Dynamics & Control · Lab
Modal Analysis
Natural frequencies, mode shapes and response of a linear multi-degree-of-freedom system M q̈ + C q̇ + K q = f(t). Enter a physical mass–spring–damper chain or the matrices directly.
Modal table
| Mode | ωₙ [rad/s] | fₙ [Hz] | ζ | mᵢ | kᵢ | Status |
|---|---|---|---|---|---|---|
| 1 | 0.1592 | 0 | 1 | 1 | residual 0.0e+0 | |
| 1.732 | 0.2757 | 0 | 1 | 3 | residual 0.0e+0 |
Modes are mass normalized: φᵢᵀMφᵢ = 1 and φᵢᵀKφᵢ = ωᵢ². Modal mass and stiffness are reported in that normalization. A mode shape has arbitrary scale and sign: φ and −φ are the same mode, and the components are not physical displacements.
Mode 1 shape
| DOF | Component | Direction |
|---|---|---|
| x1 | 0.7071 | + (in phase) |
| x2 | 0.7071 | + (in phase) |
Participation and effective mass
| Mode | Γᵢ | m_eff [kg] | Fraction | Cumulative |
|---|---|---|---|---|
| 1 | 1.414 | 2 | 100.0 % | 100.0 % |
| 2 | 0 | 0 | 0.0 % | 100.0 % |
Influence vector r = [1 … 1] (uniform motion of every DOF). Γᵢ = φᵢᵀMr, m_eff,i = Γᵢ² and the fraction uses the denominator rᵀMr = 2 kg. No universal “enough captured mass” criterion is applied.
Orthogonality
| max |ΦᵀMΦ| off-diagonal | 0.00e+0 |
|---|---|
| max |ΦᵀMΦ − I| diagonal | 2.22e-16 |
| max |ΦᵀKΦ| off-diagonal | 0.00e+0 |
| max |ΦᵀKΦ − Ω²| diagonal | 2.22e-16 |
| Quality | acceptable |
| ΦᵀCΦ off-diagonal ratio | 0.00e+0 |
What this Lab models
A linear multi-degree-of-freedom system M q̈ + C q̇ + K q = f(t): natural frequencies and mass-normalized mode shapes from the undamped eigenproblem, modal or direct-state-space forced response, participation factors and receptance FRFs. Shared ground between Dynamics & Control and Engineering Mechanics — the same eigenanalysis either domain uses for a coupled structure.