Joint Constraints and Kinematics
Theory and Physics
Joint Constraints
Professor, how are MBD joint constraints different from FEM constraints?
FEM constraints (SPC) are fixed constraints like "displacement = 0". MBD joints are dynamic constraints that "restrict degrees of freedom of relative motion". For a revolute joint, "rotation about one axis is free, others are constrained".
Gruebler-Kutzbach Equation
Number of DOF (degrees of freedom) of a mechanism:
$n$: Number of bodies (including ground), $c_i$: Number of constrained DOF for each joint. The mechanism moves if $DOF > 0$.
Summary
Constraint Stabilization for DAE Systems was Invented by Baumgarte (1972)
Constraints in multibody systems give rise to Differential Algebraic Equations (DAE). The numerical stability problem of these DAEs was solved by J. Baumgarte's 1972 proposal of the "Constraint Stabilization Method (Baumgarte Stabilization)". It was a simple idea of adding feedback gains to the constraint condition Φ and its velocity level Φ˙ to dampen errors, but "appropriate gain selection" proved difficult and became a topic of debate for many years. The Gear-Gupta-Leimkuhler (GGL) method (1985) solved this problem algebraically and significantly improved the integration accuracy of MBD solvers.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried forward" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings sway during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly enough that acceleration can be ignored". It cannot be omitted in impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So, 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". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force, 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 it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated 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 vibration 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 vibration energy to improve ride comfort. What if damping were zero? Buildings would continue swaying 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 the stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): 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 the balance 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 mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit 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 | For mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify to N for mm system, N for m system. |
Numerical Methods and Implementation
Joints in FEM/MBD
Summary
For 6-DOF Description of Rigid Body MBD, Quaternions are Better than Euler Angles
Euler angles, which represent rigid body orientation, have a singularity problem called "gimbal lock". Since the NASA Apollo 13 (1970) incident where gimbal lock became a problem in spacecraft attitude control, rotation representation using quaternions (four-element numbers) has become standard in MBD solvers. Quaternions use 4 parameters to represent 3 degrees of freedom, making them redundant, but they have no singularities and are numerically stable for computer operations. MSC Adams transitioned to internal quaternion representation in the late 1980s, drastically reducing crash reports due to singularities.
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 Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is important.
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 (e.g., ZZ estimator). 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 every iteration. Provides quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix at initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
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
Instead of applying full load at once, apply in small increments. 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) than to search sequentially from the first page (direct).
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 "flexible curves"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Joint Practice
Robot joints, vehicle suspensions, engine cranks, folding structures.
Practical Checklist
Virtual Joints for Suspensions Originated from CATIA DMU
When modeling automotive multi-link suspensions in MBD, the technique of replacing physical bushings (rubber elastic bodies) with "equivalent rigid joints" was popularized by CATIA Digital Mockup (DMU Kinematics) in the 1990s. Approximating the nonlinear characteristics of bushings with equivalent linear joints can speed up calculations by tens of times. Toyota's production engineering department published an SAE paper disclosing that they incorporated this technique into the mass production design process for stroke analysis of the Hilux (Land Cruiser series) suspension.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do the prep work (Mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's "prep work". If mesh quality is poor, results will be a mess no matter how good the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across 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 often the most important step in the entire analysis.
Software Comparison
Tools
Simpack Has Adoption Record for Shinkansen (Bullet Train) Bogie MBD
Simpack by German Simula (now Dassault Systèmes) holds market share in the industry with features specialized for railway vehicle dynamics analysis. It was adopted by JR Central for joint and wheel flange contact analysis of the Tokaido Shinkansen N700 series bogies, and contributed to the design for suppressing hunting oscillation at speeds over 350 km/h, as confirmed in publications by the Railway Technical Research Institute. SIMPACK 2022 added VR integration features enabling real-time co-simulation to check NL behavior of contact joints.
Three Most Important Questions for Selection
- "What to solve?": Does the required physical model/element type for joint constraints and kinematics have support? For example, presence of LES support for fluids, contact/large deformation capability for structures makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission (GUI) and manual transmission (script) cars.
- "How far to expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Joints
Redundant Constraints are Rank
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