Free Vibration Analysis of Beams
Theory and Physics
Beam Vibration — The Origin of Dynamic Analysis
Professor, is the free vibration of a beam the most fundamental problem in natural frequency analysis?
Yes. Free vibration of a beam is one of the few problems for which an analytical solution exists, making it essential for verifying FEM accuracy.
Governing Equation
Free vibration equation for an Euler-Bernoulli beam:
Using separation of variables $w(x,t) = W(x) e^{i\omega t}$, the spatial part becomes:
It's a 4th-order ordinary differential equation. What is the solution?
General solution:
where $\beta^4 = \rho A \omega^2 / (EI)$.
The four constants $C_1 \sim C_4$ are determined by four boundary conditions. The natural frequencies are obtained from the condition for a non-trivial solution (frequency equation).
Natural Frequencies for Various Boundary Conditions
The $n$-th natural frequency $f_n = (\beta_n L)^2 / (2\pi L^2) \sqrt{EI/(\rho A)}$:
| Boundary Condition | $(\beta_1 L)^2$ | $(\beta_2 L)^2$ | $(\beta_3 L)^2$ | Mode Shape |
|---|---|---|---|---|
| Cantilever | 3.516 | 22.03 | 61.70 | Maximum at tip |
| Simply Supported | $\pi^2 = 9.870$ | $4\pi^2 = 39.48$ | $9\pi^2 = 88.83$ | $\sin(n\pi x/L)$ |
| Fixed-Fixed | 22.37 | 61.67 | 120.9 | Zero at both ends |
| Free-Free | 22.37 | 61.67 | 120.9 | Free at both ends |
The first mode $(\beta L)^2$ is 3.516 vs. 22.37 for cantilever vs. fixed-fixed... a difference of over 6 times. Is the frequency about $\sqrt{6} \approx 2.5$ times higher?
Since $f \propto (\beta L)^2$, it's 22.37/3.516 = 6.36 times higher. The natural frequency can change by over 6 times depending on the boundary condition. If FEM results don't match theory, first suspect the boundary conditions.
Timoshenko Beam Vibration
What about Timoshenko beam vibration?
It includes shear deformation and rotary inertia, causing differences from the EB beam at higher modes:
Timoshenko beam frequencies are always lower. The difference becomes significant when $f \cdot h / c_s$ ($c_s$ = shear wave speed) is large.
Does the difference increase for higher modes?
Yes. The difference may be slight (1-2%) for the 1st mode, but can be 10-30% for the 10th mode. The Timoshenko beam is necessary if accurate results up to higher modes are required.
FEM Verification
Free vibration of a beam is ideal as an FEM benchmark problem:
1. The exact analytical solution is known.
2. Mesh convergence can be checked by varying the number of elements.
3. The difference between EB beam elements and Timoshenko beam elements can be experienced.
4. Beam elements can be compared with shell/solid elements.
It's the best practice problem for those learning FEM.
Calculate the 1st natural frequency of a cantilever beam using FEM and compare it with the theoretical value $f_1 = 3.516/(2\pi L^2) \sqrt{EI/\rho A}$. This alone allows you to verify the element accuracy, boundary conditions, and mass settings.
Summary
Let me summarize free vibration of beams.
Key points:
- $EI W'''' = \rho A \omega^2 W$ — A 4th-order ODE eigenvalue problem.
- Analytical solution exists — The best benchmark for FEM.
- Natural frequency can change by over 6 times depending on boundary conditions — Check this first.
- Timoshenko beam is needed for higher modes — EB beam is inaccurate for higher modes.
- Memorize the $(\beta_n L)^2$ values — Cantilever: 3.516, Simply supported: $\pi^2$.
History of Euler-Bernoulli Beam Theory
The "Euler-Bernoulli beam theory" is a collaboration by Leonhard Euler (1744) and Jacob Bernoulli. Initially, Bernoulli studied bending from a lying-down perspective, and Euler generalized it. This remained the only beam vibration theory until Timoshenko published an improved version in 1921 that included shear deformation and rotary inertia.
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 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 enough that 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 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. Now a question — an iron rod and a rubber band, which stretches more under 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". Stiffness is "resistance to deformation", strength is "resistance to failure" — different concepts.
- External force term (load term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire contents" (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 ending up with "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 is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational 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 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 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 equilibrium 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 deformations). |
| 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
Beam Vibration Analysis by FEM
When doing vibration analysis with beam elements, how many elements are roughly needed?
EB beam elements (Hermite interpolation) can accurately represent the 4th-order polynomial for bending, so one element was sufficient for static analysis. However, multiple elements are needed for vibration analysis to represent the mode shapes.
To represent the half-wavelength of the $n$-th mode:
- 1st mode: Minimum 4 elements
- 5th mode: Minimum 20 elements
- $n$-th mode: Minimum $4n$ elements (guideline)
So more elements are needed than for static analysis.
Vibration mode shapes are combinations of sines/cosines, with shorter wavelengths for higher modes. Finer meshes are needed to resolve shorter wavelengths. This is the same for plates and solids.
Effect of Mass Matrix
Do results change between consistent mass and lumped mass?
For EB beam elements:
- Consistent mass — Tends to overestimate $f$ (upper bound theorem for stiffness).
- Lumped mass — Tends to underestimate $f$.
- The true value lies between them.
Which is more accurate?
Generally, consistent mass is more accurate. However, the difference is small if enough elements are used. For the 1st natural frequency of a cantilever beam with 4 EB beam elements, consistent mass yields accuracy within 0.1%.
Solver-Specific Settings
```
SOL 103
CEND
METHOD = 10
BEGIN BULK
CBAR, ...
EIGRL, 10, , , 10
```
```
*BEAM SECTION, SECTION=RECT, ELSET=beam
0.01, 0.001
*STEP
*FREQUENCY, EIGENSOLVER=LANCZOS
10, ,
*END STEP
```
Natural frequency analysis with beam elements is simple.
Beam vibration is an ideal FEM exercise because it is "simple to set up and can be directly compared with analytical solutions". When starting to use a new solver, it is recommended to first perform a beam vibration analysis to verify the settings.
Summary
Let me summarize the numerical methods for beam vibration.
Key points: