On Mathieu Equations
On Mathieu Equations
by
Nikola Mišković, dipl. ing.
Postgraduate course
Differential equations and dynamic systems
Professor: prof. dr. sc. Vesna Županović
The Mathieu Equation
An interesting class of linear differential equations is the class with time variant parameters. One of the most common ones, due to its simplicity and straightforward analysis is the Mathieu equation. The Mathieu function is useful for treating a variety of interesting problems in applied mathematics, including vibrating elliptical drumheads, quadrupoles mass filters and quadrupole ion traps for mass spectrometry, exact plane wave solutions in general relativity, the Stark effect for a rotating electric dipole, the phenomenon of parametric resonance in forced oscillators, [2]. In this work we will focus on parametric resonance effects.
1 The undamped Mathieu equation
The Mathieu Equation in its original form can be described with Eq. (1).
[pic] (1) where [pic] and [pic]. This equation is also known as the undamped Mathieu equation.
Mathieu equation does not have analytical solutions, [2], [3]. However, in some cases, it is not important to know the solution of the equation, but the stability regions.
Some of the properties of the Mathieu equation are as follows: a) linear differential equation b) parameters are periodically time-variant c) solutions are bounded or unbounded – bounded solutions are in the form of limit-cycles (oscillatory) and can never be of a focus type. The type of solutions depend only on the values of a pair (δ, ε), [1]. d) if parameter ε is fixed, the solution of the Mathieu equation will alternate between stable and unstable region as parameter δ increases, [5].
Referring to c), i.e. the nature of the solution, by choosing the parameter pair (δ, ε) = (0.3, 0.5) a bounded (stable) solution which is shown in Figure 1 is obtained. The first graph
by
Nikola Mišković, dipl. ing.
Postgraduate course
Differential equations and dynamic systems
Professor: prof. dr. sc. Vesna Županović
The Mathieu Equation
An interesting class of linear differential equations is the class with time variant parameters. One of the most common ones, due to its simplicity and straightforward analysis is the Mathieu equation. The Mathieu function is useful for treating a variety of interesting problems in applied mathematics, including vibrating elliptical drumheads, quadrupoles mass filters and quadrupole ion traps for mass spectrometry, exact plane wave solutions in general relativity, the Stark effect for a rotating electric dipole, the phenomenon of parametric resonance in forced oscillators, [2]. In this work we will focus on parametric resonance effects.
1 The undamped Mathieu equation
The Mathieu Equation in its original form can be described with Eq. (1).
[pic] (1) where [pic] and [pic]. This equation is also known as the undamped Mathieu equation.
Mathieu equation does not have analytical solutions, [2], [3]. However, in some cases, it is not important to know the solution of the equation, but the stability regions.
Some of the properties of the Mathieu equation are as follows: a) linear differential equation b) parameters are periodically time-variant c) solutions are bounded or unbounded – bounded solutions are in the form of limit-cycles (oscillatory) and can never be of a focus type. The type of solutions depend only on the values of a pair (δ, ε), [1]. d) if parameter ε is fixed, the solution of the Mathieu equation will alternate between stable and unstable region as parameter δ increases, [5].
Referring to c), i.e. the nature of the solution, by choosing the parameter pair (δ, ε) = (0.3, 0.5) a bounded (stable) solution which is shown in Figure 1 is obtained. The first graph
References: 1] Galeazzi R. and Blanke M., “On the feasibility of stabilizing parametric roll with active bifurcation control” 2] Hayashi C., Nonlinear Oscillations in Physical Systems, McGraw-Hill Electrical and Electronic Engineering Series, 1964. 3] Struble R. A., Nonlinear Differential Equations, McGraw-Hill Series in Pure and Applied Mathematics, 1962. 4] Ramani D. V., Keith W. L. and Rand R. H., “Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation”, International Journal of Non-Linear Mechanics 39 (2004), 491- 502 5] Simakhina S 6] Trefethen L. N., Spectral Methods in MATLAB, 1994. 7] Shin Y. S., Belenky V. L., Paulling J. R., Weems K. M. and Lin W. M., “Criteria for parametric roll of large containerships in longitudinal seas”, ABS Technical Papers, 2004 8] Jazar G