The extended periodic motion concept for fast limit cycle detection of self-excited systems

Jahn, Martin; Stender, Merten; Tatzko, Sebastian; Hoffmann, Norbert; Grolet, Aurélien; Wallaschek, Jörg

doi:10.1016/j.compstruc.2019.106139

Cited by 12 publications

(16 citation statements)

References 36 publications

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“…Here, the damping parameters are reduced by a factor of 10 to d x = d y = d z = d lin = 0.002. This system configuration has already been studied in [13,14] where the authors found an isolated solution branch resulting from the damping variation. Figure 7 (a) displays the bifurcation diagram for the horizontal stiffness k x .…”

Section: Bi-stable Oscillator With Isolated Periodic Solutionmentioning

confidence: 94%

“…For the nonlinear joint element, a cubic stiffness nonlinearity k nl is chosen [11]. The equations of motion and parameter values are given in Appendix B and the model is displayed in branches [11,13,14]. In this study, a variation of the horizontal stiffness k x is performed.…”

Section: Bi-stable Oscillator With Mode-couplingmentioning

confidence: 99%

“…Sub-critical Hopf bifurcations [11,12] and isolated solution branches [13][14][15] are a common observation in those systems, such that bi-and multi-stable systems have been reported numerously [14,16,17]. The computation of those nonlinear responses (periodic, quasi-period orbits, chaotic trajectories) is a well-established field of research [18][19][20][21], mostly resulting in the identification of complicated bifurcation diagrams [11,13,[22][23][24]. The stability of the solutions is usually assessed by local Lyapunov-type stability metrics [25,26].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stender¹,

Hoffmann²,

Papangelo³

2020

Preprint

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Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e. they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability has been proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

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Section: Bi-stable Oscillator With Isolated Periodic Solutionmentioning

confidence: 94%

Section: Bi-stable Oscillator With Mode-couplingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stender¹,

Hoffmann²,

Papangelo³

2020

Preprint

Self Cite

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“…The E-PMC is thus an excellent choice for limit cycle detection of self-excited systems with non-conservative nonlinear forces. Details considering implementation and further results can be found in [6] and [7]. Limit cycles of non-conservative nonlinear systems can be represented by 1D-submanifolds for a certain parameter variation.…”

Section: Concept Of Nonlinear Modesmentioning

confidence: 99%

Limit cycle computation of self‐excited dynamic systems using nonlinear modes

Tatzko

Stender

Jahn

et al. 2021

Proc Appl Math and Mech

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A self‐excited dynamic system is able to oscillate periodically by itself. Corresponding solutions of the autonomous differential equation are called limit cycles or periodic attractors. To find these solutions, a simple approach would be brute‐force search for the corresponding basins of attraction. However, grid searching might become unfeasible with increasing number of degrees of freedom. Instead, solution path continuation techniques are often used to keep computational costs low. As the continuation of solution branches and their bifurcations provides only solutions which are connected to each other, isolas and detached branches are missed out. We present a method for fast limit cycle detection of self‐excited systems with isolas based on nonlinear modes. A nonlinear mode, often referred to as nonlinear normal mode, is defined as a periodic motion of the undamped and unforced mechanical system. For nonconservative systems however, e.g. with friction nonlinearity, damping cannot be neglected as it is characteristic for the oscillators nonlinear dynamics. Therefore, the Extended Periodic Motion Concept (E‐PMC) was proposed recently to find periodic solutions of nonconservative nonlinear systems. In this work, the E‐PMC is applied to self‐excited dynamic systems in order to find periodic attractors along its nonlinear modes. Zero crossings of the nonlinear damping curve indicate autonomous solutions which can be used as starting points for single parameter continuation. Thus, solutions corresponding to the main branch and detached curves in the solution space are connected by nonlinear modes. The proposed method is applied to a frictional oscillator with cubic stiffness and proves to be robust in the search for isolated periodic solutions that are already known from literature.

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“…In particular, this is the case if the mechanical system is externally driven near a well-separated primary resonance. This may also be the case under self-excitation [14,15], or combinations of self-and forced excitation [16]. When the vibration is dominated by a single nonlinear mode, the dynamics take place on a two-dimensional invariant manifold in state-space [17][18][19], and the mechanical system behaves like a single-degree-of-freedom oscillator.…”

Section: Identification Of a Nonlinear-mode Modelmentioning

confidence: 99%

Numerical Assessment of Polynomial Nonlinear State-Space and Nonlinear-Mode Models for Near-Resonant Vibrations

et al. 2020

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In the present article, we follow up our recent work on the experimental assessment of two data-driven nonlinear system identification methodologies. The first methodology constructs a single nonlinear-mode model from periodic vibration data obtained under phase-controlled harmonic excitation. The second methodology constructs a state-space model with polynomial nonlinear terms from vibration data obtained under uncontrolled broadband random excitation. The conclusions drawn from our previous work (experimental) were limited by uncertainties inherent to the specimen, instrumentation, and signal processing. To avoid these uncertainties in the present work, we pursued a completely numerical approach based on synthetic measurement data obtained from simulated experiments. Three benchmarks are considered, which feature geometric, unilateral contact, and dry friction nonlinearity, respectively. As in our previous work, we assessed the prediction accuracy of the identified models with a focus on the regime near a particular resonance. This way, we confirmed our findings on the strengths and weaknesses of the two methodologies and derive several new findings: First, the state-space method struggles even for polynomial nonlinearities if the training data is chaotic. Second, the polynomial state-space models can reach high accuracy only in a rather limited range of vibration levels for systems with non-polynomial nonlinearities. Such cases demonstrate the sensitivity to training data inherent in the method, as model errors are inevitable here. Third, although the excitation does not perfectly isolate the nonlinear mode (exciter-structure interaction, uncontrolled higher harmonics, local instead of distributed excitation), the modal properties are identified with high accuracy.

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The extended periodic motion concept for fast limit cycle detection of self-excited systems

Cited by 12 publications

References 36 publications

The Basin Stability of Bi-Stable Friction-Excited Oscillators

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Limit cycle computation of self‐excited dynamic systems using nonlinear modes

Numerical Assessment of Polynomial Nonlinear State-Space and Nonlinear-Mode Models for Near-Resonant Vibrations

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