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 ap...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.展开更多
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...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.展开更多
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 tha...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.展开更多
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 differentia...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.展开更多
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 eq...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.展开更多
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 wi...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.展开更多
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 a...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.展开更多
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 cir...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.展开更多
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....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.展开更多
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 ...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.展开更多
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 th...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.展开更多
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 curva...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.展开更多
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 ...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.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No. 60927005)the Innovation Foundation of BUAA for Ph. D. Graduates,Chinathe Fundamental Research Funds for the Central Universities,China (Grant No. YWF-10-01-A17)
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10675061 adn 11175090
文摘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.
文摘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.
基金financially supported by the National Natural Science Foundation of China(Grant Nos.51379005 and 51009093)
文摘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.
基金The project supported by National Natural Science Foundation of China under Grant No. 10405009 and Natural Science Foundation of Hebei Province of China under Grant No. 103143
文摘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.
基金supported by National Natural Science Foundation of China under Grant No.10405009the Youth Foundation of North China Electric Power University
文摘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.
基金Natural Science Foundation of Shaanxi Province,China(No. 09JK456)
文摘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.
基金National Natural Science Foundation of China(No.50275128)Natural Science Foundation of Hebei Province,China(No.A2006000190).
文摘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.
文摘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.
基金Supported by the Hunan Provincial Department of Education (19B464)the National Natural Science Foundation of China (NSFC, 12165009)。
文摘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.
基金financial support from the National Natural Science Foundation of China(Grants No.11675017 and No.11975050)。
文摘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.
