Experimental bifurcation analysis of an impact oscillator—Determining stability

Bureau, Emil; Schilder, Frank; Elmegård, Michael; Santos, Ilmar; Thomsen, Jon Juel; Starke, Jens

doi:10.1016/j.jsv.2014.05.032

Cited by 32 publications

(29 citation statements)

References 10 publications

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“…We observe the typical response curve of a spring-mass-damper system with stiffening spring, exhibiting a large window of hysteresis and multi-stability. The bifurcation diagram in Figure 3(b) was obtained with the most recent version of our toolbox, which allows assessing stability of a response [2]. Stability is indicated in gray-scale along the response curve, black corresponds to asymptotically stable responses, while gray indicates instability.…”

Section: A Mechanical Non-linear Oscillator Experimentsmentioning

confidence: 99%

“…Stability is indicated in gray-scale along the response curve, black corresponds to asymptotically stable responses, while gray indicates instability. Extensive studies of the dynamics of this experiment and of a fitted one-degree of freedom model of the oscillator were conducted in [1,2,18].…”

Section: A Mechanical Non-linear Oscillator Experimentsmentioning

confidence: 99%

“…Our interface to the special case of control based continuation implements several explicit stability tests; see [2]. These tests require significant additional measurement time and an interesting question is the development of methods that do not require additional measurements.…”

Section: A Zero Problem For Continuation In Experimentsmentioning

confidence: 99%

“…In [1,2] we have implemented and tested bifurcation analysis directly in a non-linear mechanical oscillator experiment using control based continuation [3,4,5,6], that is, tracking stable and unstable responses under variations of forcing frequency and amplitude. In these experiments we used a harmonically excited impact oscillator with control forces applied via electromagnetic actuators as a test specimen and developed methods for tuning the control parameters and for determining stability of steady-state responses.…”

Section: Introductionmentioning

confidence: 99%

“…Here, we will provide information about the details of the implementation of the toolbox CONTINEX developed for this type of experiment as well as for equation-free analysis [7,8,9,10,11]. We will motivate the algorithmic developments with new experimental results collected with the set-up used in [1,2]. The presented methods are able to extract bifurcation diagrams directly from strongly non-linear experiments, and the toolbox CONTINEX is freely available for download [12].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

Schilder

Bureau

Santos

et al. 2015

Journal of Sound and Vibration

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Noise contaminated zero problems involve functions that cannot be evaluated directly, but only indirectly via observations. In addition, such observations are affected by a non-deterministic observation error (noise). We investigate the application of numerical bifurcation analysis for studying the solution set of such noise contaminated zero problems, which is highly relevant in the context of equation-free analysis (coarse grained analysis) and bifurcation analysis in experiments, and develop specialized algorithms to address challenges that arise due to the presence of noise. As a working example, we demonstrate and test our algorithms on a mechanical non-linear oscillator experiment using control based continuation, which we used as a main application and test case for development of the the Coco compatible Matlab toolbox Continex that implements our algorithms.

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Section: A Mechanical Non-linear Oscillator Experimentsmentioning

confidence: 99%

Section: A Mechanical Non-linear Oscillator Experimentsmentioning

confidence: 99%

Section: A Zero Problem For Continuation In Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

Schilder

Bureau

Santos

et al. 2015

Journal of Sound and Vibration

Self Cite

View full text Add to dashboard Cite

show abstract

Continuation methods for lab experiments of nonlinear vibrations

2023

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In this work, we will give an overview of our recent progress in experimental continuation. First, three different approaches are explained and compared which can be found in scientific papers on the topic. We then show S‐Curve measurements of a Duffing oscillator experiment for which we derived optimal controller gains analytically. The derived formula for stabilizing PD‐controller gains makes trial and error search for suitable values unnecessary. Since feedback control introduces higher harmonics in the driving signal, we consider a harmonization of the forcing signal. This harmonization is important to reduce shaker‐structure interaction in the treatment of nonlinear frequency responses. Finally, the controlled nonlinear testing and harmonization is enhanced by a continuation algorithm adapted from numerical analysis and applied to a geometrically nonlinear beam test rig for which we measure the nonlinear forced response directly in the displacement‐frequency plane.

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Isola in a linear one‐degree‐of‐freedom feedback system with actuator rate saturation

Nguyen

Hill

Lowenberg

2023

Int Journal of Mech Sys Dyn

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This short communication uses numerical continuation to highlight the existence of an isola in a simple one‐degree‐of‐freedom harmonically forced feedback system with actuator rate limiting as its only nonlinear element. It was found that the isola (1) contains only rate‐limited responses, (2) merges with the main branch when the forcing amplitude is sufficiently large, and (3) includes stable solutions that create a second attractor in regions where rate limiting is not expected. Furthermore, the isola is composed of two solutions for a given forcing frequency. These solutions have the same amplitudes in the state (pitch rate) projection; however, they have distinct phases, and their amplitudes are also distinct when projected onto the integrator state in the controller. The rich dynamics observed in such a simple example underlines the impact of rate limiting on feedback systems. Specifically, the combination of feedback and rate limiting can create detrimental dynamics that is hard to predict and requires careful analysis.

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Experimental bifurcation analysis of an impact oscillator—Determining stability

Cited by 32 publications

References 10 publications

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

Continuation methods for lab experiments of nonlinear vibrations

Isola in a linear one‐degree‐of‐freedom feedback system with actuator rate saturation

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