Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation

Śmiechowicz, Wojciech; Loup, Théo; Olejnik, Paweł

doi:10.3390/mca24040090

Cited by 8 publications

(10 citation statements)

References 17 publications

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“…It has also been confirmed that the proposed method can be much faster than other ones [12]. This algorithm has already been successfully applied in the fields of synchronization [13], time series analysis [14], systems with delays [15] and as a benchmark for comparison of results with other algorithms [16].…”

Section: Introductionmentioning

confidence: 62%

“…The limit δ → 0 + is introduced due to the fact the perturbations to be transformed are infinitesimal. The matrix T x − , τ i presented in the expression (27) can be applied to calculate the quantities λ * * j according to the formulas (16) and (18), and thus to estimate the values of the LEs of the system (20) by means of the equation (17).…”

Section: Determining the Transition Matrix Of A Perturbationmentioning

confidence: 99%

See 1 more Smart Citation

Fast and simple Lyapunov Exponents estimation in discontinuous systems

Balcerzak

Sagan

Dąbrowski

et al. 2020

Eur. Phys. J. Spec. Top.

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Typically, to estimate the whole spectrum of n Lyapunov Exponents (LEs), it is necessary to integrate n perturbations and to orthogonalize them. Recently it has been shown that complexity of calculations can be reduced for smooth systems: integration of (n-1) perturbations is sufficient. In this paper authors demonstrate how this simplified approach can be adopted to non-smooth or discontinuous systems. Apart from the reduced complexity, the assets of the presented approach are simplicity and ease of implementation. The paper starts with a short review of properties of LEs and methods of their estimation for smooth and non-smooth systems. Then, the algorithm of reduced complexity for smooth systems is shortly introduced. Its adaptation to non-smooth systems is described in details. Application of the method is presented for an impact oscillator. Implementation of the novel algorithm is comprehensively explained. Results of simulations are presented and validated. It is expected that the presented method can simplify investigations of non-smooth dynamical systems and support research in this field.

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Section: Introductionmentioning

confidence: 62%

Section: Determining the Transition Matrix Of A Perturbationmentioning

confidence: 99%

Fast and simple Lyapunov Exponents estimation in discontinuous systems

Balcerzak

Sagan

Dąbrowski

et al. 2020

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

show abstract

“…Now, we will look at the variablelength pendulum with stiffness and damping as investigated by different authors. In [36] one finds an analysis without the damping force, while [37][38][39], and [40] include damping force in the analysis.…”

Section: If Resonance Appears At Frequencymentioning

confidence: 99%

On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

Yakubu

Olejnik

Awrejcewicz

2022

Arch Computat Methods Eng

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A comprehensive review of variable-length pendulums is presented. An attempt at a unique evaluation of current trends in this field is carried out in accordance with mathematical modeling, dynamical analysis, and original computer simulations. Perspectives of future trends are also noted on the basis of various concepts and possible theoretical and engineering applications. Some important physical concepts are verified using dedicated numerical procedures and assessed based on dynamical analysis. At the end of the review, it is concluded that many variable-length pendulums are very demanding in the modeling and analysis of parametric dynamical systems, but basic knowledge about constant-length pendulums can be used as a good starting point in providing much accurate mathematical description of physical processes. Finally, an extended model for a variable-length pendulum’s mechanical application being derived from the Swinging Atwood Machine is proposed. The extended SAM presents a novel SAM concept being derived from a variable-length double pendulum with a suspension between the two pendulums. The results of original numerical simulations show that the extended SAM’s nonlinear dynamics presented in the current work can be thoroughly studied, and more modifications can be achieved. The new technique can reduce residual vibrations through damping when the desired level of the crane is reached. It can also be applied in simple mechatronic and robotic systems.

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“…This modification would give a faster and longer oscillation, which will, in turn, result in more rapid pumping of fluid since the variable-length pendulum can undergo a quicker and longer oscillation, as presented by the following authors: Ref. [12] derived the differential equations of dynamics for both the first and the second mutation from the sum of kinetic and potential energy for a rigid pendulum's two and three degrees of freedom. It was observed that a successive expansion in the forms of representation of the energy is introduced.…”

Section: Introductionmentioning

confidence: 99%

Modeling, Simulation, and Analysis of a Variable-Length Pendulum Water Pump

2021

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Due to the long-term problem of electricity and potable water in most developing and undeveloped countries, predominantly rural areas, a novelty of the pendulum water pump, which uses a vertically excited parametric pendulum with variable-length using a sinusoidal excitation as a vibrating machine, is presented. With this, more oscillations can be achieved, reducing human effort further and having high output than the existing pendulum water pump with the conventional pendulum. The pendulum, lever, and piston assembly are modeled by a separate dynamical system and then joined into the many degrees-of-freedom dynamical systems. The present work includes friction while studying the system dynamics and then simulated to verify the system’s harmonic response. The study showed the effect of the pendulum length variability on the whole system’s performance. The vertically excited parametric pendulum with variable length in the system is established, giving faster and longer oscillations than the pendulum with constant length. Hence, more and richer dynamics are achieved. A quasi-periodicity behavior is noticed in the system even after 50 s of simulation time; this can be compensated when a regular external forcing is applied. Furthermore, the lever and piston oscillations show a transient behavior before it finally reaches a stable behavior.

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Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation

Cited by 8 publications

References 17 publications

Fast and simple Lyapunov Exponents estimation in discontinuous systems

Fast and simple Lyapunov Exponents estimation in discontinuous systems

On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

Modeling, Simulation, and Analysis of a Variable-Length Pendulum Water Pump

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