Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution

Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the approximate output of the resonant gyroscope.The method of small parameter perturbation is used to analyze the approximate solution of the Mathieu equation.The theoretical analysis and the numerical simulations show that the approximate solution of the Mathieu equation is close to the dynamic output characteristics of the resonant gyroscope.The experimental analysis shows that the theoretical curve and the experimental data processing results coincide perfectly,which means that the approximate solution of the Mathieu equation can present the dynamic output characteristic of the resonant gyroscope.The theoretical approach and the experimental results of the Mathieu equation approximate solution are obtained,which provides a reference for the robust design of the resonant gyroscope.

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 firstorder solvability condition.The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

SOME EXTENDED RESULTS OF“SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

Mathieu Equation and Elliptic Curve

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q《1 and q》1, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We make a few conjectures about the general structure of these differential operators.

Numerical solution of fractional Mathieu equations by using block-pulse wavelets

In this paper,we introduce a method based on operational matrix of fractional order integration for the numerical solution of fractional Mathieu equation and then apply it in a number of cases.For this,we use the block-pulse wavelets matrix of fractional order integration with respect to the Caputo sense.The method was tested by some numerical examples and changes occurred in the coefficients as well as in the derivative of the equation.Results prove the accuracy and computational efficiency of the proposed algorithm.©2019 Shanghai Jiaotong University.Published by Elsevier B.V.

Maximum Interval of Stability and Convergence of Solution of a Forced Mathieu’s Equation

This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu’s equation, using a combination of Frobenius method and Eigenvalue approach. The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution. Furthermore, the solution to Mathieu’s equation converges which extends and improves some results in literature.

Parametric Resonance Analyses for Spar Platform in Irregular Waves

The parametric instability of a spar platform in irregular waves is analyzed. Parametric resonance is a phenomenon that may occur when a mechanical system parameter varies over time. When it occurs, a spar platform will have excessive pitch motion and may capsize. Therefore, avoiding parametric resonance is an important design requirement. The traditional methodology includes only a prediction of the Mathieu stability with harmonic excitation in regular waves. However, real sea conditions are irregular, and it has been observed that parametric resonance also occurs in non-harmonic excitations. Thus, it is imperative to predict the parametric resonance of a spar platform in irregular waves. A Hill equation is derived in this work, which can be used to analyze the parametric resonance under multi-frequency excitations. The derived Hill equation for predicting the instability of a spar can include non-harmonic excitation and random phases. The stability charts for multi-frequency excitation in irregular waves are given and compared with that for single frequency excitation in regular waves. Simulations of the pitch dynamic responses are carried out to check the stability. Three-dimensional stability charts with various damping coefficients for irregular waves are also investigated. The results show that the stability property in irregular waves has notable differences compared with that in case of regular waves. In addition, using the Hill equation to obtain the stability chart is an effective method to predict the parametric instability of spar platforms. Moreover, some suggestions for designing spar platforms to avoid parametric resonance are presented, such as increasing the damping coefficient, using an appropriate RAO and increasing the metacentric height.

Quantum Effects of Mesoscopic Inductance and Capacity Coupling Circuits

Using the quantum theory for a mesoscopic circuit based on the discretenes of electric charges, the finitedifference Schrodinger equation of the non-dlssipative mesoscopic inductance and capacity coupling circuit is achieved. The Coulomb blockade effect, which is caused by the discreteness of electric charges, is studied. Appropriately choose the components in the circuits, the finlte-dlfference Schrodinger equation can be divided into two Mathieu equations in representation.＂ With the WKBJ method, the currents quantum fluctuations in the ground states of the two circuits are calculated. The results show that the currents quantum zero-point fluctuations of the two circuits are exist and correlated.

Quantum Effect in Mesoscopic Open Electron Resonator

The open electron resonator is a mesoscopic device that has attracted considerable attention due to its remarkable behavior： conductance oscillations. In this paper, using an improved quantum theory to mesoscopic circuits developed recently by Li and Chen, the mesoscopic electron resonator is quantized based on the fundamental fact that the electric charge takes discrete value. With presentation transformation and unitary transformation, the SchrSdinger equation becomes an standard Mathieu equation. Then, the detailed energy spectrum and wave functions in the system axe obtained, which will be helpful to the observation of other characters of electron resonator. The average of currents and square of the current are calculated, the results show the existence of the current fluctuation, which causes the noise in the circuits, the influence of inductance to the noise is discussed. With the results achieved, the stability characters of mesoscopic electron resonator are studied firstly, these works would be benefit to the design and control of integrate circuit.

Vibration Analysis of Warp on Weaving

Warp is a nonlinear viscoelastic material. The loom efficiency was reduced considerably by the increase of the frequency of weak places in the warp. Warp should vibrate with low-frequency to reduce the friction times. The objective of this research was to establish the transverse/longitudinal vibration differential equations of warp and analyze the fluctuation process of warp movement by nonlinear method. Based on variable separation method, the time variable was separated from space variable, and the numerical solutions of vibration equations were obtained by the fourth-order Runge-Kutta method. Finally, the influencing factors and variable trends on warp vibration were discussed. The methods on control warp vibration were introduced, which could guide the engineering practice. The most important way to reduce warp vibration is quick adjustment of warp tension.

MAGNETIC-ELASTIC BUCKLING OF A THIN CURRENT CARRYING PLATE SIMPLY SUPPORTED AT THREE EDGES

The magnetic-elasticity buckling problem of a current plate under the action of a mechanical load in a magnetic field was studied by using the Mathieu function. According to the magnetic-elasticity non-linear kinetic equation, physical equations, geometric equations, the expression for Lorenz force and the electrical dynamic equation, the magnetic-elasticity dynamic buckling equation is derived. The equation is changed into a standard form of the Mathieu equation using Galerkin＇s method. Thus, the buckling problem can be solved with a Mathieu equation. The criterion equation of the buckling problem also has been obtained by discussing the eigenvalue relation of the coefficients 2 and r/ in the Mathieu equation. As an example, a thin plate simply supported at three edges is solved here. Its magnetic-elasticity dynamic buckling equation and the relation curves of the instability state with variations in some parameters are also shown in this paper. The conclusions show that the electrical magnetic forces may be controlled by changing the parameters of the current or the magnetic field so that the aim of controlling the deformation, stress, strain and stability of the current carrying plate is achieved.

EFFECT OF SKIN FRICTION ON THE DIRECTIONAL STABILITY OF DRIVEN PILES DURING DRIVING

In this paper, the driving forces at a pile top are considered as a periodic load during driving and the Mathieu equation is derived. From the stability charts of this equation, we can obtain the critical length of the pile, and the effect of skin friction upon the critical length is discussed.

Primordial black hole from the running curvaton

In light of our previous study [Chin. Phys. C 44(8), 085103(2020)], we investigate the possibility of the formation of a primordial black hole in the second inflationary process induced by the oscillation of the curvaton. By adopting the instability of the Mathieu equation, one can utilize the δ function to fully describe the power spectrum.Owing to the running of the curvaton mass, we can simulate the value of the abundance of primordial black holes covering almost all of the mass ranges. Three special cases are given. One case may account for dark matter because the abundance of a primordial black hole is approximately 75%. As late times, the relic of exponential potential may be approximated to a constant of the order of a cosmological constant, which is dubbed as the role of dark energy.Thus, our model could unify dark energy and dark matter from the perspective of phenomenology. Finally, it sheds new light on exploring Higgs physics.

Instability Analysis of Mosquito Fascicle under Compressive Load with Vibrations and Microneedle Design

Mosquito has the ability to penetrate the skin with painless insertion. It has attracted the researchers to mimic the bite and develop a painless microneedle. Mosquito applies axial compressive load along with frequency on fascicle to penetrate the human skin and retract if it senses instability prior to insertion. The mechanism of mosquito bite is studied in this work which is divided into two stages for analysis considering different boundary conditions. The probing behaviour of mosquito is considered as stage I and the process of penetration as stage II. An equivalent mechanical model for stage I is proposed and a mathematical model is developed to understand the instability of fascicle in terms of frequency and magnitude of force applied. The governing equation and associated boundary conditions are simplified into Mathieu equation and regions of dynamic instability are obtained through the solution. Results confirm instability of the fascicle during stage I of insertion. The probing behaviour of mosquito is discussed in terms of applied force and vibrating frequency. Horizontal reaction forces exerted by labium on fascicle during buckling improve the stability and enable fascicle to withstand high compressive forces. The analysis and results are utilized to set design guidelines for the development of dynamically stable vibration-assisted microneedle.

Dynamics of momentum distribution and structure factor in a weakly interacting Bose gas with a periodical modulation

The momentum distribution and dynamical structure factor in a weakly interacting Bose gas with a time-dependent periodic modulation in terms of the Bogoliubov treatment are investigated.The evolution equation related to the Bogoliubov weights happens to be a solvable Mathieu equation when the coupling strength is periodically modulated.An exact relation between the time derivatives of momentum distribution and dynamical structure factor is derived,which indicates that the single-particle property is strongly related to the two-body property in the evolutions of Bose–Einstein condensates.It is found that the momentum distribution and dynamical structure factor cannot display periodical behavior.For stable dynamics,some particular peaks in the curves of momentum distribution and dynamical structure factor appear synchronously,which is consistent with the derivative relation.