2. Multibody and FE formulations in MADYMO

MADYMO is a hybrid solver: within a single model it can host rigid multibody systems, flexible finite-element structures, and the coupling interfaces that tie them together. Understanding which formulation owns which degrees of freedom — and how the two worlds exchange state — is the prerequisite for building occupant models that are both stable and physically credible. This section walks through the four building blocks (rigid bodies, joints, force elements, shell/solid elements) and then the coupling strategies that connect them.

2.1 Rigid Bodies and the Multibody State

A rigid body in MADYMO is defined by mass, centre of gravity, and an inertia tensor expressed in a body-fixed local frame. Its configuration is described by six generalized coordinates: three translational (position of the local origin in the global frame) and three rotational (orientation, typically Euler parameters or a rotation matrix internally). Because the body cannot deform, its internal energy is zero; all motion comes from the joint constraints and applied loads.

Typical occupant-model rigid bodies include the head, torso segments, pelvis, and limb segments when a lumped-parameter dummy is used. Each carries an ellipsoid or facet surface for contact, but the surface is only a geometric attachment — it does not add stiffness. Stiffness enters through force elements and contacts.

2.2 Joints: Constraining Relative Motion

Joints remove degrees of freedom between two bodies. MADYMO offers revolute, translational, spherical, universal, cylindrical, and free joints, plus bracket joints for arbitrary kinematic pairs. A revolute joint, for example, leaves one rotational DOF and constrains the other five; the constraint is enforced by the solver, not by stiff springs.

Joint placement matters. The joint location defines the instantaneous centre of rotation, and an offset joint produces a moment arm that converts force into torque. In a neck model, the occipital condyle joint is placed at the anatomical pivot; moving it 20 mm forward changes the head rotation response noticeably under the same belt load. Joints may also carry friction or damping, specified as a torque–velocity curve, which is useful for representing soft-tissue resistance without adding a separate element.

2.3 Force Elements: Springs, Dampers, and Contacts

Force elements generate loads as functions of relative displacement, velocity, or user-defined state. The common types are:

• Spring/damper elements — defined by a force–displacement curve and a damping coefficient, acting along a line between two points.
• Belt elements — slip-rings and belt segments with tension–strain behaviour, used for seatbelt routing.
• Contact interactions — penalty-based or impulse-based, between ellipsoids, facets, or FE surfaces.
• Point restraints — prescribed motion or force at a node, often used to drive a sled.

A spring element is characterised by its loading and unloading curves; hysteresis is captured by specifying separate paths. For a chest deflection model, a typical force–deflection curve might rise to 3 kN at 50 mm compression, with unloading following a lower path to represent energy dissipation. The element reports force and relative displacement, which are the quantities you compare against corridor data.

2.4 Shell and Solid Elements

When deformation matters — a deformable steering column, a crash-pad, or a biofidelic rib cage — the FE formulation takes over. Shell elements (4-node quadrilateral, 3-node triangular) carry membrane and bending stiffness and are suited to thin structures. Solid elements (hexahedral, tetrahedral) carry full 3D stress and are used for bulky components. MADYMO's FE solver supports explicit time integration, which is appropriate for impact events lasting 50–150 ms.

Material models range from linear elastic to elastoplastic, viscoelastic, and foam. A foam material, for example, is defined by a stress–strain curve in compression and a separate unloading curve; the energy absorbed is the area between them. Element size should be small enough to resolve the deformation wavelength — a common rule is at least three elements across the expected bending radius.

2.5 Coupling Strategies

The hybrid capability is what distinguishes MADYMO. Coupling connects multibody and FE domains through one of several mechanisms:

Force-based coupling is the most common in occupant safety. The multibody side computes a force from its spring or contact model; that force is applied to the FE nodes, and the resulting nodal displacement is fed back to update the relative position. The exchange happens every time step, so stability depends on the time step being small enough relative to the stiffest element. A practical check: if the coupling force oscillates or the energy balance drifts, reduce the time step or add numerical damping to the force element.

2.6 Choosing the Right Formulation

The decision is driven by what you need to observe. If the question is "how far does the head travel," a rigid multibody model is sufficient and fast. If the question is "what is the rib strain under belt load," FE is required. Hybrid models are justified when a rigid body interacts with a deformable structure and the interaction itself is the phenomenon of interest — for example, an airbag inflating against a deformable instrument panel while the occupant's head loads the airbag.

A useful rule of thumb: start with a multibody-only model to establish kinematics and timing, then replace the specific component whose deformation you need to study with an FE representation, and couple it. This staged approach keeps the model tractable and makes debugging easier, because you can compare the hybrid result against the multibody baseline.

The diagram above shows the two domains and the coupling interface that exchanges force and displacement every time step. The multibody solver computes joint and force-element loads; the FE solver computes nodal displacements; the coupling interface maps one to the other. The dashed feedback path is the key to stability — if the exchange is not consistent, the solution diverges.

In practice, the formulation choice is recorded in the model input deck. A minimal coupling definition for a belt-to-chest interaction might look like this:

FORCE_ELEMENT / BELT_TO_CHEST
  TYPE = CONTACT
  MB_SURFACE = ellipsoid_chest
  FE_SURFACE = shell_set_ribcage
  PENALTY_STIFFNESS = 1.0e5   ! N/m
  DAMPING = 0.05              ! fraction of critical
  FRICTION = 0.3
END

The penalty stiffness must be high enough to prevent penetration but low enough to keep the time step stable. A useful starting point is to set it so that the contact frequency is at least ten times the highest structural frequency of interest. The damping fraction controls the numerical oscillation; 0.05 to 0.10 is typical for occupant contact.

Once the model runs, verify the coupling by plotting the contact force on both sides — the multibody force and the sum of FE nodal forces should match within a few percent. If they diverge, check for missing nodes in the FE surface set or a mismatch in the local coordinate systems. This verification step is what separates a working hybrid model from a plausible-looking but incorrect one.