Mathieu functions and its useful approximation for elliptical waveguides

Pillay, Shamini; Kumar, Deepak

doi:10.1051/epjconf/201716201064

Cited by 2 publications

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“…LPTV systems are used to describe the motion of lunar perigee (Hill, 1886), (Hill, 1878), vibrations of stretched elliptical membranes (Mathieu, 1868), the motion of side rods of a locomotive (Meissner, 1918), elliptical waveguides (McLachlan, 1947), (Pillay & Kumar, 2017), the motion of gravitationally stabilized Earth-pointing satellites (Schechter, 1964), quadrupole mass spectrometry (Dawson, 1976), the rolling motion of ships (Jovanoski & Robinson, 2009), micromechanical tuning fork gyroscope dynamics (King, 1989), pendulum dynamics (Seyranian & Seyranian, 2006) and (Rugh, 1996) Example 5.21, helicopter rotors (Friedmann, 1986), wind turbines (Stol, Balas, & Bir, 2002), multistage DC-DC converters (Li, Guo, Ren, Zhang, & Zhang, 2017), etc. An LPTV system can be obtained by linearizing a nonlinear system that has periodic components (e.g., the vertical motion of a pendulum undergoes periodic motion, where the pendulum's angle with the vertical axis and its angular velocity is the output).…”

Section: Lptv Systems Applications and Physical Examplesmentioning

confidence: 99%

Analysis of Linear Time-Varying & Periodic Systems

Fivel¹

2021

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This thesis applies Floquet theory to analyze linear periodic time-varying (LPTV) systems, which are represented by a system of ordinary differential equations (ODEs) that depend on a time variable 𝑡 and have a matrix of coefficients with period 𝑇 > 0 (or equivalently, with frequency 𝜔 = 2𝜋 𝑇).According to Floquet theory, the transition matrix of an LPTV system represented by a square periodic-function matrix 𝐴(𝑡) = 𝐴(𝑡 + 𝑇) can be expressed as the product of a square periodic function matrix 𝑃(𝑡) = 𝑃(𝑡 + 𝑇) and an exponentiated square matrix of the form 𝑅𝑡, where 𝑅 is a constant matrix (independent of 𝑡). Despite the validity of Floquet theory, it is difficult to find an analytical closed form for the matrices 𝑃(𝑡) and 𝑅 when the transition matrix Φ 𝐴 (𝑡, 𝑡 0 ) is unknown. In essence, it is difficult to find an analytical solution for an LPTV system (i.e., a closed form for its transition matrix).The objective of this work is to characterize a sufficiently large family of periodic matrices 𝐴(𝑡) with a finite number of harmonics such that the periodic part 𝑃(𝑡) of the solution also has a finite number of harmonics. In addition, for each family, an effective procedure is required to find the matrices 𝑃(𝑡) and 𝑅.The research method involves studying periodic matrices 𝐴(𝑡) with frequency 𝜔 as a free parameter (i.e., not a fixed number) to obtain a sufficiently large family of LPTV systems so that their transition matrices are defined by the frequency 𝜔. In this work, we focus on cases in which the Fourier coefficients of 𝐴(𝑡) are finite polynomials in 𝜔, 𝑅 is a finite polynomial in 𝜔, and the Fourier coefficients of 𝑃(𝑡) do not depend on 𝜔.The research results show that for a given family of periodic matrices 𝐴(𝑡), we can compare the powers of 𝜔 that multiply the harmonics (i.e., 𝜔 is part of the coefficients multiplying the cosine [sine] factors in even [odd] representations or of the exponential factors in complex representations) to determine the matrices -ii-July 2021 𝑃(𝑡) and 𝑅. In addition, the results lead to relations between LPTV systems at frequency 𝜔 and the associated linear time-invariant system, which is defined by having zero frequency (𝜔 = 0). A buy-product of using the frequency 𝜔 as a free parameter is that allows the stability of the LPTV system to be determined based on how 𝑅 depends on the frequency 𝜔.

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Section: Lptv Systems Applications and Physical Examplesmentioning

confidence: 99%

Analysis of Linear Time-Varying & Periodic Systems

Fivel¹

2021

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Study of triaxial loading of segregated granular assemblies through experiments and DEM simulations

Pola,

Annabattula

2024

Preprint

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A simple position-dependent body force-based confinement for simulating triaxial tests using the Discrete Element Method is presented. The said method is used to perform triaxial simulations on mono-disperse and segregated assemblies of glass spheres. The macroscopic load response obtained in simulations is validated with experimental load response. A mesh construction algorithm is presented to check whether the confinement applied in the triaxial simulations is accurate. The particle displacement data obtained from triaxial simulations are used to obtain a particle-wise average strain tensor. This is further used to compare the strain localisation between the mono-disperse and segregated assemblies. It is observed that, in the segregated assembly, the interface between the two particle phases acts as a barrier for strain localisation, and the smaller particles preferentially undergo a higher degree of shear strain on average.

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Mathieu functions and its useful approximation for elliptical waveguides

Abstract: Abstract. The standard form of the Mathieu differential equation is

Cited by 2 publications

References 2 publications

Analysis of Linear Time-Varying & Periodic Systems

Analysis of Linear Time-Varying & Periodic Systems

Study of triaxial loading of segregated granular assemblies through experiments and DEM simulations

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