Mathieu Oscillator Research Articles

Thermo-magnetic convection and convective magneto-gravitational parametric resonance in a horizontal ferrofluid layer under time quasi-periodic vibrations: A balance harmonic analysis

Convective instability in a horizontal ferromagnetic liquid layer with rigid boundaries is investigated in the presence of both an uniform magnetic field and a vertical quasi-periodic forcing having two incommensurate frequencies ω1 and ω2. A linearized convective instability analysis predicting the onset of convection is performed and by means of a Galerkin projection truncated to the first order, the linear dynamic of the system is reduced to a damped quasi-periodic Mathieu oscillator. Using a balance harmonic analysis, it turns out that at sufficiently low vibration frequencies, we recover the critical instability parameters related to the unmodulated case where the convection is mainly due to the magnetic mechanism. This feature means that the buoyancy mechanism does not contribute into the convective cells formation and consequently heating from below or from above does not alter the onset of convection. However, at sufficiently high vibration frequencies, both the magnetic and buoyancy mechanisms are operative and here we are dealing with a magneto-gravitational parametric resonance convection. The transition between these convective modes is occurred via bicritical states giving rise to cusp points in the magnetic Rayleigh number besides discontinuities in its corresponding critical wave number. In addition, it is shown that a proper tuning of the irrational frequency ratio ω=ω2/ω1 provides a control of the convection onset as well as the nature of its mechanism. Moreover, special attention is devoted to highlight the effect of the Prandtl number on the dynamic of the system at different values of the frequency ratio ω.

Adaptive integral alternating minimization method for robust learning of nonlinear dynamical systems from highly corrupted data

This paper proposes an adaptive integral alternating minimization method (AIAMM) for learning nonlinear dynamical systems using highly corrupted measured data. This approach selects and identifies the system directly from noisy data using the integral model, encompassing unknown sparse coefficients, initial values, and outlier noisy data within the learning problem. It is defined as a sparse robust linear regression problem. An adaptive threshold parameter selection method is proposed to constrain model fitting errors and select appropriate threshold parameters for sparsity. The robustness and accuracy of the proposed AIAMM are demonstrated through several numerical experiments on typical nonlinear dynamical systems, including the van der Pol oscillator, Mathieu oscillator, Lorenz system, and 5D self-exciting homopolar disc dynamo. The proposed method is also compared to several advanced methods for sparse recovery, with the results indicating that the AIAMM demonstrates superior performance in processing highly corrupted data.

Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.

Analytical solution of parametrically induced payload nonlinear pendulation in offshore lifting

This paper presents a concept of analytical modelling methodology of a parametrically induced payload pendulation. A nonlinear analytical approximated solution was found for the considered 3-DOF model by simplifying to an elliptically excited Mathieu oscillator. The dynamic responses of the parametric pendulum being excited by bidirectional regular waves applied as kinematic functions at the crane tip were studied by means of the perturbation method.In order to investigate the model’s dynamic properties, an analytical first order solution was formulated through the utilisation of multiple-scale analysis. The accuracy of the solution derived was validated for two scenarios — payload oscillations outside the resonant region and on the resonance curve. The approximate analytical solution was examined for different sets of the system excitation curve parameters.The authors conducted numerical analyses (direct integration of a nonlinear governing equation) in order to give a satisfactory confirmation of the proposed methodology and evaluate the robustness of the obtained solution.Using the first order approximation, the results are found to be more conservative comparing to the simulations however, for all of the presented cases, the solution achieves a very good correspondence with the numerical results. The studies were summarised with an examination of the stability assessment of the lifted object.

On the modeling of a parametric cubic–quintic nonconservative Duffing oscillator via the modified homotopy perturbation method

Abstract This paper is devoted to obtain an approximate solution to the damped quintic–cubic nonlinear Duffing–Mathieu equation via a modified homotopy perturbation method (HPM). The modification under consideration deals with the improvement of the HPM with the exponential decay parameter. This scheme allows us to get a solution to the damped nonlinear Duffing–Mathieu equation, which the classical HPM failed to obtain. It is found that the solutions and the characteristic curves are affected by the presence of the damping force. The frequency-amplitude characteristics of a symbiotic solution are confirmed as well as the stability condition is carried out in the (non)-resonance cases. All the calculations are done via Mathematica. The comparison between both of the numerical and analytical solutions showed a very good agreement. Illustrated graphs are plotted for a superior realization of periodic motions in the Duffing–Mathieu oscillator. Nonlinear behaviors of each oscillation motion have been characterized through frequency curves.

A Mobile Mathieu Oscillator Model for Vibrational Locomotion of a Bristlebot

Abstract Terrestrial locomotion that is produced by creating and exploiting frictional anisotropy is common amongst animals such as snakes, gastropods, and limbless lizards. In this paper we present a model of a bristlebot that locomotes by generating frictional anisotropy due to the oscillatory motion of an internal mass and show that this is equivalent to a stick–slip Mathieu oscillator. Such vibrational robots have been available as toys and theoretical curiosities and have seen some applications such as the well-known kilobot and in pipe line inspection, but much remains unknown about this type of terrestrial locomotion. In this paper, motivated by a toy model of a bristlebot made from a toothbrush, we derive a theoretical model for its dynamics and show that its dynamics can be classified into four modes of motion: purely stick (no locomotion), slip, stick–slip, and hopping. In the stick mode, the dynamics of the system are those of a nonlinear Mathieu oscillator and large amplitude resonance oscillations lead to the slip mode of motion. The mode of motion depends on the amplitude and frequency of the periodic forcing. We compute a phase diagram that captures this behavior, which is reminiscent of the tongues of instability seen in a Mathieu oscillator. The broader result that emerges in this paper is that mobile limbless continuum or soft robots can exploit high-frequency parametric oscillations to generate fast and efficient terrestrial motion.

First passage time of nonlinear diffusion processes with singular boundary behavior

New theorems for moments of the first passage time of one dimensional nonlinear stochastic processes with an entrance boundary xe are formulated. This important class of one dimensional stochastic processes results among others from approximations of the energy or amplitude of second order nonlinear stochastic differential equations. Since the diffusion of a stochastic process vanishes at an entrance boundary, xe is called a singular point of the stochastic process. The theorems for the moments of the first passage times are validated based on existing analytical results. In addition, the first passage times of a forced and damped Mathieu oscillator, as well as a nonlinear stochastic differential equation, which is important for the determination of dangerous ship roll dynamics, are calculated. The proposed analytical expressions for the moments of the first passage times can be calculated very fast using standard quadrature formulas.

Non-Perturbative Treatment of Quantum Mathieu Oscillator

We study the evolution in time of the quantum Mathieu oscillator (QMO), according to the motion of a charged particle in a radio frequency Paul trap. We adopt non-perturbative treatment based on the quantized Floquet formalism together with the resonating averages method (RAM). We prove that we can develop solutions of the time-dependent Schrödinger equation of such a system, in terms of the simple harmonic oscillator wave functions. Numerical simulations of the analytical results are performed to show the coherence and the squeezed proprieties of the wave-packet of this system.

First passage time as an analysis tool in experimental wind engineering

The first passage time is the time required for a system to evolve from an initial configuration to a certain target state. This concept is of high interest for the study of transient regimes which are widely represented in wind engineering. Although the concept has been widely studied from theoretical and numerical standpoints, there are very few practical or experimental applications where the results are seen from this angle. This work is a first attempt at bringing first passage times of stochastic systems into wind engineering by suggesting the use of a first passage time map as a standard analysis tool in experimental wind engineering. The wind tunnel data related to the spinning dynamics of a rotating square cylinder in turbulent flow is processed under the frame suggested by a theoretical model for a simple linear oscillator. A specific algorithm is developed for the determination of the average first passage time as a function of initial and target conditions based on the experimental measurements. It is shown that the simple theoretical model is able to capture the different regimes of the experimental setup, so that an equivalent linear Mathieu oscillator, presenting the same evolution of energy, from a first passage time point of view, was identified. This experimental investigation provides a first link between an analytical but simplified result and a more complex reality.

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

This work focuses on the stochastic version of the linear Mathieu oscillator with both forced and parametric excitations of small intensity. In this quasi-Hamiltonian oscillator, the concept of energy stored in the oscillator plays a central role and is studied through the first passage time, which is the time required for the system to evolve from a given initial energy to a target energy. This time is a random variable due to the stochastic nature of the loading. The average first passage time has already been studied for this class of oscillator. However, the spread has only been studied under purely parametric excitation. Extending to combinations of both forcing and parametric excitations, this work provides a closed-form solution and a thorough analytical study of the coefficient of variation of the first passage time of the energy in this system. Simple asymptotic solutions are also derived in some particular ranges of parameters corresponding to different regimes.

Resonance Analysis of Fractional-Order Mathieu Oscillator

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.

Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations

A linear oscillator simultaneously subjected to stochastic forcing and parametric excitation is considered. The time required for this system to evolve from a low initial energy level until a higher energy state for the first time is a random variable. Its expectation satisfies the Pontryagin equation of the problem, which is solved with the asymptotic expansion method developed by Khasminskii. This allowed deriving closed-form expressions for the expected first passage time. A comprehensive parameter analysis of these solutions is performed. Beside identifying the important dimensionless groups governing the problem, it also highlights three important regimes which are called incubation, multiplicative and additive because of their specific features. Those three regimes are discussed with the parameters of the problem.

On the Heteroclinic Connections in the 1:3 Resonance Problem

Analytical predictions of the triangle and clover heteroclinic bifurcations in the problem of self-oscillations stability loss near 1:3 resonance are provided using the method of nonlinear time transformation. The analysis was carried out considering the slow flow of a self-excited nonlinear Mathieu oscillator corresponding to the normal form near this 1:3 strong resonance. Using the Hamiltonian system of the corresponding slow flow near this resonance, the unperturbed zero-order approximation of the heteroclinic connections is established. Conditions of persistence of homoclinic connections in the perturbed first-order approximation of the heteroclinic connections provide close analytical approximations of the triangle and clover heteroclinic bifurcation curves, simultaneously. The analytical predictions are compared to the results obtained by numerical simulations for validation.

An analytical method for Mathieu oscillator based on method of variation of parameter

A simple, but very accurate analytical method for forced Mathieu oscillator is proposed, the idea of which is based on the method of variation of parameter. Assuming that the time-varying parameter in Mathieu oscillator is constant, one could easily obtain its accurately analytical solution. Then the approximately analytical solution for Mathieu oscillator could be established after substituting periodical time-varying parameter for the constant one in the obtained accurate analytical solution. In order to certify the correctness and precision of the proposed analytical method, the first-order and ninth-order approximation solutions by harmonic balance method (HBM) are also presented. The comparisons between the results by the proposed method with those by the numerical simulation and HBM verify that the results by the proposed analytical method agree very well with those by the numerical simulation. Moreover, the precision of the proposed new analytical method is not only higher than the approximation solution by first-order HBM, but also better than the approximation solution by the ninth-order HBM in large ranges of system parameters.

Periodic and Quasi-periodic Responses of Van der Pol–Mathieu System Subject to Various Excitations

Abstract This paper addresses the steady-state periodic and quasi-periodic responses of van der Pol–Mathieu system subject to three excitations (i.e., self, parametric and external excitations). Method of multiple scales and double perturbation technique are employed to study the original system. The cases of van der Pol–Mathieu oscillator with and without external excitation are considered, and periodic and quasi-periodic solutions are obtained and discussed. In the parametric study, the effects of various parameters and self, parametric and external excitations on the system behaviors are studied. Results from method of multiple scales well agree with those from numerical method.

Accurate frequency domain measurement of the best linear time-invariant approximation of linear time-periodic systems including the quantification of the time-periodic distortions

Time-periodic (TP) phenomena occurring, for instance, in wind turbines, helicopters, anisotropic shaft-bearing systems, and cardiovascular/respiratory systems, are often not addressed when classical frequency response function (FRF) measurements are performed. As the traditional FRF concept is based on the linear time-invariant (LTI) system theory, it is only approximately valid for systems with varying dynamics. Accordingly, the quantification of any deviation from this ideal LTI framework is more than welcome. The “measure of deviation” allows us to define the notion of the best LTI (BLTI) approximation, which yields the best – in mean square sense – LTI description of a linear time-periodic LTP system. By taking into consideration the TP effects, it is shown in this paper that the variability of the BLTI measurement can be reduced significantly compared with that of classical FRF estimators. From a single experiment, the proposed identification methods can handle (non-)linear time-periodic [(N)LTP] systems in open-loop with a quantification of (i) the noise and/or the NL distortions, (ii) the TP distortions and (iii) the transient (leakage) errors. Besides, a geometrical interpretation of the BLTI approximation is provided, leading to a framework called vector FRF analysis. The theory presented is supported by numerical simulations as well as real measurements mimicking the well-known mechanical Mathieu oscillator.

Interfacial instability in a time-periodic rotating Hele-Shaw Cell

The effect of time-periodic angular velocity on the interfacial instability of two immiscible, viscous fluids of different densities and confined in an annular Hele-Shaw cell is investigated. An inviscid linear stability analysis of the viscous and time dependent basic flow leads to a periodic Mathieu oscillator describing the evolution of the interfacial amplitude. We show that the relevant parameters that control the interface are the Bond number, viscosity ratio, Atwood number and the frequency number.

Interfacial instability of two superimposed immiscible viscous fluids in a vertical Hele-Shaw cell under horizontal periodic oscillations

The effect of horizontal periodic oscillations on the interfacial instability of two immiscible, viscous fluids of different densities, confined in a vertical Hele-Shaw cell, is investigated. An inviscid linear stability analysis of the viscous basic flow leads to the periodic Mathieu oscillator describing the evolution of interfacial amplitude. We examine mainly the effect of the periodic oscillations and the influence of the viscosity on the stability of the interface. The results show that a decrease in the viscosity contrast has a stabilizing effect on the Kelvin-Helmholtz instability, which is displaced towards the long-wave region. The effects of other parameters such as the frequency number and the Weber number are also examined.

Stochastic Stability of Damped Mathieu Oscillator Parametrically Excited by a Gaussian Noise

This paper analyzes the stochastic stability of a damped Mathieu oscillator subjected to a parametric excitation of the form of a stationary Gaussian process, which may be both white and coloured. By applying deterministic and stochastic averaging, two Itô’s differential equations are retrieved. Reference is made to stochastic stability in moments. The differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. A necessary and sufficient condition of stability in the moments of order r is that the matrix Ar of the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. In the case of the first moments (averages), critical values of the parameters are expressed analytically, while for the second moments the search for the critical values is made numerically. Some graphs are presented for representative cases.

Effect of Horizontal Vibration on the Interfacial Instability in a Horizontal Hele-Shaw Cell

The e ect of periodic oscillations on the interfacial instability of two immiscible fluids, confined in a horizontal Hele-Shaw cell, is investigated. A linear stability analysis of the basic state leads to a periodic Mathieu oscillator corresponding to the amplitude of the interface. Then, the threshold of parametric instability of the interface is characterized by harmonic or subharmonic periodic solutions. We show that the relevant parameters that control the interface are the Bond number, density ratio, Weber number and amplitude and frequency of oscillations.