Chebyshev Quadrature Formula Research Articles

Оn the issue of the development of permanent magnet suspension systems for transport systems

Aim: to develop recommendations for the development of optimal design schemes of permanent magnet systems designed to create a suspension force.
 Methods: we described the electromagnetic interaction in the system using classical field theory, the principle of superposition was used, the forces were calculated using the method of mirror images and Chebyshev quadrature formulas.
 Results: an increase of the number of rows of magnets with alternating polarity leads to an increase in the resulting strength of the demagnetizing field and there is the opposite effect in the case without alternating polarities. The suspension force reaches its maximum at a certain distance between the rows of permanent magnets. The side force reaches a maximum at a certain amount of transverse displacement in the horizontal suspension system. The control of the loading capacity of the suspension by changing the size of the cross sections of permanent magnets is accompanied by an increase their mass of the magnetic material. The presence of a ferromagnetic buss significantly improves the characteristics of the suspension and has a beneficial effect on the stability of the suspension.
 Conclusion: it is advisable to use multi-row structures on the track and on the vehicle with a certain step of installing magnetic rows with alternating polarity and the distance between the rows affects the magnitude of electromagnetic forces. The choice of the cross-sectional dimensions of permanent magnets should be carried out taking into account the specific requirements for the suspension system, taking into account the reversibility of the remagnetization process.

Anti-Gaussian quadrature formulae of Chebyshev type

We prove that there is no positive measure d σ d\sigma on the interval [ a , b ] [a,b] such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure d σ ( t ) = d σ ( − t ) d\sigma (t)=d\sigma (-t) on the symmetric interval [ − a , a ] [-a,a] , for which the anti-Gaussian formula has the form ∫ − a a f ( t ) d σ ( t ) = μ 0 2 [ f ( a ) + f ( − a ) ] + R 2 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{2}[f(a)+f(-a)]+R_{2}^{AG}(f) for n = 1 n=1 and ∫ − a a f ( t ) d σ ( t ) = w 1 f ( a ) + w ∑ μ = 2 n f ( t μ ) + w 1 f ( − a ) + R n + 1 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=w_{1}f(a)+w\sum _{\mu =2}^{n}f(t_{\mu })+w_{1}f(-a)+R_{n+1}^{AG}(f) for all n ≥ 2 n\geq 2 , is the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt . It turns out that the formula for n ≥ 2 n\geq 2 is the ( n − 1 ) (n-1) -point Gauss-Lobatto quadrature formula for the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt , which is a generalization of what happens in the case of the Chebyshev measure of the first kind. Moreover, we compute the anti-Gaussian formulae for any one of the four Chebyshev measures.

Indentation load–depth relation for an elastic layer with surface tension

The load–depth relation is a fundamental requisite in nanoindentation tests for thin layers; however, the effects of surface tension are seldom included. This paper concerns micro-/nano-sized indentation by a rigid sphere of a bonded elastic layer. The surface Green’s function with the incorporation of surface tension is first derived by applying the Hankel integral transform, and subsequently used to formulate the governing integral equation for the axisymmetric contact problem. By using a numerical method based on the Gauss–Chebyshev quadrature formula, the singular integral equation is solved efficiently. Several numerical results are presented to investigate the influences of surface tension and layer thickness on contact pressure, surface deformation, and bulk stress. It is found that when the size of contact is comparable to the ratio of surface tension to elastic modulus, the contribution of surface tension to the load–depth relation becomes quite prominent. With the help of a parametric study, explicit general expressions for the indentation load–depth relation as well as the load–contact radius relation are summarized, which provide the groundwork for practical applications.

A dimension-wise method and its improvement for multidisciplinary interval uncertainty analysis

• The Chebyshev polynomial approximation is used in multidisciplinary interval uncertainty analysis. • The dimension-wise method is improved through an iterative procedure to address systematic coupling effects. • The high accuracy of the proposed iterative dimension-wise method is demonstrated by numerical examples. Considering that uncertain factors widely exist in practical engineering, this study develops an improved dimension-wise method for multidisciplinary interval uncertainty analysis, in which the extremums of each interval variable for determining systematical response bounds are solved using the Chebyshev polynomial approximation and iterative criterion. First, Chebyshev basis functions are involved to construct the approximate relation between the system output variables and initial interval parameters dimension by dimension in multidisciplinary frameworks. The Gauss–Chebyshev quadrature formulas are then utilized to confirm coefficients of the fitting function. Due to the weakness of traditional dimension-wise method, that is, it ignores the coupling effects of uncertain variables, a new iterative dimension-wise method (IDWM) where the nominal states of intervals can be updated in each iteration, is proposed. Discussions on efficiency and accuracy are further expounded. Both numerical and engineering examples are eventually given to demonstrate the usage and validity of the developed methodology, and results indicate that the presented IDWM has a superiority in uncertainty propagation problems of multidisciplinary issues.

Analytic relations for two-dimensional indentations with surface tension

Surface tension plays a significant role in micro- and nanoindentation tests. Based on the solution of a concentrated force acting on an elastic half-plane with surface tension, the two-dimensional indentations of an elastic half-plane by a cylindrical indenter, a wedge indenter and a flat-ended indenter are formulated, and by employing the Gauss–Chebyshev quadrature formula, the corresponding singular integral equations are numerically solved. For each indentation, the analytic solutions of two limit cases considering only bulk elasticity or surface tension are derived. Then through simple combinations of the results of two limit cases, the analytic relations between load and contact half-width or indent depth are formulated for the general case, and compared with the numerical results. The analytic relations established in this paper facilitate better characterization of mechanical properties of materials by micro/nanoindentation tests.

Effects of surface tension on the nanoindentation with a conical indenter

In the present paper, the nanoindentation of an elastic half-space by a conical indenter is investigated with the influence of surface tension. Based on the solution of a point force acting on an elastic half-space with surface tension, the singular integral equation of this problem is formulated and is then solved numerically by using the Gauss–Chebyshev quadrature formula. Surface tension flattens the pressure distribution in the contact region. Compared to the result of the classical elasticity model, surface tension evidently decreases the normal displacement on the surface of the half-space. The explicit relations between load and contact radius, and between load and indent depth are derived, which are helpful to characterize the mechanical properties of the materials in nanoindentation.

General Relations of Indentations on Solids With Surface Tension

The conventional contact mechanics does not account for surface tension; however, it is important for micro- or nanosized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half-space by a rigid sphere, cone, and flat-ended cylinder are investigated, and the corresponding singular integral equations are formulated. Due to the complicated structure of the integral kernel, it is difficult to obtain their analytical solutions. By using the Gauss–Chebyshev quadrature formula, the integral equations are solved numerically first. Then, for each indenter, the analytical solutions of two limit cases considering only the bulk elasticity or surface tension are presented. It is interesting to find that, through a simple combination of the solutions of two limit cases and fitting the direct numerical results, the dependence of load on contact radius or indent depth for general case can be given explicitly. The results incorporate the contribution of surface tension in contact mechanics and are helpful to understand contact phenomena at micro- and nanoscale.

Micro/Nanocontact Between a Rigid Ellipsoid and an Elastic Substrate With Surface Tension

For micro/nanosized contact problems, the influence of surface tension becomes prominent. Based on the solution of a point force acting on an elastic half space with surface tension, we formulate the contact between a rigid ellipsoid and an elastic substrate. The corresponding singular integral equation is solved numerically by using the Gauss–Chebyshev quadrature formula. When the size of contact region is comparable with the elastocapillary length, surface tension significantly alters the distribution of contact pressure and decreases the contact area and indent depth, compared to the classical Hertzian prediction. We generalize the explicit expression of the equivalent contact radius, the indent depth, and the eccentricity of contact ellipse with respect to the external load, which provides the fundament for analyzing nanoindentation tests and contact of rough surfaces.

Two-dimensional Hertzian contact problem with surface tension

In the present paper, we consider a two-dimensional contact problem of a rigid cylinder indenting on an elastic half space with surface tension. Based on the solution of a point force acting on a substrate with surface tension, we derive the singular integral equation of this problem. By using the Guass–Chebyshev quadrature formula, the integral equation is solved numerically to illuminate the influence of surface tension on the contact response. It is found that when the contact width is comparable with the ratio of surface tension to elastic modulus, surface tension significantly alters the pressure distribution in the contact region and the contact width. Compared to that of the classical Hertzian contact, the existence of surface tension decreases the displacements on the half plane and yields a continuous slope of normal stress and displacements across the contact fringe. In addition, it predicts the increase of hardness as the radius of indent cylinder decreasing. The obtained results are useful for the measurement of mechanical properties of materials based on the indentation technique.

A generalized Birkhoff–Young–Chebyshev quadrature formula for analytic functions

A generalized N -point Birkhoff–Young quadrature of interpolatory type, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some special cases of this quadrature formula are derived.

Computing rational Gauss–Chebyshev quadrature formulas with complex poles: The algorithm

We provide an algorithm to compute arbitrarily many nodes and weights for rational Gauss–Chebyshev quadrature formulas integrating exactly in spaces of rational functions with complex poles outside [−1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O ( n ) . This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions on [−1, 1] with arbitrary complex poles outside this interval.

Digital Modeling and PID Controller Design for MIMO Analog Systems with Multiple Delays in States, Inputs and Outputs

This paper presents a discrete-time state-space methodology for the digital modeling and design of an optimal digital proportional-integral-derivative (PID) plus state-feedback controller for multiple-input, multiple-output (MIMO) continuous-time systems with multiple time delays in states, inputs and outputs. To implement the digital design, first the Chebyshev quadrature formula together with a linear interpolation method is employed to obtain an extended discrete-time state-space model from the continuous-time multiple time-delay system. Then, a partially predetermined digital PID controller and the extended discrete-time state-space model are formulated as an augmented discrete-time state-space system utilizing state-feedforward and state-feedback linear-quadratic regulator (LQR) design. As a result, the parameters of the optimal PID controller and its associated state-feedback controller can be determined by tuning the weighting matrices in the LQR performance criteria. Further, an optimal discrete-time observer is jointly constructed for the multivariable system with multiple delays in states, inputs and outputs. The proposed design methodology can be applied to general MIMO continuous-time multiple time-delay systems for performance improvement and disturbance rejection. An illustrative example is given to demonstrate the effectiveness of the developed method.

Analysis of mode III crack perpendicular to the interface between two dissimilar strips

The present work is concerned with the problem of mode III crack perpendicular to the interface of a bi-strip composite. One of these strips is made of a functionally graded material and the other of an isotropic material, which contains an edge crack perpendicular to and terminating at the interface. Fourier transforms and asymptotic analysis are employed to reduce the problem to a singular integral equation which is numerically solved using Gauss–Chebyshev quadrature formulae. Furthermore, a parametric study is carried out to investigate the effects of elastic and geometric characteristics of the composite on the values of stress intensity factor.

Analysis of vertical cracking phenomena in tensile-strained epitaxial film on a substrate: Part I. Mathematical formulation

This paper presents an analysis of a single vertical crack and periodically distributed vertical cracks in an epitaxial film on a semi-infinite substrate where the cracks penetrate into the substrate. The film and substrate materials have different anisotropic elastic constants, necessitating Stroh formalism in the analysis. The misfit strain due to the lattice mismatch between the film and the substrate serves as the driving force for crack formation. The solution for a dislocation in an anisotropic trimaterial is used as a Green function, so that the cracks are modeled as the continuous distributions of dislocations to yield the singular integral equations of Cauchy-type. The Gauss–Chebyshev quadrature formula is adopted to solve the singular integral equations numerically. Energy arguments provide the critical condition for crack formation, at which the cracks are energetically favorable configurations, in terms of the ratio of the penetration depth into the substrate to the film thickness, the ratio of the spacing of the periodic cracks to the film thickness, and the generalized Dundurs parameters between the film and substrate materials.

Special moments

In this article, we show that a linear combination X ˜ of n independent, unbiased Bernoulli random variables { X k } can match the first 2 n moments of a random variable Y which is uniform on an interval. More generally, for each p ⩾ 2 , each X k can be uniform on an arithmetic progression of length p. All values of X ˜ lie in the range of Y, and their ordering as real numbers coincides with dictionary order on the vector ( X 1 , … , X n ) . The construction involves the roots of truncated q-exponential series. It applies to a construction in numerical cubature using error-correcting codes [G. Kuperberg, Numerical cubature using error-correcting codes, arXiv: math.NA/0402047]. For example, when n = 2 and p = 2 , the values of X ˜ are the 4-point Chebyshev quadrature formula.

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n − 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi-infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.

Convergence Acceleration of Gauss–Chebyshev Quadrature Formulae

The aim of this paper is to accelerate, via extrapolation methods, the convergence of the sequences generated by the Gauss–Chebyshev quadrature formula applied to functions holomorphic in ]−1,1[ and possessing, in the neighborhood of 1 or −1, an asymptotic expansion with log (1±x)(1±x)α, (1±x)α, α>−1, as elementary elements.

Discretisation of two degree-of-freedom controller and system with state, input and output delays

An interpolation method for discretisation of a two degree-of-freedom analogue controller and system with state, input and output delays is presented. A Newton centre interpolation formula is used for discretisation of a continuous-time system with state, input and output delays. Also, a Newton backward extrapolation formula and Chebyshev quadrature formula are employed to carry out output-feedback digital redesign of a two degree-of-freedom controller for the delayed analogue system. The developed digital redesign method is then extended to perform discretisation of a Smith predictor for an input-delayed analogue system. The newly developed output-feedback state-matching digital redesign method provides an alternative methodology for indirect digital design of a continuous-time system with state, input and output delays.

Discretization of cascaded continuous-time controllers for state and input delayed systems

This paper presents a new interpolation method that utilizes a Newton backward interpolation formula for the discretization of a continuous-time system with state and input delays. Also, a new output feedback state-matching digital redesign method, which utilizes the Newton extrapolation formula and the Chebyshev quadrature formula, enables the discretization of a cascaded continuous-time controller or filter for the continuous-time systems with delayed input and states, without employing a discrete time observer. The proposed method provides an alternative methodology for indirect digital design of a continuous-time unity-output feedback delayed system and enables us to implement accurately and effectively an existing or pre-designed cascaded analogue controller or filter via a digital controller.

Discretization of cascaded continuous-time controllers and uncertain systems

This paper presents a new method for the digital modeling of a continuous-time uncertain system and a new method for the digital redesign of a sampled-data uncertain system. The system matrices characterizing the state-space representation of the original uncertain system are assumed to be interval matrices. The Chebyshev quadrature formula together with the interval arithmetic are used for the digital interval modeling, and a dual concept of digital interval modeling is utilized to discretize a predesigned cascaded analog controller for robust digital control of a continuous-time uncertain system. Using the newly developed digital interval models and digitally redesigned controllers, the resulting dynamic states of the digitally controlled sampled-data uncertain systemsare able to closely match those of the originally analogously controlled continuous-time uncertain systems for a relatively longer sampling period.