A Floquet-Based Analysis of Parametric Excitation Through the Damping Coefficient

The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu’s equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.

Exact solutions of Mathieu’s equation

Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Mathieu’s eigenvalue problem −y′′(x) + 2e_0 cos(2x)y(x) = λy(x), 0 < x < ℓ is symmetric if cos(2x) = cos(2ℓ − 2x) for ℓ = k0π, k0 ∈ N, and skew-symmetric if cos(2x) = − cos(2ℓ − 2x) for ℓ = π/2. Two typical boundary conditions are considered. When the eigenfunctions are expanded by the orthonormal bases of sine functions or cosine functions, we can derive an n-dimensional matrix eigenvalue problem, endowing with a special structure of the symmetric coefficient matrix A := [a_ij], a_ij = 0 if i + j is an odd integer. Based on it, we can obtain the eigenvalues easily and analytically. When ℓ = k_0π, k_0 ∈ N, we have a_ij = 0 if |i − j| > 2k_0. Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k_0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu’s equation.

Oscillatory criteria for the second order linear ordinary differential equations

The Riccati equation method is used to establish three new oscillatory criteria for the second order linear ordinary differential equations. We show that the first of these criteria in the continuous case of the coefficient function (potential) of the equation implies the J. Deng’s oscillatory criterion. An extremal oscillatory condition for the Mathieu’s equation is obtained. The obtained results are compared with some known oscillatory criteria.

Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

This work is concerned with Mathieu's equation—a classical differential equation, which has the form of a linear second-order ordinary differential equation (ODE) 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.

CALCULATION OF PARAMETRICAL VIBRATIONS OF A SHIP SHAFT LINE SUBJECT TO RIGIDITY OF A STERN BEARING

The paper studies parametrical vibrations of the ship shaft line, which arise because of harmonical change in time of rigidity of a propeller shaft and a stern bearing. The design model of a propeller shaft shows a beam with a cross section constant along its length, which leans on hinged immovable and springy support simulating a stern bearing. At the end of a beam there is a disk simulating a propeller screw. Parametrical vibrations arise due to the external loading and as a result of amortization of the stern bearing. In the analysis of parametrical vibrations of the ship shaft line there are used Mathieu's equation and Strutt-Ince diagram. Dynamic stability of a ship shaft line is defined subject to a gap between a propeller shaft and a stern bearing.

Parametric Excitation of a Repulsive Force Actuator

Parametric resonances in a repulsive-force MEMS resonator are investigated. The repulsive force is generated through electrostatic fringe fields that arise from a specific electrode configuration. Because of the nature of the electrostatic force, parametric resonance occurs in this system and is predicted using Mathieu’s Equation. Governing equations of motion are solved using numerical shooting techniques and show both parametric and subharmonic resonance at twice the natural frequency. The primary instability tongue for parametric resonance is also mapped. This is of particular interest for MEMS sensors that require high signal-to-noise ratios due to the large oscillation amplitudes associated with parametric resonance.