properties of the constituents. These micromechanical design variables, which represent the constitutive characteristics of the laminate, are the fiber and matrix properties and the fiber volume fraction. Subsequently, these variables are incorporated in macromechanical models for the global mechanical properties of the laminate which include such design variables as ply orientations, ply thicknesses and the stacking sequence. These variables serve as manufacturing specifications for the fabrication of the laminate and also govern the dynamical response of the fabricated part. The first objective of developing a viable formulation for the elastic moduli of each ply is accomplished by utilizing the modified rule-of-mixtures approach for fibers which are transversely isotropic [4]. Graphite and aramid fibers, which account for a substantial proportion of the industrial market, are typical fibers in this category. The longitudinal Young's modulus, Ex, of a unidirectional ply is defined by Ex = V f E f + V m Em,
(I)
where Ef and E m are the elastic moduli of the fiber and matrix, respectively, and Vr and Vm are the fiber and matrix volume fractions. Assuming that the material is free of voids, these volume fractions are related by Vm = 1 - Vf.
(2)
The transverse Young's modulus, Ey, of a unidirectional ply is defined by [4], p. 397, 4 kr Gy Er - k r + nGy'
(3)
where
• 4krv~ n = 1 -t. Ex
(4)
In equation (4), vx, denotes the longitudinal Poisson's ratio which is related to the ply volume fraction by Vx = Vf vf --[- VmVm,
(5)
where vf and vm are the Poisson's ratios of the fiber and matrix, respectively. In equation (3), k r denotes the transverse plane strain bulk modulus of the ply, which is defined by
kr 1 __ (Vf+r]kVm)~kfy 1 ( V f + ~mVm "k ),
(6)
where r/k is the stress partitioning parameter which measures the relative magnitude of the average stress in the matrix as compared with the average stress in the fiber for conditions of plane strain hydrostatic uniaxial loading. The stress partitioning parameter t/k is defined by I 1+ qk = 2(1 -- vm)
(7)
and kfy, which denotes the transverse plane strain bulk modulus of the fiber, is defined by
Efy kfy = 2(1 - vf)'
(8)
where Efy is the transverse Young's modulus of the fiber. In equation (6), km represents the bulk modulus of the matrix which is defined by Gm km - 1 - 2vm'
(9)
where Gm, which denotes the shear modulus of the matrix, is defined by Em G m - 2(1 + Vm)"
(10)
824
C.K. SUNGet al.
/ h e term Gy, in equation (3) denotes the transverse shear modulus of the ply, and is defined by
l Gy -
( 1 )
[ Vf +
Vm'
Vf+/']G Vm ~Gfyfy ~]GG--mJ'
[|l}
where Gfy is the transverse shear modulus of the fiber and qG, which is the stress partitioning parameter in transverse shear, is defined by
qG=I4(llVm)](3--4Vm+~).
(12)
The shear modulus of the ply, Es, may be defined as a function of the previous expressions. Thus 1 1 Vf + qs Es Vf+qsV m where q~, which denotes the stress partitioning parameter in shear, is defined by
1( r/s=~ 1
+Gm~ Gf/"
(13)
(14)
The shear modulus of fiber, Gf is denoted by Ef Gf - 2(1 + vf)'
(15)
where vf is the transverse Poisson's ratio of the fiber. The above equations permit useful relationships to be developed for the following components of the elastic moduli of each ply by utilizing the expressions,
Qxx = mE~
(16)
Qxy = n vyE~
(17)
Q.. =
(18)
m Ey
Qrx = m vxEy
(19)
Qss = Es,
(20)
m = E1 - vxvr]- l
where
(21)
FLEXURAL MODULI The stiffness and damping characteristics of a structure are governed by the geometrical configuration and also the material properties. Having developed the necessary mathematical apparatus for representing the material properties, attention now focuses on establishing a viable model for the flexural moduli of a symmetrical laminated beam. The flexural moduli, Dab , a, b-l, 2, 6 governing the response of a symmetrical laminated beam may be expressed as a function of the micromechanical and macromechanical design variables using the foregoing expressions for each ply. These functions are 2 e/2
Dlx = ~ ~ 1 [ (z3 - z3-1 ) (U[JI + U~j' cos 20 OJ+ Uy ~cos 400~]
2 P/2
O,2 = ~Y~.=E(z~ ~ -
)(v?
v~'
z)3_ l
--
P/2
(U(~j)
2 e/2
(u~j~
2
DI6 = 3j--~t E(z3 -- z3-1 )
022 = ~j_E E(z3 - 4 - , )
(22)
cos 40 O~)-I
½sin2#J' +
U3 j)
- u ? cos20'~' +
(23)
sin4#~')] u~ j~
cos40%]
(24)
(25)
Synthesizingl a m i n a t e d
2 P/2 3s~.=, [(z 3 _ Z j3- 1 ) ~ ~,u,J, 2 ½sin20~J,
D26
066
b e a m s with o p t i m a l response
.~-
--
U3~J)sin40,J))]
2 P/2 3jE.=I [(z3 - - Z j _a 1 )(us~J,_ U3~J,cos40,J,)],
825
(26)
(27)
where the beam comprises a total of P plies, zi is the distance ofthejth ply from the mid-plane of the beam and 0 t ~ is the angle ofthejth ply relative to the longitudinal axis of the beam. The compliance [dab] may be determined by inverting matrix [Dab]. The terms U~J~, i = 1, 2, 3, 4, 5 in equations (22)-(27) are defined by ulJl
l rar~lJJ . ~otJ~ ~_ 2r~¢J~ . 4,~J~ ~xr ~Vss _1
= ~ l _ - , ~ x x -1- - ~ r r
U~2s~=
r,~'~
I "~ L ~ x x
--
~ L~,xx U(4J) U~J)
1 rs-~tJ)
,
W~ 1 ~YyA
I rr~(J)
= ~ L~
(29)
~r~yy - - - - Y ~ x y - -
~SS
()(J) d'- "--(J)
" ~¢J} ~ 4¢dss J
= g L ~ x x -I- ~ y y
(28)
O~xy
-
d
9 / ' ) (j1-4- " -(J} ± ,~yr - - - ~ x r - - 4 Q s ~ ] . , o~Jl
(31) (32)
This set of equations completes the development of the necessary expressions governing the stiffness characteristics of a symmetric laminated beam, as functions of both the micromechanical design variables and also the macromechanical design variables. Attention now focuses on utilizing these expressions in the development of relationships for the damping properties of a beam. STRUCTURAL DAMPING The model developed herein for the structural damping of a symmetrical laminated beam as a function of the fiber and matrix properties, and their volume fractions, the stacking sequences, the ply thicknesses and also the ply orientations, builds upon the theoretical arguments presented in Ref. [1] by including within the formulation the fiber volume fraction and also fibers which are transversely isotropic. This formulation generalizes the work presented in [2, 3]. The developments presented in the latter suite of publications are employed herein because of the favorable correlation between the analytical and experimental results presented in [2], thus validating the predictive capabilities of the analytical models for the structural damping. Consider the time-dependent flexural motion of an arbitrary symmetric laminated beam comprising P plies with different properties. The specific damping capacity, ~b, of the beam may be defined as AW = W'
(33)
where AW is the strain energy dissipated during each stress cycle and W is the total strain energy. The energy dissipated per cycle may be considered to comprise the sum of three components, which may be written AW = AWx + AWr + AWs,
(34)
where the subscript s denotes the shear component in the oxy plane. These three components may be written in terms of the strain energy dissipated during a single stress cycle. Thus for a beam of width b, total length I and total thickness h, AWx = 2b
AWy = 2b
do
~4)xaxexdz d/
I hI2 1 -~ qbytrrer dz dl dO
f•
(35)
(36)
C . K . SUNG et al.
826
f: (h'2 1 AWs = 2b
d,,
} 4~sasesdzd/.
(37)
These expressions contain terms defined relative to the principal fiber directions oxyz, rather than the principal geometrical axes of the beam 0123. A more convenient formulation may be established by employing orthogonal transformation expressions [4, 5] to yield
b P/2
Am
x
=
~jEI[49:, ,n~ {Q'/'ld,1 +Q':~2dl2+Q'/~d,6} (J).
2
.=
{mZda, +m~njdl6}
Mfdl
x 1- J, 3 - z 33 1)]
(38)
b P/2
AW, = 5 El.=t_.rrl-tfitJ'n2j {QTldl,+ Q':)zd12+ Q°~6d16} {n~d,l x (z~-z~_l)]
f'
-mjnjd16
M~ dl
} (39)
0
b e/2
=
.
Lq), mrnj~:z,,d,l +Q'a~2d12+Q':~d16}{2mjnjdll
E=1 '-'"'
- ( m ~ - n})d16 } (z3-z3-1)]
fo
M 2 dl,
(40)
where the integration from the beam midplane to the surface of the beam is written as a summation statement rather than an integral for the P/2 plies. In equations (38)-(40), mj = cos0 °~, n~ = sin0 I~, M1 is the bending moment per unit width of the beam, and the elastic moduli for the jth ply are written 4.- 2mj2 nj2 ()tj~ + 4 m j2n j2QsstJ)
/)i j}
~
~1/2)
~2.2,¢'~(j) i O I j p 4mj2 nj2 Qss (j~ = mj2 nj2 c.).(.j )._ I '"i "J ~rr + (m~ + n. j, *,~x~
QOl 16
DO) - - m j n j 3Q rtj~ = m3n ""j "'jr..,xx r +
--.4t'~tD
n 4 ( ' ~ {j)
(41) (42)
(mini3 _m3n~)Q~jrl
+ 2(mjn~ - "-Jm3n-J,~OIJ~ss"
(43)
Assuming that the flexural deformation of the beam is governed by the Bernoulli-Euler hypothesis, the total strain energy of the beam may be written as
W=
fo I b: M ~ dl 2 E~ I '
(44)
where Ea is the Young's modulus associated with the longitudinal axis 01 of the beam and I is the associated second moment of area. For this class of beams
bMl kl - E , I '
(45)
where kl is the curvature component associated with M~. Since the moment-curvature relationship for symmetric laminates is kl = dll M1,
(46)
b --. EI = -dll I
(47)
the following expression results
The specific damping capacity qJ, is assumed to be the sum of three components
qJ,, = AW/W, ~br = AWr/W and qJs = AWs/W. Utilizing equations (38)-(44) these three
Synthesizing laminated beamswith optimal response
827
components can be written as
=
2[~_~1 ~2 ,t,(j)__2 I~(J).s (J) (J) 2 "v~ "'j ~ t t u l l +Qt2dt2 +Qt6d16)(mjdll +mjnjd16)
× ev =32
3 ]
- z~_ 1 )
1
(48)
th,j)n2 .-j (Q]~dll +Q,l~d12 1=1
(49)
×
~Os=3 d-~xxj=l ~p°s)mjnj(Q',~d,t±t~°"tT y.g12~12 "~/"~(J)d, kgl6 16/' x (2mjnjdt 1 -- (mj2 --nj2 )dl6)(z~-zj_ 3 1) J
(50)
These three equations complete the development of a formulation for predicting the damping capacity of symmetrical laminated beams as a function of both micromechanical and also macromechanical design variables. Having developed models for the micromechanical material properties, stiffness and damping characteristics, these formulations provide the essential ingredients for predicting the elastodynamic response of symmetrically laminated beams by employing finite-element techniques. OPTIMAL DESIGN STRATEGY The mass, stiffness and damping matrices contained in the finite-element model govern the elastodynamic response of symmetrical laminated beams. The objectives of an optimal design formulation must first be specified before utilizing an optimization algorithm in order to develop a capability to synthesize optimal manufacturing specifications. Herein, the objective of the optimal design formulation is posed as one which seeks to maximize the damping capacity of a laminated beam subject to constraints imposed on the fiber volume fraction, ply thickness, ply angle, stacking sequence, number of plies and the flexural rigidity. The formulation may be written as maximize subject to 0max >~ 0j i> 0min tmax ~ tj ~ tmin
Vfm
Vo Vfmi°
El I = constant P = constant. This class of optimization problems is solved herein by employing the generalized reduced gradient algorithm. This algorithm has been shown to exhibit superior speed and robustness under a wide variety of problem conditions [6, 7]. The essential features of this non-linear programming algorithm employed are as follows. (1) Input the initial parameters. (2) Partition the design variables as basic and non-basic variables., (3) Search for a feasible starting point by using the constraint function as a temporary objective function and minimize the error. (4) Calculate the reduced gradient. (5) Calculate the search direction. (6) Check the convergence criterion: if this is satisfied the results are stored and the iterations stopped, otherwise the iterations continue. (7) Restricted line search: use the Newton-Raphson approach to determine the step size. ns
29:12-D
828
C . K . SUN~ et al.
(8) Develop a better partition if necessary: change non-basic and basic variables. (9) Update the Hessian matrix. (10) Check for the limit of the maximum number of iterations; if the total iterations are less than a user-prescribed limit, then the control returns to step (4) above and the cycle is repeated; otherwise the iterations stop. This class of problems in the optimally-tailored design of composite laminates is a non-convex programming problem, hence this optimization method will only converge to a local minimum. It is therefore evident that the several different initial points must be chosen in order that an approximate global solution be determined. ILLUSTRATIVE
EXAMPLE
The task of designing a fibrous composite cantilever beam with optimal damping which is to be fabricated in an E-glass fiber and DX-210 epoxy resin, is considered herein as being a viable illustrative example for demonstrating the capabilities of the proposed methodology. The elastic moduli and density for the fiber and matrix are written Ef = 72.0 GPa E m = 3.45 GPa
vf = 0.2 vm = 0.35 pf ---= 2.6 g cm- 3 Pm ----- 1.2 g cm- 3. The specific damping capacities q~x, ~y and ~bs are abstracted from results presented in Ref. [3]. The stacking sequence for the different plies in each laminate is defined using the notation [ + 03 / - 0a Is where the symmetry of the composite laminate with respect to the mid-plane is denoted by the subscript s. A solution was sought for the above optimal design problem using the following data set: tmin -----0 . 1 8 0 m m ,
re/2,
tma x
Omi n =
--
Omax =
Eli=
l x 1 0 6 N m m 2.
= 0.230
mm
1r/2
The manufacturing specification for an initial arbitrary design is presented in the upper portion of Table 1, which is a unidirectional laminate. Finite-element transient dynamic TABLE 1
VI
Initial design
Optimal design
Design variables zj (mm)
0j (°)
VI VI~ V~ V~ V~ V~
= = = = = =
0.55 0.55 0.55 0.55 0.55 0.55
zl z2 z3 z4 z5 z6
= = = = = =
0.185 0.370 0.550 0.740 0.925 1.110
01 02 03 04 05 06
= = = = = --
V[ V[ V~ vf, V~ V~
= = = = = =
0.6339 0.5586 0.5086 0.5242 0.5243 0.5211
zl z2 z3 z4 zs z6
= = = = = =
0.230 0.460 0.690 0.920 1.150 1.380
01 02 03 04 05 06
= -29.24 = -29.24 = - 29.24 --- 29.24 = 29.24 = 29.24
E1 = 1.0296 x 106 N m m 2
0 0 0 0 0 0
Damping capacity
Young's modulus (GPa)
0.009
44.46
0.036
17.9
Synthesizing laminated beams with optimal response
829
analyses of cantilever beams of length 800 mm and width 25.4 mm, fabricated with the initial specification and then the optimal laminate specification were undertaken in order to compare the two response histories and to evaluate the superior elastodynamic characteristics of the optimally designed beam. These simulations modelled the free vibrational response of the cantilever beams which were initially subjected to a deflection of 6.7 mm. In these simulations, Euler-Bernoulli beam elements were employed 1,8], which readily permit the flexural moduli expressions developed herein to be incorporated within the formulation. The model for the damping matrix is based on the concept of equivalent viscous damping and also the hypothesis that structural damping is a function of the strain energy dissipated during each stress cycle [9]. The damping matrix [C] is therefore written as
~=1\~i/)
[C]-2nA2
where subscript i denotes the ith mode, n denotes the number of active modes in the response of the beam, A~is the amplitude of the ith mode which responds with a natural frequency o9i. In equation (51) the term [K] represents the stiffness matrix, A represents the total energy content in the n modes and A~ = oJ? ( { C , } r i , M ] { X o } ) 2 + ( { V , } r [ M ] {.~'o }) 2,
(52)
where {Ui } is the ith modal vector, [M] is the mass matrix, and {Xo } and {)~o} are the initial displacement vector and initial velocity, respectively. It may be noted that the mass, stiffness and damping matrices are all functions of the micromechanical and macromechanical design variables of the laminate. Simulations based on the above finite-element formulation are presented in Figs 1 and 2, from which it is evident that the elastodynamic response of the beam with the optimal properties is considerably superior to the response of the beam fabricated in a unidirectional composite. The optimal laminate specification is presented in the lower portion of Table 1. This optimal manufacturing specification for a laminated beam yields a damping capacity four times larger than the initial design, while maintaining the same flexural rigidity as the initial design. CONCLUSION
A methodology for synthesizing the manufacturing specification of symmetrical laminated beams has been proposed which permits structural members with optimal damping properties to be fabricated. The design parameters employed in this methodology are the
•"~
4
E
0 "o ,o. I---4-
-8
o
i
5
i
~o
,'~
2'0
2'5
Time (s)
Original design FIG. I. A simulation of the transient response of the tip of a cantilever beam fabricated with a unidirectional laminate.
830
C.K. SUNGet al. 8-
!4 g
'i! a_
()
d 5
Io
15
21©
L
Time (s) OptimoL design
FIG. 2. A simulation of the transient response of the tip of a cantilever beam fabricated with the optimal material specification.
fiber a n d m a t r i x properties, their v o l u m e fractions, ply thicknesses, ply o r i e n t a t i o n s a n d the stacking sequence. An illustrative e x a m p l e d e m o n s t r a t e s the significance o f the p r o p o s e d m e t h o d o l o g y in laminate design, by s h o w i n g the critical d e p e n d e n c e o f d a m p i n g u p o n the various m i c r o m e c h a n i c a l a n d m a c r o m e c h a n i c a l design variables. The m e t h o d o l o g y presented here is generic in n a t u r e a n d can be clearly extended to various multi-objective multic o n s t r a i n e d o p t i m i z a t i o n p r o b l e m s in p o l y m e r i c c o m p o s i t e laminate design. Acknowledgements--This work was funded partially by the National Science Foundation, under Grant Number
MSM-8514087, the Defence Advanced Research Projects Agency under Grant Number DAAL03-87K-0018, the State of Michigan, Department of Commerce, Research Excellenceand Economic Development Fund, and also the Composite Materials and Structures Center at Michigan State University. The support is gratelully acknowledged. REFERENCES 1. D. X. LIAO,C. K. SUNGand B. S. THOMPSON,The optimal design of symmetric laminated beams considering damping. J. Composite Mater. 21, 485-501 (1986). 2. R. G. NJ and R. D. ADAMS,The damping and dynamic moduli of symmetric laminated composite beams-theoretical and experimental results. J. Composite Mater. 18, 104-121 (1984). 3. R.G. NI and R. D. ADAMS,A rational method for obtaining the dynamic mechanical properties of laminae for predicting the stiffness and damping of laminated plates and beams. Composites 15, 193 199 (1984). 4. S. W. TSA[ and H. T. HAHN, Introduction to Composite Materials. Technomic Publishing Co., Westport, Connecticut (1980). 5. R. M. JONES, Mechanics of Composite Materials. McGraw-Hill, New York (1975). 6. E. SANDGRENand K. M. RAGSDELL,The utility of nonlinear programming algorithms: a comparative study-parts 1 and 2. ASME J. Mech. Des. 102, 540-551 (1980). 7. K. SCHITTKOWSKI,Nonlinear Programming Codes: Information, Tests, Perjormance, Lecture Notes in Economics and Mathemmtical Systems, Vol. 183. Springer-Vertag, New York (1980). 8. L. MEIROVITCH,Elements of Vibration Analysis. McGraw-Hill, New York (1975). 9. L. MEIROVITCH,Analytical Methods in Vibrations. MacMillan, New York (1967).