Vibration Analysis

Vibration Analysis (vibrations)

Purpose

Compute natural frequencies and mode shapes of beam-like structures using Euler–Bernoulli FEM with optional damping. Evaluates separation margin between operating excitation frequency and resonances per ISO 10816 context.

Physics & theory

Free vibration of elastic structures satisfies $\mathbf{M}\ddot{\mathbf{u}}+\mathbf{Ku}=0$, yielding eigenvalue problem $(\mathbf{K}-{\omega }^2\mathbf{M})\mathbf{v}=0$. Natural frequencies $f_n={\omega }_n/(2\pi )$ depend on stiffness distribution, mass, and boundary conditions.

Damped natural frequency $f_d=f_n\sqrt{1-{\zeta }^2}$ for damping ratio $\zeta$. Resonance occurs when excitation frequency matches $f_n$; separation margin $SM=|f_{\mathrm{excitation}}-f_n|/f_n$ should exceed code guidance (often 10–20% for machinery).

Dynamic analysis requires careful identification of mass, stiffness, and damping distribution. Natural frequencies depend on boundary conditions — a cantilever beam has fundamentally different modes than a simply supported beam of the same dimensions.

Damping limits resonant amplification; lightly damped structures (( zeta < 0.05 )) can see transmissibility peaks exceeding 10 near resonance. Separation margin between operating excitation and natural frequency should typically exceed 15–20% for rotating machinery.

Governing equations

Numerical method

Euler–Bernoulli beam FEM (euler-bernoulli-fem solver): mesh up to 240 segments. Mass matrix from material density and cross-section. Eigenvalue extraction for first N modes; mode shapes normalized. Physics checks verify positive, monotonic frequencies.

Inputs

Outputs

• Natural frequencies (undamped and damped), mode shapes, separation margin, resonance notes, solver warnings.

Design codes & checks

• Indicative: Natural frequency, excitation separation margin
• ISO: ISO 10816 mechanical vibration severity (context)

Assumptions & limitations

• 1D beam model; no plate/shell or 3D solid modes.
• Linear modal analysis; no geometric stiffness or spin softening.
• Damping is uniform modal fraction — not frequency-dependent material damping.
• Low segment count (< 8) reduces accuracy warning issued.

References

1. Rao, S. S. Mechanical Vibrations, 6th ed. Pearson.
2. Inman, D. J. Engineering Vibration, 5th ed. Pearson.
3. ISO 10816-1:1995. Mechanical vibration — Evaluation of machine vibration.
4. Timoshenko, S. P. Vibration Problems in Engineering, 5th ed.
5. PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
6. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.