THE SIMULATION OF AERIAL MOVEMENT-II. A MATHEMATICAL INERTIA MODEL OF THE HUMAN BODY M. R. YEADON Biomechanics Laboratory. Faculty of Physical Education, University of Calgary, Canada Abtraet-A mathematical inertia model which permits the determination of personalized segmental inertia parameter values from anthropometric measurements is described. The human body is modelled using 40 geometric solids which are specitkd by 95 anthropometric measurements. A ‘stadium’ solid is introduced for modelling the torso segments using perimeter and width measurements. This procedure is more accurate than the use of elliptical dixs of given width and depth and permits a smaller number of such solids to be used. Inertia parameter values may be obtained for body models of up to ?Osegments. Errors in total body mass estimates from this and other models are discussed with reference to the unknown
ments of a living individual
ISTRODUCTION
1968: Hay, A computer movement
simulation requires
model
the input
of human
airborne
of information
1973; Hatze.
(Bouisset and Pertuzon,
1975: Tichonov.
speci-
ing moments
of inertia of a central segment such as
the pelvis, or for determining
ation during
B limb about its longitudinal
of the
human body. The initial motion and the time history of body configuration
can be described
using orien-
tation angles, while the mass distribution
of the body
can be described using length. mass and moment inertia
descriptors
of the body segments. From
input information
P simulation
Mathematical segments
Whitsett
model is able to deter-
Hatze
The values of the orientation
angles used as input movement,
from film data of an actual inertia
chosen to be representative may be personalized
(1980).
performance
uniform
as
parameters
may be
of an ‘average
man’, or
estimates for B particular
Although
piirameters
have been calculated anthropometric
that
to identify segmental
indi-
for itn average
using the mean
measurements
ol. (1954) as input
been developed
(1964).
by
Jensen (1976) and
the detail
with which
body
these assumptions
mim
given by Hertzberg
will
since there are no criterion inertia parameters
the calculated
such as
It may be
lead
to sys-
values.
also be expected
cr
the calculated
(1963).
as input
values of the
with which to compare
model presented in this paper is
also based upon simplifying
values of the
to the model of Whitsett
assumptions
tematic errors in the models. Such errors are diliicult
The mathematical inertia
have
Hanavan
density over a given cross-section.
expected
vidual. Segmental
the body solids are
values for all segmental inertia
Such models
(1963).
represent
of geometric
above models make simplifying
or may be
1 of this series (Yeadon. 1989a). The
values of the segmental
axis.
which
using a number
parameters.
this
the moment of inertia of
segments are modelled can be quite complex, all of the
may describe a hypothetical described in Part
models
capable of estimating
of
mine the resulting motion.
derived
1976). How-
ever, none of these techniques is suitable for determin-
fying the initial motion, the changes in body configurtlight and the mass distribution
lung volumes.
assumptions
to contain
systematic
and may
errors. Since
inertia values will subsequently
to a simulation
model,
be used
the effect of such
Daniels (1952) showed that if a subject is regarded as
errors can be evaluated and there exists the possibility
average for a particular
anthropometric
of adjusting
when his measurement
lies between the 35th and 65th
measurement
the calculated
values to minimize
errors. The model described
percentiles, then not one of the 4063 subjects of Hertz-
be regarded only as an initial
berg er al. (1954)
ation of accurate segmental
was average
linear measurements. mim’
may
fraction
in all of IO selected
be representative
of only
a very
As a consequence,
values.
small
THE HUMAN BODY MODEL
it will be
necessary to consider a range of inertia values when to generalize
results of computer
lations. There are a number of experimental determining
stage in the determin-
inertia parameter
This indicates that an ‘average
of a population.
attempting
these
below should therefore
the inertia
parameters
The
simu-
inertia
model
described
signed to produce segmental
below
for input into the I I segment simulation techniques for
cribed
of the body seg-
in Parts
1989b; Yeadon
111 and et al.,
IV
has been de-
values of an individual model des-
of this series (Yeadon.
1989). In the I
I segment simu-
lation model it is assumed that the segments are rigid bodies and that Receiced in
final firm February 1989.
no movement
occurs at the neck,
wrists or ankles. Such assumptions may be regarded 67
M. R. YEAOON
68
as weaknessesof the modelsince they limit the versatility of the model. However, if it can be shown that the agreement between simulations and performances of aerial movements is good despite these simplifying assumptions, the assumptions may be regarded as adequate. Although the inertia model is described in terms of the 1I segmentsof the simulation model, it should not be regarded as an I1 segment model. Each segment is composed of a number of subsegments and it is a simple matter to regard the head, hands and feet as individual segments and to output values for 16 segments. Indeed
the hands and feet may each be re-
garded as comprising
two segments so that segmental
inertia parameters for 20 body segmentscan be calcu-
lated by the method. Geometrical
representation
Earlier models (Whitsett. 1963; Hanavan. 1964; Jensen, 1976; Hatze. 1980) have used ellipses to model cross-sectionsof body segments because it was mathematically convenient to do so. Figure I compares a cross-section of a thorax obtained by Cornelis et al. (1978) with the stadium shape of Sady et af. (1978) and
a
_.-----.___.-...._
-.
.’ ,
‘.
,’
:
-, :
1 I
,
‘. --..
b
Cd
- . . . . .---.-.__..-
.-’
____. . . . _ . _ -__ I’
:
**\
,I
\ ‘1
:
I
)..
I
:
.
\ \
,
,
\ *.
; --,
,’
I
.’
1..
___.._._.
an ellipse. It is apparent that the stadium bears more resemblance to the thorax cross-section than to an ellipse of equal width and depth. Sady et al. (1978) used solids bounded by two such stadia of different dimensions to calculate volumes of body segments.Such a procedure may be expected to give a better representation than the use of cylindrical solids, since the cross-section of the stadium solid is permitted to vary. However, the formula used by Sady et al. (1978) is based upon the volume of a truncated pyramid and this gives the correct volume only when the bounding stadia have the same ratio of width to depth. A stadium may be defined as a rectangle of width 2t and depth 2r with an adjoining semi-circle of radius r at each end of its width (Fig. 2). The perimeter p = 4t + 2nr and width w = 2t + tr so that r and t may be calculated from perimeter and width using the equations: r = (p - Zw)/(Zn - 4)
and
t = (nw - p)/(2n
- 4).
It should be noted that using the perimeter and width as input measurements, rather than depth and width, reduces the error in the estimation of crosssectional arca arising from the use of a (somewhat) arbitrary geometrical shape (Yeadon, 1984). A stadium solid bounded by parallel stadia (Fig. 3) is defined in Appendix I and it is shown that crosssections which arc parallel to the two bounding stadia are also stadia. Formulae for the mass M. location of mass centre i and principal moments of inertia /,.f,,f, about the mass centre are derived for a stadium solid in Appendix 2. When the widths of the rectangles of the bounding stadia are zero a stadium solid becomes a truncated cone. In such a case the formulae for the inertia parameters become indeterminate. This difftculty is overcome by adding a very thin trapezium to a truncated cone so that the formulae for a stadium solid can be used. In the inertia model all body segments, with the exception of the cranium, are represented by a number of stadium solids or truncated corms. The cranium is modelled as a semi-ellipsoid of revolution. This solid is produced by rotating a quadrant of an ellipse with semi-axes r and h about the one axis to produce a semi-ellipsoid with base radius rand 2t 4
+
C e---w_ _.--
-.
.
-.
I’
I’
:
1 ,
\
\
\
..
5 I ,
\ \
‘.
-. --_i_-
.
.’
- *-*
of (a) thorax cross-section (adnptcd from Cornelis Y[ a/., 1978). (b) stadium. (c) ellipse.
Fig. I. Comparison
cilB31
Fig. 2. A stadium is a rectangle of width 2~ and depth 21 with an adjoining semi-circle at each end of its width.
69
Tbc simulation of aerial movement--II
B are each sectioned into seven solids oi and bi (i = 1.7) while the left leg J and right leg K are each sectioned into nine solids ji and ki (i = 1.9). The 11 segments of the simulation model of aerial movement described in Part IV of this series(Yeadon et ul., 1989) comprise the solids listed in Table 2. For each segment the mass, location of mass ccntrc. principal moments of inertia about the mass ccntre and distance between joint ccntrcs are calculated. It is assumed that the solids comprising a segment have coincident longitudinal axes. The values for the left and right limbs arc then averaged since the simulation model is designed to have symmetrical inertia values (Ycadon et al., 1989). Fig. 3. A stadium solid.
I Table 1. The kvels at which the body segments are sectioned
Torso S M Lsl Ls2 Ls3 Ls4 LsS M Ls7 Ls8
hip joint centre umbilicus lowest front rib nipple shoulder joint centre acromion beneath nose above ear top of head
Left arm A
&Cl shoulder join! centre Lul La2 Lu3 Lu4 La5 Lo6 La7
mid-arm elbow joint centrc maximum forearm perimeter wrisl joint cen(rc base of thumb knuckles fingernails
Measurements The levels in Table I are measured using anthropometric calipers so that the heights of the 40 solids can be calculated. At each level the perimeter is measured using a measuring tape. At the shoulder level Ls4 of the torso a depth measurement is taken since it is Table 2. Segment Chest-head Thorax Pelvis Lefl upper arm Left forearm-hand Right upper arm Right forearm-hand L&Cthigh Left shank-foot Right thigh Right shank-foot
Symbol
Solids
C T P
s4, 55. s6. s7. st s3 sl. s2 al. a2 a3. a4. US, u6. a7 hl, b2
A, AI B, B1
J, J, K, KZ
b3. b4. bS. h6. b7 /L/2,13
14. $.,16.rki8.
~9
k4. kS. i6, i7. k8, k9
tcrt leg J
LJQ hip joint cCntre Ljl Lj2 fj3 Lj4 LjS
crotch mid-thigh knee joint antre maximum calf perimeter ankle joint antre
Lj6 hal Lj7
arch
Lj8 ball LJ~ toe nails
height h. Formulae for the inertia parameters of a semi-ellipsoid are listed in standard enginkring texts (e.g. Meriam, 1971). Segmentation The body segments are sectioned into 40 solids by planes perpendicular to the longitudinal axes of the segments(Fig. 4). The levels at which the segmentsarc sectioned are given in Table 1. The levels Lsi (i = 0.8) section the torso S into eight solids si (i = I,@ where the solid si is bounded by the levtls Lsi and Ls(i - I). The left arm A and right arm
Fig. 4. Sectioning of the torso S. left arm A, right arm B. lcrt leg J and right leg K into 40 solids.
70
M.
R.
not possible lo measure the perimeter directly. For the shoulder levels LOO, L60 of the arms the perimeters are taken as high up the arms as possible. The solids representing the head, arms and legs are assumed lo be of circular cross-section so that no further measurements are needed. The remaining solids of the torso. hands and feet are stadium solids and so width measurements are taken at the boundary levels. At the heel levels Lj6, Lk6 anterior-posterior depths rather than medio-lateral widths are measured since these values correspond lo the widths of the stadium cross-sections.The foot arch levels Lj7. L&7 are assumed to be circular in cross-section to permit the transition from stadia which are aligned anterior-posteriorly lo stadia which are aligned medio-laterally. The 95 measurements taken comprise 34 lengths. 41 perimeters. 17 widths and three depths and require between 20 and 30 min of the subject’s time. Density values The density values used in the inertia model are taken from Dempster (1955) and are listed in Table 3 together with the corresponding solids. These values are used in preference to those obtained in the cadaver studies of Clauser et al. (1969) and Chandler et al. (1975) since only Dempster found individual values for shoulders, thorax and pelvis. It should be noted that the value for the density of Dempster’s thorax segment is used with solids s3 and s4 whereas segment T of the simulation model, which has been named the thorax segment for convenience, comprises only ~3. APPLICATION
The method was applied to three subjects (two males, one female) prior to performances of twisting somersaults on trampoline. Table 4 compares total body masses, determined by weighing, with the estimates obtained using the inertia model. DISCUSSION
YEADON Table 4. Accuracy of total body mass estimates Subject
Mass (kg)
A
60.0 60.9 64.3
Estimated
mass (kg)
58.8 62.0 65.8
Error - 2.0% 1.8% 2.3%
parable with the value 1.8% obtained by Jensen (1978) but is much greater than the value 0.5% obtained by Hatze (1980). When measuring torso perimeters there is the problem of changes due to breathing. In this study an attempt was made to minimize this effect by asking the subjects lo breathe shallowly. However, it may be expected that when the lungs contain an additional one litre of air, the volume of the torso increases by one litre and this will produce an increase in the total body mass estimate of about 1.5% for a 70 kg subject. Thus, even if the amount of air in the lungs is held approximately constant and the torso volume estimte is accurate, there appears to be an uncertainty in the torso mass estimate of the order of 1% without a method of determining the amount of air in the lungs. With this in mind, the errors in the total mass estimates of this study and of Jensen (1978) are not unreasonable. The accuracy with which the model estimates total body mass, however, does not give much indication of the accuracy with which segmental masses and inertias are estimated. The reason for considering total body mass is that it is the only quantity for which a direct measurement is available. Since the inertia model has been designed lo produce personalized segmental values for input into a simulation model, the inertia model may be considered adequate for this purpose providing there is good agreement between simulations and actual performances. Such evaluations will be made in Parts III and IV of this series (Yeadon, l989k Yeadon er al., 1989) where the effects of anthropometric and film measurement errors will also be determined.
The maximum error of the total body mass estimates given in Table 4 is 2.3%. This value is comREFERENCES Table
Segment
3.
Segmental density values (Dempster,
1955)
Segmental density values (kgl”) Solids
Density
Head-neck Shoulders Thorax Abdomen-pelvis Upper arm Forearm Hand Thigh
Lower leg Fool
~6, sl. s8 ss s3. s4 sl. s2 al. a2 a3. a4 a5, a6. a7 jl.jZ.j3
j4. iS j6. j7. j8. j9
I.11 I.@4 0.92 I.01 I .07 I.13 I.16 I .05
1.09 I.10
Bouisset. S. and Perluzon. E. (1968) Experimental determination of the moment of inertia of limb segments. Biomechunics I (Edited by Wartenwciler, J.. Jokl, E. and Hcbbelinck, M.). ppq 107-109. Karger, Bascl. Chandler. R. F.. Clauser, C. E.. McConville, J. T.. Reynolds, H. M. and Young, J. W. (1975) InvcsGgaGon of inertial orooertier of the human bodv. AMRL-TR-74-137. ADkOi6-485. DOT-HS-801-430.V Aerospace Medical Rcsearch Laboratories. Wright-Patterson Air Force Base. Ohio. Clauser. C. E., McConvillc. J. T. and Young. J. W. (1969) Weight. volume and center of mass of segments of the human body. AMRL-TR-69-70. Aerospace Medical Research Laboratory. Wright-Patterson Air Force Base, Ohio.
The simulation of aerial movement-II
Corn&. J, Van Gheluwe. B.. Nysen. M. and Van den Berghe. F. (1978) A photographic method for the tridimensional reconstruction of the human thorax. Proceedings of the NATO Symposium on Applications of Human Biostereometrks (Edited by Cobknt A. M. and Heron. R. E.). Procredincls of rhe Sociery ol Photo-opricol Instrumentot~onEn&&s I&, pp. 2&G. Danicls. G. S. (1952) The averaae man. Technical Note WCRD-53-7. AD-lb203. WrightAir Development Center, Ohio. Dempster. W. T. (1955) Space requirements of the seated operator. WADC-55-159. AD-087-892. Wright-Patterson Air Force Base. Ohio. Hanavan. E. P. (1964) A mathematical model of the human bodv. AMRL-TR-64-102. AD-608-463. Aerosnace Medical Research Laboratories, Wright-Patterson Air Force Base. Ohio. Hatzc, H. (1975) A new method for the simultaneous measurement of the moment of iner,tia. the damping coefficient and the location of the centre of mass of a body segment in situ. Eur. 1. appl. Phys. 34.217-226. Hatze, H. (1980) A mathematical model for the computational determination of parameter values of anthropomorphic segments. 1. Biomechunics13. 833-843. Hay. J. G. (1973) The center of gravity of the human body. Kinesiology 111 (Edited by Widule, C.). pp. 20-44. American Association of Health. Physical Education and Recreation, Washington. DC. Hertzberg. H. T.. Danicls. E. and Churchill. E. (1954) Anthropomctry of flying pcrsonncl. WADC-TR-52-321. Wright Air Development Center, Wright-Patterson Air Force Base, Ohio. Jensen, R. K. (1976) Model for body segment parameters. Biomuchanics Y-f3 (Edited by Komi, P.V.). pp. 380-386. University Park Press. Baltimore. Jcnscn. R. K. (1978) Estimation of the hiomcchanical propertics of three body types using a photogrammctric method. J. BiomechunicsI I. 349-358. Mcriam. J. L. (1971) Dynumics. 2nd Edn. p. 475. Wiley, New York. Sady, S.. Freedson. P.. Katch, V. L. and Reynolds, H. M. (1978) Anthropomctric model of total body volume for males of dibrcnt sizes. Ifuman Biol. 50, 529-540. Tichonov. V. N. (1976) Distribution of body masses of a sportsman. Biomechunics Y-B (Edited by Komi. P. V.). DD. 103-108. Univcrsitv Park Press. Baltimore. Wi&ett. C. E. (1963) So& dynamic r&onsc ch&actcristics of weightless man. AMRL-TDR-63-18. AD-412-541. Acrospacc Medical Rcscarch Laboratories. WrightPatterson Air Force Base, Ohio. Yeadon. M. R. (1984) The mechanics of twisting somersaults. Ph.D. dissertation, University of Loughborough. Ycadon. M. R. (1989a) The simulation of aerial movcmentI. The determination of orientation angles from film data. J. Biomechanics 23. 59-66. Yeadon. M. R. (1989b) The simulation of aerial movcmcnt111.The determination of the angular momentum of the human body. 1. Biomechanics 23, 75-83. Ycadon, M. R.. Atha. J. and Hales, F. D. (1989) The simulation of aerial movcmcnt-IV. A computer simulation model. J. Binmechonics23. 85-89.
APPENDIX 1. SlADlUhl
SOLID
A stadium solid bounded by parallel stadia (Fig. 3) is defined as follows: let the lower bounding stadium have parameters r,, and I,. the upper bounding stadium parameters r, and I, and Ict h be the distance separating the stadia. Consider a point P which lies on the boundary of the first quadrant of a stadium with parameters r and I (Fig. 5). If P
71
Y
t+r Fig. 5. A quadrant of a stadium.
has coordinates (x, y) then: forO~x~r.
.v-L1(0~1~1)
r
x=r+rcosO
for x >
and
_v=r;
and
y = rsin 0 (0 I H 5 n!2). A correspondence may now be defined between the points on the boundaries of the first quadrants of the upper and lower stadia. Points on the straight sections correspond when they have equal L values and points on the curved seclions correspond when they have qua1 0 values. In a similar way, a correspondence may be defined for the remaining quadrants. Suppose that Pn and P, arc corresponding points of the lower and upper bounding stadia (Fig. 3). if p0 and p, arc Position vectors of P, and P, relative to some origin. then the point P with position vector p = (I - :)p,, + :p, lies on the line segment P,P, when 0 s I S I and divides it in the ratio ,-:(I - z). If : is held constant and P, and P, move around the lower and upper stadia. the point P moves around the boundary of an intermediate section of the stadium solid. This procedure defines the stadium solid and it will now bc shown that each intermediate section is itself a stadium and is parallel to the bounding stadia. Let the coordinate system (x.J.:) have origin at the centrc of the lower stadium with the : axis lying along the line joining the centrcs of the bounding stadia (Fig. 3). The points PO(ir,.r,.O) and P,(lf,.r,,h) on the straight scclions correspond 10 P(( 1 - r)lf, + :i.f, .( I - -_)r0 +:r,.:h) which is quivaknt to P(i.t(z),r(:).zh) where I(Z) =(I-:)I,+:I, andr(z)=(l-:)r,+zr,. Similarly, the points PO( r,, + r0 cos 0. r0 sin 0.0) and P, (I, + r, cos&r, sin U.h) correspond to P(f(Z) + r(,_)cosO. r(z)sinU..-h) where I(:) and r(z) arc as above. Thus. as P, and P, move around the bounding stadia the point P moves around a parallel stadium which has parameters I(:) and r(z) and is at a distance ,-h from the lower bounding stadium. It should be noted that although the above treatment does not assume that the : axis is normal to the bounding stadia. only right stadium solids will be used.
APPENDlX 2. FORMULAE FOR THE lSERTlA PARAMETERS OF A STADIUM SOLID
The derivations of formulae for the location of mass ccntre. mass and principal moments of inertia of a stadium solid are given below. Prior to consideration of the solid, expressions are derived for the second moments of arca of a stadium. Srcond moments of area n/a
sradium
The calculation of the second moments of area of a stadium will require the evaluation of a number of integrals. For convenience these definite integrals will be evaluated
72
M. R.
prior to the calculation of second moments.
YEADON
ThUS
I,
= 4r’r/3
+ ~74,
and
Ix’
ydx
= [
x’ydx
+ I;,,.
x2yfd8
(I +rc~sO)~~~sin~ OdO
+2rd.x+[~=
+ cos2 2O)dO 112s
4
[I
J0
+t(l
=
-2cos20
J II
+ 2r’r[l/3]
+ r’[a/l6].
Thus J, = 4nJ/3 112s
-sin’0 [ 3 10
+ 8rJ1/3 + nr’/4.
For a lamina the theorem of pcrpcndicular axes states that
=3
J, = J, + J,.
(I - sin2 0)sin’ OdO
0 sinzOdO= t
+ w212
,
1121 cos2
I
+ r212[n/4]
sin2 O*d(sin 0)
J0
I,Z~
2r’t
= r[IS/3]
I =
sin2 OdO
sin20cosOd0 J0 Ilam cos20 sin’Od0 + I4 J0
+
I6 112.
r2P
+
112.
+cos4O)]dO
3n
sin20cosOdO=
r[x’/3yO
0
=l/2_
J
112s
I =-
Thus
0
0
J, = 4rI’/3
112m
sin2 (IdO
m I
+ arat
+ 4r’t + ~‘12.
These expressions for A. J,. J,. J, will bc used in the derivations of the inertia parameter of a stadium solid.
0 II2rn
sin4 OdO -I
0
n 3% =---a--. 4 I6
n 16
A point tics on the boundary of the first quadrant of a stadium (Fig. 5) providing either 0 s x I; I and y = r or x~1+rcosOandy=rsinO(OJO4n/Z). Thus the area of the stadium is A where: yld0 4
where
dx i =do
The muss and locarion o{mass cenlre o/a srarlium solid
It has been shown that the intermediate stadium of Fig. 3 is at a distance zh from the lower bounding stadium and has parameters r and I where r = r. + z(r, -ro) and I = 1, t z(,, -r,). Thus r and I may bc expressed as r = r,(l t a~) and I 3 Io(I + br) where a = (r, - ro)/ro and b = (t, - I,)/I,. If the intermediate stadium has density D. area A(r) and thickness h6r the mass will bc DA(r)hdr so that the mass of a stadium solid of uniform density is given by I
,,2r
M=
DA(r)hdr. I
= [Lrdx
+ r2~~‘sin20d0 The
I
I
= rl + -f’n. 4
Thus A = 4rr + nr2 as expected. Let J,. J, and J, be the second moments of area of the stadium in Fig. 5 about the x. y and : axes where the z axis is orthogonal to both the x and y axes.
first
0
moment of mass about the lower plane is
hrDA(r)hdr and the distance i of the mass ccntre from I thOelower plane is given by:
I i=
J0
Dh*rA(t)dr/M.
Defining the functions FI. F2, F3 by the equations: Fl(a.h) = I t (a + b)/2 + ah/3 the integration occurring over a single quadrant. Now
Jy~dr=J~y~d,~+J~,2,y’*do
FZ(a.b)
= I+
f3(a.b)
= f + (at
t
b)/3
t
ah/4
b)/4 + ah/S.
Area A(:) =4rr + ar* =4ro(l
=lr’dx
(a
+az)r,(l
+bz)tnri(l
x r’~~‘sin’OdO = 4r,,I,( I + (a + b)r
= ,JI
t 3nr’/l6.
+ ~$1
t
t 2ar t 0’2~)
abz2)
taz)2
73
The simulation of aerial movement-II
so that
and
I 1 + (a + b),‘?
Ad: = Jr&(
FS(a.b)
= 1+ (2~ + 2b)/2 + (a2 + 4ub + b2)/3
+ ab/3)
+ Zab(a + b)/4 + a2 b’/S.
j 0 Now
+ nrt( I + Za,‘? + a1 /3) = 4r,f,Fl(a.b)
+ nr~Fl(a,a)
rf’
1 Ad:.
M=Dh
j 0
= r&l
+a:)(1
= r,ri(l
+ a.-)(1 + 36: + 3b2:*
= r,&l
+ (a + 3b)z + (3ab + 3b’)?
+ (3ab’
Thus M = Dkr,[4t,Fl(a,b)
+ nr,Fl(a.a)J.
+ b:)’
+ b’)?
+ ab’r’).
so that
I
Now
rt’d; ,-A = 4r,f,(;
+ (a + b).?
+ b’?)
=r,fiF4(a,b).
j0
+ ab:‘)
Similarly + ffr$:
+ 2a?
+ oz.?),
I
so that
r’td.- = rif,F4(b.a), j0
:Ad:
= 4r,fo
whilst
i + (a + b)/3 + ab/4 >
0
+ =
(i+ +
nri
2al3
4r,f,F2(u.b)
rzt2 = &;(I >
+ nriF?(a.a)
Dh2
+b:)’
+ ?a: + a2:‘)(l
+ 2b: + b’?)
I + (2~ + 2h); + (a* + 4ab + b’)?
= rif$
+ _ ‘uh(u
I it
+a:)‘(1
= riri(l
aa/4
+ b)-’-
+ a* bf --’
1
so that
:Ad:/hl.
I
j 0
Thus
rzr’dz = riri$FS(a,h) = = Dh’[G,r,F2(u.h) _
j0
+ nriF2(a.a)]/M. and
Note idso :j A = 4r,r,(:’
r’ = rt(l + a:)’
+ ((I + h)? + U/J.-~)
+ nr:(:’
+ 2~’
=r~(I+4u:+Q’:‘+4aJ.-‘+u4~~)
+ a’:‘).
so that so that
I
I
r’d:
:* Ad: j I)
= riM(a.u).
j0
= 4rofa( f + (U + b)/4 + ub/S)
Thus + nr:( i + 2~14 + a’/S).
i.e.
I: = Dh[4rof~F4(u,b)/3
+ nr~f~FS(a.b)
+ 4r~f,F4(b,u)
+ nr4,F4(a.a)/2].
I :‘Ad:
= 4r,r,F3(u,b)
+ nr:F3(u,a).
j II This result will be used in conjunction with the theorem of parallel axes in the next section to calculate momrnts of inertia. Momrn~s oJ invrriu u/u sfudium
solid
In thisxxtion cxprcssions are derived for the moments of inertia I,. I,. I, of a stadium solid about principal axes through the mass ccntrc. The moment of inertia of a stadium solid about the : axis
I
(Fig. 3) is I, =
The moment of inertia of a stadium section about the p aais (Fig. 3) is given by the theorem of parallel axes as J,Dhd: +(h:)*ADhd: where Jr is the second moment of arca. D the density. htS: the thickness. h-_ the distance from the y axis and A the area. If /T dcnotrs the moment of inertia of the stadium solid about ths )’ axis (which lies in the lower fttcc) then: ,T = j;J,Dhd: i.e.
I
DJ,hd: whcrc the second moment of areas
1; =
j 0 of a stadium has been obtained as J, = 4rfJ/3
I
+ nr2fa + 4r’f
f nr’j2.
t J,d I =
+4j~r’rd:+~aj~r’d:].
j0
where:
t + nj:r’r’d:
:’ Ad:.
J,d: +
Dh
j 0
Thus I, = Dh[(4,3)j:rr’d:
+ j;(hr)‘ADhd.-.
j 0
(4rr’/3
j 0 = 4r 00f’FJ(a
+ nr2f2 + 8r3/3 + nr’/4)dr
’ b)/3 + 8r:r,F4(b.u)/3
+ nrir:F5(a.b) + nr:F4(a.a)/4
using the results obtained during the calculation of i:. and
I
Define
:‘Ad,F4(a.b)
=
I + (a +
3b)/2 + (Sob + 3bz)/3
+ (3ah’
+ b’)i4
+ ub’J5
= 4r,f,F3(u,b)
+ nriF3(a,
u)
j 0 having
been derived in the section on the calculalion off.
14
M. R. YEADON
If I, denotes the moment of inertia about a parallel axis through the mass cen~re then the theorem of parallel axes
i.e.
gives:
,.=Dhj~l.d:+Dh’j~I’Adl.
/ = p - .\f=‘. I
Similarly.
the moment
*
where
-
of inertia
1 I:
about
the x axis is
J,d: j 0
given by:
I =
(4rcJ/3 + xr’/cl)d: j 0
= 4r,r:f4(4(a.b)/3 + nr~F4(a. a)/4 ,; = j+Ihd:
+ j)tr,‘ADhd:.
and the moment of inertia I, about a parallel axis through the mass centre is given by I, = 1: - Mf’.
ments of a living individual
ISTRODUCTION
1968: Hay, A computer movement
simulation requires
model
the input
of human
airborne
of information
1973; Hatze.
(Bouisset and Pertuzon,
1975: Tichonov.
speci-
ing moments
of inertia of a central segment such as
the pelvis, or for determining
ation during
B limb about its longitudinal
of the
human body. The initial motion and the time history of body configuration
can be described
using orien-
tation angles, while the mass distribution
of the body
can be described using length. mass and moment inertia
descriptors
of the body segments. From
input information
P simulation
Mathematical segments
Whitsett
model is able to deter-
Hatze
The values of the orientation
angles used as input movement,
from film data of an actual inertia
chosen to be representative may be personalized
(1980).
performance
uniform
as
parameters
may be
of an ‘average
man’, or
estimates for B particular
Although
piirameters
have been calculated anthropometric
that
to identify segmental
indi-
for itn average
using the mean
measurements
ol. (1954) as input
been developed
(1964).
by
Jensen (1976) and
the detail
with which
body
these assumptions
mim
given by Hertzberg
will
since there are no criterion inertia parameters
the calculated
such as
It may be
lead
to sys-
values.
also be expected
cr
the calculated
(1963).
as input
values of the
with which to compare
model presented in this paper is
also based upon simplifying
values of the
to the model of Whitsett
assumptions
tematic errors in the models. Such errors are diliicult
The mathematical inertia
have
Hanavan
density over a given cross-section.
expected
vidual. Segmental
the body solids are
values for all segmental inertia
Such models
(1963).
represent
of geometric
above models make simplifying
or may be
1 of this series (Yeadon. 1989a). The
values of the segmental
axis.
which
using a number
parameters.
this
the moment of inertia of
segments are modelled can be quite complex, all of the
may describe a hypothetical described in Part
models
capable of estimating
of
mine the resulting motion.
derived
1976). How-
ever, none of these techniques is suitable for determin-
fying the initial motion, the changes in body configurtlight and the mass distribution
lung volumes.
assumptions
to contain
systematic
and may
errors. Since
inertia values will subsequently
to a simulation
model,
be used
the effect of such
Daniels (1952) showed that if a subject is regarded as
errors can be evaluated and there exists the possibility
average for a particular
anthropometric
of adjusting
when his measurement
lies between the 35th and 65th
measurement
the calculated
values to minimize
errors. The model described
percentiles, then not one of the 4063 subjects of Hertz-
be regarded only as an initial
berg er al. (1954)
ation of accurate segmental
was average
linear measurements. mim’
may
fraction
in all of IO selected
be representative
of only
a very
As a consequence,
values.
small
THE HUMAN BODY MODEL
it will be
necessary to consider a range of inertia values when to generalize
results of computer
lations. There are a number of experimental determining
stage in the determin-
inertia parameter
This indicates that an ‘average
of a population.
attempting
these
below should therefore
the inertia
parameters
The
simu-
inertia
model
described
signed to produce segmental
below
for input into the I I segment simulation techniques for
cribed
of the body seg-
in Parts
1989b; Yeadon
111 and et al.,
IV
has been de-
values of an individual model des-
of this series (Yeadon.
1989). In the I
I segment simu-
lation model it is assumed that the segments are rigid bodies and that Receiced in
final firm February 1989.
no movement
occurs at the neck,
wrists or ankles. Such assumptions may be regarded 67
M. R. YEAOON
68
as weaknessesof the modelsince they limit the versatility of the model. However, if it can be shown that the agreement between simulations and performances of aerial movements is good despite these simplifying assumptions, the assumptions may be regarded as adequate. Although the inertia model is described in terms of the 1I segmentsof the simulation model, it should not be regarded as an I1 segment model. Each segment is composed of a number of subsegments and it is a simple matter to regard the head, hands and feet as individual segments and to output values for 16 segments. Indeed
the hands and feet may each be re-
garded as comprising
two segments so that segmental
inertia parameters for 20 body segmentscan be calcu-
lated by the method. Geometrical
representation
Earlier models (Whitsett. 1963; Hanavan. 1964; Jensen, 1976; Hatze. 1980) have used ellipses to model cross-sectionsof body segments because it was mathematically convenient to do so. Figure I compares a cross-section of a thorax obtained by Cornelis et al. (1978) with the stadium shape of Sady et af. (1978) and
a
_.-----.___.-...._
-.
.’ ,
‘.
,’
:
-, :
1 I
,
‘. --..
b
Cd
- . . . . .---.-.__..-
.-’
____. . . . _ . _ -__ I’
:
**\
,I
\ ‘1
:
I
)..
I
:
.
\ \
,
,
\ *.
; --,
,’
I
.’
1..
___.._._.
an ellipse. It is apparent that the stadium bears more resemblance to the thorax cross-section than to an ellipse of equal width and depth. Sady et al. (1978) used solids bounded by two such stadia of different dimensions to calculate volumes of body segments.Such a procedure may be expected to give a better representation than the use of cylindrical solids, since the cross-section of the stadium solid is permitted to vary. However, the formula used by Sady et al. (1978) is based upon the volume of a truncated pyramid and this gives the correct volume only when the bounding stadia have the same ratio of width to depth. A stadium may be defined as a rectangle of width 2t and depth 2r with an adjoining semi-circle of radius r at each end of its width (Fig. 2). The perimeter p = 4t + 2nr and width w = 2t + tr so that r and t may be calculated from perimeter and width using the equations: r = (p - Zw)/(Zn - 4)
and
t = (nw - p)/(2n
- 4).
It should be noted that using the perimeter and width as input measurements, rather than depth and width, reduces the error in the estimation of crosssectional arca arising from the use of a (somewhat) arbitrary geometrical shape (Yeadon, 1984). A stadium solid bounded by parallel stadia (Fig. 3) is defined in Appendix I and it is shown that crosssections which arc parallel to the two bounding stadia are also stadia. Formulae for the mass M. location of mass centre i and principal moments of inertia /,.f,,f, about the mass centre are derived for a stadium solid in Appendix 2. When the widths of the rectangles of the bounding stadia are zero a stadium solid becomes a truncated cone. In such a case the formulae for the inertia parameters become indeterminate. This difftculty is overcome by adding a very thin trapezium to a truncated cone so that the formulae for a stadium solid can be used. In the inertia model all body segments, with the exception of the cranium, are represented by a number of stadium solids or truncated corms. The cranium is modelled as a semi-ellipsoid of revolution. This solid is produced by rotating a quadrant of an ellipse with semi-axes r and h about the one axis to produce a semi-ellipsoid with base radius rand 2t 4
+
C e---w_ _.--
-.
.
-.
I’
I’
:
1 ,
\
\
\
..
5 I ,
\ \
‘.
-. --_i_-
.
.’
- *-*
of (a) thorax cross-section (adnptcd from Cornelis Y[ a/., 1978). (b) stadium. (c) ellipse.
Fig. I. Comparison
cilB31
Fig. 2. A stadium is a rectangle of width 2~ and depth 21 with an adjoining semi-circle at each end of its width.
69
Tbc simulation of aerial movement--II
B are each sectioned into seven solids oi and bi (i = 1.7) while the left leg J and right leg K are each sectioned into nine solids ji and ki (i = 1.9). The 11 segments of the simulation model of aerial movement described in Part IV of this series(Yeadon et ul., 1989) comprise the solids listed in Table 2. For each segment the mass, location of mass ccntrc. principal moments of inertia about the mass ccntre and distance between joint ccntrcs are calculated. It is assumed that the solids comprising a segment have coincident longitudinal axes. The values for the left and right limbs arc then averaged since the simulation model is designed to have symmetrical inertia values (Ycadon et al., 1989). Fig. 3. A stadium solid.
I Table 1. The kvels at which the body segments are sectioned
Torso S M Lsl Ls2 Ls3 Ls4 LsS M Ls7 Ls8
hip joint centre umbilicus lowest front rib nipple shoulder joint centre acromion beneath nose above ear top of head
Left arm A
&Cl shoulder join! centre Lul La2 Lu3 Lu4 La5 Lo6 La7
mid-arm elbow joint centrc maximum forearm perimeter wrisl joint cen(rc base of thumb knuckles fingernails
Measurements The levels in Table I are measured using anthropometric calipers so that the heights of the 40 solids can be calculated. At each level the perimeter is measured using a measuring tape. At the shoulder level Ls4 of the torso a depth measurement is taken since it is Table 2. Segment Chest-head Thorax Pelvis Lefl upper arm Left forearm-hand Right upper arm Right forearm-hand L&Cthigh Left shank-foot Right thigh Right shank-foot
Symbol
Solids
C T P
s4, 55. s6. s7. st s3 sl. s2 al. a2 a3. a4. US, u6. a7 hl, b2
A, AI B, B1
J, J, K, KZ
b3. b4. bS. h6. b7 /L/2,13
14. $.,16.rki8.
~9
k4. kS. i6, i7. k8, k9
tcrt leg J
LJQ hip joint cCntre Ljl Lj2 fj3 Lj4 LjS
crotch mid-thigh knee joint antre maximum calf perimeter ankle joint antre
Lj6 hal Lj7
arch
Lj8 ball LJ~ toe nails
height h. Formulae for the inertia parameters of a semi-ellipsoid are listed in standard enginkring texts (e.g. Meriam, 1971). Segmentation The body segments are sectioned into 40 solids by planes perpendicular to the longitudinal axes of the segments(Fig. 4). The levels at which the segmentsarc sectioned are given in Table 1. The levels Lsi (i = 0.8) section the torso S into eight solids si (i = I,@ where the solid si is bounded by the levtls Lsi and Ls(i - I). The left arm A and right arm
Fig. 4. Sectioning of the torso S. left arm A, right arm B. lcrt leg J and right leg K into 40 solids.
70
M.
R.
not possible lo measure the perimeter directly. For the shoulder levels LOO, L60 of the arms the perimeters are taken as high up the arms as possible. The solids representing the head, arms and legs are assumed lo be of circular cross-section so that no further measurements are needed. The remaining solids of the torso. hands and feet are stadium solids and so width measurements are taken at the boundary levels. At the heel levels Lj6, Lk6 anterior-posterior depths rather than medio-lateral widths are measured since these values correspond lo the widths of the stadium cross-sections.The foot arch levels Lj7. L&7 are assumed to be circular in cross-section to permit the transition from stadia which are aligned anterior-posteriorly lo stadia which are aligned medio-laterally. The 95 measurements taken comprise 34 lengths. 41 perimeters. 17 widths and three depths and require between 20 and 30 min of the subject’s time. Density values The density values used in the inertia model are taken from Dempster (1955) and are listed in Table 3 together with the corresponding solids. These values are used in preference to those obtained in the cadaver studies of Clauser et al. (1969) and Chandler et al. (1975) since only Dempster found individual values for shoulders, thorax and pelvis. It should be noted that the value for the density of Dempster’s thorax segment is used with solids s3 and s4 whereas segment T of the simulation model, which has been named the thorax segment for convenience, comprises only ~3. APPLICATION
The method was applied to three subjects (two males, one female) prior to performances of twisting somersaults on trampoline. Table 4 compares total body masses, determined by weighing, with the estimates obtained using the inertia model. DISCUSSION
YEADON Table 4. Accuracy of total body mass estimates Subject
Mass (kg)
A
60.0 60.9 64.3
Estimated
mass (kg)
58.8 62.0 65.8
Error - 2.0% 1.8% 2.3%
parable with the value 1.8% obtained by Jensen (1978) but is much greater than the value 0.5% obtained by Hatze (1980). When measuring torso perimeters there is the problem of changes due to breathing. In this study an attempt was made to minimize this effect by asking the subjects lo breathe shallowly. However, it may be expected that when the lungs contain an additional one litre of air, the volume of the torso increases by one litre and this will produce an increase in the total body mass estimate of about 1.5% for a 70 kg subject. Thus, even if the amount of air in the lungs is held approximately constant and the torso volume estimte is accurate, there appears to be an uncertainty in the torso mass estimate of the order of 1% without a method of determining the amount of air in the lungs. With this in mind, the errors in the total mass estimates of this study and of Jensen (1978) are not unreasonable. The accuracy with which the model estimates total body mass, however, does not give much indication of the accuracy with which segmental masses and inertias are estimated. The reason for considering total body mass is that it is the only quantity for which a direct measurement is available. Since the inertia model has been designed lo produce personalized segmental values for input into a simulation model, the inertia model may be considered adequate for this purpose providing there is good agreement between simulations and actual performances. Such evaluations will be made in Parts III and IV of this series (Yeadon, l989k Yeadon er al., 1989) where the effects of anthropometric and film measurement errors will also be determined.
The maximum error of the total body mass estimates given in Table 4 is 2.3%. This value is comREFERENCES Table
Segment
3.
Segmental density values (Dempster,
1955)
Segmental density values (kgl”) Solids
Density
Head-neck Shoulders Thorax Abdomen-pelvis Upper arm Forearm Hand Thigh
Lower leg Fool
~6, sl. s8 ss s3. s4 sl. s2 al. a2 a3. a4 a5, a6. a7 jl.jZ.j3
j4. iS j6. j7. j8. j9
I.11 I.@4 0.92 I.01 I .07 I.13 I.16 I .05
1.09 I.10
Bouisset. S. and Perluzon. E. (1968) Experimental determination of the moment of inertia of limb segments. Biomechunics I (Edited by Wartenwciler, J.. Jokl, E. and Hcbbelinck, M.). ppq 107-109. Karger, Bascl. Chandler. R. F.. Clauser, C. E.. McConville, J. T.. Reynolds, H. M. and Young, J. W. (1975) InvcsGgaGon of inertial orooertier of the human bodv. AMRL-TR-74-137. ADkOi6-485. DOT-HS-801-430.V Aerospace Medical Rcsearch Laboratories. Wright-Patterson Air Force Base. Ohio. Clauser. C. E., McConvillc. J. T. and Young. J. W. (1969) Weight. volume and center of mass of segments of the human body. AMRL-TR-69-70. Aerospace Medical Research Laboratory. Wright-Patterson Air Force Base, Ohio.
The simulation of aerial movement-II
Corn&. J, Van Gheluwe. B.. Nysen. M. and Van den Berghe. F. (1978) A photographic method for the tridimensional reconstruction of the human thorax. Proceedings of the NATO Symposium on Applications of Human Biostereometrks (Edited by Cobknt A. M. and Heron. R. E.). Procredincls of rhe Sociery ol Photo-opricol Instrumentot~onEn&&s I&, pp. 2&G. Danicls. G. S. (1952) The averaae man. Technical Note WCRD-53-7. AD-lb203. WrightAir Development Center, Ohio. Dempster. W. T. (1955) Space requirements of the seated operator. WADC-55-159. AD-087-892. Wright-Patterson Air Force Base. Ohio. Hanavan. E. P. (1964) A mathematical model of the human bodv. AMRL-TR-64-102. AD-608-463. Aerosnace Medical Research Laboratories, Wright-Patterson Air Force Base. Ohio. Hatzc, H. (1975) A new method for the simultaneous measurement of the moment of iner,tia. the damping coefficient and the location of the centre of mass of a body segment in situ. Eur. 1. appl. Phys. 34.217-226. Hatze, H. (1980) A mathematical model for the computational determination of parameter values of anthropomorphic segments. 1. Biomechunics13. 833-843. Hay. J. G. (1973) The center of gravity of the human body. Kinesiology 111 (Edited by Widule, C.). pp. 20-44. American Association of Health. Physical Education and Recreation, Washington. DC. Hertzberg. H. T.. Danicls. E. and Churchill. E. (1954) Anthropomctry of flying pcrsonncl. WADC-TR-52-321. Wright Air Development Center, Wright-Patterson Air Force Base, Ohio. Jensen, R. K. (1976) Model for body segment parameters. Biomuchanics Y-f3 (Edited by Komi, P.V.). pp. 380-386. University Park Press. Baltimore. Jcnscn. R. K. (1978) Estimation of the hiomcchanical propertics of three body types using a photogrammctric method. J. BiomechunicsI I. 349-358. Mcriam. J. L. (1971) Dynumics. 2nd Edn. p. 475. Wiley, New York. Sady, S.. Freedson. P.. Katch, V. L. and Reynolds, H. M. (1978) Anthropomctric model of total body volume for males of dibrcnt sizes. Ifuman Biol. 50, 529-540. Tichonov. V. N. (1976) Distribution of body masses of a sportsman. Biomechunics Y-B (Edited by Komi. P. V.). DD. 103-108. Univcrsitv Park Press. Baltimore. Wi&ett. C. E. (1963) So& dynamic r&onsc ch&actcristics of weightless man. AMRL-TDR-63-18. AD-412-541. Acrospacc Medical Rcscarch Laboratories. WrightPatterson Air Force Base, Ohio. Yeadon. M. R. (1984) The mechanics of twisting somersaults. Ph.D. dissertation, University of Loughborough. Ycadon. M. R. (1989a) The simulation of aerial movcmentI. The determination of orientation angles from film data. J. Biomechanics 23. 59-66. Yeadon. M. R. (1989b) The simulation of aerial movcmcnt111.The determination of the angular momentum of the human body. 1. Biomechanics 23, 75-83. Ycadon, M. R.. Atha. J. and Hales, F. D. (1989) The simulation of aerial movcmcnt-IV. A computer simulation model. J. Binmechonics23. 85-89.
APPENDIX 1. SlADlUhl
SOLID
A stadium solid bounded by parallel stadia (Fig. 3) is defined as follows: let the lower bounding stadium have parameters r,, and I,. the upper bounding stadium parameters r, and I, and Ict h be the distance separating the stadia. Consider a point P which lies on the boundary of the first quadrant of a stadium with parameters r and I (Fig. 5). If P
71
Y
t+r Fig. 5. A quadrant of a stadium.
has coordinates (x, y) then: forO~x~r.
.v-L1(0~1~1)
r
x=r+rcosO
for x >
and
_v=r;
and
y = rsin 0 (0 I H 5 n!2). A correspondence may now be defined between the points on the boundaries of the first quadrants of the upper and lower stadia. Points on the straight sections correspond when they have equal L values and points on the curved seclions correspond when they have qua1 0 values. In a similar way, a correspondence may be defined for the remaining quadrants. Suppose that Pn and P, arc corresponding points of the lower and upper bounding stadia (Fig. 3). if p0 and p, arc Position vectors of P, and P, relative to some origin. then the point P with position vector p = (I - :)p,, + :p, lies on the line segment P,P, when 0 s I S I and divides it in the ratio ,-:(I - z). If : is held constant and P, and P, move around the lower and upper stadia. the point P moves around the boundary of an intermediate section of the stadium solid. This procedure defines the stadium solid and it will now bc shown that each intermediate section is itself a stadium and is parallel to the bounding stadia. Let the coordinate system (x.J.:) have origin at the centrc of the lower stadium with the : axis lying along the line joining the centrcs of the bounding stadia (Fig. 3). The points PO(ir,.r,.O) and P,(lf,.r,,h) on the straight scclions correspond 10 P(( 1 - r)lf, + :i.f, .( I - -_)r0 +:r,.:h) which is quivaknt to P(i.t(z),r(:).zh) where I(Z) =(I-:)I,+:I, andr(z)=(l-:)r,+zr,. Similarly, the points PO( r,, + r0 cos 0. r0 sin 0.0) and P, (I, + r, cos&r, sin U.h) correspond to P(f(Z) + r(,_)cosO. r(z)sinU..-h) where I(:) and r(z) arc as above. Thus. as P, and P, move around the bounding stadia the point P moves around a parallel stadium which has parameters I(:) and r(z) and is at a distance ,-h from the lower bounding stadium. It should be noted that although the above treatment does not assume that the : axis is normal to the bounding stadia. only right stadium solids will be used.
APPENDlX 2. FORMULAE FOR THE lSERTlA PARAMETERS OF A STADIUM SOLID
The derivations of formulae for the location of mass ccntre. mass and principal moments of inertia of a stadium solid are given below. Prior to consideration of the solid, expressions are derived for the second moments of arca of a stadium. Srcond moments of area n/a
sradium
The calculation of the second moments of area of a stadium will require the evaluation of a number of integrals. For convenience these definite integrals will be evaluated
72
M. R.
prior to the calculation of second moments.
YEADON
ThUS
I,
= 4r’r/3
+ ~74,
and
Ix’
ydx
= [
x’ydx
+ I;,,.
x2yfd8
(I +rc~sO)~~~sin~ OdO
+2rd.x+[~=
+ cos2 2O)dO 112s
4
[I
J0
+t(l
=
-2cos20
J II
+ 2r’r[l/3]
+ r’[a/l6].
Thus J, = 4nJ/3 112s
-sin’0 [ 3 10
+ 8rJ1/3 + nr’/4.
For a lamina the theorem of pcrpcndicular axes states that
=3
J, = J, + J,.
(I - sin2 0)sin’ OdO
0 sinzOdO= t
+ w212
,
1121 cos2
I
+ r212[n/4]
sin2 O*d(sin 0)
J0
I,Z~
2r’t
= r[IS/3]
I =
sin2 OdO
sin20cosOd0 J0 Ilam cos20 sin’Od0 + I4 J0
+
I6 112.
r2P
+
112.
+cos4O)]dO
3n
sin20cosOdO=
r[x’/3yO
0
=l/2_
J
112s
I =-
Thus
0
0
J, = 4rI’/3
112m
sin2 (IdO
m I
+ arat
+ 4r’t + ~‘12.
These expressions for A. J,. J,. J, will bc used in the derivations of the inertia parameter of a stadium solid.
0 II2rn
sin4 OdO -I
0
n 3% =---a--. 4 I6
n 16
A point tics on the boundary of the first quadrant of a stadium (Fig. 5) providing either 0 s x I; I and y = r or x~1+rcosOandy=rsinO(OJO4n/Z). Thus the area of the stadium is A where: yld0 4
where
dx i =do
The muss and locarion o{mass cenlre o/a srarlium solid
It has been shown that the intermediate stadium of Fig. 3 is at a distance zh from the lower bounding stadium and has parameters r and I where r = r. + z(r, -ro) and I = 1, t z(,, -r,). Thus r and I may bc expressed as r = r,(l t a~) and I 3 Io(I + br) where a = (r, - ro)/ro and b = (t, - I,)/I,. If the intermediate stadium has density D. area A(r) and thickness h6r the mass will bc DA(r)hdr so that the mass of a stadium solid of uniform density is given by I
,,2r
M=
DA(r)hdr. I
= [Lrdx
+ r2~~‘sin20d0 The
I
I
= rl + -f’n. 4
Thus A = 4rr + nr2 as expected. Let J,. J, and J, be the second moments of area of the stadium in Fig. 5 about the x. y and : axes where the z axis is orthogonal to both the x and y axes.
first
0
moment of mass about the lower plane is
hrDA(r)hdr and the distance i of the mass ccntre from I thOelower plane is given by:
I i=
J0
Dh*rA(t)dr/M.
Defining the functions FI. F2, F3 by the equations: Fl(a.h) = I t (a + b)/2 + ah/3 the integration occurring over a single quadrant. Now
Jy~dr=J~y~d,~+J~,2,y’*do
FZ(a.b)
= I+
f3(a.b)
= f + (at
t
b)/3
t
ah/4
b)/4 + ah/S.
Area A(:) =4rr + ar* =4ro(l
=lr’dx
(a
+az)r,(l
+bz)tnri(l
x r’~~‘sin’OdO = 4r,,I,( I + (a + b)r
= ,JI
t 3nr’/l6.
+ ~$1
t
t 2ar t 0’2~)
abz2)
taz)2
73
The simulation of aerial movement-II
so that
and
I 1 + (a + b),‘?
Ad: = Jr&(
FS(a.b)
= 1+ (2~ + 2b)/2 + (a2 + 4ub + b2)/3
+ ab/3)
+ Zab(a + b)/4 + a2 b’/S.
j 0 Now
+ nrt( I + Za,‘? + a1 /3) = 4r,f,Fl(a.b)
+ nr~Fl(a,a)
rf’
1 Ad:.
M=Dh
j 0
= r&l
+a:)(1
= r,ri(l
+ a.-)(1 + 36: + 3b2:*
= r,&l
+ (a + 3b)z + (3ab + 3b’)?
+ (3ab’
Thus M = Dkr,[4t,Fl(a,b)
+ nr,Fl(a.a)J.
+ b:)’
+ b’)?
+ ab’r’).
so that
I
Now
rt’d; ,-A = 4r,f,(;
+ (a + b).?
+ b’?)
=r,fiF4(a,b).
j0
+ ab:‘)
Similarly + ffr$:
+ 2a?
+ oz.?),
I
so that
r’td.- = rif,F4(b.a), j0
:Ad:
= 4r,fo
whilst
i + (a + b)/3 + ab/4 >
0
+ =
(i+ +
nri
2al3
4r,f,F2(u.b)
rzt2 = &;(I >
+ nriF?(a.a)
Dh2
+b:)’
+ ?a: + a2:‘)(l
+ 2b: + b’?)
I + (2~ + 2h); + (a* + 4ab + b’)?
= rif$
+ _ ‘uh(u
I it
+a:)‘(1
= riri(l
aa/4
+ b)-’-
+ a* bf --’
1
so that
:Ad:/hl.
I
j 0
Thus
rzr’dz = riri$FS(a,h) = = Dh’[G,r,F2(u.h) _
j0
+ nriF2(a.a)]/M. and
Note idso :j A = 4r,r,(:’
r’ = rt(l + a:)’
+ ((I + h)? + U/J.-~)
+ nr:(:’
+ 2~’
=r~(I+4u:+Q’:‘+4aJ.-‘+u4~~)
+ a’:‘).
so that so that
I
I
r’d:
:* Ad: j I)
= riM(a.u).
j0
= 4rofa( f + (U + b)/4 + ub/S)
Thus + nr:( i + 2~14 + a’/S).
i.e.
I: = Dh[4rof~F4(u,b)/3
+ nr~f~FS(a.b)
+ 4r~f,F4(b,u)
+ nr4,F4(a.a)/2].
I :‘Ad:
= 4r,r,F3(u,b)
+ nr:F3(u,a).
j II This result will be used in conjunction with the theorem of parallel axes in the next section to calculate momrnts of inertia. Momrn~s oJ invrriu u/u sfudium
solid
In thisxxtion cxprcssions are derived for the moments of inertia I,. I,. I, of a stadium solid about principal axes through the mass ccntrc. The moment of inertia of a stadium solid about the : axis
I
(Fig. 3) is I, =
The moment of inertia of a stadium section about the p aais (Fig. 3) is given by the theorem of parallel axes as J,Dhd: +(h:)*ADhd: where Jr is the second moment of arca. D the density. htS: the thickness. h-_ the distance from the y axis and A the area. If /T dcnotrs the moment of inertia of the stadium solid about ths )’ axis (which lies in the lower fttcc) then: ,T = j;J,Dhd: i.e.
I
DJ,hd: whcrc the second moment of areas
1; =
j 0 of a stadium has been obtained as J, = 4rfJ/3
I
+ nr2fa + 4r’f
f nr’j2.
t J,d I =
+4j~r’rd:+~aj~r’d:].
j0
where:
t + nj:r’r’d:
:’ Ad:.
J,d: +
Dh
j 0
Thus I, = Dh[(4,3)j:rr’d:
+ j;(hr)‘ADhd.-.
j 0
(4rr’/3
j 0 = 4r 00f’FJ(a
+ nr2f2 + 8r3/3 + nr’/4)dr
’ b)/3 + 8r:r,F4(b.u)/3
+ nrir:F5(a.b) + nr:F4(a.a)/4
using the results obtained during the calculation of i:. and
I
Define
:‘Ad,F4(a.b)
=
I + (a +
3b)/2 + (Sob + 3bz)/3
+ (3ah’
+ b’)i4
+ ub’J5
= 4r,f,F3(u,b)
+ nriF3(a,
u)
j 0 having
been derived in the section on the calculalion off.
14
M. R. YEADON
If I, denotes the moment of inertia about a parallel axis through the mass cen~re then the theorem of parallel axes
i.e.
gives:
,.=Dhj~l.d:+Dh’j~I’Adl.
/ = p - .\f=‘. I
Similarly.
the moment
*
where
-
of inertia
1 I:
about
the x axis is
J,d: j 0
given by:
I =
(4rcJ/3 + xr’/cl)d: j 0
= 4r,r:f4(4(a.b)/3 + nr~F4(a. a)/4 ,; = j+Ihd:
+ j)tr,‘ADhd:.
and the moment of inertia I, about a parallel axis through the mass centre is given by I, = 1: - Mf’.
