Comparison of different harmonic balance based methodologies for computation of nonlinear modes of non-conservative mechanical systems

Jahn, Martin; Tatzko, Sebastian; Scheidt, Lars Panning‐von; Wallaschek, Jörg

doi:10.1016/j.ymssp.2019.03.005

Cited by 24 publications

(7 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The artificial negative damping term −2DωMq compensates the natural dissipation (in average over a vibration cycle). The EPMC simplifies to the conventional periodic motion concept in the conservative case (where D = 0), and is consistent with the linear case under modal damping [29]. If, at the same time, more than one linear mode shape contributes strongly to the nonlinear mode, and damping is not light (i. e. |D| not 1), the artificial term will generally distort the modal coupling [28].…”

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

confidence: 56%

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

Krack

2021

Journal of Sound and Vibration

View full text Add to dashboard Cite

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.

show abstract

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

confidence: 56%

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

Krack

2021

Journal of Sound and Vibration

View full text Add to dashboard Cite

show abstract

“…The artificial negative damping term −2DωMq compensates the natural dissipation (in average over a vibration cycle). The EPMC simplifies to the conventional periodic motion concept in the conservative case, and is consistent with the linear case under modal damping [21]. If, at the same time, more than one linear mode contributes strongly to the nonlinear mode, and damping is not light (i.e., |D| not 1), the artificial term will generally distort the modal coupling [20].…”

Section: Identification Of a Nonlinear-mode Modelmentioning

confidence: 68%

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

et al. 2020

View full text Add to dashboard Cite

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.

show abstract

“…The almost-period vibration is also analysed using HBM in [6][7][8]. The HBM is described in a number of papers [9][10][11][12] and monographs [13,14]. Due to the abundance of literature on the subject, only selected and most recent items are mentioned above.…”

Section: Introductionmentioning

confidence: 99%

“…The exponential version of HBM together with the continuation method was used in [11] to analyse periodic responses of rotor/stator dynamics. The discussed method was also utilized in papers [9,12] to discuss some problems of nonlinear dynamics. Using the exponential version of HBM, the steady state vibration is also analysed in [4].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

Lewandowski

2022

Meccanica

View full text Add to dashboard Cite

This paper presents the application of an exponential version of the harmonic balance method to the analysis of steady state vibration of geometrically nonlinear systems. A detailed description of the method and of the corresponding numerical procedure is provided. The von Karman theory is used to describe the effects of geometric nonlinearity. The material of the beams is modelled with the help of the Zener model using the fractional calculus. The problem is solved using an exponential version of the harmonic balance method. In the above-mentioned version, the complex calculus is used in contrast to the ordinary harmonic balance method, where the steady state vibrations are described with the help of the trigonometric functions. It significantly simplifies derivation of the amplitude equations. Moreover, the exponential version of the harmonic balance method is more elegant in comparison with the ordinary one. A detailed derivation of the amplitude equations is presented. The modified continuation method is proposed to solve the nonlinear amplitude equations and to determine the response curves. Moreover, the results of the exemplary calculation are presented and compared with known results in order to justify the efficiency and the correctness of the proposed approach.

show abstract

Comparison of different harmonic balance based methodologies for computation of nonlinear modes of non-conservative mechanical systems

Cited by 24 publications

References 6 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

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

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

Contact Info

Product

Resources

About