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Journal articles on the topic 'Un-damped and damped oscillations'

Author: Grafiati

Published: 4 June 2025

Last updated: 6 July 2025

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1

Getachew, Kuma Watiro. "Simulation of Un-damped and Damped Oscillations in RLC Circuit using MATLAB Computer program." J. of Advancement in Engineering and Technology 7, no. 3 (2020): 07. https://doi.org/10.5281/zenodo.3751733.

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In this paper, we elaborate the characteristics of oscillations in RLC circuit using MATLAB computer program. To study the characteristics we apply second order differential equations to obtain the expression of electric charge resulted from Kirchhoff’s loop rule and use the solution of it to determine the expressions for electric current (i), energy stored in capacitor and energy stored in the inductor. In the simplest case, the resonant circuit consists only of a capacitor C and inductor L, and characterizes un-damped electrical oscillations.  An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor(C), connected in series or in parallel, and characterizes damped oscillations. For both cases, the un-damped and damped oscillations, we compare the graphs of circuit without resistance with the graphs of circuit with resistance obtained using MATLAB computer program.

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2

Csernovszky, Z., M. Hömöstrei, and K. Kurucz. "Study of damped oscillations using Phyphox and Arduino controlled Hall-sensor." Journal of Physics: Conference Series 2693, no. 1 (2024): 012004. http://dx.doi.org/10.1088/1742-6596/2693/1/012004.

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Abstract The paper presents physics education activities organized around the topic of damped oscillations. We used the Phyphox smartphone application for secondary school physics classes. These activities served as a basis for a physics education workshop, where an Arduino-controlled Hall-sensor and the Phyphox Magnetometer were presented. The problem of a damped pendulum, a vertical oscillation in water, and an LCr oscillating circuit was examined as part of a Phyphox project. Mechanical and electromagnetic damped oscillations can be demonstrated with our devices. Using our data, we could compare Hall-sensors of different devices, estimate some characteristics of the waves and help plan an LCr oscillating circuit. Activities for secondary school physics classes are suggested, based on the pedagogical goals.

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3

Jordan, Thomas F., E. C. G. Sudarshan, and Prashant Valanju. "Measurement-damped oscillations." Physical Review A 44, no. 5 (1991): 3340–42. http://dx.doi.org/10.1103/physreva.44.3340.

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4

Feireisl, Eduard, and Leopold Herrmann. "Oscillations of a nonlinearly damped extensible beam." Applications of Mathematics 37, no. 6 (1992): 469–78. http://dx.doi.org/10.21136/am.1992.104525.

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5

Rohit, Gupta, Gupta Rahul, and Rajput Sonica. "ANALYSIS OF DAMPED HARMONIC OSCILLATOR BY MATRIX METHOD." International Journal of Research and Analytical Reviews (IJRAR) 5, no. 4 (2023): 479–84. https://doi.org/10.5281/zenodo.7716591.

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The motion of damped harmonic oscillator capable of oscillating in a medium such as air, water etc. is always opposed by the frictional (or damping) forces which arise from the resistance of the medium or from within the system itself and act opposite to direction of the motion of the body and result in the decrease in the amplitude of the oscillations with time. As a result of the decrease in the amplitude of the oscillations, there is a dissipation of energy of the oscillations and ultimately, the body stops oscillating. The analysis of damped harmonic oscillator is generally done by adopting the classical method. In this paper, we analyze the response (behaviour) of a damped harmonic oscillator by solving its differential equation by the matrix method. The response obtained will provide an expression for the displacement of the damped oscillator from the equilibrium position at any instant. The behaviour of the displacement obtained is determined by the damping factor and the stiffness factor.

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6

Wadhwa, Adya, and Ajay Wadhwa. "Investigating damped oscillations in simple pendulum." Physics Education 58, no. 6 (2023): 065016. http://dx.doi.org/10.1088/1361-6552/acf433.

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Abstract We describe an experimental set-up of a simple pendulum for the measurement of damping coefficient of oscillations. We have used an Arduino microcontroller with infrared sensor interface to detect the oscillating object in its path. The Arduino program records the instants of time to calculate the time-period and the total number of oscillations. The dependence of damping coefficient on the size, shape and mass of the pendulum bob as well as on the thickness of the pendulum string is investigated.

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7

Murayama, Yoriko, Hiroshi Kori, Chiaki Oshima, Takao Kondo, Hideo Iwasaki, and Hiroshi Ito. "Low temperature nullifies the circadian clock in cyanobacteria through Hopf bifurcation." Proceedings of the National Academy of Sciences 114, no. 22 (2017): 5641–46. http://dx.doi.org/10.1073/pnas.1620378114.

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Cold temperatures lead to nullification of circadian rhythms in many organisms. Two typical scenarios explain the disappearance of rhythmicity: the first is oscillation death, which is the transition from self-sustained oscillation to damped oscillation that occurs at a critical temperature. The second scenario is oscillation arrest, in which oscillation terminates at a certain phase. In the field of nonlinear dynamics, these mechanisms are called the Hopf bifurcation and the saddle-node on an invariant circle bifurcation, respectively. Although these mechanisms lead to distinct dynamical properties near the critical temperature, it is unclear to which scenario the circadian clock belongs. Here we reduced the temperature to dampen the reconstituted circadian rhythm of phosphorylation of the recombinant cyanobacterial clock protein KaiC. The data led us to conclude that Hopf bifurcation occurred at ∼19 °C. Below this critical temperature, the self-sustained rhythms of KaiC phosphorylation transformed to damped oscillations, which are predicted by the Hopf bifurcation theory. Moreover, we detected resonant oscillations below the critical temperature when temperature was periodically varied, which was reproduced by numerical simulations. Our findings suggest that the transition to a damped oscillation through Hopf bifurcation contributes to maintaining the circadian rhythm of cyanobacteria through resonance at cold temperatures.

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8

Khan, Kamran-ul-Haq, and Suhaib Masroor. "Numerical simulation along with the experimental work for an underdamped oscillator using fourth order Runge–Kutta method. An undergraduate experiment." Physics Education 58, no. 6 (2023): 065006. http://dx.doi.org/10.1088/1361-6552/acede4.

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Abstract Experiments based on oscillatory motion are an essential part of the curricula for students of physics and engineering in their undergraduate studies, such as the determination of spring constant for a well-known classical oscillator i.e., spring-mass system. Moreover, it is important for students to understand the physics of damped oscillators because, in real-world scenarios, a system involving oscillations cannot be completely analysed without understanding conditions of damped oscillations i.e., underdamped, overdamped, and critical damped. In this work, we design a computational physics lab for the simulation of the underdamped oscillator using an Excel spreadsheet by employing 4th order Runge–Kutta method. Further, we construct a simple experimental setup to observe the damped oscillation and obtaindata for the underdamped system. Final analysis was based on comparison between the experimental data and the numerically estimated data.

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9

Cherentsov, D. A., S. P. Pirogov, S. M. Dorofeev, and S. A. Cherentsova. "RESEARCH OF DAMPED OSCILLATIONS MANOMETRIC SPRING WITH HARD TIP." Oil and Gas Studies, no. 1 (March 1, 2017): 116–20. http://dx.doi.org/10.31660/0445-0108-2017-1-116-120.

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The main results of the calculations of damped oscillations ofmanometric tubular springs (MTS) considering the hard tip are presented. The calculations are based on previously developed mathematical model. An assessment of the impact of the MTS mass on damped oscillation parameters (frequency and oscillation damping parameter) was made, the change of the limit value of viscosity of the damping fluid at which the aperiodic motion begins, as well as the value of the tip mass at which the system ceases to oscillate.

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10

Grebenakova, Elena, and Stojan Manolev. "Demonstration of Damped Electrical Oscillations." Natural Science and Advanced Technology Education 30, no. 1 (2021): 26–33. http://dx.doi.org/10.53656/nat2021-1.02.

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Introducing mechanical oscillations in schools is a fairly simple and easy experimental feasible task. To demonstrate electromagnetic oscillations, we have difficulty in understanding by students. The explanation of electromagnetic circuits is more abstract. We offered an experiment where we make electromagnetic oscillations obvious and understandable to students. In our experiment we used the software and interface of the AMSTEL Institute (AMSTEL Institute – Amsterdam Mathematics, Science and Technology Education Laboratory) as well as elements from the sets of experimental tasks from the Physics Olympiads organized by the Sofia branch of physicists.

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11

Houdek, G., B. Chaplin, and Y. Elsworth. "How are solar oscillations damped?" Astronomy & Geophysics 39, no. 4 (1998): 4.30–4.32. http://dx.doi.org/10.1093/astrog/39.4.4.30.

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12

Chen, James, and Peter W. Schuck. "Damped Oscillations of Coronal Loops." Solar Physics 246, no. 1 (2007): 145–64. http://dx.doi.org/10.1007/s11207-007-9011-9.

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13

Lezcano, Marciano Santamaría, Evgeni Svenk Cruz de Gracia, and Lucio Strazzabosco Dorneles. "Damped Oscillations—A Smartphone Approach." Physics Teacher 62, no. 2 (2024): 123–26. http://dx.doi.org/10.1119/5.0063290.

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14

Arregui, I., J. Terradas, R. Oliver, and J. L. Ballester. "Damped oscillations of two interacting coronal loops." Proceedings of the International Astronomical Union 3, S247 (2007): 133–39. http://dx.doi.org/10.1017/s1743921308014786.

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AbstractWe present results on the oscillatory properties (periods, damping rates, and spatial distribution of perturbations) for resonantly damped oscillations in a system of two inhomogeneous coronal slabs and compare them to the properties found in single slab loop models. A system of two identical coronal loops is modelled, in Cartesian geometry, as being composed by two density enhancements. The linear magnetohydrodynamic (MHD) wave equations for oblique propagation of waves are solved and the damping due to resonant absorption is computed. Due to the interaction between the loops, the normal modes of oscillation present in a single slab split into symmetric and antisymmetric oscillations when a system of two identical slabs is considered. The frequencies of these solutions may differ from the single slab results when the distance between the loops is of the order of a few slab widths. Oblique propagation of waves weakens this interaction, since solutions become more confined to the edges of the slabs. The damping is strong for surface-like oscillations, while sausage body-like solutions are unaffected.

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15

Yartsev, B. A., V. M. Ryabov, and L. V. Parshina. "Dissipative properties of three-layered composite structures. 4. Numerical experiment." Transactions of the Krylov State Research Centre 3, no. 401 (2022): 58–70. http://dx.doi.org/10.24937/2542-2324-2022-3-401-58-70.

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Object and purpose of research. The study is concerned with a three-layer plate formed by two rigid anisotropic layers and a soft medium isotropic layer of viscoelastic polymer. Each rigid layer presents anisotropic structure formed by a finite number of randomly oriented orthotropic viscoelastic layers of composites. The paper is intended to study the influence of reinforcement orientation of rigid layers, relative thickness of the soft layer of an isotropic viscoelastic polymer and the ambient temperature on values values of natural frequencies and mechanical loss factors of the coupled damped oscillations in symmetric and assymetric plates. Materials and methods. Numerical experiment using a computer program implementing the previously proposed method for solving coupled differential equations of damped oscillations in anisotropic three-layer plates [2]. Main results. It was shown that in unsupported globally monoclinic symmetrical three-layer rectangular plate a bendingtorsional interaction occurs, generating mutual transformations of the eigenforms of the coupled oscillation modes if at least in one of the directions of the plate one of the eigenforms is characterized by an even number of quarters of the wave, and the other eigenform is characterized by an odd number of quarters of the wave. In unsupported globally orthotropic asymmetric three-layer rectangular plate interaction of bending modes of oscillations occurs in two mutually orthogonal planes, if both eigenforms are characterized by either an even or an odd number of wave quarters in main directions of the plate. It was found that each mode of natural oscillations of both symmetric and asymmetric three-layer plates has its own effective relative thickness of the soft layer of an isotropic viscoelastic polymer corresponding to the maximum level of dissipative properties. A further increase in the relative thickness is often accompanied by a decrease in values of the mechanical loss factors. The significant influence of ambient temperature on natural frequencies values and mechanical loss factors of all considered oscillation modes of symmetric and asymmetric unsupported rectangular three-layer composite plates is demonstrated. Conclusion. It was found that coupled damped oscillations of a symmetric three-layer plate are described by two systems of differential equations, with structures close to that of the systems of corresponding differential equations describing the damped oscillations of a quasi-homogeneous monoclinic plate. At the same time, the coupled damped oscillations of an asymmetric three-layer plate are described by two systems of differential equations that coincide with the systems of corresponding differrential equations describing the damped oscillations of a globally orthotropic three-layer plate.

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16

S. O., Essang,, Kolawole, O. M., Francis, R. E., et al. "On The Novel Damped Oscillatory Logistic Growth Model: A Hybrid Approach." African Journal of Mathematics and Statistics Studies 8, no. 2 (2025): 48–66. https://doi.org/10.52589/ajmss-wnrxbc1z.

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The Damped Oscillatory Logistic Growth (DOLG) Model is introduced as a novel hybrid framework that integrates oscillatory dynamics, damping, and logistic growth into a single differential equation. This model extends classical systems such as the harmonic oscillator, logistic growth equation, and damped systems by combining their key features into a unified framework. Numerical solutions reveal rich dynamical behaviors, including damped oscillations, stabilization to carrying capacity, and phase-dependent growth patterns. The system’s stability is analytically and numerically confirmed, with trajectories converging to the non-trivial equilibrium (x,v) = (K,0) for all parameter regimes, and by extension, the trivial solution. The effects of damping, growth rate, and oscillation frequency are explored through time series and phase portraits, demonstrating the model’s versatility in capturing complex phenomena. Potential applications span ecology, economics and engineering, offering new insights into oscillating populations, cyclical growth, and mechanical systems with growth constraints. This study lays the groundwork for future research on hybrid dynamical systems and their interdisciplinary applications.

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17

Radionoff, Anatoly A. "On Small Oscillations Inside a Volcano Feeding System." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 1 (205) (March 31, 2020): 78–84. http://dx.doi.org/10.18522/1026-2237-2020-1-78-84.

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A simple analytical model representing one of the possible mechanisms for the occurrence of low-frequency oscillations in a feeding system of volcano has been developed. The model is presented for a cylindrical chamber filled with magma with Maxwell rheology. It is shown that damped harmonic oscillations in the magma flow velocity can occur in the volcanic chamber. These damped harmonic oscillations can appear as a reaction to remote seismic events or seismic events in the volcanic feeding system. The dependence of the oscillation frequency on the physical characteristics of the magmatic melt and the geometric dimensions of the volcano chamber is shown. The occurrence of magmatic oscillations can be observed near the surface as a volcanic tremor. The model is applied to the measurements result of low-frequency oscillations for the magma chamber of the Elbrus volcanic center.

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18

Knizhnik, Kalman, Manuel Luna, Karin Muglach, Holly Gilbert, Therese Kucera, and Judith Karpen. "Observational Study of Large Amplitude Longitudinal Oscillations in a Solar Filament." Proceedings of the International Astronomical Union 8, S300 (2013): 428–29. http://dx.doi.org/10.1017/s174392131301140x.

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AbstractOn 20 August 2010 an energetic disturbance triggered damped large-amplitude longitudinal (LAL) oscillations in almost an entire filament. In the present work we analyze this periodic motion in the filament to characterize the damping and restoring mechanism of the oscillation. Our method involves placing slits along the axis of the filament at different angles with respect to the spine of the filament, finding the angle at which the oscillation is clearest, and fitting the resulting oscillation pattern to decaying sinusoidal and Bessel functions. These functions represent the equations of motion of a pendulum damped by mass accretion. With this method we determine the period and the decaying time of the oscillation. Our preliminary results support the theory presented by Luna and Karpen (2012) that the restoring force of LAL oscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism is the ongoing accumulation of mass onto the oscillating threads. Following an earlier paper, we have determined the magnitude and radius of curvature of the dipped magnetic flux tubes hosting a thread along the filament, as well as the mass accretion rate of the filament threads, via the fitted parameters.

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19

Monsia, M. D., and Y. J. F. Kpomahou. "Simulating Nonlinear Oscillations of Viscoelastically Damped Mechanical Systems." Engineering, Technology & Applied Science Research 4, no. 6 (2014): 714–23. http://dx.doi.org/10.48084/etasr.518.

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The aim of this work is to propose a mathematical model in terms of an exact analytical solution that may be used in numerical simulation and prediction of oscillatory dynamics of a one-dimensional viscoelastic system experiencing large deformations response. The model is represented with the use of a mechanical oscillator consisting of an inertial body attached to a nonlinear viscoelastic spring. As a result, a second-order first-degree Painlevé equation has been obtained as a law, governing the nonlinear oscillatory dynamics of the viscoelastic system. Analytical resolution of the evolution equation predicts the existence of three solutions and hence three damping modes of free vibration well known in dynamics of viscoelastically damped oscillating systems. Following the specific values of damping strength, over-damped, critically-damped and under-damped solutions have been obtained. It is observed that the rate of decay is not only governed by the damping degree but, also by the magnitude of the stiffness nonlinearity controlling parameter. Computational simulations demonstrated that numerical solutions match analytical results very well. It is found that the developed mathematical model includes a nonlinear extension of the classical damped linear harmonic oscillator and incorporates the Lambert nonlinear oscillatory equation with well-known solutions as special case. Finally, the three damped responses of the current mathematical model devoted for representing mechanical systems undergoing large deformations and viscoelastic behavior are found to be asymptotically stable.

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20

Monsia, M. D., and Y. J. F. Kpomahou. "Simulating Nonlinear Oscillations of Viscoelastically Damped Mechanical Systems." Engineering, Technology & Applied Science Research 4, no. 6 (2014): 714–23. https://doi.org/10.5281/zenodo.14692.

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The aim of this work is to propose a mathematical model in terms of an exact analytical solution that may be used in numerical simulation and prediction of oscillatory dynamics of a one-dimensional viscoelastic system experiencing large deformations response. The model is represented with the use of a mechanical oscillator consisting of an inertial body attached to a nonlinear viscoelastic spring. As a result, a second-order first-degree Painlevé equation has been obtained as a law, governing the nonlinear oscillatory dynamics of the viscoelastic system. Analytical resolution of the evolution equation predicts the existence of three solutions and hence three damping modes of free vibration well known in dynamics of viscoelastically damped oscillating systems. Following the specific values of damping strength, over-damped, critically-damped and under-damped solutions have been obtained. It is observed that the rate of decay is not only governed by the damping degree but, also by the magnitude of the stiffness nonlinearity controlling parameter. Computational simulations demonstrated that numerical solutions match analytical results very well. It is found that the developed mathematical model includes a nonlinear extension of the classical damped linear harmonic oscillator and incorporates the Lambert nonlinear oscillatory equation with well-known solutions as special case. Finally, the three damped responses of the current mathematical model devoted for representing mechanical systems undergoing large deformations and viscoelastic behavior are found to be asymptotically stable.

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21

Quiroga, G. D., and P. A. Ospina-Henao. "Dynamics of damped oscillations: physical pendulum." European Journal of Physics 38, no. 6 (2017): 065005. http://dx.doi.org/10.1088/1361-6404/aa8961.

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22

Shakur, Asif, and Jeffrey Emmert. "Damped Oscillations with a Smart Cart." Physics Teacher 57, no. 7 (2019): 490–92. http://dx.doi.org/10.1119/1.5126833.

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23

Mungan, Carl E., and Trevor C. Lipscombe. "Oscillations of a quadratically damped pendulum." European Journal of Physics 34, no. 5 (2013): 1243–53. http://dx.doi.org/10.1088/0143-0807/34/5/1243.

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24

Bayliss, Alvin. "Forced oscillations in quadratically damped systems." Communications on Pure and Applied Mathematics 31, no. 1 (2010): 69–88. http://dx.doi.org/10.1002/cpa.3160310104.

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25

Gorczyca, P., K. Kołek, R. Rosół, and A. Turnau. "Semi-Active Suspension Laboratory System." Solid State Phenomena 147-149 (January 2009): 350–55. http://dx.doi.org/10.4028/www.scientific.net/ssp.147-149.350.

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The paper describes the semi-active suspension laboratory system (SAS), built to demonstrate and test a number of control algorithms. The heart of the system is the automotive engineering damper with the restoring force controlled by magnetic field. We use in our apparatus the Lord 1097 magnetorheological (MR) damper manufactured by the American company. MR devices benefit from the ability of MR fluids to rapidly change rheological properties upon exposure to a magnetic field. The main advantage of the SAS is its portability as a demonstrative experimental rig to test control damping algorithms. The damped less oscillations are compared to the On-off damped oscillations of the apparatus. The MR damper provides an effective solution to SAS control in a variety of applications.

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26

Alyousef, Haifa A., Alvaro H. Salas, Sadah A. Alkhateeb, and S. A. El-Tantawy. "Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators." Journal of Mathematics 2022 (May 5, 2022): 1–10. http://dx.doi.org/10.1155/2022/2715767.

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Some novel exact solutions and approximations to the damped Duffing–Mathieu-type oscillator with cubic nonlinearity are obtained. This work is divided into two parts: in the first part, some exact solutions to both damped and undamped Mathieu oscillators are obtained. These solutions are expressed in terms of the Mathieu functions of the first kind. In the second part, the equation of motion to the damped Duffing–Mathieu equation (dDME) is solved using some effective and highly accurate approaches. In the first approach, the nonintegrable dDME with cubic nonlinearity is reduced to the integrable dDME with linear term having undermined optimal parameter (maybe called reduced method). Using a suitable technique, we can determine the value of the optimal parameter and then an analytical approximation is obtained in terms of the Mathieu functions. In the second approach, the ansatz method is employed for deriving an analytical approximation in terms of trigonometric functions. In the third approach, the homotopy perturbation technique with the extended Krylov–Bogoliubov–Mitropolskii (HKBM) method is applied to find an analytical approximation to the dDME. Furthermore, the dDME is solved numerically using the Runge–Kutta (RK) numerical method. The comparison between the analytical and numerical approximations is carried out. All obtained approximations can help a large number of researchers interested in studying the nonlinear oscillations and waves in plasma physics and many other fields because many evolution equations related to the nonlinear waves and oscillations in a plasma can be reduced to the family of Mathieu-type equation, Duffing-type equation, etc.

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27

Sawkmie, Ivan Skhem, and Mangal C. Mahato. "Free Oscillations of a Damped Simple Pendulum: An Analog Simulation Experiment." Physics Educator 01, no. 04 (2019): 1950015. http://dx.doi.org/10.1142/s266133951950015x.

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The frequency of free oscillation of a damped simple pendulum with large amplitude depends on its amplitude unlike the amplitude-independent frequency of oscillation of a damped simple harmonic oscillator. This aspect is not adequately emphasized in the undergraduate courses due to experimental and theoretical difficulties. We propose an analog simulation experiment to study the free oscillations of a simple pendulum that could be performed in an undergraduate laboratory. The needed sinusoidal potential is obtained approximately by using the available AD534 IC by suitably augmenting the electronic circuitry. To keep the circuit simple enough we restrict the initial angular amplitude of the simple pendulum to a maximum of [Formula: see text]. The results compare well qualitatively with the theoretical results. The small quantitative discrepancy is attributed to the inexact nature of the used “sinusoidal potential”.

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Kontomaris, Stylianos-Vasileios, and Anna Malamou. "An elementary proof of the amplitude’s exponential decrease in damped oscillations." Physics Education 59, no. 2 (2024): 025030. http://dx.doi.org/10.1088/1361-6552/ad26d4.

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Abstract A damped oscillation is characterized by the diminishing amplitude of an oscillating system resulting from the dissipation of energy. A crucial example of damped oscillations involves a block of mass attached to the end of a linear spring, experiencing a damping force proportional to the object’s velocity and acting in opposition to the direction of its motion. For small values of the damping constant, the amplitude decreases exponentially over time. Typically, this behavior is introduced at the secondary education level without providing a justification. In this paper, a new approach tailored for the secondary education level is introduced to explain the aforementioned exponential decrease without relying on advanced mathematical tools. In addition, utilizing this analysis simplifies the demonstration of why the exponential decrease in amplitude is applicable only for small damping forces. It is also worth noting that the methods used to derive this result can be applied in various areas, including the derivation of the equation that connects the root mean square value of the AC sinusoidal current to its maximum value.

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29

Jadhav, Changdev P., Tanisha B. Dale, Vaijanath L. Chinchane, et al. "On solutions of fractional differential equations for the mechanical oscillations by using the Laplace transform." AIMS Mathematics 9, no. 11 (2024): 32629–45. http://dx.doi.org/10.3934/math.20241562.

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<p>In this article, we employ the Laplace transform (LT) method to study fractional differential equations with the problem of displacement of motion of mass for free oscillations, damped oscillations, damped forced oscillations, and forced oscillations (without damping). These problems are solved by using the Caputo and Atangana-Baleanu (AB) fractional derivatives, which are useful fractional derivative operators consist of a non-singular kernel and are efficient in solving non-local problems. The mathematical modelling for the displacement of motion of mass is presented in fractional form. Moreover, some examples are solved.</p>

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30

Liu, Jin, and Shu Qiang Zhao. "Study on Sub-Synchronous Control Interaction (SSCI) of Wind-Power Generator." Advanced Materials Research 953-954 (June 2014): 518–21. http://dx.doi.org/10.4028/www.scientific.net/amr.953-954.518.

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Large-scale wind turbine generators with power electronic converters that operate near series compensated transmission lines are susceptible to un-damped sub-synchronous oscillations. This sub-synchronous oscillation is called Sub-synchronous Control Interaction (SSCI). With the rapid development of wind power technology, SSCI emerges as a new sub-synchronous oscillation phenomenon. The first SSCI event occurred for a wind farm in America in2009, which resulted in damage to the wind turbines. The origin of SSCI is presented and its characteristics of different types of wind turbines are summarized. Then the research prospect on this field is explored.

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31

Robertson, D., and M. S. Ruderman. "Resonantly damped oscillations of two coronal loops." Astronomy & Astrophysics 525 (November 26, 2010): A4. http://dx.doi.org/10.1051/0004-6361/201015525.

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Terradas, J., R. Oliver, and J. L. Ballester. "Damped Coronal Loop Oscillations: Time‐dependent Results." Astrophysical Journal 642, no. 1 (2006): 533–40. http://dx.doi.org/10.1086/500730.

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33

Soler, R., M. S. Ruderman, and M. Goossens. "Damped kink oscillations of flowing prominence threads." Astronomy & Astrophysics 546 (October 2012): A82. http://dx.doi.org/10.1051/0004-6361/201220111.

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34

HANKS, RICHARD W. "DAMPED OSCILLATIONS IN A BINGHAM PLASTIC FLUID." Chemical Engineering Communications 89, no. 1 (1990): 187–94. http://dx.doi.org/10.1080/00986449008940569.

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35

Soler, Roberto, and Manuel Luna. "Damped transverse oscillations of interacting coronal loops." Astronomy & Astrophysics 582 (October 2015): A120. http://dx.doi.org/10.1051/0004-6361/201526919.

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36

Dias, Frédéric, and Gérard Iooss. "Capillary-gravity solitary waves with damped oscillations." Physica D: Nonlinear Phenomena 65, no. 4 (1993): 399–423. http://dx.doi.org/10.1016/0167-2789(93)90064-8.

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37

Bryant, Peter J., and John W. Miles. "On a periodically forced, weakly damped pendulum. Part 3: Vertical forcing." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 32, no. 1 (1990): 42–60. http://dx.doi.org/10.1017/s0334270000008201.

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AbstractWe consider the phase-locked solutions of the differential equation governing planar motion of a weakly damped pendulum forced by a prescribed, vertical acceleration εg sin ωt of its pivot, where ω and t are dimensionless, and the unit of time is the reciprocal of the natural frequency. Resonance curves and stability boundaries are presented for downward and inverted oscillations of periods T, 2T, 4T, …, where T (≡ 2π/ω) is the forcing period. Stable, downward oscillations are found to occur in distinct regions of the (ω, ε) plane, reminiscent of the regions of stability of the Mathieu equation (which describes the equivalent undamped, parametrically excited pendulum motion). The regions are dominated by oscillations of frequencies , each region being bounded on one side by a vertical state at rest in stable equilibrium and on the other side by a symmetry-breaking, period-doubling sequence to chaotic motion. Stable, inverted oscillations are found to occur also in distinct regions of the (ω, ε) plane, the principal oscillation in each region being symmetric with period 2T.

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38

Bryant, Peter J. "Chaotic breakdown of a periodically forced, weakly damped pendulum." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 34, no. 2 (1992): 153–73. http://dx.doi.org/10.1017/s0334270000008705.

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AbstractAn investigation is made of the transition from periodic solutions through nearly-periodic solutions to chaotic solutions of the differential equation governing forced coplanar motion of a weakly damped pendulum. The pendulum is driven by horizontal, periodic forcing of the pivot with maximum acceleration Є g and dimensionless frequency ω As the forcing frequency ω is decreased gradually at a sufficiently large forcing amplitude Є, it has been shown previously that the pendulum progresses from symmetric oscillations of period T (= 2 π/ω) into a symmetry-breaking, period-doubling sequence of stable, periodic oscillations. There are two related forms of asymmetric, stable oscillations in the sequence, dependent on the initial conditions. When the frequency is decreased immediately beyond the sequence, the oscillations become unstable but remain in the neighbourhood in (θ,) phase space of one or other of the two forms of periodic oscillations, where θ(t) is the pendulum angle with the downward vertical. As the frequency is decreased further, the oscillations move intermittently between the neighbourhoods in (θ,) phase space of each of the two forms of periodic oscillations, in paired nearly-periodic oscillations. Further decrease of the forcing frequency leads to time intervals in which the motion is strongly unstable, with the pendulum passing intermittently over the pivot, interspersed with time intervals when the motion is nearly-periodic and only weakly unstable. The strongly-unstable intervals dominate in fully chaotic oscillations. Windows of independent, stable, periodic oscillations occur throughout the frequency range investigated. It is shown in an appendix how the Floquet method may be interpreted to describe the linear stability of the periodic and nearly-periodic solutions, and the windows of periodic oscillations in the investigated frequency range are listed in a second appendix.

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39

KAUFFMAN, LOUIS H. "PENDULUM TRACED CURVES AND DAMPED OSCILLATIONS IN THE PLANE." Journal of Knot Theory and Its Ramifications 16, no. 10 (2007): 1451–57. http://dx.doi.org/10.1142/s0218216507005907.

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40

Danylchuk, Hanna, Liubov Kibalnyk, and Olexandr Serdiuk. "Study of critical phenomena in economic systems using a model of damped oscillations." SHS Web of Conferences 65 (2019): 06008. http://dx.doi.org/10.1051/shsconf/20196506008.

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The article describes the construction of a model for the analysis and forecasting of critical phenomena in economic systems based on the equation of the damped oscillations. The model of the damped oscillations based on the analysis of wavelet coefficient energy allows identifying critical phenomena, in the first place, crashes. Two parameters of the model, the initial phase and the damping coefficient, are the most appropriate for the analysis and prediction of the critical events in the economic systems. The sequence of steps for conducting research is presented and the possibility to automate the process of predicting critical phenomena is described. Critical phenomenon can be predicted based on the initial phase and the damping coefficient, the prediction horizon depends on the scale at which the model of the damped oscillations was constructed. The study of the results of the model is based on the known crashes and shocks given in the work.

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41

Konarasinghe, W.G.S, and K.M.U.B. Konarasinghe. "Model Development for Damped and Forced Type of Oscillations in Time Series." Journal of New Frontiers in Mathematics and Statistics 2, no. 2 (2021): 20——35. https://doi.org/10.5281/zenodo.5704703.

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The periodic motion defines as the motion which repeats after a regular interval of time. The periodic motion in which there is the existence of a restoring force and the body moves along the same path to and fro about a definite point called equilibrium position/mean position is called oscillatory motion. The oscillatory motion could be either linear oscillation or circular oscillation. For example, the oscillation of strings of musical instruments is linear oscillation whilst the oscillation of the simple pendulum of a clock is circular oscillation. A wave is a correlated collection of oscillations. For example, in a wave traveling along a string, each point in the string oscillates back and forth in the transverse direction (perpendicular to the direction of the string); in sound waves, each air molecule oscillates back and forth in the longitudinal direction (the direction in which the sound is traveling). Therefore understanding oscillatory motions is the basis of understanding waves. Oscillatory motions and wave-like patterns are common in time series data as well. For example, the number of infected cases of a disease in epidemiology; species migration in ecology, human blood sugar or blood pressure levels in biology; the harvest of crops in agriculture; behavior of consumer price index in economics; share returns in finance; the number of arrivals to a cultural landscape in tourism management, etc. follow regular or irregular wave-like patterns. The Auto-Regressive Integrated Moving Average (ARIMA), Seasonal Auto-Regressive Integrated Moving Average (SARIMA), Circular Model (CM), and Sama Circular Model (SCM) were successful in modeling such series. The literature revealed that the daily infected cases of Covid 19 show irregular wave-like patterns with; increasing amplitudes, decreasing amplitudes, or both, but none of the existing time series forecasting techniques are capable of capturing them. The pattern of these series is somewhat similar to the pattern of Damped oscillation and Forced oscillation described in Physics. Hence the authors of the study intended to develop suitable forecasting techniques to model such time series and developed two new stochastic models named; Damped Circular Model (DCM)  and Forced Circular Model (FCM). The development of the models was based on the Circular model (which was based on Simple harmonic motion), the theory of Damped and Forced Oscillations, and the Second-order Differential Equations. It is recommended to test the DCM and FCM on real-life data in the fields of epidemiology and others.  

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Irwanto, M., Norfadilah, N. Gomesh, M. Irwan, and M. R. Mamat. "Improvement of Dynamic Electrical Power System Stability Using Riccati Matrix Method." Applied Mechanics and Materials 793 (September 2015): 29–33. http://dx.doi.org/10.4028/www.scientific.net/amm.793.29.

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Power system stability means the ability to develop restoring forces equal to or greater than the disturbing forces to maintain the state of equilibrium. Successful operation of a power system depends largely on providing reliable and uninterrupted service to the loads by the power utility. The stability of the power system is concerned with the behavior of the synchronous machines after they have been disturbed. If the disturbance does not involve any net change in power, the machines should return to their original state. Due to small disturbances, power system experience these poorly damped low frequency oscillations. The dynamic stability of power systems are also affected by these low frequency oscillations. This paper presents to analyze and obtain the optimum gain for damping oscillation in SMIB by using Riccati matrix method to improve dynamic power system stability. The result shows that with suitable gain which is act as a stabilizer that taken from Riccati matrix, the oscillations of rotor speed and rotor angle can be well damped and hence the system stability is enhanced.

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43

Visschers, Jim C., Emma Wilson, Thomas Conneely, Andrey Mudrov, and Lykourgos Bougas. "Rapid parameter determination of discrete damped sinusoidal oscillations." Optics Express 29, no. 5 (2021): 6863. http://dx.doi.org/10.1364/oe.411972.

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44

Chikrii, Greta Ts. "On One Problem of Approach for Damped Oscillations." Journal of Automation and Information Sciences 41, no. 10 (2009): 1–9. http://dx.doi.org/10.1615/jautomatinfscien.v41.i10.10.

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45

Tomme, E. B., B. M. Annaratone, and J. E. Allen. "Damped dust oscillations as a plasma sheath diagnostic." Plasma Sources Science and Technology 9, no. 2 (2000): 87–96. http://dx.doi.org/10.1088/0963-0252/9/2/301.

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46

Hinrichsen, Peter F. "Acceleration, Velocity, and Displacement for Magnetically Damped Oscillations." Physics Teacher 57, no. 4 (2019): 250–53. http://dx.doi.org/10.1119/1.5095384.

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47

Dymova, M. V., and M. S. Ruderman. "Resonantly damped oscillations of longitudinally stratified coronal loops." Astronomy & Astrophysics 457, no. 3 (2006): 1059–70. http://dx.doi.org/10.1051/0004-6361:20065051.

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48

Dymova, M. V., and M. S. Ruderman. "Resonantly damped oscillations of longitudinally stratified coronal loops." Astronomy & Astrophysics 463, no. 2 (2006): 759. http://dx.doi.org/10.1051/0004-6361:20065051e.

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49

Poshusta, R. D. "Data Analysis (Damped Oscillations) Using the Genfit Function." Journal of Chemical Education 82, no. 7 (2005): 1101. http://dx.doi.org/10.1021/ed082p1101.1.

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Matsuo, Shigemasa, Toshiyuki Fujii, Norihito Kosugi, and Noriyuki Hatakenaka. "Theory of damped Rabi oscillations at finite temperatures." Journal of Physics: Conference Series 150, no. 2 (2009): 022056. http://dx.doi.org/10.1088/1742-6596/150/2/022056.

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