Period-doubling bifurcation in two-stage power factor correction converters using the method of incremental harmonic balance and Floquet theory

In this paper, period-doubling bifurcation in a two-stage power factor correction converter is analyzed by using the method of incremental harmonic balance （IHB） and Floquet theory. A two-stage power factor correction converter typically employs a cascade configuration of a pre-regulator boost power factor correction converter with average current mode control to achieve a near unity power factor and a tightly regulated post-regulator DC-DC Buck converter with voltage feedback control to regulate the output voltage. Based on the assumption that the tightly regulated postregulator DC-DC Buck converter is represented as a constant power sink and some other assumptions, the simplified model of the two-stage power factor correction converter is derived and its approximate periodic solution is calculated by the method of IHB. And then, the stability of the system is investigated by using Floquet theory and the stable boundaries are presented on the selected parameter spaces. Finally, some experimental results are given to confirm the effectiveness of the theoretical analysis.

Floquet instability of a large density ratio liquid-gas coaxial jet with periodic fluctuation

By numerical simulation of basic flow, this paper extends Floquet stability analysis of interracial flow with periodic fluctuation into large density ratio range. Stability of a liquid aluminum jet in a coaxial nitrogen stream with velocity fluctuation is investigated by Chebyshev collocation method and the Floquet theory. Parametric resonance of the jet and the influences of different physical parameters on the instability are discussed. The results show qualitative agreement with the available experimental data.

FLOQUET STABILITY ANALYSIS OF TWO-LAYER FLOWS IN VERTICAL PIPE WITH PERIODIC FLUCTUATION

Based on linear stability theory, parametric resonance phenomenon of a liquidgas cylindrical flow in a vertical pipe with periodic fluctuation was discussed with the help of Floquet theory and Chebyshev spectral collocation method. The effects of different physical parameters were investigated on the properties of parametric resonance and the stability characteristics of flow field.

ASYMPTOTIC THEORY FOR THE CIRCUIT ENVELOPE ANALYSIS

Asymptotic theory for the circuit envelope analysis is developed in this paper.A typical feature of circuit envelope analysis is the existence of two significantly distinct timescales:one is the fast timescale of carrier wave,and the other is the slow timescale of modulation signal.We first perform pro forma asymptotic analysis for both the driven and autonomous systems.Then resorting to the Floquet theory of periodic operators,we make a rigorous justification for first-order asymptotic approximations.It turns out that these asymptotic results are valid at least on the slow timescale.To speed up the computation of asymptotic approximations,we propose a periodization technique,which renders the possibility of utilizing the NUFFT algorithm.Numerical experiments are presented,and the results validate the theoretical findings.

Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure

Impulsively coupled systems are high-dimensional non-smooth systems that can exhibit rich and complex dynamics.This paper studies the complex dynamics of a non-smooth system which is unidirectionally impulsively coupled by three Duffing oscillators in a ring structure.By constructing a proper Poincare map of the non-smooth system,an analytical expression of the Jacobian matrix of Poincare map is given.Two-parameter Hopf bifurcation sets are obtained by combining the shooting method and the Runge-Kutta method.When the period is fixed and the coupling strength changes,the system undergoes stable,periodic,quasi-periodic,and hyper-chaotic solutions,etc.Floquet theory is used to study the stability of the periodic solutions of the system and their bifurcations.

Application of Arnoldi method to boundary layer instability

The Arnoldi method is applied to boundary layer instability, and a finite difference method is employed to avoid the limit of the finite element method. This modus operandi is verified by three comparison cases, i.e., comparison with linear stability theory（LST） for two-dimensional（2D） disturbance on one-dimensional（1D） basic flow, comparison with LST for three-dimensional（3D） disturbance on 1D basic flow, and comparison with Floquet theory for 3D disturbance on 2D basic flow. Then it is applied to secondary instability analysis on the streaky boundary layer under spanwise-localized free-stream turbulence（FST）. Three unstable modes are found, i.e., an inner mode at a high-speed center streak, a sinuous type outer mode at a low-speed center streak, and a sinuous type outer mode at low-speed side streaks. All these modes are much more unstable than Tollmien–Schlichting（TS） waves, implying the dominant contribution of secondary instability in bypass transition. The modes at strong center streak are more unstable than those at weak side streaks, so the center streak is ‘dangerous＇ in secondary instability.

Investigation for Electromechanical Coupling Unstability of Rolling Mill

The unsteady condition of rolling Mill vibration that was caused by flexural vibration of strip was investigated. The parametric flexural vibration equation of rolled strip was established. The parametric flexural vibration stability of rolled strip was studied and region of stability and unstability was determined based on Floquet theory and perturbation method. The flexural-vibration of strip was unstable if the frequency of variable tension was twice as the natural frequency of flexural-vibration strip. The characteristics of electric current in a temp driving motor’s main loop was studied and tested, and approved that there were 6 humorous current ponderance and 12 humorous current ponderance in main circuit of driving motor. Vertical vibration of working roller was tested, the test approved that there were running unsteady caused by parametric vibration. It attached importance to the parametric vibration of rolling mill.

Global instability of Stokes layer for whole wave numbers

The study on the global instability of a Stokes layer, which is a typical unsteady flow, is usually a paradigm for understanding the instability and transition of unsteady flows. Previous studies suggest that the neutral curve of the global instability obtained by the Floquet theory is only mapped out in a limited range of wave numbers （0.2 ≤ a ≤ 0.5）. In this paper, the global instability is investigated with numerical simulations for all wave numbers. It is revealed that the peak of the disturbances displays irregularity rather than the periodic evolution while the wave number is beyond the above range. A ＂neutral point＂ is redefined, and a neutral curve of the global instability is presented for the whole wave numbers with this new definition. This work provides a deeper understanding of the global instability of unsteady flows.

Photoinduced Weyl semimetal phase and anomalous Hall effect in a three-dimensional topological insulator

Motivated by the fact that Weyl fermions can emerge in a three-dimensional topological insulator on breaking either time-reversal or inversion symmetries,we propose that a topological quantum phase transition to a Weyl semimetal phase occurs under the off-resonant circularly polarized light,in a three-dimensional topological insulator,when the intensity of the incident light exceeds a critical value.The circularly polarized light effectively generates a Zeeman exchange field and a renormalized Dirac mass,which are highly controllable.The phase transition can be exactly characterized by the first Chern number.A tunable anomalous Hall conductivity emerges,which is fully determined by the location of the Weyl nodes in momentum space,even in the doping regime.Our predictions are experimentally realizable through pump-probe angle-resolved photoemission spectroscopy and raise a new way for realizing Weyl semimetals and quantum anomalous Hall effects.

BIFURCATION IN A NONLINEAR DUFFING SYSTEM WITHMULTI-FREQUENCY EXTERNAL PERIODIC FORCES

By introducing nonlinear frequency, using Floquet theory and referring to the characteristics of the solution when it passes through the transition boundaries, all kinds of bifurcation modes and their transition boundaries of Duffing equation with two periodic excitations as well as the possible ways to chaos are studied in this paper.

Nonlinear Vibration and Stability Analysis of Axially Accelerating Beam in Axial Flow

The dynamics of an axially accelerating beam subjected to axial flow is studied.Based on the Floquet theory and the Runge-Kutta algorithm,the stability and nonlinear vibration of the beam are analyzed by considering the effects of several system parameters such as the mean speed,flow velocity,axial added mass coefficient,mass ratio,slenderness ratio,tension and viscosity coefficient.Numerical results show that when the pulsation frequency of the axial speed is close to the sum of first-and second-mode frequencies or twice the lowest two natural frequencies,instability with combination or subharmonic resonance would occur.It is found that the beam can undergo the periodic-1 motion under subharmonic resonance and the quasi-periodic motion under combination resonance.With the change of system parameters,the stability boundary may be widened,narrowed or drifted.In addition,the vibration amplitude of the beam under resonance can also be affected by changing the values of system parameters.

Effective Hamiltonian of the Jaynes–Cummings model beyond rotating-wave approximation

The Jaynes–Cummings model with or without rotating-wave approximation plays a major role to study the interaction between atom and light. We investigate the Jaynes–Cummings model beyond the rotating-wave approximation. Treating the counter-rotating terms as periodic drivings, we solve the model in the extended Floquet space. It is found that the full energy spectrum folded in the quasi-energy bands can be described by an effective Hamiltonian derived in the highfrequency regime. In contrast to the Z_(2) symmetry of the original model, the effective Hamiltonian bears an enlarged U(1)symmetry with a unique photon-dependent atom-light detuning and coupling strength. We further analyze the energy spectrum, eigenstate fidelity and mean photon number of the resultant polaritons, which are shown to be in accordance with the numerical simulations in the extended Floquet space up to an ultra-strong coupling regime and are not altered significantly for a finite atom-light detuning. Our results suggest that the effective model provides a good starting point to investigate the rich physics brought by counter-rotating terms in the frame of Floquet theory.

Floquet analysis of fundamental,subharmonic and detuned secondary instabilities of Grtler vortices

Nonlinear parabolized stability equations are employed in this work to investigate the nonlinear development of the G6rtler insta- bility up to the saturation stage. The perturbed boundary layer is highly inflectional both in the normalwise and spanwise directions and receptive to the secondary instabilities. The Floquet theory is applied to solve the fundamental, subharmonic and detuned secondary instabilities. With the Gortler-vortices-distorted base flow, two classes of secondary disturbances, i.e. odd modes and even modes, are identified according to the eigenfunctions of the disturbances. These modes may result in different patterns in the late stages of the transition process. Li and Malik [ 1 ] have shown the sinuous and varicose types of breakdown originating from the odd and even modes. The current study focuses on the four most amplified modes termed the even modes I ＆ Ⅱ and odd modes I ＆ lI. Odd mode II was missing in the work of Li and Malik [1] probably due to their inviscid simplifeation. The detuned modes are confirmed to be less amplifed than the fundamental （for the odd mode I） and subharmonic modes （for even modes I ＆ II and the odd mode II）.

Close relative motion on distant retrograde orbits

Distant Retrograde Orbits(DROs)in the Earth-Moon system have great potential to support varieties of missions due to the favorable stability and orbital positions.Thus,the close relative motion on DROs should be analyzed to design formations to assist or extend the DRO missions.However,as the reference DROs are obtained through numerical methods,the close relative motions on DROs are non-analytical,which severely limits the design of relative trajectories.In this paper,a novel approach is proposed to construct the analytical solution of bounded close relative motion on DROs.The linear dynamics of relative motion on DRO is established at first.The preliminary forms of the general solutions are obtained based on the Floquet theory.And the general solutions are classified as different modes depending on their periodic components.A new parameterization is applied to each mode,which allows us to explore the geometries of quasi-periodic modes in detail.In each mode,the solutions are integrated as a uniform expression and their periodic components are expanded as truncated Fourier series.In this way,the analytical bounded relative motion on DRO is obtained.Based on the analytical expression,the characteristics of different modes are comprehensively analyzed.The natural periodic mode is always located on the single side of the target spacecraft on DRO and is appropriate to be the parking orbits of the rendezvous and docking.On the basis of quasi-periodic modes,quasi-elliptical fly-around relative trajectories are designed with the assistance of only two impulses per period.The fly-around formation can support observations to targets on DRO from multiple viewing angles.And the fly-around formation is validated in a more practical ephemeris model.

Parametric Vibration Stability Analysis of a Pyramid Lattice Sandwich Plate Subjected to Pulsatile External Airflow

Lattice sandwich structures are broadly used in aerospace,navigation,and high-speed rail engineering.In engineering practice,the airflow outside the vehicle or aircraft always exhibits the pulsatile property,which makes the elastic structural components and the external airflow a parametric excitation system.In this paper,the parametric vibration stability analysis and dynamic characteristics of a lattice sandwich plate interacting with the pulsatile external airflow are studied.The equation of motion is derived using Hamilton’s principle and discretized using the assumed mode method.The linear potential flow theory is applied to derive the perturbation aerodynamic pressure.The stability of the system is analyzed using the Floquet theory and validated by numerical simulations.The effects of design parameters of the lattice sandwich plate on the stability of the system are discussed.From the simulations and discussions,some practical principles for the optimal design of lattice sandwich structures in the aerodynamic environment are proposed.

Milling stability analysis using the spectral method

This paper focuses on the development of an efficient semi-analytical solution of chatter stability in milling based on the spectral method for integral equations. The time-periodic dynamics of the milling process taking the regenerative effect into account is formulated as a delayed differential equation with time-periodic coefficients, and then reformulated as a form of integral equation. On the basis of one tooth period being divided into a series of subintervals, the barycentric Lagrange interpolation polynomials are employed to approximate the state term and the delay term in the integral equation, respectively, while the Gaussian quadrature method is utilized to approximate the integral tenn. Thereafter, the Floquet transition matrix within the tooth period is constructed to predict the chatter stability according to Floquet theory. Experimental-validated one-degree-of-freedom and two-degree-of-freedom milling examples are used to verify the proposed algorithm, and compared with existing algorithms, it has the advantages of high rate of convergence and high computational efficiency.

Periodic response analysis of a misaligned rotor system by harmonic balance method with alternating frequency/time domain technique

A dynamic model is established for an offset-disc rotor system with a mechanical gear coupling, which takes into consideration the nonlinear restoring force of rotor support and the effect of coupling misalignment. Periodic solutions are obtained through harmonic balance method with alternating frequency/time domain(HB-AFT) technique, and then compared with the results of numerical simulation. Good agreement confirms the feasibility of HB-AFT scheme. Moreover, the Floquet theory is adopted to analyze motion stability of the system when rotor runs at different speed intervals. A simple strategy to determine the monodromy matrix is introduced and two ways towards unstability are found for periodic solutions: the period doubling bifurcation and the secondary Hopf bifurcation. The results obtained will contribute to the global response analysis and dynamic optimal design of rotor systems.

A multi-order method for predicting stability of a multi-delay milling system considering helix angle and run-out effects

In this paper, a multi-delay milling system considering helix angle and run-out effects is firstly established. An exponential cutting force model is used to model the interaction between a work-piece and a cutting tool, and a new approach is presented for accurately calibrating exponential cutting force coefficients and cutter run-out parameters. Furthermore, based on an implicit multi-step Adams formula and an improved precise time-integration algorithm, a novel stability prediction method is proposed to predict the stability of the system. The involved time delay term and periodic coefficient term are integrated as a comprehensive state term in the integral response which is approximated by the Adams formula. Then, a Floquet transition matrix with an arbitraryorder form is constructed by using a series of matrix multiplication, and the stability of the system is determined by the Floquet theory. Compared to classical semi-discretization methods and fulldiscretization methods, the developed method shows a good performance in convergence, efficiency,accuracy, and multi-order complexity. A series of cutting tests is further carried out to validate the practicability and effectiveness of the proposed method. The results show that the calibration process needs a time of less than 5 min, and the stability prediction method is effective.

Bifurcation and Chaos in a Dynamical System of Rub-impact Rotor

Nonlinear vibration characteristics of a rub impact Jeffcott rotor are investigated. The system is two dimensional, nonlinear, and periodic. Fourier series analysis and the Floquet theory are used to perform qualitative global analysis of the dynamical system. The governing ordinary differential equations are also integrated using a numerical method to give the quantitative result. This preliminary study revealed the chaotic feature of the system. After the rub impact, as the rotating speed is increased three kinds of routes to chaos are found, that is, from a stable periodic motion through period doubling bifurcation, grazing bifurcation, and quasi periodic bifurcation to chaos.

Flutter analysis of an airfoil with multiple nonlinearities and uncertainties

An original method for calculating the limit cycle oscillations of nonlinear aeroelastic system is presented.The problem of detemining the maximum vibration amplitude of limit cycle is transfomed into a nonlinear optimization problem.The hamonic balance method and the Floquet theory are selected to construct the general nonlinear equality and inequality constraints.The resulting constrained maximization problem is then solved by using the MultiStart algorithm.Finally,the proposed approach is validated and used to analyse the limit cycle oscillations of an airfoil with multiple nonlinearities and uncertainties.Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation.