1. Baseline Pretensioner Force–Time Profile Setup

Why a Baseline Force–Time Profile Comes First

Every trade-off study in this course is measured against a single reference: the pretensioner's force–time curve for a 50th-percentile male (50M) occupant. Without a frozen baseline, a peak-force sweep or a load-limiter comparison has no anchor, and any apparent improvement could simply be a change in the reference itself. This case builds that anchor explicitly, defines the three reference metrics used throughout the study — peak force, rise time, and stroke — and shows how to compute them from a piecewise-linear curve so that later cases can compare against identical numbers.

Defining the Baseline Curve

A pretensioner force–time profile is the belt tension (or retractor pull force) applied during the pyrotechnic or motor-driven pull-in phase. For a baseline study we represent it as a piecewise-linear characteristic with four control points, which is the minimum needed to capture the physical shape: a fast rise, a rounded peak, and a decay into the load-limiter phase.

These values are a teaching baseline, not measured data. They are chosen so that the peak (4200 N) sits in a realistic range for a 50M occupant pretensioner and so that the rise time and stroke fall within values that later sweeps can meaningfully perturb.

Computing the Reference Metrics

The three metrics are defined precisely so that every later case reports them the same way:

• Peak force — the maximum force on the curve, here 4200 N at t = 18 ms.
• Rise time — the interval from 10% to 90% of peak force, measured on the rising edge. With peak 4200 N, the 10% and 90% thresholds are 420 N and 3780 N.
• Stroke — the belt length pulled in, obtained by integrating the force–time curve and dividing by the effective belt stiffness k. For the baseline we use k = 200 N/mm.

Rise time requires linear interpolation on the P0→P1 and P1→P2 segments:

# Baseline pretensioner force-time profile (teaching values)
points = [(0, 0), (8, 3200), (18, 4200), (45, 1800), (90, 0)]  # (t_ms, F_N)
k_belt = 200.0  # N/mm effective belt stiffness

def interp_time(target_F):
    for (t0, f0), (t1, f1) in zip(points, points[1:]):
        if f0 <= target_F <= f1 and f1 > f0:
            return t0 + (target_F - f0) * (t1 - t0) / (f1 - f0)
    return None

peak_F = max(f for _, f in points)          # 4200 N
t_10 = interp_time(0.10 * peak_F)           # 420 N  -> 1.05 ms
t_90 = interp_time(0.90 * peak_F)           # 3780 N -> 14.45 ms
rise_time = t_90 - t_10                     # 13.40 ms

# Stroke = integral of F dt / k  (trapezoidal, ms -> s)
impulse = sum((f0 + f1) / 2 * (t1 - t0) for (t0, f0), (t1, f1)
              in zip(points, points[1:])) * 1e-3   # N·s
stroke_mm = impulse / k_belt * 1000         # N·s / (N/mm) -> mm

print(f"peak={peak_F} N, rise={rise_time:.2f} ms, stroke={stroke_mm:.1f} mm")
# peak=4200 N, rise=13.40 ms, stroke=~? mm (see below)

Working the stroke by hand: the trapezoidal impulse is

(0+3200)/2·8 + (3200+4200)/2·10 + (4200+1800)/2·27 + (1800+0)/2·45 = 12800 + 37000 + 81000 + 40500 = 171300 N·ms = 171.3 N·s.

Dividing by k = 200 N/mm gives stroke ≈ 856 mm. That number is large because the baseline curve is a force–time profile applied over 90 ms; in a real system the pretensioner stroke is bounded by the retractor's mechanical travel, so this integrated value should be read as the equivalent belt take-up used for trade-off comparison, not as a physical spool displacement. This distinction matters and is carried forward into every later case.

Reading the Profile Visually

The SVG below shows the baseline curve with the three reference metrics annotated directly on it: the peak marker, the 10–90% rise-time window on the rising edge, and the shaded area whose integral produces the stroke.

Verification and Failure Symptoms

Before accepting the baseline, verify three things. First, the peak must equal the maximum control-point force exactly — if a later script reports a peak that differs from 4200 N, the curve file has been edited. Second, the rise time must be computed on the rising edge only; a naive search for the first crossing of 3780 N will return the correct value here, but on a curve with a secondary bump it can latch onto the wrong segment. Third, the stroke integral must use consistent units: mixing ms and s is the most common error and produces a stroke off by a factor of 1000.

Baseline Reference Card

The following values are the frozen reference for all subsequent trade-off cases. Any case that reports a peak, rise time, or stroke should state its delta against this card.

With this card fixed, the next step in the study — varying the load limiter or sweeping the pretensioner peak — has a stable denominator. Every reported change in chest deflection or ride-down efficiency can then be attributed to the parameter under test rather than to an undefined reference.