Near the resonance behavior of a periodicaly forced partially dissipative three-degrees-of-freedom mechanical system

This paper considers three dynamic systems composed of a mathematical pendulum suspended on a sliding body subjected to harmonic excitation. A comparative dynamic analysis of the studied parametric mutations of the rigid pendulum with inertial suspension point and damping was performed. The examined system with parametric mutations is solved numerically, where phase planes and Poincaré maps were used to observe the system response. Lyapunov exponents were computed in two ways to classify the dynamic behavior at relatively early stage of forced responses using two proven methods. The results show that with some parameters three systems exhibit a very similar dynamic behavior, i.e., quasi-periodic and even chaotic motions. Math. Comput. Appl. 2019, 24, 90 2 of 15 the full spectrum of Lyapunov exponents is presented in [3], the calculation of the largest Lyapunov exponent is given in [5,6], and the determination of the spectrum of Lyapunov exponents can be found in [7].From a practical point of view, Lyapunov exponents are used in various fields of science, such as: rotor systems [8], electricity systems [9], aerodynamics [10]. The Parametric Pendulum and Its Mutations

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“…Similar derivation for the three-degree-of-freedom system was performed in [12]. The equation for kinetic energy was found follows:…”

Section: Mathematical Model Of the Second Mutation Of The Rigid Pendulummentioning

confidence: 93%

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

Śmiechowicz

Loup

Olejnik

2019

MCA

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“…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%

“…A pendulum-spring system with variable-length is shown in Fig. 14 [39]. The equations of motion describe a threedegrees-of-freedom system, of which kinetic energy has the form:…”

Section: Variable-length Pendulum With Stiffness and Dampingmentioning

confidence: 99%

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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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“…Refs. [13,14], the Euler-Lagrange equation, and the Rayleigh dissipation function are used to derive the equation for a three-degree of freedom pendulum system. The numerical results reveal that a variable-length spring pendulum hung from the occasionally forced slider can demonstrate quasi-periodicity and chaotic motions in a resonance condition.…”

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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Near the resonance behavior of a periodicaly forced partially dissipative three-degrees-of-freedom mechanical system

Cited by 8 publications

References 9 publications

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

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

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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