Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features

Ivana Kovacic; Richard H. Rand; Si Mohamed Sah

doi:10.1115/1.4039144

Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features

Ivana Kovacic, Richard H. Rand, Si Mohamed Sah

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Abstract

This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.

Original language	English
Article number	020802
Journal	Applied Mechanics Reviews
Volume	70
Issue number	2
Number of pages	22
ISSN	0003-6900
DOIs	https://doi.org/10.1115/1.4039144
Publication status	Published - 2018

Keywords

Parametric excitation
Stability chart
Transition curves
Perturbation method
Floquet theory
Harmonic balancing
Geometric nonlinearity;
Damping nonlinearity
Fractional derivative
Delay
Quasiperiodic excitation
Elliptic-type excitation

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title = "Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features",

abstract = "This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.",

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year = "2018",

doi = "10.1115/1.4039144",

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volume = "70",

journal = "Applied Mechanics Reviews",

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AU - Kovacic, Ivana

AU - Rand, Richard H.

AU - Sah, Si Mohamed

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N2 - This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.

AB - This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.

KW - Parametric excitation

KW - Stability chart

KW - Transition curves

KW - Perturbation method

KW - Floquet theory

KW - Harmonic balancing

KW - Geometric nonlinearity;

KW - Damping nonlinearity

KW - Fractional derivative

KW - Delay

KW - Quasiperiodic excitation

KW - Elliptic-type excitation

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DO - 10.1115/1.4039144

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VL - 70

JO - Applied Mechanics Reviews

JF - Applied Mechanics Reviews

IS - 2

M1 - 020802

ER -

Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features

Abstract

Keywords

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