Injury Mitigation By Cushioning Impact

Introduction

The cushioning of impact is a very important method of preventing (or reducing) the adverse consequences. Humans are important, and much work has been done to protect them in transport, occupational, sporting, military, and other contexts. Hard impacts causing immediate injury receive most attention, but repeated weak impacts sometimes gradually cause injury, as with footwear and keyboards. Things are usually less important than humans, but protective packaging of manufactured items and the handling of fruits and vegetables to avoid bruising are nevertheless major industries themselves. And a search will quickly turn up other contexts for impact mitigation that may be unexpected, such as robots colliding with humans, rocks weighing several tonnes falling on to roads, collisions of structures moving in an earthquake, and raindrops striking soil or plants.

The context of the paper by Saunders et al [1] was the use of visco-elastic polyurethane (VEPU) foams in aircraft seating. They make the point that the great majority of aircraft crashes are considered potentially survivable. For improved survivability, deceleration needs to be spread over a long time and a long distance, and this is achieved by such mechanisms as landing gear stroke, collapse of the sub-floor structure, stroking of the seat structure, and compression of the seat cushion. According to Saunders et al ([1] p. 11), “the load limiting function is designed such that it allows the seat and occupant to move (stroke), at loads just under the humanly tolerable limit, over the maximum distance available”. The key phrases here are maximum distance and humanly tolerable.

• A cushion gives only a certain distance in which to change the human’s velocity. The term bottoming out is used to refer to the impacted structure (such as the foam) deforming so much that it contacts (or almost contacts) some very stiff structure (such as a steel block). Acceleration, and hence injury, greatly increases when this occurs. A plot of stress against strain (force against deflection) steepens very much (Figure 2 of [1]). Even more sudden than the compression of a foam until it can compress no more is deformation of a flexible structure until it contacts something very stiff: for example, a car bonnet that is struck by a pedestrian’s head may deform until the engine is contacted.
• The quotation does not, however, address what is meant by humanly tolerable limit. Suppose only a certain distance is available, and it is desired to prevent any injury at all from occurring. Minor injury is typically much more common than serious injury, so a relatively soft cushion will be selected that reduces some minor injuries to no injury. There will be little reduction of serious and fatal injuries, because bottoming out will occur. Alternatively, suppose it is desired to prevent fatal injury from occurring. In this case, a relatively stiff cushion will be selected. In low-energy impacts, the stiff cushion will not be as effective as a soft cushion; in effect, it will itself be injurious.
• Thus it seems likely that foams for which stiffness is nonlinear or is velocity-dependent may be advantageous; but appropriate selection of a single intermediate stiffness may also be an improvement on both too soft and too stiff cushions.

There will be some further comment on this dilemma in the Discussion section below.

Saunders et al [1] used a test rig having a mass, with an accelerometer on it, that falls on to the VEPU foam. They reported peak (maximum) acceleration and other quantities relevant to the cushioning of impact. Surprisingly, it is not very common to encounter datasets like that in [1], in which various independent variables (such as speed, thickness of foam, and nature of foam) are manipulated and accelerations and other variables are observed. A possible reason is that even simple theories such as that in the next section are not well known, and thus there is not a strong tradition that helps with the organisation and presentation of the data. Another possibility is that once an experimental method is shown to be practicable, it quickly becomes standardised for comparative purposes, with experimental variation of the independent variables no longer occurring.

The theory below was developed for tests in which a headform, with an accelerometer inside it, is projected at the exterior of a car: this tests how dangerous or relatively safe is the car if involved in a collision with a pedestrian [2]. The experimental set up used in [1] was not very different from this. Theory is useful for the usual reasons—to organise, to understand, and to extrapolate from empirical data. This paper will first outline how maximum acceleration might be affected by speed of impact. The theory will be used to comment on the results in [1], in particular by summarising the dependence using the exponent of a power-law relationship.

Theory

A differential equation describing impact

Consider an impact in which a mass with an accelerometer is decelerated. At any instant, the force is assumed to depend on the displacement (distance travelled) after first contact and on velocity. The acceleration of the headform is the ratio of force to mass. Velocity is the first differential of displacement, and acceleration is the second. Hence the differential equation is m.y’’ = some function of y and y’, where mass is m and displacement is y, with differentials y’ and y’’ (velocity and acceleration). The differential equation, if it is sufficiently near correct, represents causation, and so will permit inputs such as speed of impact v and mass to be connected to outputs such as maximum acceleration and the Head Injury Criterion HIC.

The function of y and y’ to be used here is (1). Equations (2)–(4) are special cases (as described below).

The term k.yⁿ represents the static response of a spring, and the product of powers of y and y’ represents damping. It is understood that the constants of proportionality b and k, and the exponents n, p, q (reflecting the nonlinearity of the dependencies) remain constant, not only as time t passes, but also as v changes. It is not clear to what extent (1) is plausible: it might be objected that initial velocity v should not appear in the equation. Thus (2)–(4), which are special cases of (1), have been given as examples, and are discussed in the following paragraphs.

The special case of (1) that has no damping (b = 0) is (2). This represents an undamped spring, linear if n is 1.

The special case of (1) that has p = n, q = 1 is given as (3). Having a damping term proportional to y^p.(y’)^q was proposed by Hunt and Crossley [3], and they gave particular attention to the case p = n, q = 1; the v^–1 factor for this case is due to Herbert and McWhannell [4]. The justification for choosing the damping term to be proportional to a product of power functions of y and y’ is that then the term is zero at initial contact (y = 0) as well as at maximum displacement (y’ = 0). If it were proportional to a sum of power functions of y and y’, for example, then at initial contact there would be step change from zero to some finite value, which is considered unrealistic. The v^–1 term in (3) is appropriate as coefficient of restitution is approximately independent of v and for other empirical reasons: see [4–6].

Equation (4) is suggested as being a good approximation for the behaviour of y’’ when y’’ is close to its maximum (this is the region of greatest importance to injury). This maximum occurs before maximum displacement does, that is, when displacement is increasing and speed is decreasing. Consequently, it seems likely that some example of (4) will be a good approximation, as the damping term is the product of an increasing term and a decreasing term (provided n exceeds p), and n and p can be adjusted so that this product has a maximum. (However, this argument is not claimed to be relevant to maximum displacement, as this is not determined largely by the behaviour of the equation when y’’ is close to its maximum.) It might be thought that n = 1.5, p = 0.25, for example, is a reasonable pair of values and that the corresponding exponent of y’ of 2(n-p)/(n+1) = 1 is also reasonable.

If, despite these theoretical and empirical considerations, it is still considered that v should not appear in the equation, then Equations (2) and (4) might be considered the most important, with (1) and (3) merely showing that the results are likely to be approximately true more broadly.

In the second term of (1), the exponent of v has been chosen in such a way that the shape of the acceleration pulse remains constant, except for horizontal and vertical linear stretching, when v changes. And if that is true, then maximum acceleration A is proportional to a power function of the initial velocity v:

The following quantities are also power functions of v: maximum displacement; HIC; and any injury response function based on [av(a)]^u.(t₂-t₁)^w, where av(a) means average acceleration during the time interval from t₁ to t₂, and this interval is chosen so that the result is as large as possible (this is a generalisation of HIC, for which u is 2.5 and w is 1). The claims in this paragraph may be proved by the argument in [7]. As (2)–(4) are special cases of (1), these results apply to those equations as well.

This work was done in the context of testing car exteriors with projected headforms, and the work of Saunders et al was concerned with an aircraft crash, but various forms of cushioning or padding are widely used in protecting people from injury and things from damage, and advances in models for describing impacts and drawing conclusions from the models may correspondingly have wide application.

Bottoming out

A limitation of (1)–(4) is that they do not fully reflect the possibility of bottoming out. Although stiffness of the spring can increase with displacement (if n > 1), this may not be quickly enough to represent bottoming out. So-called tangent elasticity, with tan(y/clearance) replacing yⁿ as the stiffness term (where clearance is the distance available before bottoming out occurs—such as the distance between the bonnet and a stiff structure underneath), might be appropriate instead. Proportionality results such as (5) would no longer be valid, and it would be necessary to obtain relationships via numerical simulation instead.

For small speeds of impact and small maximum displacements, a cushion of low stiffness will lead to lower maximum accelerations than a cushion of high stiffness. However, low stiffness implies greater displacement and hence a greater likelihood of bottoming out: this will occur at lower speeds if stiffness is low than if it is high. Consequently, at high speeds a cushion of high stiffness is desirable.

For (1)–(4), the exponent for the dependence of maximum acceleration on speed is 2n/(n+1)—see (5)—and will be denoted c below. The spring exponent n is positive, and thus c increases with n. However, it cannot exceed 2 (for large n, the denominator is only a little greater than n and thus c is a little less than 2). If c is empirically found to exceed 2, (1) cannot be valid. An obvious possible reason is the occurrence of bottoming out. Without some sort of theory, there would be no baseline like this against which results could be judged. It is not clear whether n and c should be expected to be less than 1 (the foam may have been designed to be a spring that gets less stiff with increasing displacement); or to be greater than 1 (displacement may be approaching the point of bottoming out).

Comments

Reports of experimental studies of impact protection commonly note the difficulties associated with obtaining data at high speeds (such as the risk of damage to accelerometers). This reinforces the need for theory.

In relation to (5), four points may be made:

• The theory here suggests that a power function, not a straight line, should be used to fit empirical data relating A to v, and to extrapolate.
• If it is thought that a qualitative change such as bottoming out might occur, extrapolation is impossible.
• It may be possible to measure a thickness or a clearance distance in order to determine how great displacement can be before bottoming out occurs [8].
• It may be possible to estimate the exponent n by achieving greater energies of impact with a larger mass impactor, if higher speeds are impracticable [7].

Quantitative results in saunders et al.

Peak acceleration, energy absorbed, and maximum displacement will be discussed.

Peak acceleration

Table 1 gives most of the results in Tables 3–9 of Saunders et al [1].

• The quantity shown is peak acceleration.
• The rows refer to seven types of foam. (Baseline-Yellow and Sample A are not VEPU foams, the others are.)
• The columns refer to four speeds of impact, and parts (a) and (b) refer to two thicknesses of foam.

The theory of the previous section permits the effects of speed on peak acceleration shown in Table 1 to be summarised, as in Table 2. It is assumed that peak acceleration A depends on speed v via a power-law relationship, A = k.v^c. In terms of (1)–(5), c = 2n/(n+1).

• The quantity shown in Table 2 is the exponent c. It is calculated from the change in ln(A) divided by the corresponding change in ln(v).
• The columns of Table 2 refer to a comparison of speeds 8.2 and 2.8 m/sec for the 3” thick foams, a comparison of speeds 2.8 and 2.0 m/sec for the 3” thick foams, and a comparison of speeds 2.8 and 2.0 m/sec for the 1” thick foams.
• For many foams, peak acceleration did not vary very much over the speed range 2.0 to 2.8 m/sec (see Table 1). And in some cases, presumably as a result of random variation, peak accelerations were not in exactly the same order as speeds. The exponent c at these lower speeds has thus been estimated from the change in ln(A) over the whole range 2.0 to 2.8 m/sec, the data at 2.4 m/sec being omitted. Even so, there must be some inaccuracy in the estimates.

It was suggested earlier that if the exponent c is found to exceed 2, this is likely to indicate bottoming out. Several cases of this are evident in Table 2. As an example, Table 1 shows that an increase of speed from 2.8 to 8.2 m/sec increases A from 64.9 to 898.8 in the case of the Baseline-Yellow foam. Such a steep increase might itself suggest bottoming out. Working out that the corresponding exponent is 2.45 (Table 2), and knowing that this is incompatible with power-law stiffness, considerably reinforces this interpretation. Beyond indicating probable bottoming out, it is not known how to interpret c when it is larger than 2. However, some comments will be made later about the Baseline-Yellow foam.

(a) 3” thick foam

(b) 1” thick foam

Energy absorbed

Saunders et al report the energy absorbed as a percentage of the original kinetic energy. However, if maximum acceleration is the appropriate indicator of likely injury, the energy absorption ratio is irrelevant. It may be that Saunders et al feel that maximum acceleration is not a perfect indicator, and that the energy absorption ratio may also affect the likely injury.

The energy absorption ratio, and the closely-related coefficient of restitution, are outputs from the differential equation—such as (1)—that describes the impact.

However, in the present state of knowledge, it might be better to assume a differential equation only applies up to occurrence of the maximum displacement. At this time, velocity change is v. At the end of the impact, velocity change is (1+R).v, where R is the coefficient of restitution. By analogy, the effect of what happens after maximum displacement might be represented by multiplying maximum acceleration by 1+R. In this case, R is used as an input: both maximum acceleration and R are very simple quantities derived from the acceleration pulse. However, this may give too much weight to R, as maximum acceleration and similar quantities such as HIC are designed to emphasise the highest accelerations, and the accelerations subsequent to maximum displacement are lower. This would suggest replacing R with some fraction of R (perhaps half) in the multiplying factor. Limited knowledge about what really causes injury means that this is somewhat speculative.

Maximum displacement

High maximum displacement might be taken as an indicator of high impact speed, of likely bottoming out, and (for both reasons) of probable high severity of injury. However, for a given speed of impact and in the absence of bottoming out, high maximum displacement is desirable: the necessary speed reduction is spread over a long distance and a long time.

Although Saunders et al do not report maximum displacements occurring in their tests, it is often desirable to do so, as it may suggest whether bottoming out is occurring, or would occur at only slightly higher speed.

In some contexts, it might be thought possible that what is impacted is initially very weak. In that case, the maximum displacement becomes decoupled from quantities more directly linked to injury such as maximum acceleration and HIC. Thus the proportionality results for maximum acceleration—see (5)—and also corresponding results for HIC, may be valid in circumstances where those for maximum displacement are not. This may be the case for some of the foams tested by Saunders et al. It appears from Figures 6 and 7 of Saunders et al (referring to impacts at 8.2 and 2.8 m/sec) that the Baseline-Yellow foam was crushed by 50 or 60 mm before much deceleration at all took place. If the consequences of (1) were incorrect in respect of predictions of maximum displacement, the equation could possibly be modified by assuming that there is some initial distance d over which acceleration is small and some small amount of energy E is absorbed, and only thereafter does (1) apply.

This distinction between prediction of maximum displacement and of maximum acceleration could be taken further. Ideally, a single differential equation would be valid both when there is no bottoming out and when there is. However, an equation of such wide validity is unlikely to be tractable, and so one will be chosen that is valid only in the absence of bottoming out. Provided bottoming out is not a danger, the chief region of interest of the differential equation is where y’’ is close to its maximum (because high y’’ is associated with injury); displacement y as such is unimportant. But y is still important for determining that bottoming out is not occurring: prediction of y could be separated from prediction of y’’, the two tasks using different differential equations.

Discussion

Reasons for improved protection

In the experimental programme of Saunders et al, several properties of the VEPU foams differed from those of the standard foams. Saunders et al did not attribute the advantages of the VEPU foams to one difference rather than another. In terms of the very simple theory represented by (1), three types of adjustment can be made to a cushioning material.

• Stiffness k can be increased.
• Response to deformation can be made nonlinear (concave downwards, as in the initial part of Figure 2 of Saunders et al): n should be less than 1.
• Velocity-dependence b can be increased. (Changing the nature of velocity-dependence by changing p and q is conceivable, too.)

It is probable that in practical situations, k should be high, n should be low, and b should be high. These give good protection in severe (high speed) impacts. This is likely to be at the expense of less protection in low speed impacts. And considerations of comfort limit the choices available.

Without further evidence, the advantages of VEPU foams reported by Saunders et al should not be attributed to the advanced or sophisticated properties (such as velocity-dependence) rather than to the simple properties (such as stiffness). It was estimated earlier that the Baseline-Yellow foam was crushed by 50 or 60 mm before much deceleration occurred: that seems to be a waste of available distance, and better protection would be given simply by greater stiffness.

Concluding comment

Saunders et al note that it is highly desirable that test conditions are relevant to real-world impacts, and that variation in conditions implies there may be variation in which foam is best.

As already mentioned, an important specific issue is that high stiffness is preferable for high speed impacts, but low stiffness is preferable for low speed impacts. Suppose it is asked whether it is appropriate for the Baseline-Yellow foam to crush so much at low speeds. For 3’’ thick foam, Table 1(a) shows, for example, that low-speed performance is better (lower accelerations) than with the Sunmate XX foam, but high-speed performance is worse. The question implies attempting to optimise stiffness. This would require information on the likely practical consequences of different levels of acceleration (for example, probability of death, probability of incapacitation, or average dollar cost of injury) and on the relative frequencies of different speeds of impact. My colleagues and I have recently made this point in the context of road crashes [9,10]. There are three steps to what we propose. The first is to generalise the quantity observed in test conditions (such as HIC or maximum acceleration) to what would be observed if (for example) speed were different. The second is to convert the test quantity to something that can be meaningfully averaged (such as the dollar cost or the probability of death corresponding to a given HIC or maximum acceleration). The third is to obtain the average cost, or average probability of death, by integration over what varies from crash to crash (such as speed).

Acknowledgements

The Centre for Automotive Safety Research, University of Adelaide, is supported by both the South Australian Department of Planning, Transport and Infrastructure and the South Australian Motor Accident Commission. The views expressed in this paper are those of the author, and do not necessarily represent those of the University of Adelaide, or CASR’s funding organizations.

References

Author

Dr Paul Hutchinson is a Senior Research Fellow in the Centre for Automotive Safety Research, University of Adelaide, South Australia 5005, Australia. email paul@casr.adelaide.edu.au, phone +61 8 8313 5997, fax +61 8 8232 4995.