2. Airbag deployment physics

An airbag is not a cushion that waits passively for a body to fall into it. It is a gas-driven structure that must inflate, pressurize, vent, and collapse within roughly 30–100 ms, and it must do so while a human body is already moving toward it. This chapter treats the restraint as a coupled physical system: the inflator as a gas source, the cushion as a deformable pressure vessel, and the occupant as an inertial mass that may be only 100–250 mm from the module cover at the moment of deployment. Understanding this coupling is the foundation for every simulation and test that follows.

Inflator Output: The Gas Source

Pyrotechnic inflators burn a solid propellant; hybrid inflators release stored compressed gas and add heat from a small pyrotechnic charge; cold-gas inflators release only stored gas. Regardless of type, the output that matters to the cushion is the mass flow rate ṁ(t) and the temperature of the gas entering the bag. A useful engineering approximation for a pyrotechnic inflator is a burn that ramps up quickly, peaks, then decays:

# Example: simplified pyrotechnic mass-flow model (illustrative parameters)
# t in ms, mdot in g/ms
def mdot(t, m_total=120.0, t_rise=8.0, t_peak=25.0, t_end=60.0):
    if t < 0 or t > t_end:
        return 0.0
    if t < t_rise:                      # ignition ramp
        return (m_total / (0.5*t_rise + (t_peak-t_rise) + 0.5*(t_end-t_peak))) * (t/t_rise)
    if t < t_peak:                      # plateau
        return m_total / (0.5*t_rise + (t_peak-t_rise) + 0.5*(t_end-t_peak))
    return m_total / (0.5*t_rise + (t_peak-t_rise) + 0.5*(t_end-t_peak)) * (t_end-t)/(t_end-t_peak)

With these illustrative values, total delivered mass is 120 g over 60 ms, and the peak flow is about 3.1 g/ms. Real inflators are characterized on a tank test: the inflator fires into a fixed-volume pressure vessel, and the measured pressure–time trace is converted to an equivalent mass flow using the ideal-gas relation. That tank curve, not the propellant datasheet, is the input a simulation should use. Two parameters dominate occupant outcome: time-to-first-gas (TTFG, typically 2–10 ms) and peak mass flow. A short TTFG with a high peak fills the bag fast but also produces a violent initial membrane motion — exactly the hazard in close-proximity cases.

Bag Kinematics: From Folded Pack to Pressurized Cushion

The cushion begins as a folded pack behind the module cover. Deployment is a sequence of unfolding, membrane stretching, and internal pressurization. Three kinematic quantities govern the interaction with an occupant:

  • Face-of-bag position xbag(t) — how far the cushion front has traveled from the module.
  • Face velocity vbag(t) — the speed of the deploying membrane; this is the primary injury driver in close proximity.
  • Internal pressure p(t) — set by the balance of inflow, venting through the fabric, and volume change.

A first-order model treats the bag as a control volume with venting:

# Lumped-parameter bag pressure (illustrative)
# p: gauge pressure [kPa], V: bag volume [L], m: gas mass [g]
# R_eff: effective gas constant, T: gas temp [K], A_vent: vent area [mm^2]
dp/dt = (R_eff*T/V) * mdot(t)  -  (p/V) * dV/dt  -  k_vent * A_vent * sqrt(p)

The first term is inflator inflow, the second is the pressure drop caused by the bag expanding, and the third is venting. In a close-proximity event, the occupant is already inside the bag's intended deployment volume, so dV/dt is small and the vent term cannot relieve pressure quickly. The result is a stiff, high-pressure cushion whose face is still moving outward — the worst combination for a small-stature or out-of-position occupant.

Key insight: In normal seating, the occupant arrives after the bag has largely finished deploying, so the bag acts as a energy-absorbing cushion. In close proximity, the occupant arrives during deployment, so the bag acts as a moving piston. The same inflator that is safe at 300 mm can be injurious at 100 mm.

Occupant Interaction in Close Proximity

The occupant is modeled as a lumped mass mocc with an initial position and velocity. The interaction force depends on whether contact occurs before or after the bag is fully deployed. Define the gap g(t) = xbag(t) − xocc(t). Contact begins when g = 0. The contact force is then:

F_contact = k_bag * max(0, g_overlap) + c_bag * v_rel

where g_overlap is the penetration of the occupant into the bag face, and v_rel is the relative velocity. In close proximity, v_rel is dominated by the bag face velocity, which can reach 15–25 m/s in the first 20 ms. This is why the injury metric in OOP testing is not just force but force × time and, for the head/neck, moment about the occipital condyle.

Consider an illustrative case: a 5th-percentile female occupant, head 120 mm from the module cover, no seat belt pretensioner firing. The inflator delivers first gas at 4 ms. By 12 ms the bag face has traveled 90 mm and is moving at 18 m/s. The head, initially at rest, is struck at 12 ms. The resulting head acceleration in this illustrative simulation peaks near 180 g over 3 ms — well above the 80 g threshold used in many OOP criteria. Reducing inflator peak flow by 30% and adding a tether to slow the face extends time-to-contact to 18 ms and drops peak acceleration to about 95 g. This is the trade space: slower deployment for OOP safety versus faster deployment for normal-position energy absorption.

Close-Proximity Deployment: Bag Face vs. Occupant Timeline 0 ms 10 ms 20 ms 30 ms 40 ms Inflator mass flow ṁ(t) peak ≈ 3.1 g/ms TTFG ≈ 4 ms Bag face position x_bag(t) fully deployed ≈ 28 ms Bag face velocity v_bag(t) peak ≈ 18 m/s Occupant head x_occ(t) head starts moving at contact Contact ≈ 12 ms Legend Inflator flow Bag face position Bag face velocity

What This Means for Simulation and Test

Three modeling decisions follow directly from the physics above. First, the inflator must be represented by a measured tank curve, not a constant mass flow, because the shape of ṁ(t) controls the bag face velocity peak. Second, the bag must be modeled with venting and fabric permeability; a rigid, sealed bag will over-predict pressure and under-predict the benefit of tethers. Third, the occupant must be positioned and given an initial velocity consistent with the OOP scenario being studied — a static occupant at 120 mm is not the same as a braking occupant at 120 mm with 5 m/s forward velocity.

For a reproducible check, run a baseline simulation with the parameters in the table below, then perturb one parameter at a time and record the peak head acceleration and the time of contact. The expected qualitative behavior is: reducing peak mass flow by 30% delays contact by 4–6 ms and reduces peak acceleration by 30–45%; adding a tether that limits face travel to 80% of free deployment reduces peak acceleration by 15–25% with minimal effect on normal-position performance.

Parameter Baseline (illustrative) Perturbation Expected effect on peak head accel.
Peak mass flow 3.1 g/ms −30% −30% to −45%
TTFG 4 ms +3 ms −10% to −20%
Tether length free deployment 80% travel −15% to −25%
Vent area 2000 mm² +50% −5% to −15%
Occupant initial velocity 0 m/s 5 m/s forward +20% to +40%
Practical check: If your simulation shows contact before 10 ms with a bag face velocity above 15 m/s, the model is in the OOP hazard regime. Verify the tank curve, the vent model, and the occupant initial conditions before trusting any injury metric. A common failure symptom is a bag that inflates too fast and too stiff — the pressure trace rises monotonically with no vent plateau. That usually means the vent term is missing or the bag volume is not updating with face position.

The physics in this chapter sets the boundary conditions for everything that follows: the simulation setup, the test matrix, and the injury criteria. The next step is to translate these coupled equations into a solver-ready model, which is the subject of the following chapter.

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