4. Impact kinematics

In an airbag pendulum impact test, the pendulum is the energy source. Everything downstream — the airbag's internal pressure rise, the fabric stress distribution, the seam loading, and the occupant-relevant output metrics — is determined by how much kinetic energy the pendulum delivers and how that energy is transferred through the contact interface. This chapter develops the kinematic relationships that connect pendulum mass, release height, impact velocity, kinetic energy, and contact geometry, and shows how these parameters are selected to produce representative loading conditions for a given airbag application.

4.1 The Pendulum as a Controlled Energy Source

A pendulum impactor converts gravitational potential energy into kinetic energy during its swing. For a rigid pendulum released from rest at a height h above its lowest point, the impact velocity at the bottom of the arc is:

v = sqrt(2 * g * h)

where:
  v = impact velocity at the lowest point (m/s)
  g = gravitational acceleration (9.81 m/s^2)
  h = release height of the pendulum center of gravity (m)

The corresponding kinetic energy at impact is:

E_k = 0.5 * m * v^2 = m * g * h

where:
  E_k = kinetic energy at impact (J)
  m = effective pendulum mass (kg)

These two equations define the fundamental trade-off in pendulum test design. A given kinetic energy can be achieved with a heavy pendulum at low velocity or a light pendulum at high velocity. The choice matters because the airbag responds differently to each combination: a heavy, slow impact produces a longer, more distributed pressure pulse, while a light, fast impact produces a sharper, more localized loading event. For airbag validation, the pendulum mass and velocity must be selected to match the energy and the loading rate of the real-world scenario being represented.

4.2 Representative Loading Conditions

Representative loading means that the pendulum impact reproduces the essential mechanical characteristics of the event the airbag is designed to manage. For driver and passenger airbags, the relevant events include:

• Occupant contact during deployment: the airbag is still inflating when the occupant's head or torso arrives. The loading is a combination of membrane tension and internal pressure against a moving boundary.
• Out-of-position (OOP) loading: a small-stature occupant or a child in close proximity to the module. The airbag contacts the occupant early in deployment, producing high local pressure and seam stress.
• Static deployment against a rigid wall: the airbag inflates against a fixed surface, producing a pressure spike and a characteristic force–time curve used for module-level validation.

Each of these events has a characteristic energy level and loading rate. The pendulum test is configured so that the impactor mass and velocity produce a kinetic energy and a contact duration that fall within the range of the target event. Table 4.1 gives representative parameter sets for three common airbag test configurations. These values are illustrative examples for teaching purposes and should not be treated as measured or certified test data.

The contact duration is not set directly by the pendulum; it emerges from the interaction between the pendulum's inertia, the airbag's internal pressure, and the venting characteristics. A stiffer airbag (higher internal pressure, smaller vents) produces a shorter contact duration and a higher peak force. A softer airbag produces a longer contact duration and a lower peak force. This is why the same pendulum can produce very different force–time histories depending on the airbag configuration under test.

4.3 Contact Geometry and Its Effect on Loading

The contact geometry determines how the pendulum's kinetic energy is distributed over the airbag surface. Three geometric parameters are most influential:

• Impactor face shape: a flat cylindrical face produces a broad, uniform contact patch; a hemispherical face produces a concentrated contact point that expands as the airbag deforms. The hemispherical face is more representative of a human head, while the flat face is more representative of a rigid barrier.
• Contact area: the initial contact area A_0 sets the initial pressure P_0 = F / A_0. A small contact area produces high local pressure and high fabric stress, which is the critical condition for seam and fabric burst testing.
• Angle of attack: the angle between the pendulum's velocity vector and the airbag surface normal. A normal impact (0°) transfers the full kinetic energy into the airbag. An oblique impact (e.g., 30°) transfers only the normal component, E_n = E_k * cos^2(θ), and introduces a tangential component that can cause the airbag to slide or rotate on its mounting.

For a hemispherical impactor of radius R contacting a flat airbag surface, the contact radius a at a given penetration depth δ is:

a = sqrt(2 * R * delta - delta^2)

For small penetration (delta << R):
  a ≈ sqrt(2 * R * delta)

Contact area:
  A = pi * a^2 ≈ 2 * pi * R * delta

This relationship shows that the contact area grows with the square root of the penetration depth. In the early stage of impact, the contact area is small and the local pressure is high. As the airbag deforms and the penetration increases, the contact area grows and the pressure distributes over a larger region. The peak local pressure occurs at or near the moment of initial contact, which is why the first few milliseconds of the impact are the most critical for fabric and seam integrity.

4.4 From Pendulum Kinematics to CAE Input

In a CAE validation workflow, the pendulum is typically represented as a rigid body with a prescribed initial velocity. The airbag is modeled as a membrane or shell structure with internal pressure applied via a control-volume or uniform-pressure method. The contact between the pendulum and the airbag is defined using a penalty-based or kinematic contact algorithm.

The key inputs that must be extracted from the physical test and transferred to the CAE model are:

• Pendulum mass and initial velocity: used to define the rigid body's mass and initial condition. The effective mass at the contact point must be used, not the total physical mass.
• Impactor geometry: the face shape, radius, and any chamfers or edges that affect the contact patch.
• Initial gap: the distance between the pendulum face and the airbag surface at the moment of release. This sets the time of first contact and must be consistent between test and simulation.
• Airbag internal pressure: the initial pressure and the gas inflow rate, which determine how the airbag resists the pendulum's motion.

A common validation check is to compare the pendulum's velocity–time history from the CAE model against the physical test. If the CAE model shows the pendulum decelerating too quickly, the airbag is too stiff (over-pressurized or under-vented). If the pendulum decelerates too slowly, the airbag is too soft. The area under the force–time curve should match the pendulum's initial kinetic energy, minus any energy lost to friction or venting.

4.5 Worked Example: Sizing a Pendulum for a Driver Airbag OOP Test

Suppose the target loading condition is a 6.8 kg effective mass impacting at 2.4 m/s, with a hemispherical impactor of radius 50 mm. The required release height is:

h = v^2 / (2 * g) = (2.4)^2 / (2 * 9.81) = 5.76 / 19.62 = 0.294 m

The kinetic energy at impact is:

E_k = 0.5 * 6.8 * (2.4)^2 = 0.5 * 6.8 * 5.76 = 19.6 J

If the airbag is inflated to 150 kPa gauge and the initial contact area is estimated as 200 mm², the initial contact pressure is:

P_0 = F / A_0

The force at initial contact is not simply the weight of the pendulum; it is the force required to decelerate the pendulum. For a first estimate, assume the pendulum decelerates over 10 ms:
  F_avg = m * v / t = 6.8 * 2.4 / 0.010 = 1632 N
  P_0 = 1632 / 0.0002 = 8.16 MPa

This local pressure is well above the airbag's internal pressure, which is why the fabric and seams must be designed to withstand the contact pressure, not just the inflation pressure. The CAE model should be checked to ensure that the peak local pressure at the contact patch does not exceed the fabric's tensile strength or the seam's peel strength.

The SVG above summarizes the kinematic chain from release height to airbag response. The pendulum's potential energy at release is converted to kinetic energy at impact, which is then transferred to the airbag through the contact patch. The contact geometry — impactor radius, penetration depth, and angle of attack — determines how that energy is distributed over the airbag surface. The key parameters on the right side of the diagram are the ones that must be measured or estimated before a CAE model can be built.

4.6 Summary of Design Relationships

The following relationships govern the selection of pendulum parameters for representative loading:

• Velocity is set by release height: v = √(2gh). Doubling the height increases velocity by a factor of √2.
• Energy is set by mass and height: E_k = mgh. Doubling either mass or height doubles the energy.
• Contact pressure is set by force and area: P = F/A. A small contact area produces high local pressure, which is the critical condition for fabric and seam testing.
• Oblique impacts reduce normal energy transfer: E_n = E_k·cos²θ. A 30° angle of attack transfers only 75% of the kinetic energy into the airbag.
• Contact duration emerges from the airbag's stiffness and venting: it is not a directly controlled input but a result of the airbag's pressure–volume–venting characteristics.

When setting up a pendulum test or building a CAE model, the first step is to define the target loading condition in terms of mass, velocity, and contact geometry. The second step is to verify that the chosen parameters produce a kinetic energy and contact duration that fall within the range of the real-world event. The third step is to check the CAE results against the physical test data, using the force–time curve and the pendulum velocity–time history as the primary validation metrics. If the energy balance does not close — that is, if the area under the force–velocity curve does not match the initial kinetic energy minus losses — the model's contact definition or airbag pressure characteristics need to be revisited.