• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.23 no.8
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• pp.734-741
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• 2013

In contrast of external excitations, parametric excitations can produce a large response when the excitation frequency is away from the linear natural frequencies. The Mathieu equation is the simplest differential equation with periodic coefficients, which lead to the parametric excitation. The Mathieu equation may have the unbounded solutions. This work conducted the stability analysis for the Mathieu equation, using Floquet theory and numerical method. Using Lindstedt's perturbation method, harmonic solutions of the Mathieu equation and transition curves separating stable from unstable motions were obtained. Using Floquet theory with numerical method, stable and unstable regions were calculated. The numerical method had the same transition curves as the perturbation method. Increased stable regions due to the inclusion of damping were calculated.

• Journal of IKEEE
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• v.17 no.4
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• pp.559-563
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• 2013

This paper analytically proves the equivalence between two main oscillator phase noise theories, which are based on the ISF, and Floquet vector, respectively. For this purpose, this study obtains the power spectral density matrix from the ISF-based phase noise theory. As a result, one can prove that the power spectral density matrix obtained from the ISF-based phase noise theory is essentially equivalent to the power spectral density matrix presented by the Floquet vector-based phase noise theory, which manifests the equivalence of the two main theories. This study is intended to provide deeper insight into the relations between the two main theories.

• Ocean Systems Engineering
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• v.4 no.3
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• pp.243-261
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• 2014

Aqua is an underwater biomimetic vehicle designed and built at McGill University that uses six paddles to produce control and propulsion forces. It has the particularity of having time-periodic thrust due to its oscillating paddles. Using an existing model of the vehicle, two types of controller were developed: a PD controller and a Floquet controller. The Floquet controller has the advantage of explicitly addressing the time-periodicity of the system. The performance of the controllers was assessed through simulation and experimentally in the Caribbean Sea. We find that the vehicle was able to follow the prescribed trajectories with relative accuracy using both controllers, though, the Floquet controller slightly outperforms the PD controller. Furthermore, a key advantage of the Floquet controller is that it requires no tuning while the PD controller had to be tuned by trial and error.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.10 s.241
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• pp.1321-1328
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• 2005

Based upon the Floquet theory, the complex modal solution for general rotor systems with periodically time-varying parameters is newly derived. The complete modal response can be obtained from the orthonormality condition between the time-variant eigenvectors and the corresponding adjoint vectors. The harmonic solutions such as the response and directional special a patterns are then derived in terms of harmonic modes whose coefficients are obtained from the modal analysis. The stability analysis by the Floquet's transition matrix and the eigen-analysis is also performed.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.3
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• pp.323-329
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• 2017

In this paper, by rigorous calculations, we consider $L^2({\mathbb{R}})-spectrum$ of linear differential operators with periodic coefficients. These operators are usually seen in linearization of nonlinear partial differential equations about spatially periodic traveling wave solutions. Here, by using the solution operator obtained from Floquet theory, we prove rigorously that $L^2({\mathbb{R}})-spectrum$ of the linear operator is determined by the eigenvalues of Floquet matrix.

• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.115-119
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• 1991

In this paper we obtain the asymptotic behavior of solutions of the perturbed system x' = (A(t) + B(t))x of x' = A(t)x by using the Floquet theorem.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2013.04a
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• pp.267-268
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• 2013

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.40 no.4
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• pp.27-33
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• 2003

Circular Distributed-feedback (DFB) guiding structures can be incorporated in most of the semiconductor laser devices because of the frequency-selective property applicable as an optical filter in optical communications. In this paper, we present a novel and simple modal transmission-line theory (MTLT) using Floquet-Babinet's principle to analyze the optical filtering characteristics of Bragg gratings with cylindrical profile. The numerical results reveal that this method offers a simple and convenient algorithm to analyze the filtering characteristics of circular DFB configurations as well as is extended conveniently to evaluate the guiding problems of circular multi-layered periodic structures.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.877-882
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• 2005

Dynamic stability of an axially accelerating beam stucture is investigated in this paper. The equations of motion of a fixed-free beam are derived using the hybrid deformation variable method and the assumed mode method. Unstable regions due to periodical acceleration are obtained by using the Floquet's theory. Stability diagrams are presented to illustrate the influence of the dimensionless acceleration, amplitude, and frequency. Also, buckling occurs when the acceleration exceeds a certain value. It is found that relatively targe unstable regions exist around the first bending natural frequency, twice the first bending natural frequency, and twice the second bending natural frequency. The validity of the stability diagram is confirmed by direct numerical integration of the equations of motion.

• Journal of KSNVE
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• v.9 no.6
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• pp.1157-1165
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• 1999

Dynamic stability of a flying structure undertaking constnat and pulsating thrust force is investigated in this paper. The equations of motion of the structure, which is idealized as a free-free beam, are derived by using the hybrid variable method and the assumed mode method. The structural system includes a directional control unit to obtain the directional stability. Unstable regions due to periodically pulsating thrust forces are obtained by using the Floquet's theory. Stability diagrams are presented to illustrate the influence of the constant force, the location of gimbal, and the frequency of pulsating force. The validity of the diagrams are confirmed by direct numerical simulations of the dynamic system.