A rotating disk linked by a pair of springs

Han, Ning; Cao, Qingjie; Tao, Liang

doi:10.1007/s11071-014-1742-1

Cited by 10 publications

(4 citation statements)

References 32 publications

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“…Some other similar nonlinear geometrical oscillators have also been proposed and analyzed, such as an oscillator with two geometrical parameters, seen in [30,31], and an oscillator with three geometrical parameters, seen in [32]. Some nonlinear geometrical oscillators with characteristics of cylindrical system are also studied, such as a novel oscillator consisting of two oblique springs and a rotating disk, seen in [33], and a rotating SD oscillator, seen in [34]. Similar to SD oscillator, their dynamical behaviours are also controlled by geometrical parameter, causing a change from smoothness to discontinuity.…”

Section: Introductionmentioning

confidence: 99%

The complicated dynamical behaviours of a geometrical oscillator with a mass parameter

Huang

Cao

2023

Preprint

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In this paper, we consider a special kind of geometrical nonlinear oscillator with a mass parameter admitting two different dynamical states leading to a double-valued potential energy. A cylindrical manifold is introduced to formulate the equation of motion to describe the distinguished dynamical behaviours. With the help of Hamiltonian, the complex bifurcations are demonstrated with the varying of parameters including periodic solutions, the steady states and the blowing up phenomenon near θ = ± π/2 to infinity. A toroidal manifold is introduced to map the infinities into (0, ±2, 0) on the torus exhibiting saddle-node-like behaviour, where the uniqueness of solution is failed, for which a special ‘collision’ parameter is introduced to define the possible motion leaving from the infinities. A numerical method which is proposed to get solution near the infinity where Runge-Kutta method fails, is employed to get the bifurcation diagrams using Poincaré sections for the perturbed system to exhibit the complex dynamics including the co-existence of periodic solutions, the chaos from the coexisted periodic doubling and also the instant chaos from the coexisted periodic solutions. The results demonstrated herein this paper provide a brand new insight into the understanding of enriched nonlinear dynamics and an essential explanation about ‘collision’ of mechanical system with both the geometrical and mass parameters.

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Section: Introductionmentioning

confidence: 99%

The complicated dynamical behaviours of a geometrical oscillator with a mass parameter

Huang

Cao

2023

Preprint

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“…All these student works were performed in the framework of current activities of the chair MMD at TU Berlin on applications of the SINDy and other data analysis methods, for example, on bistable oscillators [6]. The setup itself follows classical ideas of easy‐to‐built‐up rotatory oscillators with translatory (not rotatory) springs, for example, described in [7].…”

Section: Introductionmentioning

confidence: 99%

Sparse identification of the dynamics of a nonlinear multistable oscillator

Kamecke,

Wulff,

Gräbner

et al. 2023

Proc Appl Math and Mech

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Recently, data‐driven modeling approaches are getting increasingly examined regarding their applicability for nonlinear mechanical or mechatronic systems. With a high data availability and often insufficiently accurate descriptions of complex behavior of real systems using established physical models, statistical models provide promising alternatives. Alongside machine learning techniques like deep neural networks, sparse regression is increasingly used to obtain models from measurement data. With sparse regression, governing equations are estimated from a given function space so that the data are explained with as few terms as possible while maintaining a low model error. This method is implemented in a framework called sparse identification of nonlinear dynamics. This paper demonstrates the application of this method on free and forced vibrations of a two degree of freedom nonlinear rotary oscillator with two stable equilibrium positions. The setup and data acquisition as well as the application of sparse identification are described. The selected function space containing the candidate functions is essential for an accurate representation of the system at hand. Monomials form a commonly used function space because they can approximate a wide variety of nonlinear characteristics. However, in this work, it is shown that monomials alone are insufficient when dry friction appears. Therefore, to account for Coulomb friction, a sign‐function is added as a candidate to the function space of monomials up to the fifth order. Adaptions of the common optimization algorithms turned out to be necessary for the inclusion of Coulomb friction. As a result, it is found that the addition of the sign‐function for Coulomb friction increases the model quality significantly.

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“…The authors introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the rotating pendulum system without barrier of the irrationalities (Han et al, 2013). Then the chaotic boundary of discontinuous case with a pair of heteroclinic-like orbits (Han et al, 2015) are obtained by using semi-analytical method. The bifurcation of cylindrical dynamical system with double well can be investigated in (Han et al, 2016) by means of Melnikov method and numerical simulation.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

Han¹,

Liu²

2018

MAS

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This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor' structures and the Lyapunov characteristics between the original system and approximate system.

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A rotating disk linked by a pair of springs

Cited by 10 publications

References 32 publications

The complicated dynamical behaviours of a geometrical oscillator with a mass parameter

The complicated dynamical behaviours of a geometrical oscillator with a mass parameter

Sparse identification of the dynamics of a nonlinear multistable oscillator

Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

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