Kinematics: Velocity, Acceleration, and Time Integration in FEM

1. What Kinematics Is (and Why It Matters for Drop Tests)

Kinematics is the branch of mechanics that describes how objects move — position, velocity, and acceleration as functions of time — without caring about the forces causing that motion. Once you have the kinematic description of a motion event, you can set up your FEM simulation correctly and interpret the results in physical terms.

The most common place this comes up in practice: drop test simulations. Whether you're testing a smartphone hitting the floor, a military electronics package surviving a 1.5m drop, or a packaged battery pack for shipping qualification, you need to know the impact velocity to set the simulation's initial conditions correctly.

2. Position, Velocity, Acceleration: The Calculus Chain

The three kinematic quantities are connected by differentiation and integration:

Going the other way (integration):

This is exactly what FEM time integration algorithms do: starting from known initial conditions ($\mathbf{u}_0$, $\dot{\mathbf{u}}_0$), they numerically integrate the acceleration (computed from Newton's 2nd Law) forward in time to get velocity and displacement at the next time step.

The Velocity-Position Relationship in Accelerometer Data

In physical testing, accelerometers measure $\ddot{u}(t)$. To get velocity and displacement, you integrate numerically — the same thing FEM does. This means you can directly compare FEM simulation results against test data by integrating the acceleration channel from a physical crash test or drop test.

3. Three Equations of Constant Acceleration (SUVAT)

When acceleration $a$ is constant, three powerful equations apply:

Where $u$ = initial velocity, $v$ = final velocity, $s$ = displacement, $a$ = acceleration, $t$ = time. These seem like high school physics, but engineers use them constantly for quick sanity checks before running a full simulation.

4. Worked Example: Drop Test Impact Velocity

This is the most common kinematic calculation in product durability testing. An object is released from height $h$ and falls under gravity ($g = 9.81$ m/s²) with no initial velocity.

Using Equation (3) with $u=0$:

Time to Impact

Using equation (2) with $u=0$: $s = \frac{1}{2}gt^2$, so the time of flight from height $h$ is:

For a 1.2 m drop: $t = \sqrt{2 \times 1.2 / 9.81} = 0.495$ s. This is useful for physical test setup — you need high-speed cameras running at this frame rate to capture the impact event.

5. Projectile Motion and Oblique Impact

Not all drops are perfectly vertical. A component slipping off a workbench may have horizontal velocity; a vehicle hitting a guardrail hits at an angle. Projectile motion handles 2D kinematics where horizontal and vertical motions are independent:

Oblique Impact Angle in FEM

For an oblique drop test, the contact angle matters. A 30° oblique drop onto a corner concentrates energy very differently than a flat drop. When setting up the simulation, you need to resolve the impact velocity into components normal and tangential to the contact surface, then consider whether friction is important for the tangential component.

6. Numerical Time Integration in FEM

When acceleration is not constant (which is almost always the case in structural dynamics), we must numerically integrate the equations of motion. FEM time integration algorithms differ in stability, accuracy, and computational cost.

Explicit vs. Implicit Time Integration

The fundamental distinction:

• Explicit: Compute acceleration at time $t_n$ from known state; step forward. No matrix inversion needed but time step must be very small for stability.
• Implicit: Simultaneously solve for state at $t_{n+1}$. Requires matrix inversion (expensive per step) but can use large time steps.

Central Difference Method (Explicit — LS-DYNA default)

Simple and cheap per step, but requires $\Delta t \leq \Delta t_{\text{crit}} = \frac{2}{\omega_{\max}}$. For crash simulations with millisecond events and millions of elements, this often means ~1 microsecond time steps — meaning millions of steps, but each step is very fast.

7. Newmark-β Method in Detail

The Newmark-β method is the most widely used implicit time integration scheme in structural dynamics (Abaqus Standard, NASTRAN dynamic). It predicts displacement and velocity at time $t_{n+1}$ using:

Parameters $\beta$ and $\gamma$ control stability and accuracy:

For Abaqus Standard dynamic analyses, the default $\beta=0.25$, $\gamma=0.5$ (average acceleration) is unconditionally stable — you can take large time steps without the solution blowing up. The trade-off is solving a large system of equations at each step.

8. Setting Initial Conditions in FEM Solvers

Abaqus Explicit: Drop Test Setup

In Abaqus/CAE, initial velocity is applied as:

*INITIAL CONDITIONS, TYPE=VELOCITY
ALL_PARTS, 1, 0.0    ! X-velocity = 0
ALL_PARTS, 2, -4.85  ! Y-velocity = -4.85 m/s (downward)
ALL_PARTS, 3, 0.0    ! Z-velocity = 0

LS-DYNA: Initial Velocity Card

*INITIAL_VELOCITY_GENERATION
$#  nsid  styp  omega    vx     vy     vz    ivatn
     1     2    0.0    0.0  -4.85   0.0      0

Common Mistake: Missing Gravity During Impact

After applying the impact initial velocity, you still need gravity acting during the simulation, otherwise the structure "floats" during the post-impact phase. Apply gravity as a body load:

*DLOAD
ALL_PARTS, GRAV, 9.81, 0., -1., 0.

9. Cross-Topics