A simple occupant dynamics model

J. Biomechonics,

Vol. 1, pp. 185-191.

Pergmwn Press, 1968.

A SIMPLE

Printedin Great Britain

OCCUPANT

DYNAMICS

MODEL*

J. R. WEAVER

Impact Dynamics Department, Automotive Safety Research Office, Ford Motor Company, Dearborn, Michigan Abstract- A simple, accurate, two mass, two-degree-of-freedom mathematical model has been developed to simulate lap belted vehicle occupants subjected to frontal collision loadings. This model was designed as a substitute for more complex models when only seat belt loads and head impact velocities are being studied. The model has been programed for digital computers and tested against the Cornell eleven-degree-of-freedom model for comparison. Differences in results between the two models were less than 5 per cent. BACKGROUND

modeling a compromise must be made between the accuracy of the simulation and the complexity of the model. It would be fortuitous if the simplest model provided the best simulation, but this is seldom the case. For simulating occupant dynamics during vehicle collision, the Cornell Aeronautical Laboratory developed an elevendegree-of-freedom model (McHenry and Naab, 1966) which provides the most complete simulation available. But it can be very cumbersome to use. Widely used singledegree-of-freedom models are simple and convenient but inherently inaccurate. The described two-degree-of-freedom model below retains much of the simplicity of the single-degree-model but provides an improved simulation of occupant dynamics. It was designed as a substitute for the Cornell model when the complexity of that model is not needed. IN

MATHEMATICAL

lower mass to the foundation. The excitation to the system, in the form of deceleration versus time, is applied at the foundation. The seat belt load is applied tangent to a hip circle and the belt is “wound up” on the hip circle as the upper mass rotates. The belt load is thus a function of the displacement of the lower mass and the rotation of the upper mass. This method of attaching the belt to the body is very similar to that used by Cornell in the eleven-degree model. A constant friction force, F, applied to the lower mass simulates the energy dissipation by seat contact forces and body joint restraints in the Cornell model. The equations of motion of MI and Mz were obtained from Lagrange’s equation (Langhaar, 1962) and the following procedure: 1. Write equations for the kinetic and potential energy of the system. (For complex systems, the energy equations are easier to obtain than the motion equations .) 2. Differentiate the energy equations as indicated by the Lagrange equation. 3. Equate the resulting differential equations to the corresponding dissipative force. 4. Solve for the accelerations gi;, and 8.

DISCUSSION

The two-degree-of-freedom model consists of two masses connected by a rigid link (Fig. 1). The pelvic or lower mass, MI, is free to translate horizontally, and the upper mass, Mz, is free to rotate about the link pivot. The seat belt, a linear spring, connects the *First

received

8 November

1967; in revisedform

11 January 18.5

1968.

J. R. WEAVER

Fig. 1. Two-degree-of-freedom

The equations of motion for M, and MZ are shown in Appendix A. They are complex, non-linear differential equations. Since an analytical solution was not possible, they were numerically integrated on a digital computer to obtain the velocity and displacement time histories. The BASIC (Kemeny and Kurtz, 1965) program used and sample data are shown in Appendix B. The required input is 1. The number of pairs of time/deceleration values desc.ribing the foundation deceleration (X0). 2. The pairs of time/deceleration values. 3. Vehicle initial velocity. While the seat belt stiffness, belt anchor location, initial torso angle and the magnitudes of MI and MZ are not required input data, the program user may easily change them. The belt anchor location may be varied to yield belt angles between 290” from horizontal. The initial torso angle, belt stiffness and mass magnitudes may be any desired value. With minor program changes, non-linear belts may be simulated.

model.

As output, the program prints belt load, torso link rotation and relative head velocity, each as a function of time. Head velocity is measured perpendicular to the end of the link and relative to the foundation. This velocity simulates a head to instrument panel impact velocity. RESULTS

The two-degree-of-freedom model was tested for accuracy against the Cornell elevendegree-of-freedom model. Twelve matched runs were made, using four geometric deceleration pulses and three initial velocities. Belt stiffness and inclination were 12,000 lblft and 38” from horizontal for both models. The Cornell model used a 150 lb articulated body,while the two-degree model used weights of 65 lb for M, and 60 lb for MZ. The peak belt loads from the twelve runs are shown in Fig. 2. The average difference between the two models is 1.9 per cent and the maximum is 4-7 per cent. The head impact velocities (defined relative to the vehicle) are compared in Fig. 3. These are the velocities

A SIMPLE OCCUPANT

DYNAMICS

MODEL

187 INITIALVELOCITY (MPH)

INITIALVELOCITY (MPH) PULSE SHAPE

PULSE SHAPE

20

30

4"

38.9

50.5

39.0

1 I /

DEGREE BELT LOAD (LBS)

51.0 I

61.6 I

CORNELL MODEL HEAD IMPACT VELOCITY (FThEC)

r\/

~1.4%

-2.3%

-1.4%

- .6%

-1.6%

-3.1%

-4.7%

+2.4%

+ .4%

P nIIFFERENCE

IIFFERENCE -

-1.4%

Fig. 2. Comparison of peak belt loads.

occurring when the head center intersects a plane at 45” from horizontal passing through the hip center. The average difference in head velocities is 1.7 per cent and the maximum is 3.8 per cent. SAMPLE APPLICATION

As an example of possible usage of the program, a study was conducted to determine the effect of deceleration onset rate on the vehicle occupant. (Onset rate is the initial slope or rate of change of the deceleration pulse.) A triangular deceleration pulse with an initial velocity of 44 ft/sec for 0.120 set duration and a peak value of 22.8 g (g = 32.2 ft/sec2) was selected. The time to the peak was varied between 0.01 and OXI set, which yielded onset rates between 2280 glsec and 22800 gfsec. The resulting peak seat belt loads and head impact velocities are shown below. The total variation in seat belt load is 1a8 per

B.M. VOL.

I NO.

3-C

Fig. 3. Comparison of head impact velocity.

Onset rate (g/set)

Peak belt load (lb)

Head impact velocity (ftlsec)

2280 3260 4560 5680 7600 11380 22800

3129 3112 3092 3090 3086 3082 3074

45.4 45.2 45.2 45.0 44.9 44.8 44.8

cent and in heat impact velocity is 1.3 per cent. Thus, within the range studied, onset rate has a negligible effect on system dynamics. This result corroborates the findings of other investigators (Kornhauser, 1964). CONCLUSIONS

The two-degree-of-freedom following desirable features: 1. It provides

model has the

an accurate estimate of the restrained occupant response to vehicle deceleration.

J. R. WEAVER

188

2. It can be used by people with little knowledge of digital computers. 3. It is easier and cheaper to use than the more complex Cornell eleven-degree-offreedom model.

APPENDIX A EQUATIONS OF MOTION* 1.

Kineticenergy equation T = &+!f,B,~ + 3M*(1,2 + 2X&i cos I9+ &)

.

2. Potential energy equation

model is of The two-degree-of-freedom necessity less general than the Cornell model: 1. There are no provisions for upper torso restraints. 2. The torso is not articulated and there are no arms or legs. 3. There are no head or chest impact surfaces. REFERENCES McHenry, R. R. and Naab, K. N. (1966) Computer simulation of the automobile crash victim in a frontal collision-A validation study. Repor t No. YB-2126V- 1, Cornell Aeronautical Laborator ‘y, Buffalo,New York. Langhaar, H. L. (1962) Energy met1hods in applied mechanics. p. 247. Wiley, New York. Kemeny, J. G. and Kurtz, T. E. (1965) BASIC. Dartmouth College Computation Center, Hanover, New Hampshire. Kornhauser, M. (1964) Structural effects of impact. p. 94. Spartan Books, Baltimore, Md.

V=~~((6)‘-M,gL,(l-cos8) where 8 = seat belt stretch. IZ= 32.2 ftlse?. 3. Dissipative forces R=-F

4. LaGrange’s equation

$(g)-$($)-z+g=Ri. 5 Equations motion of

2, =

1 M, sin2 8 + M,

$M,L

sin 8-F

Ri cos 6+ h

-K8(_-- >1 \

e=

-cos 0 L(M,

sin2 tI+ M,)

$M,L

_,I” where

APPENDIX B

1 2 3 4 5 6 7 8 9 10 II 1; 14 15 16 17

THURS.

R: 56

OY/28/67

REM REM FEN REM REM PEN REN PEM PEN REM PEN ;;;

PR0GRAN DEFINITI0NS V A.R I P B L E MI AND M2rLBWER AND UPPER MASSES (SLUGS) JO = INITIAL T0RS(? ANGLE FRBN VERTICAL.(RAD) L = LINK LENGTH TO UPPER T0RS0 NASS, LI q LINK LENGTH T0 HEAD CENTER.(FT) RI - RADIUS 0F HIP CIRCLE,(FT) . K z SEAT BELT STIFFNESS,(LE/FT) CO - INITIAL BELT ANGLE WITH H0RIZ0NTAL,(RAD) CY = BELT LENGTH FRBM ANCHaR T0 HIP CIRCLE,(FT) TYPE

N P U T FBRNAT ” 1000 CATA N ”

REM PEN

TYPE

”

1001 (TINE

REM

TYPE

”

1005

I

(N-NUMBER 0F TINE AND DECELERATIQN PBINTS BEING INPUT) CATA H(I),F(I),H(2),A(Z) H(N) ,A(N) ” (..... AND DECELERATI0N CBORD’S IN SEC ANC FT/SEC/SEC) VEHICLE VEL0CITy,FT/SEC) CATA VO ” (IFIITIAL

REM

,g

;;;

;f;;;;;gO’

30

;‘lAT

AtZER(50)

*The variables and constants used are defined by Fig. 1.

_

sin l3-- F

TWO DEGREE-OF-FREEDOM BASIC LANGUAGE PROGRAM TYODF

- M,g sin 0 cos 0

RI(M,+MI) M,L cos 8 >

+A

1

A SIMPLE 40 50

60

70

PO 90

100

110 120 130 140 15c 154 156 15P 160 162 164 166 170 Ia0 I90 200 210 220 2;10 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 420 490 500 510 520 530 540 550 560 570 5RO 590 600 610 620 630 640 650

REAC N F0R J=l TD N READ H(J),F(J)

OCCUPANT

DYNAMICS

MODEL

STEP 1

NEXT J PEAC vo LET Ml-65/32.2 LET 1'12~60/32.2 LET PI--.33 LET Lz1.3 LET JG=-15/57.3 LET LI-2.2 LET K:12CO0 LET [?9-1.71 LET CO=38/57.3 LET AlzSBR(D9t2+Rlt2) LET CZ:FTN(FI/D9) LET C3rCO-C2 LET 5t=AluSIN(C3) LET G:Al*CGS(C3) DIM X(2,R),Y(2,8),2(2,S) MAT X?ER(2,F0 BAT YzZER(2,8) MAT Z-ZER(z,B) PRI NT PRINT PRINT PRINT PRINT PRINT" TIME BELT LBAD R0TATI3N IlEAD VELr:CITY" PRINT" (SEC) (GEG) (LBS) (FT/SEC)" LET B=O LET ZI=O LET z2=0 LET H-.OOf LET T-H LET Y(O,O)=VO LET Y(I,O):VO LET X(2,0) -Jc: LET J:l LET I=1 LET Z~O,I~~A~J~+~A~J+l~-A~J~~*~T-H~J~~/~W~J+I~-lI~J~~ LET Y~O,I~=Y~O,I-l~+~t~O,I~+Z~O,I-I))*H/2 LET X~O,I~=X~O,I-I~+Y~O,I-I~*H+Z~O,I-l~*~?2/3+Z~O,I~~Ht2/6 GBSUF 960 LET Z(l,I)=ZI LET LET

YI-Y~l,I-1~+~Z~I,I~+Z~1,I-l~~*H/2

XI-X~1,I-I~+Y~1,I-l~*t~+7~I,I-l~*Ht~~/3+Z~t,I~~Ht2/6 LET Z(2,1)-22 LET Y2-Y(2,I-I)+(Z(2,I)+Z~2,I-l))*ti/2 LET X2~X~2,I-l~+Y~2,I-l~*H+Z~2,1-1~*t(t2/3+2~2,1~*ht2/6 LET A:YZTZ*M~*L*SIN(X~) G0SUE P70 IF E!>O THEN 540 LET ErO LET FI-0 LET C=tMI+M2*SIN(XZ)t2 LET D~.33*K*L5*(M1+~2)/(~~2*L*C~S(~2)) IF Da0 THEN 580 LET C=O LET Zl=(A-E-F+.33+FI*C~S(X2)/L-32.2?M2*SIN(X2)*C0S(X2))/C LET Z~I-CBS(X~)*(A-B+D-F-~~.%*O*TAN(X~))/(L*C) IF Zl=O THEN 620 IF ABS((ZI-Z(l,I))/ZI)~.OOOl THEN 430 IF 22~0 THEN 640 IF ((Z2-Z(2,1))/Z2)>.0001 THEN 430 LET Z(I,I)-Zl LET Y(I,I) -Yl

189

190 660 670 @O 690 700 710 720 730 740 750 760 770 780 790 eoo 810 isi 840 es0 860 870 880 890 900 910 920 930 940 950 960 970 980 990 1000 99%

J. R. WEAVER

LET x(1,1)-x1 LET z(2.1) =ZZ

LET Y t2;I 1 :Y2 LET X(2,1)=X2 IF I-4 THEN 730 LET I=I+I GBT0 820 LET V9=~Y~l,I~-Y~o,I~~*c0s~X~2,I~~+LI~Y~2,I~ PRINT T,Fl,j7.3*X(i,I),VS F0R QzO T@ 2 STEP 1 LET Z(Q,O)=Z(F,Il LET Y(Q,O)=Y(Q,I) LET X(Q,O)=X(Q,I) NEXT Q IF X2745/57.3 THEN 10 LET 1x1. IF INT~T+IOOOO,~INT~H~J+I~*l0000~ THEN 840 G0T0 850 LET J-J+1 LET T-T+H GDTB 390 LET L4:SQR~~G+XI-X~O,I~~t2+81t2-Rlt2~ LET Cl-ATN~Rl/L4~+ATN~~I/~G+Xl-X~0,1~~~ LET LET LET LET LET LET RETURN LET F-350 THEN 990 IF WI-Y(O,I)+RliY2)>0 LET F-O RETURN GATP END BASIC PROGRAM

SAMPLE

INPUT

Total input for 30 mph barrier simulation 1000 DATA 17,O, 0, .007,-675, .Ol,-289, .015,-610, .019,-257, ,025,~837 1001 DATA .03, -386, .034, -675, ,045, -514, .O49,-869, .055,-514, .O62,-708 1002 DATA .072,-289, .079, -322, .082,-193, .O97, -96.4, .O98,0 1003 DATA 44.6 Number of deceleration data points

LOOO

DATA

17,O,O,OO7,-675,

.Ol.-289,

.015,-610,

.019,-257,

.025,-837

1 I

Deceleration table (set, ft/se?) 1001

DATA

.03, -386, .034, -675, .O45, -514, .O49,--869, .055, -514, .O62, -708

Deceleration table 1002

DATA

.072, -289, .079, -322, .082, -193, .O97,-96.4,

.O98,0 T

Deceleration table 1003

DATA

44.6 T IInitial

velocity (Ftlsec)

A SIMPLE

OCCUPANT

DEFINITION Al

DYNAMICS

OF PROGRAM

Initial distance from belt anchor to center of hip circle, ft. A CJ) Foundation deceleration at time H (J ) , ft/sec2. Dummy variables used to calculate d,B.C.D,E accelerations Z I and 22. Bl Vertical distance from belt anchor to center of hip circle, ft. CO Initial angle of belt from horizontal, radians. Angle of seat belt from horizontal, Cl radians. c2 Initial angle between belt and anchor/ hip-center line, radians. Initial angle of anchor/hip-center line c3 with horizontal, radians. D9 Initial belt length from anchor to hip circle, ft. Friction applied to MI, pounds. F Spring force, lb. Fl distance from belt G Initial horizontal anchor to hip center, ft. Time coordinates of deceleration table, H(J) sec. H Integration time increment. sec. I Printing index. J Deceleration table index. JO Initial link angle, radians. K Belt stiffness, Ib/ft.

MODEL

191

VARIABLES L

Distance from link pivot to upper mass, ft. Ll Distance from link pivot to head center, ft. L4 Distance from belt anchor to tangent point on hip circle, ft. L5 Belt elongation, ft. Dummy variables. L6, Ll L8 Partial derivative of L5 with respect to Xl, ft/ft. Ml Lower mass, slugs. M2 Upper mass, slugs. N Number of A(J ) and Hf J ) points. Q Transfer index. Rl Radius of hip circle. ft. T Time, sec. vo Initial foundation velocity, ftlsec. v9 Tangential head velocity relative to foundation, ft/sec. X(0, 1) Displacement of foundation, ft. X(1, I) Displacement of lower mass. ft. X(2,1) Angular position of link, radians. Interim values ofX( 1. I) andX(2, I). x1,x2 Y(O.I),Y(i,I),

Y(2, I) Yl. Y2 Z(0, I), Z( I, I). Z(2. I) 21.22

Corresponding velocities. Interim values of Y(1, I), and Y(2. I). Corresponding accelerations. Interim values of Z( 1, I), and Z(2, I).