2. Deployment physics

Once the inflator begins releasing gas, the airbag stops being a static bag and becomes a transient pressure vessel with moving boundaries. The cushion's final shape is not simply "the shape of the fabric"; it is the integrated result of how fast mass enters each chamber, how quickly pressure can redistribute between chambers, and how the fabric and venting resist that redistribution. This section examines the three coupled mechanisms that govern that process: gas dynamics at the inflator, pressure equalization through internal passages, and chamber-to-chamber interaction during the inflation transient.

Gas Dynamics at the Inflator Interface

An inflator is a mass-flow device, not a pressure source. Its output is characterized by a mass flow rate curve ṁ(t) and a gas temperature, both of which vary strongly over the first tens of milliseconds. For a pyrotechnic inflator, most of the propellant burns within roughly 20–40 ms; for a hybrid or stored-gas design, the release profile is flatter but still transient. The cushion's internal pressure at any instant is therefore a balance between inflow and outflow:

dP/dt = (γ · R · T / V) · ṁ_in  −  (γ · P / V) · dV/dt  −  (γ · P / V) · ṁ_out/ρ

where γ is the ratio of specific heats, R the specific gas constant, T the gas temperature, V the instantaneous chamber volume, ṁ_in the inflator mass flow, and ṁ_out the mass leaving through vents or fabric porosity. The first term drives pressure up; the second term (volume growth) and third term (venting) pull it down. In a multi-chamber cushion, this equation applies to each chamber separately, with the coupling term being the mass flow through the internal passage between chambers.

A practical consequence: the inflator's flow area and the cushion's inlet throat must be matched. If the throat is too small, the inflator's output is choked at the inlet and the cushion fills slowly, producing a soft, late deployment. If the throat is too large, the first chamber pressurizes almost instantly and the fabric experiences a sharp pressure spike that can cause seam rupture or "bag slap." Designers typically target a throat area that passes the inflator's peak flow with a modest pressure drop, so the first chamber reaches its target pressure within the desired time window.

Pressure Equalization Between Chambers

In a multi-chamber airbag, chambers are connected by internal passages—sewn channels, cutouts in the internal baffle, or one-way valves. The driving force for equalization is the pressure difference ΔP = P₁ − P₂ across the passage. The mass flow through a passage of effective area A_p can be approximated by an orifice relation:

ṁ_passage = C_d · A_p · √(2 · ρ · ΔP)   for incompressible flow
ṁ_passage = C_d · A_p · P₁ · √(γ/(R·T)) · f(P₂/P₁)   for compressible flow

where C_d is the discharge coefficient (typically 0.6–0.8 for sewn fabric passages) and f(P₂/P₁) is the compressible flow function that accounts for choking when the pressure ratio exceeds the critical value. The key insight is that equalization is not instantaneous. A passage that is too restrictive leaves the first chamber over-pressurized and the second chamber under-inflated during the critical first 30–50 ms, which is exactly when occupant contact occurs. A passage that is too open allows the first chamber to dump its pressure into the second before the first has developed useful restraint force.

The equalization time constant for a two-chamber system can be estimated as:

τ ≈ V₂ / (C_d · A_p · √(2·ΔP/ρ))

For a typical passenger-side cushion with V₂ ≈ 60 L, A_p ≈ 0.004 m², ΔP ≈ 50 kPa, and ρ ≈ 1.2 kg/m³, the time constant is on the order of 10–15 ms. This is comparable to the inflator burn time, meaning the two chambers are strongly coupled and cannot be treated independently. In simulation, this coupling must be resolved with a small enough time step to capture the pressure wave; explicit solvers typically require Δt < 0.1 ms for stable coupling.

Chamber Interaction and the Inflation Sequence

The interaction between chambers is not merely a matter of pressure equalization. Each chamber's volume changes as the fabric moves, and that volume change feeds back into the pressure equation. When the first chamber inflates, it pushes against the internal baffle, which in turn compresses or displaces the second chamber. This mechanical coupling means that the second chamber's pressure can rise before significant mass has flowed into it, simply because its volume is being reduced by the first chamber's expansion.

This effect is particularly important in "dual-depth" or "dual-chamber" passenger airbags, where the rear chamber acts as a reaction surface for the front chamber. If the rear chamber is too soft, the front chamber's pressure pushes it backward instead of forward, delaying occupant restraint. If the rear chamber is too stiff, it transmits the inflator's pressure spike directly to the occupant. The design target is a rear chamber that is stiff enough to react the front chamber's pressure but soft enough to absorb the initial spike.

Venting and the Pressure Balance

Vents are not just a safety feature; they are an active part of the pressure management system. A vent in the first chamber reduces its peak pressure and slows the equalization to the second chamber. A vent in the second chamber controls the final restraint stiffness. The interaction between venting and chamber coupling is nonlinear: increasing the first chamber's vent area reduces its pressure, which reduces the driving force for equalization, which in turn reduces the second chamber's pressure. This can create a "soft" cushion that deploys quickly but provides insufficient restraint.

In simulation, venting is typically modeled as a pressure-dependent orifice with a discharge coefficient that varies with fabric porosity and vent geometry. The vent flow must be solved simultaneously with the chamber pressures, because the vent's flow rate depends on the pressure difference between the chamber and the environment, which itself depends on the chamber pressure.

Simulation Implications

For a multi-chamber airbag simulation, the gas dynamics model must include: (1) a time-dependent mass flow input from the inflator, (2) a separate control volume for each chamber, (3) a compressible flow model for the internal passages, (4) a venting model for each chamber, and (5) a mechanical coupling model that updates each chamber's volume based on the fabric's motion. The time step must be small enough to resolve the fastest pressure transient, typically 0.05–0.1 ms. The most common failure mode in simulation is treating the chambers as independent, which underestimates the pressure spike in the first chamber and overestimates the deployment speed of the second.

The figure above summarizes the coupled system: the inflator delivers mass to Chamber 1, the internal passage transfers mass to Chamber 2, both chambers vent to atmosphere, and the mechanical coupling between chambers modifies each volume as the fabric moves. The pressure-time curves at the bottom illustrate the characteristic behavior: Chamber 1 rises first and peaks earlier, while Chamber 2 lags and reaches a lower peak, with the pressure difference ΔP driving the equalization flow. In a well-designed cushion, the two curves converge within 40–60 ms; in a poorly designed one, they remain separated, and the occupant sees an uneven restraint surface.

Key Parameters and Their Coupling

These parameters are not independent. Increasing A_p reduces the equalization time but also reduces the peak pressure in Chamber 1, which may reduce the restraint force. Increasing A_v1 reduces Chamber 1's pressure but also reduces the driving force for equalization, which delays Chamber 2's inflation. The design process is therefore iterative: adjust the passage and vent areas until the pressure-time histories of both chambers meet the restraint targets at the occupant contact time.

In practice, the deployment physics of a multi-chamber airbag is validated by comparing simulated pressure-time histories at instrumented points in each chamber against sled or static deployment test data. The key validation metrics are the peak pressure in each chamber, the time to peak, and the time for the two chamber pressures to converge within 10%. If the simulated convergence time is more than 20% different from the measured value, the passage discharge coefficient or the vent model needs recalibration. This calibration is typically performed at a single temperature (e.g., 23°C) and then verified across the operating range (−35°C to +85°C), since gas viscosity and fabric permeability both vary with temperature.