2. Restraint system fundamentals in MADYMO

MADYMO models a restraint system as a set of discrete, physically parameterized components that interact with the occupant through contact and force exchange. Before any optimization study can be meaningful, the analyst must understand how each hardware element — seat belt, airbag, pretensioner, and load limiter — is represented in the solver, which parameters govern its behavior, and how those parameters map to real-world design variables. This section establishes that mapping and shows how the components are assembled into a working restraint model.

Seat Belt Modeling: FE Belts and Multibody Belts

MADYMO offers two fundamentally different belt representations. The multibody belt discretizes the webbing into a chain of point masses connected by spring-damper elements. Each segment carries a stiffness and damping characteristic, and the belt slides through sliprings and retractor anchors. This approach is computationally cheap and well suited to early-phase concept studies where the belt routing and force level matter more than local webbing deformation.

The finite element belt uses membrane or shell elements with a defined thickness and material law. It captures webbing stretch distribution, local folding, and interaction with the airbag surface far more accurately, at a significantly higher computational cost. In practice, FE belts are preferred when belt-to-airbag interaction or belt routing over the pelvis is a design concern.

Key belt parameters that appear repeatedly in optimization studies include:

  • Belt stiffness — the force-versus-elongation slope of the webbing, typically expressed in N per unit strain or as a force-elongation curve.
  • Slipring friction — the Coulomb friction coefficient between webbing and slipring, which controls how much force is transferred between belt segments.
  • Retractor behavior — locked, unlocked, or with a defined payout characteristic.
  • Belt routing geometry — anchor points, slipring positions, and the resulting belt path across the occupant.

A common modeling error is to define belt stiffness as a single linear value when the actual webbing exhibits progressive stiffening. MADYMO allows a piecewise-linear force-elongation curve, and using it typically changes peak belt force predictions by 10–20% in frontal loading.

Airbag Modeling: Uniform Pressure and Gas Flow

MADYMO represents airbags through the gas flow module. The bag geometry is defined as a folded or unfolded reference mesh, and the solver computes the internal pressure from mass flow in and out of the bag, temperature, and volume change. The two dominant modeling choices are:

  • Uniform pressure (UP) bag — a single control volume with one pressure value. Fast, stable, and adequate for driver frontal airbags where pressure gradients are small.
  • Gas flow (GF) bag — the bag is divided into multiple chambers connected by flow passages. Required for side airbags, curtain airbags, and any bag with significant internal pressure gradients.

The inflator is characterized by a mass flow rate curve and a gas temperature curve, both functions of time. These curves are typically derived from tank tests. The bag fabric is defined by a leakage coefficient (permeability) and a vent hole area. Vent hole sizing is one of the most sensitive parameters in occupant restraint optimization: a 10% change in vent area can shift peak chest deflection by several millimeters.

When converting a tank test mass flow curve into a MADYMO inflator definition, verify that the total delivered gas mass matches the tank test within 2%. A mismatch here propagates directly into bag pressure and occupant loading.

Pretensioners and Load Limiters

A pretensioner removes slack from the belt early in the crash event. In MADYMO it is modeled as a retractor with a defined payout-versus-time or force-versus-payout characteristic. The pretensioner stroke is typically 40–80 mm, and the firing time is a critical parameter — too early and the occupant is pulled into the deploying airbag, too late and the benefit is lost.

A load limiter caps the belt force transmitted to the occupant. It is modeled as a force-limiting element in the retractor or as a torsion bar with a defined force-payout curve. The load limiter force level is the single most influential parameter for chest deflection in frontal impacts. Typical values range from 2.5 kN to 4.5 kN, with the optimum depending on occupant size and airbag aggressiveness.

The interaction between pretensioner and load limiter is sequential: the pretensioner fires first to remove slack, then the load limiter engages once belt force reaches its threshold. In MADYMO, this is captured by defining the retractor force-payout curve with an initial low-force pretensioning segment followed by a plateau at the load limit level.

System Integration and Parameter Mapping

The figure below shows how the four restraint components connect in a MADYMO model and which parameters flow between them.

Restraint System Component Mapping in MADYMO Occupant Multibody / FE Seat Belt MB chain or FE membrane Stiffness · Friction · Routing Retractor + Pretensioner Payout vs. time Firing time · Stroke Load Limiter Force-payout curve Limit level (2.5–4.5 kN) Airbag UP or Gas Flow Inflator · Vent · Leakage belt force payout force cap contact pressure belt-bag interaction Key Coupling Pretensioner timing vs. airbag deployment vs. load limiter level Modeling Choice MB belt → concept phase FE belt → detail phase GF bag → side/curtain

Practical Modeling Sequence

A reproducible workflow for building a frontal restraint model in MADYMO proceeds as follows. First, define the occupant model — either a Hybrid III multibody dummy or an FE human body model — and position it according to the seating reference point. Second, route the belt across the occupant using slipring coordinates that match the vehicle geometry. Third, attach the retractor and define the pretensioner payout curve. Fourth, define the load limiter force-payout curve. Fifth, position and inflate the airbag, ensuring the folded geometry does not intersect the occupant at time zero.

After the model runs, the key verification outputs are belt force at the retractor, chest deflection, head acceleration, and airbag internal pressure. If belt force exceeds the load limiter plateau, the load limiter is not engaging correctly — check that the force-payout curve is defined in the correct units and that the retractor is not locked. If airbag pressure spikes unrealistically, verify the vent hole area and leakage coefficient against the tank test data.

Do not tune pretensioner firing time and load limiter force independently. They interact strongly: a shorter pretensioner stroke requires a lower load limiter force to achieve the same chest deflection, and vice versa. Always evaluate them as a coupled pair in any optimization study.

With these component models in place, the restraint system is ready for parameter studies. The next step is to define which parameters are design variables, which are fixed, and what objective function — typically a weighted combination of injury criteria — will drive the optimization.

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