固有振動数解析
Theory and Physics
What is Natural Frequency?
Professor, what is "natural frequency"?
It is the natural vibration frequency of a structure when it vibrates without any external force. All structures have their own natural frequency, and when the frequency of an external force matches this, resonance occurs.
What's the problem with resonance?
During resonance, the amplitude increases rapidly. Bridge collapses (Tacoma Narrows Bridge, 1940), building damage (during earthquakes), machine failures (dangerous speeds of rotating bodies)... all are caused by resonance. Understanding the natural frequency is the most basic requirement of structural design.
Governing Equation
Equation for undamped free vibration:
Assuming a solution of $\{u\} = \{\phi\} e^{i\omega t}$, we get the eigenvalue problem:
It has the same form as the buckling analysis equation $([K] + \lambda [K_\sigma])\{\phi\} = \{0\}$!
Exactly the same mathematical structure. In buckling, $[M]$ takes the place of $[K_\sigma]$. Therefore, the same eigenvalue solver (Lanczos method, etc.) can be used.
$\omega_i$ is the angular frequency (rad/s), $f_i = \omega_i / (2\pi)$ is the natural frequency (Hz), and $\{\phi_i\}$ is the mode shape.
Natural Frequency of a Single-Degree-of-Freedom System
Before FEM, the most basic case:
It's determined solely by the spring constant $k$ and mass $m$. Higher stiffness and smaller mass lead to higher frequency.
This simple relationship is very useful for sanity checking FEM results. Estimate the equivalent stiffness and equivalent mass of the structure to get an order-of-magnitude estimate for $f$.
Natural Frequency of Beams
Fundamental (first) natural frequency of basic beams:
| Condition | $f_1$ | Note |
|---|---|---|
| Cantilever Beam | $\frac{3.516}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$ | Lowest |
| Simply Supported Beam | $\frac{\pi^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$ | |
| Fixed-Fixed Beam | $\frac{22.37}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$ | Highest |
The natural frequency changes quite a bit depending on the boundary conditions.
There's a difference of over 6 times between cantilever and fixed-fixed. If FEM results deviate significantly from theoretical values, the boundary condition settings should be the first thing to suspect.
Natural Frequency Analysis in FEM
What is the procedure for finding natural frequencies with FEM?
1. Modeling — Mesh, material ($E, \rho$), boundary conditions
2. Execute Eigenvalue Analysis — Use Lanczos method to find the lower $n$ modes
3. Check Results — Natural frequencies and mode shapes
4. Verification — Compare with theoretical solutions or experimental modal analysis
Unlike static analysis, the material constant $\rho$ (density) is required here.
Correct. Static analysis only needs $E$, but natural frequency analysis requires $\rho$. Forgetting to set the density will result in natural frequencies being zero or infinite. This is a very common mistake.
Summary
Let me organize the theory of natural frequency analysis.
Key points:
- $([K] - \omega^2 [M])\{\phi\} = \{0\}$ — Same eigenvalue problem as buckling
- $f = (1/2\pi)\sqrt{k/m}$ — The foundation of everything. Use it for sanity checks
- Boundary conditions significantly change natural frequency — 6x difference between cantilever vs. fixed-fixed
- Density $\rho$ setting is mandatory — Results become meaningless if forgotten
- Avoiding resonance is the design goal — Separate the external force frequency from the natural frequency
So natural frequency analysis is the most basic dynamic analysis to know "at what Hz a structure vibrates".
Exactly. Natural frequency analysis is the foundation for all dynamic analyses (response analysis, time history analysis). Dynamic design cannot be established without it.
From Hooke's Law to Vibration Theory
Robert Hooke published the proportional relationship between elastic force and displacement (Hooke's law) in 1678, but he himself did not reach the theory of vibration frequency. It was Christian Huygens who completed it, deriving the period formula for a simple pendulum T=2π√(L/g) from its isochronism in 1673. This formula is a special case of the current single-degree-of-freedom frequency formula fn=1/(2π)√(k/m), a fundamental equation that has remained unchanged for over 300 years.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced your body being thrown forward when slamming on the brakes? 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, which is the assumption that "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 pull a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, 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 band. This "resistance to stretching" is 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 force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). 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 it becomes "compression"—it 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 away. That's because the vibration energy is converted into 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 shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behaviors like plasticity and creep require 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 to N in mm system, N in m system |
Numerical Methods and Implementation
Eigenvalue Solver
How is the eigenvalue problem for natural frequencies solved?
The same solver used for buckling analysis can be used. The Lanczos method is the industry standard.
| Method | Characteristics | Application |
|---|---|---|
| Lanczos Method | Optimal for large-scale sparse matrices. Efficiently extracts lower modes | Industry standard |
| Subspace Iteration Method | More stable than Lanczos but slower | Closely spaced eigenvalues |
| AMLS (Automated Multi-Level Substructuring) | Handles extremely large-scale problems | Millions of DOF |
What is AMLS?
AMLS is Nastran's large-scale eigenvalue analysis method. It automatically partitions the structure into substructures, finds the eigenvalues of each substructure individually, and then assembles the whole. It can efficiently find hundreds of modes for models with millions of DOF.
Nastran
```
SOL 103
CEND
METHOD = 10
BEGIN BULK
EIGRL, 10, , , 20 $ Find 20 modes
```
Abaqus
```
*STEP
*FREQUENCY, EIGENSOLVER=LANCZOS
20, ,
*END STEP
```
Ansys
```
/SOLU
ANTYPE, MODAL
MODOPT, LANB, 20 ! 20 modes using Lanczos method
SOLVE
```
The Lanczos method is the default in all solvers.
In modern FEM solvers, it's fair to say that eigenvalue analysis = Lanczos method. The setting is simply specifying "the number of modes to find".
Mass Matrix Selection
Which should we use, consistent mass or lumped mass?
The accuracy of natural frequencies depends on the mass matrix:
- Consistent mass — High accuracy. Recommended default
- Lumped mass — Faster computation but slightly lower accuracy. Differences appear in higher modes
The default for Nastran and Abaqus is consistent mass.
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