2. Multibody and finite element foundations in MADYMO

MADYMO is a hybrid solver: it combines a multibody (MB) formulation for rigid and jointed systems with an explicit finite element (FE) formulation for deformable structures. Occupant restraint modeling sits precisely at the interface between these two worlds. An airbag is a folded FE membrane inflated by gas; a belt is a multibody element with stiffness and slip; a Hybrid III dummy is a chain of rigid bodies connected by joints and restrained by force models. Understanding how these three modeling primitives — rigid bodies, joints, and force models — behave, and how they interact with FE parts, is the prerequisite for building a restraint model that converges and produces physically meaningful injury metrics.

Rigid bodies and the multibody state

In MADYMO, a rigid body is defined by mass, center of gravity, and an inertia tensor expressed in a body-fixed local coordinate system. Each body carries six degrees of freedom (three translations, three rotations) unless constrained. The global position and orientation are updated by integrating the equations of motion in time. For restraint work, the most important rigid bodies are the dummy segments (head, thorax, pelvis, femur, etc.) and the vehicle reference bodies (seat, floor, steering column).

A practical point that is often overlooked: the inertia tensor must be defined in the body's local frame, and the local frame origin should coincide with the center of gravity. If the origin is offset, MADYMO will still compute the dynamics correctly only if the parallel-axis terms are handled by the pre-processor; manual input of an inertia tensor about a non-CG origin is a common source of unrealistic rotation. For a Hybrid III 50th percentile head, a typical input is mass ≈ 4.54 kg and principal moments of inertia on the order of 0.01–0.02 kg·m². These values are illustrative and should be taken from the dummy manufacturer's calibration data, not assumed.

Joints: kinematic constraints between bodies

Joints remove degrees of freedom and transmit forces between bodies. MADYMO supports revolute, translational, spherical, universal, cylindrical, and free joints, plus bracket and cardan joints. In a dummy model, the neck is typically represented by a combination of revolute and translational joints with nonlinear stiffness, while the lumbar spine uses a spherical or universal joint with rotational stiffness.

The choice of joint type determines which load paths exist. A revolute joint transmits forces in all three directions but no moments about its axis; a spherical joint transmits forces but no moments at all. If you model the neck as a single spherical joint, you lose the bending stiffness that governs head rotation in a frontal impact. The correct approach is to add a force model (a rotational spring-damper) across the joint to restore the missing stiffness. This is the standard MADYMO idiom: joints define kinematics, force models define kinetics.

Force models: the constitutive behavior of restraint systems

Force models in MADYMO are user-defined relationships between relative displacement (or velocity) and force, applied between two bodies or between a body and the ground. They are the workhorse for restraint modeling because they let you represent belt webbing, seat cushion compression, airbag contact, and dummy joint stiffness without meshing every component.

Common force model types include:

• Spring-damper (linear or nonlinear): force as a function of relative displacement and velocity. Used for belt elongation, seat foam, and joint stiffness.
• Contact force model: penalty-based contact between MB bodies or between MB and FE surfaces. Used for dummy-to-airbag and dummy-to-seat contact.
• Belt model: a specialized force model that includes slip, retractor force, and pretensioner characteristics.
• Airbag gas flow model: defines the mass flow and energy input from the inflator into the FE airbag chamber.

For a belt force model, a typical input is a loading-unloading curve. For example, a 50 mm width polyester webbing might be specified with a loading stiffness of 200 N/mm up to 4 kN, then a yield plateau, and an unloading path with hysteresis. These numbers are illustrative; actual values come from component tests. The key is that the force model must be defined in the correct local coordinate system and with the correct sign convention. MADYMO uses a right-handed coordinate system; tension is positive when the belt is pulled away from the anchor. Reversing the sign will cause the belt to push instead of restrain, and the simulation will diverge or produce nonsensical kinematics.

FE parts and their coupling to the multibody system

FE parts in MADYMO are used for airbags, folded membranes, and deformable interior trim. They are meshed with shell or membrane elements and solved with an explicit time integration scheme. The coupling between FE and MB is handled through contact interfaces and tied nodes. A dummy surface (MB) contacts an airbag surface (FE) via a penalty contact; the contact force is computed from penetration depth and a contact stiffness. The airbag itself is inflated by a gas flow model that adds mass and energy to the FE chamber.

The critical modeling decision is the time step. FE parts require a stable time step based on element size and material wave speed. A typical airbag mesh with 10 mm elements and a wave speed of 50 m/s gives a stable step around 0.2 µs. The multibody solver can take much larger steps, so MADYMO subcycles the FE solver within the MB step. If the FE mesh is too coarse or the material stiffness too high, the simulation will terminate with a negative volume or time step error. In restraint modeling, this usually means the airbag mesh needs refinement near the fold lines or the contact stiffness needs to be reduced.

Putting it together: a belt-restraint modeling sequence

Consider a frontal impact model with a Hybrid III dummy, a seat belt, and a driver airbag. The modeling sequence in MADYMO is typically:

1. Define the dummy multibody chain. Import or build the rigid bodies with correct mass and inertia. Connect them with joints. Add joint force models for neck, lumbar, and knee stiffness.
2. Define the belt system. Create belt segments as force models between the dummy and the vehicle anchors. Specify the retractor force, pretensioner stroke, and slip behavior. The belt path must be defined with the correct routing points; a missing routing point will cause the belt to cut through the dummy.
3. Define the airbag FE model. Mesh the folded airbag, assign membrane properties, and attach the gas flow model. The inflator mass flow curve is a function of time and is usually provided by the airbag supplier.
4. Set up contact. Define contact between the dummy and the airbag, the dummy and the seat, and the belt and the dummy. Contact stiffness should be high enough to prevent penetration but low enough to avoid instability. A common starting value is 10–50 N/mm² for dummy-to-airbag contact.
5. Run and check. Monitor the energy balance. The total energy (kinetic + internal + contact) should remain approximately constant. A sudden increase in internal energy indicates hourglassing or a bad element. A negative volume error indicates the airbag mesh is too coarse or the contact stiffness is too high.

Sign conventions and coordinate systems

MADYMO uses a right-handed Cartesian coordinate system. The default global frame has X forward, Y left, and Z up. All force models and joints must be defined consistently with this frame. When you define a belt force model, the relative displacement is computed along the belt direction; a positive displacement means the belt is stretched. When you define a contact force, the penetration depth is positive when the surfaces overlap. Reversing these signs will produce a model that pushes instead of restrains, and the dummy will fly out of the seat. Always verify the sign convention by running a simple component test — for example, a single belt segment with a known force — before assembling the full model.

Summary

Rigid bodies provide the inertia, joints provide the kinematics, and force models provide the constitutive behavior. FE parts add deformable structures such as airbags and trim. The art of MADYMO restraint modeling lies in choosing the right level of detail for each component: use multibody elements where the deformation is not critical, and FE where the deformation drives the injury outcome. The coupling between the two domains is handled by contact and subcycling, and the stability of the simulation depends on the time step, mesh quality, and contact stiffness. Mastering these foundations is what allows you to build a restraint model that is both computationally efficient and physically accurate.