Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator

Cusumano, Joseph P.; Bai, Baojun

doi:10.1016/0960-0779(93)90003-j

Cited by 70 publications

(51 citation statements)

References 15 publications

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“…POD is known in other fields as Principal Components Analysis (PCA) [84] (with the mean subtracted out), the Hotelling transform [85], Empirical Orthogonal Functions (EOF) [86,87], and/or the Karhunen-Loéve (KL) decomposition [88]. The method has been utilized in a large and diverse set of scientific and engineering applications [2][3][4][5][6][7][8][9][10][11][12][13].…”

Section: The Proper Orthogonal Decomposition and Galerkin Projectionmentioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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Abstract. The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without physics-based governing equations. We review standard model reduction techniques such as Proper Orthogonal Decomposition (POD) with Galerkin projection and Balanced POD (BPOD). Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory. These data-driven methods can mitigate the challenge of not having a well-characterized set of governing equations. We illustrate that this equation-free approach that is being applied to measurement data from complex systems can be extended to include inputs and control. Three specific research examples are presented that extend current equation-free architectures toward the characterization and control of complex systems. These examples motivate a potentially revolutionary shift in the characterization of complex systems and subsequent design of objective-based controllers for data-driven models.

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Section: The Proper Orthogonal Decomposition and Galerkin Projectionmentioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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“…In particular, the bifurcation behaviour of two degree of freedom impact system with a single impact constraint has been studied in detail [6][7][8]. More general studies of multiple degree of freedom impact oscillators have also been carried out [9][10][11].…”

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confidence: 99%

Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator

Wagg

2004

Chaos, Solitons & Fractals

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We consider the rising phenomena which occur in sticking solutions of a two-degree of freedom impact oscillator. We describe a mathematical formulation for modelling such a systems during both free flight and during sticking solutions for each of the masses in the system.Simulations of the sticking solutions are carried out, and rising events are observed when the forcing frequency parameter is varied. We show how the time of sticking reduces significantly as a rising event occurs. Then within the sticking region we show how rising is qualitatively similar to the multi-sliding bifurcation for sliding orbits.

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“…Higher degree of freedom impact systems have also been considered by Cusumano & Bai [1993], who consider the dynamics associated with a ten degree of freedom impact oscillator and Babitsky [1998] who considers multi-degree of freedom and elastic systems subject to vibro-impact. Periodic impacting motions which occur in multi-degree of freedom impact systems with a single impact stop have been studied by Natsiavas [1993] and Pun et al [1998].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Two Degree of Freedom Vibro-Impact System With Multiple Motion Limiting Constraints

Wagg

Bishop

2004

Int. J. Bifurcation Chaos

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We consider the dynamics of impact oscillators with multiple degrees of freedom subject to more than one motion limiting constraint or stop. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom system are considered, along with definitions of the impact map for multiply constrained systems.We consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, and discuss the computational issues related to computing such solutions.Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. Numerical examples of sticking orbits for this system are shown and we discuss identifying the region, S in phase space where these orbits exist. We use bifurcation diagrams to indicate differing regimes of vibro-impacting motion for two different cases; firstly when the stops are both equal and on the same side (i.e. the same sign) and secondly when the stops are unequal and of opposing sign. For these two different constraint configurations we observe

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Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator

Cited by 70 publications

References 15 publications

Including inputs and control within equation-free architectures for complex systems

Including inputs and control within equation-free architectures for complex systems

Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator

Dynamics of a Two Degree of Freedom Vibro-Impact System With Multiple Motion Limiting Constraints

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