Coupled Mathieu Equations: γ-Hamiltonian and μ-Symplectic

2020 ◽  
Author(s):  
Miguel Ramírez Barrios ◽  
Joaquín Collado ◽  
Fadi Dohnal
Keyword(s):  
Mathieu Equations

Stability diagrams for coupled Mathieu-equations

1985 ◽  
Vol 55 (6) ◽  
pp. 463-473 ◽  
Author(s):  
J. Hansen
Keyword(s):  
Stability Diagrams ◽  
Mathieu Equations

scholarly journals Stability criteria for generalized Mathieu equations

1976 ◽  
Vol 34 (3) ◽  
pp. 301-304 ◽  
Author(s):  
E. Infeld
Keyword(s):  
Stability Criteria ◽  
Mathieu Equations

Interplays between Harper and Mathieu equations

2001 ◽  
Vol 64 (5) ◽  
Author(s):  
E. Papp ◽  
C. Micu
Keyword(s):  
Mathieu Equations

scholarly journals Liquid Vibrations in Cylindrical Tanks with and Without Baffles Under Lateral and Longitudinal Excitations

2020 ◽  
Vol 25 (3) ◽  
pp. 117-132
Author(s):  
E. Strelnikova ◽  
D. Kriutchenko ◽  
V. Gnitko ◽  
A. Tonkonozhenko
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Domain Boundary ◽  
Forced Vibrations ◽  
Boundary Element Methods ◽  
Parametric Vibrations ◽  
Harmonic Excitations ◽  
Basic Functions ◽  
Cylindrical Tanks ◽  
Mathieu Equations

AbstractThe paper is devoted to issues of estimating free surface elevations in rigid cylindrical fluid-filled tanks under external loadings. The possibility of baffles installation is provided. The liquid vibrations caused by lateral and longitudinal harmonic loadings are under consideration. Free, forced and parametrical vibrations are examined. Modes of the free liquid vibrations are considered as basic functions for the analysis of forced and parametric vibrations. The modes of the free liquid vibrations in baffled and un-baffled cylindrical tanks are received by using single-domain and multi-domain boundary element methods. Effects of baffle installation are studied. The problems of forced vibrations are reduced to solving the systems of second order ordinary differential equations. For parametric vibrations the system of Mathieu equations is obtained. The numerical simulation of free surface elevations at different loadings and baffle configurations is accomplished. Beat phenomena effects are considered under lateral harmonic excitations. The phenomenon of parametric resonance is examined under longitudinal harmonic excitations.

Pattern Formation in a Parametrically-Excited P.D.E.

2001 ◽  
Author(s):  
Michael D. Stubna ◽  
Richard H. Rand
Keyword(s):  
Fourier Series ◽  
Pattern Formation ◽  
Boundary Conditions ◽  
Limit Cycle ◽  
Neumann Boundary ◽  
Slow Flow ◽  
Rectangular Region ◽  
Parametrically Excited ◽  
Type Of Behavior ◽  
Mathieu Equations

Abstract We investigate the dynamics of the parametrically-excited P.D.E.(1)∂2u∂t2-c2(∂2u∂x2+∂2u∂y2)+εβ∂u∂t+(∂+εγcos⁡t)u=εαu3 with Neumann boundary conditions on a rectangular region:∂u∂x=0forx=0,π and ∂u∂y=0fory=0,πμ where 0 < μ ≤ 1. Our approach involves expanding u(x, y, t) in a 3-term Fourier series truncation:(2)u=f0(t)+f1(t)cos⁡x+f2(t)cos⁡μy By substituting (2) into (1) we obtain a system of 3 coupled nonlinear Mathieu equations which we analyze using averaging in the neighborhood of 2 : 1 resonance. By varying the parameters c and δ we obtain bifurcation curves which divide the cδ-plane into more than forty regions, each containing a distinct slow flow. Individual regions are found to differ from one another with respect to such features as the number and character of slow flow equilibria, and the presence or absence of a limit cycle. When interpreted in the original variable u, these regions account for a variety of patterns which may be classified as stationary, traveling or rotating. This type of behavior is comparable to various experimental observations made by other investigators on vertically driven fluids or sand.

scholarly journals On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

2019 ◽  
Vol 96 (1) ◽  
pp. 737-750 ◽  
Author(s):  
Subhadip Biswas ◽  
Jayanta K. Bhattacharjee
Keyword(s):  
Higher Order ◽  
Quantum Oscillator ◽  
Mathieu Equations

Free Vibration of Thin Shallow Elliptical Shells

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
April Bryan
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Transverse Motion ◽  
Transverse Strain ◽  
Compatibility Equation ◽  
New Approach ◽  
Strain Compatibility ◽  
Mathieu Equations

This research presents a study of the free vibration of thin, shallow elliptical shells. The equations of motion for the elliptical shell, which are developed from Love's equations, are coupled and nonlinear. In this research, a new approach is introduced to uncouple the transverse motion of the shallow elliptical shell from the surface coordinates. Through the substitution of the strain-compatibility equation into the differential equations of motion in terms of strain, an explicit relationship between the curvilinear surface strains and transverse strain is determined. This latter relationship is then utilized to uncouple the spatial differential equation for transverse motion from that of the surface coordinates. The approach introduced provides a more explicit relationship between the surface and transverse coordinates than could be obtained through use of the Airy stress function. Angular and radial Mathieu equations are used to obtain solutions to the spatial differential equation of motion. Since the recursive relationships that are derived from the Mathieu equations lead to an infinite number of roots, not all of which are physically meaningful, the solution to the eigenvalue problem is used to determine the mode shapes and eigenfrequencies of the shallow elliptical shell. The results of examples demonstrate that the eigenfrequencies of the thin shallow elliptical shell are directly proportional to the curvature of the shell and inversely proportional to the shell's eccentricity.

scholarly journals Instability maps of a system of Mathieu-Equations using different perturbation methods

PAMM ◽  
2011 ◽  
Vol 11 (1) ◽  
pp. 319-320
Author(s):  
Stefan Hubinger ◽  
Hubert Gattringer ◽  
Hartmut Bremer ◽  
Karl Mayrhofer
Keyword(s):  
Perturbation Methods ◽  
Mathieu Equations

Spatial bifurcation problems, solved by Mathieu equations

1977 ◽  
Vol 4 (1) ◽  
pp. 1-4
Author(s):  
H. Schmieg ◽  
P. Vielsack
Keyword(s):  
Bifurcation Problems ◽  
Mathieu Equations
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