Damped Mathieu Equation Research Articles

The stability analysis based on viscous theory of Faraday waves in Hele–Shaw cells

The linear instability of Faraday waves in Hele–Shaw cells is investigated with consideration of the viscosity of fluids after gap-averaging the governing equations due to the damping from two lateral walls and the dynamic behavior of contact angle. A new hydrodynamic model is thus derived and solved semi-analytically. The contribution of viscosity to critical acceleration amplitude is slight compared to other factors associated with dissipation, and the potential flow theory is sufficient to describe onset based on the present study, but the rotational component of velocity can change the timing of onset largely, which paradoxically comes from the viscosity. The model degenerates into a novel damped Mathieu equation if the viscosity is dropped with two damping terms referring to the gap-averaged damping and dissipation from dynamic contact angle, respectively. The former increases when the gap size decreases, and the latter grows as frequency rises. When it comes to the dispersion relation of Faraday waves, an unusual detuning emerges due to the imaginary part of the gap-averaged damping.

Primary Resonance in a Weakly Forced Oscillator With Both Parametric Damping and Stiffness

Abstract We study the primary resonance of a parametrically damped Mathieu equation with direct excitation. Potential applications include wind-turbine blade vibration with cyclic stiffening and aeroelastic effects, which may induce parametric damping, and devices with designed cyclic damping for resonance manipulation. The parametric stiffness, parametric damping, and the direct forcing all have the same excitation frequency, with phase parameters between these excitation sources. The parametric amplification at primary resonance is examined by applying the second-order method of multiple scales. With parametric stiffness and direct excitation, it is known that there is a primary parametric resonance that is an amplifier under most excitation phases, but can be a slight suppressor in a small range of phases. The parametric damping is shown to interact with the parametric stiffness to further amplify, or suppress, the resonance amplitude relative to the resonance under parametric stiffness. The effect of parametric damping without parametric stiffness is to shift the resonant frequency slightly, while inducing less significant resonance amplification. The phase of the parametric damping excitation, relative to the parametric stiffness, has a strong influence on the amplification or suppression characteristics. There are optimal phases of both the direct excitation and the parametric damping for amplifying or suppressing the resonance. The effect of the strength of parametric damping is also studied. Numerical simulations validate the perturbation analysis.

Responses of a Strongly Forced Mathieu Equation—Part 1: Cyclic Loading

AbstractThis work concerns the response of a damped Mathieu equation with hard cyclic excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds resonances and stability. Superharmonic and subharmonic resonances are analyzed and the effect of different parameters on the responses are examined. While superharmonic resonances of order two have been captured by a first-order analysis, the second-order analysis improves the prediction of the peak frequency. Superharmonic resonances of order three are captured only by the second-order analysis. The order-two superharmonic resonance amplitude is of order ε0, and the order-three superharmonic amplitude is of order ε. As the parametric excitation level increases, the superharmonic resonance amplitudes increase. An nth-order multiple-scales analysis will indicate conditions of superharmonic resonances of order n + 1. At the subharmonic of order one-half, there is no steady-state resonance, but known subharmonic instability is unfolded consistently. Analytical expressions for resonant responses are presented and compared with numerical results for specific system parameters. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric resonators.

Responses of a Strongly Forced Mathieu Equation—Part 2: Constant Loading

AbstractThe present study deals with the response of a damped Mathieu equation with hard constant external loading. A second-order perturbation analysis using the method of multiple scales (MMS) unfolds resonances and stability. Non-resonant and low-frequency quasi-static responses are examined. Under constant loading, primary resonances are captured with a first-order analysis, but are accurately described with the second-order analysis. The response magnitude is of order ϵ0, where ϵ is the small bookkeeping parameter, but can become arbitrarily large due to a small denominator as the Mathieu system approaches the primary instability wedge. A superharmonic resonance of order two is unfolded with the second-order MMS. The magnitude of this response is of order ϵ and grows with the strength of parametric excitation squared. An nth-order multiple scales analysis under hard constant loading will indicate conditions of superharmonic resonances of order n. Subharmonic resonances do not produce a non-zero steady-state harmonic, but have the instability property known to the regular Mathieu equation. Analytical expressions for predicting the magnitude of responses are presented and compared with numerical results for a specific set of system parameters. In all cases, the second-order analysis accommodates slow time-scale effects, which enable responses of order ϵ or ϵ0. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric amplifiers.

Rayleigh-Taylor instability of viscous liquid film under an inclined substrate with time modulated temperature

The Rayleigh-Taylor instability of viscous liquid film under a temperature-modulated inclined substrate is studied. Long wave approximation and weighted residual model are used respectively to derive the nonlinear evolution equations of liquid film, and it is shown that the two evolution equations by the above two approaches are equivalent. The recurrence relation and damped Mathieu equation are obtained by employing Floquet theory and linear analysis, respectively. When amplitude a and oscillation frequency ω of the temperature exceed critical values, the film surface loses its stability. The stability diagram is obtained by linear analysis based on Floquet theory. Moreover, based upon the line approach, the nonlinear evolution equation is numerically simulated to give the spatiotemporal evolution of the free surface. Heated capillary tension might accelerate the production of suspended droplets due to the appearing of Marangoni number Ma produced by heating inclined wall. The results reveal that the larger Reynolds number Re, Marangoni number Ma and Biot number Bi will result in the instability of free surface. Moreover, the liquid film becomes more unstable at the smaller Froude number Fr and Weber number We.

To Integration of the Damped Mathieu Equation in the Monograph of N. N. Bogoliubov and Y. A. Mitropolsky “Asymptotic Methods in the Theory of Nonlinear Oscillations”

To Integration of the Damped Mathieu Equation in the Monograph of N. N. Bogoliubov and Y. A. Mitropolsky “Asymptotic Methods in the Theory of Nonlinear Oscillations”

Effect of the odd viscosity on Faraday wave instability

Faraday waves arise in fluid systems with free surfaces subject to vertical oscillations of sufficient strength due to parametric resonance. The odd viscosity is a peculiar part of the viscosity stress tensor that does not result in dissipation and is allowed when parity symmetry is broken spontaneously or due to external magnetic fields or rotations. The effect of the odd viscosity on the classic Faraday instability of thin liquid films in infinite horizontal plates is investigated by utilizing both linear Floquet theory and nonlinear lubrication theory based on the weighted residual model. This work derives the nonlinear evolution equations about the flow rate and free surface height, and linear stability analysis is performed to achieve the damped Mathieu equation. The results show that the neutral stability curves derived from the Mathieu equation agree well with those obtained from the linear Floquet analysis, especially for lower viscosity ratios μ. The nonlinear numerical results simulated by the method of lines indicate interesting results where the odd viscosity gives rise to a “sliding” of the wave configuration parallel to the wall, and the interface wave then translates into a traveling wave.

Quasi-Periodic Solutions of a Damped Nonlinear Quasi-Periodic Mathieu Equation by the Incremental Harmonic Balance Method With Two Time Scales

Abstract Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method with two time scales. The damped nonlinear QP Mathieu equation contains two incommensurate harmonic excitation frequencies, one is a small frequency while the other nearly equals twice the linear natural frequency. It is found that Fourier spectra of QP solutions of the equation consist of uniformly spaced sidebands due to cubic nonlinearity. The IHB method with two time scales, which relates to the two excitation frequencies, is adopted to trace solution curves of the equation in an automatical way and find all frequencies of solutions and their corresponding amplitudes. Effects of parametric excitation are studied in detail. Based on approximation of QP solutions by periodic solutions with a large period, Floquet theory is used to study the stability of QP solutions. Three types of QP solutions can be obtained from the IHB method, which agree very well with results from numerical integration. However, the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions since the ratio of the small frequency to the linear natural frequency of the first reduced-modulation equation is nearly 1 in the second perturbation procedure, while the other two types of QP solutions from the IHB method with two time scales do not need the ratio. Furthermore, the results from the double-step MMS are different from those numerical integration and the IHB method with two time scales.

Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method

In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the first-order solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

Role of dissipation in the stability of a parametrically driven quantum harmonic oscillator

We study the dissipative dynamics of a single quantum harmonic oscillator subjected to parametric driving within an effective Hamiltonian approach. Using the Liouville–von Neumann approach, we show that the time evolution of a parametrically driven dissipative quantum oscillator has a strong connection with the classical, damped Mathieu equation. Based on the numerical analysis of the Monodromy matrix, we demonstrate that the dynamical instability generated by the parametric driving is reduced by the effect of dissipation. Furthermore, we obtain a closed relationship between the localization of the Wigner function and the stability of the damped Mathieu equation.

On the convergence of WKB approximations of the damped Mathieu equation

The form of the fundamental set of solutions of the damped Mathieu equation is determined by Floquet theory. In the limit as m → 0, we can apply WKB theory to get first order approximations of the fundamental set. WKB theory states that this approximation gets better as m → 0 and T is fixed. However, convergence of the periodic part and characteristic exponent is not addressed. We show that they converge to those predicted by WKB theory. We also provide a rate of convergence that is not dependent on T.

Generalized Numerical Differentiation Method for Stability Calculation of Periodic Delayed Differential Equation: Application for Variable Pitch Cutter in Milling

This paper proposes a generalized numerical differentiation method for stability prediction of the non-autonomous delayed differential equations (DDEs) with periodic coefficients and discrete delays. Firstly, the periodic DDE is described in state-space form and the period of a system is equally discretized. Then, the discrete first derivatives versus time are approximated by a linear combination of the state function values at multiple neighboring sampling grid points based on the finite-difference formulas. Such that, the original DDE is approximated as a series of algebraic equations and the Floquet transition matrix can be constructed on one period. At last, the system stability is determined according to the Floquet theory by checking the eigenvalues. The delayed damped Mathieu equation is regarded as a typical case to verify the effectiveness and efficiency of the presented method. The stability diagrams and rate of convergence are computed in comparison with those via the benchmark algorithm (the semi-discretization method). As an application, the presented method is used to predict the stability of milling with variable pitch cutter, and the computational result agrees well with the experimentally verified example.

Nonlinear Dynamics and Parametric Excitation of Piezo-Micro Biological Sensor

In this work, nonlinear dynamics and parametric excitation of piezoelectric micro beam subjected a combination of DC and AC voltages is investigated for micro biological sensor applications. The governing differential equation of the motion is derived by the Hamiltonian principle and then by using the Galerkin approach, this partial differential equation (PDE) is simplified into an ordinary differential equation (ODE) in terms of forced damped Mathieu equation with cubic nonlinearity. A perturbation technique, i.e. multiple time scales approach is used for investigation of superharmonic resonances of orders 1/2 and 1/3 and obtaining of the modulation equations in the amplitude and phase. Numerical results of different parameters effects on the piezoelectric micro beam behavior are presented graphically.

Study on lateral stability of levitation modules for low- and medium-speed maglev trains

In order to study the lateral dynamics stability of the levitation module of low- and medium-speed maglev train, the motion equation is simplified by reasonable assumptions, and the simplest lateral dynamics model is established under the premise of retaining the lateral dynamic characteristics of the levitation module. The model can be characterized by a non-autonomous differential equation with periodic coefficients, which can be transformed into a damped Mathieu equation under certain conditions, with its stability equivalent to that of its Poincare mapping. The stability of the system can be determined by whether the largest modulus of the eigenvalues of the mapped coefficient matrix is > 1. Then, the sensitivity frequency of the vertical electromagnetic force is obtained by calculating the stability domain of the equation in the parameter space. When the variation frequency of the vertical electromagnetic force is approximately equivalent to the sensitivity frequency, the lateral dynamics model is more likely to be unstable. Besides, the variation amplitude and variation frequency of the vertical electromagnetic force, and other factors such as vehicle load and system damping will also change the stability of the lateral dynamics model. However, due to the uncontrollability of these factors, it is necessary to install a lateral damper for the levitation module to improve the lateral stability.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

Dynamic characteristics of resonant vibratory gyroscopes based on the damped Mathieu equation

PurposeThis paper aims to develop resonant vibratory gyroscopes for high sensitive detection. The dynamic characteristics of resonant vibratory gyroscopes are investigated.Design/methodology/approachFirstly, the working principle and the dynamic output characteristics of the resonant vibratory gyroscope could be described by the damped Mathieu equation. Moreover, an approximate analytical method based on the small parameter perturbation has been used for the purpose of investigating the approximate solution of the damped Mathieu equation. Finally, to verify the feasibility of the approximate analytical method of the damped Mathieu equation, dynamic output characteristics’ experiments of the resonant vibratory gyroscope are built.FindingsThe theoretical analysis and numerical simulations show that the approximate solution of the damped Mathieu equation is close to the dynamic output characteristics of the resonant vibratory gyroscope. On the other hand, it is concluded from the tested result that there exists a correlation between the theoretical curve and the experimental data processing result, meaning the damped dynamics analytical method is effective in building resonant vibratory gyroscopes.Originality/valueThis paper seeks to establish a foundation for optimizing and testing the performance of the resonant vibratory gyroscope. To this end, the approximate analytical method of the damped Mathieu equation was discussed. The result of this research has proved that the dynamic characteristics based on the damped Mathieu equation is an effective approach and is instructional in the practical resonant sensor design.

Finite Difference Computational Method for Trajectory Controllability of a Delayed Damped System Governed by Fractional Differential Equation

In this paper, the trajectory controllability (T-controllability) of a nonlinear fractional-order damped system with time delay is studied. Existence and uniqueness of solution are obtained by using the Banach fixed point theorem and Green's function. Necessary and sufficient conditions of trajectory controllable for the nonlinear system are formulated and proved under a predefined trajectory. Modified fractional finite difference method is applied to the system for numerical approximation of its solution. The applicability of this technique is demonstrated by numerical simulation of two scientific models such as neuromechanical interaction in human snoring and fractional delayed damped Mathieu equation.

Stability Analysis of Parametrically Excited Systems Using the Energy-Growth Exponent/Coefficient

In this paper, the energy exponent is used to study the instability of parametrically excited systems governed by the damped Mathieu equation. Through the numerical tests, an energy-growth exponent (EGE) is adopted to evaluate the instability intensity and instability boundary of the system. The EGE can be expressed as a product of the modal frequency and a dimensionless coefficient, defined as the energy-growth coefficient (EGC). Based on the Floquet theory, the relationship between the EGE, Floquet exponent and Lyapunov exponent are derived. An energy criterion of parametric instability is proposed by using the EGE. Using a simple pendulum as an example, the geometric characteristics of the EGC are investigated, and approximate analytical formulae of the EGC/EGE for three different unstable patterns are developed. The EGC/EGE formulae are applicable to the parametrically excited systems governed by the damped Mathieu equation. The unstable behaviors and properties of parametric vibrations are analyzed and discussed in details with EGE/EGC for three systems including a triple pendulum, two-dimensional sloshing fluid, and a two-span continuous beam. The stability boundaries established by using EGE/EGC agree well with those by the conventional theory and experiment. As a practical tool, the EGE/EGC formulae can be easily applied to analyzing the unstable intensities and boundaries of the parametrically excited systems.

Updated numerical integration method for stability calculation of Mathieu equation with various time delays

The determination of the stability of systems with time delays is of high importance in many industrial and research applications. In this study, we improve the numerical integration method (NIM) by using the Lagrange form interpolating polynomial to approximate the delayed terms and construct a periodic discrete dynamical map for the damped Mathieu equation with time delays. Hence, we can obtain the Floquet transition matrices without matrix multiplication, which can reduce calculation time when the matrix inversed has a small bandwidth according to the computation based on MATLAB. To compare the calculation efficiency and computational accuracy between the updated and original methods, we compare the stability charts calculated by using three NIM methods for a damped Mathieu equation with multiple delays. To further confirm the efficiency of the presented method, we calculate the stability of the Mathieu equation with distributed delays and time-periodic delays, and then we compare the computational accuracy and calculation time with those of the semi-discretization method.

Stochastic stability of the non-homogeneous damped Mathieu equation

Many physical phenomena might be modeled with the well-known Mathieu equation. Engineering applications, such as wind turbine blade dynamics, and helicopter rotor dynamics, lead to equations of motions with frequency-dependent coefficients. In such applications, the rotations cause the periodic parametric excitations. In addition, (1) there is external excitation, (2) damping must be included in the model and (3) uncertainties must be taken into account. The aim of this article is to analyze the dimensionless form of the stochastic non-homogeneous damped Mathieu equation. Uncertainties are modeled using the probability theory and are propagated using the Monte Carlo method. The Floquet theory is employed to assess the deterministic and stochastic stability of the system.