NASA Technical Memorandum 102343

An Integrated Methodology

for Optimizing Structural

Composite Damping

D.A. Saravanos and C.C. Chamis

Lewis Research Center

Cleveland, Ohio

Prepared for the

1989 Winter Annual Meeting

of the American Society of Mechanical Engineers

San Francisco, California, December 10-15, 1989

' ,- v -_ L:;(,

An Integrated Methodology For Optimizing

Structural Composite Damping

D.A. Saravanos*, and C.C. Cbamis**

National Aeronautics and Space Administration

Lewis Reserch Center

Cleveland, Ohio 44135

ABSTRACT

A method is presented for tailoring plate and shell composite structures for optimal

forced damped dynamic response. The damping of specific vibration modes is optimized

with respect to dynamic performance criteria including placement of natural frequencies

and minimization of resonance amplitudes. The structural composite damping is synthesized from the properties of the constituent materials, laminate parameters, and structural

geometry based on a specialty finite eIement. Application studies incIude the optimization of laminated composite beams and composite shei1s with/_ber voIume ratios and ply

angles as design variabIes. The results illustrate the signit_cance of damping tailoring to

the dynamic performance of composite structures, and the effectiveness of the method in

optimizing the structural dynamic response.

INTRODUCTION

Fiber composite materials are broadly utilized in light-weight structures, as they readily provide superior specific modulus and strength. In addition to stiffness and strength,

polymer-matrix composite materials provide higher material damping than most metals because of their "viscoelastic" matrix and heterogeneity. High specific stiffness and strength

are sufficient conditions for improved static performance, but they do not always ensure

improved dynamic performance. Passive structural damping is also a crucial dynamic property in vibration and sound control, as it generalJy improves resonance phenomena, settling

times, and fatigue life. Composite materials are primarily targeted for structures requiring good dynamic performance, such as engine, aircraft, and space structures. Therefore,

the inherent damping capacity of composites becomes a significant design factor, making

polymer-matrix fiber composites even more attractive as structural materials.

*National Research Council--NASA Research Associate.

**Senior Research Scientist.
Research on the damping capacity of composite materials, laminates, and beams [1-6],

has shown that laminate damping is highly tailorable with respect to constitutive properties, volume fractions, and ply-orientation angles. The same work suggests that composite

structures should be tailored for optimal combinations of damping and stiffness in order

to obtain improved dynamic performance. To the authors' best knowledge, formal methods for tailoring general plate/shell composite structures for optimal damping and optimal

damped dynamic response are not presently available. Research has been reported on the

tailoring of plate/shell composite structures for optimum static and/or undamped dynamic

performance. Composite plates have been optimized [7-10] based on static performance

criteria. Composite plates have been also optimized either with a constraint on the first

natural frequency [11], or with performance criteria based on the undamped frequency

response [12]. Refs. [13,14] present the optimal tailoring of composite beams and links

for maximum damping capacity based on static constraints. Most reported methods can

produce designs with improved integrity and stiffness, having natural frequencies within

desirable ranges. The inclusion of the structural composite damping into the present

method provides a new dimension regarding the optimal tailoring of composite structures,

as trade-offs between damping, weight, stiffness, and placement of natural frequencies are

now possible. Furthermore, to the authors' best knowledge, damping micromechanics have

not been included so far into the optimal design of composite structures, and in most reported eases the analysis starts from the laminate level. The present method incorporates

such a damping micromechanics theory, hence, it provides the capabifity for tailoring also

the constituent materials and fiber volume ratios.

This paper presents an integrated computational methodology for simulating the dynamic response of composite structures, and optimizing structural damping in conjunction

to other static and dynamic design criteria. The method is targeted for composite structures under forced excitation. The dynamic performance criteria axe based on modal

damping capacities, resonance dynamic amplitudes, and natural frequencies. The method

can produce designs having natural frequencies inside the desirable frequency domain, and

2
also minimized resonance amplitudes. The optimal tailoring includes tailoring of the basic composite material(s), and tailoring of the laminate configuration. The modeling of

composite damping and other elastic properties is based on micromechanics and laminate

theories [4,5,15]. The global damping capacity of composite structures is simulated based

on a specialty finite element. Finite element damping matrices have been developed, based

on previous theories, for a triangular plate element. The optimal design problem is formulated, and solved with the feasible-directions non- linear programing algorithm. The

method is demonstrated by applying it to a composite beam and a composite cylindrical

shell.

METHOD

This section summarizes the methodology. The method includes: (1) simulation of

the structural composite damping, (2) simulation of the damped frequency response of

composite structures, and (3) formulation and solution of the optimal design problem.

Structural Composite Damping

Local Laminate Damping

Approximate micromechanics equations are utilized, which have been derived based

on hysteretic damping. On-axis specific damping capacities (SDC's) are represented in

terms of elastic and dissipative properties of the fibers and matrix, interface properties,

temperature, and moisture. Off-axis SDC's, ie. the SDC's of a ply loaded at an off-axis

angle, are related to the on-axis SDC's with proper transformations, and subsequently, the

specific damping capacity ($DC) of the composite laminate is synthesized. The laminate

SDC is related to constituent properties, volume fractions, interface properties, hygrothermal parameters, fiber orientation, laminate configuration, and local deformation. The

micromechanics and laminate damping theories are presented in refs. [4,5]. Additional

micromechanics and laminate theories [15] are utilized for other mechanical properties of

the composite material.

3
Structural Damping

The laminate damping capacity is a local structural property. The global damping

capacity of the structure at a given deformation shape would be the integrated action

of local laminate damping over the structural volume. For plate and shell structures, the

structural SDC es would be the ratio of the integrals of the local laminate damping energy

6WL and local strain energy WL respectively, over the structural area A,

42, = f A 6WL dA

fA WLdA (1)

ttence, the local laminate damping can only provide a rough estimate of the global structural damping of simple structural components subjected to forced vibration loading conditions. In case of complex structures or structural components, global structural damping

is recommended. Numerical integration of laminate damping over the structural volume is

performed based on finite element discretization. The developed procedure is summarized

in the following paragraphs.

The local laminate damping and strain energies are first integrated over the volume

of the finite element based on numerical quadrature. The finite element damping and

stiffness matrices, [C,] and [Ke] respectively, are obtained from the laminate damping

matrices ([Ad], [Cd], [Dd] )[5] and stiffness matrices ([A,], [Cs], [De])as follows:

Ec.l I ')Ic,l I ,ljfc'JI

[C,l [D.]

where [B] is the strain shape function matrix, and Ae the element area. After the integration, the damping and strain energi,'es of the finite element, $1,,l,\ and We, are directly

related to the respective element matrices and nodal displacements u..

6W, = lu, T[C,]u, (4)

1UeT[K_]u, (5)

t,
Summation of the damping and strain energiesof the individual elementsprovides

the global damping and strain energiesof the structure, _Ws and W. respectively, for the

specific deformation state.

we/

i=l

nel

i=1

The SDC of the structure at a specific deformation state is the ratio of damping and stored

strain energies.

¢, = (8)

ws

In order to establish a coherent measure regarding the damping capacity of a structure,

specification of representative deformation shapes is required. The mode shapes of the

structure are the natural choice for such representative deformation shapes, since any

small vibrational deformation can be expressed as a linear combination of mode shapes.

The structural damping associated with an individual mode shape is defined as modal

structural damping.

The previously described structural damping theory has been incorporated into a triangular plate element with three nodes, six degrees of freedom per node (3 deflections

u=, u u and uz; and 3 rotations ¢_, Cu and ¢.), linear shape functions for the in-plane

(membrane) deformations, and cubic shape functions for the out-of-plane (flexural) deformations. The same element is also utilized for modal analysis and simulation of the

dynamic response.

Structural Dynamic Response

Main emphasis is focused on the forced (frequency) response of composite structures.

Using finite element discretization, the dynamic response of the composite structure may

be approximated by the following system of n discrete dynamic equations,

[M]{_} + [C]{_} + [K]{u} = {F(t)} (9)

5
where [M], [C], and [K] are the mass, damping and stiffness matrices respectively, and {u}

is the vector of the n discretized degrees of freedom.

A typical excitation force {F(t)} would involve: surface tractions, body forces, and

hygrothermal forces. In the present study, the effects of body forces and hygrothermal

forces have been neglected. The study is limited to structures operating in room temperature dry environment. Hygrothermal effects may be significant for the design of composite

structures, and the theory has provisions to model them. However, the design problem is

complicated beyond the scope of the current paper, because the damping and stiffness of

the composite depend also on temperature and moisture variations [5].

The dynamic system in eq. (9) is transformed to the m x m modal space through the

linear transformation:

{u} = [¢]{q} (10)

where q isthe modal displacement vector,and [01isensembled from the firstm normalized

undamped eigenvectors of eq. (9). Assuming proportional damping, the transformation

yields the reduced uncoupled dynamic system,

[m]{4}+ [c]{0}+ [k]{q}= {f(t)}

Ira]= [¢]r[M][O]

[<: [¢]T[C1[¢]

[k]= [¢]r[K][¢]

{f(t))= [o]T{F(t)}

where,

(11)

The damping matrix [c] is formulated from the first m modal SDC's in accordance to

eqs. (2-8).

Forced Dynamic Response

The frequency response of the j-th nodal displacement would be,

w(_) = Z cJ_q_(_) (12)

k----1

6
where q_(_) is the frequency response of the k-th mode in modal space. In general, uj(w)

would be a complex number describing both amplitude and phase. The dynamic amplitude

would be:

lluj( )ll (13)

The resonance amplitude of the k-th mode, would be:

(14)

where wd,k is the damped natural frequency of the k-th mode. The resonance amplitudes

Ufk are used as dynamic performance measures.

Optimal Design

The structural damping methodology aims to optimize the forced response of composite structures. The objective is to tailor the composite materials and the laminate

configurations for optimal combination of stiffness and damping such that selected natural

frequencies are placed within desirable frequency ranges, and selected resonance peaks are

minimized. Compared to other methods which cnly ensure proper placement of natural

frequencies, the present method is far more powerful, as it drives all resonance frequencies in the feasible frequency domain, and concurrently minimizes the resonance peaks of

selected modes at selected structural sites. Hence, the present methodology is even applicable when placement of all resonant frequencies into the desirable frequency domain

is infeasible. The later problem is frequently encountered in design and is further complicated by the fact that the frequency bounds are usually stiff, in that, we have not

control on them during the design of a given structural component. The present method

readily includes the capacity to identify the most critical natural frequencies within the

infeasible frequency range and to minimize their resonance amplitudes by increasing the

respective modal damping values. The optimization problem is formulated in non-linear

programming form, as described bellow.

Design Vector: The design vector may include: (1) the ply orientation angles 8i of

each sublaminate (group of +0 angle plies), and (2) the fiber volume ratios (FVR's) k],_

of each sublaminate.
The structural dimensions are not altered. The effect of shape optimization on the

damped dynamic responseof composite structures and its interaction with material tailoring will be addressedin a future study.

Objective Function: In general, the method provides the flexibility to minimize

selected resonance amplitudes, based on the particular design requirements. The following

objective function will minimize the maximum resonance amplitude of the first m vibration

modes for qd displacements:

min(maz{Uf_(Si,kf,i)}) j = 1,...,qa, n = 1,...,m (15)

Constraints: The minimization of the objective function in Eq. (15) is subject to

the following constraints:

Upper and lower bounds on design variables:

-90.0 ° _ Oi < 902 °

(16)

o.o _<k_,i < 0.70 (17)

Ts,U s

Upper bounds [j on qs static displacements uj:

u; < U; '_ : = 1,...,q, (18)

Frequency constraints described in general by k upper and lower bounds, _u and flL

respectively, on m natural frequencies:

,,,._ a_', _. > a2 (19)

n=l,...,m and j=l,...,k

Upper bounds Uf_ U on m resonance amplitudes for qd displacements :

v;_ < u;__ j = 1,...,q_, and , = 1,...,m (20)
it is pointed out that the objective function (15) and constraints (18) and (20) may

include either deflections or rotations. Stress failure constraints are not included in the

present paper, but they will be included in future work.

Optimization Algorithm

As already mentioned, the optimization problem has been formulated in non- linear

programming form and is solved with the method of feasible directions. The feasible

directions algorithm is a primal optimization method, performing a direct search in the

feasible design space based on first order sensitivity for the objective function and active

constraints. Primal methods are more suitable for non-linear programing problems which

have computationally expensive constraints, since they typically require fewer constraint

evaluations than do the penalty transformation methods. The feasible directions method

used in this study incorporated active constraint set and line-search strategies for improved

computational efficiency.

APPLICATION STUDIES

Assumptions

The present section presents applications of the previously described method on the

two composite structures shown in Fig. 1:

(1) a cantilever laminated composite beam, 6in (152.4mm) long, lin (25.4mm) wide, and

0.2in (5.08ram) uniform thickness, and

(2) a cantilever cylindrical laminated composite shell , 16in (406.4mm) long, 16in

(406.4mm) wide, of 10in (254mm) radius, and uniform 0.2in (5.08ram) thickness.

Both structures were modeled with 80 triangular plate elements. The basic composite

material system for each case was HM-S (high modulus surface treated) graphite fiber in

an epoxy matrix. The mechanical properties of the fiber and matrix are shown in Table 1.

Each ply is 0.01in (0.254mm) thick (equal to 2 preimpregnated tapes of 0.005in), hence,

the composite laminate has 20 plies through the thickness. The applications were limited

to symmetric laminate configurations, shown in Fig. 2, consisting by either one or three

sublaminates at each symmetric side. As seen in Fig. 2, each sublaminate is a set of

regular 4-8 angle-plies. The fiber orientation angle and the fiber volume ratio (FVR) of
eachsublaminate wereconsideredasdesignvariables. The effectsof multiple sublaminates

with different fiber orientation anglesand FVR's are investigated with the following 4

laminates incorporating:

(a) one sublaminate of fixed 50% FVR with the ply angleas designvariable.

(b) one sublaminate with both the ply angleand FVR asdesignvariables.

(c) three sublarrSnatesof fixed 50% FVR with their ply anglesas designvariables.

(d) three sublaminateswith the ply anglesand FVR's as designvariables.

Case 1: Beams

The assumedloading conditions of the beam werea static load of 5 lb/in (876 N/m)

applied along the free edge, and a cyclic load of 0.1 lb/in (17.5 N/m) also applied along

the free edge. The design objective was to minimize the maximum resonance(z-axis)

amplitude of the first five modesat the middle of the tip. The first natural frequencywas

constrained to be less than 350Hz. An upper bound of 0.050in (1.27rnm) was imposedon

the z-axis static deflection (u-) of the free edge.

For comparison purposes, an additional optimization study was performed for one

sublaminate with varying ply orientation angle and FVR without optimizing damping.

In the later case,the ma_nimumstatic deflection of the beam was minimized subject to

an identical frequency constraint. The performancecriteria did not include any damped

resonanceamplitudes, hence,the effectof compositedamping on the designwasneglected.

The optimum designs for each of the previously described subcasesare presented

in Table £. The same table presents the reference design of a 50% FVR unidirectional

composite beam. The unidirectional beam was selected as a reference design, because is

known to exhibit the maximum static bending stiffness. As seen in Table 2, the first mode

has the higher resonance amplitude, consequently, this resonance peak was minimized. The

present method has produced optimum designs (Cases la, lb, lc, and ld) having lower

resonance amplitudes for the bending modes by factor of 2, compared to the unidirectional

design. In addition, the unidirectional design violates the frequency constraint, while all

optimum designs have a first natural frequency less than 350 Hz. The trade-off for the

improvements in dynamic performance was a reduction in static stiffness.

10
The superiority of the current method to other methods which do not optimize damping is also demonstrated in Table 2. The optimum design without considering damping in

the performance criteria satisfies the frequency constraint marginally, but has a higher first

resonance amplitude than the reference design. In contrast, all optimum designs produced

by the present method have reduced first resonance amplitudes by factor of 2, and have

first natural frequencies substantially below the upper bound constraint.

Additional comparison of the optimum designs produced by the current methodology

for various laminate configurations indicates that the optimization of FVR has produced an

additional 11% improvement in the objective function. This seems a rather insignificant

improvement compared to the higher manufacturing costs related with the production

of composites with customized FVR. Laminates with multiple sublaminate systems have

virtually produced no additional reductions in the objective function.

The frequency response functions (FRF's) of the optimum design for Case ld (three

sublaminates with varying fiber volume), and the reference design are shown in Fig. 3.

The z-axis dynamic deflection at the center of the tip induced by the previously described

cyclic load is plotted as a function of frequency. Clearly, the optimum design has a better

frequency response.

Case 2: Open Cylindrical Shells

The current section covers applications on a more complicated structure, such as the

cylindrical laminated composite shell shown in Fig. lb. The assumed loading conditions

are: (1) a uniform static force of 0.31 lb/in (54.3 N/m) applied along the free-edge in the

z-direction, and (2) a uniform cyclic force of 0.0625 lb/in (10.9 N/m) also applied along

the free-edge in the z-direction. The objective function was set to minimize the maximum

z-axis resonance amplitude of the first 5 natural frequencies at 3 sites (the middle and two

corners of the free-edge). The dynamic deflections Qf both corners were incorporated into

the design criteria, in addition to the deflections of the midpoint, in order to take into

account the bending in transverse direction (y-axis) and the material coupling between

bending and torsion. Frequency constraints were imposed on the first 5 natural frequencies

11
such that:

wn _< 150Hz, or w,_ _ 350Hz, n = 1,...,5 (21)

The z-axis static deflections along the free-edge were restricted to be less that 0.050in

(1.27ram). The z-axis resonance amplitudes of the first five modes at the three sites were

also restricted to be less than 0.050in (1.27mm). The reference design was a unidirectional

shell (0)_0 of 50% FVlZ.

The resultant optimum designs for the 4 different laminate configurations are shown

in Table 3. Table 3 also shows the maximum z-axis static deflection along the tip, the

z-axis resonance dynamic amplitudes at the site with the maximum resonance peak, the

natural frequencies, and the modal SDC's. The unidirectional reference design is also

shown in Table 3. The reference design violates the static displacement constraint and the

upper frequency bound of 350 Hz, since the 3rd and 4th natural frequencies are less than

350 Hz. All optimum designs have produced significant improvements to all performance

measures, as they are within the feasible design space and exhibit superior dynamic and

static stiffness. Contrary to Case 1 (beam), the consideration of multiple sublaminates

and varying FVR's had a definite impact on the resultant optimal designs.

Subcase 2a: The optimum design for the simplest laminate configuration has

produced a 39% decrease in the maximum resonance amplitude, in addition to a 59%

decrease in static deflection, and ha_ nx',ural frequencies within the feasible frequency

domain. The resultant optimum design is a +23.3 degrees angle-ply symmetric laminate.

Subcase 2b: The introduction of the FVR into the design vector has produced an

optimum design of _=25.8 degrees angle-plies with 0.62 FVlZ. With respect to the reference design, the optimum design has reduced the maximum resonance peak by 50%, the

maximum static deflection by 69% and satisfies all frequency constraints. The optimum

design has reduced the modal SDC's of the shell, hence, the additional improvements were

mostly accomplished by the increased stiffness due to the higher FVR.

Subcase 2c: Multiple sublaminates produced further reductions in the objective

function. In contrast to the beam in Case 1, the open cylindrical shell is a three-dimensional

12
structure, therefore, multiple sublaminate systems are expected to provide better tailoring capacity. Indeed, the consideration of three sublaminates with equal and fixed fiber

volume ratios has produced an optimum design with 53% reduction in objective function,

66% reduction in static deflections, and natural frequencies Within the feasible frequency

domain.

Subcase 2d: Three sublaminates with varying FVR's have produced the best

improvement in the objective function, as the resultant optimum design exhibits a 64%

reduction in the maximum resonance amplitude, decreased static compliance by 63%, and

satisfies all frequency constraints. Interestingly, sublaminate 2 has been reduced to a

passive damping layer. The outer sublaminate with -T26.5 degrees angle-plies and 0.691

fiber volume ratio provides most of the stiffness. Sublaminate 2 with 0.010 FVR is virtually

pure matrix and provides most of the damping. The inner sublaminate 3 with -T12.8

degrees angle-plies and 0.245 FVR provides both in-plane stiffness and damping. This

optimum design produced a constrained layer damping structure. The high first modal

SDC indicates that the reduction in the dynamic amplitude was mostly accomphshed due

to increased damping.

In all previous subcases, both maximum dynamic and static deflections occurred at

the corners of the free-edge. In contrast to the unidirectional design, all optimal designs

have asymmetric bending modes due to couphng between torsion and bending. The mode

shapes of the first four modes for the reference and optimum (subcase 2d) designs are

shown in Figs. 4 and 5 respectively. The first mode in both designs is the first transverse

bending mode. The second bending and twisting modes of the optimum design have been

switched. Couphng between torsion and flexure was observed in the modes of the optimum

designs.

The FRF's of the reference and optimum (Subcase 2d) designs are shown in Fig. 6.

The z-axis dynamic deflections of the corner with the maximum resonance amphtude are

plotted. The superior frequency response of the optimal design is apparent.

13
SUMMARY

An integrated formal method for the tailoring of structural composite damping was

described. The method is based on static and dynamic performance criteria and its primary objective is optimization of the forced damped dynamic response of the candidate

composite structure. The damping capacities of individual modes were optimized such

that selected resonance amplitudes were minimized subject to constraints on static and

dynamic deflections, and natural frequencies. The method incorporates unified micromechanics and laminate theories for composite damping and other mechanical properties,

therefore, can be utilized for the simultaneous tailoring of basic composite systems and

laminate configurations. The simulation of structural composite damping is based on

finite-element analysis, for this reason, the method is applicable to a wide array of plate

and shell composite structures.

Applications of the method included: (1) the structural tailoring of a composite beam,

and (2) the structural tailoring of a composite shell. The effect of laminates with multiple

sublaminate systems on the optimal designs was also investigated. The more important

conclusions are summarized in the following paragraphs.

1. Both application cases have illustrated the effectiveness of the method. In both cases

the tailoring of structural composite damping produced optimum designs with superior

dynamic performance.

2. Tailoring of composite structures based only on static performance criteria and/or

frequency constraints, and neglecting the effects of composite damping, may produce

designs with poor dynamic performance.

3. The optimum tailoring of composite beams with laminates of one or multiple angleply sublaminates produced optimum designs with virtually equivalent dynamic performance. Compared to a unidirectional beam, the optimum designs have reduced the

maximum resonance amplitudes by factor of 2, and have natural frequencies in the

feasible frequency domain.

4. In the structural tailoring of the composite shell, laminates with multiple angle-plied

14
sublaminates and variable fiber content produced optimum designs with significantly

improved dynamic performance. In view of the attained improvements, any higher

costs that may result in the fabrication of more complex laminate configurations seem

justified. With respect to a unidirectional shell, the optimum designs exhibited lower

resonance amplitudes by factor of 2.76, lower static compliance by factor of 2.70, and

natural frequencies within the feasible frequency domain.

5. The optimization of shells with multiple sublaminates of variable fiber volume ratio

produced laminates with passive damping layers of pure matrix. These laminate

configurations have provided the best dynamic performance of all other optimum

designs, and resemble a constrained layer damping structure.

15
REFERENCES

1. Adams, R. D., and Bacon, D. G. C., "Effect of Fibre Orientation and Laminate

Geometry on the Dynamic Properties of CFRP," Journal of Composite Materials,

Vol. 7, Oct. 1973, pp. 402-428.

2. Siu, C. C., and Bert, C. W., "Sinusoidal Response of Composite-Material Plate with

Material Damping," ASME Journal of Engineering for Industry, May 1974, pp. 603-

610.

3. Suarez, S. A., Gibson, R. F., Sun, C. T., and Chaturvedi, S. K., "The Influence of

Fiber Length and Fiber Orientation on Damping and Stiffness of Polymer Composite

Materials," Ezperimental Mechanics, Vol. 26, No. 2, 1986, pp. 175-184.

4. Saravanos, D.A., and Chamis, C.C., "Unified Micromechanics of Damping for Unidirectional Fiber Composites," NASA TM-102107, 1989.

5. Saravanos, D. A., and Chamis, C. C., "Mechanics of Damping for Fiber Composite

Laminates Including Hygro-Thermal Effects," $Oth Structures, Structural Dynamics,

and Materials Conference, Paper No. 89-1191-CP, Mobile, Alabama, Apr. 3-5, 1989.

6. Ni, R.G., and Adams, R.D., " The Damping and Dynamic Moduli of Symmetric

Laminated Composite Beams - Theoretical and Experimental Results," Journal of

Composite Materials, Vol. 18, 1984, pp. 104-121.

7. Schmit, L.A., and Farhsi, B. "Optimum Design of Laminated Fibre Composite Plates,"

International Journal for Numerical Methods in Engineering, Vol. 11, pp. 623-640,

1977.

8. Schmit, L.A., and Mehrinfar, M., "Multilevel Optimum Design of Structures with

Fiber-Composite Stiffened-Panel Components," AIAA Journal, Vol. 20, No. 1, pp.

138-147, 1982.

9. Eschenauer, H.A., and Fuchs, W., "Fiber-Reinforced Sandwich Plates Under Static

Loads - Proposals for their Optimization," ASME paper 85-Det-8_, 1985.

10. Tauchert, T.R., and Adibhatla, S., "Design of Laminated Plates for Maximum Stiffness," Journal of Composite Materials, Vol. 18, pp. 58-69, 1984.

16
11. Soni, P.J., and Iyengar, N.G.R., "Optimal Designof Clamped Laminated Composite

Plates," Fibre Science and Technology, Vol. 19, pp. 281-296, 1983.

12. Adali, S.,"Multiobjective Design of an Antisymmetric Angle-Ply Laminate by Nonlinear Programming," ASME Journal of Mechanisms, Transmissions, and Automation

in Design, Vol. 105, pp 214-219, 1983.

13. Liao, D.X., Sung, C.K., and Thompson, B.S., "The Optimal Design of Laminated

Beams Considering Damping," Journal of Composite Materials, Vol. 20, pp. 485-501,

1986.

14. Sung, C.K., and Thompson, B.S., "A Methodology for Synthesizing High - Performance Robots Fabricated with Optimally Tailored Composite Materials," ASME Paper No. 86-DET-8, 1986.

15. Murthy, P.L.N., and Chamis, C.C., "ICAN: Integrated Composite Analyzer," AIAA

Paper 84-0974, May 1984.

17
Table 1. Mechanical properties of HM-S/epoxy system.

Epoxy HM-S Graphite 50% HM-S/Epoxy(ref. [6])

Era= 0.500 Mpsi

(3.45 GPa)

Gm= 0.185 Mpsi

(1.27 GPa)

Cm,_ = 10.30 %

¢_s = 11.75 %

E.faa = 55.0 Mpsi

(379.3 GPa)

El22 = 0.9 Mpsi

( 6.2 GPa)

Gfa2 = 1.1 Mpsi

( 7.6 GPa)

vj'12 : 0.20

_'fl1 = 0.4 %

Wf22 = 0.4 %

¢y_2 = 0.4 %

¢tll = 0.45 %

¢/22 : 4.22 %

Ca_ = 7.05 %

18
Table 2. Optimum designsfor the beam (Case1).

Unidirectional Optimum Designs

Design (Case la) (Caselb) (Case lc) (Caseld) (w/o damping

optimization)

Ply Angles, degrees

61 0.0

82 0.0

03 0.0

Fiber volume ratios

kfa 0.50

k f, 0.50

k f3 0.50

26.69 26.09 26.46 25.97 5.66

- - 26.60 26.18 -

- - 0.02 29.99 -

D

0.700 - 0.700 0.408

- - 0.700 -

- - 0.495 -

Max. z-axis Static deflection (tip), (10 -3 in)

u_ 19.99 48.94 34.56

Resonance z-axis Amplitudes (tip), (10 -3 in)

U_ a 426.9 235.1 209.4

U -2 0.0 0.0 0.0

0.0 5.4 4.8

[_ 9.3 0.1 0.1

U_ 0.0 0.7 0.6

Natural Frequencies, (Hz)

wa 383.2 254.1 289.2

w2 1088.7 1369.5 1530.9

w3 1565.1 1587.1 1802.2

w4 2395.5 2945.6 3272.5

w5 3854.0 5072.6 5770.2

Modal SDC's, (%)

_ba 0.57 2.55 2.02

¢2 5.57 3.85 3.12

_b3 3.43 2.69 2.15

0.66 0.83 0.70

4.54 2.19 1.74

47.54 34.49 25.56

237.3 209.3 481.9

0.3 0.0 0.1

5.5 4.8 0.0

0.1 0.1 10.2

O.7 0.5 0.0

257.5 292.2 350.0

1558.6 1492.5 1219.9

1608.0 1820.7 1577.7

2924.0 3293.6 2193.2

5130.9 5828.2 4167.6

2.45 2.01 0.65

2.28 3.28 4.15

2.60 2.14 2.05

0.84 0.70 0.73

2.13 1.74 3.68

19
,..-_,< ._...._ _LC:_. _

"I / .'I\'/ //.'-"." .'I, ._ " / "" _-:;- _"

; /" ir,, / 1" -

y "_.',. I f

.... .V._: t ,'"/'

I_DE _ (F!_EI_ - 8S.9SI SDC =_,284gE-I_I )

22
x

6

(a) PLATE BEAR.

Z

4 y

(b) OPEN CYLINDRICAL SHELL.

FIGURE I. - CANDIDATE COMPOSITE STRUCTURES• DIMENSIONS ARE IN INCHES.

_ -_(01/-01)5" kfl

(a) 1 ANGLE-PLY SUBLARINATE.

/,'-(011-01) 2.

_/////////////////////////////d.'-(°2'-eP2,

_-r(03/-e 3)

_///////////////////1

(b) 3 ANGLE-PLY SUBLARINATES.

FIGURE 2, - CANDIDATE LAMINATE CONFIGURATIONS.

kfl

kf2

kf3

•450

.375

• 300

.225

• 150

.0750

OPTIMUM (SUBCASE ld)

UNIDIRECTIONAL

|

Ii

I _,_.IL__J l I I

167 1;33 500 667 833 1000

FREQUENCY, HZ

FIGURE 3. - FREQUENCY RESPONSE FUNCTION OF THE COMPOSITE

BEAM (CASE I). UNIDIRECTIONAL AND OPTIMUM (SUBEASE I(I)

DESIGNS.

21
22
MODE 1

Z

MODE 2

-i

8

/

(FREQ = 149.97, 5DC =0.27gSE Bl) MODE 4

FIGURE 5. - RODE SHAPES OF THE OPTIMIZED (SUBCASE 2d) COP@OSITE SHELL.

i//

(FREQ = 447.81 , SDC =e. 1673E-01

Z

b,..

• 420

• 350

.280

.210

.140

.0700

OPTIMUM (SUBCASE 2d)

UNIDIRECTIONAL

150

I

300 450 600

FREQUENCY, Hz

I I

750 900

FIGURE 6. - FREQUENCY RESPONSE OF THE COMPOSITE SHELL AT

FHE SIIE OF MAXIMUM RESONANCE AMPLITUDE, UNIDIRECIIONAL

AND OPFIMUM (SUBCASE 2d) DESIGNS.

23
National Aeronautics and

Space Administration

Report Documentation Page

1. Report No. l

J

NASA TM- 102343

4. Title and Subtitle

2. Government Accession No, 3. Recipient's Catalog No.

5. Report Date

An Integrated Methodology for Optimizing

7. Author(s)

D.A. Saravanos and C.C. Chamis

9, Performing Organization Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

2. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

Structural Composite Damping

6. Performing Organization Code

8. Performing Organization Report No,

E-5051

10. Work Unit No.

505-63- I i

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15, Supplementary Notes

Prepared for the 1989 Winter Annual Meeting of the American Society of Mechanical Engineers, San Francisco,

California, December 10-15, 1989. D.A. Saravanos, National Research Council--NASA Research Associate.

16. Abstract

A method is presented for tailoring plate and shell composite structures for optimal forced damped dynamic

response. The damping of specific vibration modes is optimized with respect to dynamic performance criteria

including placement of natural frequencies and minimization of resonance amplitudes. The structural composite

damping is synthesized from the properties of the constituent materials, laminate parameters, and structural

geometry based on a specialty finite element. Application studies include the optimization of laminated composite

beams and composite shells with fiber volume ratios and ply angles as design variables. The results illustrate the

significance of damping tailoring to the dynamic performance of composite structures, and the effectiveness of the

method in optimizing the structural dynamic response.

17. Key Words (Suggested by Author(s))

Composite structures; Damping optimization; Off-axis

damping; Laminate damping; Structural damping; Forced

dynamic response; Optimal design method

18. Distribution Statement

Unc!assified- Unlimited

Subject Category 24

NASAFORM162SOCT_ *For sale by the National Technical Information Service, Springfield, Virginia 22161

National Aeronautics and

SDace Administration

Lewis Reseerch Center

Cleveland, Ohio 44135

om:ld sumlll

Pemty_ p_m_ IS00

SECOND CLASS MAIL

ADDRESS CORRECTION REQUESTED IIIIII

PoaageandFeesPJ_I

NalWW lWonaut_

Spec*Adminmra_n

N,4SA-4Sl