Chapter Contents

• Applications of Integration
• 1. Applications of the Indefinite Integral
• 2. Area Under a Curve
• 3. Area Between 2 Curves
• 4. Volume of Solid of Revolution
• 5. Centroid of an Area
• 6. Moments of Inertia
• 7. Work by a Variable Force
• 8. Electric Charges
• 9. Average Value
• Head Injury Criterion
• a. Normal Braking
• b. Crash Tests
• c. Data
• d. Severity Index
• e. Head Injury Criterion

• f. Modelling the HIC

• g. Conclusions
• 10. Force by Liquid Pressure
• 11. Arc Length of a Curve
• 12. Arc Length of a Curve in Parametric or Polar Coordinates
• Applications of Integration Problem Solver

• Comments, Questions?

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Modelling the HIC

Our approach is to model the deceleration curve with a function. We recognise that the shape of the crash data graph is basically like the curve:

`f(x)=1/(x^2+1)`

(This similar to the 'bell-shaped curve' in statistics.)

Let's look at the curve using LiveMath:

LIVEMath

Model for the Acceleration

By modifying the function above, we can get curves very close to the Mercedes experimental data.

`a(t)=16400/((t-68)^2+400)+1480/((t-93)^2+18)`

[This model was obtained by observing 2 peaks in the deceleration graph, centred at 68 ms and 93 ms. By adding the 2 bell-like curves, we get a model very close to the required graph.]

The computer time to calculate the HIC expression is high, so we need to simplify things a bit...

We can simplify that part of the HIC formula in {} for different values of d = t₂ − t₁. We define a family of curves:

`H_(t,d)=d(1/dint_t^(t+d)a(T)dT)^2.5`

We then vary the value of d. The value of the highest peak of the family of curves obtained gives us the HIC.

HIC without Airbag

We now use the model for a(t) from above and maximise the value of H for different values of d.

We see that the highest peak occurs when `d=50` and reading from the graph we see that the HIC is approximately `725`. This is reasonably close to the Mercedes Benz data. (Other values for d are shown, but not all, of course. The value d = 50 did in fact give the highest curve.)

HIC with Airbag

The model is simpler for the airbag case, as the deceleration is smoother and is almost bell-shaped. Some modelling achieves the following expression for the acceleration:

`a(t)=22000/((t-74)^2+500)`

The graph is as follows (drawn with the same vertical scale as the non-airbag case):

Now to apply the formula for H again. We get a family of curves and once again, `d = 50` gives us the maximum value, hence the HIC.

The HIC for the airbag case is around `310`, close to the Mercedes Benz data.

Our main aim here is to reduce the effects of rapid deceleration. If the curve is low and flat, injury to the head is reduced.

Let's get LiveMath to draw these models for us:

LIVEMath