The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The a...The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.展开更多
The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to ...The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to complex symbolic operations.Many variants have been developed to improve the HB method,among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation.So far,there are two problems remaining to be addressed.i)A dozen of different versions of HB-like methods,in frequency domain or time domain or in hybrid,have been developed;unfortunately,misclassification pervades among them due to the unclear borderlines of different methods.ii)The time domain HB-like methods suffer from non-physical solutions,which have been shown to be caused by aliasing(mixture of the high-order into the low-order harmonics).Although a series of dealiasing techniques have been developed over the past two decades,the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community.This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences.In particular,the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.展开更多
Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we rep...Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.展开更多
Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical ha...Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical harmonics function is used to expand the angular flux. A set of differential equations about the spatial variable, which are coupled with each other, can be obtained. They are solved iteratively by using the finite element method on un- structured-meshes. A two-dimension transport calculation program is coded according to the model. The numerical results of some benchmark problems demonstrate that this method can give high precision results and avoid the ray effect very well.展开更多
A versatile analytical method(VAM) for calculating the harmonic components of the magnetomotive force(MMF) generated by diverse armature windings in AC machines has been proposed, and the versatility of this method ha...A versatile analytical method(VAM) for calculating the harmonic components of the magnetomotive force(MMF) generated by diverse armature windings in AC machines has been proposed, and the versatility of this method has been established in early literature. However, its practical applications and significance in advancing the analysis of AC machines need further elaboration. This paper aims to complement VAM by augmenting its theory, offering additional insights into its conclusions, as well as demonstrating its utility in assessing armature windings and its application of calculating torque for permanent magnet synchronous machines(PMSM). This work contributes to advancing the analysis of AC machines and underscores the potential for improved design and performance optimization.展开更多
A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A ...A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In this paper,we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes.展开更多
Poloidal field(PF) converters provide controlled DC voltage and current to PF coils. The many harmonics generated by the PF converter flow into the power grid and seriously affect power systems and electric equipment....Poloidal field(PF) converters provide controlled DC voltage and current to PF coils. The many harmonics generated by the PF converter flow into the power grid and seriously affect power systems and electric equipment. Due to the complexity of the system, the traditional integral operation in Fourier analysis is complicated and inaccurate. This paper presents a piecewise method to calculate the harmonics of the ITER PF converter. The relationship between the grid input current and the DC output current of the ITER PF converter is deduced. The grid current is decomposed into the sum of some simple functions. By calculating simple function harmonics based on the piecewise method, the harmonics of the PF converter under different operation modes are obtained.In order to examine the validity of the method, a simulation model is established based on Matlab/Simulink and a relevant experiment is implemented in the ITER PF integration test platform.Comparative results are given. The calculated results are found to be consistent with simulation and experiment. The piecewise method is proved correct and valid for calculating the system harmonics.展开更多
Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time(HB-AFT) domain technique has some advantages in dealing with nonlinear probl...Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time(HB-AFT) domain technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.展开更多
In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman pl...In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlineaT frequency response (F-R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases:(1) semi-linear vibration;(2) weakly nonlinear vibration;(3) highly non linear vibration, are validated by comparing with the numerical solutio ns. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano /microelectromechanical transducers such as microphones and pressure sensors.展开更多
The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability c...The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.展开更多
Comparisons of wave reflection, transmission and harmonics due to different types of sub merged structures are investigated by a numerical method, the boundary-fitted coordinate (BFC) method. The types of submerged st...Comparisons of wave reflection, transmission and harmonics due to different types of sub merged structures are investigated by a numerical method, the boundary-fitted coordinate (BFC) method. The types of submerged structures include a submerged horizontal plate, submerged breakwa ters (rectangular and trapezoidal) and a step-type structure (topography). First, the BFC method is ver ified by comparing the computed results with the experimental data, including wave surface elevations, reflected and transmitted wave heights, and amplitudes of higher harmonics, showing that the method is a reasonable one to predict wave deformations due to the submerged structures. Secondly, the wave sur face elevations and the higher harmonics over different submerged structures are compared. Thirdly, re flected and transmitted waves due to different submerged structures are investigated.展开更多
One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying li...One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying linear system and a small number of extra terms that exhibit the non-linear effects.In this paper,the method is illustrated in a simulated system and an experimental structure.The main objective of the non-linear resonant decay method is to identify the non-linear dynamic systems based on the use of a multi-shaker excitation using appropriated excitation which is obtained from the force appropriation approach.The experimental application of the method is indicated to provide suitable estimates of modal parameters for the identification of non-linear models of structures.展开更多
Dielectric elastomer(DE) is suitable in soft transducers for broad applications,among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices cal...Dielectric elastomer(DE) is suitable in soft transducers for broad applications,among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices call for an efficient and reliable method to analyze the dynamic response of DE. This remains to be a challenge since the resultant vibration equation of DE, for example, the vibration of a DE balloon considered here is highly nonlinear with higher-order power terms and time-dependent coefficients. Previous efforts toward this goal use largely the numerical integration method with the simple harmonic balance method as a supplement. The numerical integration and the simple harmonic balance method are inefficient for large parametric analysis or with difficulty in improving the solution accuracy. To overcome the weakness of these two methods,we describe formulations of the incremental harmonic balance(IHB) method for periodic forced solutions of such a unique system. Combined with an arc-length continuation technique, the proposed strategy can capture the whole solution branches, both stable and unstable, automatically with any desired accuracy.展开更多
We have deduced an iteration scheme in theincremental harmonic balance (IHB) method using the harmonicbalance plus the Newton-Raphson method. Since theconvergence of the iteration is dependent upon the initial valuesi...We have deduced an iteration scheme in theincremental harmonic balance (IHB) method using the harmonicbalance plus the Newton-Raphson method. Since theconvergence of the iteration is dependent upon the initial valuesin the iteration, the convergent region is greatly restrictedfor some cases. In this contribution, in order to enlarge theconvergent region of the IHB method, we constructed thezeroth-order deformation equation using the homotopy analysismethod, in which the IHB method is employed to solvethe deformation equation with an embedding parameter asthe active increment. Taking the Duffing and the van derPol equations as examples, we obtained the highly accuratesolutions. Importantly, the presented approach renders a convenientway to control and adjust the convergence.展开更多
A transmission grating coupled with an X-ray charge coupled device(CCD)is used to quantitatively measure the proportion of high-order harmonics of the soft-X-ray source of beam line 4B7B.The results show that the mono...A transmission grating coupled with an X-ray charge coupled device(CCD)is used to quantitatively measure the proportion of high-order harmonics of the soft-X-ray source of beam line 4B7B.The results show that the monochromatic X-ray has third-order and second-order harmonics.The proportion of second-order harmonic of 4B7B is less than 9.0%and the thirdorder harmonic is below 0.7%when no suppressing method is applied.When suppression methods are used,the proportion of second-order harmonic is less than 1.7%and the third-order harmonic is ignorable.展开更多
In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve ...In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.展开更多
In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Conver...In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.展开更多
The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered...The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.展开更多
This article aims to provide a brief overview of the relevance of new findings about the Fibonacci Life Chart Method (FLCM) for understanding synchronicity. The FLCM is reviewed first, including an exposition of the g...This article aims to provide a brief overview of the relevance of new findings about the Fibonacci Life Chart Method (FLCM) for understanding synchronicity. The FLCM is reviewed first, including an exposition of the golden section model, and elaboration of a new harmonic model. The two models are then compared to illuminate several strengths and weaknesses in connection with the following four major criteria regarding synchronicity: explanatory adequacy;predictability of future synchronicities;simplicity of the model;and generalizability to other branches of knowledge. The review indicates that both models appear capable of simulating nonlinear and fractal dynamics. Hybrid approaches that combine both models are feasible and necessary for projects that aim to experimentally address synchronicity.展开更多
We review three derivative-free methods developed for uncertainty estimation of non-linear error propagation, namely, MC(Monte Carlo), SUT(scaled unscented transformation), and SI(sterling interpolation). In order to ...We review three derivative-free methods developed for uncertainty estimation of non-linear error propagation, namely, MC(Monte Carlo), SUT(scaled unscented transformation), and SI(sterling interpolation). In order to avoid preset parameters like as these three methods need, we introduce a new method to uncertainty estimation for the first time, namely, SCR(spherical cubature rule), which is no need for setting parameters. By theoretical derivation, we prove that the precision of uncertainty obtained by SCR can reach second-order. We conduct four synthetic experiments, for the first two experiments, the results obtained by SCR are consistent with the other three methods with optimal setting parameters, but SCR is easier to operate than other three methods, which verifies the superiority of SCR in calculating the uncertainty. For the third experiment, real-time calculation is required, so the MC is hardly feasible. For the forth experiment, the SCR is applied to the inversion of seismic fault parameter which is a common problem in geophysics, and we study the sensitivity of surface displacements to fault parameters with errors. Our results show that the uncertainty of the surface displacements is the magnitude of ±10 mm when the fault length contains a variance of 0.01 km^(2).展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11972129 and12372008)the National Major Science and Technology Projects of China(No.2017-IV-0008-0045)+3 种基金the Natural Science Foundation of Heilongjiang Province of China(No.YQ2022A008)the Fundamental Research Funds for the Central Universities of China(No.HIT.OCEF.2023006)the Polish National Science Centre of Poland under the OPUS 18 grant(No.2019/35/B/ST8/00980)the Tianjin University Independent Innovation Foundation of China(No.2023XJS-0038)。
文摘The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.
基金supported by National Key Research and Development Program of China(No.2021YFA0717100)NationalNatural Science Foundation of China(Nos.12072270,U2013206).
文摘The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to complex symbolic operations.Many variants have been developed to improve the HB method,among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation.So far,there are two problems remaining to be addressed.i)A dozen of different versions of HB-like methods,in frequency domain or time domain or in hybrid,have been developed;unfortunately,misclassification pervades among them due to the unclear borderlines of different methods.ii)The time domain HB-like methods suffer from non-physical solutions,which have been shown to be caused by aliasing(mixture of the high-order into the low-order harmonics).Although a series of dealiasing techniques have been developed over the past two decades,the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community.This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences.In particular,the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.
基金supported by the National Natural Science Foundation of China(Grant No.61372046)the Research Fund for the Doctoral Program of Higher Education of China(New Teachers)(Grant No.20116101120018)+6 种基金the China Postdoctoral Science Foundation Funded Project(Grant Nos.2011M501467 and 2012T50814)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2011JQ1006)the Fundamental Research Funds for the Central Universities(Grant No.GK201302007)Science and Technology Plan Program,in Shaanxi Province of China(Grant Nos.2012 KJXX-29 and 2013K12-20-12)the Science and Technology Plan Program in Xian of China(Grant No.CXY1348(2))the.GraduateInovation Project of Northwest University(Grant No.YZZ12093)the Seience and Technology Program of Educational Committee,of Shaanxi Province of China(Grant No.12JK0729).
文摘Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.
基金Supported by pre-research fund of State Key Laboratory (51479080201 JW0802)
文摘Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical harmonics function is used to expand the angular flux. A set of differential equations about the spatial variable, which are coupled with each other, can be obtained. They are solved iteratively by using the finite element method on un- structured-meshes. A two-dimension transport calculation program is coded according to the model. The numerical results of some benchmark problems demonstrate that this method can give high precision results and avoid the ray effect very well.
基金supported by the Natural Science Foundation of China under Grant U22A20214 and Grant 51837010。
文摘A versatile analytical method(VAM) for calculating the harmonic components of the magnetomotive force(MMF) generated by diverse armature windings in AC machines has been proposed, and the versatility of this method has been established in early literature. However, its practical applications and significance in advancing the analysis of AC machines need further elaboration. This paper aims to complement VAM by augmenting its theory, offering additional insights into its conclusions, as well as demonstrating its utility in assessing armature windings and its application of calculating torque for permanent magnet synchronous machines(PMSM). This work contributes to advancing the analysis of AC machines and underscores the potential for improved design and performance optimization.
基金support from the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No.2019319)support from the Start-up Foundation of Suzhou Institute of Nano-Tech and Nano-Bionics,CAS,Suzhou (Grant No.Y9AAD110)。
文摘A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In this paper,we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes.
文摘Poloidal field(PF) converters provide controlled DC voltage and current to PF coils. The many harmonics generated by the PF converter flow into the power grid and seriously affect power systems and electric equipment. Due to the complexity of the system, the traditional integral operation in Fourier analysis is complicated and inaccurate. This paper presents a piecewise method to calculate the harmonics of the ITER PF converter. The relationship between the grid input current and the DC output current of the ITER PF converter is deduced. The grid current is decomposed into the sum of some simple functions. By calculating simple function harmonics based on the piecewise method, the harmonics of the PF converter under different operation modes are obtained.In order to examine the validity of the method, a simulation model is established based on Matlab/Simulink and a relevant experiment is implemented in the ITER PF integration test platform.Comparative results are given. The calculated results are found to be consistent with simulation and experiment. The piecewise method is proved correct and valid for calculating the system harmonics.
基金supported by the National Natural Science Foundation of China(No.10632040)
文摘Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time(HB-AFT) domain technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.
文摘In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlineaT frequency response (F-R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases:(1) semi-linear vibration;(2) weakly nonlinear vibration;(3) highly non linear vibration, are validated by comparing with the numerical solutio ns. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano /microelectromechanical transducers such as microphones and pressure sensors.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11872254 and 11672191)
文摘The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.
文摘Comparisons of wave reflection, transmission and harmonics due to different types of sub merged structures are investigated by a numerical method, the boundary-fitted coordinate (BFC) method. The types of submerged structures include a submerged horizontal plate, submerged breakwa ters (rectangular and trapezoidal) and a step-type structure (topography). First, the BFC method is ver ified by comparing the computed results with the experimental data, including wave surface elevations, reflected and transmitted wave heights, and amplitudes of higher harmonics, showing that the method is a reasonable one to predict wave deformations due to the submerged structures. Secondly, the wave sur face elevations and the higher harmonics over different submerged structures are compared. Thirdly, re flected and transmitted waves due to different submerged structures are investigated.
文摘One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying linear system and a small number of extra terms that exhibit the non-linear effects.In this paper,the method is illustrated in a simulated system and an experimental structure.The main objective of the non-linear resonant decay method is to identify the non-linear dynamic systems based on the use of a multi-shaker excitation using appropriated excitation which is obtained from the force appropriation approach.The experimental application of the method is indicated to provide suitable estimates of modal parameters for the identification of non-linear models of structures.
基金the National Natural Science Foundation of China(Nos.11702215 and11972277)the Natural Science Basic Research Plan in Shaanxi Province of China(Nos.2017JQ5062 and 2018JQ1029)。
文摘Dielectric elastomer(DE) is suitable in soft transducers for broad applications,among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices call for an efficient and reliable method to analyze the dynamic response of DE. This remains to be a challenge since the resultant vibration equation of DE, for example, the vibration of a DE balloon considered here is highly nonlinear with higher-order power terms and time-dependent coefficients. Previous efforts toward this goal use largely the numerical integration method with the simple harmonic balance method as a supplement. The numerical integration and the simple harmonic balance method are inefficient for large parametric analysis or with difficulty in improving the solution accuracy. To overcome the weakness of these two methods,we describe formulations of the incremental harmonic balance(IHB) method for periodic forced solutions of such a unique system. Combined with an arc-length continuation technique, the proposed strategy can capture the whole solution branches, both stable and unstable, automatically with any desired accuracy.
基金supported by the National Natural Science Foundation of China (10772202)Doctoral Program Foundation of Ministry of Education of China (20050558032)Guangdong Province Natural Science Foundation (07003680, 05003295)
文摘We have deduced an iteration scheme in theincremental harmonic balance (IHB) method using the harmonicbalance plus the Newton-Raphson method. Since theconvergence of the iteration is dependent upon the initial valuesin the iteration, the convergent region is greatly restrictedfor some cases. In this contribution, in order to enlarge theconvergent region of the IHB method, we constructed thezeroth-order deformation equation using the homotopy analysismethod, in which the IHB method is employed to solvethe deformation equation with an embedding parameter asthe active increment. Taking the Duffing and the van derPol equations as examples, we obtained the highly accuratesolutions. Importantly, the presented approach renders a convenientway to control and adjust the convergence.
文摘A transmission grating coupled with an X-ray charge coupled device(CCD)is used to quantitatively measure the proportion of high-order harmonics of the soft-X-ray source of beam line 4B7B.The results show that the monochromatic X-ray has third-order and second-order harmonics.The proportion of second-order harmonic of 4B7B is less than 9.0%and the thirdorder harmonic is below 0.7%when no suppressing method is applied.When suppression methods are used,the proportion of second-order harmonic is less than 1.7%and the third-order harmonic is ignorable.
基金supported in part by NSF of China N.10871131The Science and Technology Commission of Shanghai Municipality,Grant N.075105118+1 种基金Shanghai Leading Academic Discipline Project N.T0401Fund for E-institute of Shanghai Universities N.E03004.
文摘In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.
基金the Natural Science Foundation of China(Grant Nos.61673169,11301127,11701176,11626101,and 11601485)The Natural Science Foundation of Huzhou City(Grant No.2018YZ07).
文摘In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear equations.Convergence analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is nine.Our main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU time-methodology.Dynamical planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative methods.Convergence behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test examples.Numerical test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.
基金Project supported by the Ph. D. Programs Foundation of Ministry of Education of China (No.20050558032) the Natural Science Foundation of Guangdong Province of China (No.05003295) the Foundation of Sun Yat-sen University Advanced Research Center (No.06M8) the Young Teacher Scientific Research Foundation of Sun Sat-sen University (No.1131011)
文摘The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.
文摘This article aims to provide a brief overview of the relevance of new findings about the Fibonacci Life Chart Method (FLCM) for understanding synchronicity. The FLCM is reviewed first, including an exposition of the golden section model, and elaboration of a new harmonic model. The two models are then compared to illuminate several strengths and weaknesses in connection with the following four major criteria regarding synchronicity: explanatory adequacy;predictability of future synchronicities;simplicity of the model;and generalizability to other branches of knowledge. The review indicates that both models appear capable of simulating nonlinear and fractal dynamics. Hybrid approaches that combine both models are feasible and necessary for projects that aim to experimentally address synchronicity.
基金supported by the National Natural Science Foundation of China (41721003, 41974022, 41774024, 41874001)Open Research Fund Program of the Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, China(20-02-05)
文摘We review three derivative-free methods developed for uncertainty estimation of non-linear error propagation, namely, MC(Monte Carlo), SUT(scaled unscented transformation), and SI(sterling interpolation). In order to avoid preset parameters like as these three methods need, we introduce a new method to uncertainty estimation for the first time, namely, SCR(spherical cubature rule), which is no need for setting parameters. By theoretical derivation, we prove that the precision of uncertainty obtained by SCR can reach second-order. We conduct four synthetic experiments, for the first two experiments, the results obtained by SCR are consistent with the other three methods with optimal setting parameters, but SCR is easier to operate than other three methods, which verifies the superiority of SCR in calculating the uncertainty. For the third experiment, real-time calculation is required, so the MC is hardly feasible. For the forth experiment, the SCR is applied to the inversion of seismic fault parameter which is a common problem in geophysics, and we study the sensitivity of surface displacements to fault parameters with errors. Our results show that the uncertainty of the surface displacements is the magnitude of ±10 mm when the fault length contains a variance of 0.01 km^(2).