Results 141 to 150 of about 71,208 (194)

Applications of the Mathieu equation

American Journal of Physics, 1996
The properties of the Mathieu equation are reviewed in order to discuss some of the applications that have appeared in recent years. Those mentioned are: vibrations in an elliptic drum, the inverted pendulum, the radio frequency quadrupole, frequency modulation, stability of a floating body, alternating gradient focusing, the Paul trap for charged ...
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Fractional Mathieu equation

Communications in Nonlinear Science and Numerical Simulation, 2010
After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE: x″+(δ+εcost)x+cDαx=0 where Dαx is the order α derivative of x(t), where 0 < α < 1.
Richard H. Rand, Si M. Sah, Meghan K. Suchorsky   +2 more
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Asymmetric Mathieu equations

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2006
An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible).
Amol Marathe, Anindya Chatterjee
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A Quasiperiodic Mathieu–Hill Equation

SIAM Journal on Applied Mathematics, 1980
A study is made of a generalized Mathieu–Hill equation \[u'' + [ \delta + \varepsilon (\cos 2t + \alpha \cos 2(\lambda + \varepsilon )t) ]u = 0,\] where $\delta$, $\varepsilon$, $\alpha$, $\lambda$ are constants, with $| \varepsilon | \ll 1$ and $\lambda $ a rational fraction.
Davis, Stephen H., Rosenblat, S.
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A Quasiperiodic Mathieu Equation

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems, 1995
Abstract In this work we investigate the following quasiperiodic Mathieu equation: x ¨ + ( δ + ϵ cos ⁡ t + ϵ cos ⁡ ω t ) x = 0 We use numerical integration to determine regions of stability in the δ–ω plane for fixed ϵ. Graphs of these stability regions are presented, based on extensive computation.
Richard Rand, Rachel Hastings
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The stochastic Mathieu's equation

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
In this manuscript, we consider generalizations of the classical Mathieu's equation to stochastic systems. Unlike previous works, we focus on internal frequencies that vary continuously between periodic and stochastic variables. By numerically integrating the system of equations using a symplectic method, we determine the Lyapunov exponents for a wide ...
Poulin, Francis J., Flierl, Glenn R.
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Interplays between Harper and Mathieu equations

Physical Review E, 2001
This paper deals with the application of relationships between Harper and Mathieu equations to the derivation of energy formulas. Establishing suitable matching conditions, one proceeds by inserting a concrete solution to the Mathieu equation into the Harper equation.
E, Papp, C, Micu
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Matrix solution of periodic mathieu equations

Journal of Computational Physics, 1973
Abstract The application of matrix methods to periodic Mathieu equations is discussed, and it is shown that accurate solutions may be found for any real value of the parameter, including the asymptotic case.
Ewig, Carl S., Harris, David O.
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Slow Passage through Resonance in Mathieu's Equation

Journal of Vibration and Control, 2002
We investigate slow passage through the 2:1 resonance tongue in Mathieu's equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions.
Ng, Leslie, Rand, Richard, O'Neil, Michael   +2 more
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THE NONLINEAR MATHIEU EQUATION

International Journal of Bifurcation and Chaos, 1994
The purpose of this paper is to classify the different sequences of bifurcation that can occur for small amplitude solutions to the nonlinear Mathieu equation near to the Mathieu regions of instability. We do this by using the Lindstedt-Poincare perturbation method to construct a vector field which interpolates the successive iterations of the ...
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