nonlinear-vibration

name	nonlinear-vibration
description	Nonlinear vibration — Duffing oscillator (hardening/softening), jump phenomena, backbone curves, multiple scales method, harmonic balance, Melnikov criterion for chaos, van der Pol oscillator, parametric excitation (Mathieu equation), nonlinear normal modes (NNMs), and identification methods for engineering structures.
metadata	{"priority":7,"promptSignals":{"phrases":["nonlinear vibration","Duffing oscillator","jump phenomenon","backbone curve","nonlinear normal modes","parametric resonance"],"minScore":3}}

Nonlinear Vibration — Complete Skill

Fundamentals of Nonlinear Dynamics

Sources of Nonlinearity in Engineering

Geometric nonlinearity: Large deflections; Euler-Bernoulli assumption breaks down when δ > 0.1 × L Restoring force: F = kx + k₃x³ [cubic stiffness; k₃ > 0 hardening; k₃ < 0 softening] Examples: stretched cables, beams with large deflection, snap-through structures

Material nonlinearity: Hysteresis in elastomers; plasticity; shape memory alloys Restoring force: not unique function of displacement (depends on history)

Contact nonlinearity: Clearance/backlash: F = 0 for |x| < δ; F = k(x - δ) for x > δ (bilateral clearance) Hertzian contact: F = k_c × x^(3/2) (nonlinear stiffness)

Friction nonlinearity: Coulomb friction: F_friction = μN × sign(ẋ) [discontinuous at ẋ = 0; stick-slip] Bouc-Wen hysteresis model: ż = Aẋ - β|ẋ||z|^(n-1)z - γẋ|z|^n

Amplitude-dependent damping: Viscous: F_d = cẋ [linear] Quadratic (aerodynamic): F_d = c₂ẋ|ẋ| [significant at high velocity] Coulomb: F_d = μN × sign(ẋ) [constant magnitude]

Duffing Oscillator

Equation of Motion

Standard Duffing equation: mẍ + cẋ + kx + k₃x³ = F₀ cos(Ωt)

Non-dimensional form: ẍ + 2ζẋ + x + εαx³ = f cos(Ωt) [ε = small parameter; α = nonlinearity coefficient] ω₀ = √(k/m); ζ = c/(2mω₀); f = F₀/(mω₀²)

Hardening spring: k₃ > 0; resonance frequency increases with amplitude Softening spring: k₃ < 0; resonance frequency decreases with amplitude

Frequency Response and Jump Phenomenon

Backbone curve (undamped amplitude-frequency relation): For hardening (k₃ > 0): ω_r = ω₀ × √(1 + 3k₃A²/(4k)) [A = response amplitude; resonance shifts up] For softening (k₃ < 0): ω_r = ω₀ × √(1 - 3|k₃|A²/(4k)) [resonance shifts down]

Jump phenomenon: Frequency sweep up (hardening): response follows high-amplitude branch → sudden drop at f_jump_up Frequency sweep down (hardening): response follows low-amplitude branch → sudden jump up at f_jump_down Hysteresis between up-sweep and down-sweep responses

Stability: Points on vertical portion of frequency response curve: unstable (two stable branches; one unstable) Jump occurs when sweeping through saddle-node bifurcation (fold) in frequency space Identify fold: ∂F/∂A = 0 simultaneously with F(A, Ω) = 0 (amplitude-frequency curve)

Harmonic Balance Method (HBM)

Assume single harmonic response: x(t) = A cos(Ωt + φ)

Substitute into EOM; equate harmonics of same frequency: For Duffing with cubic: (k + 3k₃A²/4 - mΩ²)A = F₀ cos(φ); cΩA = F₀ sin(φ)

Amplitude equation: [(k + 3k₃A²/4 - mΩ²)² + (cΩ)²] × A² = F₀²

Solve for A as function of Ω (implicit): Iterate or solve cubic in A² for given Ω → frequency response curve Multiple solutions at some frequencies → jump phenomenon appears naturally

Multi-harmonic balance: x(t) = Σ_n [a_n cos(nΩt) + b_n sin(nΩt)] [superharmonics; important for strong nonlinearity] 3rd harmonic appears for Duffing (cubic); 5th if quintic terms present HBM accuracy: include enough harmonics (n = 5–20 for strong nonlinearity)

Method of Multiple Scales (MMS)

Perturbation Expansion

Small-parameter expansion: x(t, ε) = x₀(T₀, T₁) + ε × x₁(T₀, T₁) + ... T₀ = t (fast time); T₁ = εt (slow time; amplitude/phase evolution)

Time derivatives: d/dt = D₀ + εD₁ + ... [D_n = ∂/∂T_n] d²/dt² = D₀² + 2εD₀D₁ + ...

Order ε⁰ (leading): D₀²x₀ + ω₀²x₀ = f cos(ΩT₀) [linear; solution: x₀ = A(T₁)e^(iω₀T₀) + cc + particular]

Order ε¹ (first correction): Secular term condition → amplitude/phase modulation equations (solvability condition) Eliminate terms proportional to e^(±iω₀T₀) on right side

Primary resonance (Ω ≈ ω₀): amplitude-frequency relation: a² [(σ - 3αa²/8)² + ζ²] = f²/4 [σ = Ω - ω₀ = detuning; a = slow amplitude; ζ = c/(2m)] Same backbone curve recovered; closed-form; valid for ε << 1

Superharmonic resonance (Ω ≈ ω₀/3): Third-order term excites resonance at Ω = ω₀/3 → third superharmonic resonance MMS solvability condition: σ̂ = Ω×3 - ω₀; amplitude modulation equation derived similarly

Parametric Excitation — Mathieu Equation

Mathieu Equation

Standard form: ẍ + (δ + ε cos(2τ)) × x = 0 [τ = ωt/2; δ = (ω₀/ω)²; ε = amplitude of parametric excitation]

Stability (Strutt / Ince-Strutt diagram): Unstable regions (Arnold tongues) centered at δ = 1/4, 1, 9/4, 4, ... (i.e., ω = 2ω₀/n, n = 1, 2, 3, ...) First tongue (δ ≈ 1, ω ≈ ω₀): principal parametric resonance; widest instability region; most dangerous At δ = 1: instability begins at ε_crit ≈ 2ζ (damping threshold for parametric instability)

Engineering examples: Rotating shaft with asymmetric cross-section: parametric excitation from gravity Pendulum with oscillating pivot: Kapitza pendulum (stabilized inverted if ε large) Cable under varying tension (bridge stay cables in wind): parametric excitation Blade vibration from rotor (parametric from rotation speed): fan/turbine blade stability

Response in unstable region: x(t) grows exponentially until nonlinearity limits amplitude Saturation amplitude from Duffing + Mathieu combined equation

Van der Pol Oscillator (Self-Excited)

Equation: ẍ - ε(1 - x²)ẋ + x = 0 [ε = nonlinear damping coefficient]

Negative damping for |x| < 1: energy input → oscillation grows Positive damping for |x| > 1: energy dissipated → oscillation decays Limit cycle: stable; amplitude ≈ 2; frequency ≈ ω₀ = 1 (for small ε)

Engineering analogy: Aerodynamic flutter: negative aerodynamic damping → limit cycle oscillation Brake squeal: friction-velocity slope negative → self-excited vibration (van der Pol type) Stick-slip friction oscillation: relaxation oscillator (large ε limit)

Limit cycle amplitude (energy balance): ε(1 - A²/4) = 0 → A = 2 (exact for MMS/HBM)

Chaos in Nonlinear Systems

Melnikov Criterion

For slowly forced, lightly damped Duffing oscillator: Transition to chaos when Melnikov function M(t₀) has simple zeros: M(t₀) = Σ residues - γ/ε × Integral of sech²

Practical criterion: F₀/c > threshold(ω) from Melnikov → chaos possible (homoclinic tangency) In practice: use Poincaré sections, Lyapunov exponents, bifurcation diagrams

Routes to chaos: Period-doubling cascade (Feigenbaum): period-1 → period-2 → period-4 → ... → chaos Quasiperiodicity → chaos: two incommensurate frequencies → torus breakdown Intermittency: near-periodic bursts of chaotic behavior

Identification: Lyapunov exponent λ > 0 → chaotic (exponential divergence of nearby trajectories) Correlation dimension D₂ < integer → strange attractor Recurrence plot: block-like structures vs. scattered = regular vs. chaotic

Nonlinear Normal Modes (NNMs)

Definition (Shaw-Pierre)

NNM: periodic orbit of the conservative nonlinear system; generalizes linear normal mode Shaw-Pierre NNM: invariant manifold in phase space; 2D surface for each mode pair On manifold: dynamics reduce to 2D (amplitude + phase evolve on manifold)

NNM frequency-energy plot (backbone curve): Frequency varies with energy (amplitude); each mode has its own backbone Modal interactions: two NNMs can interact (internal resonance at specific energy levels)

Internal resonance: ω_n = n × ω_m (for integer n): modal coupling → energy exchange between modes 1:3 internal resonance: third harmonic of mode m = fundamental of mode n Result: localization, anomalous frequency response, modal saturation

Computation

Time integration: continuation from low-amplitude (linear) to high amplitude; track periodic orbits Shooting method: find initial conditions that return to same state after one period T Newton-Raphson: F(x₀) = x(T, x₀) - x₀ = 0 → iterate on x₀

Software: MATCONT, AUTO, MANLAB (ANM continuation), NNM toolbox (Peeters/Kerschen)

Experimental Identification

Nonlinear Identification Methods

Restoring force surface (RFS): mẍ = F_external - F_nl(x, ẋ) [measure ẍ, x, ẋ → infer F_nl surface] Plot F_nl vs. (x, ẋ) → identify nonlinear stiffness and damping characteristics

Hilbert transform: Instantaneous amplitude A(t) and frequency ω_inst(t) from analytic signal Backbone: A(t) vs. ω_inst(t) from free-decay → directly gives backbone curve

Wavelet transform: Time-frequency representation; track frequency as function of amplitude during decay Reveals frequency-energy dependence of each mode

CONCERTO / NIFO (Nonlinear Identification Force-Output): Frequency-domain identification; accounts for higher harmonics Useful for multi-DOF systems with localized nonlinearities

Standards and References

Source	Scope
Nayfeh & Mook "Nonlinear Oscillations" (1979)	Classic MMS and HBM reference
Guckenheimer & Holmes "Nonlinear Dynamics and Chaos"	Bifurcation theory
Kerschen et al. (2009) Mech. Syst. Signal Process.	NNM review paper
Ewins "Modal Testing"	Experimental modal analysis (including NL ID)
Thomsen "Vibrations and Stability"	Engineering nonlinear vibration textbook

Output

Provide: nonlinearity type (geometric/contact/friction/aerodynamic) and location in structure, equation of motion (with nonlinear term; dimensional form; parameters m, c, k, k₃, F₀, Ω), analysis method (HBM/MMS/numerical integration; justification: ε magnitude), backbone curve equation (ω_r vs. A; [rad/s] and [mm] units), frequency response: amplitude A [mm] vs. Ω/ω₀ (identify jump frequencies if applicable), jump frequency (up-sweep: Ω_jump_up/ω₀; down-sweep: Ω_jump_down/ω₀), check for parametric resonance (if time-varying stiffness: Mathieu stability; critical ε at operating δ), check for chaos (Melnikov criterion or Lyapunov exponent estimate), NNM energy-frequency backbone at relevant amplitude [J], internal resonance check (integer ratio between modes), identification method recommended (RFS/Hilbert/wavelet for experimental), and applicable reference (Nayfeh 1979, Kerschen 2009, Guckenheimer 1983).