Random Vibration of Multi-Degree-of-Freedom Systems
Theory and Physics
Random Vibration of Multi-Degree-of-Freedom Systems
Professor, how do we handle random vibration in multi-degree-of-freedom systems?
In multi-degree-of-freedom systems, correlation between modes (cross-mode terms) is important. The response PSD is:
If modes are sufficiently separated, SRSS combination is sufficient. For closely spaced modes, CQC (Complete Quadratic Combination) is necessary.
Multi-Point Input
Correlated random input at multiple support points:
$[S_{in}]$ is the cross-spectral matrix. Diagonal = auto PSD, off-diagonal = cross PSD.
Summary
Key Points:
- Cross-Mode Terms — Important for closely spaced modes. Combine using CQC.
- Cross-Spectral Matrix — Describes correlation in multi-point input.
- SRSS vs. CQC — Use SRSS if modes are separated, CQC if they are closely spaced.
- FEM PSD analysis automatically includes cross-mode terms.
Fundamental Theory of Multi-Degree-of-Freedom Random Vibration
Multi-degree-of-freedom (MDOF) random vibration is handled by decomposing the system into individual natural modes using modal superposition, finding the random response for each modal coordinate, and then superimposing them. Methods to consider correlation between modes (cross PSD) include SRSS (Square Root of Sum of Squares) and CQC (Complete Quadratic Combination). CQC is essential when natural frequencies are close (ratio <10%). This was formulated in the 1970s by E.L. Wilson and others at the University of California, Berkeley.
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 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 enough that acceleration can be ignored". It absolutely cannot be omitted in impact loading 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 here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. 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 forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., 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 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 pitfall here: getting the load direction wrong. Intending "tension" but modeling "compression"—it 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 vibration energy is converted to heat through 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 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 the 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 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 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) | In mm system: N, in m system: N (consistent). |
Numerical Methods and Implementation
Multi-Degree-of-Freedom PSD in FEM
Nastran's SOL 111 + RANDOM automatically calculates all mode auto-terms + cross-terms.
Multi-point input:
```
RANDPS, 1, 1, 1, 100, 0.0 $ Auto PSD for input 1
RANDPS, 1, 2, 1, 101, 0.5 $ Cross PSD for input 1-2
RANDPS, 2, 2, 1, 102, 0.0 $ Auto PSD for input 2
```
If correlation is unknown, evaluate the range with two cases: fully correlated + uncorrelated.
Summary
Solution Using Cross-PSD Matrices
The rigorous solution for MDOF random vibration involves calculating the output cross-PSD matrix S_XX(ω) = H(ω)S_FF(ω)H*(ω)^T from the input cross-power spectral density matrix S_FF(ω) and the frequency response function matrix H(ω), and then obtaining the response variance through frequency integration. This formulation is essential when there are multiple input points (e.g., 4-point suspension). In Python, this can be implemented numerically by combining scipy.signal.coherence and numpy.fft.
Linear Elements (1st-Order Elements)
Linear interpolation between nodes. Low computational cost but lower 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 (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 the tangent stiffness matrix each iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is lower, 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 the full load at once, apply it 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 can take 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 it to an estimated location and adjusting forward/backward (iterative) is more efficient than searching sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
First-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. Second-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
Practical Checklist
Actual Application to Automotive Chassis
In the automotive industry, road input is treated as random loads at multiple tires (4-point input) to evaluate chassis fatigue durability. Ford established the Road Simulation Table (LST) test standard in the 1990s based on road surface PSD data from Belgium's Pave test course. Analysis uses MSC Nastran's random vibration SOL 111 to solve models with up to 10,000 degrees of freedom, evaluating fatigue damage from stress RMS using the Rainflow counting method.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question—in cooking, which step 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 excellent 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 can 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 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 truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraints is often the most critical step in the entire analysis.
Software Comparison
Tools
Selection Guide
MDOF Capability Comparison by Solver
Comparison of major solvers for MDOF random vibration: Ansys Mechanical supports GPU parallelization for large models (8x faster with RTX 4090), MSC Nastran has over 50 years of track record and abundant NASA verification cases, Abaqus does not support random vibration (use transient implicit + time history as alternative), SIMcenter Nastran enables probabilistic design optimization integrated with DAKOTA. In HPC environments, Nastran and Ansys Mechanical have nearly equivalent scaling performance.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the required physical models/element types for multi-degree-of-freedom random vibration? For example, presence of LES support for fluids, contact/large deformation capability for structures can be differentiating factors.
- "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 (GUI) and manual (script) transmission in cars.
- "How far will you expand?": Choosing with future expansion in mind—scaling up analysis (HPC support), deployment to other departments, integration with other tools—leads to long-term cost reduction.
Advanced Technologies
Advanced Research
Equivalent Linearization for Nonlinear MDOF
For nonlinear MDOF random vibration including rubber bushings or friction dampers, the Equivalent Linearization (EL) method is effective. The stochastic EL method proposed by Bogoliubov in 1954 replaces nonlinear restoring forces with equivalent linear stiffness and equivalent damping, converging through iterative calculation. A 2018 paper by Simulia's research team reported suppressing response RMS error to within 5% for a nonlinear automotive suspension model.
Troubleshooting
Troubles
Underestimation Caused by Mode Truncation
For NASA's Cassini probe launched in 1997, vibration analysis used over 200 modes, but an initial model truncated at 150 modes underestimated response in a specific frequency band by 18%. For multi-degree-of-freedom random analysis, MIL-STD-810G recommends including eigenvalues up to twice the target frequency.
When You Think "The Analysis Doesn't Match"
- First, take a deep breath—Panicking and randomly changing settings will only complicate the problem further.
- Create a minimal reproducible case
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