Multibody Dynamics · Module 10 of 10
Impacts, Contact, and Simulation Validation
Machines collide, latch, and stop suddenly. Impulsive dynamics handles those instants, and energy and constraint checks decide whether the whole simulation can be trusted.
Readiness check
Tick only what you can do closed-notes before starting.
- State conservation of linear momentum for a collision.
- Recall the coefficient of restitution.
- Compute kinetic energy before and after an event.
- Recall energy conservation for a frictionless system.
- State why a simulation result needs an independent check.
The core idea
An impact is an impulse acting over a negligible time, so positions do not change but velocities jump. Momentum is conserved and a coefficient of restitution sets the rebound. A finished simulation is trusted only after momentum, energy, and constraint checks pass.
momentum: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'restitution: v₂' − v₁' = e (v₁ − v₂)validate: check energy and Φ against physicsTwo things close this course: the last piece of physics and the habit of proof. The physics is impact. A collision applies a large force over a very short time, an impulse, so the bodies barely move but their velocities change abruptly. Across the impact, momentum is conserved because internal impulses cancel; the outcome is fixed by a coefficient of restitution e, which relates the separation speed after to the approach speed before. A perfectly elastic impact (e = 1) keeps kinetic energy; a plastic one (e = 0) sticks. Wittenburg treats these instantaneous velocity changes and the internal joint impulses they cause. The habit of proof is validation. A simulation is a prediction, and predictions must earn trust: conserved momentum in a collision, conserved energy in a frictionless run, and constraint values Φ near zero throughout are independent checks the physics must pass. When a simulation drifts in energy or violates a constraint, the number it reports is not yet evidence. This closes the loop back to verification and validation.
The skills, taught in order
Five skills handle sudden events and prove the whole model.
10.1 Impulse and momentum
Treat an impact as an impulse over negligible time: positions frozen, velocities jumping. Conservation of momentum is the master equation across the event.
10.2 Coefficient of restitution
Use e to set the outcome: separation speed equals e times approach speed. It ranges from 1 (elastic) to 0 (plastic) and captures the energy lost in the contact.
10.3 Contact and internal impulses
In a jointed system an impact propagates as internal impulses through the joints, changing several bodies at once, an effect Wittenburg links by analogy to elastostatics.
10.4 Energy accounting
Compute kinetic energy before and after. In conservative motion it must hold; across an impact the loss must match the restitution. Energy is the most sensitive validation signal.
10.5 Validating the simulation
Confirm momentum in collisions, energy in frictionless runs, and constraint values near zero throughout. Only a model that passes these becomes engineering evidence.
| Check | Should hold when | Signals |
|---|---|---|
| Momentum conservation | collisions, no external impulse | impact handled correctly |
| Energy conservation | frictionless, no dissipation | numerical damping or gain |
| Constraint value Φ | always | drift off the geometry |
| Restitution energy loss | across an impact | correct rebound model |
Each check is a physical law the simulation must obey; a failed check is a found bug.
Engineering connection: a crash or drop simulation lives or dies on its impact model and its energy accounting, which is why restitution and energy checks are the first things a reviewer asks for.
Worked example 1: a one-dimensional impact with restitution
A 2 kg block moving at 3 m/s strikes a stationary 1 kg block. The coefficient of restitution is e = 0.5. Find the velocities after impact and the kinetic energy lost.
- ProblemFind the post-impact velocities and energy loss in Figure 1.
- Given / findm₁ = 2 kg, v₁ = 3 m/s, m₂ = 1 kg, v₂ = 0, e = 0.5. Find v₁', v₂', and the energy lost.
- AssumptionsCentral, one-dimensional impact; external impulses negligible during contact.
- ModelMomentum: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'. Restitution: v₂' − v₁' = e(v₁ − v₂).
- Equations2(3) + 0 = 2 v₁' + 1 v₂' = 6v₂' − v₁' = 0.5(3) = 1.5
- SolveSolving the pair: v₁' = 1.5 m/s, v₂' = 3.0 m/s. KE before = ½(2)(3²) = 9 J; KE after = ½(2)(1.5²) + ½(1)(3²) = 2.25 + 4.5 = 6.75 J.
- CheckMomentum after = 2(1.5) + 1(3.0) = 6 kg·m/s, matching before. Energy lost is 9 − 6.75 = 2.25 J, a quarter, consistent with e = 0.5.
- ConclusionThe blocks leave at 1.5 and 3.0 m/s, momentum conserved and 2.25 J dissipated. Restitution plus momentum fully determines an impact.
Worked example 2: validating a simulation by energy
A frictionless pendulum (m = 1 kg, L = 0.5 m) is released from 30 degrees. Find the kinetic energy it should have at the bottom, and judge a simulation that reports 0.72 J there.
- ProblemFind the expected bottom energy in Figure 2 and validate the simulation.
- Given / findm = 1 kg, L = 0.5 m, release angle 30°, g = 9.81. Simulation reports 0.72 J at the bottom. Find the expected value and the error.
- AssumptionsFrictionless, conservative motion, so energy is exactly conserved from release to the bottom.
- ModelEnergy conservation: KE_bottom = m g L (1 − cosθ₀).
- EquationsKE = m g L (1 − cos30°)= 1(9.81)(0.5)(1 − 0.866)
- SolveKE = 4.905 × 0.134 = 0.657 J, so the speed is √(2·0.657) = 1.15 m/s. The simulation’s 0.72 J is high by (0.72 − 0.657)/0.657 = 9.6%.
- CheckA frictionless run must conserve energy, so a 9.6% gain is not physics; it is numerical error, likely too large a time step or a poor integrator.
- ConclusionThe model should read 0.657 J at the bottom. A simulation off by 9.6% fails the energy check and must be refined before its results are trusted.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Assuming impacts conserve energy | Elastic result used for a real, lossy collision | Is the restitution really one? | Use the measured e; energy is conserved only at e = 1. |
| Moving positions during an impact | Geometry changed across an instantaneous event | Did positions stay fixed through the impulse? | Impacts change velocities only; positions are frozen. |
| Reporting results without validation | A number presented as evidence with no check | Which conservation law did I verify? | Confirm momentum, energy, and constraints first. |
| Blaming physics for numerical drift | Calling an energy gain a real effect | Should this frictionless run conserve energy? | An energy drift in a conservative model is numerical error. |
Practice ladder
A 3 kg block at 4 m/s hits a 3 kg block at rest with e = 1. Find the post-impact velocities.
Show answer
Equal masses, elastic: they exchange velocities, so v₁' = 0, v₂' = 4 m/s.
For the Level 1 impact, confirm kinetic energy is conserved.
Show answer
Before = ½(3)(16) = 24 J; after = ½(3)(16) = 24 J; conserved, as e = 1 requires.
A pendulum (m = 1 kg, L = 1 m) is released from 60 degrees. Find its expected kinetic energy at the bottom.
Show answer
KE = m g L (1 − cos60°) = 1(9.81)(1)(0.5) = 4.905 J.
For a simulation you have run or plan to run, list the conservation and constraint checks you would use to validate it.
What good work looks like
A good answer names momentum checks for impacts, energy checks for conservative motion, and constraint-value monitoring throughout, with a tolerance and a step-refinement plan.
Working with AI, and proving it yourself
Ask AI to check your validation, not to vouch for the result
Portfolio task
Take one simulation, run the momentum, energy, and constraint checks, and write a short validation note stating what passed, what drifted, and what you would fix.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What is conserved across an impact?
Linear momentum, since internal impulses cancel.
2. What does the coefficient of restitution set?
The separation speed as a fraction of the approach speed.
3. When must a simulation conserve energy?
In frictionless, non-dissipative motion.
4. What does constraint drift reveal?
That the simulation is leaving the geometry the joints define.
5. When does a result become evidence?
After it passes momentum, energy, and constraint checks.
Textbook mapping
Impacts and the internal impulses they cause are Wittenburg’s final chapter. Use these references to read further.
| Topic in this module | Where to read more |
|---|---|
| Collisions and instantaneous velocity changes | Wittenburg, Dynamics of Multibody Systems, ch. 6 |
| Internal joint impulses | Wittenburg, Dynamics of Multibody Systems, ch. 6 |
| Validating simulation results | the MechCompass VVUQ course and Oberkampf and Roy, Verification and Validation in Scientific Computing |
Chapter references are to Wittenburg, Dynamics of Multibody Systems (Springer); validation practice connects to the VVUQ course.