From Decoupled Linear Modes to Nonlinear Mode Interaction

Posted on Sep 8, 2025

Why superposition breaks, energy starts to slosh, and what to look for in data

TL;DR. In the linear regime, each mode is an independent damped oscillator—clean, decoupled, and predictable. As amplitudes grow (or when damping/forcing aren’t “nice”), cross terms re-couple the modal equations. Near internal resonance (e.g., 1:1, 2:1, 3:1 ratios), those cross terms become near-resonant drives, and you see energy exchange, sidebands, and new frequencies. That’s “mode interaction.” Our benchmark builds controlled simulations that span both regimes and labels interaction strength automatically.

I) The linear world in 90 seconds

Start with a standard MDOF system in physical coordinates:

$$M\ddot{x} + C\dot{x} + Kx = f(t),$$

solve the generalized eigenproblem

$$K\phi_i = \omega_i^2 M\phi_i,$$

and use mass-normalized modes $\mathbf{\Phi} = [\phi_1 \ldots \phi_n]$ so

$$\mathbf{\Phi}^\top \mathbf{M} \mathbf{\Phi} = \mathbf{I}, \qquad \mathbf{\Phi}^\top \mathbf{K} \mathbf{\Phi} = \mathbf{\Omega}^2.$$

With the modal transform $\mathbf{x} = \mathbf{\Phi}\mathbf{q}$ and proportional (Rayleigh) damping $\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}$, the linear part decouples:

$$\ddot{q}_i + 2\zeta_i\omega_i\dot{q}_i + \omega_i^2 q_i = \Gamma_i f(t).$$

Interpretation: each $q_i$ is its own grounded SDOF oscillator. Superposition holds. No interaction.

II) Where does “mode interaction” come from?

As soon as anything non-diagonal in modal space appears, the modes re-couple:

A) Nonlinear restoring/damping (most common)

Compact 2-DOF modal:

$$\ddot{q}_i + 2\zeta_i\omega_i\dot{q}_i + \omega_i^2 q_i + \sum_{j,k} a_{ijk} q_j q_k + \sum_{j,k,\ell} b_{ijk\ell} q_j q_k q_\ell = \Gamma_i f(t), \quad i = 1, 2.$$

Terms like $a_{112}q_1 q_2$, $b_{1122}q_1 q_2^2$ are cross-terms. They vanish in the linear limit, but at finite amplitude they couple the modal equations.

B) Damping that isn’t proportional

If $\mathbf{C}$ isn’t Rayleigh, the projected $\mathbf{C}_m = \mathbf{\Phi}^\top \mathbf{C} \mathbf{\Phi}$ has off-diagonals → linear coupling even without nonlinearities.

C) Forcing/parametric effects

Multiple inputs, non-collocated actuation/sensing, time-varying stiffness, gyroscopic effects—all can couple modes.

III) Why internal resonance supercharges interaction

Consider the quadratic coupling and a 2:1 ratio $\omega_1 \approx 2\omega_2$. If $q_2 \approx A_2\cos(\omega_2 t)$, then

$$q_2^2 = \frac{A_2^2}{2}\left[1 + \cos(2\omega_2 t)\right].$$

The $\cos(2\omega_2 t)$ term drives mode 1 at frequency $2\omega_2$. When $\omega_1 \approx 2\omega_2$, that drive is near-resonant, so $q_1$ grows and draws energy from $q_2$. Symmetrically, terms like $q_1 q_2$ contain components near $\omega_2$ (since $\omega_1 - \omega_2 \approx \omega_2$), feeding energy back. Net effect: periodic energy exchange (beating)—the hallmark of interaction.

Selection rules (handy intuition):

Quadratic terms → strong effects at 1:1, 2:1, and create sum/difference lines.
Cubic terms → 1:1 backbone bending, 3:1 interactions, and combination tones.

IV) What it looks like in data

Time domain (energies).

Modal energies $E_i(t) = \frac{1}{2}(\dot{q}_i^2 + \omega_i^2 q_i^2)$.

Interacting: envelopes of $E_1, E_2$ swing in out of phase (phase difference $\approx \pi$) (energy sloshing).
Non-interacting: envelopes flat/monotone.

Frequency domain.

Sidebands/combination lines: components at $\omega_i \pm \omega_j$, sub/superharmonics.
Bicoherence: large $b^2(\omega_1, \omega_2)$ shows quadratic phase coupling.

Practical tip: you don’t need to eyeball—compute an Energy-Transfer Index (out-of-phase envelope correlation), a Sideband/Combination Score, and bicoherence. Two out of three high → interacting.

V) Common myths

“Modal analysis means modes never interact.” Only true for the linear part with proportional damping. Nonlinearities (or non-proportional damping/forcing) re-couple them.
“You need exact frequency ratios to see interaction.” Near-commensurate is enough. Detuning just sets how strong/clean the energy exchange is.
“You can spot interaction from time traces alone.” Sometimes, but robust diagnostics (energy envelopes, spectral features, bicoherence) are far more reliable—especially with noise.

VI) Bridging to our benchmark (what we’ll publish next)

We build a large, labeled dataset that spans both regimes:

Linear/near-linear cases where modes stay independent.
Weak/strong/deep nonlinear cases with cross-terms and near-commensurate ratios (1:1, 2:1, 3:1) so interactions actually manifest.
Multiple excitations (harmonic, two-tone, chirp, impulse, band-limited noise), amplitudes, and SNRs.

Automatic labels (no eyeballing):

ETI (energy exchange), SCS (sidebands/combination lines), BIC (bicoherence).
Thresholds decide non / weak / strong /deep interaction per sample.

This lets you: (1) study physics transparently, and (2) train/evaluate ML models on a clean, reproducible benchmark.

Appendix — Derivations & Worked Details

A.1 Physical equations (MCK form)

$$\mathbf{M}\ddot{\mathbf{x}}(t) + \mathbf{C}\dot{\mathbf{x}}(t) + \mathbf{K}\mathbf{x}(t) + \mathbf{g}(\mathbf{x}, \dot{\mathbf{x}}) = \mathbf{f}(t), \qquad \mathbf{x} \in \mathbb{R}^n.$$

A.2 Modes and normalization

Solve the generalized eigenproblem

$$\mathbf{K}\phi_i = \omega_i^2 \mathbf{M}\phi_i, \quad i = 1, \ldots, n,$$

stack $\mathbf{\Phi} = [\phi_i, \ldots \phi_n]$, and mass-normalize:

$$\mathbf{\Phi}^\top \mathbf{M} \mathbf{\Phi} = \mathbf{I}, \qquad \mathbf{\Phi}^\top \mathbf{K} \mathbf{\Phi} = \mathbf{\Omega}^2 = \mathrm{diag}(\omega_1^2, \ldots, \omega_n^2).$$

A.3 Modal transform and projection

With $\mathbf{x} = \mathbf{\Phi}\mathbf{q}$ (so $\dot{\mathbf{x}} = \mathbf{\Phi}\dot{\mathbf{q}}$, $\ddot{\mathbf{x}} = \mathbf{\Phi}\ddot{\mathbf{q}}$):

$$\underbrace{\mathbf{\Phi}^\top \mathbf{M} \mathbf{\Phi}}_{\mathbf{I}}\ddot{\mathbf{q}} + \underbrace{\mathbf{\Phi}^\top \mathbf{C} \mathbf{\Phi}}_{\mathbf{C}_m}\dot{\mathbf{q}} + \underbrace{\mathbf{\Phi}^\top \mathbf{K} \mathbf{\Phi}}_{\mathbf{\Omega}^2}\mathbf{q} + \underbrace{\mathbf{\Phi}^\top \mathbf{g}(\mathbf{\Phi}\mathbf{q}, \mathbf{\Phi}\dot{\mathbf{q}})}_{\mathbf{N}(\mathbf{q},\dot{\mathbf{q}})} = \underbrace{\mathbf{\Phi}^\top \mathbf{f}(t)}_{\mathbf{\Gamma} f(t)}$$

A.4 Damping

If $\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}$ (Rayleigh),

$$\mathbf{C}_m = \alpha\mathbf{I} + \beta\mathbf{\Omega}^2 = 2\mathbf{\Xi}\mathbf{\Omega}, \quad \zeta_i = \frac{\alpha}{2\omega_i} + \frac{\beta\omega_i}{2}.$$

The linear part decouples mode-by-mode:

$$\ddot{q}_i + 2\zeta_i\omega_i\dot{q}_i + \omega_i^2 q_i + N_i(\mathbf{q}, \ddot{\mathbf{q}}) = \Gamma_i f(t).$$

(If damping is non-proportional, $\mathbf{C}_m$ has off-diagonals → linear coupling remains.)

A.5 Forcing map and participation

If $\mathbf{f}(t) = \mathbf{B}u(t)$, then modal forcing is

$$\mathbf{\Phi}^\top \mathbf{f}(t) = \mathbf{\Gamma}u(t), \quad \mathbf{\Gamma} = \mathbf{\Phi}^\top \mathbf{B},$$

whose entries $\Gamma_i$ measure how strongly the input excites each mode.

A.6 Where the modal nonlinear coefficients come from

If $\mathbf{g}$ admits a local polynomial expansion,

$$\mathbf{g}(\mathbf{x}, \dot{\mathbf{x}}) \approx \mathbf{H}^{(2)}:(\mathbf{x} \otimes \mathbf{x}) + \mathbf{H}^{(3)}\vdots(\mathbf{x} \otimes \mathbf{x} \otimes \mathbf{x}) + \ldots,$$

then with $\mathbf{x} = \mathbf{\Phi}\mathbf{q}$,

$$a_{ijk} = \phi_i^\top \mathbf{H}^{(2)}:(\phi_j \otimes \phi_k), \qquad b_{ijk\ell} = \phi_i^\top \mathbf{H}^{(3)}\vdots(\phi_j \otimes \phi_k \otimes \phi_\ell).$$

B. How Nonlinear Cross Terms Create Mode Interaction

Consider a 2-DOF modal model with quadratic/cubic terms:

$$\ddot{q}_i + 2\zeta_i\omega_i\dot{q}_i + \omega_i^2 q_i + \sum_{j,k} a_{ijk} q_j q_k + \sum_{j,k,\ell} b_{ijk\ell} q_j q_k q_\ell = \Gamma_i f(t), \quad i = 1, 2.$$

B.1 Frequency content via product-to-sum

For a quadratic cross term $q_1 q_2$ with $q_j \approx A_j\cos(\omega_j t)$,

$$q_1 q_2 = \frac{A_1 A_2}{2}\left[\cos((\omega_1 - \omega_2)t) + \cos((\omega_1 + \omega_2)t)\right].$$

→ drives components at $\omega_1 \pm \omega_2$ (sum/difference lines).

For a quadratic self term $q_2^2$,

$$q_2^2 = \frac{A_2^2}{2}\left[1 + \cos(2\omega_2 t)\right],$$

→ DC + component at $2\omega_2$.

B.2 Internal resonance “selection rules”

If $\omega_1 \approx 2\omega_2$ (2:1),

The $\cos(2\omega_2 t)$ in $q_2^2$ is near $\omega_1$ → it resonantly drives $q_1$.
The $q_1 q_2$ term contains a component near $\omega_2$ (since $\omega_1 - \omega_2 \approx \omega_2$) → feeds back into $q_2$.

Result: energy exchange between modes (beating).

B.3 Slow-flow (averaged) equations for 2:1 case (structure)

Assume small nonlinearity $\varepsilon$ and detuning $\omega_1 = 2\omega_2 + \varepsilon\sigma$. Write

$$q_1 = A_1(T)\cos(\omega_1 t + \phi_1(T)), \quad q_2 = A_2(T)\cos(\omega_2 t + \phi_2(T)), \quad T = \varepsilon t.$$

Averaging/multiple-scales yields the modulation system (schematic form):

$$\dot{A}_1 = -\zeta_1\omega_1 A_1 + \kappa_1 A_2^2 \sin\psi + \text{(drive terms)},$$$$\dot{A}_2 = -\zeta_2\omega_2 A_2 + \kappa_2 A_1 A_2 \sin\psi + \text{(drive terms)},$$$$\dot{\psi} = \sigma + \lambda_1 A_1^2 + \lambda_2 A_2^2 + \kappa_3\!\left(\frac{2A_1}{A_2} - \frac{A_2}{A_1}\right)\cos\psi + \text{(drive terms)},$$

with phase combination $\psi = 2\phi_2 - \phi_1$. The constants $\kappa_i, \lambda_i$ are algebraic functions of the nonlinear coefficients $a_{ijk}$, $b_{ijk\ell}$ and $\omega_i$.

Key takeaway: the $\sin\psi$ coupling terms are the engine of energy exchange.

(You can specialize to other ratios (1:1, 3:1) and get analogous slow flows with the appropriate phase combinations.)

C. Worked Mini-Example (No Numbers Needed)

Take the unforced, lightly damped pair:

$$\ddot{q}_1 + \omega_1^2 q_1 + \alpha q_2^2 = 0, \qquad \ddot{q}_2 + \omega_2^2 q_2 + \beta q_1 q_2 = 0, \qquad \omega_1 \approx 2\omega_2.$$

Let $q_2 \approx A_2\cos(\omega_2 t)$. Then $q_2^2$ contains $\cos(2\omega_2 t)$. If $\omega_1 = 2\omega_2$, the $q_1$-equation is a resonantly forced SDOF:

$$\ddot{q}_1 + \omega_1^2 q_1 \approx -\frac{\alpha A_2^2}{2}\cos(2\omega_2 t) \quad \text{(near resonance)}.$$

Thus $q_1$ grows (limited by damping and higher-order terms), which in turn feeds back through $\beta q_1 q_2$ to modulate $q_2$. That mutual drive is the mechanism of mode interaction.

D. Why Linear Modes “Decouple” Yet Interaction Appears

The modal transform diagonalizes the linear part (when damping is proportional).
Nonlinear terms $N_i(\mathbf{q}, \dot{\mathbf{q}})$ are not diagonal in modal space → they re-couple the modal equations.
At small amplitude, those terms are weak; near internal resonance, they become near-resonant and dominate the slow dynamics.
In strongly nonlinear regimes, the true vibration shapes are amplitude-dependent (Nonlinear Normal Modes); linear modes are just a convenient basis.

E. Diagnostics We’ll Use Later (with precise math)

E.1 Energy-Transfer Index (ETI)

Bandpass each mode around $\omega_k(\pm 5\text{–}10\%)$, compute analytic signal $z_k = q_k^{bp} + i\mathcal{H}\{q_k^{bp}\}$, envelope $A_k = |z_k|$, modal energy $E_k = A_k^2$. Define the most negative sliding correlation:

$$\text{ETI}_{12} = \min_{t \in \text{windows}} \mathrm{corr}(E_1(t), E_2(t)).$$

Interpretation: strong anti-phase (out of phase by $\pi$ rad) envelope correlation $(e.g.,\; \leq -0.5)$ $\Rightarrow$ energy exchange.

E.2 Sideband / Combination Score (SCS)

Compute STFT power $P(\omega, t)$. Around modal lines and expected nonlinear lines $\nu \in \{\omega_i \pm \omega_j,\, 2\omega_j,\, \frac{1}{2}\omega_{exc}, \ldots\}$, define

$$\text{SCS} = \max_\nu \frac{\mathrm{median}_t\, P(\nu, t)}{\max\!\left(\mathrm{median}_t\, P(\omega_\text{main}, t),\; \mathrm{median}_{\omega,t}\, P(\omega, t)\right)}.$$

Prominent nonlinear lines ($\geq 10\text{–}20\%$ of main) $\Rightarrow$ interaction.

E.3 Bicoherence (quadratic phase coupling)

With DFT $X(\omega)$ and segment averaging $\mathbb{E}[\cdot]$,

$$b^2(\omega_1, \omega_2) = \frac{\left|\mathbb{E}\{X(\omega_1)X(\omega_2)X^*(\omega_1+\omega_2)\}\right|^2}{\mathbb{E}\{|X(\omega_1)|^2\}\,\mathbb{E}\{|X(\omega_2)|^2\}\,\mathbb{E}\{|X(\omega_1+\omega_2)|^2\}}.$$

High $b^2$ at relevant pairs $\Rightarrow$ strong quadratic coupling (e.g., 2:1, sum lines).

E.4 Decision (no eyeballing)

All three above threshold $\Rightarrow$ deep; two out of three $\Rightarrow$ strong; one $\Rightarrow$ weak; none $\Rightarrow$ non.

F. Grounded vs Non-Grounded (clarifying the model)

In modal coordinates with mass normalization, each $q_i$ has a restoring term $\omega_i^2 q_i$: it’s a grounded oscillator.

Interaction does not require non-grounded (free-free) DOFs; it comes from the nonlinear cross terms (and possibly off-diagonal damping/forcing), as shown above.