2. MADYMO solver fundamentals for crash pulse analysis

Before any restraint parameter can be tuned against a crash pulse, the analyst must understand how MADYMO actually advances a model through time, how it decides that two bodies are touching, and how the model must be assembled so those two mechanisms behave as intended. This section covers the three foundations that determine whether a pulse-sensitivity study produces physically meaningful numbers or numerical noise: explicit time integration, contact algorithms, and model setup.

Explicit Time Integration

MADYMO's core solver is an explicit, conditionally stable integrator. The equations of motion are not solved simultaneously at each step; instead, accelerations are computed from the current force state, then integrated forward to obtain velocities and displacements. For a lumped-mass degree of freedom i, the update sequence within one time step Δt is:

a_i(t) = F_i(t) / m_i
v_i(t+Δt/2) = v_i(t-Δt/2) + a_i(t) · Δt
x_i(t+Δt) = x_i(t) + v_i(t+Δt/2) · Δt

This is the central-difference (leapfrog) scheme. Because no stiffness matrix is inverted and no iteration is performed, each step is cheap, which is why MADYMO can handle large contact-heavy occupant models. The price is conditional stability: the step size must satisfy the Courant–Friedrichs–Lewy (CFL) condition, approximately Δt ≤ 2/ωmax, where ωmax is the highest natural frequency in the model. In practice MADYMO derives a critical step from the stiffest spring and the smallest mass in the system, then applies a safety factor.

For crash pulse work this has a direct consequence. A stiff seatbelt webbing element or a rigid barrier contact can drive the critical step down to the order of 1 µs, so a 120 ms frontal event may require 120,000 or more steps. If the analyst forces a larger step to save runtime, the classic symptom is energy growth: the hourglass and total energy reports diverge, contacts chatter, and the occupant kinematics become erratic. The correct response is to soften the offending stiffness (for example, by using a realistic belt force–strain curve rather than a rigid approximation) or to accept the smaller step, not to override the stability limit.

Always check the solver's reported critical time step and the energy balance at the end of a run. A stable explicit solution keeps total energy approximately constant in the absence of external work, and the ratio of hourglass energy to internal energy should stay well below a few percent.

Contact Algorithms

MADYMO handles interaction through two distinct mechanisms, and choosing the wrong one is a common source of pulse-sensitivity error.

Kinematic contact is used for surface-to-surface interaction between ellipsoids, planes, and facet meshes. At each step the solver computes penetration depth along the contact normal and applies a penalty force proportional to that penetration, typically through a nonlinear stiffness and a damping term. The force is a function of penetration, not of a pre-defined spring, so the contact stiffness must be chosen to be stiff enough to prevent visible interpenetration but soft enough not to collapse the time step. A useful rule of thumb is to set the contact stiffness so that maximum expected penetration stays below roughly 1–2 mm for occupant-scale contacts.

Force-based contact (belt elements, springs, and prescribed force–penetration curves) is used where the physical interaction is better described by a measured characteristic than by a geometric penalty. Seatbelt elements are the canonical example: the belt is modelled as a series of one-dimensional elements whose force–strain behaviour is taken from a dynamic sled test. This matters for pulse sensitivity because the belt's force–strain curve, not the pulse alone, governs how much occupant excursion occurs.

Friction is applied within the contact definition using a Coulomb model with a friction coefficient µ. For occupant-to-seat and occupant-to-airbag contacts, µ in the range 0.3–0.5 is typical; belt-to-clothing friction is often higher. The friction coefficient interacts strongly with pulse sensitivity: a small change in µ can shift peak occupant forward excursion by several millimetres, so it should be treated as a calibration parameter, not a fixed constant.

Explicit Integration Loop and Contact Evaluation in MADYMO State at t x, v known Contact detection penetration δ, normal n friction µ Force assembly F = k·δ + c·δ̇ + belt force curve Integrate a = F/m → t+Δt next time step (Δt ≤ 2/ω_max) Stability constraint Δt_crit = 2 / ω_max (stiffest spring, smallest mass) Typical occupant model: Δt ≈ 0.5–2 µs 120 ms event → 60,000–240,000 steps Symptom of violation: energy growth, contact chatter, erratic kinematics Contact type selection Kinematic: ellipsoid / plane / facet → penalty force from penetration δ Force-based: belt, spring, prescribed curve → force from measured characteristic Friction µ: 0.3–0.5 (occupant–seat)

Model Setup for Pulse Sensitivity Studies

A pulse-sensitivity study is only as good as the model it runs on. The following setup sequence reflects what actually has to be in place before a parameter sweep is meaningful.

  1. Define the pulse as a prescribed acceleration or velocity boundary condition. In MADYMO this is typically applied to the vehicle body or to a prescribed-motion rigid body. The pulse should be entered as a time–acceleration table, and the corresponding velocity and displacement should be checked by integration to confirm the pulse is physically consistent (zero net velocity change at the end of the event for a full-stop crash).
  2. Build the occupant model with consistent mass and inertia. A Hybrid III 50th percentile male dummy model is standard. Verify total mass against the dummy specification (approximately 78 kg) and check that the centre of gravity location matches the reference.
  3. Define contacts explicitly. Every surface pair that can interact must have a contact definition. Missing contacts are the most common cause of an occupant passing through a structure without resistance.
  4. Route the belt through the correct anchorage points and retractor. The belt path determines the effective restraint geometry; a belt that slips off the shoulder or routes over the wrong point will produce non-physical excursion.
  5. Set the output requests before running. At minimum, request occupant head and chest acceleration, belt forces at each segment, and contact forces. These are the quantities that reveal pulse sensitivity.
Do not tune the pulse and the restraint system simultaneously. Fix the pulse first, run a baseline, then vary one restraint parameter at a time. Changing both at once makes it impossible to attribute a change in occupant response to either input.

Worked Example: Baseline Frontal Pulse Run

Consider a 50 km/h full-frontal pulse applied to a vehicle body, with a belted Hybrid III 50th percentile occupant and a driver airbag. The pulse is a 120 ms triangular deceleration with a peak of approximately 40 g at 60 ms. The model uses kinematic contact between the occupant and the seat, force-based belt elements with a peak force of about 8 kN, and a facet-mesh airbag with a vent coefficient calibrated to a tank test.

After running the baseline, the analyst should inspect three things. First, the energy balance: total energy should remain approximately constant, and hourglass energy should be a small fraction of internal energy. Second, the belt force time history: it should rise smoothly to a peak and then decay, without spikes that indicate contact chatter. Third, the occupant kinematics: head forward excursion should peak at a plausible value (typically 300–500 mm for this configuration) and the chest acceleration should show a single dominant peak rather than multiple oscillations.

If the belt force shows high-frequency spikes, the likely cause is a contact stiffness that is too high relative to the time step, or a belt element that is being compressed rather than tensioned. If the occupant passes through the seat, the contact definition is missing or the penetration stiffness is too low. If the energy balance diverges, the time step is too large for the stiffest element in the model.

Once the baseline is stable and physically plausible, the sensitivity study can proceed: vary the pulse peak by ±10%, vary the pulse duration by ±10 ms, and record the change in peak head excursion and chest acceleration. The ratio of output change to input change is the sensitivity that the restraint design must accommodate. A restraint system that produces large output changes for small pulse changes is poorly conditioned and will be difficult to calibrate across the range of real-world crash pulses.

Key Takeaways

  • MADYMO uses explicit central-difference integration; the time step is limited by the stiffest element and smallest mass in the model.
  • Contact is handled either kinematically (penalty force from penetration) or through force-based elements (belt, spring, prescribed curve); the choice affects how pulse changes propagate to occupant response.
  • Model setup must include a physically consistent pulse, correct mass and inertia, complete contact definitions, and a properly routed belt before any sensitivity sweep is meaningful.
  • Energy balance, belt force smoothness, and kinematic plausibility are the three checks that confirm a baseline is ready for parameter variation.
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