CMS法（Component Mode Synthesis）
Theory and Physics
What is the CMS Method?
Professor, what is the CMS method?
CMS (Component Mode Synthesis) is a technique that divides a large-scale structure into substructures (components), reduces the dynamic characteristics of each component using modal coordinates, and then assembles the whole.
"Divide, reduce, and assemble"?
For example, directly solving a full-vehicle model (millions of DOF) requires enormous computation. With the CMS method:
1. Divide into substructures like body, engine, suspension, etc.
2. Reduce each substructure using eigenmodes + constraint modes (to hundreds of DOF)
3. Combine the reduced substructures and solve the whole system
Millions of DOF are reduced to thousands!
Computation time can become less than 1/100. Particularly effective when many substructures are used repeatedly (e.g., different vehicle body variations).
Craig-Bampton Method
The most widely used CMS method is the Craig-Bampton method (1968). Each substructure is represented by two types of modes:
- Fixed-Interface Eigenmodes — Internal eigenmodes with the interface fixed
- Constraint Modes — Static deformation when each interface DOF is given a unit displacement
The most widely used CMS method is the Craig-Bampton method (1968). Each substructure is represented by two types of modes:
.
So eigenmodes represent internal vibration, and constraint modes represent interface deformation.
Displacement of each substructure:
$\{q\}$ are modal coordinates (tens to hundreds), $\{u_b\}$ are interface DOFs. Total DOFs are reduced to $\sum (n_{modes} + n_{boundary})$.
Advantages of the CMS Method
| Advantage | Explanation |
|---|---|
| Significant reduction in computation time | Reduces total DOFs to 1/10 to 1/1000 |
| Parallelization | Each substructure can be computed independently |
| Efficient design changes | Changing one substructure does not require recalculating others |
| Intellectual property protection | Suppliers can provide only reduced models without disclosing internal structure |
Interesting that it can be used for IP protection.
In the automotive supply chain, suspension manufacturers provide CMS reduced models to OEMs. OEMs can perform full-vehicle vibration analysis without seeing the internal structure.
Summary
Let me organize the CMS method.
Key points:
- Divide large-scale structures into substructures and reduce — DOFs to 1/10 to 1/1000
- Craig-Bampton method is standard — Eigenmodes + Constraint modes
- Significant reduction in computation time — Essential for full-vehicle NVH analysis
- Flexible for design changes — Changes possible per substructure
- IP protection — Share reduced models without disclosing internal structure
Hurty and Craig's Invention of "Solving by Parts"
The CMS (Component Mode Synthesis) method was developed by Hurty (1960) and Craig & Bampton (1968). It's a technique that divides large-scale models into substructures, reduces them using the eigenmodes of each part, and then couples them. The Craig-Bampton method combines fixed-interface modes (internal DOF reduction) and constraint modes (boundary DOF retention) and remains the most widely used CMS formulation today.
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 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 "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly so acceleration is negligible". It absolutely cannot be omitted for 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", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So a question—an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is 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"—different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces). 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 the road 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. That's because vibrational energy turns into heat due to air resistance and internal friction in the string. Car shock absorbers work on the same principle—intentionally absorbing vibrational energy for a better ride. 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, stress-strain relationship is linear
- Isotropic material (unless specified otherwise): Material properties are direction-independent (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considers only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity/creep requires constitutive law extension
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting mm, unify loads/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note 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 to N in mm system, N in m system |
Numerical Methods and Implementation
Implementation of the Craig-Bampton Method
Please explain the implementation steps of the Craig-Bampton method.
Step 1: Substructure Definition
- Divide the structure into logical components
- Define interface DOFs (connection points between components)
Step 2: Reduction of Each Component
- Eigenfrequency analysis with fixed interface → Fixed-interface eigenmodes $[\Phi_f]$
- Static analysis giving unit displacement to each interface DOF → Constraint modes $[\Psi_c]$
- Calculate reduced mass/stiffness matrices
Step 3: Assembly of the Whole
- Combine reduced matrices of each component via interface DOFs
- Solve the overall eigenvalue problem
Nastran
```
SOL 103
CEND
SUBCASE 1
METHOD = 10
BEGIN BULK
$ Superelement definition
SELOC, 100, ... $ Substructure definition
SECONM, 100, ... $ Reduction
```
Nastran's superelement functionality is the industry standard for CMS.
Abaqus
```
*SUBSTRUCTURE GENERATE
*RETAINED NODAL DOFS
interface_nodes, 1, 6
*FREQUENCY
50, ,
*END STEP
```
Ansys
Substructuring analysis type in Workbench allows CMS reduction setup via GUI.
Is Nastran's superelement the industry standard?
For automotive and aerospace CMS analysis, Nastran's superelement has overwhelming track record. There is an industry-standard format (OP2/OP4 files) for exchanging reduced matrices.
How to Determine the Number of Modes to Retain
How many fixed-interface eigenmodes should be retained?
Retain modes up to 1.5 to 2 times the frequency of the range of interest. For example, for full-vehicle analysis up to 500 Hz, retain modes up to 750~1000 Hz for each substructure.
Summary
Let me organize the numerical methods of CMS.
Key points:
- Craig-Bampton method — Reduction using fixed-interface eigenmodes + constraint modes
- Nastran superelement is industry standard — OP2/OP4 format
- Number of modes = up to 1.5~2 times the frequency of interest — Balance accuracy and computational cost
- Interface DOF definition is key — Appropriate interface selection affects accuracy
Craig-Bampton Reduction Procedure and Accuracy Verification
The Craig-Bampton method involves three steps: ① Eigenmode calculation for internal DOFs (typically 0~f_max range) ② Constraint mode calculation by applying unit displacement to boundary DOFs ③ Assembly of reduced matrices. The standard procedure is to incorporate the reduced model into the full system, calculate eigenvalues, and verify accuracy within ±1% compared to the full FEM. The cutoff frequency for internal eigenmodes uses 1.5 to 2 times the evaluation upper limit.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but DOFs increase 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.
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 each iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$~$10^{-6}$). 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 trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately by hand calculation"—reliable but too time-consuming for large-scale problems. The iterative method is like "making educated guesses...
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