Rigid Body Dynamics
Rigid Body Dynamics: Theoretical Foundations
Rigid Body Dynamics
Professor, what is the difference between rigid body dynamics and FEM dynamic analysis?
FEM dynamic analysis deals with the elastic deformation of structures. Rigid body dynamics tracks the motion (translation + rotation) of non-deforming bodies. The goal is motion simulation of mechanisms.
Equations of Motion
Newton-Euler Equations:
$m$: mass, $[I]$: inertia tensor, $\mathbf{F}$: force, $\mathbf{M}$: moment.
Constraints (Joints)
Joints constrain relative motion between rigid bodies:
| Joint | Degrees of Freedom | Example |
|---|---|---|
| Fixed (Weld) | 0 | Welded connection |
| Revolute | 1 (Rotation) | Hinge, bearing |
| Prismatic | 1 (Translation) | Slider |
| Cylindrical | 2 | Piston |
| Spherical | 3 (Rotation 3) | Ball joint |
| Free | 6 | No constraint |
Summary
Euler's Rigid Body Equations Were Proposed in 1758
The "Euler equations" describing the rotational motion of rigid bodies were published by Leonhard Euler in 1758 in 'Novi Commentarii academiae scientiarum Petropolitanae'. These equations, which concisely express the relationship between the inertia tensor and angular velocity vector in the principal axes system, are at the core of numerical integration in modern multibody rigid dynamics. Precessional motion of tops and gyroscopes can also be directly derived from Euler's equations, and they still appear in the first chapter of textbooks as the foundation for spacecraft attitude stability analysis.
Computational Methods for Rigid Body Dynamics
MBD Solvers
Summary
Verlet Integration Changed Game Engines and Molecular Dynamics
Among numerical integration methods for rigid body MBD, the "Störmer-Verlet method" (Stormer's paper in 1907, re-applied to molecular dynamics by Verlet in 1967) is a representative symplectic integrator with high energy conservation. Its low computational cost and resistance to error accumulation in long-term integration led to the VelocityVerlet variant being adopted in physics engines for Unity and Unreal Engine. In CAE rigid body MBD, it remains effective for conservative systems without contact, but higher-order Runge-Kutta methods are superior in accuracy for problems with significant non-conservative forces (damping).
Rigid Body Dynamics in Practice
Rigid Body Dynamics in Practice
Automotive suspension kinematics, robot joint motion, engine crank mechanisms, deployment structure motion.
Practical Checklist
Drop Test Analysis for Transport Packaging is Fastest with Rigid Body MBD
For drop test analysis in transport packaging design for electronics and appliances (free fall from 1.2m height), rigid body MBD is mainstream in practice because finite element method is computationally expensive. In Apple iPhone packaging design (iPhone 12 and later), design to reduce impact acceleration from 300G to below 120G using rigid body MBD + nonlinear spring model for polyurethane foam is suggested by Apple engineers' conference presentations to keep error within ±15% compared to drop tests at Apple Park's internal testing facility.
Rigid Body Dynamics: Software & Solver Comparison
Tools
Summary
Verlet Integration Changed Game Engines and Molecular Dynamics
Among numerical integration methods for rigid body MBD, the "Störmer-Verlet method" (Stormer's paper in 1907, re-applied to molecular dynamics by Verlet in 1967) is a representative symplectic integrator with high energy conservation. Its low computational cost and resistance to error accumulation in long-term integration led to the VelocityVerlet variant being adopted in physics engines for Unity and Unreal Engine. In CAE rigid body MBD, it remains effective for conservative systems without contact, but higher-order Runge-Kutta methods are superior in accuracy for problems with significant non-conservative forces (damping).
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