4. Seatbelt system modeling

A seatbelt system in MADYMO is not a single belt element but a chain of interacting components: the webbing path, the slipring at the D-ring or buckle, the retractor spool, the pretensioner, and the load limiter. Each component changes how force and displacement propagate to the occupant. If any one of them is modeled incorrectly, the chest deflection and head excursion predicted by the simulation can deviate substantially from sled or vehicle test data. This chapter explains how to build each component in MADYMO, how they connect, and how to verify that the assembled system behaves as intended.

Belt Routing and the Belt Element Chain

Belt routing defines the geometric path the webbing follows from the retractor anchor, across the occupant, through the slipring, to the buckle anchor. In MADYMO, this path is discretized into a chain of BELT elements. Each belt element is a one-dimensional, tension-only, visco-elastic member that can slide through sliprings and interact with the occupant's FE or facet surface through contact.

The routing must be defined in the same coordinate system as the occupant model. A common convention is the vehicle coordinate system: X forward, Y to the occupant's left, Z upward, with the origin at the vehicle datum. The belt anchor points are then specified as fixed points in this system. For a typical driver-side three-point belt, the anchor coordinates might be:

Retractor anchor (fixed):   X = -0.15 m, Y = -0.55 m, Z = 1.05 m
D-ring / slipring:          X = -0.10 m, Y = -0.50 m, Z = 1.35 m
Buckle anchor:              X =  0.05 m, Y =  0.35 m, Z = 0.10 m
Lap anchor (outboard):      X = -0.05 m, Y = -0.50 m, Z = 0.05 m

These values are illustrative and must be replaced with the actual vehicle geometry. The belt is then routed from the retractor, through the D-ring, across the shoulder and chest, through the buckle slipring, across the lap, and back to the lap anchor. The number of belt elements along each segment controls the trade-off between contact accuracy and computational cost. A practical starting point is 8–12 elements across the shoulder segment and 6–10 across the lap segment, with finer discretization near the slipring where curvature is high.

Belt material properties are defined by the stiffness and damping of the webbing. A typical polyester webbing has a stiffness on the order of 200–400 N per percent elongation, but this is strongly rate-dependent. In MADYMO, the belt element uses a force–strain curve that can be specified as a piecewise linear function. For a load-limiting belt, the webbing itself may be designed to stretch, but more commonly the load limiter is a separate mechanism at the retractor or buckle.

Sliprings: Modeling Friction and Direction Change

A slipring is a point constraint that allows the belt to slide through it while transmitting tension. In MADYMO, a slipring is defined as a node with a specified friction coefficient. The D-ring and the buckle tongue are the two most important sliprings in a three-point belt. The D-ring changes the belt direction from the retractor to the shoulder, and the buckle slipring changes direction from the shoulder to the lap.

Friction at the slipring is modeled by a Coulomb friction coefficient, typically 0.1–0.3 for a steel D-ring with polyester webbing. This friction causes a tension drop across the slipring: the tension on the retractor side is lower than on the occupant side during loading, and the difference is proportional to the friction coefficient and the wrap angle. In MADYMO, the slipring friction is specified as a single value, but the effective tension ratio depends on the wrap angle, which is determined by the routing geometry.

If the slipring is omitted or assigned zero friction, the belt will slide too freely, and the predicted shoulder belt force will be lower than in reality. Conversely, excessive friction can lock the belt and over-constrain the occupant. A useful check is to compare the simulated belt force at the retractor and at the shoulder with test data; the ratio should match the expected friction loss.

Retractors: Spool, Locking, and Belt Pay-Out

The retractor is modeled as a belt element with a spool that can pay out or retract webbing. In MADYMO, this is typically implemented as a BELT element with a prescribed pay-out curve or as a retractor node with a locking mechanism. The retractor has three states: free pay-out, locked, and load-limited pay-out.

During the initial phase of a crash, the retractor is unlocked and the belt pays out as the occupant moves forward. The pay-out is governed by the spool inertia and the retractor spring. In MADYMO, this can be modeled by a force–displacement curve at the retractor node. Once the retractor locks (typically within 10–20 ms of the crash pulse), the belt becomes a fixed length, and further occupant motion is resisted by the belt stiffness and the load limiter.

The locking time is a critical parameter. If the lock time is too late, the occupant moves forward excessively before the belt engages. If it is too early, the belt may be too tight and increase chest loading. A typical lock time for a pyrotechnic pretensioner is 5–15 ms after the crash pulse reaches the threshold. In MADYMO, the lock time is specified as a time-dependent switch on the retractor node.

Pretensioners: Removing Slack

A pretensioner removes slack from the belt by pulling the webbing into the retractor or buckle. In MADYMO, this is modeled as a prescribed displacement or force at the retractor or buckle anchor. The pretensioner stroke is typically 50–150 mm, and the stroke time is 10–30 ms. The pretensioner force is applied in the direction that shortens the belt path.

For a retractor pretensioner, the belt is pulled into the spool, which reduces the effective belt length. For a buckle pretensioner, the buckle anchor moves downward, which also shortens the belt path. In MADYMO, the pretensioner can be modeled as a BELT element with a time-dependent rest length, or as a prescribed motion at the anchor point. The key parameters are the stroke, the stroke time, and the force limit. A typical retractor pretensioner might have a stroke of 80 mm over 15 ms, with a peak force of 2–3 kN.

The pretensioner must be synchronized with the retractor lock. If the pretensioner fires before the retractor locks, the belt may pay out again. If it fires after the occupant has already moved forward, the benefit is reduced. In MADYMO, the pretensioner and retractor lock are typically triggered by the same crash pulse, with the pretensioner firing slightly before or at the same time as the lock.

Load Limiters: Controlling Belt Force

A load limiter caps the belt force by allowing the belt to pay out once a threshold force is reached. In MADYMO, this is modeled as a force-limiting element at the retractor or buckle. The load limiter is defined by a force–displacement curve: the belt force rises linearly with displacement until the limit force is reached, after which the force remains constant (or increases slightly) while the belt pays out.

Typical load limiter forces are 2–4 kN for a driver-side belt. The force level is chosen to balance chest deflection against head excursion. A lower force reduces chest injury but allows more forward motion, which can increase head excursion and the risk of head contact with the steering wheel. A higher force reduces head excursion but increases chest loading. In MADYMO, the load limiter is specified as a force–displacement curve at the retractor node, with the force plateau representing the limit force.

The load limiter interacts with the pretensioner: the pretensioner removes slack, and the load limiter controls the force after the belt is taut. The combination of pretensioner and load limiter is often called a "force-limiting retractor." In MADYMO, this is modeled as a single retractor node with both a pretensioner stroke and a load limiter curve.

Verification and Troubleshooting

After assembling the belt system, the model should be verified against a simple sled test or a component test. Key outputs to check include belt force at the retractor, belt force at the shoulder, occupant chest deflection, and head excursion. If the belt force is too low, check the slipring friction and the load limiter force. If the belt force is too high, check the pretensioner stroke and the retractor lock time. If the belt is not engaging the occupant properly, check the routing and the contact definition.

A common failure mode is the belt sliding through the slipring without generating tension. This can happen if the slipring friction is too low or if the belt elements are not properly connected. Another common issue is the belt penetrating the occupant's surface, which can be caused by insufficient contact stiffness or too few belt elements. In MADYMO, the contact between the belt and the occupant is defined by a contact stiffness and a friction coefficient; these should be tuned to match the observed belt–occupant interaction.

The diagram above shows the complete modeling chain. The retractor, pretensioner, and load limiter all act on the belt path, which is routed through the D-ring and buckle sliprings before contacting the occupant. The dashed feedback loop represents the force interaction between the occupant and the belt, which in turn affects the retractor pay-out and the load limiter force. This closed-loop behavior is what makes the seatbelt system a coupled problem: changing the load limiter force changes the occupant motion, which changes the belt force, which changes the load limiter pay-out.

In summary, a well-built seatbelt model in MADYMO requires careful attention to the routing geometry, the slipring friction, the retractor lock time, the pretensioner stroke, and the load limiter force. Each component has a clear physical role, and each can be verified independently before the full system is assembled. The next step is to integrate this belt model with the occupant model and the crash pulse, which is covered in the following chapter.