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Nonlinear modal analysis of nonconservative systems: Extension of the periodic motion concept

Malte Krack

arXiv:2101.00949v1nlin.CDcs.CE

TL;DR

Nonconservative autonomous motions are typically not periodic, limiting the classical periodic-motion definition of nonlinear modes. The paper enforces periodicity with added mass-proportional damping, and reports that the resulting modal properties reflect resonant vibration behavior over a wide range.

  • Problem

    Nonconservative autonomous motions are typically unsteady rather than periodic, so the classical periodic-motion definition does not directly characterize their resonant vibration behavior.

    Source: §1 · p. 1, §2.1 · p. 5

  • Method

    The paper defines nonlinear modes as periodic motions enforced in nonconservative systems by mass-proportional damping or self-excitation.

    Source: §2.1 · p. 6, §6 · p. 31

  • Results

    The extracted modal properties accurately describe vibration behavior near forced resonances and self-excited limit cycles across a wide range.

    Source: Modal characteristics · p. 16, §6 · p. 32

  • Takeaways & Limitations

    The energy-dependent modal properties provide relevant measures for nonlinear structural design, including a direct measure of damping effectiveness.

    Source: §6 · p. 32

    The approach is restricted to isolated modes without strong modal interactions and does not capture quasi-periodic or chaotic motions.

    Source: §2.2 · p. 7

Abstract

from arXiv · show

As the motions of nonconservative autonomous systems are typically not periodic, the definition of nonlinear modes as periodic motions cannot be applied in the classical sense. In this paper, it is proposed 'make the motions periodic' by introducing an additional damping term of appropriate sign and magnitude. It is shown that this generalized definition is particularly suited to reflect the periodic vibration behavior induced by harmonic external forcing or negative linear damping. In a large range, the energy dependence of modal frequency, damping ratio and stability is reproduced well. The limitation to isolated or weakly-damped modes is discussed.

1. Introduction

The paper extends periodic-motion nonlinear modes to dissipative systems, targeting persistent resonant vibrations and assessing how the added damping affects modal properties and scope.

1. Introduction

  • Nonlinear modal analysis characterizes energy-dependent vibration behavior and can support understanding modal interactions and model reduction, but established definitions lose useful properties when nonlinear effects matter.The introduction contrasts periodic motions with invariant manifolds; periodic-motion modes do not generally apply to nonconservative autonomous systems.Source: §1 · pp. 1–3

2.1. Periodic vs. damped motion concept

  • The periodic-motion concept targets persistent vibrations at resonance, including responses induced by harmonic forcing or negative modal damping.Nonconservative autonomous motions are typically unsteady, and their frequency-energy characteristic need not match the frequency-response backbone.Source: §2.1 · pp. 5–6

2.2. A new definition of nonlinear modes

  • An added mass-proportional damping term compensates nonconservative forces to enforce periodicity, but may cause artificial modal coupling and limits use to isolated modes without strong interactions.The concept is restricted to periodic motions and is not useful for capturing quasi-periodic or chaotic motions.Source: §2.2 · pp. 6–7

2. Extension of the periodic motion concept to dissipative systems

  • The extension defines nonconservative nonlinear modes as periodic motions enforced by mass-proportional damping or self-excitation.The paper outlines Shooting and Harmonic Balance implementations and assesses capabilities and limitations through numerical examples.Source: §1 · pp. 3–4, §2.2 · p. 6

2.3. Comparison to force appropriation

  • Unlike force appropriation, the method needs no iterative adjustment of forcing phase or frequency and directly yields a damping measure.The paper attributes automatic resonance tuning to positive velocity feedback.Source: §2.3 · p. 7

3. Computational procedures

The computational procedures use Harmonic Balance to represent periodic modal motion with truncated Fourier components and solve for the modal properties across amplitudes.

3. Computational procedures

  • The study considers Harmonic Balance and Shooting as procedures for analyzing the limit cycles defined by the extended nonlinear modes.The two procedures are based on standard limit-cycle analysis methods.Source: §3 · p. 8

3.1. Harmonic Balance

  • Harmonic Balance truncates a Fourier-series representation at harmonic order H, with base frequency ω0, and solves for modal amplitude, frequency, harmonics, and damping.The complex-valued harmonics allow phase lag between coordinates, and the Fourier basis makes the approximation periodic.Source: §3.1 · p. 8
  • Mass and phase normalization constrain the Harmonic Balance eigenproblem, while the formulation used considers super-harmonics but not sub-harmonics.Kinetic-energy normalization is noted as an alternative suited to internal resonances.Source: §3.1 · pp. 8–9

Evaluation of the forces

The alternating-frequency-time scheme evaluates nonlinear force harmonics through repeated transformations between frequency and time domains.

Evaluation of the forces

  • Because nonlinear forces generally lack a closed-form relationship to coordinate harmonics and frequency, the scheme computes their harmonics numerically.The force harmonics depend on the coordinate harmonics and eigenfrequency.Source: Evaluation of the forces · p. 9
  • Each solver iteration transforms coordinate harmonics to time-domain coordinates and velocities, evaluates forces there, and transforms the forces back into harmonics.The scheme applies IFFT before time-domain force evaluation and FFT afterward.Source: Evaluation of the forces · p. 10
  • The alternating-frequency-time scheme accommodates arbitrary nonlinear and non-smooth forces.The force law is evaluated directly in the time domain.Source: Evaluation of the forces · p. 10

Numerical solution

The numerical procedures solve the modal equations with a nonlinear solver, assess stability from Hill eigenvalues, and use a boundary-value formulation for Shooting.

Numerical solution

  • The Harmonic Balance problem is solved with a Newton-method variant using an analytically evaluated Jacobian, with sparsity exploitable for localized nonlinearities.Amplitude is logarithmically scaled, and path continuation addresses turning points.Source: Numerical solution · pp. 10–11

Stability analysis

  • A nonlinear mode is unstable when any Hill eigenvalue has a positive real part.Stability is analyzed a posteriori through a quadratic eigenvalue problem whose coefficient matrices depend on the solution of Eq. (4).Source: Stability analysis · pp. 11–12

3.2. Shooting

  • The Shooting method determines periodic solutions by solving a boundary-value problem, prescribing the initial displacement and velocity of one coordinate.The method treats velocity as the time derivative of displacement.Source: §3.2 · p. 12

Numerical solution

The numerical solution uses nonlinear algebraic equations, path continuation, and time integration; periodic-motion stability is assessed using Floquet theory.

Numerical solution

  • Eq. (7) is solved by Newton iteration with path continuation, integrating states over each period with the implicit average-acceleration Newmark scheme.The passage notes that Runge–Kutta also works and that temporal symmetries can reduce integration cost.Source: Numerical solution · p. 12
  • Fourier-transforming the shooting solution makes its modal quantities comparable with Harmonic Balance results, whose amplitudes are not directly comparable.The procedure also derives quantities such as maximum coordinate displacement and kinetic energy from frequency- and time-domain results.Source: Numerical solution · pp. 12–13

Stability analysis

  • Floquet stability analysis uses monodromy-matrix eigenvalues, with the matrix available from the Jacobian computed during integration.Source: Stability analysis · p. 13

4. Numerical examples

The examples compare nonlinear-mode concepts in a van-der-Pol oscillator, emphasizing where their modal characteristics differ and the behaviors each concept is designed to represent.

4. Numerical examples

  • The methodology is tested on a van-der-Pol oscillator, a viscously damped two-degree-of-freedom system, and a friction-damped rod.The van-der-Pol case is used to contrast periodic-motion and damped-motion concepts and examine strong-damping limitations.Source: §4 · p. 13, §4.1 · p. 13

Modal characteristics

  • For a single-degree-of-freedom oscillator, Harmonic Balance reduces to harmonic linearization at H = 1, and the methods coincide at D = 0.Source: Modal characteristics · p. 14
  • At large amplitudes, modal frequency-amplitude characteristics and phase curves differ between concepts, especially for D > 20%.The comparison uses Harmonic Balance results at H = 1 and H = 13 and damped-motion results; Fig. 3 shows eigenfrequency, damping ratio, and phase curves.Source: Modal characteristics · pp. 14–15
  • Periodic-motion modes target nearly periodic responses under forcing or self-excitation, whereas damped-motion modes target autonomous decay or growth.Source: Modal characteristics · p. 16

Free decay

The examples show that the periodic-motion concept reproduces limit cycles under varying linear damping, while free-decay accuracy depends on initial energy; a two-springs system illustrates damping-sensitive nonlinear behavior.

Free decay

  • Free-decay envelopes agree reasonably at lower initial energy but are inaccurate at higher energy, regardless of which modal characteristics are used.The amplitude equation captures approximate slow dynamics because averaging is applied to the faster oscillation.Source: Free decay · p. 16

Limit cycles for varying linear damping

  • Periodic-motion predictions match reference limit-cycle amplitude, frequency, and phase curves, while damped-motion predictions agree poorly for α ≫ 0.The comparison assumes nonlinear and linear damping can be superimposed and uses modal characteristics computed for α = 0; the damped-motion frequency dependence is not qualitatively correct.Source: Limit cycles for varying linear damping · pp. 17–18

4.2. Two-springs system with viscous damping

  • In the two-springs system, small-amplitude orthogonal motions are decoupled, while geometric nonlinearity couples them as amplitudes grow; damping can shift hardening to softening behavior.Source: §4.2 · p. 19

Modal characteristics

In the two-springs system, the first mode’s frequency first rises slightly and then falls with kinetic energy, while damping and interaction with the second mode shape its qualitative behavior. Harmonic Balance and Shooting agree when enough harmonics are retained, and folding poses a parameterization issue for invariant-manifold methods but no special difficulty for the proposed procedure.

Modal characteristics

  • The first mode’s eigenfrequency rises slightly and then decreases with kinetic energy, while damping strongly influences its qualitative behavior.The modal damping ratio is comparatively small, and the damping distribution appears more important than its extent.Source: Modal characteristics · pp. 20–21, Figure 7 · p. 21
  • At larger energies, near-2:1 interaction between the first two modes makes higher harmonics important to the evolution of modal properties.The linear frequencies are given as ω1 = 1.13 and ω2 = 2; u2 has two maxima and minima per cycle while u1 has one of each.Source: Modal characteristics · pp. 20–21
  • Harmonic Balance and Shooting become indistinguishable with enough time steps and harmonics, while retaining only H = 1 predicts softening alone.Several harmonics are needed for asymptotic convergence of the modal characteristics.Source: Modal characteristics · p. 21
  • Invariant-manifold folding can cause parameterization problems, but it requires no special treatment in the proposed procedure.Figure 7 identifies the invariant manifolds before and beyond the fold.Source: Modal characteristics · pp. 21–22, Figure 7 · p. 21

Forced response

For the two-springs system, forced-response maxima track the nonlinear-mode backbone across depicted excitation levels, while stability can be lost on response branches. The section also introduces a friction-damped rod model and the performance measures relevant to its forced and self-excited use.

Forced response

  • Frequency-response maxima match the nonlinear-mode backbone across depicted excitation levels, despite high-level resonant instability and invariant-manifold folding.With comparatively small modal damping, the frequency difference between forced-response maxima and undamped eigenfrequency is not visible in the plotted curves.Source: Forced response · p. 22, Figure 8 · p. 23
  • The response branch between its two turning points is unstable; a Floquet eigenvalue crossing at +1 indicates a saddle-node bifurcation.Above a certain excitation level, a −1 crossing indicates period doubling; transients from points A and B reach different outcomes, including a 2Ωe orbit and a possibly quasi-periodic attractor.Source: Forced response · pp. 23–24
  • The nonlinear-mode stability analysis agrees with forced-response stability loss near (Ωe ≈ 1.1, u1,max ≈ 0.24), supporting its use for 1:1 external resonances.Floquet stability analysis is described as an inexpensive post-processing step with Shooting.Source: Forced response · pp. 22, 24

4.3. Friction-damped rod

  • For friction-damped design, resonance-frequency shift matters under external forcing, while effective damping ratio is assessed under self-excitation.The proposed method is presented as an alternative to widely used forced-response simulations for determining these measures.Source: §4.3 · pp. 24–25
  • The rod example models a slender rod clamped at one end and coupled to an elastic Coulomb friction element at the other, using small longitudinal vibrations and linear-elastic material assumptions.Its friction force is written in differential form, with contact stiffness kt and limit friction force µN; the structural dynamics are approximated with 20 equal-length finite elements.Source: §4.3 · p. 25

Modal characteristics

In the friction-damped rod, sticking gives a constant frequency and zero damping at low energy; slipping then produces softening and damping, while high-energy frequency approaches that of the friction-free linear rod. The invariant-manifold orbits also show higher harmonics and energy-dependent mode shape, with Shooting and Harmonic Balance in quantitative agreement.

Modal characteristics

  • Low-energy sticking produces constant eigenfrequency and zero damping; once slipping begins, friction causes softening and damping.Kinetic energy is normalized by its value at the first occurrence of slipping.Source: Modal characteristics · pp. 26–27
  • At large energies, modal frequency approaches that of the linear rod without friction contact, while the effective damping ratio tends toward zero.Sliding friction dominates, and dissipated energy grows linearly with displacement amplitude, unlike the quadratic growth for a linear viscous damper.Source: Modal characteristics · p. 27
  • Invariant-manifold orbits depart from ellipses and planes across energy levels, indicating higher harmonics and an energy-dependent mode shape.The behavior is influenced by the nonsmooth dry-friction forces; the manifolds are shown in a three-dimensional subspace of the 40-dimensional phase space.Source: Modal characteristics · pp. 26–27, Figure 10 · p. 26
  • Shooting and Harmonic Balance agree quantitatively, and frequency- and damping-energy characteristics do not significantly depend on harmonic order H.The invariant manifold still reveals the motions’ multi-harmonic character.Source: Modal characteristics · p. 27

Free decay

The study compares free-decay and forced-response behavior with nonlinear-mode predictions, then evaluates the computational effort of the analysis procedures. The results include good agreement in much of the free decay, forced-response trends consistent with the modal backbone, and lower computational cost for Harmonic Balance than Shooting.

Free decay

  • The nonlinear-mode approximation gives reasonably good free-decay agreement, but deviations arise around p > 35 as stick-slip transitions excite frequencies the single-mode approximation cannot resolve.The comparison uses synthesized tip displacement and full-equation time histories; the authors conjecture that non-smooth stick-slip transitions are important in this regime.Source: Free decay · pp. 27–28, Figure 11 · p. 28

Forced response

  • The forced-response maxima agree quantitatively with the nonlinear mode’s frequency-amplitude backbone, while the response shows softening and narrower resonance peaks at higher energy.The narrower peaks accompany a reported decrease in damping effect.Source: Forced response · p. 29, Figure 12 · p. 29

5. Evaluation of the computational performance

  • For both test cases, sufficiently accurate results were produced within several seconds, and almost all analyses remained below one minute.Harmonic Balance was considerably less expensive than Shooting in the calculations carried out.Source: §5 · pp. 30–31

6. Conclusions

The paper defines nonconservative nonlinear modes as self-excited limit cycles induced by mass-proportional damping, designed to capture periodic dynamics associated with resonant excitation. The resulting modal properties describe vibration behavior in stated regimes, but artificial damping and non-periodicity limit the approach.

6. Conclusions

  • The approach uses artificial mass-proportional damping to define nonconservative nonlinear modes as self-excited limit cycles and applies standard periodic-motion analysis methods.The described procedures use Harmonic Balance and Shooting and accommodate generic nonlinearities, including non-smooth forces.Source: §6 · pp. 31–32
  • Across a wide range, extracted modal frequency, damping ratio, deflection shape, and asymptotic stability accurately describe vibration near forced resonances and self-excited limit cycles.The concept is fully consistent with conservative nonlinear and nonconservative linear cases.Source: §6 · p. 32
  • Artificial damping may induce modal coupling in nonconservative nonlinear systems, restricting the concept to isolated modes or regimes with reasonably low modal damping; non-periodic regimes are generally not captured.Source: §6 · p. 32
  • Energy-dependent modal properties provide measures for structural-dynamic design, including damping effectiveness, and can support model reduction for vibration prediction across operating conditions.The stated model-reduction use is supported by previous works.Source: §6 · pp. 32–33
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