2. Occupant kinematics fundamentals

Airbag timing decisions only make sense against the motion they are meant to intercept. Before any discussion of squib timing, crash pulse selection, or suppression logic, an engineer must be able to describe where the occupant is, how fast they are moving, and how their body loads the restraint system at each millisecond of a frontal event. This section establishes that kinematic baseline for both belted and unbelted occupants, defines the metrics used to compare their motion, and connects those metrics to the injury criteria that ultimately justify a deployment decision.

Reference Frame and Sign Convention

All motion in this chapter is described in the vehicle coordinate system: +X forward (direction of travel), +Y to the driver's left, +Z upward, with the origin at the vehicle CG at t = 0. Occupant displacement is expressed relative to the vehicle interior, so a positive Δx means the occupant moves forward relative to the cabin. Crash pulse acceleration is defined as positive when directed rearward (deceleration of the vehicle), which is the SAE J211 convention with CFC 60 filtering. Keeping this convention consistent matters: a sign error in a pulse or a displacement trace will invert every downstream timing conclusion.

The Restrained Occupant: Three Phases of Coupled Motion

A belted occupant does not travel freely; the restraint system progressively couples them to the decelerating vehicle. The motion divides into three observable phases.

Phase 1 — Free travel (0 to ~15–25 ms). The occupant continues forward at approximately the pre-crash velocity while the vehicle begins to decelerate. Relative displacement grows almost linearly, and belt force is still near zero because the retractor has not locked and webbing slack has not been taken up. For a 50 km/h barrier impact, the occupant is still moving at roughly 13.9 m/s while the vehicle has already lost 1–2 m/s.

Phase 2 — Belt engagement and ride-down. The retractor locks, the load limiter begins to pay out webbing, and the belt force rises. The occupant decelerates over a longer distance than the vehicle does — this is the ride-down effect. A well-matched system stretches the occupant's deceleration over 60–90 ms instead of the 80–100 ms vehicle pulse, reducing peak chest acceleration. The key observable is the relative displacement plateau: forward motion peaks and reverses as the belt and seat interact.

Phase 3 — Rebound and secondary contact. Stored belt energy and seat cushion rebound push the occupant rearward into the seat back. Rebound velocity is typically 10–25% of the initial forward velocity; excessive rebound indicates an over-stiff restraint or a load limiter set too high.

The Unrestrained Occupant: Ballistic Until Contact

An unbelted occupant follows a nearly ballistic path. With no belt force, the only decelerating influence before contact is friction with the seat cushion — typically a coefficient of 0.3–0.5, contributing only a few g. The occupant therefore closes on the interior at nearly the full delta-v. In a 50 km/h frontal impact, an unbelted 50th-percentile male travels approximately 300–400 mm forward relative to the cabin within 60–80 ms, arriving at the airbag and knee bolster at 12–14 m/s. This is the motion the airbag must be inflated to meet: the bag must be fully deployed and venting before the occupant arrives, which is why unbelted occupants drive earlier required deployment times (RDT) and larger bag volumes.

Key Metrics for Comparing Motion

Four metrics allow restrained and unrestrained motion to be compared quantitatively and fed into deployment logic:

• Relative forward displacement Δx(t) — occupant-to-cabin travel; determines the free-flight distance the bag must span.
• Occupant velocity relative to the cabin v_rel(t) — the closing speed at the moment of airbag contact; drives bag pressure and membrane loading.
• Ride-down distance and time — how far and how long the restraint stretches the deceleration; a proxy for chest loading severity.
• Time-to-contact t_c — the instant the occupant reaches the airbag module face or knee bolster; the primary input to required deployment time.

From Kinematics to Injury Criteria

Kinematic metrics are only useful because they map onto injury criteria. The restrained occupant's ride-down directly controls chest deflection and the resulting viscous criterion VC (chest deflection rate × deflection, limit 1.0 m/s for the 50th-percentile male) and chest compression (limit ~35% for a 40-year-old, rising with age). The unrestrained occupant's high closing velocity at contact produces large HIC15 values on head-to-bag and head-to-interior contact, and the knee bolster loads drive femur axial force (limit ~10 kN) and tibia index. Airbag timing therefore has two distinct targets: for the belted occupant, the bag must be in place before the ride-down phase brings the chest forward; for the unbelted occupant, it must be fully deployed before the ballistic arrival at t_c. A single fixed deployment time cannot optimize both, which is precisely why occupant classification and belt-buckle status feed the restraint control module.

Worked Example: Estimating Time-to-Contact

Consider a simplified example (illustrative values, not measured data). Vehicle pulse: 35 g average deceleration over 70 ms, delta-v = 50 km/h. Occupant initial velocity v0 = 13.9 m/s. For the unbelted case, assume the occupant decelerates only at 0.4 g due to seat friction. Relative acceleration a_rel ≈ 35g − 0.4g ≈ 34.6g ≈ 339 m/s². Solving Δx = v0·t − ½·a_rel·t² for Δx = 350 mm gives t_c ≈ 27 ms — far earlier than the 60–80 ms often quoted, because this simplified model ignores occupant rotation and the fact that the pulse is not constant. The lesson is methodological: always integrate the actual filtered pulse rather than using an average g, and validate t_c against a sled or CAE run before setting RDT.

Practical Checks Before Using Kinematic Outputs

Three checks catch most errors. First, confirm the pulse is filtered at CFC 60 and that the integration to delta-v closes to the intended value within 2%. Second, verify that occupant displacement is measured relative to a fixed cabin reference, not to the moving seat frame, unless seat motion is explicitly modeled. Third, confirm that t_c is defined at the same surface the airbag actually presents — module cover, not the undeployed bag envelope. With these checks in place, the kinematic baseline becomes a reliable input to the deployment timing analysis that follows.