How to Perform a Modal Analysis

1. The Engineering Task

Extract the natural frequencies (eigenvalues) and mode shapes (eigenvectors) of a structure, and verify that the model is correct and the results are physically meaningful.

2. When to Use This Method

Modal analysis is the first step in any dynamic assessment — frequency response, random vibration, shock response, flutter or acoustic analysis. It is also used to check that a structure's natural frequencies are away from excitation frequencies.

3. What You Need Before Starting

• The FE model with correct mass distribution — include non-structural mass (fuel, equipment, payload)
• Boundary conditions representing the actual support conditions in service
• Correct material density values — errors directly affect frequency predictions
• The number of modes to extract — enough to capture >90% of the effective mass in each direction

4. Step-by-Step Method

1. Apply boundary conditions that represent the actual support conditions. For a free-free analysis, do not constrain any DOF — the first six modes will be rigid-body modes at zero frequency
2. Ensure mass distribution is correct: include all non-structural mass as point masses, distributed mass or mass proportional to element volume
3. Remove or linearise any nonlinear features (contact, plasticity, gap elements) — modal analysis is linear
4. Request enough modes to capture >90% of the effective mass in each direction. Typically 20–100 modes for a moderately complex model
5. Run the eigenvalue extraction (Lanczos or subspace method)
6. Check for rigid-body modes: for a constrained model, there should be no zero-frequency modes. For a free-free model, the first six modes should be rigid-body at approximately zero frequency
7. Examine the mode shapes: verify that bending, torsion and local modes appear in the expected order and that the shapes are physically sensible
8. Extract modal participation factors and effective mass for each mode in each direction

5. What to Check

• Effective mass: does the sum of effective mass across all extracted modes exceed 90% of the total mass in each direction? If not, extract more modes
• Rigid-body modes: for a constrained model, any zero-frequency mode indicates an unconstrained DOF — check for disconnected nodes or missing constraints
• Mode shape order: do the modes appear in a physically sensible order? A local mode appearing before the first global bending mode may indicate a modelling issue
• Frequency range: are the natural frequencies in the expected range? Compare with hand calculations (beam frequency, plate frequency) for order-of-magnitude verification

A model with unexpected zero-frequency modes has unconstrained degrees of freedom. Check for disconnected nodes, missing constraints or improperly defined contact.

6. How to Interpret the Result

The modal results provide the dynamic characteristics of the structure. The natural frequencies identify resonance conditions — the structure will amplify any excitation at or near these frequencies. The mode shapes show how the structure deforms at each frequency. The effective mass indicates how strongly each mode is excited in each direction — modes with low effective mass contribute little to the response and may be excluded from subsequent dynamic analyses.

Modal Assurance Criterion (MAC):
              (φ_A^T · φ_X)²
MAC  =   ─────────────────────────────────
           (φ_A^T · φ_A)(φ_X^T · φ_X)

MAC > 0.9: same mode   |   MAC < 0.1: different modes

7. Common Mistakes

• Not including non-structural mass — equipment and fuel can significantly change natural frequencies
• Applying nonlinear contact in a modal analysis — modal is linear; use tie constraints or linearised contact
• Not extracting enough modes — missing high-frequency modes can miss significant effective mass
• Density errors — a factor of 1000 error in density changes frequencies by √1000 ≈ 31×
• Not checking effective mass — the mode count may look sufficient but still miss significant mass

8. Further Reading

See the Dynamics, Vibration & Shock Knowledge category for modal theory and modal analysis fundamentals. See How to Check Modal Effective Mass for the effective mass verification and How to Perform a Frequency Response Analysis for the subsequent dynamic analysis.

• Modal Analysis Example (Worked Example)