NASA Technical Memorandum 102094 

Tailoring of Composite 

Links for Optimal Damped 

Elasto-Dynamic Performance 

D.A. Saravanos and C.C. Chamis 

Lewis Research Center 

Cleveland, Ohio 

Prepared for the 

1989 Design Automotive Conference 

sponsored by the American Society of Mechanical Engineers 

Montreal, Canada, September 17-20, 1989 9 

.. .. . __& 

(NASA-TB-102094) TAILORING OF COClPOSIlE N89-269 12 

LINKS FOR OPTIflAL DAHFED ELBSTO-DYAAl¶IC 

PBRPOBfl$ICE INASA, Levis Research Center) 

30 P CSCL 11D Unclas 

G3/2U 0222721 

!I 
TAILORING OF COMPOSITE LINKS 

FOR OPTIMAL DAMPED ELASTO-DYNAMIC PERFORMANCE 

D. A. Saravanos", and C. C. Chamis 

N a t,i on a1 -4 eron aut i cs and S p ace ,4 dmini strati on 

Lewis Research Center 

Cleveland, OH 44135 

ABSTRACT 

A method is developed for the optimal design of composite links based on dynamic 

performance criteria directly related to structural modal damping and dynamic stiffness. 

An integrated mechanics theory correlates structural composite damping to the parameters of basic composite material systems, laminate parameters, link shape, and modal 

deformations. The inclusion of modal properties allows the selective minimization of vibrations associated with specific modes. Ply angles and fiber volumes are tailored to obtain 

optimal combinations of damping and stiffness. Applications to simple composite links indicate wide margins for trade-offs and illustrate the importance of various design variables 

to the optimal design. 

INTRODUCTION 

Fiber reinforced composites show great potential as candidate materials for high-speed 

mechanisms and high-speed robotic manipulators. Continuous fiber composites provide 

superior specific modulus and specific strength allowing the fabrication of stiff, strong, and 

light-weight links. In thiy 1 ner, positioning errors associated with the static or clwpniir 

~ 

* h'ational Research Council - A'ASA Research -4 ssociaie 

1 
response are readily decreased, since the elastic deflections are reduced, the strength is 

increased, and the mass of the links is also decreased resulting in lower inertial loading. 

Composites also provide the unique capability to hailor the anisotropy of the mateerial to 

meet specific design requirements. For the case of robotic manipulators, tailoring of the 

material may result in robotic arms with superior and more uniform properties within 

their working envelope [l]. 

In addition to high specific modulus and strength, composite materials will incorporate higher material damping into the structure, since most polymer- matrix composites 

exhibit higher damping fact,ors by an order of magnitude compared to commonly used 

metallic alloys. Properly designed composite links can provide higher passive damping, 

and they may furhher improve accuracy: strength: and fatigue endurance by attenuating 

undesirable elasto-dynamic phenomena such as overshooting, long settling times and structural resonances. Recent work [2,3] has shown that structural composite damping is highly 

tailorable, and is primarily controlled by constituent parameters (fiber/matrix properties, 

volume fraction), laminate parameters (ply angles/thicknesses, stacking sequence), and the 

deformation of the articulated structure (mode shapes). The same study has also demonstrated that any increase in damping usually results in decreased stiffness and possibly 

strength, therefore, any tailoring of the composite material for optimal damped response 

will be based on trade-offs between damping, stiffness, and strength. In order to realize 

full benefits from the damping capacity of composite materials, integrated methodologies 

for the optimal design of composite links should be developed entailing: (1) capability for 

tailoring the basic composite materials and/or laminate for maximum damping, stiffness, 

and strength; (2) dynamic design criteria based, if possible, on the performance of the 

articulated structure; (3) three-dimensional modeling capacity, as the vibrational response 

of robotic manipulators and mechanisms is in general three-dimensional. The current paper presents the development of such a formal design method allowing the tailoring of 

composite links for optimal elasto-dynamic performance. 

Limited research has been reported on the optimal design of composite links includ2 
ing damping as design criterion. The flexural damping capacity of laminated beams [4] 

and tubular composite manipulator links [5,6] has been optimized with bounds on design 

parameters and bending rigidity. Ply angles, ply thicknesses, fiber volume fractions, and 

cross-sectional dimensions were utilized as design parameters. The present optimal design 

method is targeted to three-dimensional kinematic configurations. It implements criteria based on global struct,ural dynamic properties rather than on local properties, and it 

readily incorporates micromechanics for all properties of the composite material, including 

damping. The method can be used to perform each or combinations of the following functions: (1) prediction of optimal laminate configurations (ply angles and ply thicknesses); 

(2) optimal t,ailoring of the composite material in terms of volume fraction, matrix and 

fiber properties; and (3) prediction of optiinal link shapes. The method provides the unique 

feature to model and optimize the structural damping of individual vibration modes, therefore, superior tailoring of the composite material may result by selective minimization of 

positioning errors associated with the more predominant modes. In view of the later 

feature, the design method may be readily combined with active vibration control, such 

that the elasto-dynamic effects of higher modes are passively minimized by tailoring the 

composite links, while vibrations from lower modes are actively suppressed. 

The performance criteria used in this work are directly related to the damping, static 

deflections, and dynamic deflections of the composite structure. These properties are 

modeled using finite element analysis. Composite damping simulation capabilities have 

been incorporated into a three-dimensional triangular plate element. The optimal design 

problem is formulated in standard non-linear programming form and is numerically solved 

with the feasible directions method. For a preliminary assessment regarding the merits of 

the method, the optimal design of simple links is investigated. 

NOMENCLATURE 

3 x 3 Extensional laminate damping matrix. 

3 x 3 Extensional laminate stiffness matrix. 

Strain shape function matrix. 

3 
Y 

Global and modal damping matrices. 

3 x 3 Coupling laminate damping matrix. 

3 x 3 Coupling laminate stiffness matrix. 

3 x 3 Flexural laininate damping matrix. 

3 x 3 Flexural laminate stiffness matrix. 

Normal modulus. 

On-axis and off-axis ply stiffness matrices. 

Shear modulus. 

Volume ratio. 

Global and modal stiffness matrices. 

Global and modal mass matrices. 

Number of plies in a laminate. 

Modal vect.or. 

Time. 

Ply thickness. 

Displacement. vector. 

Undamped dynamic amplitude. 

Deflections along x,y,z structural axes. 

Stored and dissipated specific strain energies. 

Stored and dissipated strain energies. 

Distance from laminate mid-surface. 

Engineering strain. 

Fiber orientation angle. 

Curvature. 

Poisson's ratio. 

Stress. 

Specific damping capacity. 

t Bold characters indicate vectors 

4 
Subscripts 

b 

C 

e 

2 

1 

L 

m 

n 

TP 

t 

Superscripts 

d 

L 

0 

s 

U 

Modal matrix. 

Rotations about x,y,z struct,ural axes. 

Angular velocity. 

Ply bot. t om. 

Ply (off-axis). 

Finite element. 

Fiber. 

ru’ode. 

Pl~7 (on- axis). 

L ami n at, e. 

Matrix. 

n-th mode. 

Ply interface. 

Ply t.0p. 

D y n ami c . 

Lower. 

Laminate inid-surface. 

Static. 

Upper. 

Direction 

11 

22 

12 

22 

YY 

Normal longitudinal. 

Normal in-plane transverse. 

Shear in-plane. 

In- pl ane nor mal, off - axis , x- di rec t i on. 

In-plane normal, off-axis, y-direction. 

5 
ss In-plane shear, off-axis: y-direction. 

COMPOSITE STRUCTURAL DAMPING 

I 

In this section an integrated theory for modeling structural composite damping is described. Analogous theories were utilized for other mechanical properties [ 71. The damping 

of the composite link is simulated based on an upward synthesis scheme. First, the on-axis 

damping of each composite material (damping along the material axes) is synthesized from 

the constituent fiber/matrix properties and volume fractions. Then, the damping capacities of plies subject.ed to off-axis loading (off-axis plies) are calculated, and subsequently 

the local laminate damping capacit,y is formulat.ed. Integration of local laminate damping 

over the whole structure provides the global structural damping capacit,y at a particular 

deformation state. Finite element discretization is utilized for the numerical integration 

of local laminate damping. The same finit,e element mesh is also utilized to simulate the 

static and dynamic response. 

On-Axis Composite Damping 

A simplified micromechanics theory [a] has been utilized for the synthesis of hysteretic 

on-axis composite damping. The micromechanics equations for on-axis damping are presented in the Appendix. The specific strain energy released per cycle due to in-plane 

on-axis cyclic loading is: 

Off-Axis Composite Damping 

Off-axis composite specific damping capacities (SDC’s) are synthesized from the onaxis properties and the ply angle. The transformation law is given in the Appendix. The 

off-axis damping matrix [+,I is fully populated due to in-plane coupling between extension 

and shear. As has been demonstrated in ref. [2] coupling provides an additional physical 

mechanism for strain energy dissipation, and consequently, higher damping. The specific 

strain energy released per cycle during cyclic in-plane off-axis loading is, 

6 
Laminate Damping 

The laminate damping is ensembled from the damping contributions of each off-axis 

ply. As shown in the Appendix and ref. [3], the damping capacity of a general composite laminate is formally described by three 3 x 3 fully-populated matrices: (I) the 

extensional damping matrix [Ad] associated with extensional deformations only; (2) the 

flexural damping matrix [Dd] associated with bending and torsional deformations; and 

(3) the extension-flexure coupling damping matrix [Cd] associated with coupled extensionflexure deformations. The coupling damping matrix [ Cd] is identically zero for symmetric 

laminates. In addition to hysteretic ply damping, the laminate damping includes also 

contributions from hysteretic interlaminar strain energy dissipated in the interply matrix 

layer, as a result of the interlaminar shear stress developed between two adjacent plies 

in order to restrict their relative local movement (scissors effect). The dissipated strain 

energy through-the- thickness of the laminate per unit area per cycle is: 

~ 

-4s seen in eq. (3), the laminate damping is a local structural property, depending on the 

local deformation, the basic constituent composite materials, and laminate configuration. 

Correlation with experimental measurements has yielded excellent agreement [3]. 

Structural Modal Damping 

The local laminate damping provides rough estimates regarding the global damping 

capacity of simple structural components and loading conditions where the local strain and 

curvature remain mostly uniform over the structure. However: most composite structures 

of practical interest would exhibit more complicated and coupled deformations, and their 

global damping capacit,y would be the integrated contribution of local damping over the 

whole volume at a particular deformation shape. 

Since the elasto-dynamic response of any structure is a linear combination of individual 

vibration modes, structural damping associated with each mode would provide a natural 

and meaningful measure regarding the overall damping capacity of the composite structure. 

7 
In the remaining sections, the global damping capacity of the structure for a particular 

mode shape will be identified as structural modal damping. In this manner, an array 

of modal damping factors is generated, each one representing the damping capaciby of 

the structure vibrating solely at. the respective mode shape. The modal specific damping 

capacity (SDC) associated to the n-th vibration mode would be: 

where Sw, and wn represent the dissipated and stored specific strain energies per cycle. 

Utilizing eq. (3), the modal damping of plate/shell composite st,ructures becomes, 

The integrations in eq. (5) are performed numerically using finite element discretization. Such a finite element scheme has been developed for an isoparametric triangular 

plate element with 3 nodes, 6 degrees of freedom per node ui = {u,,z~~,u,,~,,~~,~,}, 

linear shape functions for membrane deflections, and cubic shape functions for flexure. 

Integration is first performed over the area of the triangular finite element. Given the 

shape functions for membrane and flexural deflections, the mid-plane strain and curvature 

at a point is related to the nodal displacement vector, 

Based on this relation, the damping and strain energies per cycle, over the area of the 

triangular element would be respectively: 

8 
Both integrals are calculat,ed numerically using three-point, quadrature. Once the dissipated and stored strain energies are calculated for each element, t,hen their summation 

provides the total dissipat,ed and stored energies. The modal SDC of the n-th node q!~~ is 

then assembled according to equation (5). 

ELASTO-DYNAMIC RESPONSE 

The dynamic elastic response of cornpositme links is simulat,ed with finite element analysis. The p1at.e triangular element, mentioned previously is used for this purpose. This t,ype 

of element was preferred in this initial study, because it provides more accurate modeling 

and insight: than the beam finite elements used in previous studies [1,4-61. In this manner, 

a reliable data-base of results is assembled for future assessments of the accuracy of less 

expensive beam elements. Using finite element discretization, the elasto-dynamic response 

of the composite link may be simulated by the following system of n discrete dynamic 

equations, 

[hf]{G,} + [C]{k> + [h’]{u} = {F(t)} (9) 

The damping matrix is formulated in the modal space. A typical exciting force for 

composite robotic manipulators would involve: surface tractions F,(t), body forces Fb(t), 

inertial forces due to local, centrifugal and Coriolis acceleration F;( F), and hygrothermal 

forces FHT(t). 

{F(i)} = {Fs(i)> + {Fb(i)) + {FT(?)} + {FHT(f)} (10) 

This study neglects the effects of body and hygrothermal forces, hence, is limited to links 

operating in room temperature and dry environment. Hygrothermal effects may be significant in the design of composite links, and the method outlined here has provisions to 

model them, however, the design problem is complicated beyond the scope of the current 

9 
paper, as the damping and stiffness of the composite depend also on temperature and 

moisture variations [3]. 

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

the linear transformation [8]: 

{u) = Pl{q) (11) 

where q is the modal displacement vector: and I@] contains the first m normalized undamped eigenvectors. For proportional damping, t.he transformation yields the reduced 

uncoupled dynamic syst,em, 

where, 

Th 

[mj = [@]'[Mj I@] 

[c] = pjT[c][@j 

[k] = p]T[K][q 

{f(t>) = i@lTW)I 

decoupled damping matrix is formulated from th first m modal SDC's from 

equations (5-8). For general load histories, the differential equations are numerically solved. 

Impulse Response 

The response of the composite link to an impact force {F}h(t), where h(t) is the unit 

impulse, would be [8]: 

where Wd,n is the damped natural angular velocity of the n-th mode, j any nodal displacement of the dynamic system (9), and {f}h(t) = [@]T{F}h(t). Eq. (13) is rewritten in the 

form: 

10 
Eq. (14) illustrates the effect of modal undamped amplitudes Utn and modal damping 

ratios on the impulse response. The undamped amplitudes of each mode, 

are used as performance measures for the dynamic stiffness of composite links in the 

following section. 

OPTIMAL DESIGN METHOD 

Most common transient vibration problems in high-speed manipulation are usually induced by impulsive excitation. During minimal time (bang-bang) control, the actuat#ors are 

mostly operating at their maximum torque capacities. The minimal time movement of the 

end-effector between two points typically involves an acceleration phase under maximum 

positive torque, immediately followed by a deceleration phase under maximum negative 

torque. More complicated maneuvers of articulated robotic arms with multiple degrees 

of freedom exhibit the same fundament>al pattern of switchings between acceleration and 

deceleration. Impact excitations are applied during the initiation of the motion, each 

switching from acceleration to deceleration and at the end of the rigid body motion. 

It is reasonable, therefore, to tailor the composite links for optimal elasto-dynamic 

response in impulsive excitation. As seen in eq. (14), a feasible way to control the elastodynamic positioning error would be by restraining each undamped modal amplitude within 

acceptable limits. One advantage of using dynamic amplitudes, in addition to static stiffness, is that the modal mass is also introduced into the design. In this manner, the 

elasto-dynamic vibration will remain within acceptable bounds, while the maximization of 

modal damping will ensure minimal settling times. The static stiffness is also bounded, 

such that, acceptable static deflections under the maximum static loads are ensured. 

Two different optimal design cases are formulated, each one representing different 

design considerations and requirements. Both cases are presented in standard non-linear 

programming form. 

Case 1: 
The present design case aims to maximize the damping capacity of the first mode, in 

conjunction with other constraints. The first, mode has t,he lower frequency and requires 

the higher settling time. Preliminary invest,igations regarding the vibration of off-axis 

composite beams have shown that the first mode is a bending mode. Case (1) is most 

suitable when active control is infeasible or undesirable, and t,he only remaining option 

is passive vibration control. In this context, the optimal design problem is formulated as 

follows: 

m.uz( $1 ) (16) 

subjected to: 

( i.) Upper and lower bounds on design variables: 

-90.0" 5 8; 5 90.0" 

0.0 5 k,,; 5 0.70 

(17.1) 

(17.2) 

where 8; and IC,?i are respectively the ply angle and the fiber volume ratio (FTTR) of 

the i-th sublaminate. 

( ii.) Constraints on qs static displacements at the tip of the link: 

21s 3- < ,f'" j = 1, ..., qs (18) 

( iii.) Constraints on m undamped modal dynamic amplitudes as provided by eq. (15) 

for qd displacements at the tip of the link: 

lrd 3rn < -u;f j=1, ..., qd, and n=l, ..., m (19) 

Case 2: 

This case is suitable for manipulators with active vibration controllers. Active controllers are most. effective up to a frequency limit, therefore, they successfully suppress 

the static deflections and the elasto-dynamic deflections from the first vibration modes. 

Passive attenuation of higher modes is then desirable. The present methodology provides 

the capability to design composite links such that the passive damping of higher modes 

is maximized and the respective undamped dynamic amplitudes remain within acceptable 

limits. 

12 
Assuming that passive vibration control is desirable for modes ml t,o m, the following 

proposed performance criteria will maximize the modal damping of the mode with t.he 

higher dynanlic amplitude: 

where the mode n is selected such that: 

d I'd . U: = nzax{Uj,ml, ..., j,m? J' = 1, e.*, qd) 

subjected to: 

( i.) Upper and lower bounds on design variables: 

-90.0" 5 8; 5 90.0" (21.1) 

(21 .2) 

( ii.) Constraints on undamped modal dynamic amplitudes for qd displacements at the 

~ 

tip of the link and for modes ml to m: 

Design Vector 

In both cases the design vector includes: (1) the ply orientation angle 8; of each 

sublaminate (group of rt8; plies), and (2) the fiber volume ratios (FVR.'s) k,,i of each 

sublaminate. By considering only composite material parameters, the current study is 

mainly focused on the tailoring of composite links. The effect of shape optimization and 

its interaction with mat,erial tailoring will be addressed in a future study. 

Optimization Algorithm 

13 
Both design cases have been formulated in non-linear programming form and are 

numerically solved with the method of feasible directions. The feasible directions algorithm 

is a primal optimization method, and performs a direct search of the feasible design space 

based on first order sensitivity for objective function and constraints. Primal met,hods 

are more suitable for designs involving computationally expensive constraint,s, like the 

present one, since they typically require fewer constraint evaluations than the penalty 

transformation methods. The feasible directions method used in this study incorporated 

active constraint set, and line-search strategies for improved efficiency. 

CASE STUDIES 

Assumptions 

A4pplication stmudies were performed regarding the optimal tailoring of a plate composite link 6 in long, 1 in wide, and 0.2 in thick. This simple composite link was selected in 

order to obtain an initial assessment of t.he merits of the present design methodology and 

to identify the important design parameters. A finite element mesh of 55 nodes and 80 

plate triangular elements was used as shown in Figure 1. The link was assumed clamped at 

one end (nodes 1 to 5) and free at the other (nodes 51 to 55). The composite material was 

high-modulus treated graphite (HM-S) fiber in an epoxy matrix. The mechanical properties of the matrix and the fiber are shown in Table 1. The material system was selected 

to match the .50% FVR HM-S/DX210 graphite/epoxy composite for which experimental 

damping measurements are provided [9]. The damping properties of the fiber and matrix 

for this material system were back-calculated from the reported on- axis damping values 

of the 50% FVR graphite/epoxy composite (see Table 1) based on the micromechanics 

equations in the Appendix. 

The basic laminate configurations are shown in Figure 2. Both configurations are symmetric and consist of regular angle-ply sublaminates (Ok/ - ek/Ok ...). Configuration (a) 

involves one sublaminate, and configuration (b) three. Regardless the number of sublaminates, the link consists of 20 plies through the thickness, each 0.Olin thick, such that each 

ply may be manufactured from 0.005in thick unidirectional preimpregnated tape. Each 

14 
sublaminate may have a different FVR, consequently, the i- th sublaminate will introduce 

2 design variables into the global design vector: the ply angle 02, and the FVR kf,t. 

The static load is a uniform transverse (z-axis) force of 5.0 lb/in applied at the tip of 

the link (nodes 51-55), and the dynamic load a uniform transverse (z-axis) impulse of 0.1 

lb/in applied at the same location. The reference design was taken to be a unidirectional 

composite link of 50% HM-S/epoxy, since a unidirectional composite provides very high 

rigidity but low damping in bending. Unless otherwise stated. the static (z-axis) deflections 

along the tip were restricted to be less that 0.05 in (2.5 times the maximum static deflection 

of the unidirectional link), and each individual modal amplitude of the transient (z-axis) 

response at the middle of the tip (node 53) to be less than 1.87 in (2.0 times the maximum 

amplitude for the unidirectional link). 

Case 1 

Three different subcase studies were performed for this type of problem for: (a) one 

sublaminate with fixed 50 % FJTR: (b) dhree sublaminat,es of 50 % FVR, and (c) three 

sublaminates with varying FVR's. 

Subcase la. The current link was assumed to consist of one sublaminate with fixed 

fiber volume ratio. This is the simplest tailoring that can be performed on the composite 

link, and the design vector involves only the ply angle O1. The optimal design is shown 

in Table 2. The optimal, i27" angle-ply symmetric laminate provides 4.5 times higher 

damping for the fundamental mode with respect. to the reference unidirectional link and 

4.2 times higher damping for the second bending mode. The trade-offs for these damping 

improvements are higher static compliance by 2.5 times, higher dynamic amplitudes by 

1.5 times and lower natural frequencies. 

Subcase lb. The current subcase involves three sublaminates of the same basic 

material system. Sublaminates 1 and 2 have 4 plies, and sublaminate 3 has 2 plies. The 

design vector now involves three ply angles. The optimal design is shown in Table 2. The 

utilization of three sublaminates has not improved the optimal damping capacit,y of the 

fundamental mode compared to the optimal design of Subcase la. The resultant optimal 

15 
ply angles demonstrate that the outer sublaminates are critical to the damping and stiffness 

of the composite link, while the inner sublaminate had a rather insignificant effect. Such 

type of arrangement was indeed expected, because the first bending mode controls the 

design. However, the 15" plies in sublaminate 3 provide higher in-plane stiffness in xdirection than the optimal design of Subcase la. The effect of higher in-plane stiffness is 

indicated by the higher value of the second natural frequency. 

Subcase IC. In this subcase, the fiber volume ratio of each sublaminate system is 

incorporated into the design vector, which now consists of 3 ply angles and 3 fiber volume 

ratios. As seen in Table 2, the introduction of FI'R's as design variables has produced a 

sandwich type of laminate. in that. the fiber volume ratio of the outer sublaminate has 

reached its upper bound. The outer sublaminate provides more bending stiffness but less 

damping. The FI'R's of the other two sublaminates have been slightly changed. This 

laminate arrangement has produced almost 8% higher SDC for the first modal damping 

and higher SDC for the second bending mode, without any additional penalties on the 

static stiffness. dynamic amplitudes. and natural frequencies. 

All subcases demonstrate wide margins for trade-offs between various modal damping 

capacities, static and dynamic stiffness, and natural frequencies. Additional insight into 

the dynamics of the problem is provided by Figures 3 and 4, which illustrate the mode 

shapes for the unidirectional reference design and the optimum design of Subcase IC . The 

bending-torsion coupling is minimal. The optimum design has also switched the order of 

modes and has affected the SDC's of the torsional modes. Torsional modes may also induce 

elasto-dynamic deflections at the end-effector of robotic manipulators. Although the current applications are restricted to transverse bending deflections, the torsional deflections 

may be significant for manipulators with multiple links. 

Case 2 

A link with the previous laminate configuration of three sublaminates and three different material syst.ems was optimized. The basic assumption of the current case is that the 

fundamental mode will be eliminated with active vibration control, hence, the optimum 

16 
tailoring is now solely focused on modes 2 to 5. In this context, the damping capacity 

of the most predominant mode between modes 2 and 5 was maximized. The imposed 

upper bound on the modal dynamic amplitudes of modes 2 to 5 was 0.29 in (2 times the 

maximum modal amplitude for modes 2 to 5 of the unidirectional link). The optimum 

design is shown in Table 3. The mode with the higher undamped amplitude is mode 3 

(2nd bending), while at the reference design the most predominant mode was mode 4 

(2nd bending). The respective modal damping has increased by a factor of 5, while the 

respective modal amplitude has been increased by a factor of 2. The damping capacity of 

the fundamental mode 1 (1st bending) has been also drastically increased, although was 

not incorporated into the design criteria. The current case illustrates that the utilization 

of active controllers in composite robotic manipulators, will not only eliminate the elastodynamic effects of the lower modes. but it will also facilitate the design of composite links 

with better passive vibration suppression for the higher modes. 

The transient responses of the unidirectional and optimum (subcase IC) designs are 

shown in Figure 5. Each graph presents the tip vibration (node 53) of the respective 

design induced by the z-axis impulsive load of 0.1 lb/in applied at the tip of the link. The 

superiority of the optimum design is clearly illustrated. Initially, the dynamic deflections of 

the optimum design were higher than the dynamic deflections of the unidirectional design, 

however, because of the higher damping the deflections of the optimum design were quickly 

reduced to lower values compared to the unidirectional design. At 0.6 secs, the deflections 

of the optimum design were virtually reduced to 40 % of the respective deflections of the 

unidirectional link. 

SUMMARY 

A formal design method was presented which automates the design of composite 

links for optimal elasto-dynamic response. The method was based on static and dynamic 

performance criteria. The proposed dynamic design criteria were directly related to the 

damping and amplitudes of selected vibration modes. The elasto-dynamic response was 

simulated with finite element analysis. Simplified, design oriented micromechanics and 

17 
laminate mechanics models for composite damping correlated the performance criteria 

with laminate and composite material parameters. The present method can be utilized for 

basic material: laminate, and structural shape optimization. Two distinct design problems 

were formulated aiming to manipulators with and without active controllers. 

The applications illustrate the effect of ply angles, multiple angle-ply sublaminates, 

and fiber volume fractions on the optimal design. Based on the case studies, the following 

conclusions are summarized: 

1. The resultant optimal laminate configurations and composite material systems 

significantly improved the damping of bending modes by factors of 5 with respect to the 

reference unidirectional design, while the undamped dynamic amplitudes only degraded 

by factors of 1.5. Wide margins for trade-offs exist between damping, and static/dynamic 

stiffness. These trade-offs can be achieved without changing the geometric shape of the 

links, and consequently, their mass. 

2. Ply angles were the most critical design parameters. 

3. Laminates with single or multiple angle-ply sublaminates produced optimal designs 

of comparable performance. 

4. The optimization of fiber volume fractions has produced small improvements in 

the damping of bending modes. 

5. Dynamic properties associated with torsional modes where also found sensitive to 

variations of laminate and composite material parameters. The actual effect of torsional 

modes on the end-effector deflections is yet unclear. Minimization of the torsional dynamic effects will be better accomplished with design criteria based on the elasto-dynamic 

response of the articulated robotic arm, rather than on individual links. 

18 
REFERENCES 

I . Saravanos D.A., 1988, “Structural Optimization of Robotic Manipulators with Fiber Reinforced 

Composite Materials,” Ph. D 77zesis, The Pennsylvania State University. 

Saravanos D.A., and Chamis C.C., 1989, “Unified Micromcchanics of Damping for Composite 

Plies,” Journal of Cotnposites Technology and Research, in review. 

Saravanos D.A., and Chamis C.C., 1989, “Mechanics of Damping for Fiber Composite 

Laminates Including Hygro-Thermal Effects, ” 30th Structures, Structural Dynamics, and 

Materials Conference, Part 1, AIAA, New York, pp. 296-303. 

~ 

4. Liao, D.X., Sung, C.K., and Thompson, B.S., 1986, “The Optimal Design of Laminated 

Beams Considering Damping,” Journal of Composite Materials, Vol. 20, No. 5, pp. 485-501. 

5 . Sung, C.K., and Thompson, B.S., 1986, “A Methodology for Synthesizing High-Performance 

Robots Fabricated with Optimally Tailored Composite Laminates, ” Journal of Mechanisms, 

Transmissions, and Automation in Design, Vol. 109, No. 1, pp. 74-86. 

6. Liao, D.X., Sung, C.K., and Thompson, B.S., .1987, “The Design of Flexible Robotic 

Manipulators with Optimal Arm Geometries Fabricated from Composite Laminates with Optimal 

Material Properties,” The International Journal of Robotics Research, Vol. 6, No. 3, 

pp. 116-130. 

7. Murthy, P.L.N., and Chamis, C.C., 1984, “ICAN: Integrated Composites Analyzer,” AIM 

Paper 84-0974. 

8. Meirovitch, L., 1975, Elements of Vibration Analysis, McGraw-Hill, New YorK. 

9. Ni, R.G., and Adams, R.D., 1984, “The Damping and Dynamic Moduli of Symmetric 

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

Maferials, Vol. 18, No. 2, pp. 104-121. 

19 
APPENDIX 

The present Appendix summarizes the nlicromechanics and laminate mechanics theories for composite damping implemented in this work [2,3]. Analogous micromechanics and 

laminate mechanics theories ['i] have been ut,ilized for the remaining mechanical properties 

of polymer matrix composites. 

On-axis Composite damping: 

Longitudinal normal ply SDC @111: 

Transverse normal ply SDC's $/22: 

4122 = d7f224+ Ef?2 ?- 7)mn(l - fi)E?? Em 

where. 

In-plane shear ply SDC +112: 

where. 

Off-Axis Composite Damping: 

where, 

cos? 0 sin28 sin28 

-0.5sin20 0.5sin20 cos20 

c0s28 -sin20 

20 
Laminate Damping: 

The SDC is expressed in the form of ext,ensional, coupling, and flexural 3 x 3 damping 

matrices. 

IVL NL -1 

[-Ad] = (21 - =b)t[Ec]i[$~]i + H;[S]i['$,]; (AW 

i=l 2=1 

A'L A-L -1 

[GI = 2(4 - =b2)i[~cli[$c]i + &p,iHi[S]i[$m]i (,411) 

i= 1 z=1 

(,412) " i= 1 i= 1 

[$,I is the matrix damping, Hj the interply distortion energy coefficient [7], and [E,] the 

off-axis ply stiffness matrix, 

21 
Table 1 Mechanical properties of the HM-S/epoxy system. 

EPOXY HM-S Graphite 50% HM-S/Epoxy 

22 
Table 2 Optimum designs for Case 1. 

Property Unidirectional Optimum Optimum Optimum 

(Case la) (Case lb) (Case IC) 

Ply Angles, degrees 

81 

62 

83 

0.0 

0.0 

0.0 

27.00 

- 

27.16 

27.30 

5.21 

30.24 

30.49 

29.76 

Fiber volume ratios 

kjl 

kf2 

bf3 

0.50 

0.50 

0.50 

0.69 

0.53 

0.50 

Modal SDC‘s. (%) 

0.5-1 

5.566 

3.426 

0.6.56 

4.540 

0.0199 

2.60 

3.90 

2.74 

0.82 

2.32 

0.05 0 0 

2.58 

2.41 

2.71 

0.82 

2.20 

0.0500 

2.78 

3.87 

2.81 

0.63 

2.1s 

0.0499 

d4 

w5 

Max. static deflection (tip), (in) 

Modal amplitudes (tip), (in) 

0.934 1.536 

0.000 0.000 

0.000 0.244 

0.145 0.000 

0.000 0.091 

1.534 

0.000 

0.213 

0.000 

0.091 

1.51s 

0.000 

0.240 

0.000 

0.08; 

Katural Frequencies, (Hz) 

fl 

f? 

f3 

f4 

f5 

383. 

1089. 

1565. 

2395. 

3854. 

252. 252. 

1360. 1534. 

1572. 1572. 

2964. 2968. 

5029. 5027. 

24s. 

1335. 

1558. 

3518. 

50.53. 

23 
Table 3 Optimum design for Case 2. 

Property Unidirectional 0 p t imum 

Ply Angles, degrees 

01 

82 

03 

Fiber volume ratios 

kfl 

kf2 

kf3 

Modal SDC'*s. (%) 

t' 1 

u72 

2/73 

d4 

v5 

Max. Static deflection (tip), (in) 

Modal amplitudes (tip), (in) 

U!] 

LTf2 

624 . 

id 

z5 

Katural Frequencies, (Hz) 

fl 

f? 

f3 

f4 

f5 

0.00 

0.00 

0.00 

0.50 

0.50 

0.50 

0.571 

5.566 

3.436 

0.656 

4.540 

0.0199 

0.934 

0.000 

0.000 

0.145 

0.000 

383. 

1089. 

1565. 

2395. 

38.54. 

31 24 

31.74 

30.02 

0.53 

0.52 

0.52 

3.3.3 

4.10 

3.34 

0.70 

2.59 

0.0660 

1.775 

0.000 

0.280 

0.000 

0.101 

220. 

1275. 

1383. 

3274. 

4488. 

24 
(A) LINK GEOMETRY. 

(B) FINITE ELEMENT NESH. 

FIGURE 1. - TYPICAL COMPOSITE LINK. 

(A) ONE SUBLAMINATE. 

(el/-el )2, kf 

(B) THREE SUBLAMINATES. 

FIGURE 2. - TYPICAL LAMINATE CONFIGURATIONS. 

25 
Y X 

FIGURE 3. - NODE SHAPES OF THE UNIDIRECTIONAL COMPOSITE LINK, 

26 
1.2 "8 c 

, 

vqv-. . -1.2 L 

c 

g -1.8 

W u 

z 

(A) UNlDlRECTlONAL COMPOSITE LINK. U 

-1.8 I I 

0 .1 .2 .3 .4 .5 .6 

TIME. SEC 

(6) OPTIMALLY TAILORED COMPOSITE LINK. 

CASE (IC). 

FIGURE 5. - TRANSIENT IMPULSE RESPONSE. 

28 
NODE 1 

(FREQUENCY = 249.03. 

SDC = 0.278%-01) 

SDC = 0.387%-01) --.=\- 

MDE .~~ 3 \\--a 

(FREQUENCY = 

1562.32, 

SDC = 0.7812E-01) 

Y X 

FIGURE 4. - NODE SHAPES OF THE OPTIMUM DESIGN. CASE (IC). 

27 
1. Report No. 

NASA TM- 102094 

7. Author@) 

D.A. Saravanos and C.C. Chamis 

2. Government Accession No. 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

7. Key Words (Suggested by Author@)) 

Composites; Composite links; Off-axis damping; 

Laminate damping; Structural damping; Dynamic 

response; Impulsive response; Optimum design method 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 24 

3. Recipient's Catalog No. 

9. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages 

Unclassified Unclassified 30 

5. Report Date 

22. Price' 

A03 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-4875 

10. Work Unit No. 

505-63- 1 1 

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 Design Automation Conference sponsored by the American Society of Mechanical 

Engineers, Montreal Canada, September 17-20, 1989. D.A. Saravanos, National Research Council-NASA 

Research Associate. 

16. Abstract 

A method is developed for the optimal design of composite links based on dynamic performance criteria directly 

related to structural modal damping and dynamic stiffness. An integrated mechanics theory correlates structural 

composite damping to the parameters of basic composite material systems, laminate parameters, link shape, and 

modal deformations. The inclusion of modal properties allows the selective minimization of vibrations associated 

with specific modes. Ply angles and fiber volumes are tailored to obtain optimal combinations of damping and 

stiffness. Applications to simple composite links indicate wide margins for trade-offs and illustrate the importance 

of various design variables to the optimal design. 

*For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA FORM 1626 OCT 86