剛体動力学
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass is "left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly enough to ignore acceleration". It absolutely cannot be omitted in impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is not correct. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but ending up with "compression"—sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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).
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (ZZ estimator, etc.). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Achieves quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Lower cost per iteration but linear convergence speed.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can track beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "calculation ran = results are correct"? This is actually the most common trap for CAE beginners. The solver always returns "some answer" for the given mesh. But if the mesh is too coarse, that answer is far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software 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).
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (ZZ estimator, etc.). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Achieves quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Lower cost per iteration but linear convergence speed.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load in small increments rather than all at once. The arc-length method (Riks method) can track beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—re
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