Techniques were developed for presenting an analytical and graphical means for selecting an optimum flywheel system design, based on system requirements, geometric constraints, and weight limitations. The techniques for creating an analytical solution are formulated from energy and structural principals. The resulting flywheel design relates stress and strain pattern distribution, operating speeds, geometry, and specific energy levels. The design techniques incorporate the lowest stressed flywheel for any particular application and achieve the highest specific energy per unit flywheel weight possible. Stress and strain contour mapping and sectional profile plotting reflect the results of the structural behavior manifested under rotating conditions. This approach toward flywheel design is applicable to any metal flywheel, and permits the selection of the flywheel design to be based solely on the criteria of the system requirements that must be met, those that must be optimized, and those system parameters that may be permitted to vary

Nizza, R. S.

English

NASA Technical Reports Server

N76128276
SHAPE OPTIMIZATION OF DISC-TY PE FLYWHEELS
By Richard S. Nizza
Lockheed Missiles & Space Company, Inc.
ABSTRACT
Recent developments in the field of flywheel powered electrical
energy storage systems has prompted the need for a better understanding
of the varied design and analytical criteria that must be considered in the
selection of a flywheel. Techniques have been developed for presenting
an analytical and graphical means for selecting an optimum flywheel sys-
tem design, based on system requirements, geometric constraints and
weight limitations. The techniques for creating an analytical solution are
formulated from energy and structural principals. The resulting flywheel
design relates stress and strain pattern distribution, operating speeds,
geometry, and specific energy levels. The design techniques incorporate
the lowest stressed flywheel for any particular application and achieve
the highest specific energy per unit flywheel weight possible. Stress and
strain contour mapping and sectional p.rofile plotting reflect the results
of the structural behavior manifested under rotating conditions, This
approach toward flywheel design is applicable to any metal flywheel, and
permits the selection of the flywheel design to be based solely on the "_
criteria of the system requirements that must be met, those that must
be optimized, and those system parameters that may be permitted to vary.
INTRODUCTION
This paper describes a procedure for designing an optimum flywheel
shape based on the constraints of geometry, speed and stress so as to
maximize energy density. The design procedure described relies on the
application of linear elastic structural mechanics and the laws of con-
servation of energy and momentum. Little work has been reported in
maximizing the energy density of solid disc flywheels. Much work however
has gone into the design of turbine blades and discs, and electric gene-
rators and motors, which are perhaps the closest entity to the energy
storage flywheel. The basic structural laws under which flywheels, tur-
bine blades, generators and motors behave are the same but their func-
tions, based on different design objectives, are different.
The energy density of a flywheel is represented by the simple
relationship:
E=K -_ (1)
SO
:i_i'_ 38 ffPL Technical Memorandum 33-777 _'
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1976021184-045
https://ntrs.nasa.gov/search.jsp?R=19760021188 2020-03-20T02:30:34+00:00Z
where E = energy density ,
K S = flywheel shape factor (dimensionless)
= material working stress
P : rr, aterial density
Flywheel shape factors for several geometries are shown in Table 1.
For disc flywheels the shape factor can approach 1.00. The disc shaped
flywheel that produces this high a shape factor has constant stresses
throughout. This is attributed to the fact that each unit volume of material
is stressed equally to a predetermined working stress level and therefore
produces the largest amount of energy possible. The flywheel shape that
produces this constant stress is exponential in profile.
Equation (2) expresses the summation of forces in a flywheel
(Reference 2).
d(XYo r)
dX Y°t + -_wZxZY = 0 (2)g
For uniform strength the tangential and radial stresses must be
equal and of constant value throughout.
Therefore,
0 t = Or = ¢Y = constant
Equation (Z) can be restated as
d(XV) Y + P--w3xZY = 0 (3)
dx go
and by integrating,
y WZXZ0
In T = - _go (4)
Applying the boundary conditions, at X = XR, Y = YR
_w z l
Y
_ _ =X_RR In_ (5) _
Substituting Equation (5) into Equation (4),
o
IPL Technical Memorandum 33-777 39
_l i
1976021184-046
where X = radius _
: Y - thickness at radius X
Y = hub thickness
0
YR = tip thickness _ -,
I
X R = tip radius
Equation (6) gives the normalized thickness as a function of the normalized
radius and the hub-to-tip thickness ratio. This equation represents the
profile configuration for the constant stress flywheel geometry. According
to Equation (6), the disc, even though with infinitely decreasing thickness,
is prolonged to infinity. But practically, the disc is limited by a cylin-
drical boundary or radius X_ at which it has a thickness YR" Although =
the theoretical flywheel of in_nite diameter would have a shape factor
of 1.00, the practical flywheel of finite radius X R would have a shape
factor less than 1.00. In order to improve the shape factor of this ex-
ponentiaUy shaped finite diameter flywheel, the author has chosen to take
i :
some of the material that theoretically existed between the finite diameter
and infinity and place it near the rim of the flywheel producing a constant /
thickness section running from a point on the surface to the rim. The
utilization of a flat tip as the means for improving the flywheel shape •
factor, in _ieu of an exponentially flaired tip, was chosen for two reasons:
number one, the shape factor difference between a flaired and a flat tip ._
was found to be insignificant, and secondly, the manufacturing and machin-
ing operations are considerably simplfied by having a flat tip rather than
a flaired tip. The question of how much constant thickness material
should be added to the flywheel must now be determined. Since it is the
objective to improve the shape factor as much as possible, it is necessary
to solve Equation (I) in terms of the shape factor, K_, for the various
stresses and energy densities associated with each flatted tip flywheel
that is generated for each hub-to-tip thickness ratio used in Equation (b).
: An analytical evaluation must first be performed to evaluate the resulting
stresses and energy densities for each flywheel geometry.
One such analytical method developed at Lockheed Missiles & Space
Company utilizes a computer program based on two dimensional stres ses
and strains developed in rotating machinery. These relationships were
then expanded (Reference 3) and culminated in the computer program.
A typical set of results are shown in Table 2. Tangential stresses,
: radial stresses and flywheel thicknesses are presented for various radii
starting at the rim and extending to the hub. Once the stresses are deter-
mined the program then utilizes Equation (I) to calculate the shape factor
I from the calculated kinetic energy and maximum flywheel stress. A maxi-
• mum shape factor is then obtained for each hub-to-tip thickness ratio by Itera-
tively evaluating different flatted tips that begin at different percent radii.
Figure I represents the results of the relationship between the fly-
wheel shape factor and the hub-to-tip ratio. The point at which the optimum
_ 40 J"PL Technical Memorandum 33-777 : _
4
I i
J
1976021184047
flatbegins has an optimum value which varies with the hub-to-tip thickness :
ratio. By applying the __ppropriateflattipas the ,neans for optimizing the
flywheel shape factor for a particular hub-to-tip ratio, itcan be recognized
that for hub-to-tip ratios less than I.00, the appropriate flattip would ex-
tend from the tipto the hub, and the flywheel shape would be that of a flat
unpierced disc having a shape factor of 0.606.
FLYWHEEL PARAMETRIC TRANSFORMATIONS , :
By increasing the hub-to-tip ratio, the shape factor is improved and
becomes I.00 at a hub-to-tip ratio of infinity.There are instances when
a high hub-to-tip ratio may not be practical, such as when geometric con-
straints limit the axial length of the hub. It is therefore desirable to
determine the effectsof changing parameters on the rest of the system.
A set of parametric relationshipsfor relatingflywheel diameters, speeds,
weights, kinetic energy levels, operating stresses and thicknesses permits
an easy determination of the effectson each parameter when one or two
are changed. Equations (7), (8),and (9)express these relationships, and
can be used for the flywheel shapes generated by Equation (6)having fixed
hub-to-tip ratios.
\Wo/ (7)
:: THKN /W _2/Do X4(KEN_
WTN /THENX/DN
- (9)
where o = working stress
D = flywheel diameter
W = flywheel speed
KE = kinetic energy
THK = flywheel thickness
WT = flywheel weight
JPL Technical Memorandum 33-777 41
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1976021184-048
IA specific example is used to demonstrate the application of
these equations for a flywheel having a kinetic energy of 12 kilowatt-hours.
This is shown in Figure 2 for several flywheels from three to four feet
in diameter. The stagger of the data points is caused bythe quantizing
error in selection of either 75 or 80 percent flat value for the optimum
hub-to-tip ratio. Since the optimum flat lies between these two values a
• smooth curve in actuality joins the optimum points.
I
The curves represent a family of varying ci_ameters for a particular
kinetic energy level. It is reasonable to assume that the kinetic energy
= requirements are already known for a desired application. The require-
ments for the operating speed, or the maximum flyv,heel stress will
further restrict the number of available flywheel design selections. If we
permit the flywheel speed to vary, we can superimpose a family of fly-
; wheel speeds on Figure 2 indicative of a specific kinetic energy and operat-
ing stress level. This was done for an operating stress level of 100 ksi.
By utilizing a series of plots similar to Figure 2, reflecting various operat-
ing stress levels, a more complete selection of flywheel geometries is
' possible. These curves represent a means for selecting an optimum :
geometry flywheel based on the kinetic energy requirements, volumetric
lin,itations, and desired flywheel life (reflected through operating stress i
level). Once a selection is made, a stress profile may be performed
using the two dimensional stress program of Reference 3, which produces
results similar to those of Table 2. If upon examination of the results of
this initial computer run, the flywheel selected is found to be satisfactory,
a much more rigorous, three dimensional, stress-strain examination can
be performed using a finite element computer program. A process of con-
tour mapping of the stresses and strains developed under rotating con-
: ditions for each and every point within the flywheel can then be made.
Figures 3 through 6 show contours for radial, tangential, axial, and axial
shear stresses for a quarter section view of a 12 kilowatt-hour flywheel.
Figures 7 and 8 show the radial, tangential, axial and axle| shear stress
distribution along an axis of symmetry, perpendicular to the axis of
rotation. The graphs are plotted from right to left. The radial and tan-
gential stresses are maintained at the maximum for almost 80% of the
radius and decrease only at the tip.
These techniques permit a very accurate determination of all stresses
throughout a homogeneous flywheel, and provide all the quantitative in-
formation necessary to perform sensitivity tradeoff studies. This allows
a flywheel to be geometrically oF_tlmized for a given application in a pre-
cise, quick, and economical fashion. The desirability of a constant stress
and homogeneous material was asqumed; however, in manufacturing thick
forgings the metallurgical composition can vary considerably from the
core to the surface as well as from the hub to the rim. The effects of the
resulting stress pattern variations, developed within the material, must
be taken into account and applied to the optimization procedure presented.
Such methods have been developed at Lockheed Missiles & Space Company
in the form of additional computer programs that evaluate the effects of
non-homogeneity of the flywheel material.
42 SPL Technical Memorandum 33-777 ,
1976021184-049
REFERENCES
1. Lawson, L. J. , "Design and Testing of High Energy Density Flywheels
for Application to Flywheel/Heat Engine Hybrid Vehicle Drives,"
Intersociety Energy Conference, P-38, Paper 719150, New York:
Society of Automotive Engineers, Inc., 1971.
J
Z. Stodola, H., Steam and Gas Turbines, Vol. 1. McGraw-Hill Book
Co., Inc., New York, 19Z7.
3. Gilbert, R. R., et al., Clywheel Feasibility Study and Demonstration,
Final Report to Environn,cntal Protection Agency/Air Pollution
Control Office, under contract EHS 70-104, April 50, 1971.
°:
i JPL Technical Memorandum 33-777 43
.11 iiii i _ . I _,
1976021184-050
w 1
, t
Table 1. F'lywheel shape facters for various geometries
Flywheel Geometry 5},at, e Factor
KS".'
Constant-stress disc (OI)_o_) 1 00 '
Modified constant-stress disc {typical} 0 931
Truncated conical disc' 0 806
Flat unpierced disc 0 606
Thin rim (ID/OD---_ 1.0) 0 500
Shaped bar (OD---_ =) 0 500
Rim with web {typical} 0 400
Single filament (about transverse axis) 0 553
Flat pierced disc 0 305
:::Fru=,_ Ref. I. ._
44 TPL Technical Memorandum 33-777
1976021184-051
Table 2. Flywheel geometry and stress distribution
FLYWhEeL PANA_T_RS
SPEED ............. - ......... 13_ _AD/5EC
H_TRPIAL D_N_ITY ............ .Pm5 LBSICU-IN
POISSONs RATIO .............. ,30
YOUN(?_ MODULUS *- ............ _O_GO.OE_n3
T_MP. _XP, COEFFICIENT ...... 13n0,_-09
T@TAL TEMP. D|_F[REMCE ...... *0 DEG*F "-
OUTSIDE RAOltlS --*- .......... _1,0_0 INCHES i
FLYWH[KL DESIGq FACTORS
MKAN TANGENTIAL STRESS ...... InR?PI PSI
MOMENT OF INERTIA ........... 400.453 LE*IN-SECt2
KINETIC ENEROT ............ *- 3B_411805 IN-L_5
I_,00_P KWH
WEIGHT ...................... 106P*_l U_
[N_PB¥ DENSITY .... - ......... 359954.0_ IN-bBS#LB
11,3 #H/bB
SHAPE FACTOR ...... -'- .... **" .9280
RADIAL GR0_TH ............... *463714[*01 INCHES
J
_A_IUS THICKNESS TENPERATUR[ TANGENTIAL RADIAl.
(IN) (IN) (DIG*F) STRESS STRESS
(PSI) (PSl}
Pl.O000 1._658 *0 66_48 0
P0.50_0 I*_ASR *0 7el39 13118
PO*_O00 1._658 *0 77789 860_5 ,_
1_.5000 I._5_ *0 _3184 381g9
19.0000 I*PAS8 *0 88314 51_40
I_.5_00 I-P65_ *0 9316_ 63569
I_*000_ 1*_65q *0 97734 TSTeT
IT,ShOO I._65_ .0 101999 811_q _.,11.0000 I._65_ *0 105946 99589
16.S000 1.3466 *0 I071_9 104940
16-0000 1.4914 .G 108093 105571
15.5000 1.6461 *0 I08_85 I061_8
15.0000 1.81P4 *0 108519 I066ei
14.5000 I*9885 *0 !081_9 101058
1400000 _*ITA8 *0 108911 107446
13,50_0 _.3110 *O 109086 101791
13.0000 _.577P *0 109638 108099
1_,5000 _.79_3 ,0 109315 I08313
!_*0000 3.0158 *0 109499 I08619
II.50_0 3._471 ,0 109610 I08838
I1.0000 3.4_50 ,0 109111 109035
10.50_ 3.7P87 .0 !0980_ 109_11.
I0.0000 3.9169 *0 !098A5 109369
- 9*SflO0 4,_883 .0 109959 109511
9*0000 4.4915 *0 II00_1 109639
8.500_ 4*7350 *Q I10008 109153
_,0000 4.9_10 *0 110144 109856
1.5000 S.P_60. *0 110194 |0994_
1*flO00 S*480P *0 110839 110031
6,5flflO S.7170 *0 llOe?9 110104
6,0000 5.9410 *0 110316 IIOI10
5*5000 6 1659 *0 110346 110889
_,0fl00 6.31_R *0 110311 110881
4,5000 6.5660 *0 110403 1103_6
; 4*0000 6.143_ *0 110495 110366
3*5000 6.9fl41 "0 110444 110400
3.0000 7.041_ *0 110461 I10A_9
_- t.5_0 1.1701 *0 110414 :10454
_'_ _*flO00 7.1vlrEP .0 110484 110414
_* !.5000 ?*35_1 *0 1104_1 110490
4 1,0000 1.4101 *O 110493 110506 '
*5000 9.4451 ,0 110465 110543
i *0000 I*4514 *0
5PL Technical Memorandum 33-777 45
1976021184-052
Ii
1.0
A8YMPTOTE
0,5 _
I I i I I I i I _.1 I
4 6 S I0 _ 14 16 18 20
HUB/Tn=RATIO
Fig. I. Flywheel shape factor vs. flywhee! hub/tip
thickness ratio
46 3PL Technical Memorandum 33-777
/
1976021184-053
CONrrANT&
lfl[Nrt_ ENERGY - 1Jr
FLYWHKKLHT_ - 1N KI
JPL Technical Memorandum 33-777 47 ,t I.
1976021184-054
_"_-'-----"--_-N-'---'_ _TOUR IDE'NT
2 u_IE_:=,.._'t_,,"_,7 ,__ s.ooxlo
I,OOXIO œ1.50 l $2.0 Xl0 +04
2._OXIO +04
_._ 300XlO +048 3.50XI0 *04
.OOXlO ¬A 4.50X10 +04
B 500XlO +04
C 5,50Xl0 0D 6.0 O ´E x10 +04
F 7.00XlO *04
G 7.50X10 @H 8 0 lO ÀI +04
J 9.00XIO _04
K 9.50X10 @HL I.OOXIO ÌH 1.05 l PN I.I XlO Ð0 1 lO Fig. 3. Contour map of R-stress
\ k)/ ,Pf,(,I I ]
vvvv
CONTOUR |D[NT
I O.OOXlO *00
2 5,00XlO *03
3 I.OOXlO +04
4 I .50XlO *Oq
_.OOXlO +04
§ _,50Xl 0 '.04
7 3.00X|O œ%8 50 iO*O_
9 4.00XlO *u't
A 4.50X10 *04
B S.OOXlO *04
C 5,50X10 *04
O 6.OOXlO *04
[ fi.SOXlo*O. _
F 7.00XlO *04
0 '/.50XlO +04
H §.OOXlO *Oq
I 11.50x10 *04
J Sl.OOXl0*0.*
K 9.50X10 _04
L I .OOXlO_05
H I.OSXlO .*'Os
N l.lOXlo++O__
0 I. 15XlO .*'0_
Fig. 4. Contour map of T-stress
48 /PL Technical Memorandum 33-777
)5
1976021184-055
0+0_
I -2._QXl 0+0_ ,2 -2 30}([ -,.
3 -_aoxI°+"7
-2.00xlO
+-,.,o_,o:°_
"7 t BOXIO -,
9 I 60XtO i
- " '*++O_
A -I 50XtO -.
,,-,.'+°,,°++o:
C I 30X10
- ' c 0 I aOxtO -.
F t OOXlO -
° -+.00,,0:,_
m 8 ooxlo -
- . 0_,03!-"t.OOXt +03
j -60oxtQ --
+ L "_ OOXtO__
N 2 OOXtO --
. . 4.U .Ii
0-100XtO ..
o ,.oo,.o_,;
R 200XtO ..
• |o,UZS 300X
, ,.00,,0:_;
U _ OOXiO ..
,, +.00_,0;;;
t4 "1 OOXlO_._
,f g OOXIO -.
z __00x_°'-°-_
I. IGx IO',+v*
ELg. 5. ContOUr map o£ Z.stress
, !
or)
N
I
............................................................... •..................................................................... I
R-AXIS
IOQQQQ
rY ." I
I
' 1
'" '1 J-AXIS
....... - ........................ 1 I 1
100000 • 1
1
L..- l
1
I
I
I
60000 i
LJ.J / I
(.¢)1-"_oooo / ]1
I i/: 1
C]_ _0000 / 1] i
.; [ I 1 I
sO 3o L=O IO ;_
I .........'-,r J_^xis
Fig. 7. Radialand tangentialstressdistribution
50 JPL TechnicalMemorandum 33-777
?
1976021184-057
oo _
x
I
N
i
.................................................................................................. ?T,.t........................... T..t *_
Fig. 8. Axial stress and axial shear stress distribution
JPL Technical _4emorandum 33-777 51
Y
1976021184-058


Shape optimization of disc-type flywheels

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