Optimization of composite flywheel design

Int. J mech. Sci., Vol. 19, pp. 69-78.

P e r g a m o n Pres s 1977.

Printed in Great Britain

O P T I M I Z A T I O N OF C O M P O S I T E F L Y W H E E L D E S I G N EDWARD L. DANFELT, SAMUEL A. HEWES a n d T s u - W E I CHOU Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, Delaware 19711, U.S.A. (Received

30 June 1975, and in revised f or m 26 October 1976)

Summary--Composite flywheels are effective energy storage devices. The multi-rimmed flywheel configuration is first chosen for this study because of its superior operating characteristics and versatility. Then the Kevlar-49/epoxy system is adopted as the basic composite material, which is sandwiched between thin layers of rubber. A general stress analysis procedure is developed for the multi-rimmed structure and a computer routine is established to investigate the effects of various material and geometric parameters on the internal stress levels. A maximum stress criterion is used for failure of the composite flywheel. The basic goal of optimization is to achieve a stress distribution such that each ring in the multi-ringed structure will fail at approximately the same angular speed. The parameters varied in the optimization process include the thickness, Poisson's ratio, Young's modulus and density of the inter-ring material, the density and thickness of the composite material, and the thickness of the flywheel in the axial direction. The optimization process demonstrated that this procedure can be applied in general when other failure criteria or performance characteristics (such as maximum kinetic energy, kinetic energy per weight and kinetic energy per volume) are preferred.

NOTATION r,O

u to 19 o'~e erE~ E, Eo

polar co-ordinates radial displacement angular velocity of flywheel mass density radial and tangential stresses radial and tangential strains elastic modulus in r-direction elastic modulus in 0-direction

A = (E,,/E,) '~2 VOr Poisson's ratio

which characterizes contraction in r-direction due to tension in O-direction v,0 Poisson's ratio which characterizes contraction in O-direction due to tension in r-direction o-. ultimate tensile strength

I. INTRODUCTION RECENT r e s e a r c h a c t i v i t i e s in f l y w h e e l s w e r e s t i m u l a t e d b y t h e c o n c e r n o v e r the p r o b l e m of e n e r g y s t o r a g e . T h e k i n e t i c e n e r g y s t o r e d in a f l y w h e e l is l i n e a r l y p r o p o r t i o n a l to t h e m o m e n t o f i n e r t i a o f t h e b o d y w i t h r e s p e c t to t h e r o t a t i o n axis a n d is also p r o p o r t i o n a l to t h e s q u a r e o f t h e r o t a t i o n a l s p e e d . F o r a g i v e n g e o m e t r i c s h a p e it is d e s i r a b l e , f r o m t h e e n e r g y s t o r a g e v i e w p o i n t , to spin the f l y w h e e l at high s p e e d . On t h e o t h e r h a n d , t h e r a d i a l a n d c i r c u m f e r e n t i a l s t r e s s e s i n d u c e d in a r o t a t i n g d i s k are p r o p o r t i o n a l to t h e d e n s i t y o f t h e m a t e r i a l a n d to t h e s q u a r e o f its a n g u l a r s p e e d . 8 A s a r e s u l t o f the s t r e s s d i s t r i b u t i o n , t h e a n g u l a r s p e e d a n d h e n c e t h e e n e r g y s t o r e d in a f l y w h e e l a r e l i m i t e d b y t h e t e n s i l e s t r e n g t h o f t h e m a t e r i a l o f w h i c h t h e f l y w h e e l is m a d e . T h e d e s i r a b l e m a t e r i a l s f o r f a b r i c a t i n g f l y w h e e l s a r e t h e o n e s t h a t h a v e high specific s t r e n g t h a n d c a n s t o r e a large a m o u n t o f e n e r g y p e r unit weight. M a t e r i a l s t h a t a r e c h a r a c t e r i z e d b y b o t h high s t r e n g t h a n d low d e n s i t y a r e fiber c o m p o s i t e s . C o n t r a r y to all the c o n v e n t i o n a l s t r u c t u r a l m a t e r i a l s , fiber c o m p o s i t e s a r e u n i q u e in t h a t t h e i r s t r e n g t h a n d d e n s i t y c a n b e d e s i g n e d b y v a r y i n g t h e c o m b i n a t i o n s o f fiber a n d r e s i n s y s t e m s . F l y w h e e l s f a b r i c a t e d f r o m fiber c o m p o s i t e m a t e r i a l s c a n also g r e a t l y r e d u c e t h e h a z a r d o f c a t a s t r o p h i c f a i l u r e o f t e n o c c u r r e d in m e t a l f l y w h e e l s . It is t h e i n t e n t o f this r e s e a r c h e f f o r t to i n v e s t i g a t e t h e f l y w h e e l d e s i g n s p r o p o s e d explicitly for use of composite materials, select one which appears the most promising, a n d to i n v e s t i g a t e a n a l y t i c a l l y its p e r f o r m a n c e c h a r a c t e r i s t i c s . T h e f l y w h e e l c o n f i g u r a t i o n w h i c h a p p e a r s to b e s u p e r i o r to all o t h e r s in b o t h its o p e r a t i n g M S V O l . . 19 NO. 2 - - A

69

70

EDWARD [~. DANFEI '~ ef ol,

characteristics and versatility is the multi-rimmed flywheel design. Based upcm ~hi choice in configuration, a computer routine has been generated to investigate the slre~, level in the spinning flywheel. Then a general survey of readily available composite: materials is made. The Kevlar-49/epoxy composite system is chosen as the ba:,,ic material to use for the flywheel. The computer routine is then used to investigate the.. effects of various material and geometric parameters on the internal stres~ levels. U ~i~Jg the criterion of desiring failure throughout the flywheel at nearly the same level ,,~ spinning velocity, illustrative examples of flywheel design are performed~ l'he~,~ examples do not, by any means, comprise a complete design analysis, but they do poir~ out the significant characteristics in optimizing the design of multi-rimmed flywheei~ ;~, energy storage devices. 11. C H O I C E

OF FLYWHEEL

GEOMETRY

Fiber-reinforced composites are attractive structural materials because of their high strength and low density.' Flywheels fabricated from composite materials provide higher energy storage per unit weight and per unit volume then those fabricated from any other materials. Furthermore, fiber composites can be tailor-made to fit the stress and energy storage requirements in flywheels. There are a number of flywheel designs which have been proposed specifically with composite materials in mind. One of the popularly known composite flywheel is the rimless multi-spoke "superflywheel".~ The configuration of this design consists of one or more relatively thin solid bars of undirectional composite lamina. By orienting the fibers in the radial direction, they are designed to bear the centrifugal force. This design suffers several drawbacks? These include the low volume efficiency of the flywheel, the uneven stress distribution along the lamina, and the stress concentration induced in anchoring the ends of the rods at the hub~ Another novel flywheel design is the brush-like rotors composed of thin rods or filaments suitably bonded into a hub. 2 Some of the shortcomings of this design are similar to those of the multi-spoke superflywheel. In addition, since the filaments remain unprotected by any matrix material, they are more susceptible to cracking. A circular disk of composite material, where fabrication takes place by winding the filaments about the axis of rotation, has also been suggested. "'5 Similar to the previous designs, this configuration also possesses an uneven stress distribution under centrifugal loading. In addition, the stored kinetic energy per unit weight of such a design is significantly less than the previous two designs. A special case of a disk-shaped flywheel would be the single-ring design which consists of a thin ring of composite material connected by spokes to a hub. Such a configuration yields excellent weight effectiveness, yet quite poor swept volume effectiveness. The flywheel which appears capable of incorporating many advantages of the above mentioned configurations is the multi-rimmed design. 3"6"7 The multi-rimmed design consists of a concentric set of axially wound composite material rings separated by thinner rings of some softer materials. Various design optimizations indicate that multi-rimmed flywheels possess weight and volume effectiveness equivalent to or better than all the other designs. In addition, the versatility of this configuration give~ the designer the ability to choose the performance characteristic desirable to ~,. particular application. Theoretical analysis and experimental study on multi-rimmed flywheels also have shown that all natural resonant frequencies of the rotor lie well above the highest operating speed? The above considerations seem to suggest that the multi-rimmed flywheel design has great potential for energy storage, therefore, it is; chosen as a basis for further investigation.

lIl. STRESS

ANALYSIS

A generalized stress analysis is developed for the multi-ringed composite flywheel. First, for a single ring of cyclindrically anisotropic material, the analysis is a simple extension of the isotropic case. ~ The displacement-strain relations for this axisym-

Optimization of composite flywheel design

71

metric problem are du

E, = ~

(1)

u eo = - .

(2)

r

The stress-strain relations in polar co-ordinates are or, E, - E , --

ee

O'0

E0

re,o%

(3)

Eo l~rOO'r

Er "

(4)

For a single ring of uniform thickness and rotating at constant speed ~o, the force equilibrium equation in the radial direction is given by d

dr

(5)

(ro~,) - o'e + pa~2r 2 = O.

Equation (5) can be rewritten in terms of the radial displacement as Er r 2d2u ~ + E r r - ~d ru - E o u

= - pro2(1 - v,ov~r)r 3.

(6)

In deriving equation (6) we employed the relation E r v o , = Eovro, which is a direct consequence of the s y m m e t r y of the stiffness coefficients in the stress-strain relations, l The general solution of equation (6) can be expressed as u = c~r ~ + c2r A _

pro2(1

- VrOVOrJr3 9 E , -- 17,8

(7)

where h = ( E d E r ) "2, and c, and c2 are constants to be determined f r o m the b o u n d a r y conditions. The corresponding stress c o m p o n e n t s are:

or, =

¢1

E,(A + vo,) r ~ 1_4_ C2 l - - VOrVrO

E , ( - X + vo~) r_~_ , 1 ~ V0,V~

Eo(1 + Arm) r~_~ + c2 Eo(l - hv~) r_~_ , o'o = c ,

1 - re, v , ,

1 - VerY,,

(3 + u~.)E,o~O2r ~ 9E~ E0

(8)

(l + 3 v ~ ) E o P O 2 r 2 .

(9)

9Er - Eo

In the multi-ringed flywheel, the a b o v e displacement and stress expressions can be applied to each c o m p o n e n t ring. By assuming a perfect bonding, the following boundary conditions can be specified at each interface: ,r,' = ~ / + '

(10)

u'=

(11)

u '÷'

where the superscripts i and i + 1 are used to designate two neighboring rings. It is also assumed in this analysis that the inner and outer boundaries of the flywheel are free of stresses. The stress free inner surface m a y simulate the boundary condition of filament composite wound onto a hub. There are two unknown constants needed to be solved for the elastic field of each ring. The total number of available equations, as seen f r o m equations (10) and (11), is equal to twice the number of rings. Consequently, the a b o v e formulation can be used to

72

EDWARD L. DANFEL'r et al.

solve the elastic field in composite flywheels made of any number of rings. The analysi~ can be easily modified to take into account other types of boundary conditions A computer routine has been set up to carry out the calculations. The input to the routine consists of the values of E,, Eo, uor, u,o and p of each ring material, the positions of the interfaces and free surfaces, and the angular velocity of the flywheel~ The stre~s distribution in the flywheel and its stored kinetic energy can then be readily calculated IV. SELECTION OF COMPOSITE MATERIALS

There is a wide variety of choices of composite material systems available for flywheel constructions. The selection of materials can be made based upon the kinetic energy storage per unit weight, kinetic energy storage per dollar or kinetic energy per swept volume. A flywheel cannot be optimized for all these considerations simultaneously. Therefore, decision must be made as to which of the above considerations is most suitable to a particular problem. The value of the stored kinetic energy per unit weight of a flywheel is proportional to the ratio of composite ultimate tensile strength and density (cr./p). In Table 1~ the values of KE/lb and KE/$ fo~ various composite systems are calculated and listed according to their relative magnitude. The largest value is taken as one unit. It is seen that Kevlar/epoxy system leads the list in KE/lb and is also very satisfactory from the standpoint of KE/$. It is evident that K e v l a r / e p o x y system is one of the most promising material systems available and is, therefore, chosen for our study. V. OPTIMIZATION PROCEDURES

Now that a material has been chosen, it will be shown how the computer generated stress distributions can be used in optimizing and detailing the flywheel design. As can be seen from earlier considerations, it is impossible to design a multi-rimmed flywheel which is optimum for all applications. What may be of most use to those designing a multi-rimmed flywheel is a general approach to the optimization process. Therefore, it will be shown which parameters strongly affect the performance characteristics of a flywheel. Also, typical optimization examples will be demonstrated. Hopefully, then such examples will provide the designer with ideas on what are the important considerations in a multi-rimmed flywheel design and typical methods of attack° H o w e v e r , final weighting assessments of various performance characteristics are then left to the designer and his particular application. Throughout the following discussion the basic design goal considered here is to create a stress distribution such that each ring will fail at approximately the same angular speed. Thus strength is not " w a s t e d " on rings which otherwise experience proportionately smaller stress levels. This means that for any given angular speed, either the radial or circumferential (and possibly both) stress level in a certain ring is of the same proportion of its ultimate stress as in any other ring. This, of course, assumes a simple maximum stress failure criterion. The above criterion assumes the flywheel will fail radially and circumferentially at approximately the same angular speed. H o w e v e r , a radial failure mode has been shown to be somewhat more desirable than a circumferential one, ~ hence, the designer may choose to keep the radial stresses slightly TABLE 1. COMPARISONOF COMPOSITE MATERIALSYSTEMS Material Kevlar/Ep. Gr/Ep. B/Ep. S-Glass/Ep. E-Glass/Ep. B/AI

o', ksi (109N/m 2)

p Ib/in 3(g/cm3)

KE/kg*

200(I.38) 186(1.28) 188 (1-30) 275 (1.90) 149 (1.03) 215 (1-48)

0.05 (I.38) 0-054(1.49) 0-073 (2.02) 0.079 (2.19) 0.073 (2-02) 0.1 (2.77)

1.00 0.87 0.65 0-87 0-51 0-54

*In units of the KE/kg value of Kevlar/Ep. t i n units of the KE/$ value of E-Glass/Ep

$/kg 1.76 154-00 440-00 22.00 6.60 462.00

KE/$t 0-74 0-07 0.02 0.51 1.00 0.02

3.7 3-5 3.4 2.05 1.28

Optimization of composite flywheel design

73

larger in proportion to the radial strength, so if failure occurs, it will occur in the radial mode. Although the local stress levels are considered with respect to the ultimate strengths throughout the rest of the text, the designer is free to assign some safety factors so local stresses do not rise above a given percentage of the ultimate strength. Finally, the computer generated stress distributions can also be used in designs based upon other failure criteria. 9"~° It also needs to be noted that the maximum stress criterion for failure of composite materials should be applied to the fiber and transverse directions separately. This is due to the fact that the strength of fiber composite materials is highly directional. Strength along the fiber direction in glass fiber-resin systems, for instance, could be more than two orders of magnitude higher than that along the transverse direction. Furthermore, the strengths in the radial and hoop directions are physically uncoupled: one is a measurement of the fiber strength and other relies on either the matrix strength or the fiber-matrix interfacial bonding strength. In this design analysis an overall geometry is chosen. The dimensions of the configuration used are: an outer diameter of 3ft, an axial thickness of l in., and an inner hole diameter of 2 in. The flywheel geometry consists of six rings of composite material with five rings of softer material sandwiched in between. This configuration choice is fairly arbitrary, but it is kept constant throughout the analysis so when one parameter, such as a material property, is varied, any changes in the stress distribution can be traced directly to this parameter change. When the bulk composite material already chosen, the next logical step is to investigate the softer inter-ring material. It has been suggested that rubber or some similar materials may be used for this purpose. 3 In the following analysis the mechanical properties of a typical rubber compound 11 are first employed (Young's modulus = 85 psi; Poisson's ratio = 0.5; and density = 0.041b/in ~) in the computer routine. Several parameters of this inter-ring material are then altered to see what changes in the stress distribution they produce. The first variable examined is the thickness of the rubber rings. Radial thicknesses of 0.1, 0.5 and 1.0 in. are tried. The results show that as the thickness of the rubber rings is made smaller, the maximum circumferential stresses in the composite rings also become smaller. The lowest radial stress observed in the composite material occurs for a radial thickness of 0.5 in. Since the minimization of radial stress is considered most important a thickness of 0.5 in. for the inter-ring material is chosen. Another parameter that is varied is the Poisson's ratio of the inter-ring material. Values of 0.4 and 0.3 are compared to 0.5 (that of rubber). It is found by examining the stress distribution for u -- 0.3 and u -- 0.5 that the only thing that varies is the " s h a p e " of the stress distribution in the inter-ring material. Maximum stresses in the inter-rings and the overall stress distribution in the composite rings are identical through the fourth significant digit. Hence, the Poisson's ratio for rubber ( u - - 0 . 5 ) appears acceptable. The next parameter varied is Young's modulus of the inter-ring material. The value of E is varied from 85 psi (that of rubber) to 500,000 psi (a value comparable to that of the epoxy matrix used in the composites). The variation in the circumferential stress resulted from the change in E is relatively small. The variation in the maximum radial stress is significant and the result is given in Table 2. It is obvious that the radial stress increases as the inter-ring material Young's modulus is raised. Therefore, it can be concluded that rubber's low value of Young's modulus is another reason why it would make a good inter-ring material. It also needs to be pointed out that the values of the maximum radial stress given in Table 2 change with rotation speed and, hence, do not have too muOh significance. It is only that the trend of variation in stress with modulus is important. The above discussion gives a good example of how one might go about choosing the inter-ring material. From this discussion it appears that rubber with a low Young's modulus and a radial thickness of about 0.5 in. for the chosen overall geometry is a good choice. Now that parameters involving the inter-ring material have been established, changes elsewhere can be made.

74

EDWARD L. DANFEI ! et 01, TABLE 2. T H E EFFECTS OF VARYING Y()UNG~S MODULUS OF THE INTER-RING MATERIAl ()N STRESS DISTRIBUTION

Inter-ring material Young's modulus psi (N/m~;

Maximum radial ~tress p~i !N/m:}

85 (5-86 x 500(3.45 x 50t)0(3.45 × 50,000 (3.45 × 500,000 (3"45 ×

129 (8-89 x 131 (9.03x 152 (1.05 x 293 (2.02 x 632 (4"36 x

10'1 10~) 10") 10") 109)

t0") 10') t0 '~) I(P) 106)

Refer now to Fig. 1. This shows the stress distribution of the flywheel c o m p o s e d of six Kevlar-49/epoxy c o m p o s i t e rings separated by five 0.5 in thick rubber rings. This stress distribution is anything but level. It can easily be seen that the o u t e r m o s t composite ring will fail long before the inner rings do. Hence, optimization processes need to be carried out to flatten the stress distribution. Before moving on, it is important to examine the increments of the two vertical axes in Fig. 1. It is noted that the ultimate longitudinal strength of Kevlar-49/epoxy composite is about 40 times the ultimate transverse strength. 'z Hence, the radial and circumferential stress axes are made to this same proportion so that radial and circumferential stresses appearing at the same physical height on the graph c o r r e s p o n d to equal percentages of respective ultimate strengths. This also implies that radial and circumferential failure will occur at a p p r o x i m a t e l y the same angular speed. Therefore, the aim now is to bring both radial and circumferential stresses to the same physical level on such a graph, thereby ensuring o p t i m u m utilization of available strength. One solution suggested by several sources 3'6 was to add ballast material, such as lead chips or iron powder, to the inner rings, thereby increasing their stress levels. Such a procedure would m o s t assuredly have some effect on the mechanical properties of the composite. H o w e v e r ascertaining these changes is beyond the scope of this paper. So to get at least a gross estimate of what the effects would be, all physical p a r a m e t e r s are held constant while density alone is varied. After some trial and error, the results shown in Fig. 2 were obtained. Densities of the composite rings range f r o m 0.05 ib/in 3 in the 200

- - - - - - - ] 8000

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i

i ooo

150

-"

I00

oooo

°b

tO 0,.

;5

/ ~

b~ 50

it"\ /i~ / " , ,,

,..

,

I/

I

~ ,#-- ~,1! v

-~ t 2000 i

II

V

i

,",

I

o

I !

-50 o

4

8

12

16

20

r

FIG. I. Stress distribution in a flywheel composed of six composite rings separated by five 0,5 in, thick rubber rings:

Optimization of composite flywheel design 200

8000

I 50

6000

I

I

I

/\

1,00 U3 n

75

4OOO



50

I I

I

P

I I I I I I

I

II I

2000

#,%

/

II II

I

II

I

II

I

-50 0

I

I

4

8

'

I I

I/

I

L!

12

16

-2000

F

FIG. 2. Stress distribution in a flywheel obtained by varying the densities of the composite rings.

outer ring (no ballast) to 0.45 lb/in 3 at the inner ring. The result of the trial and error procedure indicates that it is impossible to get both radial and circumferential stresses to the same level on the graph while varying density alone. Recall that it was determined earlier that if one material were to be used in between the composite rings, a material similar to rubber with a low Young's modulus might be the best choice. However, if the Young's modulus of the inter-ring material could be varied in steps, perhaps a smoother stress distribution would result. Such a procedure has been tried and the results are shown in Fig. 3. Everything, including density, is constant except the Young's moduli of the inter-ring materials. They vary from 800,000 psi (the same as pure composite matrix material) at the inner ring, to 85 psi (that of rubber) at the outer ring. The stress distributions thus obtained appear to be more level than those observed in Fig. 1. Another parameter which could affect stress distribution is the thickness of the individual rings measured parallel to the axis of rotation. Such a possibility might at first

150

6000

.~. ^/

,oo

/

b~

/

/ V

/

:~,

'

!!

!

/ ~

li

Ii ,

Iri I1!

i~-< °

,

II

11 .,,

hi/ Id I

fll/

4ooo

~ ' b~

0

/

0

!

I

I/ - 50

II

f

0

4

8

12

16

-2000 20

r

FIG. 3. Stress distribution in a flywheel obtained by varying the Young's moduli of the inter-ring materials.

76

EDWARD L. DANFEL'I el al.

seem well founded since variation of axial thickness in a single-ringed disk can ha,,c p r o f o u n d effects on the levels of stress. H o w e v e r , in the multi-rimmed flywheel, ~he stress levels o b s e r v e d are fairly independent of the axial thickness as !ong ~; the a~Jai thickness across a ring is constant, and the inter-ring material p o s s e ~ e ~ ;~ ~maii Young's modulus. This is true because for an inter-ring material such as r u b b e r ~e~~ little stress is transmitted f r o m one c o m p o s i t e ring to the next; hence, ~he c o m p o s n e rings act almost independent of one another. Numerical results of the stres~ distribu~io~ have been obtained for just a single ring of composite located at the same radi~i position as the o u t e r m o s t ring in the multi-rimmed configuration of Fig. 1. h is found that the circumferential stresses show a m a x i m u m difference of less than 8%. The radiai stress distributions vary s o m e w h a t more. Overall, the results imply that rings do in fac~ act fairly independently and that changes in axial thickness of one ring would no,~ be e x p e c t e d to greatly change the stress distributions in neighboring rings, Hence, change~ in axial thickness would not help much in leveling off the multi-rimmed stres.,, distribution, but m a y be used for storing more mass in the outer rings and thereby increasing total kinetic energy. N o w that all the important p a r a m e t e r s have been examined, and materials have been chosen, an overall optimization process may take place. First, the designer should decide which, if not all, methods he would like to use to produce the soughtafter level stress distribution. Then he should proceed to incorporate together these methods. Fig. 4 shows an example of such a process where both composite ring thickness in ~he radial direction and their densities are varied to attain the desired stress distribution (density ranged f r o m 1.0 lb/in 3 in the inner ring to 0.061 lb/in 3 in the outer ring and radial thickness of the c o m p o s i t e rings ranged f r o m 1.5 in. in the inner ring to 4.0 in. in the outer ring). The method used is basically trial and error, and it can easily be seen that the whole process can b e c o m e quite complicated once more than two p a r a m e t e r s are varied. It also needs to be pointed that the stress profiles depicted in Figs. 1-4 are obtained for o~ = 400 rad/sec. The choice of ~o is arbitrary and should not affect the characteristics of the results. Since the stress c o m p o n e n t s are proportional 002 the stress magnitude fo~ other values of 00 can be easily derived f r o m these figures. The a b o v e analysis provides the stress distribution e v e r y w h e r e in a flywheei including that at the layer interface. For a given material combination the interfaciat strength can be m e a s u r e d in simple tests and c o m p a r e d with the m a x i m u m interracial stress level allowed b y the theoretical design. An inspection of Fig. 4 indicates that the 8000

200 A

150

6000

!, I

"3

I00

/

I \

\

4ooo

OD O_

i,

b~ 50

I

II II I

I

Jc

II

V - 2000

I

-50 0

I

I

4

8

I . . . . 12

[6

20

FIG. 4. Stress distribution in a flywheel obtained by varying both the radial thickness and density of the composite rings.

Optimization of composite flywheel design

77

interfacial stress b e t w e e n the fiber rings and the rubber inter-ring material is either in small tension or compression, and m a y not pose a problem in flywheel design. It also needs to be mentioned that the present analysis is for flywheels rotating at constant speed and the effect of acceleration is not included. The contributions to stress due to acceleration and deceleration are relatively unimportant especially for flywheels used for long term energy storage rather than transient type loadings. As mentioned earlier, stress distribution is not the only criterion of interest. Depending upon the application, p e r f o r m a n c e characteristics such as m a x i m u m kinetic energy, kinetic energy per unit weight (weight effectiveness), or kinetic energy per unit volume (volume effectiveness) may have varied degrees of importance. To give the potential designer an idea how the various p a r a m e t e r changes affect p e r f o r m a n c e , Table 3 is presented showing how each p e r f o r m a n c e characteristic varies with design change. As can be seen f r o m this table, various methods of obtaining a level stress distribution affect the p e r f o r m a n c e characteristics in various ways. Hence, tables generated similar to Table 3 can give the designer a better idea which " s t r e s s - l e v e l e r " m e c h a n i s m best fits his particular application. Another way one could theoretically optimize the multi-rimmed design would be to construct a hybrid model. Such a design would consist of rings made of differing composite materials. P e r h a p s the strongest (and most expensive) composite material would be placed in the o u t e r m o s t ring, while weaker composites would be placed in the rings closer to the hub. Optimization process for hybrid composite systems is not carried out, however, the c o m p u t e r routine developed could readily be applied to such a scheme. TABLE 3.

Figure number corresponding KEm,~ KE/wt KE/vol KE/$ to the flywheelconstruction W-hr (106J) W-hr/lb(10~J/kg) W-hr/in3(108J/m3) W-hr/$(10"J/$) I 2 3 4 *

1590 (5.724) 2270 (8.172) 1810 (6-516) 2660 (9-576) 2650 (9.540)

32.4 (2.566) 21.8 (1.727) 37.0 (2.930) 21.3 (1.687) 37.0 (2.930)

1.57 (3.449) 2.24 (4.921) 1.78 (3.910) 2.62 (5.756) 1-31 (2.878)

4.05 (1.46) 2.73 (0.983) 4-63 (1.67) 2.66 (0.958) 4.63 (1.67)

*Axial t h i c k n e s s e s of the two outer rings in Fig. 1 are doubled.

VI. CONCLUSIONS It can be seen f r o m the foregoing discussion, the multi-rimmed flywheel configuration does possess versatility in design. As was mentioned earlier, the intent of this research is not to actually present a detailed multi-rimmed design for a given application; instead, it is to provide an insight into the broad spectrum of variables associated with this flywheel configuration and a possible procedure of optimizing these variables according to a chosen criterion. This research work has d e m o n s t r a t e d that the multi-rimmed configuration is desirable, the method of analysis is workable, and the optimization procedure can be applied in general when other failure criteria and p e r f o r m a n c e characteristics are preferred.

REFERENCES I. J. R. VINSONand T.-W. CHOU, Composite Materials and Their Use in Structures. Elsevier-Applied Science, London (1975). 2. D. W. RABENHORST,Intersociety Energy Conversion Engng Conf. Proc. p. 38, Society of Automotive Engineers, Inc., pp. 1118--1128 (1971). 3. R. F. POST and S. F. POST, Sci. Am. 229, 17 (1975). 4. S. TANG,Int. J. mech. Sci. 11, 509 (1969). 5. G. F. MORGANTHALERand S. P. BONK, 12th National S A M P E Syrup. D-5. 6. D. W. RAIaENHORST,14th Ann. Syrup. on Engineering for the Materials/Energy Challenge. New Mexico Sections of the ASME and ASM and the University of New Mexico, February (1974). 7. A. LEVY,M. KESSELMAN,A. TOB1N,G. GESCHWIND,S. TANG,M. SANDLERand H. PINNICK,Grumman Research Department Memorandum, RM-598 (1975).

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EDWARD L. DANFEL'| e[ at,

S. P. TIMOSHENKO and J. N. GOODIER, Theory o/ Elasticity. McGraw-Hill, New York (197(L~ T. W. CHOU and A. KELLY, Mat. Sci. Engng 25, 35 (1976). S. W. TSAI and E. M. W u , J. Comp. Mat. 5, 58 (19717. Mark's Standard H a n d b o o k for Mechanical Engineer~, '71h edil. {Editor-ill-chlei I l~,'~t ~.,~-~ili~ McGraw-Hill, N e w York (1%7). 12. " K e v l a r " 49 data manual, E. I. D u P o n t de N u m o u r s and Co., Wilmington, Delaware. 8. 9. 10. II.