4. Restraint system modeling

In MADYMO, the restraint system is not a single object but a coordinated set of FE and multibody components whose timing, geometry, and force–displacement behavior determine the occupant's forward excursion and the resulting injury metrics. This chapter explains how to define seat belts, retractor pretensioners, force-limiting torsion bars, and airbag deployment in a coupled MADYMO model, and how to verify that each subsystem behaves as intended before running a full sled or vehicle crash simulation.

Seat Belt Modeling: FE Belt vs. Multibody Belt

MADYMO offers two principal belt representations. The multibody belt is a chain of belt segments connected by joints, with point-to-surface contact against ellipsoids. It is computationally cheap and well suited to concept studies and parameter sweeps. The FE belt uses membrane or shell elements with a defined width, thickness, and material law, and contacts the occupant through the FE or facet surface. FE belts capture belt roll-out, local pressure distribution, and interaction with the airbag cushion, which matters for rib fracture and submarining assessment.

For an FE belt, the essential inputs are the belt path (a sequence of nodes or a curve), the belt width (typically 46–50 mm for a 3-point belt), the thickness, and the material model. A common choice is a fabric material with orthotropic behavior: stiff along the belt axis and compliant in shear. A simplified linear-elastic belt with a Young's modulus of approximately 2–5 GPa along the axis and a Poisson's ratio near 0.3 is often used for initial correlation, but a nonlinear stress–strain curve derived from a tensile test is preferred for production-level models.

Belt routing must respect the actual hardware: the D-ring (shoulder anchor), the buckle, the lap anchor, and the retractor. In MADYMO, these are represented by belt segments passing over rigid or deformable surfaces. A frequent modeling error is to route the belt through the occupant's torso without a slip surface at the D-ring, which artificially increases shoulder belt tension and reduces forward excursion. Always define a low-friction contact at the D-ring and buckle to allow belt slip.

Pretensioners: Retractor and Buckle Pretension

A pretensioner removes slack from the belt early in the crash, typically within 10–20 ms. MADYMO models this with a belt pretensioner element that applies a prescribed retraction stroke or a force–time curve to a belt segment. Two common types are the retractor pretensioner (pulls the shoulder belt into the retractor) and the buckle pretensioner (pulls the buckle downward, tensioning both lap and shoulder belts).

The pretensioner is defined by a stroke–time or force–time characteristic. For example, a retractor pretensioner might retract 60–80 mm of belt over 15 ms, with the force rising to 2–3 kN and then dropping as the gas generator pressure decays. In MADYMO, this is entered as a PRETENSIONER with a FUNCTION table:

PRETENSIONER
  ID        = 1
  TYPE      = RETRACTOR
  BELT_ID   = 1
  SEGMENT   = 3
  FUNCTION  = 1001   ! stroke [m] vs time [s]
  DIRECTION = 1      ! retraction direction along belt
END

The function table 1001 might contain points such as (0.0, 0.0), (0.005, 0.02), (0.010, 0.05), (0.015, 0.07), (0.020, 0.075). The exact values come from the supplier's gas generator data or from a bench test. A common pitfall is to apply the pretensioner stroke too late; if the occupant has already moved forward, the pretensioner cannot recover the slack and the belt will bottom out on the load limiter prematurely.

Load Limiters: Force-Limiting Torsion Bar

The load limiter caps the belt force transmitted to the occupant, trading increased forward excursion for reduced chest deflection. In MADYMO, this is modeled by a force-limiting belt segment or a retractor with a force-limiting function. The belt force is limited to a plateau value, typically 3–5 kN for a front-row occupant, after which the belt pays out additional webbing.

The implementation uses a force–displacement curve for the torsion bar. A representative curve for a 4 kN limiter might be: (0 mm, 0 kN), (10 mm, 3.5 kN), (30 mm, 4.0 kN), (60 mm, 4.0 kN), (80 mm, 4.5 kN) as the bar work-hardens. This curve is assigned to the retractor element, and the belt segment connected to it is allowed to extend according to the curve. The key verification is to plot belt force at the retractor versus time and confirm the plateau is reached and maintained. If the force exceeds the plateau significantly, the limiter curve may be applied to the wrong segment or the belt may be locking at the D-ring.

Airbag Deployment Definitions

Airbag modeling in MADYMO uses the gas flow or uniform pressure approach. The uniform pressure method is simpler and adequate for driver and passenger frontal airbags when the cushion shape is well defined. The gas flow method, using GASFLOW and CHAMBER definitions, is preferred when the airbag has multiple chambers, vents, or interacts with the occupant before full inflation.

An airbag definition requires: the folded geometry (often imported as an FE mesh), the inflator mass flow rate and temperature, the vent area, and the contact definition with the occupant. The inflator is defined by a GAS property and a MASSFLOW function. For a typical driver airbag, the mass flow might peak at 2–3 kg/s at 10 ms and decay to zero by 40 ms, with a gas temperature of 800–1000 K.

AIRBAG
  ID          = 10
  TYPE        = UNIFORM_PRESSURE
  MESH        = driver_bag
  INFLATOR    = 1
  VENT_AREA   = 0.002   ! m^2
  GAS         = 1
END

INFLATOR
  ID          = 1
  MASSFLOW    = 2001   ! kg/s vs time
  TEMPERATURE = 900    ! K
END

Deployment timing is critical. The airbag must be fully inflated before the occupant contacts it, but not so early that the occupant "punches through" a deflating bag. In MADYMO, the TIME of inflator activation is set relative to the crash pulse. A common starting point is 10–15 ms after the crash pulse onset for a 50 km/h frontal impact. The occupant contact with the airbag is defined through a CONTACT between the occupant surface and the airbag mesh, with a friction coefficient of 0.3–0.5.

Integrated Restraint Timing and Verification

The restraint system must be verified as a whole. The figure below shows the timing chain from crash pulse to occupant contact, with the key parameters at each stage.

After the run, verify the belt force at the retractor and buckle, the airbag pressure, and the occupant kinematics. A common failure mode is the "belt slack" problem: if the pretensioner stroke is insufficient or the belt routing has too much friction, the occupant loads the belt before the pretensioner completes, and the load limiter engages early, increasing forward excursion. Another is airbag "punch-through": if the inflator mass flow is too low or the vent area too large, the bag deflates before the occupant contacts it, and the occupant strikes the steering wheel or IP.

Finally, always check the energy balance: the sum of kinetic energy absorbed by the belt, airbag, and structural deformation should match the initial kinetic energy within a few percent. A large discrepancy indicates a modeling error such as missing contact, incorrect mass, or a non-physical force curve. With the restraint system defined and verified, the model is ready for occupant injury assessment using criteria such as HIC, chest deflection, and femur force.