Orthogonal Polynomial Series Research Articles

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

Dynamic response of H-FGS plates integrated with Pasternak foundation subjected to harmonic loading in the thermal environment using Chebyshev-FEM

The main goal of this article is to introduce a finite element method (FEM) based on Chebyshev polynomials, referred to as Chebyshev-FEM, to investigate the dynamic response of functionally graded sandwich (FGS) plates with a honeycomb core and two functionally graded (FG) skin layers, denoted as H-FGS plates integrated with Pasternak foundation (PF) subjected to harmonic loading (HL) within a thermal environment. Chebyshev polynomials are a series of orthogonal polynomials defined recursively. The value of the polynomials belongs to the range [–1] and vanishes at Gauss points. The equation of motion for H-FGS plates is derived using Hamilton’s principle. Comparative examples are implemented to validate the accuracy and performance of the proposed method. Additionally, the effect of input parameters on both free and forced vibrations of H-FGS plates integrated with PF subjected to HL within a thermal environment is studied in detail.

Transverse vibration of plate with multiple curved through cracks

A modeling method for transverse vibration of cracked plate containing multiple internal curved through cracks is presented and experimental studies are performed to validate its effectiveness. The cracked plate is assumed to be divided into several rectangular and curved trapezoidal domains in the global coordinates system. For each sub-domain, the transverse displacement is expanded as Chebyshev orthogonal polynomials series in local coordinates system. As for the shared edges between neighboring two sub-domains, the local collocation method is adopted and a number of Chebyshev zero points are selected to ensure the displacement continuity condition. To verify the convergence and correctness of the presented model and solutions, the natural frequencies, mode shapes and dynamic responses for several cases are compared with reference results and experimental results, with good agreement observed. In addition, the influence of several significant parameters of cracks are investigated, including the number, location and geometrical shape of the cracks. It is shown that the crack inside the plate presents less influence than that at the boundary on the mode change of the plate. The investigations in the study implies that the present model method is effective and convenient for plate with multiple curved through cracks.

Рациональные коэффициенты ортогональных разложений некоторых функций

Expansions of many elementary and special functions in series of orthogonal polynomials have coefficients known explicitly. However, almost always these coefficients are irrational. Thus any numerical method would give these coefficients approximately with computations in any arithmetic. This is also true for the spectral methods, which give effective approximations for holonomic functions. However, in some exceptional cases the coefficients of expansions obtained with spectral methods turn out to be rational, and are computed exactly in rational arithmetic. We consider these expansions in series of some classical orthogonal polynomials. We demonstrate that, in this way, an infinite number of linear forms can be obtained for some irrationalities, in particular, for the Euler constant.

Vibration control of interconnected composite beams: Dynamical analysis and experimental validations

This paper presents a comprehensive theoretical investigation and experimental validation of vibration and nonlinear vibration control in interconnected composite beams with fixed ends operating within a thermal environment. The study employs the Rayleigh-Ritz method, utilizing a series of orthogonal polynomials as interpolation basis functions to derive the vibration mode functions of the composite structure under thermal conditions. Furthermore, the motion control equation for the system is derived based on Lagrange's equation. These equations of motion are then simplified using the Galerkin method, leading to ordinary differential equations. The response of the system is subsequently computed using the harmonic balance method and validated through the Runge-Kutta method. Finally, experimental results confirm that the NiTiNOL-steel wire rope effectively controls the vibration of interconnected composite beams in a thermal environment and is consistent with the theoretical results.

Aerothermoelastic Analysis of Conical Shell in Supersonic Flow

The aerothermoelastic behavior of a conical shell in supersonic flow is studied in the paper. According to Love’s first approximation shell theory, the kinetic energy and strain energy of the conical shell are expressed and the aerodynamic model is established by using the linear piston theory with a curvature correction term. By taking the characteristic orthogonal polynomial series as the admissible functions, the mode function of conical shell under different boundary conditions can be obtained using the Rayleigh–Ritz method. Then, the dynamic model of the conical shell is derived by using the Lagrange equation. Based on the model, variations in the natural frequencies with respect to temperature and free-stream static pressure are analyzed. Additionally, the effects of the length-to-radius ratio, the thickness-to-radius ratio, and semi-vertex angle, as well as the thermal and aerodynamic loads on the aerothermoelastic stability of the structure are investigated in detail.

Some Properties of Wigner Polynomials

Such well-known scientists as Legendre, Gegenbauer, Jacobi, Lager and others were engaged in the study of various properties of orthogonal polynomials. They introduced the concept of various polynomials and determined their properties. With these polynomials the series in orthogonal polynomials are composed and the issue of representing a function with these series is studied; the convergence and summability of these series are also studied by various methods. In the presented paper, the so-called Wigner polynomials are considered. These polynomials are involved in the definition of generalized spherical functions. Therefore, determining the properties of these polynomials would be useful for studying the convergence and summability of Fourier series with respect to generalized spherical functions. In this paper, some basic properties of Wigner polynomials are studied. In particular, asymptotic formulas and some estimates for these polynomials are determined.

Vibration isolation performance of a rectangular panel with high-static-low-dynamic stiffness supports

In this paper, the nonlinear vibration of a corner-supported rectangular thin panel with high static low dynamic stiffness (HSLDS) isolator supports is investigated. The dynamic model of the panel clamped at corners is established and the frequency equations are derived by employing the Rayleigh-Ritz procedure. The characteristic orthogonal polynomial series is introduced as the assumed modes and the equations of motions are obtained. The dynamic response of the system is analytically calculated for the first time by applying the complexification-averaging method. And the approximation solutions are verified with the numerical and finite element method. The isolation performance of the system is evaluated by the displacement transmissibility and the effects of the parameters are investigated for optimal design. The comparison with the isolation performance of linear elastic-supported panel illustrates that introducing the HSLD isolators has a superior effect on reducing the peak transmissibility in the low-frequency band and provides a wide isolation range for the system.

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

Application of Nonlinear Differential Equation in Electric Automation Control System

Abstract This article uses fifth-order nonlinear differential equations to describe the dynamic process of electrical automation control systems. This method first derives the equivalent system of the nonlinear fuzzy global system and then uses the orthogonal polynomial series expansion technique and its integral operation matrix. The local manifold at the dominant unstable equilibrium point of a single-machine infinite-bus system after a failure described by a two-dimensional quadratic nonlinear differential equation is calculated, and the stability boundary of the power system is obtained. The research results show that the output frequency fluctuation of the electrical automation control system is small after the algorithm is adopted, and the intelligent control system can accurately diagnose and warn the electrical faults. The system can meet the requirements of online voltage coordinated control.

Solution of generalized Abel’s integral equation using orthogonal polynomials

This research presents the solution of the generalized version of Abel’s integral equation, which was computed considering the first and second kinds. First, Abel’s integral equation and its generalization were described using fractional calculus, and the properties of Orthogonal polynomials were also described. We then developed a technique of solution for the generalized Abel’s integral equation using infinite series of orthogonal polynomials and utilized the numerical method to approximate the generalized Abel’s integral equation of the first and second kind, respectively. The Riemann-Liouville fractional operator was used in these examples. Our technique was implemented in MAPLE 17 through some illustrative examples. Absolute errors were estimated. In addition, the occurred errors between using orthogonal polynomials for solving Abel’s integral equations of order \(0\ <\ \alpha \ <\ 1\) and the exact solutions show that the orthogonal polynomials used were highly effective, reliable and can be used independently in situations where the exact solution is unknown which the numerical experiments confirmed.

Development and Research of Method in the Calculation of Transients in Electrical Circuits Based on Polynomials

Long electromagnetic transients occur in electrical systems because of switching and impulse actions As a result, the simulation time of such processes can be long, which is undesirable. Simulation time is significantly increased if the circuit in the study is complex, and also if this circuit is described by a rigid system of state equations. Modern requests of design engineers require an increase in the speed of calculations for realizing a real-time simulation. This work is devoted to the development of a unified spectral method for calculating electromagnetic transients in electrical circuits based on the representation of solution functions by series in algebraic and orthogonal polynomials. The purpose of the work is to offer electrical engineers a method that can significantly reduce the time for modeling transients in electrical circuits. Research methods. Approximation of functions by orthogonal polynomials, numerical methods for integrating differential equations, matrix methods, programming and theory of electrical circuits. Obtained results. Methods for calculating transients in electrical circuits based on the approximation of solution functions by series in algebraic polynomials as well as in the Chebyshev, Hermite and Legendre polynomials, have been developed and investigated. The proposed method made it possible to convert integro-differential equations of state into linear algebraic equations for images of time-dependent functions. The developed circuit model simplifies the calculation method. The images of true current functions are interpreted as direct currents in the proposed equivalent circuit. A computer program for simulating the transient process in an electrical circuit was developed on the basis of the described methods. The performed comparison of methods made it possible to choose the best method and a way to use it. The advantages of the presented method over other known methods are to reduce the simulation time of electromagnetic transients (for the considered examples by more than 6 times) while ensuring the required accuracy. The calculation of the process in the circuit over a long time interval showed a decrease and stabilization of errors, which indicates the prospects for using research methods for calculating complex electrical circuits.

Analysis of vibration characteristics of stators of electrical machines under actual boundary

This paper presents a general method for the studies of the vibration characteristics of an electrical machine stator with encased construction under practical boundary conditions. In this paper, artificial springs are introduced at both ends of the stator, and arbitrary boundary conditions can be simulated by adjusting the spring stiffness, including the support conditions in practical applications of the stator. In order to cooperate with the solution of the vibration characteristics of the stator with arbitrary edges, the Gram-Schmidt method is employed to construct characteristic orthogonal polynomial series to simulate the mode shape functions under arbitrary boundary conditions. The analysis is based on the Novozhilov shell theory, which is more applicable to the construction and actual operating environment of stators of electric vehicle drive motors, and takes into account moment of inertia neglected in the past. Further, the effects of teeth, windings, frame and cooling ribs are incorporated in the analysis, and the teeth with complicated shapes are considered as longitudinal ribs of unlimited cross-section, with bending and torsional coupling motion and warping deformation considered. The Rayleigh-Ritz method is used to build a unified dynamic analysis model of the stator of an electric machine, which can provide information on the radial, circumferential and axial vibrations of the stator with higher time efficiency, and a series of comparison studies and experiments are performed to demonstrate the general validity and accuracy of the model. On this basis, the possibility of construction optimization of the stator is discussed to improve the vibration and noise performance while reducing the cost.

Vibration Analysis of Rotating Combined Thin-Walled Shells With Multiple Conical Segments

Abstract In this paper, vibration analysis of rotating combined thin-walled shells with multiple conical segments has been carried out. Considering the centrifugal force, Coriolis force, and initial hoop tension due to rotation, the elastic strain energy and kinetic energy of a single rotating conical shell are expressed based on Love’s first approximation theory. The artificial springs are introduced to simulate the connections of adjacent conical shells and the boundaries of the rotating combined thin-walled shells. Taking characteristic orthogonal polynomial series as the admissible functions, the Rayleigh–Ritz method is employed to derive the frequency equations of the combined shell and corresponding vibration characteristics are then obtained. Given that the cylindrical shell and annular plates can be regarded as conical shells with semi-vertex angles of 0 deg and 90 deg, respectively, the solution given is also available for the vibration analysis of rotating combined thin-walled shells comprised of any segments of cylindrical, conical shell, and annular plate. As examples of rotating combined thin-walled shells with two and five segments, vibrations of rotating conical–conical joined shell and cylindrical–conical–cylindrical–conical–cylindrical joined shell are investigated in the paper. Traveling wave frequencies and corresponding mode shapes are shown, and the effects of rotating speed, circumferential wave number, spring stiffness, and semi-vertex angles on the vibration behavior are given in detail.

Uncertainty analysis of MSD crack propagation based on polynomial chaos expansion

Widespread fatigue damage (WFD) is one of the main damages in aging aircraft structures, and the prediction of multi-site damage (MSD) crack propagation plays an important role in the safety assessment of aging aircraft. However, the existing multi-crack propagation stochastic models need to calculate multiple integrals, and the calculation efficiency is low. Based on the Polynomial Chaos Expansion (PCE) method, this paper proposes a new uncertainty analysis method to effectively solve the problem of multi-crack mutual interference. The rivet area of aircraft skin mostly appears randomly in the form of short cracks. Therefore, 30 groups of multi-crack fatigue propagation experiments are carried out to obtain the data of multi-cracks at different stages. On the basis of the existing multi-crack mutual interference model, a new semi-analytical multi-crack propagation model is designed by considering the uncertainty of multi-crack initiation and the interaction factors between cracks. The random variables are substituted into the improved multi-crack propagation model, and the Hermite polynomials are introduced and expanded into orthogonal polynomial series, and the time-varying polynomial coefficients, moments and probability densities are solved by stochastic regression method. The accuracy and effectiveness of the method are verified by comparing with the experimental and Monte Carlo methods.

An adaptive polynomial dimensional decomposition method and its application in reliability analysis

PurposeThe polynomial dimensional decomposition (PDD) method is a popular tool to establish a surrogate model in several scientific areas and engineering disciplines. The selection of appropriate truncated polynomials is the main topic in the PDD. In this paper, an easy-to-implement adaptive PDD method with a better balance between precision and efficiency is proposed.Design/methodology/approachFirst, the original random variables are transformed into corresponding independent reference variables according to the statistical information of variables. Second, the performance function is decomposed as a summation of component functions that can be approximated through a series of orthogonal polynomials. Third, the truncated maximum order of the orthogonal polynomial functions is determined through the nonlinear judgment method. The corresponding expansion coefficients are calculated through the point estimation method. Subsequently, the performance function is reconstructed through appropriate orthogonal polynomials and known expansion coefficients.FindingsSeveral examples are investigated to illustrate the accuracy and efficiency of the proposed method compared with the other methods in reliability analysis.Originality/valueThe number of unknown coefficients is significantly reduced, and the computational burden for reliability analysis is eased accordingly. The coefficient evaluation for the multivariate component function is decoupled with the order judgment of the variable. The proposed method achieves a good trade-off of efficiency and accuracy for reliability analysis.

Bayesian calibration of multi-level model with unobservable distributed response and application to miter gates

Bayesian calibration plays a vital role in improving the validity of computational models’ predictive power. However, the presence of unobservable distributed responses and uncertain model parameters in multi-level models poses challenges to Bayesian calibration, due to the lack of direct observations and the difficulty in identifying the hidden and distributed model discrepancy under uncertainty. This paper proposes a Bayesian calibration framework for multi-level simulation models to calibrate an unobservable distributed model using measurements of an observable model. In the proposed framework, the distributed model discrepancy of an unobservable model with distributed response is first represented as a series of orthogonal polynomials, with the polynomial coefficients modelled by surrogate models with unknown hyper-parameters. A two-phase machine learning method is then developed to construct surrogate models of the polynomial coefficients based on measurements of an observable model. The constructed model discrepancy is finally used to update the uncertain model parameters by following a modularized Bayesian calibration scheme. The developed framework is applied to the joint Bayesian calibration of the uncertain gap length and unobservable and distributed boundary condition model for a miter gate problem. Results of the miter gate application demonstrate the efficacy of the proposed framework.

Dynamic Analysis of a Rotating Double-Tapered FGM Beam with Various Shear Models Using a Constrained Variational Method

A constrained variation modeling method for free vibration analysis of rotating double tapered functionally graded beams with different shear deformation beam theories is proposed in this paper. The material properties of the beam are supposed to continuously vary in the width direction with power-law exponent for different indexes. The mathematical formulation is developed based on the geometrically exact beam theory for each beam segment, the admissible functions denoting motion quantities are then expressed by a series of Chebyshev orthogonal polynomials. The governing equations are eventually derived using the constrained variational method to involve the continuity conditions of adjacent segments. Different shear deformation beam theories have been incorporated in the formulations, and the nonlinear effect of bending–stretching coupling vibration together with the Coriolis effect is taken into account. Comparison of dimensionless natural frequencies is performed with the existing literature to ensure the accuracy and reliability of the proposed method. Comparative discussions are performed on the vibration behaviors of the double tapered rotating functionally graded beam with first-order shear deformation beam theory and other higher-order shear deformation beam theories. The effect of material property graduation, power-law index, rotation speed, hub radius, slenderness ratio, and taper ratios is scrutinized via parametric studies, respectively.

On Series of Orthogonal Polynomials and Systems of Classical Type Polynomials

On Series of Orthogonal Polynomials and Systems of Classical Type Polynomials

A general approach for free vibration analysis of spinning joined conical–cylindrical shells with arbitrary boundary conditions

This paper presents a general approach for free vibration analysis of spinning local elastic-foundation supported joined conical–cylindrical shells (JCCSs) with arbitrary boundary conditions. The Donnell’s shell theory is employed in the modeling of the structures, and all the effects of centrifugal forces, Coriolis forces as well as initial hoop tensions are considered. The approach utilizes the concept of artificial springs which permit convenient joining of the conical and cylindrical shells. Particularly, it can efficiently simulate arbitrary boundary conditions of JCCSs. The spinning JCCSs are locally or completely surrounded by elastic foundation, formulated by the Pasternak model. By using the orthogonal polynomial series as the admissible functions, the frequency equations of spinning JCCSs with arbitrary boundary conditions are derived using the Rayleigh–Ritz method. Then, some numerical examples are compared with the available literature and finite element results to validate the present approach. Finally, the effects of key parameters on free vibration characteristics are investigated including spring stiffness, spinning speed, cone angle and local elastic foundation.