Calculator · Dynamics & Control
2DOF Natural Frequencies Calculator
Undamped natural frequencies and mode shapes of two masses connected by springs to ground and to each other, from the generalized eigenproblem K φ = ω² M φ.
Chain parameters
Topology: ground —[k₁]— m₁ —[k₂]— m₂ —[k₃]— ground. Setting k₁ = k₃ = 0 gives a free–free chain with one rigid-body mode.
Modes
| Mode | ωₙ [rad/s] | fₙ [Hz] | φ (q₁, q₂) | Character |
|---|---|---|---|---|
| 1 | 1 | 0.1592 | (0.7071, 0.7071) | masses move in phase |
| 2 | 1.732 | 0.2757 | (0.7071, -0.7071) | masses move out of phase |
Assumptions
- Linear, time-invariant, planar chain with lumped masses and massless springs.
- Frequencies are UNDAMPED natural frequencies of the eigenproblem K φ = ω² M φ; damping is ignored here.
- Mode shapes are mass normalized (φᵀMφ = 1). Their scale and overall sign are arbitrary: φ and −φ describe the same mode.
Technical reference
M q̈ + K q = f(t)
K φᵢ = ωᵢ² M φᵢ (undamped generalized eigenproblem)
- M
- mass matrix (diag(m₁, m₂) for this chain)[kg]
- K
- stiffness matrix assembled from k₁, k₂, k₃[N/m]
- ωᵢ
- natural (angular) frequency of mode i[rad/s]
- φᵢ
- mode shape of mode i, mass-normalized (φᵢᵀMφᵢ = 1)[normalization-dependent]
Solved as the symmetric eigenproblem after a Cholesky factorization of M; no damping term appears, so ωᵢ are undamped natural frequencies, not damped ones.
Eigenvalues are classified against a scale-aware tolerance: λ above +tol is an elastic mode, |λ| within the tolerance is a rigid-body mode with ω = 0, and λ below −tol is reported as a negative eigenvalue for which no natural frequency is invented.
The numerical scale of φᵢ follows from the mass normalization used here, so it carries no fixed unit; the relative shape is the physical content, and φ and −φ represent the same mode.
Continue in the Lab
Open this exact chain in the Modal Analysis Lab for mode-shape plots, damping models, modal superposition, participation factors and receptance FRFs.