Global detection of detached periodic solution branches of friction-damped mechanical systems

Heinze, Torsten; Scheidt, Lars Panning‐von; Wallaschek, Jörg

doi:10.1007/s11071-019-05425-4

Cited by 16 publications

(13 citation statements)

References 87 publications

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“…It is next assumed that the response of the structure with external forces (f (1) , c (1) ) is dominated by a single nonlinear mode (single-nonlinear-mode theory) of the structure without external forces. As mentioned before, this scenario is typical under near-resonant excitation, self-excitation or combinations thereof [17], provided that strong nonlinear modal interactions are absent. In these cases, the structure behaves like a nonlinear single-degree-of-freedom oscillator.…”

Section: Nonlinear Modal Analysis and Reduction To A Single Modal Oscillatormentioning

confidence: 79%

“…In particular, this is the case if the mechanical system is externally driven near a well-separated primary resonance, which can be formally argued using perturbation theory [14]. This may also be the case under self-excitation [15,16], or combinations of self-and forced excitation [17]. When the vibration is dominated by a single nonlinear mode, the dynamics take place on a two-dimensional invariant manifold in state space [18,19,20], and the mechanical system behaves like a single-degree-of-freedom oscillator.…”

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confidence: 99%

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Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Krack

2021

Journal of Sound and Vibration

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Under certain conditions, the dynamics of a nonlinear mechanical system can be represented by a single nonlinear modal oscillator. This holds, in particular, under external excitation near primary resonance or under self-excitation by negative damping of the respective mode. The properties of the modal oscillator can be determined by computational or experimental nonlinear modal analysis. The simplification to a singlenonlinear-mode model facilitates qualitative and global analysis, and substantially reduces the computational effort required for probabilistic methods and design optimization. Important limitations of this theory are that only purely mechanical systems can be analyzed and that the respective nonlinear mode has to be recomputed when the system's structural properties are varied. With the theoretical extension proposed in this work, it becomes feasible to attach linear subsystems to the primary mechanical system, and to approximate the dynamics of this coupled system using only the nonlinear mode of the primary mechanical system. The attachments must be described by linear ordinary or differential-algebraic equations with time-invariant coefficient matrices. The attachments do not need to be of purely mechanical nature, but may contain, for instance, electric, magnetic, acoustic, thermal or aerodynamic models. This considerably extends the range of utility of nonlinear modes to applications as diverse as model updating or vibration energy harvesting. As long as the attachments do not significantly deteriorate the host system's modal deflection shape, it is shown that their effect can be reduced to a complex-valued modal impedance and an imposed modal forcing term. In the present work, the proposed approach is computationally assessed for the analysis of exciter-structure interaction. More specifically, the force drop typically encountered in frequency response testing is revisited. A cantilevered beam with cubic spring and an attached electro-dynamical shaker serves as benchmark. The proposed approach shows excellent accuracy. Mainly the already known limitations of single-nonlinear-mode theory reappear. In particular, higher harmonics should not be too pronounced. In the transient case, the time scales of vibration and amplitude-phase modulation should be well separated, and the attachment dynamics should be in quasi-steady state.

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Section: Nonlinear Modal Analysis and Reduction To A Single Modal Oscillatormentioning

confidence: 79%

mentioning

confidence: 99%

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Krack

2021

Journal of Sound and Vibration

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“…Therefore, the crossing solution between the forced response and the underlying dNNM can be effectively predicted. A situation with multiple crossing solutions might also happen in the emergence of the isola [4,6], the existence of isola is not considered in this work. Predicting the isola is also achievable for E-EBM [29].…”

Section: Nonlinear Modal Synthesismentioning

confidence: 99%

Comparison of different methodologies for the computation of damped nonlinear normal modes and resonance prediction of systems with non-conservative nonlinearities

et al. 2021

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The nonlinear modes of a non-conservative nonlinear system are sometimes referred to as damped nonlinear normal modes (dNNMs). Because of the non-conservative characteristics, the dNNMs are no longer periodic. To compute non-periodic dNNMs using classic methods for periodic problems, two concepts have been developed in the last two decades: complex nonlinear mode (CNM) and extended periodic motion concept (EPMC). A critical assessment of these two concepts applied to different types of non-conservative nonlinearities and industrial full-scale structures has not been thoroughly investigated yet. Furthermore, there exist two emerging techniques which aim at predicting the resonant solutions of a nonlinear forced response using the dNNMs: extended energy balance method (E-EBM) and nonlinear modal synthesis (NMS). A detailed assessment between these two techniques has been rarely attempted in the literature. Therefore, in this work, a comprehensive comparison between CNM and EPMC is provided through two illustrative systems and one engineering application. The EPMC with an alternative damping assumption is also derived and compared with the original EPMC and CNM. The advantages and limitations of the CNM and EPMC are critically discussed. In addition, the resonant solutions are predicted based on the dNNMs using both E-EBM and NMS. The accuracies of the predicted resonances are also discussed in detail.

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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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Global detection of detached periodic solution branches of friction-damped mechanical systems

Cited by 16 publications

References 87 publications

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Comparison of different methodologies for the computation of damped nonlinear normal modes and resonance prediction of systems with non-conservative nonlinearities

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

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