4. Pulse metrics that drive restraint performance

A crash pulse is not a single number. It is a time history — acceleration versus time — and restraint performance is governed by how that history distributes velocity change, deceleration, and energy across the event. Two pulses with identical delta-v can produce completely different occupant outcomes because they differ in peak deceleration, duration, centroid time, slope, and energy content. This chapter defines the six metric families that matter most for restraint tuning, explains how each one physically influences airbag and seatbelt behavior, and shows how to compute them from a digitized pulse.

Sign Convention and Reference Frame

Before any metric is meaningful, fix the sign convention. In this course, the vehicle body is the reference frame. A frontal impact decelerates the vehicle, so the body-mounted accelerometer records a negative longitudinal acceleration in the SAE vehicle axis system (x forward, y right, z down). For readability, most pulse tables report deceleration magnitude as positive. Delta-v is reported as the magnitude of velocity change, also positive. Whenever a sign appears in a formula below, it follows the SAE convention; when a table reports a positive number, it is a magnitude. Keep this convention consistent across all six metrics and across every figure in this chapter.

Delta-v: The Velocity Budget

Delta-v is the integral of acceleration over the pulse:

Δv = ∫ a(t) dt   (evaluated from t0 to the end of the pulse)

It is the total velocity the occupant must be brought to match, and it sets the energy the restraint system must absorb. A 56 km/h frontal barrier test corresponds to roughly 15.6 m/s of delta-v. If the restraint system only manages 12 m/s of occupant velocity change before the occupant contacts structure, the remaining 3.6 m/s is absorbed by the occupant — the classic "bottoming out" failure. Delta-v is therefore the primary sizing input for airbag volume, vent sizing, and pretensioner stroke.

Peak Deceleration and Pulse Duration

Peak deceleration is the maximum magnitude of a(t). Duration is the time from pulse onset to the point where acceleration returns to zero. These two are coupled through delta-v: for a roughly triangular pulse, Δv ≈ ½ · a_peak · Δt. A 15.6 m/s delta-v delivered over 100 ms implies an average deceleration near 15.6 g; delivered over 60 ms it implies roughly 26 g. Higher peak deceleration with shorter duration loads the occupant earlier and harder, which favors earlier airbag deployment and higher pretensioner force. Lower peak with longer duration gives the restraint system more time but demands more stroke and vent authority. Neither is universally better — the pulse family determines which trade is available.

Centroid Time: Where the Pulse Sits

Centroid time t_c is the first moment of the acceleration history normalized by delta-v:

t_c = ∫ t · a(t) dt / ∫ a(t) dt

It answers a question peak and duration cannot: is the velocity change delivered early or late? An early-centroid pulse (t_c near 30–40 ms for a 100 ms event) front-loads deceleration, so the occupant begins moving relative to the vehicle sooner. Airbag deployment must complete before that relative motion reaches the occupant. A late-centroid pulse (t_c near 60–70 ms) gives more time before occupant loading but concentrates the hardest deceleration later, which can catch a deploying airbag mid-inflation. Centroid time is the single best scalar for deciding deployment timing thresholds.

Slope: How Fast the Pulse Ramps

Slope is the rate of change of acceleration, typically measured on the rising edge in g/ms. A steep slope (high jerk) means the occupant's chest and head are loaded abruptly; a shallow slope spreads the onset. Slope drives two restraint decisions: pretensioner firing time (a steep slope requires earlier belt take-up) and load-limiter threshold (a steep slope may require a lower threshold to avoid chest deflection spikes). Slope is also the metric most sensitive to filtering — a 1000 Hz raw signal and a 300 Hz filtered signal can differ by a factor of two in peak slope. Always report the filter class (e.g., CFC 60 per SAE J211) alongside any slope value.

Energy Equivalent Metrics

Two pulses can share delta-v, peak, duration, and centroid yet differ in energy. The energy equivalent velocity, sometimes called the "equivalent square wave" or HIC-equivalent velocity, captures this:

v_eq = sqrt( (1/Δt) · ∫ a(t)² dt ) · Δt

More commonly, engineers use the energy spectral density or the root-mean-square acceleration a_rms = sqrt( (1/Δt) · ∫ a(t)² dt ). A pulse with a high a_rms concentrates energy in a short window; a pulse with the same delta-v but lower a_rms spreads it. For restraint tuning, a_rms correlates better with chest deflection than peak alone, because the chest responds to accumulated loading, not just the maximum. When two candidate pulses have equal delta-v and centroid but different a_rms, the higher-a_rms pulse will generally require a more aggressive load limiter.

Computing the Metrics from a Digitized Pulse

Given a sampled acceleration array a[i] at uniform time step dt, the metrics are computed as follows. This is the exact procedure used in restraint CAE post-processing and in test-data reduction.

# a[i] in g, dt in seconds, SAE vehicle frame (deceleration positive here)
g = 9.80665                      # m/s^2 per g
dv = sum(a[i] * g * dt for i in range(n))          # delta-v, m/s
a_peak = max(abs(a[i]) for i in range(n))          # peak, g
t_c = sum(t[i] * a[i] * dt for i in range(n)) / sum(a[i] * dt for i in range(n))
a_rms = sqrt(sum((a[i]**2) * dt for i in range(n)) / (n * dt))
slope = (a[k+1] - a[k]) / dt     # evaluated on the rising edge, g/s

Two practical cautions. First, filter before differentiating: slope computed on an unfiltered 10 kHz signal is dominated by sensor noise. Apply the SAE J211 CFC 60 filter for occupant-relevant slope, CFC 180 for structural work. Second, define pulse onset and end explicitly. A common convention is onset at the last zero crossing before 0.5 g and end at the first zero crossing after the pulse falls below 0.5 g. Without a fixed convention, duration and centroid shift by several milliseconds between analysts.

Worked Example: Two Pulses, Same Delta-v

Consider two candidate pulses for a 56 km/h frontal sled, both delivering 15.6 m/s. Pulse A is a 60 ms triangular pulse with a_peak = 53 g and t_c = 30 ms. Pulse B is a 120 ms triangular pulse with a_peak = 26.5 g and t_c = 60 ms. Both satisfy the delta-v target. Their restraint implications differ sharply:

For Pulse A, a typical tuning response is to advance the airbag trigger by 8–12 ms, raise pretensioner force, and lower the load-limiter threshold to keep chest deflection within target. For Pulse B, the trigger can be later, but the airbag must remain inflated longer, which means larger vent area or a dual-stage inflator with a delayed second stage. The metrics tell you which knobs to turn; the pulse family tells you how much authority each knob has.

Metric Selection by Restraint Decision

Not every metric drives every decision. The mapping below reflects standard restraint engineering practice and is the reference used throughout the rest of this course.

• Airbag volume and vent sizing — delta-v and a_rms.
• Deployment trigger time — centroid time and rising slope.
• Pretensioner fire time and force — rising slope and peak deceleration.
• Load-limiter threshold — a_rms and peak deceleration.
• Occupant kinematics timing (KTH, chest deflection) — centroid time and duration.
• Pulse family classification — centroid time and duration ratio.

With these six metrics defined and computable, the next step is to use them to classify pulses into families and to predict which restraint strategies will transfer across families. That classification is the subject of the following chapter.