Flexible Multibody Dynamics
Theory and Physics
What is Flexible MBD?
Professor, what does flexible MBD add to rigid body dynamics?
A combination of rigid bodies + CMS-reduced flexible bodies. Handles large rigid body motion + small elastic deformation simultaneously.
CMS + MBD
1. Reduce FEM model via CMS using the Craig-Bampton method
2. Integrate the reduced flexible body into the MBD solver
3. Joint connection + dynamic analysis
Summary
The theory of flexible body MBD is based on NASA's Shabana paper
The rigorous formulation of flexible multibody systems is based on the "Floating Frame of Reference (FFR)" method established by Ahmed Shabana (University of Illinois) in his 1982 PhD thesis. NASA's Goddard Space Flight Center's implementation of Shabana's theory for analyzing the deployment of solar array paddles on space structures was one of the earliest practical applications. The current ANCF (Absolute Nodal Coordinate Formulation) is an extended version published by Shabana himself in 1996, which significantly improved the analysis accuracy of large-deformation flexible bodies.
Physical meaning of each term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? That "feeling of being pulled" 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 "gets left behind". In static analysis, this term is set to zero, which assumes "forces are applied slowly so acceleration can be ignored". It absolutely cannot be omitted for 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 $F=kx$, which is the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External force term (load term): Body force $f_b$ (gravity, etc.) 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 tire 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 typical mistake here: getting the load direction wrong. Intending "tension" but it becomes "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 away. That's because the 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 keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and application 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: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed
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
Implementation of Flexible MBD
Summary
The Craig-Bampton method is the world standard for mode reduction
In flexible body MBD (MFBD), the "Craig-Bampton method" for reducing finite element models to a small number of modes is the de facto standard. This method, published by Roy Craig Jr. and Mervyn Bampton in the AIAA Journal in 1968, combines fixed-interface modes and constraint modes, resulting in high mode accuracy at connection points. For automotive crankshaft analysis, Craig-Bampton reduction is standardly used to reduce FEM models with over 100,000 total modes to tens or hundreds of modes for integration into MBD.
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 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. Shows 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
Applies full load not all at once, but in small increments. The arc-length method (Riks method) can trace 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: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching 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 "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
Flexible MBD Practice
Stress evaluation of automotive suspension components, crane boom deflection, robot accuracy evaluation.
Practical Checklist
MFBD is essential for satellite solar array paddle deployment
Deployable space structures (solar array paddles, antennas) are typical applications of MFBD analysis. The deployment analysis for the solar array paddles of JAXA's Hayabusa2, launched in 2014, was conducted using coupled RecurDyn and Nastran (indirect information from JAXA press release). MFBD analysis was used to pre-evaluate the risk of deployment vibration interfering with attitude control thrusters, reportedly aiding in optimizing the deployment sequence. The importance of MFBD analysis in zero-gravity environments, where ground testing is difficult, will only increase.
Analogy: 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 the "prep work". If mesh quality is poor, the 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 mesh densities — 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 exam question". If the question 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
Flexible MBD Tools
MSC Adams, founded in 1977, is the industry's oldest MBD solver
The predecessor of MSC Adams, "DADS (Dynamic Analysis and Design System)", was developed by Professor Orlandea at the University of California, Davis in 1977 and later transferred to MSC. Flexible body functionality (AdamsFlexible) was added in the late 1990s, and the workflow automating Craig-Bampton reduction in conjunction with Nastran became the industry standard. Currently under HexagonMSC, Adams 2023 features enhanced Python script APIs, making integration into ML optimization loops easier.
The 3 Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for flexible multibody dynamics? For example, presence of LES support for fluids, contact/large deformation capability for structures can be differentiators.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far will you 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 Flexible MBD
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