Momentum and Impulse — Airbag Physics and Explicit FEM Impact Analysis

1. Momentum and Impulse in Engineering

Momentum ($p = mv$) and its time-rate-of-change version — impulse ($J = F \cdot \Delta t$) — are the most direct tools for analyzing impact, collision, and crash events. While energy methods tell you how much energy is involved, momentum methods tell you about the forces during impact and the velocities before and after.

For CAE engineers, momentum thinking is most useful in three contexts: designing energy-absorbing structures (crumple zones, airbags), analyzing impact events in explicit FEM, and interpreting test data from crash labs where force-time histories are measured.

2. Linear Momentum: p = mv

Linear momentum is the product of mass and velocity. It is a vector quantity — direction matters:

Newton's Second Law can be written in terms of momentum:

For constant mass, this reduces to $\mathbf{F} = m\mathbf{a}$ as expected. But for variable-mass systems — a rocket expelling propellant, a conveyor belt accumulating material — the full form $\mathbf{F} = \frac{d(m\mathbf{v})}{dt}$ must be used.

3. Impulse: J = FΔt = Δp

Impulse is the change in momentum produced by a force acting over time:

For a constant force: $J = F \cdot \Delta t$. The impulse-momentum theorem is just Newton's Second Law integrated over time. Its power lies in relating peak force to contact duration — the same total impulse (momentum change) can be delivered by a small force over a long time, or a large force over a short time.

Force-Time Curves in Crash Testing

In physical crash tests, load cells measure the force-time history at the barrier. The area under the F(t) curve equals the total impulse. Engineers compare FEM-predicted F(t) curves against test data — if the area (impulse) matches but the peak differs, it suggests incorrect crumple zone timing, not wrong total energy absorption.

4. Airbag Design: Impulse in Action

An automotive airbag is an engineering masterpiece of impulse management. The system must:

1. Detect the crash within ~15ms (accelerometer threshold)
2. Ignite the inflator propellant
3. Fully inflate the bag before the occupant reaches it (~25ms after impact)
4. Begin controlled deflation through vent holes during occupant loading (25–60ms)

Impulse Calculation for a 70 kg Occupant at 50 km/h

This impulse must be absorbed through the airbag contact. The airbag provides a contact time of about 25ms:

Compare to a steering wheel impact (Δt ≈ 2ms): $F = 973/0.002 = 487$ kN — over twelve times higher, and certainly fatal. The airbag's vent holes are designed to maintain a constant pressure (and thus constant force) on the occupant throughout the deceleration — this is actually the optimal impulse delivery for minimizing HIC (Head Injury Criterion).

FEM Simulation of Airbag Deployment

Airbag simulation is one of the most complex FEM problems in automotive engineering. It requires:

• Folded mesh: The bag starts folded; the mesh must unfold realistically (contact self-interaction)
• Gas dynamics: Ideal gas law $PV = nRT$ governs pressure evolution as gas flows in and out
• Fabric material: Highly nonlinear, anisotropic, no compression stiffness (wrinkling)
• Occupant interaction: Contact between inflating bag and moving occupant

LS-DYNA's *AIRBAG_UNIFORM_PRESSURE and Abaqus's fluid-filled cavity approaches both use the impulse framework — the gas pressure force history integrated over the bag surface must equal the impulse required to decelerate the occupant.

5. Conservation of Momentum

When no external forces act on a system, total momentum is conserved:

This is Newton's Third Law in integral form — internal forces between bodies in a system always cancel in pairs, leaving total momentum unchanged. Conservation of momentum is the foundation for analyzing multi-body impact events.

Application: Two-Vehicle Collision Analysis

For accident reconstruction (forensic engineering), conservation of momentum allows calculating pre-impact speeds from post-impact skid patterns. For a head-on collision between car A (1500 kg, 50 km/h east) and car B (2000 kg, 30 km/h west):

The combined wreckage moves east at 1.2 m/s — car A "won" the momentum battle because its momentum slightly exceeded car B's.

6. Elastic and Inelastic Collisions

Collisions are classified by how much kinetic energy is conserved:

For elastic collisions between two bodies:

The coefficient of restitution $e$ quantifies how elastic a collision is: $e = 1$ (perfectly elastic), $e = 0$ (perfectly inelastic).

Typical values: hardened steel on steel ≈ 0.6, rubber ball on concrete ≈ 0.8, car crash ≈ 0.1–0.2.

7. Momentum Methods in Explicit FEM

Explicit FEM (LS-DYNA, Abaqus Explicit) is the tool of choice for impact and crash simulation precisely because it naturally handles the momentum dynamics of fast events. The central difference time integration is essentially a direct numerical application of the impulse-momentum theorem:

At each time step, the net nodal force $F_n$ is computed from element stresses and contact forces. Dividing by the nodal mass (diagonal mass matrix) gives the acceleration increment, which updates velocity and then position. No matrix inversion — just a scalar division for each node.

Momentum Flux in SPH and Meshfree Methods

For extreme impact problems — hypervelocity impact, explosions, penetration — Smooth Particle Hydrodynamics (SPH) represents the material as particles with mass and velocity. Conservation of momentum is enforced particle-to-particle through kernel-weighted interactions, avoiding mesh distortion that destroys regular FEM elements in these extreme events.

8. Angular Momentum and Rotating Structures

For rotating machinery (turbines, flywheels, motor rotors), angular momentum $\mathbf{L}$ is the rotational analog:

Where $I$ is the moment of inertia tensor and $\boldsymbol{\omega}$ is angular velocity. Conservation of angular momentum explains why a spinning gyroscope resists reorientation and why a suddenly unbalanced turbine rotor creates extreme bearing forces.

In FEM of rotating machinery (cyclic symmetry models, rotating frame analyses), the angular momentum terms appear as Coriolis and centrifugal body forces in the equation of motion:

The gyroscopic matrix $[\mathbf{C}_{\text{Gyro}}]$ (proportional to spin speed $\Omega$) and spin-softening matrix $[\mathbf{K}_{\text{Spin}}]$ are the FEM expressions of angular momentum effects.

9. Cross-Topics