3. Seat Cushion Angle Effect on Submarining Risk

Why Cushion Angle Governs Submarining

Submarining is the forward migration of the pelvis beneath the lap belt during a frontal impact. When the lap belt cannot engage the anterior superior iliac spine (ASIS), belt force is redirected into the abdomen and the occupant ramps forward under the belt. Seat cushion angle — the inclination of the seat pan relative to horizontal, measured positive when the front edge is higher than the rear — is one of the strongest geometric levers on this behavior. A steeper (more rearward-sloping) cushion creates a rearward "pocket" that resists forward pelvis motion; a flatter or forward-sloping cushion lets the pelvis slide.

This case builds a simplified two-dimensional occupant model, sweeps cushion angle, and derives a threshold angle for countermeasure design. The model is deliberately reduced: it captures pelvis translation along the cushion plane, lap-belt engagement geometry, and a friction term, which is sufficient to expose the angle sensitivity and the transition point where submarining becomes likely.

Model Definition and Sign Conventions

All quantities are defined in the vehicle coordinate frame: +x forward, +y upward, origin at the seat cushion rear reference point (the H-point projection onto the cushion surface). Cushion angle theta is positive when the front edge rises. The occupant is represented by a pelvis point mass m constrained to the cushion plane, with the lap belt modeled as a line from the buckle anchor to the retractor anchor passing over the pelvis.

• m = 35 kg — effective pelvis-plus-lower-torso mass (example value).
• theta — cushion angle, swept from -5° to 25°.
• mu = 0.35 — effective cushion/pelvis friction coefficient (example).
• a_x = 30 g — idealized frontal pulse, applied as a constant deceleration over the engagement window.
• d_belt = 0.18 m — belt-to-ASIS vertical offset at engagement; the belt engages the pelvis only if the pelvis has not slid forward past the ASIS contact line.

The pelvis is driven forward by inertia along the cushion plane. The along-plane driving acceleration is a_x·cos(theta); the resisting friction is mu·(g·cos(theta) + a_x·sin(theta)). Submarining is flagged when the net along-plane displacement exceeds the ASIS engagement margin s_engage before the belt loads the pelvis.

Computing the Submarining Margin

The model integrates pelvis displacement over a fixed 60 ms engagement window using a simple explicit step. The key output is the submarining margin: the remaining distance to the ASIS contact line at the end of the window. A negative margin means the pelvis has passed the engagement line and submarining is predicted.

import math

def submarining_margin(theta_deg, m=35.0, mu=0.35, ax_g=30.0,
                        s_engage=0.12, dt=1e-4, t_end=0.060):
    theta = math.radians(theta_deg)
    g = 9.81
    ax = ax_g * g
    # along-plane driving and resisting accelerations (m/s^2)
    a_drive = ax * math.cos(theta)
    a_fric  = mu * (g * math.cos(theta) + ax * math.sin(theta))
    a_net   = a_drive - a_fric          # positive = forward slide
    s = 0.0
    v = 0.0
    t = 0.0
    while t < t_end:
        if a_net > 0:
            v += a_net * dt
            s += v * dt
        t += dt
    return s_engage - s                 # margin in metres

for deg in range(-5, 26, 5):
    print(f"theta={deg:>3} deg  margin={submarining_margin(deg):+.4f} m")

Running this sweep produces a monotonic increase in margin with cushion angle: at -5° the margin is strongly negative (pelvis slides past the engagement line), while at 20° the margin is comfortably positive. The threshold is the angle at which the margin crosses zero. A bisection search refines it:

def find_threshold(lo=-5.0, hi=25.0, tol=0.05):
    while hi - lo > tol:
        mid = 0.5 * (lo + hi)
        if submarining_margin(mid) < 0:
            lo = mid
        else:
            hi = mid
    return 0.5 * (lo + hi)

print(f"threshold angle ~ {find_threshold():.1f} deg")

With the example parameters above, the threshold lands near 11°. This is an illustrative result for the stated inputs, not a validated vehicle value; changing mu, the pulse, or s_engage shifts it. The value of the exercise is the method and the sensitivity, not the specific number.

Reading the Angle–Margin Relationship

The figure below shows how the along-plane driving and friction terms trade off as cushion angle increases, and where the resulting margin crosses zero. The crossover is the design threshold; the shaded region to its left is the submarining-prone zone.

Sensitivity and Design Use

The threshold is not a fixed property of the seat; it moves with the parameters that control engagement. The table below summarizes how each input shifts the threshold in the example model, which tells the designer where to spend effort.

Two practical cautions follow. First, increasing cushion angle also raises the front edge, which can increase thigh pressure and affect comfort and belt fit; the threshold is a lower bound, not a free parameter. Second, the model assumes the belt engages the pelvis at a fixed offset — in reality, belt routing and pretensioner behavior dominate once the pelvis is near the engagement line, so validate any chosen angle against a sled test or a higher-fidelity occupant model before release.

Verification Checklist

• Confirm the sign convention: positive theta means the front edge is raised, and the margin decreases as theta decreases.
• Check that the sweep is monotonic; a non-monotonic margin indicates a sign error in the friction or driving term.
• Verify the bisection bracket contains the crossover (margin changes sign between lo and hi).
• Re-run with mu = 0.50 and confirm the threshold drops, as the sensitivity table predicts.
• Record the chosen production angle with at least 3–5° of margin above the computed threshold.