Series Of Trigonometric Functions Research Articles

On the problem of a Gurtin–Murdoch cylindrical material surface embedded in an infinite matrix

The problem of an infinite isotropic elastic matrix subjected to uniform far-field load and containing a Gurtin–Murdoch material surface of cylindrical shape is considered in plane strain setting. The governing equations and the boundary conditions for the problem, reduced to that of an infinite plane containing a material curve along a circular arc, are reviewed. The displacements inside the matrix are sought in the complex variables form of a single layer elastic potential whose density represents the jump in complex tractions across the curve. Exact complex integral representations for the elastic fields everywhere in the material are provided and the problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the strain and rotation components associated with the curve. The components are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in the series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters with the main focus on the study of curvature-induced effects.

Two-dimensional problem of an infinite matrix reinforced with a Steigmann–Ogden cylindrical surface of circular arc cross-section

The plane strain problem of an elastic matrix subjected to uniform far-field load and containing a Steigmann–Ogden material surface with circular arc cross-section is considered. The governing equations and the boundary conditions for the problem are reviewed. Exact complex integral representations for the elastic fields everywhere in the material are provided. The problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the first component of the surface stress tensor (surface stress) and the remaining component of that tensor and its second derivative, along with various problem parameters. The two unknowns are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters. The connection of the problem with that of rigid circular arc is discussed.

МЕТОД РИТЦА В НЕЛИНЕЙНЫХ РАСЧЕТАХ ОДНОПРОЛЕТНЫХ ЖЕЛЕЗОБЕТОННЫХ БАЛОК.

The paper considers the calculations of single-span reinforced concrete bent beams, taking into account the nonlinear relationship between stresses and strains. Deflections of beams according to the Ritz method are given as a series of trigonometric functions with undetermined coefficients. Nonlinearity is taken into account by the "moment-curvature" dependence in the form of a cubic parabola. The functional of the total energy of the beam and the load acting on it has a non-linear form with respect to indefinite coefficients, which are found with the help of a computer mathematics package from the condition of the minimum of this functional. Two examples are given for a statically determinate reinforced concrete articulated beam under the action of a uniformly distributed load and a statically indefinite reinforced concrete beam under the action of a concentrated force. A comparison with an elastic solution is made. The coordinate functions of the Ritz method are given for single-span beams with other boundary conditions.

Data-driven design and optimization of ultra-tunable acoustic metamaterials

This paper presents a data-driven design and optimization of acoustic metamaterials with three-phase materials for highly tunable wave transmission. The geometry of representative unitcell is defined by the trigonometric series function to describe an arbitrary shape with symmetry, which enables the unitcell to achieve a large sub-wavelength bandgap. We propose a lightweight and efficient algorithm, ‘decoupled gradient decent (DGD)’, to search for the optimal design and uncover the ‘best’ shape features—the interface curvature—in tuning the wave transmission. As a result, the host composite can partly overlap the individual cell’s bandgap and achieve a wide frequency gap that forbids wave transmission, namely a passive tunability. Another advantage of the trigonometric series designed shape is the high flexibility. A slight surface pressure obviously deforms the unitcell and shifts its band structure. Our simulation shows that a moderate pressure dramatically changes the frequency forbidding gap for both traversal and longitudinal wave transmissions, which indicates an active tunability. The surface deformation can be applied by either a mechanical pressure or external electric field if the composite uses a dielectric substrate. Therefore, this study opens a sandbox of manipulating wave transmission through the topology and structure optimization in applications such as seismic damping (Hz), noise insulating (kHz) and ultrasound imaging (MHz).

Enhanced neural network model for regional ionospheric modeling and evaluation under different solar-geomagnetic conditions

Monitoring spatiotemporal variations of ionospheric vertical total electron content (VTEC) is crucial for space weather and satellite positioning. In the present study, an enhanced neural network (ENN) model is proposed to capture the changing characteristics of ionospheric VTEC and compared with the traditional mathematical models, i.e. the POLYnomial (POLY) model, generalized trigonometric series function and spherical harmonic function (SHF) model. The ionospheric VTEC data obtained from 31 permanent global positioning system stations in the southwest region of China on 26 August and 8 September, 2017, were used to test the performance of the mentioned models under different Solar-geomagnetic conditions. The ENN model is derived from the ensemble learning method, and the disadvantage that simple backpropagation neural network learners that are not robust enough is weakened by the ENN model. After statistical analysis and single-frequency precise point positioning (SF-PPP) experiments, it is demonstrated that the ENN model is superior to the above three mathematical models, regardless of the solar-geomagnetic conditions. In terms of mean absolute error, root mean square error, standard deviation, and mean absolute percentage error, the ENN model outperforms the SHF model, which is the best mathematical model in the analysis, by 40.7%, 30.20%, 29.88%, 38.04% under quiet solar-geomagnetic conditions, and by 37.66%, 29.93%, 30.96%, 32.01% under active solar-geomagnetic conditions. In addition, the accuracy of the SF-PPP is greatly affected by the error caused by ionosphere. In the static SF-PPP experiment of this study, the ENN model can better correct ionospheric error. Under quiet and active solar-geomagnetic conditions, the SF-PPP accuracy can be improved by 85.1% and 85.2% with the ionosphere delay correction from the ENN model.

Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems II: Large Amplitude Oscillations

This work is the second in a series of articles that deal with analytical solutions of nonlinear dynamical systems under oscillatory input that may exhibit harmonic frequencies. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods, because in most cases analytical solutions such as the harmonic balance series solution turn out to be difficult, if not impossible, as they are based on an infinite series of trigonometric functions with harmonic frequencies. The key contribution of the analytic matrix methods reported in the present series of articles is to work with the invariant submanifold of the problem and to propose the solution as infinite power series of the oscillatory input; this procedure is a direct one that speeds up the computations compared to traditional series solution methods. The method reported in the first contribution of this series allows for the computation of the analytical solution only for small and medium amplitudes of the oscillatory input, because these series may diverge when large amplitudes are applied. Therefore, the analytic matrix method reported here, which is a reconfiguration of the method proposed in the first contribution in this series, allows the solving of problems in the regime of large-amplitude oscillations where the contributions of the high order harmonics affect the amplitudes of the low order harmonics, leading to amplitude- and frequency-dependent coefficients for the infinite series of trigonometric function expansion.

Evolving shapes of nuclei via evolutionary algorithm: Dynamics of the shape of 25Al nucleus in single-particle states

We present a new approach that enables us to extract shapes of the self-consistent field of a nucleus in the ground and single-particle excited states directly from the experimental data on the energies, spins, and parities of these states, as well as the measured probabilities of electromagnetic transitions between them. The approach is based on the use of an evolutionary algorithm and a series of trigonometric functions that simulates the shape of a nucleus. With help of this approach we show that the 25Al nucleus is differently deformed in its single-particle states and the contribution of the hexadecapole deformation is not small compared to the contribution of the quadrupole deformation. Our approach indicates that the abnormally weakly deformed ground state of the 25Al nucleus is a shape phase transition point from the spherical state of the nucleus into a deformed state.

Bending analysis of two-directional functionally graded beams using trigonometric series functions

Bending analysis of two-directional functionally graded beams using trigonometric series functions

Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations

This is the first on a series of articles that deal with nonlinear dynamical systems under oscillatory input that may exhibit harmonic and non-harmonic frequencies and possibly complex behavior in the form of chaos. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods because, most of the time, analytical solutions turn out to be difficult, if not impossible, since they are based on infinite series of trigonometric functions. The analytic matrix method reported here is a direct one that speeds up the solution processing compared to traditional series solution methods. In this method, we work with the invariant submanifold of the problem, and we propose a series solution that is equivalent to the harmonic balance series solution. However, the recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method. This method can be applied to nonlinear dynamic systems under oscillatory input to find the analog of a usual Bode plot where regions of small and medium amplitude oscillatory input are well described. We found that the identification of such regions requires both the amplitude as well as the frequency to be properly specified. In the second paper of the series, the method to solve problems in the field of large amplitudes will be addressed.

Haar wavelet method for frequency analysis of the combined functionally graded shells with elastic boundary condition

In this paper, the Haar wavelet discretization method (HWDM), one of the effective, convenient and accurate numerical solution methods is presented for analyzing the free vibration of combined functionally graded shells (CFGSs). The combined shells consist of elliptic, hyperbolic, parabolic, and cylindrical shells, and each shells are composed of the same functionally graded material (FGM) determined by four parameters. The theoretical model of the CFGSs is formulated by the application of the first-order shear deformation theory (FSDT). The displacement components at any point of the CFGS are expressed as Haar wavelet series in the meridian direction and as a trigonometric function series in the circumferential direction. The boundary and the continuous condition are modeled by the spring stiffness technique, by adding boundary condition equations to the equation of the main system, the constant expressed in the integral form of the Haar wavelet series is satisfied. Then, by solving the characteristic equation of the whole system, the vibration characteristics of the CFGS such as the natural frequency and the corresponding mode shape can be obtained. Through comparison with previous literature and finite element method (FEM) results, it is shown that current method has high accuracy, reliability, and good convergence for free vibration analysis of CFGS. Finally, new free vibration results of the various CFGS, which can be used as benchmark data for researchers in this field, are reported along with parameter studies.

Vibration of a simply supported graphene sheet with uncertain small scale parameter based on nonlocal theory

Small scale parameter of graphene sheet is considered as uncertain one, vibration equation of a simply supported graphene sheet with uncertainty is established based on nonlocal theory. Trigonometric function series solution and interval operator are used to obtain the upper and lower bound of response of the simply supported graphene sheet. the uncertainty level of response for the different dimension is investigated. The numerical result shows that for the same uncertainty level of small scale parameter, the uncertainty level of the response will decrease with increase of the graphene sheet dimension, and a small uncertainty level of the small scale parameter can cause much greater uncertainty level of the response before the small scale effect disappears.

Approximate Solution of Mixed Problem for Telegrapher Equation with Homogeneous Boundary Conditions of First Kind Using Special Functions

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

Ionospheric VTEC and satellite DCB estimated from single-frequency BDS observations with multi-layer mapping function

The ionospheric delays and satellite differential code biases (DCBs) act as the significant error sources in the global navigation satellite system (GNSS) positioning, navigation and timing (PNT) services, and are still challenging to estimate correctly. In this study, the ionospheric vertical total electron content (VTEC) and satellite DCBs are estimated by a refining single-frequency precise point positioning (SFPPP) method based on the multi-layer ionosphere mapping function (MF), as well as the dual-frequency methods, including the carrier-to-code leveling (CCL) and dual-frequency PPP (DFPPP). The solutions isolate the ionospheric VTEC values from the slant ionospheric delay with the generalized trigonometric series function (GTSF) and precisely estimate the satellite DCB with a zero-mean condition. The SFPPP-derived VTEC estimates are validated and evaluated by comparing with the International GNSS Service (IGS) products and using ionosphere-corrected (IC) SFPPP in both static and kinematic scenarios. Using the 74 experimental stations collected from the multi-GNSS experiment (MGEX) network from January to March 2020, the results show that the VTEC estimation precision by applying the multi-layer MF is improved for the SFPPP approach. The positioning performances of the static and kinematic BDS IC SFPPP with the ionospheric correction derived from the multi-layer MF SFPPP are better when compared to the single-layer MF. The estimated BDS DCB with the SFPPP is stable and of high accuracy. The SFPPP approach with multi-layer MF is demonstrated as a promising and reliable method to retrieve the VTEC and satellite DCB with the low-cost property for the GNSS users.

An accurate beam theory and its first-order approximation in free vibration analysis

An infinite system of one-dimensional differential equations is derived from the two-dimensional theory of elasticity by expanding the displacement field in a series of trigonometrical functions together with a linear term. Since the trigonometrical functions are pure thickness-vibration modes of infinite plates or beams with the top and bottom surfaces being free, the differential equations and the corresponding boundary conditions serve as the basis of an accurate beam theory for vibration analysis, named Lee's beam theory (LBT). Naturally, a high-order set of the infinite system should be quite useful in the analysis of beams at high frequencies. With the objective of vibration analysis of beams, this paper focuses on the first-order approximation, which leads to a first-order shear deformation beam theory for flexural vibrations (LBT1st). The differential equations in LBT1st are equivalent to those in Timoshenko's beam theory (TBT). The most important difference between LBT1st and TBT is the different field displacements. For the assessment of the accuracy of LBT1st, the numerical results of frequencies of free vibrations, frequency spectra and mode shapes of beams with classical boundary conditions are obtained and compared with those by TBT and plane stress problem of elasticity. Considering the plane stress problem as a reference, LBT1st is slightly more accurate in describing the field shapes of beams than TBT. Therefore, LBT1st, as well as LBT, is an addition to the existing beam theories with improved accuracy for the vibration analysis of beams and their combinations.

Use of trigonometric series functions in free vibration analysis of laminated composite beams

In this study, free vibration analysis of layered composite beams is performed by using an analytical method based on trigonometric series. Based on the first-order shear deformation beam theory, the governing equations are derived from the Lagrange’s equations. Appropriate trigonometric series functions are selected to satisfy the end conditions of the beam. Navier-type solution is used to obtain natural frequencies. Natural frequencies are calculated for different end conditions and lamina stacking. It was seen that the slenderness, E11/E22 and fiber angle have a significant effect on natural frequency. The results of the study are quite compatible with the literature.

Bifurcation, chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.

Spatial and Temporal Variations of Polar Ionospheric Total Electron Content over Nearly Thirteen Years.

It is of great significance for the global navigation satellite system (GNSS) service to detect the polar ionospheric total electron content (TEC) and its variations, particularly under disturbed ionosphere conditions, including different phases of solar activity, the polar day and night alternation, the Weddell Sea anomaly (WSA) as well as geomagnetic storms. In this paper, four different models are utilized to map the ionospheric TEC over the Arctic and Antarctic for about one solar cycle: the polynomial (POLY) model, the generalized trigonometric series function (GTSF) model, the spherical harmonic (SH) model, and the spherical cap harmonic (SCH) model. Compared to other models, the SCH model has the best performance with ±0.8 TECU of residual mean value and 1.5–3.5 TECU of root mean square error. The spatiotemporal distributions and variations of the polar ionospheric TEC are investigated and compared under different ionosphere conditions in the Arctic and Antarctic. The results show that the solar activity significantly affects the TEC variations. During polar days, the ionospheric TEC is more active than it is during polar nights. In polar days over the Antarctic, the maximum value of TEC always appears at night in the Antarctic Peninsula and Weddell Sea area affected by the WSA. In the same year, the ionospheric TEC of the Antarctic has a larger amplitude of annual variation than that of the TEC in the Arctic. In addition, the evolution of the ionization patch during a geomagnetic storm over the Antarctic can be clearly tracked employing the SCH model, which appears to be adequate for mapping the polar TEC, and provides a sound basis for further automatic identification of ionization patches.

Correcting Wavefront Distortion of Dual-Wavelength Beams Due to Atmospheric Turbulence with a Correction Coefficient

In order to improve the correction effect of the adaptive optical system in coherent optical communication, we investigate the relative distortion between the wavefronts of different wavelengths of the beams transmitted on the near-ground horizontal atmospheric turbulent links emitted by coherent optical communication system. And the situation is analyzed when the wavelength corresponding to the wavefront detected by the wavefront detector and the wavelength corrected by the deformed mirror are different, the influence of the wavelength factor on the adaptive optical system correction. We use a series of trigonometric functions and the Hankel transformation to derive the corrected residual variance and the Strehl ratio between the wavefront distortions of the wavelengths of the dual-wavelength combined beam in atmospheric turbulence. In relation to the parameters of the turbulent environment, the ensemble average of the wavefront difference corresponding to different wavelengths the derived is proposed as the coefficient to correct the dual-wavelength adaptive optical system. The results show that the statistic of the turbulence internal scale has a major influence on the difference between the wavefronts. By adding the correction coefficient, the signal light’s wavefront of the coherent optical communication system can be corrected more effectively by the dual-wavelength adaptive optical closure.

About the mechanisms of fibre reinforced plastic behaviour under cycling loading

In order to study the peculiarities of the behavior of fiber reinforced plastics under cyclic loading, tests of specimens made of ELUR-P carbon tape and cold curing binder XT-118 with a stacking sequence 2 [+45°/−45°]2s (s - the number of laminas) for cyclic tension were carried out. It is established that stabilization of the secant modulus of elasticity is observed as the number of cycles increases. The experimental data were compared with the results of numerical modeling of the deformation processes under study. Based on numerical calculations and analytic solutions found, for a number of types of loads and creep kernels, the value of the secant modulus of elasticity does not depend on the load amplitudes, when using relations with the Abel creep kernel, the ratio of viscous parts of strain shifted by values, does not depend on the period of cyclic load, the amplitude of the load and the parameter that determines the degree of viscosity of the material, but depends only on the parameter that determines the degree of attenuation of the creeping process. By using the method of decomposing the load in a series of trigonometric functions with the same cyclic periods, it is shown that the effect is manifested in the general case of cyclic loading using relations with the Abel creep kernel. There is established the independence of secant and tangential moduli of elasticity from the absolute values of maximum and minimum stresses for any kinds of loads and creep kernels using the linear theory of hereditarily elastic body. A method has been developed for identifying creep kernel parameters, relations for viscoplastic and elastic strain based on the analysis of experimental data on cyclic loading. For the separation of hereditarily elastic and viscoplastic strain, the hypothesis was introduced that their rates at large times are very different. This made it possible, on the basis of theoretical results obtained for the Abel kernel, to obtain ratios from which, independently of other mechanical characteristics, one can find the attenuation parameter from the strain values, measured in the experiment in times shifted by an integer number of periods. The method is applied to processing the data of real experiments conducted by the authors.

Application of Polylogarithms to the Approximate Solution of the Inhomogeneous Telegraph Equation for the Distortionless Line

The paper deals with a mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortion. This problem is reduced to the analogous one for the one-dimensional inhomogeneous wave equation. Its solution can be found as the sum of the solution for a mixed homogeneous boundary value problem for the corresponding homogeneous wave equation and for the solution of a non-homogeneous wave equation with homogeneous boundary data and zero initial conditions. Solutions to both problems can be found by separating the variables in the form of a series of trigonometric functions of the line point with time-dependent coefficients. Such solutions are inconvenient for real application because they require calculation of a large number of integrals, and it is difficult to estimate the miscalculation. An alternative method for solving this problem is proposed, based on the use of special functions, viz. polylogarithms, which are complex power-series with power coefficients converging in a unit circle. The exact solution of the problem is expressed in the integral form via the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one is expressed in the form of a finite sum via the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All these parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding powers on the segment of the length equal to the period. This makes it possible to effectively find an approximate solution to the problem. Also, a simple and convenient error estimate of the approximate solution of the problem is found. It is linear with respect to the step of splitting the line and the step of splitting the time range in which the problem is considered. The score is uniform along the length of the line at each fixed point of time. A concrete example of solving the problem according to the proposed mode is presented; graphs of exact and approximate solutions are constructed.