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162 TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 17, No. 4, 1998

Crash Tests and the Head Injury CriterionHANS-WOLFGANG HENN

Submitted January 1998, accepted February 1998

AbstractThe Head Injury Criterion (HIQ model has been

developed to measure quantitatively tbe head injuryrisk in crash situations. Using a computer algebrasystem (here MAPLE), students are able to analyseresults reported from a real crash test carried outwith Mercedes-Benz cars. The discussion of themodel provides first an excellent motivation for theRiemann integral, serves secondly as a piece of realconsumer enlightenment, and demonstrates thirdlythat the need to (oversimplify reality may producehardly meaningful results, despite considerablemathematical expenditure.

The negative acceleration during a crashFORTUNATELY, today many consumers pay moreattention to the safety of their cars than to fancyspoilers and aluminium rims. Crash tests (Fig. 1)can give valuable hints on advantages or dis-advantages of construction.

A number, denoted HIC, is often quoted in testsand reports without it being clear what thatnumber is. To clarify this idea is, on the one hand,a case of real consumer enlightenment, and on theother an excellent motivation for the Riemanintegral.

The HIC (Head Injury Criterion) is intended tojudge the head injury risk quantitatively. In thecase of an accident, the head load results from toohigh (negative) acceleration (or better, decelera-tion) values during the crash. Constructionfeatures such as crumple zones and airbags areemployed to prolong the period of braking on thedriver's body and thus to lower the decelerationduring the crash below critical values. The dum-mies used in crash tests have several sensors fixedto the head area which record the absolute value ofthe deceleration and its dependence on time. It isplausible that this head load is higher the larger thevalue of the deceleration and the longer thedeceleration lasts. Some empirical values areknown from tests with animals (not ethically with-out question), with corpses, and—especially re-markable—from the evaluation of injuries to

boxers. The goal is to get a quantitative evaluationof the head load from the recorded a-t curve.

Velocity and decelerationIf you show students a qualitative a-t diagram

like that in Fig. 2, they will usually propose as afirst model to measure the area below the curve(they even do this before the introduction of theintegral). However, the deceleration curve of Fig.3, which contains the same area, but obviouslylooks "more dangerous", shows that this firstproposal is not useful. It should be discussed withthe students what the deceleration curve measuredduring the crash actually describes. If a carchanges its velocity during the time At from V]to vj, the quotient

a = •At

is the average acceleration (<0 if the car slowsdown, > 0 if it speeds up). As usual, the quotient ofdifferences leads to the derivative in the limitAt —* 0, going from the average rate of changeto the local rate of change; here the instantaneousacceleration a{t) = v'(t).

The value a(t), measured in m/s2, shows in anintuitive way how much (more or less) the velocity,measured in m/s, will increase or decrease duringthe next second. Conversely, given the accelerationcurve (as from measurement in the crash test) thev-t curve can be reconstructed from the rates ofchange. Accelerating with velocity vo starting attime to, we have t > to

v(0 = v0 + f a(r) dr.

Deceleration during the crashWith "normal" braking it is quite acceptable to

calculate using a constant deceleration (cf. [1;p. 28], [2]). Modern cars reach values from 8 m/s2

to 11 m/s2, that is, about the terrestrial accelera-tion due to gravity, g. Racing cars reach values upto 5g (which is possible using special mixtures of

© The Institute of Mathematics and its Applications 1998

TEACHING MATHEMATICS AND ITS APPLICATIONS Volume 17, No. 4, 1998 163

km m m m m™D*CFO1292

Fig. 1. Source: ACAC Motorwelt 3/1993

y -f

Fig. 2 Fig. 3

the tyre's rubber and the high values of the down-ward pressure caused by the aerodynamic shapingof racing cars. For crash tests, one starts withvelocities of 30mph (« 48.3 km/h « 13.4m/s). In"normal" braking, the process lasts about 1.5—2 swith constant deceleration. Let your studentsabsorb that fact for future reference! During acrash, the braking process lasts about 100-200 ms, and the life-threatening deceleration peaks

have a duration of about 10 ms. The process ofdeceleration of the head area needs to be changedfrom the dangerous form of Fig. 3 to the regularform of Fig. 2 with a values clearly less than lOOg.Crumple zones and other constructive devices areused to achieve this. To evaluate the completeprocess of deceleration one tries to weight thedecelerations appropriately. Figures 4 and 5 showthe result of a crash test. Both are of the same car


ISO

160

I4O

120

100

go

60

40

20

0

BeschleunigungKopf Res.AAAR

HIC =681.85A-3ms = 89.65 g

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300ms

Fig. 4

180

160

140

120

100

80

60

40

20

0

BeschleunigungKopf Res.AAAR

HIC = 307.84A-3ms = 43.47 g

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300ms

Fig. 5

type, the former E-class of Mercedes-Benz (modelW 123). Figure 4 shows the crash performedwithout an airbag, Fig. 5 with an airbag. Themeanings of the marked rectangles and the valuesof HIC and A-3ms will be explained in thefollowing. Ask your students to describe the twocurves.

Severity Index and the Head Injury CriterionThe first model used in practice was the Severity

Index SI. The value of the deceleration wasweighted by a power n, the value of which dependedon the part of the body according to empiricalexperience. For the head, n = 2.5 was chosen. TheSI was defined in mathematical form by

SI= a(t)"dt,Jo

with n = 2.5 for the head, and with T equal to thecomplete duration of the deceleration having aneffect on the head (cf. [3] for the definition of theSI and for the following definition of the HIC).This formula is already understandable before theintroduction of the integral and can be used to leadto the integral.

The validation of the model was not satisfactoryfor comparison of different car types and differentaccident situations. Adaptation of the empiricaldata led to the currently used HIC value as arevised model. First, we look at the mean accel-eration between two times t\ and ^ (this simul-


s!K

Fig. 6

taneously prepares for integration as taking themean, cf. Fig. 6):

1 ['•a =

h - h J,,

a(t)dt.

The factor (t2 - t\) • a25 takes into account bothduration and weighted value of the decelerationfor the time interval t\ to t2. The maximum of allsuch numerical values is the Head Injury Criterion

HIC = max\(t2 - h)(—^— f'a{t)dt]

'..* [ V'2-r,J,, JApart from the condition 0 < t\ < t2 < T, the

starting point t\ and the time period At = t2- t\are arbitrary at first. To be able to evaluate themeasured curve practically, one makes some sim-plifying model assumptions:

• We demand t2 — t\ < 36 ms. Longer decelera-tion times do not increase the injury risk,according to experience.

• Peak acceleration values must last 3 ms. Thisrequirement has reasons of measurement tech-nique and is supported by the assumption thatdecelerations of shorter duration do not haveany effect on the brain.

The computation of the HIC from a measuredcurve, which actually consists of a huge number offinally interpolated points, is still very laboriousand is not possible without high-speed computers.The calculation is not done using calculus methods(one does not have an equation of the curve; laterwe will do it nevertheless!) but with numericalmethods. In both Figs 4 and 5, the markedrectangle corresponds to the ///C-determiningmaximum-interval (cf. Fig. 6). The value A-3msdescribes the maximal deceleration value lasting atleast 3 ms.

From the two measurements, we calculate asfollows.

• Fig. 4: HIC = 681.85; the rectangle correspond-ing to Fig. 6 has the basis t\ = 63 ms, t2 = 99 ms,At = 36 ms and a = 89.65g.

• Fig. 5: HIC — 307.84; the rectangle correspond-ing to Fig. 6 has the basis ri = 58 ms, t2 = 94 ms,At = 36 ms and a = 43.47g.

The airbag decreases the HIC from 682 to 308.Incidentally, what is the unit of the HIC1 It is usedas an unitless number in the publications of themotor press.) The A-3ms value decreases fromapproximately 90g to approximately 43g. In bothcases, the maximum appears for At = 36 ms. Thesame holds true for all other crash tests of which Ihave seen the measurements up to now.

Model calculations for the HICThe aim of the following is a better under-

standing of the complicated formula for theHIC. We take the deceleration as a non-negativereal function a : R -* R, t —» a(t), which should ofcourse have all "desirable" characteristics such ascontinuity and differentiability. To estimate theHIC, first we calculate the average deceleration afor the time interval from t to t + d for arbitraryvalues t € R and d € R. Then we weight it with thepower 2.5 and multiply it with the effective time dto d<?5. The maximum of all such values is theHIC. We simulate this analytically with a functionH:

a(r)dT2.5

This formula gives the unit of the HIC if the time tis measured in seconds and the deceleration a in g,then the unit of the HIC is sg25 (which is usuallyomitted).

As is common in school, we can understand Has a family of functions with variable / and par-ameter d. Of course, it is more useful to imagineH(t,d) as a surface of which the highest pointabove the t-d plane is the HIC. To establish theexistence of that point, one considers (as a goodrepetition of the basic ideas) the following:

H(t, d) > 0 for any te R and d > 0.For fixed t one has H(t, d) —> 0 for d —> 0 andfor d —» co.For fixed d one has H(t, d) —• 0 for r -+ -coand for t —* co.

A possible way to calculate the HIC is estima-tion of the global maximum H(d) of H (for fixedd) and then estimation of the maximal H(d) ford > 0. Of course, that is only possible for verysimple functional forms of H. A useful treatment


for any .//-function is the following graphicalmethod in which the estimation of the maximumof the three-dimensional t-d-H(t,d) surface re-duces to a two-dimensional problem. One drawsfamilies of functions H(t,d) with variable t andparameter d (or with variable d and parameter 0and with their help estimates with given accuracythe desired maximum HIC. In any case, the use ofa computer algebra system is necessary. We havebeen working with MAPLE.

Model functions for the Mercedes-Benz crash testsThe essential process of deceleration in Figs 4

and 5 lies between 0 and 160 ms. A curve like theone in Fig. 5 can be represented by a rationalfunction a defined as

U (t-c)2 + d

One needs the adequate addition of two of thoseterms for the curve in Fig. 4. To adapt the respect-ive coefficients, we magnified Figs 4 and 5 andmade tables of the values at intervals of 5 or 2.5 ms(cf. Table I; aw means with airbag, a,, meanswithout airbag). Using these values, we plottedthe measured curves as polygons with the help ofMAPLE.

The a functions have been fitted to the polygonsthrough suitable modification of the coefficients.Figure 7 shows the result for the crash with airbag,Fig. 9 for the crash without airbag; the measuredpoints are connected with thin polygons, themodel curves aw and ao are drawn with heavylines.

We choose aw as a fitting function for the crashwith airbag:

t/ms

a .

r/ms

Oo

t/ms

Ow

"o

20

00.1

65

35.540

100

2428

25

1.52

70

42.543

105

17.514.5

30

33.5

75

44.541

110

14.511

Table

35

3.5

80

41.537.5

115

98

1

40

10.58.5

85

35.540.5

120

7.57.5

45

10.514

90

3156

125

7.59.5

50

1720

92.5

98

130

6.54.5

55

2330

95

28.572

135

53

60

32.537.5

97.5

49

50

40

30

20

10

0 20 40 60 80 100 120 140t

Fig. 7

aw{t) =22000

(t - 74)' + 500

(/ measured in ms, a in g; cf. Fig. 7).For the crash without airbag, we first modelled

the two parts of the curve with functions/ and g ofour type. This is given in Fig 8. We took

/(') =21500

(t - 70)' + 5001800

(r-92.5)2 + 18

100

80

60

40

20

0 20 40 60 80 100 120 140t

Fig. 8


100

80

60

40

20

00 20 40 60 80 100 120 140

Fig. 9

Superposition and repeated fitting lead to themodel function OQ drawn in Fig. 9, with

16400• + •

1480

( r - 6 8 ^ + 4 0 0 (1-93)' + 18

Graphical estimation of the HIC for the modelfunctions

The manipulation of such terms is relativelylaborious, so that one has to use a computeralgebra system (CAS) like MAPLE or DERIVE.

With the simpler term am, it is possible to discussthe curve with the help of the derivatives and so toget the HIC. A CAS such as MAPLE is powerfulenough, though one should compare the requiredpower and the gain!

In any case, it is more useful to inspect thefunctions Hv(t,d) and H0(t,d) with graphicalmethods and thus to calculate the HIC for ourmodel functions aw and Oo- For this, we use theability of MAPLE (or any other CAS) to draw 2-Dand 3-D graphs. The graphs of the HIC functionshw and h0 can be understood as surfaces above thet-d plane. Figures 10 and 11 represent thosesurfaces.

One recognises clearly the distinct maxima ofthe surfaces. To be able to calculate them moreexactly, MAPLE has additionally drawn the re-spective two-dimensional parameter curves ofHw(t,d) and H0(t,d) for the parameters d = 3,14, 25, 36, 47, 58, 69, 80 and 91. This is representedin Figs 12 and 13. Your students should carefullyinspect and compare the 3-D and the 2-D graphs!With increasing parameter d, the curves shift theirmaximum from right to left and become broader.

The discussion of the families of curves makes iteasier to understand the integral terms Hw(t,d)and H0(t, d). In both cases, the global maximum isachieved at d > 36 ms. It follows that the maxi-mum for d = 36 ms defines the HIC; the parametercurves show the respective values. Table II demon-strates that our model results correspond very wellwith the Mercedes-Benz measurements.

140

Fig. 10. Hw(t,d) for the crash with airbag.


2 0 40 60 8050 d

100 120 140 0

Fig. 11 . H0(t,d) for the crash without airbag.

20 40 60 80 100 120 140-40 -20 0

Fig. 12. Family of curves with parameter d of Hw(t,d) for the crash with airbag.

With airbagWithout airbag

Model

HIC

301683

Table II

calculation

Time interval

56-92ms63-99 ms

Mercedes-Benz vaJue

HIC

308682

Time interval

58-94 ms63-99 ms

Critical review of the modelThe complicated formula, the huge computer

used for the calculation and the exact numerical

result feign a meaningfulness that crash-modellingusing the HIC does not have. Reality was simpli-fied very strongly in the modelling. Only theacceleration curve, measured with the help ofdummies, has influenced the mathematical model.Probably, the formula defining the HIC wascreated by accident: someone tried to improvethe SI, got more or less satisfactory results, andthis formulation became the general norm. Thepower of 2.5 means that doubling the decelerationproduces the same result as muliplying the relevanttime of the deceleration by the factor 5.7. Suchthings "smell" like a rule of thumb: a power of 2means a factor 4 for the time, which is perhaps a


20 40 60 80 100 120 140-HO -20 0

Fig. 13. Family of curves with pmrameter d of H0{t,d) for the crash without airbag.

bit too small; a power of 3 means a time factor of9, which seems too large; so one takes "some valuein between".

Experts agree that HIC values above 1000 areabsolutely life threatening. With respect to thevarious incalculabilities of crash tests (and, as Isuspect, in order not to annoy producers who donot reach acceptable values) one adds an additionof 25 per cent, and speaks of a low injury risk up toan HIC value of 1250, of a medium rise from 1251to 1500, and of a high injury risk only for valuesabove 1500. Similarly, one estimates the result ofthe maximal, at least 3 ms duration, head load asbeing low up to 60g, medium from 61 to lOg, andhigh only above 70#. Indeed, the range of vari-ation of the HIC is very large as Figs 14-17 fromdifferent crash tests show (one can see such meas-urement curves in the motor press). The first threecars did not have an airbag. The Citroen and Fiatcars were tested in 1992, the Golf was tested in

50 100

Time in Milliseconds

150

Fig. 15. Fiat Tipo: HIC = 818, a = 83g.

1991. The Mercedes tested in 1993 was equippedwith an airbag and the influence of the airbag isobvious! It also becomes clear how the HIC valuecan be reduced solely by use of constructionfeatures. The given deceleration values are themaximal values for at least 3 ms duration.

150

"oo•E 100

»

Head Impact„ Force

y50 100

Time in Milliseconds150

150

.E 100

Head ImpactForce

50 100


150

Fig. 14. Citroen ZX: HIC = 1368. a = 11 Op. Fig. 16. VW Golf III: HIC = 455, a=60g.


150

.= 100

I

50

Head ImpactForce

50 100


150

Fig. 17. Mercedes C 18a HIC = 307, a = 51 g.

Today, modern construction reaches bettervalues. For example, a crash test of the Audi 8(in summer 1995, with airbag) yielded an HICvalue of 142 and an A-3ms value of 33g. Unfortu-nately, such good values cannot be reached bysmall cars so far.

Just as with students' examination marks, theHIC does not provide an interval scale thatenables comparisons. The HIC cannot make anystatement about kind and severity of eventual

injuries but it gives an initial orientation for anestimation of the general injury risk.

References1. Herui, H.-W., Auto und Verkehr: Beispiele zum realilats-

nahen Mathematilcunterricht. Berichte fiber Mathematikund Unterricht. ETH Zurich, Bericht No. 94-04, 1994.

2. Meyer, J., Geschwindigkeit und Anhalteweg. In: "Material-ien fur einen realitatsnahen Mathematikunterricht", vol. 2,Franzbecker, Bad Salzdetfurth, 1995, pp. 22-29.

3. SAE Information Report J 885, pp. 220-222 (with thanks tothe car development divison of the Mercedes-Benz AG).

Hans- Wolfgang Henn studied mathematics andphysics at Karlsruhe University, and was a lecturerthere from 1971 to 1979. From 1979 to 1989 hetaught mathematics, physics and computer scienceat a Karlsruhe highschool, and holds a lectureshipin education for mathematics high school teachersat the Karlsruhe Studientseminar since 1989.

Address for correspondence: Rulaenderweg 16, D-76356 Weingarten Germany. (Wolfgang. [email protected])

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