This paper covers the concept of Fourier series and its application for a periodic signal. A periodic signal is a signal that repeats its pattern over time at regular intervals. The idea inspiring is to approximate a ...This paper covers the concept of Fourier series and its application for a periodic signal. A periodic signal is a signal that repeats its pattern over time at regular intervals. The idea inspiring is to approximate a regular periodic signal, under Dirichlet conditions, via a linear superposition of trigonometric functions, thus Fourier polynomials are constructed. The Dirichlet conditions, are a set of mathematical conditions, providing a foundational framework for the validity of the Fourier series representation. By understanding and applying these conditions, we can accurately represent and process periodic signals, leading to advancements in various areas of signal processing. The resulting Fourier approximation allows complex periodic signals to be expressed as a sum of simpler sinusoidal functions, making it easier to analyze and manipulate such signals.展开更多
In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approxim...In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Kiemann -Lebesgue matrix property and the bessel-parseval matrix inequality are proved. The concept of total set vjith respect to a positive definite matrix functional is introduced , and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional, is established.展开更多
The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classi...The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.展开更多
Permafrost,being an important component of the cryosphere,is sensitive to climate change.Therefore,it is necessary to investigate the change of temperature within permafrost.In this study,we proposed a Fourier series ...Permafrost,being an important component of the cryosphere,is sensitive to climate change.Therefore,it is necessary to investigate the change of temperature within permafrost.In this study,we proposed a Fourier series model derived from the conduction equation to simulate permafrost thermal behavior over a year.The boundary condition was represented by the Fourier series and the geothermal gradient.The initial condition was represented as a linear function relative to the geothermal gradient.A comparative study of the different models(sinusoidal model,Fourier series model,and the proposed model)was conducted.Data collected from the northern Da Xing’anling Mountains,Northeast China,were applied for parameterization and validation for these models.These models were compared with daily mean ground temperature from the shallow permafrost layer and annual mean ground temperature from the bottom permafrost layer,respectively.Model performance was assessed using three coefficients of accuracy,i.e.,the mean bias error,the root mean square error,and the coefficient of determination.The comparison results showed that the proposed model was accurate enough to simulate temperature variation in both the shallow and bottom permafrost layer as compared with the other two Fourier series models(sinusoidal model and Fourier model).The proposed model expanded on a previous Fourier series model for which the initial and bottom boundary conditions were restricted to being constant.展开更多
The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust c...The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.展开更多
Purpose–With the development of economy,China’s OFDI constantly increase in recent year.Meanwhile,OFDI hasspillovereffectoneconomicdevelopmentandtechnologicaldevelopmentofhomecountry.Thus,accurateOFDI prediction is ...Purpose–With the development of economy,China’s OFDI constantly increase in recent year.Meanwhile,OFDI hasspillovereffectoneconomicdevelopmentandtechnologicaldevelopmentofhomecountry.Thus,accurateOFDI prediction is a prerequisite for the effective development of international investment strategies.The purpose of this paper is to predict China’s OFDI accurately using a novel multivariable grey prediction model with Fourier series.Design/methodology/approach–This paper applied a multivariable grey prediction model,GM(1,N),to forecast China’s OFDI.In order to improve the prediction accuracy and without changing local characteristics of grey model prediction,this paper proposed a novel grey prediction model to improve the performance of the traditionalGM(1,N)modelbycombiningwithresidualmodificationmodelusingGM(1,1)modelandFourierseries.Findings–The coefficients indicate that the export and GDP have positive influence on China’s OFDI,and,according to the prediction result,China’s OFDI shows a growing trend in next five years.Originality/value–This paper proposed an effective multivariable grey prediction model that combined the traditionalGM(1,N)modelwitharesidualmodificationmodelinordertopredictChina’sOFDI.Accurateforecasting of OFDI provides reference for the Chinese Government to implement international investment strategies.展开更多
The perishable nature of tourism products and services makes forecasting an important tool for tourism planning,especially in the current COVID-19 pandemic time.The forecast assists tourism organizations in decision-m...The perishable nature of tourism products and services makes forecasting an important tool for tourism planning,especially in the current COVID-19 pandemic time.The forecast assists tourism organizations in decision-making regarding resource allocations to avoid shortcomings.This study is motivated by the need to model periodic time series with linear and nonlinear trends.A hybrid Polynomial-Fourier series model that uses the combination of polynomial and Fourier fittings to capture and forecast time series data was proposed.The proposed model is applied to monthly foreign visitors to Turkey from January 2014 to August 2020 dataset and diagnostic checks show that the proposed model produces a statistically good fit.To improve the model forecast,a Monte Carlo simulation scheme with 100 simulation paths is applied to the model residue.The mean of the 100 simulation paths within±2σbounds from the model curve was taken and found to give statistically acceptable results.展开更多
A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier seri...A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier series,which ensures the continuity of the derivative at the boundary and enhances the convergence.The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation.The efficiency and accuracy of the present method are validated in comparison with the finite element method(FEM)and other methods.The effects of the boundary spring stiffness,the number of spans,the spinning velocity,and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied.The results show that the present method can freely simulate any boundary constraints without modifying the solution process.The elastic range of linear springs is larger than that of torsion springs,and it is not affected by the number of spans.With an increase in the axial compressive force,the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger,while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary.展开更多
Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quoti...Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .展开更多
An absolute gravimeter is a precision instrument for measuring gravitational acceleration, which plays an important role in earthquake monitoring, crustal deformation, national defense construction, etc. The frequency...An absolute gravimeter is a precision instrument for measuring gravitational acceleration, which plays an important role in earthquake monitoring, crustal deformation, national defense construction, etc. The frequency of laser interference fringes of an absolute gravimeter gradually increases with the fall time. Data are sparse in the early stage and dense in the late stage. The fitting accuracy of gravitational acceleration will be affected by least-squares fitting according to the fixed number of zero-crossing groups. In response to this problem, a method based on Fourier series fitting is proposed in this paper to calculate the zero-crossing point. The whole falling process is divided into five frequency bands using the Hilbert transformation. The multiplicative auto-regressive moving average model is then trained according to the number of optimal zero-crossing groups obtained by the honey badger algorithm. Through this model, the number of optimal zero-crossing groups determined in each segment is predicted by the least-squares fitting. The mean value of gravitational acceleration in each segment is then obtained. The method can improve the accuracy of gravitational measurement by more than 25% compared to the fixed zero-crossing groups method. It provides a new way to improve the measuring accuracy of an absolute gravimeter.展开更多
The major advantage of grey system theory is that both incomplete information and unclear problems can be processed precisely. Considering that the modeling of grey model(GM) depends on the preprocessing of the origin...The major advantage of grey system theory is that both incomplete information and unclear problems can be processed precisely. Considering that the modeling of grey model(GM) depends on the preprocessing of the original data,the fractional-order accumulation calculus could be used to do preprocessing. In this paper, the residual sequence represented by Fourier series is used to ameliorate performance of the fractionalorder accumulation GM(1,1) and improve the accuracy of predictor. The state space model of optimally modified GM(1,1)predictor is given and genetic algorithm(GA) is used to find the smallest relative error during the modeling step. Furthermore,the fractional form of continuous GM(1,1) is given to enlarge the content of prediction model. The simulation results illustrated that the fractional-order calculus could be used to depict the GM precisely with more degrees of freedom. Meanwhile, the ranges of the parameters and model application could be enlarged with better performance. The method of modified GM predictor using optimal fractional-order accumulation calculus is expected to be widely used in data processing, model theory, prediction control and related fields.展开更多
The modified atomic transformations are constructed and proved.On their basis the new complex analytic wavelets are obtained.The proof of the Fourier transforms existence in L1 and L2 on the basis of the theory of ato...The modified atomic transformations are constructed and proved.On their basis the new complex analytic wavelets are obtained.The proof of the Fourier transforms existence in L1 and L2 on the basis of the theory of atomic functions(AF)are presented.The numerical experiments of digital time series processing and physical analysis of the results confirm the efficiency of the proposed transforms.展开更多
An effective approach for optimizing the rotor contour for variable reluctance(VR)resolver is presented.Using this approach,the procedure for optimizing the rotor is divided into two parts:the establishment of initial...An effective approach for optimizing the rotor contour for variable reluctance(VR)resolver is presented.Using this approach,the procedure for optimizing the rotor is divided into two parts:the establishment of initial shape curve,and then computation for the optimization.In order to simplify the process of the former,a shape function is constructed.And the latter is carried out by Taguchi optimization method and finite element method(FEM).An example of a 3-10 VR resolver is used to present the procedure of the optimization,and the testing results confirmed the effectivity of the approach.展开更多
Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying...Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.展开更多
The aeroelastic stability of rotating beams with elastic restraints is investigated.The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling.The Greenberg a...The aeroelastic stability of rotating beams with elastic restraints is investigated.The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling.The Greenberg aerodynamic model is used to describe the unsteady aerodynamic forces.The additional centrifugal stiffness effect and elastic boundary conditions are considered in the form of potential energy.A modified Fourier series method is used to assume the displacement field function and solve the governing equation.The convergence and accuracy of the method are verified by comparison of numerical results.Then,the flutter analysis of the rotating beam structure is carried out,and the critical rotational velocity of the flutter is predicted.The results show that the elastic boundary reduces the critical flutter velocity of the rotating beam,and the elastic range of torsional spring is larger than the elastic range of linear spring.展开更多
For non-asymmetrical bending problems of elastic annular plates, the exact solutions are not fond. To bending problems of infinite annular plate with two different boundary conditions, based on the boundary integral f...For non-asymmetrical bending problems of elastic annular plates, the exact solutions are not fond. To bending problems of infinite annular plate with two different boundary conditions, based on the boundary integral formula,the natural boundary integral equation for the boundary value problems of the biharmonic equation and the condition of bending moment in infinity,bending solutions under non-symmetrical loads are gained by the Fourier series and convolution formulae. The formula for the solutions has nicer convergence velocity and high computational accuracy, and the calculating process is simpler. Solutions of the given examples are compared with the finite element method. The textual solutions of moments near the loads are better than the finite element method to the fact that near the concentrative loads the inners forces trend to infinite.展开更多
The first proof (Sections 2 - 4), applies Doppler shifts to the Fourier time sine series. It shows if <i>K</i> = <i>f'/f</i> (frequency ratio of the shift, <i>f'</i> the shi...The first proof (Sections 2 - 4), applies Doppler shifts to the Fourier time sine series. It shows if <i>K</i> = <i>f'/f</i> (frequency ratio of the shift, <i>f'</i> the shifted frequency), then the value of the series at time = t occurs in the shifted series at time = <i>t/K</i>. That is because in each harmonic the sin(<i>f't</i>) and cos(<i>f't</i>) became sin(<i>Kft</i>) and cos(<i>Kft</i>). The original series could be the number of photons in area of a beam with encoded information. Therefore the number of observed photons and information has the same resultant Doppler shift as frequency. Resultant is total effect of axial, transverse and gravitational shifts. Mass and energy of light do not have the same Doppler shift which may indicate missing parts that Doppler shift and that mass traveling at the speed of light is different from other mass and energy. The second proof (Section 5) the vector equations of space time require observed time to have 3 dimensions if the speed of light is constant in all directions, but the Doppler shift in each direction is not the same. The blue shift (compression) of time has paradoxes. If time has many dimensions, that would solve the paradoxes, but break conservation laws. No solution of that is given here. It is not expected that radical a solution will have any followers.展开更多
As we have stated in conclusion of PART IV of these series including PART V,here in wrapping up and ending these series,we are producing with PART VI,which is nothing more than continuation of PART V.
As we have stated in conclusion of PART IV of these series,here in PART V,we will show how to find the solution for the governing equation of heat conduction as it was setup in PART IV,given the boundary and initial c...As we have stated in conclusion of PART IV of these series,here in PART V,we will show how to find the solution for the governing equation of heat conduction as it was setup in PART IV,given the boundary and initial conditions for Eq.(156)by means of exact and numerical methods.The different sections provided in here as PART V is consisting of a discussion of the approximate solution of the problem using mathematical tools and divided into four other sections parts as illustrated in this part.First four section namely 2.0,3.0,4.0 and 5.0 present an analytical method of the solution of the general governing equation using the Fourier theory.Section 6.0 is considering interaction of laser energy with materials using very short laser pulses and introduces electron-phonon theory approach to solve the heat transfer problem of the interaction of ultra-short pulses with the matter.Section 7.0 describes heating analysis with time-dependent pulse intensity and where evaporation is considered as the exclusive phenomenon taking place during the ablation process.Section 8.0 presents the heating analysis with pulsed laser heating process by considering both Fourier conduction and electron-phonon kinetic theory approaches.Finally,Section 9.0 consists of a discussion of the approximate solution of the problem using the Finite Difference Method(FDM)and Finite Element Method(FEM),and presents the computer solutions developed.展开更多
文摘This paper covers the concept of Fourier series and its application for a periodic signal. A periodic signal is a signal that repeats its pattern over time at regular intervals. The idea inspiring is to approximate a regular periodic signal, under Dirichlet conditions, via a linear superposition of trigonometric functions, thus Fourier polynomials are constructed. The Dirichlet conditions, are a set of mathematical conditions, providing a foundational framework for the validity of the Fourier series representation. By understanding and applying these conditions, we can accurately represent and process periodic signals, leading to advancements in various areas of signal processing. The resulting Fourier approximation allows complex periodic signals to be expressed as a sum of simpler sinusoidal functions, making it easier to analyze and manipulate such signals.
文摘In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Kiemann -Lebesgue matrix property and the bessel-parseval matrix inequality are proved. The concept of total set vjith respect to a positive definite matrix functional is introduced , and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional, is established.
文摘The key objective of this paper is to improve the approximation of a sufficiently smooth nonperiodic function defined on a compact interval by proposing alternative forms of Fourier series expansions. Unlike in classical Fourier series, the expansion coefficients herein are explicitly dependent not only on the function itself, but also on its derivatives at the ends of the interval. Each of these series expansions can be made to converge faster at a desired polynomial rate. These results have useful implications to Fourier or harmonic analysis, solutions to differential equations and boundary value problems, data compression, and so on.
基金founded by the Key Joint Program of National Natural Science Foundation of China(NSFC)and Heilongjiang Province for Regional Development(No.U20A2082)National Natural Science Foundation of China(No.41971151)Natural Science Foundation of Heilongjiang Province(No.TD2019D002)。
文摘Permafrost,being an important component of the cryosphere,is sensitive to climate change.Therefore,it is necessary to investigate the change of temperature within permafrost.In this study,we proposed a Fourier series model derived from the conduction equation to simulate permafrost thermal behavior over a year.The boundary condition was represented by the Fourier series and the geothermal gradient.The initial condition was represented as a linear function relative to the geothermal gradient.A comparative study of the different models(sinusoidal model,Fourier series model,and the proposed model)was conducted.Data collected from the northern Da Xing’anling Mountains,Northeast China,were applied for parameterization and validation for these models.These models were compared with daily mean ground temperature from the shallow permafrost layer and annual mean ground temperature from the bottom permafrost layer,respectively.Model performance was assessed using three coefficients of accuracy,i.e.,the mean bias error,the root mean square error,and the coefficient of determination.The comparison results showed that the proposed model was accurate enough to simulate temperature variation in both the shallow and bottom permafrost layer as compared with the other two Fourier series models(sinusoidal model and Fourier model).The proposed model expanded on a previous Fourier series model for which the initial and bottom boundary conditions were restricted to being constant.
基金supported in part by the National Natural Science Foundation of China(Nos.11702072 and 11672093)。
文摘The process of formation reconfiguration for close-range satellite formation should take into account the risk of collisions between satellites.To this end,this paper presents a method to rapidly generate low-thrust collision-avoidance trajectories in the formation reconfiguration using Finite Fourier Series(FFS).The FFS method can rapidly generate the collision-avoidance threedimensional trajectory.The results obtained by the FFS method are used as an initial guess in the Gauss Pseudospectral Method(GPM)solver to verify the applicability of the results.Compared with the GPM method,the FFS method needs very little computing time to obtain the results with very little difference in performance index.To verify the effectiveness,the proposed method is tested and validated by a formation control testbed.Three satellite simulators in the testbed are used to simulate two-dimensional satellite formation reconfiguration.The simulation and experimental results show that the FFS method can rapidly generate trajectories and effectively reduce the risk of collision between satellites.This fast trajectory generation method has great significance for on-line,constantly satellite formation reconfiguration.
文摘Purpose–With the development of economy,China’s OFDI constantly increase in recent year.Meanwhile,OFDI hasspillovereffectoneconomicdevelopmentandtechnologicaldevelopmentofhomecountry.Thus,accurateOFDI prediction is a prerequisite for the effective development of international investment strategies.The purpose of this paper is to predict China’s OFDI accurately using a novel multivariable grey prediction model with Fourier series.Design/methodology/approach–This paper applied a multivariable grey prediction model,GM(1,N),to forecast China’s OFDI.In order to improve the prediction accuracy and without changing local characteristics of grey model prediction,this paper proposed a novel grey prediction model to improve the performance of the traditionalGM(1,N)modelbycombiningwithresidualmodificationmodelusingGM(1,1)modelandFourierseries.Findings–The coefficients indicate that the export and GDP have positive influence on China’s OFDI,and,according to the prediction result,China’s OFDI shows a growing trend in next five years.Originality/value–This paper proposed an effective multivariable grey prediction model that combined the traditionalGM(1,N)modelwitharesidualmodificationmodelinordertopredictChina’sOFDI.Accurateforecasting of OFDI provides reference for the Chinese Government to implement international investment strategies.
文摘The perishable nature of tourism products and services makes forecasting an important tool for tourism planning,especially in the current COVID-19 pandemic time.The forecast assists tourism organizations in decision-making regarding resource allocations to avoid shortcomings.This study is motivated by the need to model periodic time series with linear and nonlinear trends.A hybrid Polynomial-Fourier series model that uses the combination of polynomial and Fourier fittings to capture and forecast time series data was proposed.The proposed model is applied to monthly foreign visitors to Turkey from January 2014 to August 2020 dataset and diagnostic checks show that the proposed model produces a statistically good fit.To improve the model forecast,a Monte Carlo simulation scheme with 100 simulation paths is applied to the model residue.The mean of the 100 simulation paths within±2σbounds from the model curve was taken and found to give statistically acceptable results.
基金Project supported by the National Science Fund for Distinguished Young Scholars of China (No.11925205)the National Natural Science Foundation of China (Nos.51921003 and 12272165)。
文摘A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier series,which ensures the continuity of the derivative at the boundary and enhances the convergence.The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation.The efficiency and accuracy of the present method are validated in comparison with the finite element method(FEM)and other methods.The effects of the boundary spring stiffness,the number of spans,the spinning velocity,and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied.The results show that the present method can freely simulate any boundary constraints without modifying the solution process.The elastic range of linear springs is larger than that of torsion springs,and it is not affected by the number of spans.With an increase in the axial compressive force,the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger,while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary.
文摘Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .
基金Project supported by the National Key R&D Program of China (Grant No. 2022YFF0607504)。
文摘An absolute gravimeter is a precision instrument for measuring gravitational acceleration, which plays an important role in earthquake monitoring, crustal deformation, national defense construction, etc. The frequency of laser interference fringes of an absolute gravimeter gradually increases with the fall time. Data are sparse in the early stage and dense in the late stage. The fitting accuracy of gravitational acceleration will be affected by least-squares fitting according to the fixed number of zero-crossing groups. In response to this problem, a method based on Fourier series fitting is proposed in this paper to calculate the zero-crossing point. The whole falling process is divided into five frequency bands using the Hilbert transformation. The multiplicative auto-regressive moving average model is then trained according to the number of optimal zero-crossing groups obtained by the honey badger algorithm. Through this model, the number of optimal zero-crossing groups determined in each segment is predicted by the least-squares fitting. The mean value of gravitational acceleration in each segment is then obtained. The method can improve the accuracy of gravitational measurement by more than 25% compared to the fixed zero-crossing groups method. It provides a new way to improve the measuring accuracy of an absolute gravimeter.
基金supported by the National Natural Science Foundation of China(61174145)
文摘The major advantage of grey system theory is that both incomplete information and unclear problems can be processed precisely. Considering that the modeling of grey model(GM) depends on the preprocessing of the original data,the fractional-order accumulation calculus could be used to do preprocessing. In this paper, the residual sequence represented by Fourier series is used to ameliorate performance of the fractionalorder accumulation GM(1,1) and improve the accuracy of predictor. The state space model of optimally modified GM(1,1)predictor is given and genetic algorithm(GA) is used to find the smallest relative error during the modeling step. Furthermore,the fractional form of continuous GM(1,1) is given to enlarge the content of prediction model. The simulation results illustrated that the fractional-order calculus could be used to depict the GM precisely with more degrees of freedom. Meanwhile, the ranges of the parameters and model application could be enlarged with better performance. The method of modified GM predictor using optimal fractional-order accumulation calculus is expected to be widely used in data processing, model theory, prediction control and related fields.
文摘The modified atomic transformations are constructed and proved.On their basis the new complex analytic wavelets are obtained.The proof of the Fourier transforms existence in L1 and L2 on the basis of the theory of atomic functions(AF)are presented.The numerical experiments of digital time series processing and physical analysis of the results confirm the efficiency of the proposed transforms.
文摘An effective approach for optimizing the rotor contour for variable reluctance(VR)resolver is presented.Using this approach,the procedure for optimizing the rotor is divided into two parts:the establishment of initial shape curve,and then computation for the optimization.In order to simplify the process of the former,a shape function is constructed.And the latter is carried out by Taguchi optimization method and finite element method(FEM).An example of a 3-10 VR resolver is used to present the procedure of the optimization,and the testing results confirmed the effectivity of the approach.
基金Supported by the National Natural Science Foundation of China(51276017)
文摘Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.
基金Project supported by the National Science Fund for Distinguished Young Scholars(No.11925205)the National Natural Science Foundation of China(Nos.51921003 and 51805250)the Natural Science Foundation of Jiangsu Province of China(No.BK20180429)。
文摘The aeroelastic stability of rotating beams with elastic restraints is investigated.The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling.The Greenberg aerodynamic model is used to describe the unsteady aerodynamic forces.The additional centrifugal stiffness effect and elastic boundary conditions are considered in the form of potential energy.A modified Fourier series method is used to assume the displacement field function and solve the governing equation.The convergence and accuracy of the method are verified by comparison of numerical results.Then,the flutter analysis of the rotating beam structure is carried out,and the critical rotational velocity of the flutter is predicted.The results show that the elastic boundary reduces the critical flutter velocity of the rotating beam,and the elastic range of torsional spring is larger than the elastic range of linear spring.
基金Project supported by the National Basic Research Program of China (No. 2007CB209400)the National Nature Fond (No. 50774077 and 50774081)the National Fond of Author of Doctor Thesis (100760)
文摘For non-asymmetrical bending problems of elastic annular plates, the exact solutions are not fond. To bending problems of infinite annular plate with two different boundary conditions, based on the boundary integral formula,the natural boundary integral equation for the boundary value problems of the biharmonic equation and the condition of bending moment in infinity,bending solutions under non-symmetrical loads are gained by the Fourier series and convolution formulae. The formula for the solutions has nicer convergence velocity and high computational accuracy, and the calculating process is simpler. Solutions of the given examples are compared with the finite element method. The textual solutions of moments near the loads are better than the finite element method to the fact that near the concentrative loads the inners forces trend to infinite.
文摘The first proof (Sections 2 - 4), applies Doppler shifts to the Fourier time sine series. It shows if <i>K</i> = <i>f'/f</i> (frequency ratio of the shift, <i>f'</i> the shifted frequency), then the value of the series at time = t occurs in the shifted series at time = <i>t/K</i>. That is because in each harmonic the sin(<i>f't</i>) and cos(<i>f't</i>) became sin(<i>Kft</i>) and cos(<i>Kft</i>). The original series could be the number of photons in area of a beam with encoded information. Therefore the number of observed photons and information has the same resultant Doppler shift as frequency. Resultant is total effect of axial, transverse and gravitational shifts. Mass and energy of light do not have the same Doppler shift which may indicate missing parts that Doppler shift and that mass traveling at the speed of light is different from other mass and energy. The second proof (Section 5) the vector equations of space time require observed time to have 3 dimensions if the speed of light is constant in all directions, but the Doppler shift in each direction is not the same. The blue shift (compression) of time has paradoxes. If time has many dimensions, that would solve the paradoxes, but break conservation laws. No solution of that is given here. It is not expected that radical a solution will have any followers.
文摘As we have stated in conclusion of PART IV of these series including PART V,here in wrapping up and ending these series,we are producing with PART VI,which is nothing more than continuation of PART V.
文摘As we have stated in conclusion of PART IV of these series,here in PART V,we will show how to find the solution for the governing equation of heat conduction as it was setup in PART IV,given the boundary and initial conditions for Eq.(156)by means of exact and numerical methods.The different sections provided in here as PART V is consisting of a discussion of the approximate solution of the problem using mathematical tools and divided into four other sections parts as illustrated in this part.First four section namely 2.0,3.0,4.0 and 5.0 present an analytical method of the solution of the general governing equation using the Fourier theory.Section 6.0 is considering interaction of laser energy with materials using very short laser pulses and introduces electron-phonon theory approach to solve the heat transfer problem of the interaction of ultra-short pulses with the matter.Section 7.0 describes heating analysis with time-dependent pulse intensity and where evaporation is considered as the exclusive phenomenon taking place during the ablation process.Section 8.0 presents the heating analysis with pulsed laser heating process by considering both Fourier conduction and electron-phonon kinetic theory approaches.Finally,Section 9.0 consists of a discussion of the approximate solution of the problem using the Finite Difference Method(FDM)and Finite Element Method(FEM),and presents the computer solutions developed.
