J. Btomechanlcs. 1976. Vol. 9. pp. 607413
Perpmon
Press.
Pnnted in Great Britain
ON THE KINEMATICS OF THE HEAD USING LINEAR ACCELERATION MEASUREMENTS* C. C. CHOU and S. C. SINHA Biomechanics Research Center. Dept. of Mechanical Engineering. Wayne State University. Detroit. MI 48202. U.S.A. paper deals with an application of the nine accelerometer scheme proposed by Padgaonkar ef at. (1975) to the analysis of the head kinematics including the head angular acceleration and velocity using linear acceleration measurements. The computed results have been compared with the Abstract-The
film analysis and with the actual measured quantities in the case of the dummy experiments. The study reveals an excellent aaeement among, these results and indicates the feasibility of the application of the method to biomecha>ical studies. -
set of three simultaneous, coupled non-linear differen-
INTRODUCTION
In many biomechanics experiments, anthropomorphic dummies, cadavers and human volunteers have been used to study the severity of head injuries to different impact conditions under various environments. The determination of whether the head exceeds the maximum tolerable severity, or injury threshold, subjected to a given impact requires the evaluation of the Head Injury Criterion (HIC) based upon the head C.G. resultant acceleration according to Federal Motor Vehicle Safety Standard (FMVSS 208). Therefore, an assessment of head injuries needs the knowledge of a time history of the head C.G. resultant acceleration profile for the duration of the impact. When an anthropomorphic dummy is used, the resultant head C.G. acceleration can be obtained by measuring three acceleration components using a triaxial accelerometer mounted at the C.G. of the dummy head. In the cadaver and volunteer experiments, however, the measurement of acceleration at the C.G. of the cadaver and volunteer heads cannot be made directly and therefore must be restored to alternative methods or techniques using linear accelerometers to measure exteriorly. Mertz (1967), Clarke et al. (1971) and Ewing er al. (1973) have demonstrated the feasibility of using linear accelerometers in analyzing the head dynamics in planar motion, particularly in the case of the motion taking place in the mid-sagittal plane. The angular acceleration and the angular velocity of the head were obtained using four linear acceleration measurements. The kinematic principles involved have been extended to a more general analysis for various studies at WSU (1973) and by Alem (1974). The analysis in these studies was based on a three dimensional rigid body motion for which the kinematics are completely defined by six linear acceleration measurements at three different points on the head. Generally, the six accelerometer scheme results in a *Received 2 February 1976.
1976; in revised form 25 April
tial equations of the form i = f(o, M’),
(1)
where a : measured linear accelerations (g’s) w: angular velocity vector (rad/sec) i: angular acceleration vector (rad/sec’). Mathematically. equation (1) can be solved numerically using a Runge-Kutta, a predictor-corrector, or finite difference method for a given a measured exteriorly at the dummy and volunteer heads. Upon solving equation (1) by stepwise integration procedures, one can see that the calculation of the angular acceleration components require utilization of the angular velocities from the previous time step. The aforementioned numerical procedures in general introduce round-off errors which effect the accuracy of i through numerical computations. Figure 1 shows the result of calculated acceleration which increases without bound at some time step due to the cumulative errors as well as the measurement errors as pointed out by King et al. (1974) and Padgaonkar et al. (1975). Moreoever. for a particular choice of six acceleration measurements, equation (1) reduces to the form given by equations (Sa-5c). It is easily seen that the Jacobian determinant of this system is zero. Hence, the system is inherently unstable as indicated by Lapidus and Seinfeld (1971) and AIem (1974). It can be further asserted that no amount of additional manipulation of the six accelerometers would yield a set of equations which will be stable. Difficulties encountered using numerical integration methods prevented the six acceleration analysis from yielding reliable and useful results. To circumvent the difficulties, an alternative method has been proposed by Padgaonkar et al. (1975) using nine linear accelerations measured at four points on the head. With a special arrangement of the nine accelerometers, the angular accelerations can be obtained directly and algebraically from the measured linear accelerations at each time step without utilizing the values of w and icl computed at the
607
c. c. CHOU
608
and s. c. SINHA
40.00
Velocity
0 _;
29.1 mph.
Calculated Cal. - 3 Comp
30 00
E :: g
2000
,” 5 : a
IO.00
7:
0.00
20.00 Time
from
60.00 onset
of
100.00 sled
140.00
160.00
1
MS
deceleration,
Fig. 1. Unbounded response using six accelerometer scheme.
previous time step. The reliability of this approach was established by means of validation studies using both the hypothetical data and the experimental results obtained from a special experimental set-up (see Padgaonkar et al., 1975). The nine-accelerometer scheme has been applied to the analysis of the head kinematics for both the dummy and volunteer experiments. In this paper, only the dummy experiments will be described and the results are discussed. ANALYSIS OF THE NINE-ACCELEROMETER SCHEME The analysis of the nine accelerometer scheme proposed by Padgaonkar er al. (1975) is briefly redescribed herein since it serves as the basic concern of this paper. From the kinematics of a rigid body, the acceleration of a point P on the body is given by
ap = 110+ wx(wxpp) + 3xp,
-
a rP
=
Go
w~@w,.P +
-
WchPxP
-
wr(wyPzP
-
+
W,P,P
-
W,P~P
W
+
W&P
-
WyPxP.
(34
WXPZP) W&P)
Using Padgaonkar’s scheme, the nine linear accelerometers are arranged in the configuration shown in Fig. 2 consisting of a set of triaxial accelerometers located at the origin 0 and three biaxial ones mounted at locations 1, 2, and 3. At the origin, the triaxial accelerometers are oriented with their sensitive axes in the three orthogonal directions of the body-fixed axes. At locations 1,2, and 3, the sensitive axes of a pair of accelerometers are in the directions perpendicular to the axis on which they are mounted. Referring to the configuration shown in Fig. 2, the
7
(2)
Referring to the body-fixed frame, ap = axPi + a,.,$ + arpk = acceleration of the point P relative to the body-fixed frame; a0 = axOi + a,,$ + arOk = acceleration of the bodyfixed axes at the origin; w = rt:i + w).j + wzk = angular velocity of the body; i = wxi + w,j + wzk = angular acceleration of the body ; pp = pXpi = pyaj + Pzpk = position vector of point P relative to the body-fixed frame,
W,AP)
-
A 2 I I
A*2 2
where i, j and k are the unit vectors for this frame. From equation Q), the scalar components of ap along the body-fixed axes can be expressed as (Inertia frame) X Fig. 2. Nine accelerometer scheme configuration.
h04
Linear acceleration measurements position vectors of points 1. 2. and 3 with respect to the origin are Pl = P,d
(4a)
p2
(4b)
=
Px2i
(4c)
~3 = Pz&.
Equations (3a-c) can now be applied to express the acceleration components at point P in terms of the linear acceleration components at the origin, the angular velocities and accelerations, and the relative position of point P with respect to the origin for P = 1, 2 and 3. Using equations (4a-c) and after a straight forward manipulation, the following equations are obtained: 0 (54
wx = (% - GJYPyl - wywz WY= -(azz - hJYPx2 + wxwz
W
w:
WPg
(54
+
(54
=
(a,2
-
“yoYPx2
wx
=
-by3
wy
=
bx3
-
GoMP;3
-
w:wx
(54
k4
-
4OUPyl
+
Wxfi’y
WI
WL =
-
-
~,cJ/Pz,
wyw:
Elimination of the cross-product terms of the angular velocity components yields the following algebraic equations for the angular accelerations: wx =
-
%0)/‘&Q
-
(‘$3
$0)/&%3
(64
WY = (43
-
~xoY@z3
-
k&z -
%om%z
(6b)
wz = ($3
-
~,oY%%z
-
@,I
~xOY2Pyl.
69
b&l
-
-
where Uy with i = x, y and z, and j = 0, 1, 2 and 3, denotes the i-th acceleration component measured at the j-th location; pij denotes the distance in the i-th direction of the j-th location relative to the origin. By means of equation (6), the angular acceleration along a set of body-fixed axes can be computed with measured acceleration data obtained from an experimental sled test. The angular velocity w is then obtained by a simple integration. Having the values of & and w, the linear acceleration components at the head C.G. along the body-fixed axes can be obtained by transforming acceleration measurements at the origin through equations (3a-c). More explicitly, let acg = u,,i + uycgj + uzcgk be the acceleration vector at the head C.G. with respect to the body-fixed axes and pEe= pxcei + pwej + p,,,k the position vector of the head C.G. relative to the origin 0. Using equations (3a-c), the acceleration components of ocg can be calculated as: %cg = a,0 + wY(wxPYce - w&Xcg) - wz(wzPX,o- wXPzcp)+ w,P,,, - wzpYep (7a) $rg = $0 + wz(uVzrg - w&c,) - %(wJ’,, - w&c,) + w,P,,~ - w,P,,,
(7b)
azcg= %o + wX(wzPX,(I - K&J - %(wYP.cg- w&e) + %Pycs - wypxcg, (7c) where all the quantities on the right hand side of equations (7) are known.
The resultant acceleration 1~~~1 at the head C.G. is computed from 7 (8) beg/= \ d, + Gy + ‘Ii,,. The calculated resultant acceleration la@ from equation (8) is compared with the resultant acceleration computed using measurements at the head C.G. In addition. the HIC and Gadd Severity Index (GSI) are also computed for comparison based upon the measured and calculated head resultant accelerations. Moreoever. because of the nature of the experimental set-up and the greater relative magnitudes of w, and ~3).over M’,. M’,. wr and ivr (cf. Fig. 5). the motion during these runs can be considered nearly as a planar one. Hence it would be interesting to compare M‘,and wVwith a planar 2-D film analysis on an approximate basis. METHODOLOG1
In the experiments. a HYB 11 dummy was used as a test subject. The dummy was restrained in a rigid seat mounted in the direction of motion of the sled of WHAM
III. The sled was accelerated to a pre-determined velocity over a distance of 60ft. after which it was stopped with a given stopping distance. The deceleration pulse shape approximates a square wave.
The seat was rigidly constructed using steel angles for the main structural components and plywood coverings for the seat back and bottom. The dummy was seated on the rigid seat and restrained with a series of chest. shoulder and lap belts to keep his torso nearly in the upright position during the run. The feet were also strapped in position to the impact seat. The head accelerations for the dummy were measured with linear accelerometers mounted on a frame attached to the head. A head mount was fabricated for the nine accelerometer configuration. The head mount for the accelerometer was then attached to the top of the dummy head by screws. with s-axis being parallel to the AP axis. The experimental runs were performed at various barrier equivalent impact velocities of IO. 20 and 30m.p.h. with 10” stopping distance. The instrumentation on the dummy includes (a) a triaxial accelerometer (Endevco 2264) mounted at the C.G. of the dummy head and (b) nine uniaxial accelerometers (Endevco 2264) mounted on the 9-accelerometer head mount, The photographic instrumentation consisting of three high speed cameras: offboard lateral. onboard lateral and oRboard front provides photographic coverage of the motion of the dummy during the impact event. Timing marks were put on the film throughout the run to permit accurate calculation of the framing rate. In addition, a flash bulb was located, in the field-of-view of each camera to permit synchronization of the high-speed film with the transducer records. Films from the lateral offboard camera were utilized for quantitative analysis of head rotations. The transducer outputs were recorded on a Bell & Howell FM tape recorder. and were digitized by the A/D system at a conversion rate of 2000 samples/set/channel. The digitized data were put on a digital tape by the PDP-8 computer and were processed on an IBM 360/67 computer. The binary data were read and converted to true voltages which were then multiplied by a calibration factor to obtain physical units for use in the nine-accelerometer head analysis described in the previous Section, RESCLTS
AND
DISCUSSIONS
Results of the experiment are obtained by means of a transducer record analysis (using the A/D converter) and the high-speed movie analysis.
C. C. CHOU and S. C.
610 2000
1
SINHA
-
Run no. 128 Veloaty
1000
-3O.d 0.00
9.4 mph.
16QCO
6clcil
240.00
-
PA-cGuvlea) IS-CG (Med
+-+
RL-CG.(Mesl
32090
40000
MS
Time from onset of sled decelerotlon,
Fig. 3. Acceleration components measured at the head C.G. as a function of time.
Some typical iments
are shown
records obtained from the experin Figs. 3 and 4. These are the
Calcomp plots of the digitized data from the analog tape recorder for the run made at 9.4m.p.h. with a 10” stopping distance. Figure 3 shows the acceleration components in the AP, SI and LR directions as measured at the C.G. of the dummy head, respectively. The nine acceleration components measured externally at four different points on the head mount are shown in Fig. 4. Application of the nine acceleration component analysis to the data shown in Fig. 4 yields the angular acceleration, as a function of time from onset of sled deceleration, shown in Fig 5(a). A simple integration of the angular acceleration gives the angular velocity shown in Fig. S(b). Examples of the head kinematics from the high speed 2-D film analysis are displayed in Figs. 6-8. Figure 6 shows the head rotation about the axis perpendicular to the sagittal plane as a function of elapsed time from the onset of sled deceleration. The angular velocities and angular accelerations from the
-
L
P
e ._ z
-1000
000
3 ’
-
Au, AY,
-
AY,
0
80.00 77ma
24000
Isow
?2oca
from onset 01 sled durlrroOon,
Fig. 4. Nine linear acceleration components
of time.
Run no. 128 Velocity
$
WOO*
Ax, Ax, Ax.,
OCWt
2ooc I
40.w~
2
-
9.4 mph.
-
Ang.vel. X Ang.vel. Y Ang. vel. Z
C-W, -
Ang. occel. X Ang. occel. Y Ang. occel. Z
IO.00
0.00~
9 a -W.OO -
(b)
0.00
IEOCKJ 320.00 80.00 240.00 Time from onset of sled deceleration, MS
4oo.ccl
Fig. 5. Angular accelerations (a) and velocities (b) obtained from nine acceleration scheme.
4cOco
MS
as a function
Linear acceleration measurements 3ow-
w
Run no I28 Veioc~ty 94
Head mtatvm
mph
ZOCO.
-20001 000
6000
160.00
24ODo
Time from cnset of sled decelerofion,
32000
-301 0
4cQcQ
40
MS
film analysis are obtained using the harmonic analysis described by Haut et al. (1974). The synthesized data are then differentiated with respect to time to obtain velocities and accelerations. As previously mentioned, the angular velocity and acceleration obtained from the film analysis are compared with wY and wY from the nine acceleration analysis as shown in Figs. 7 and 8, respectively. It is seen that there is a good qualitative agreement, but quantitatively, the film analysis yields lower values for the angular velocity and acceleration as compared to the nine accelerometer scheme. This may be due to the fact that the comparison was made on an approximate basis. The acceleration components calculated at the C.G. of the dummy head are shown in Fig. 9(a). Fig. 9(b) shows the calculated head resultant acceleration using calculated head acceleration components. The comparison between the calculated and measured head resultant accelerations are presented in Fig. 10 along. with the resultant acceleration measured at the origin 0 as indicated by the curve ‘RES-Base’. The HIC and GSI computed for the measured and calculated resultant accelerations are tabulated in Table 1. The
200
240
260
329
360
400
MS
1 reveal that excellent correlations between these quantities have been achieved for dummy runs at various velocities. CONcLUSlONS
The feasibility and reliability of the nine-accelerometer scheme has been demonstrated to obtain the head kinematics of a dummy when considered as a rigid body. Comparison of the head kinematics -
60
. Ix)
Time
from
a-150 0
40
160
. 200
. 2w
9Ati
280
. 320
c+wet of sled deceleration.
scheme
. 360
20.00
10.00
MS
od
PA-CG
-
IS-CG.
(CAL.) (CAL.)
-
RL-CC.
(CAL.)
m
RES-CG
(CAL.)
(b)
-00 zj L 0.00
40.00 Time
120.00 from
onset
200.00 of
+ 4M)
Fig. 8. Comparison of angular accelerations obtained from nine accelerometer scheme and film analysis.
-
‘; 3:: 5
from
results shown in Fig. 10 and in Table
Run No. 128 velocity 9.4 mph.
V al
EO
Fig. 7. Comparison of angular velocities obtained nine accelerometer scheme and film analysis.
Fig. 6. Head rotation as a function of time.
Vzl
120
60
Time from onsef of sled decelemr~on.
sled
deceleration,
240.00
360 00 MS
Fig. 9. Acceleration components (a) and resultant acceleration (b) calculated at the head C.G. vs time.
C. C. CHOU and S. C. SINHA
612
Run No. I28 velocity 9.4 m.p.h.
PO.00
-
to.00
b
3
IO.03
2
40.00
30.00
s g 2 2
Run No. I29 velocity 18.0 mph. -RES-C.G.(CAL.) -RES-CI;(MES.) RES- BASE
20.00
P P c
RES-C.G. (CAL.) RES-C.G. (MES.1 RES-BASE
-
Run No. 137 velocity 31.0 mph RES-C.G.(CAL.)
-
RES-BASE RES-C.G.(MES.)
20.00
10.00
0
120.00
40.00 Time
from
200.00
onset
of
260.00 sled
360.00
deceleration,
MS
Fig. 10. Comparisons of measured and calculated resultant head acceleration at C.G. for various impact velocity runs.
Table 1. Comparison
of the HIC and GSI computed for the calculated and measured head resultant accelerations
Run no.
Vel. (m.p.h.)
stop. dist. (in.)
128 129 137
9.4 18.0 31.0
10 10 30
HIC GSI Using head resultant accel. Cal. at Mea. at Cal. at Mea. at C.G. C.G. C.G. C.G.
obtained from the film and the nine acceleration analysis and comparison of the calculated and the measured accelerations at the C.G. of the dummy’s head show that excellent correlations of the results have been obtained. Based upon this comparison, it is concluded that the nine accelerometer scheme provides a useful means of using linear acceleration measurements to obtain kinematic information required in many biomechanics studies. The method has been applied to the volunteer experiments in studying responses of the human neck in flexion, extension and lateral tlexion and the results will be reported in the future. Acknowledgements-The authors would like to thank the reviewers for their valuable comments which led to the
5 62 168
5 64 159
7 76 218
7
final version of this manuscript. They also express their gratitude to Dr. Haut for using his harmonic analysis program. REFERENCES
Alem, N. M. (1974) The measurement of 3-D rigid body motion. Proc. 21ul Ann. int. Meeting of the Ad Hoc Committee on Human
Subjects for Biomechanical Research.
(Ed. by Hirsh, A. E.) pp. 60-64. Clarke, T. D., Gragg, C. D., Zimmerman, R. M. and Muuy, W. H. (1971) Human head linear and angular accelerations during impact. Proc. 15th Stapp Car Crash Conf., Coronado, CA. DOT Contract 83-146-3-753 (1973) Injury assessment of belted cadavers. Progress Report No. 6. Biomechanics Research Center, Wayne State University (WSU). Ewing, C. L. and Thomas, D. J. (1973) Torque versus angular displacement response of human head to -Gx im-
Linear acceleration measurements pact acceleration. Proc. lit/l Stapp Car Crash ConjI. Oklahoma City. OK. Ham. R. C., Remmers. E. P. and Meyer. W. W. (1974) A numerical method of film analysis with differentiation with application in biomechanics. Proc. Sot. Photo-oprical Instrumentation Engineers. 57, pp. 5361. King. A. I.. Padgaonkar. A. J. and Krieger. K. W. (1974) Measurement of angular accelerations of a rigid body using linear accelerations. Proc. 2nd Ann. Inr. Meeting
ofthe Ad chanical
Hoc Committee on Human Subjects for BiomeResearch. (Ed. by Hush. A. E.) pp. 48-59.
613
Lapidus. L. and Seinfeld, J. H. (1971) Numerical Solution of Ordinary Difirenriul Equations, pp. 128-1.51. Academic Press. NY. Mertz. H. J. (1967) Kinematics and kinetics of whiplash. Ph.D. dissertation. Wayne State University. Detroit. Ml. Padgaonkar. A. J.. Krieger. K. W. and King, A. 1. (1975) Measurement of angular acceleration of a rigid body using linear acceleration. J. oppl. Mech. 97(3) pp. 552-556.
Perpmon
Press.
Pnnted in Great Britain
ON THE KINEMATICS OF THE HEAD USING LINEAR ACCELERATION MEASUREMENTS* C. C. CHOU and S. C. SINHA Biomechanics Research Center. Dept. of Mechanical Engineering. Wayne State University. Detroit. MI 48202. U.S.A. paper deals with an application of the nine accelerometer scheme proposed by Padgaonkar ef at. (1975) to the analysis of the head kinematics including the head angular acceleration and velocity using linear acceleration measurements. The computed results have been compared with the Abstract-The
film analysis and with the actual measured quantities in the case of the dummy experiments. The study reveals an excellent aaeement among, these results and indicates the feasibility of the application of the method to biomecha>ical studies. -
set of three simultaneous, coupled non-linear differen-
INTRODUCTION
In many biomechanics experiments, anthropomorphic dummies, cadavers and human volunteers have been used to study the severity of head injuries to different impact conditions under various environments. The determination of whether the head exceeds the maximum tolerable severity, or injury threshold, subjected to a given impact requires the evaluation of the Head Injury Criterion (HIC) based upon the head C.G. resultant acceleration according to Federal Motor Vehicle Safety Standard (FMVSS 208). Therefore, an assessment of head injuries needs the knowledge of a time history of the head C.G. resultant acceleration profile for the duration of the impact. When an anthropomorphic dummy is used, the resultant head C.G. acceleration can be obtained by measuring three acceleration components using a triaxial accelerometer mounted at the C.G. of the dummy head. In the cadaver and volunteer experiments, however, the measurement of acceleration at the C.G. of the cadaver and volunteer heads cannot be made directly and therefore must be restored to alternative methods or techniques using linear accelerometers to measure exteriorly. Mertz (1967), Clarke et al. (1971) and Ewing er al. (1973) have demonstrated the feasibility of using linear accelerometers in analyzing the head dynamics in planar motion, particularly in the case of the motion taking place in the mid-sagittal plane. The angular acceleration and the angular velocity of the head were obtained using four linear acceleration measurements. The kinematic principles involved have been extended to a more general analysis for various studies at WSU (1973) and by Alem (1974). The analysis in these studies was based on a three dimensional rigid body motion for which the kinematics are completely defined by six linear acceleration measurements at three different points on the head. Generally, the six accelerometer scheme results in a *Received 2 February 1976.
1976; in revised form 25 April
tial equations of the form i = f(o, M’),
(1)
where a : measured linear accelerations (g’s) w: angular velocity vector (rad/sec) i: angular acceleration vector (rad/sec’). Mathematically. equation (1) can be solved numerically using a Runge-Kutta, a predictor-corrector, or finite difference method for a given a measured exteriorly at the dummy and volunteer heads. Upon solving equation (1) by stepwise integration procedures, one can see that the calculation of the angular acceleration components require utilization of the angular velocities from the previous time step. The aforementioned numerical procedures in general introduce round-off errors which effect the accuracy of i through numerical computations. Figure 1 shows the result of calculated acceleration which increases without bound at some time step due to the cumulative errors as well as the measurement errors as pointed out by King et al. (1974) and Padgaonkar et al. (1975). Moreoever. for a particular choice of six acceleration measurements, equation (1) reduces to the form given by equations (Sa-5c). It is easily seen that the Jacobian determinant of this system is zero. Hence, the system is inherently unstable as indicated by Lapidus and Seinfeld (1971) and AIem (1974). It can be further asserted that no amount of additional manipulation of the six accelerometers would yield a set of equations which will be stable. Difficulties encountered using numerical integration methods prevented the six acceleration analysis from yielding reliable and useful results. To circumvent the difficulties, an alternative method has been proposed by Padgaonkar et al. (1975) using nine linear accelerations measured at four points on the head. With a special arrangement of the nine accelerometers, the angular accelerations can be obtained directly and algebraically from the measured linear accelerations at each time step without utilizing the values of w and icl computed at the
607
c. c. CHOU
608
and s. c. SINHA
40.00
Velocity
0 _;
29.1 mph.
Calculated Cal. - 3 Comp
30 00
E :: g
2000
,” 5 : a
IO.00
7:
0.00
20.00 Time
from
60.00 onset
of
100.00 sled
140.00
160.00
1
MS
deceleration,
Fig. 1. Unbounded response using six accelerometer scheme.
previous time step. The reliability of this approach was established by means of validation studies using both the hypothetical data and the experimental results obtained from a special experimental set-up (see Padgaonkar et al., 1975). The nine-accelerometer scheme has been applied to the analysis of the head kinematics for both the dummy and volunteer experiments. In this paper, only the dummy experiments will be described and the results are discussed. ANALYSIS OF THE NINE-ACCELEROMETER SCHEME The analysis of the nine accelerometer scheme proposed by Padgaonkar er al. (1975) is briefly redescribed herein since it serves as the basic concern of this paper. From the kinematics of a rigid body, the acceleration of a point P on the body is given by
ap = 110+ wx(wxpp) + 3xp,
-
a rP
=
Go
w~@w,.P +
-
WchPxP
-
wr(wyPzP
-
+
W,P,P
-
W,P~P
W
+
W&P
-
WyPxP.
(34
WXPZP) W&P)
Using Padgaonkar’s scheme, the nine linear accelerometers are arranged in the configuration shown in Fig. 2 consisting of a set of triaxial accelerometers located at the origin 0 and three biaxial ones mounted at locations 1, 2, and 3. At the origin, the triaxial accelerometers are oriented with their sensitive axes in the three orthogonal directions of the body-fixed axes. At locations 1,2, and 3, the sensitive axes of a pair of accelerometers are in the directions perpendicular to the axis on which they are mounted. Referring to the configuration shown in Fig. 2, the
7
(2)
Referring to the body-fixed frame, ap = axPi + a,.,$ + arpk = acceleration of the point P relative to the body-fixed frame; a0 = axOi + a,,$ + arOk = acceleration of the bodyfixed axes at the origin; w = rt:i + w).j + wzk = angular velocity of the body; i = wxi + w,j + wzk = angular acceleration of the body ; pp = pXpi = pyaj + Pzpk = position vector of point P relative to the body-fixed frame,
W,AP)
-
A 2 I I
A*2 2
where i, j and k are the unit vectors for this frame. From equation Q), the scalar components of ap along the body-fixed axes can be expressed as (Inertia frame) X Fig. 2. Nine accelerometer scheme configuration.
h04
Linear acceleration measurements position vectors of points 1. 2. and 3 with respect to the origin are Pl = P,d
(4a)
p2
(4b)
=
Px2i
(4c)
~3 = Pz&.
Equations (3a-c) can now be applied to express the acceleration components at point P in terms of the linear acceleration components at the origin, the angular velocities and accelerations, and the relative position of point P with respect to the origin for P = 1, 2 and 3. Using equations (4a-c) and after a straight forward manipulation, the following equations are obtained: 0 (54
wx = (% - GJYPyl - wywz WY= -(azz - hJYPx2 + wxwz
W
w:
WPg
(54
+
(54
=
(a,2
-
“yoYPx2
wx
=
-by3
wy
=
bx3
-
GoMP;3
-
w:wx
(54
k4
-
4OUPyl
+
Wxfi’y
WI
WL =
-
-
~,cJ/Pz,
wyw:
Elimination of the cross-product terms of the angular velocity components yields the following algebraic equations for the angular accelerations: wx =
-
%0)/‘&Q
-
(‘$3
$0)/&%3
(64
WY = (43
-
~xoY@z3
-
k&z -
%om%z
(6b)
wz = ($3
-
~,oY%%z
-
@,I
~xOY2Pyl.
69
b&l
-
-
where Uy with i = x, y and z, and j = 0, 1, 2 and 3, denotes the i-th acceleration component measured at the j-th location; pij denotes the distance in the i-th direction of the j-th location relative to the origin. By means of equation (6), the angular acceleration along a set of body-fixed axes can be computed with measured acceleration data obtained from an experimental sled test. The angular velocity w is then obtained by a simple integration. Having the values of & and w, the linear acceleration components at the head C.G. along the body-fixed axes can be obtained by transforming acceleration measurements at the origin through equations (3a-c). More explicitly, let acg = u,,i + uycgj + uzcgk be the acceleration vector at the head C.G. with respect to the body-fixed axes and pEe= pxcei + pwej + p,,,k the position vector of the head C.G. relative to the origin 0. Using equations (3a-c), the acceleration components of ocg can be calculated as: %cg = a,0 + wY(wxPYce - w&Xcg) - wz(wzPX,o- wXPzcp)+ w,P,,, - wzpYep (7a) $rg = $0 + wz(uVzrg - w&c,) - %(wJ’,, - w&c,) + w,P,,~ - w,P,,,
(7b)
azcg= %o + wX(wzPX,(I - K&J - %(wYP.cg- w&e) + %Pycs - wypxcg, (7c) where all the quantities on the right hand side of equations (7) are known.
The resultant acceleration 1~~~1 at the head C.G. is computed from 7 (8) beg/= \ d, + Gy + ‘Ii,,. The calculated resultant acceleration la@ from equation (8) is compared with the resultant acceleration computed using measurements at the head C.G. In addition. the HIC and Gadd Severity Index (GSI) are also computed for comparison based upon the measured and calculated head resultant accelerations. Moreoever. because of the nature of the experimental set-up and the greater relative magnitudes of w, and ~3).over M’,. M’,. wr and ivr (cf. Fig. 5). the motion during these runs can be considered nearly as a planar one. Hence it would be interesting to compare M‘,and wVwith a planar 2-D film analysis on an approximate basis. METHODOLOG1
In the experiments. a HYB 11 dummy was used as a test subject. The dummy was restrained in a rigid seat mounted in the direction of motion of the sled of WHAM
III. The sled was accelerated to a pre-determined velocity over a distance of 60ft. after which it was stopped with a given stopping distance. The deceleration pulse shape approximates a square wave.
The seat was rigidly constructed using steel angles for the main structural components and plywood coverings for the seat back and bottom. The dummy was seated on the rigid seat and restrained with a series of chest. shoulder and lap belts to keep his torso nearly in the upright position during the run. The feet were also strapped in position to the impact seat. The head accelerations for the dummy were measured with linear accelerometers mounted on a frame attached to the head. A head mount was fabricated for the nine accelerometer configuration. The head mount for the accelerometer was then attached to the top of the dummy head by screws. with s-axis being parallel to the AP axis. The experimental runs were performed at various barrier equivalent impact velocities of IO. 20 and 30m.p.h. with 10” stopping distance. The instrumentation on the dummy includes (a) a triaxial accelerometer (Endevco 2264) mounted at the C.G. of the dummy head and (b) nine uniaxial accelerometers (Endevco 2264) mounted on the 9-accelerometer head mount, The photographic instrumentation consisting of three high speed cameras: offboard lateral. onboard lateral and oRboard front provides photographic coverage of the motion of the dummy during the impact event. Timing marks were put on the film throughout the run to permit accurate calculation of the framing rate. In addition, a flash bulb was located, in the field-of-view of each camera to permit synchronization of the high-speed film with the transducer records. Films from the lateral offboard camera were utilized for quantitative analysis of head rotations. The transducer outputs were recorded on a Bell & Howell FM tape recorder. and were digitized by the A/D system at a conversion rate of 2000 samples/set/channel. The digitized data were put on a digital tape by the PDP-8 computer and were processed on an IBM 360/67 computer. The binary data were read and converted to true voltages which were then multiplied by a calibration factor to obtain physical units for use in the nine-accelerometer head analysis described in the previous Section, RESCLTS
AND
DISCUSSIONS
Results of the experiment are obtained by means of a transducer record analysis (using the A/D converter) and the high-speed movie analysis.
C. C. CHOU and S. C.
610 2000
1
SINHA
-
Run no. 128 Veloaty
1000
-3O.d 0.00
9.4 mph.
16QCO
6clcil
240.00
-
PA-cGuvlea) IS-CG (Med
+-+
RL-CG.(Mesl
32090
40000
MS
Time from onset of sled decelerotlon,
Fig. 3. Acceleration components measured at the head C.G. as a function of time.
Some typical iments
are shown
records obtained from the experin Figs. 3 and 4. These are the
Calcomp plots of the digitized data from the analog tape recorder for the run made at 9.4m.p.h. with a 10” stopping distance. Figure 3 shows the acceleration components in the AP, SI and LR directions as measured at the C.G. of the dummy head, respectively. The nine acceleration components measured externally at four different points on the head mount are shown in Fig. 4. Application of the nine acceleration component analysis to the data shown in Fig. 4 yields the angular acceleration, as a function of time from onset of sled deceleration, shown in Fig 5(a). A simple integration of the angular acceleration gives the angular velocity shown in Fig. S(b). Examples of the head kinematics from the high speed 2-D film analysis are displayed in Figs. 6-8. Figure 6 shows the head rotation about the axis perpendicular to the sagittal plane as a function of elapsed time from the onset of sled deceleration. The angular velocities and angular accelerations from the
-
L
P
e ._ z
-1000
000
3 ’
-
Au, AY,
-
AY,
0
80.00 77ma
24000
Isow
?2oca
from onset 01 sled durlrroOon,
Fig. 4. Nine linear acceleration components
of time.
Run no. 128 Velocity
$
WOO*
Ax, Ax, Ax.,
OCWt
2ooc I
40.w~
2
-
9.4 mph.
-
Ang.vel. X Ang.vel. Y Ang. vel. Z
C-W, -
Ang. occel. X Ang. occel. Y Ang. occel. Z
IO.00
0.00~
9 a -W.OO -
(b)
0.00
IEOCKJ 320.00 80.00 240.00 Time from onset of sled deceleration, MS
4oo.ccl
Fig. 5. Angular accelerations (a) and velocities (b) obtained from nine acceleration scheme.
4cOco
MS
as a function
Linear acceleration measurements 3ow-
w
Run no I28 Veioc~ty 94
Head mtatvm
mph
ZOCO.
-20001 000
6000
160.00
24ODo
Time from cnset of sled decelerofion,
32000
-301 0
4cQcQ
40
MS
film analysis are obtained using the harmonic analysis described by Haut et al. (1974). The synthesized data are then differentiated with respect to time to obtain velocities and accelerations. As previously mentioned, the angular velocity and acceleration obtained from the film analysis are compared with wY and wY from the nine acceleration analysis as shown in Figs. 7 and 8, respectively. It is seen that there is a good qualitative agreement, but quantitatively, the film analysis yields lower values for the angular velocity and acceleration as compared to the nine accelerometer scheme. This may be due to the fact that the comparison was made on an approximate basis. The acceleration components calculated at the C.G. of the dummy head are shown in Fig. 9(a). Fig. 9(b) shows the calculated head resultant acceleration using calculated head acceleration components. The comparison between the calculated and measured head resultant accelerations are presented in Fig. 10 along. with the resultant acceleration measured at the origin 0 as indicated by the curve ‘RES-Base’. The HIC and GSI computed for the measured and calculated resultant accelerations are tabulated in Table 1. The
200
240
260
329
360
400
MS
1 reveal that excellent correlations between these quantities have been achieved for dummy runs at various velocities. CONcLUSlONS
The feasibility and reliability of the nine-accelerometer scheme has been demonstrated to obtain the head kinematics of a dummy when considered as a rigid body. Comparison of the head kinematics -
60
. Ix)
Time
from
a-150 0
40
160
. 200
. 2w
9Ati
280
. 320
c+wet of sled deceleration.
scheme
. 360
20.00
10.00
MS
od
PA-CG
-
IS-CG.
(CAL.) (CAL.)
-
RL-CC.
(CAL.)
m
RES-CG
(CAL.)
(b)
-00 zj L 0.00
40.00 Time
120.00 from
onset
200.00 of
+ 4M)
Fig. 8. Comparison of angular accelerations obtained from nine accelerometer scheme and film analysis.
-
‘; 3:: 5
from
results shown in Fig. 10 and in Table
Run No. 128 velocity 9.4 mph.
V al
EO
Fig. 7. Comparison of angular velocities obtained nine accelerometer scheme and film analysis.
Fig. 6. Head rotation as a function of time.
Vzl
120
60
Time from onsef of sled decelemr~on.
sled
deceleration,
240.00
360 00 MS
Fig. 9. Acceleration components (a) and resultant acceleration (b) calculated at the head C.G. vs time.
C. C. CHOU and S. C. SINHA
612
Run No. I28 velocity 9.4 m.p.h.
PO.00
-
to.00
b
3
IO.03
2
40.00
30.00
s g 2 2
Run No. I29 velocity 18.0 mph. -RES-C.G.(CAL.) -RES-CI;(MES.) RES- BASE
20.00
P P c
RES-C.G. (CAL.) RES-C.G. (MES.1 RES-BASE
-
Run No. 137 velocity 31.0 mph RES-C.G.(CAL.)
-
RES-BASE RES-C.G.(MES.)
20.00
10.00
0
120.00
40.00 Time
from
200.00
onset
of
260.00 sled
360.00
deceleration,
MS
Fig. 10. Comparisons of measured and calculated resultant head acceleration at C.G. for various impact velocity runs.
Table 1. Comparison
of the HIC and GSI computed for the calculated and measured head resultant accelerations
Run no.
Vel. (m.p.h.)
stop. dist. (in.)
128 129 137
9.4 18.0 31.0
10 10 30
HIC GSI Using head resultant accel. Cal. at Mea. at Cal. at Mea. at C.G. C.G. C.G. C.G.
obtained from the film and the nine acceleration analysis and comparison of the calculated and the measured accelerations at the C.G. of the dummy’s head show that excellent correlations of the results have been obtained. Based upon this comparison, it is concluded that the nine accelerometer scheme provides a useful means of using linear acceleration measurements to obtain kinematic information required in many biomechanics studies. The method has been applied to the volunteer experiments in studying responses of the human neck in flexion, extension and lateral tlexion and the results will be reported in the future. Acknowledgements-The authors would like to thank the reviewers for their valuable comments which led to the
5 62 168
5 64 159
7 76 218
7
final version of this manuscript. They also express their gratitude to Dr. Haut for using his harmonic analysis program. REFERENCES
Alem, N. M. (1974) The measurement of 3-D rigid body motion. Proc. 21ul Ann. int. Meeting of the Ad Hoc Committee on Human
Subjects for Biomechanical Research.
(Ed. by Hirsh, A. E.) pp. 60-64. Clarke, T. D., Gragg, C. D., Zimmerman, R. M. and Muuy, W. H. (1971) Human head linear and angular accelerations during impact. Proc. 15th Stapp Car Crash Conf., Coronado, CA. DOT Contract 83-146-3-753 (1973) Injury assessment of belted cadavers. Progress Report No. 6. Biomechanics Research Center, Wayne State University (WSU). Ewing, C. L. and Thomas, D. J. (1973) Torque versus angular displacement response of human head to -Gx im-
Linear acceleration measurements pact acceleration. Proc. lit/l Stapp Car Crash ConjI. Oklahoma City. OK. Ham. R. C., Remmers. E. P. and Meyer. W. W. (1974) A numerical method of film analysis with differentiation with application in biomechanics. Proc. Sot. Photo-oprical Instrumentation Engineers. 57, pp. 5361. King. A. I.. Padgaonkar. A. J. and Krieger. K. W. (1974) Measurement of angular accelerations of a rigid body using linear accelerations. Proc. 2nd Ann. Inr. Meeting
ofthe Ad chanical
Hoc Committee on Human Subjects for BiomeResearch. (Ed. by Hush. A. E.) pp. 48-59.
613
Lapidus. L. and Seinfeld, J. H. (1971) Numerical Solution of Ordinary Difirenriul Equations, pp. 128-1.51. Academic Press. NY. Mertz. H. J. (1967) Kinematics and kinetics of whiplash. Ph.D. dissertation. Wayne State University. Detroit. Ml. Padgaonkar. A. J.. Krieger. K. W. and King, A. 1. (1975) Measurement of angular acceleration of a rigid body using linear acceleration. J. oppl. Mech. 97(3) pp. 552-556.
