Academic literature on the topic 'Mathieu stability diagram'
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Journal articles on the topic "Mathieu stability diagram"
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Kidambi, R. "Inviscid Faraday waves in a brimful circular cylinder." Journal of Fluid Mechanics 724 (May 8, 2013): 671–94. http://dx.doi.org/10.1017/jfm.2013.178.
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AbstractWe study inviscid Faraday waves in a brimful circular cylinder with pinned contact line. The pinning leads to a coupling of the Bessel modes and leads to an infinite system of coupled Mathieu equations. For large Bond numbers, even though the stability diagrams and the subharmonic and harmonic resonances for the free and pinned contact lines are similar, the free surface shapes can be quite different. With decreasing Bond number, not only are the harmonic and subharmonic resonances very different from the free contact line case but also interesting changes in the stability diagram occur with the appearance of combination resonance tongues. Points on these tongue boundaries correspond to almost-periodic states. These do not seem to have been reported in the literature.
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Azimi, Mohsen. "Parametric Frequency Analysis of Mathieu–Duffing Equation." International Journal of Bifurcation and Chaos 31, no. 12 (September 25, 2021): 2150181. http://dx.doi.org/10.1142/s0218127421501819.
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The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.
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Butikov, Eugene I. "Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation." American Journal of Physics 86, no. 4 (April 2018): 257–67. http://dx.doi.org/10.1119/1.5021895.
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Orszaghova, J., H. Wolgamot, S. Draper, P. H. Taylor, and A. Rafiee. "Onset and limiting amplitude of yaw instability of a submerged three-tethered buoy." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2235 (March 2020): 20190762. http://dx.doi.org/10.1098/rspa.2019.0762.
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In this paper the dynamics of a submerged axi-symmetric wave energy converter are studied, through mathematical models and wave basin experiments. The device is disk-shaped and taut-moored via three inclined tethers which also act as a power take-off. We focus on parasitic yaw motion, which is excited parametrically due to coupling with heave. Assuming linear hydrodynamics throughout, but considering both linear and nonlinear tether geometry, governing equations are derived in 6 degrees of freedom (DOF). From the linearized equations, all motions, apart from yaw, are shown to be contributing to the overall power absorption. At higher orders, the yaw governing equation can be recast into a classical Mathieu equation (linear in yaw), or a nonlinear Mathieu equation with cubic damping and stiffness terms. The well-known stability diagram for the classical Mathieu equation allows prediction of onset/occurrence of yaw instability. From the nonlinear Mathieu equation, we develop an approximate analytical solution for the amplitude of the unstable motions. Comparison with regular wave experiments confirms the utility of both models for making relevant predictions. Additionally, irregular wave tests are analysed whereby yaw instability is successfully correlated to the amount of parametric excitation and linear damping. This study demonstrates the importance of considering all modes of motion in design, not just the power-producing ones. Our simplified 1 DOF yaw model provides fundamental understanding of the presence and severity of the instability. The methodology could be applied to other wave-activated devices.
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Li, Q. M., and M. X. Shi. "Intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2096 (April 2008): 1941–53. http://dx.doi.org/10.1098/rspa.2007.0253.
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Based on an equivalent continuum cylindrical shell model, we have predicted the intermittent transformation between radial breathing and flexural vibration modes in a single-walled carbon nanotube. It is found that the radial breathing and flexural vibration modes may appear intermittently, in certain circumstances, when the dominant parameters of the problem are in the instable region of the Mathieu stability diagram. The coupled nonlinear differential equations of the radial breathing and the flexural vibration modes are presented to a fourth-order approximation, which is then solved using a finite series expansion method. Intermittent transformation between radial breathing and flexural vibration modes is predicted, which may influence the physical properties of the carbon nanotube and the Raman spectroscope measurements.
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Shi, M. X., Q. M. Li, and Y. Huang. "Internal resonance of vibrational modes in single-walled carbon nanotubes." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2110 (July 22, 2009): 3069–82. http://dx.doi.org/10.1098/rspa.2009.0147.
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We show, by molecular dynamics simulations, that 2:1 internal resonance may occur between a radial breathing mode (RBM) and a circumferential flexural mode (CFM) in single-walled carbon nanotubes (SWCNTs). When the RBM vibration amplitude is greater than a critical value, automatic transformations between the RBM and CFM with approximately half RBM-frequency are observed. This discovery in the discrete SWCNT atom assembly is similar to the 2:1 internal resonance mechanism observed in continuum shells. A non-local continuum shell model is employed to determine the critical conditions for the occurrence of observed 2:1 internal resonance between the RBM and CFMs based on two non-dimensional parameters and the Mathieu stability diagram.
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Orszaghova, J., H. Wolgamot, S. Draper, R. Eatock Taylor, P. H. Taylor, and and A. Rafiee. "Transverse motion instability of a submerged moored buoy." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2221 (January 2019): 20180459. http://dx.doi.org/10.1098/rspa.2018.0459.
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Wave energy converters and other offshore structures may exhibit instability, in which one mode of motion is excited parametrically by motion in another. Here, theoretical results for the transverse motion instability (large sway oscillations perpendicular to the incident wave direction) of a submerged wave energy converter buoy are compared to an extensive experimental dataset. The device is axi-symmetric (resembling a truncated vertical cylinder) and is taut-moored via a single tether. The system is approximately a damped elastic pendulum. Assuming linear hydrodynamics, but retaining nonlinear tether geometry, governing equations are derived in six degrees of freedom. The natural frequencies in surge/sway (the pendulum frequency), heave (the springing motion frequency) and pitch/roll are derived from the linearized equations. When terms of second order in the buoy motions are retained, the sway equation can be written as a Mathieu equation. Careful analysis of 80 regular wave tests reveals a good agreement with the predictions of sub-harmonic (period-doubling) sway instability using the Mathieu equation stability diagram. As wave energy converters operate in real seas, a large number of irregular wave runs is also analysed. The measurements broadly agree with a criterion (derived elsewhere) for determining the presence of the instability in irregular waves, which depends on the level of damping and the amount of parametric excitation at twice the natural frequency.
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Hiroki, Seiji, Tetsuya Abe, and Yoshio Murakami. "Separation of helium and deuterium peaks with a quadrupole mass spectrometer by using the second stability zone in the Mathieu diagram." Review of Scientific Instruments 63, no. 8 (August 1992): 3874–76. http://dx.doi.org/10.1063/1.1143286.
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Белоусова, Елена Петровна. "OPTIMIZATION OF ULTRASOUND MEDICAL PARAMETERS TOOLS." СИСТЕМНЫЙ АНАЛИЗ И УПРАВЛЕНИЕ В БИОМЕДИЦИНСКИХ СИСТЕМАХ, no. 1 (April 19, 2021): 147–54. http://dx.doi.org/10.36622/vstu.2021.20.1.020.
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Для многих видов медицинских вмешательств требуется применение ультразвуковых инструментов с различными характеристиками. Используются инструменты, совершающие продольные колебания, значительно реже - инструменты с изгибами и крутильными колебаниями, либо достаточно длинные ультразвуковые медицинские инструменты, либо короткие, но тонкие. В таких инструментах часто наблюдается так называемая динамическая потеря устойчивости, когда прямолинейный инструмент, совершающий продольные колебания, внезапно начинает совершать изгибные колебания, амплитуда которых бывает настолько высока, что приводит к разрушению инструмента. Такое явление также называют параметрическим резонансом ультразвуковых инструментов. Цель статьи - анализ условий и параметров, позволяющих минимизировать травматичность применения ультразвуковых медицинских инструментов, исследование в динамике устойчивости ультразвуковых низкочастотных медицинских инструментов. Для определения оптимального набора параметров динамической устойчивости изгибных колебаний ультразвуковых низкочастотных медицинских инструментов используется уравнение Матье-Хилла. В этом аспекте решение задачи сводится к определению: 1) границ областей неустойчивости уравнения Матье; 2) границ областей неустойчивости при разных значениях коэффициента возбуждения; 3) границ областей неустойчивости с применением метода малого параметра. Для исследования динамической устойчивости уравнения колебаний прямолинейного стержня переменного сечения достаточно выполнить расчет коэффициентов уравнения Матье и использовать диаграмму Айнса-Стретта для нахождения точек попадания в область устойчивости. Результаты расчетов показали, что инструменты, изготовленные из титана, обладают высокой динамической устойчивостью, что практически исключает вероятность их разрушения при проведении медицинских операций. Полученные характеристики медицинских инструментов указывают на эффективность их применения в медицинской практике Many types of medical interventions require the use of ultrasound instruments with different characteristics. Instruments that perform longitudinal vibrations are used, much less often-instruments with bends and torsional vibrations, or rather long ultrasound medical instruments, or short, but thin. In such instruments, the so-called dynamic loss of stability is often observed, when a straight-line tool that performs longitudinal vibrations suddenly begins to make bending vibrations, the amplitude of which is so high that it leads to the destruction of the tool. This phenomenon is also called parametric resonance of ultrasonic instruments. The purpose of the article is to analyze the conditions and parameters that allow minimizing the traumaticity of the use of ultrasonic medical instruments, to study the dynamics of the stability of ultrasonic low-frequency medical instruments. The Mathieu-Hill equation is used to determine the optimal set of parameters for the dynamic stability of bending vibrations of ultrasonic low-frequency medical instruments. In this aspect, the solution of the problem is reduced to the definition of: 1) the boundaries of the instability regions of the Mathieu equation; 2) the boundaries of the instability regions at different values of the excitation coefficient; 3) the boundaries of the instability regions using the small parameter method. To study the dynamic stability of the equation of oscillations of a rectilinear rod of variable cross-section, it is sufficient to calculate the coefficients of the Mathieu equation and use the Ains-Strett diagram to find the points of falling into the stability region. The results of the calculations showed that the instruments made of titanium have a high dynamic stability, which practically eliminates the possibility of their destruction during medical operations. The obtained characteristics of medical instruments indicate the effectiveness of their use in medical practice
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WRIGHT, JEFF, STEVE YON, and C. POZRIKIDIS. "Numerical studies of two-dimensional Faraday oscillations of inviscid fluids." Journal of Fluid Mechanics 402 (January 10, 2000): 1–32. http://dx.doi.org/10.1017/s0022112099006631.
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The dynamics of two-dimensional standing periodic waves at the interface between two inviscid fluids with different densities, subject to monochromatic oscillations normal to the unperturbed interface, is studied under normal- and low-gravity conditions. The motion is simulated over an extended period of time, or up to the point where the interface intersects itself or the curvature becomes very large, using two numerical methods: a boundary-integral method that is applicable when the density of one fluid is negligible compared to that of the other, and a vortex-sheet method that is applicable to the more general case of arbitrary densities. The numerical procedure for the boundary-integral formulation uses a global isoparametric parametrization based on cubic splines, whereas the numerical method for the vortex-sheet formulation uses a local boundary-element parametrization based on circular arcs. Viscous dissipation is simulated by means of a phenomenological damping coefficient added to the Bernoulli equation or to the evolution equation for the strength of the vortex sheet. A comparative study reveals that the boundary-integral method is generally more accurate for simulating the motion over an extended period of time, but the vortex-sheet formulation is significantly more efficient. In the limit of small deformations, the numerical results are in excellent agreement with those predicted by the linear model expressed by Mathieu's equation, and are consistent with the predictions of the Floquet stability analysis. Nonlinear effects for non-infinitesimal amplitudes are manifested in several ways: deviation from the predictions of Mathieu's equation, especially at the extremes of the interfacial oscillation; growth of harmonic waves with wavenumbers in the unstable regimes of the Mathieu stability diagram; formation of complex interfacial structures including paired travelling waves; entrainment and mixing by ejection of droplets from one fluid into the other; and the temporal period tripling observed recently by Jiang et al. (1998). Case studies show that the surface tension, density ratio, and magnitude of forcing play a significant role in determining the dynamics of the developing interfacial patterns.
Dissertations / Theses on the topic "Mathieu stability diagram"
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Pothula, Sunil George. "Dynamic Response of Composite Cylindrical Shells Under External Impulsive Loads." University of Akron / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=akron1248097987.
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Vernier, Arnaud. "Développement instrumental en spectrométrie de masse pour le diagnostic in vitro en microbiologie clinique." Phd thesis, Université Claude Bernard - Lyon I, 2014. http://tel.archives-ouvertes.fr/tel-00986856.
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La spectrométrie de masse, en particulier le couplage HPLC/MRM3, est un outil bien adapté au diagnostic in vitro, particulièrement en microbiologie clinique. L'utilisation en routine de cette technologie est cependant tributaire de sa sensibilité et de sa spécificité. Ce travail de thèse a pour objectif d'étudier la possibilité d'éjecter et de détecter simultanément et de façon sélective des ions de ratio masse/charge donnés, ceux-ci étant confinés dans un piège ionique quadrupolaire. Cette approche permet de supprimer les étapes de balayage en masse et d'intégration mathématique du signal en mode MRM3 ce qui permet de gagner à la fois en sensibilité et en spécificité (en diminuant le temps de cycle et en diminuant le rapport signal sur bruit). Cet objectif a été poursuivi premièrement par une étude théorique approfondie des équations du mouvement des ions dans un piège radiofréquence ; deuxièmement par une étude numérique de la stabilité de ces équations et enfin troisièmement par une validation expérimentale de ces résultats théoriques. La présentation de ces trois approches fait l'objet du présent mémoire
Conference papers on the topic "Mathieu stability diagram"
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Mamontov, Eugenie V., Roman N. Dyatlov, Alexander A. Dvagilev, and Olga V. Melnik. "The Modeling of Ion Oscillations in Rapidly Oscillating Quadrupole Fields when Crossing the Mathieu Diagram Stability Boundaries." In 2021 10th Mediterranean Conference on Embedded Computing (MECO). IEEE, 2021. http://dx.doi.org/10.1109/meco52532.2021.9460205.
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Osada, Reiko, and Chikara Sato. "Stabilization of Inverted Coupled Pendula Using Parametric Excitation." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4008.
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Abstract Parametric stabilization of a single inverted pendulum has been extensively studied using the Mathieu equation and its corresponding stability diagram. The inverted single pendulum may be stabilized using parametric excitation at a specified frequency and amplitude given by a narrow stable region in the Mathieu diagram. Coupled pendula with parametric excitation or corresponding resonant systems have been studied from mathematical view point (Cesari, 1959; Gambill, 1955; Richards, 1983), from electrical view point (Sato, 1962a; Sato, 1962b; Sato, 1971; Sato, 1975) and from mechanical view point (Sato, 1995). Coupled pendula with parametric excitation have been studied within a limited region by some researchers, including the authors. A study of inverted coupled pendula with parametric excitation has not been performed as far as the authors know. Usually it is assumed that inverted coupled pendula are unstable in the absence of any other stabilizing mechanism such as feedback. One question is whether the inverted coupled pendula could be stabilized only by parametric excitation? The present paper gives an affirmative answer to this question in a limited and finite region. The stability is also examined using the differential equations and other methods.
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Tweten, Dennis J., Genevieve M. Lipp, Firas A. Khasawneh, and Brian P. Mann. "Comparative Study of Semi-Analytical Methods for the Stability Analysis of Delay Differential Equations." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47519.
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This paper describes and compares the zeroth-order semi-discretization, spectral element, and Legendre collocation methods. Each method is a technique for solving delay differential equations (DDEs) as well as determining regions of stability in the DDE parameter space. We present the necessary concepts, assumptions, and equations required to implement each method. To compare the relative performance between the methods, the convergence rate achieved and computing time required by each method are determined in two numerical studies consisting of a ship stability example and the delayed damped Mathieu equation. For each study, we present a stability diagram in parameter space and a convergence plot. The spectral element method is demonstrated to have the quickest convergence rate while the Legendre collocation method requires the least computing time. The zeroth-order semi-discretization method on the other hand has both the slowest convergence rate and requires the most computing time.
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Sharma, Ashu, and Subhash C. Sinha. "An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory." In ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/detc2017-68041.
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Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.
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Gavassoni, Elvidio, Paulo Batista Gonçalves, and Deane M. Roehl. "Nonlinear Dynamics of a Spar Platform." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70384.
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Spar floating platforms have been largely used for deep water drilling, oil and natural gas production and storage of these fluids. In extreme weather conditions such structures may exhibit a highly nonlinear dynamic behavior. In this paper a 2-DOF model is used to study the heave and pitch coupled response in free and forced vibration. Special attention is given to the determination of the nonlinear vibration modes (NNMs). Non-similar and similar NNMs are obtained analytically by direct application of asymptotic methods and the results show important NNM features such as instability and multiplicity of modes. The NNMs are used to generate reduced order models that consist of 1-DOF nonlinear oscillators. It facilitates the parametric analysis and the derivation of important features of the system such as the frequency-amplitude relation associated to each nonlinear mode and resonance curves. The stability is analyzed by the Floquet theory. Bifurcation diagrams and Mathieu charts are used to identify the unstable regions in the force parameter space. The analytical results show good agreement with the numerical solution obtained by direct integration of the equation of motion.