Series Of Trigonometric Functions Research Articles

An accurate two-dimensional theory of vibrations of isotropic, elastic plates

An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin's first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin's without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented.

New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity via modified trigonometric function series method

In this paper, the modified trigonometric function series method is employed to solve the perturbed nonlinear Schrödinger’s equation (NLSE) with Kerr law nonlinearity. Exact traveling wave solutions are obtained.

A solution of torsional problem by energy method in case of anisotropic cross-section

We proposed an original method to investigate the problem of torsion of anisotropic cross-section. We implemented an energy method to calculate the stress function represented by infinite series of trigonometric functions adapted to rectangular cross-section. After validation, we implemented a parametric sensitivity study to investigate the influence of the cross-section aspect ratio and the anisotropy level on the stress function, the strain energy density and the torsion stiffness. The process showed a fast convergence with a very good accuracy. The model showed a potential interest for the experimental identification of anisotropic material properties.

Reduction of the Sharma–Tasso–Olver equation and series solutions

This paper considers series solutions of the Sharma–Tasso–Olver (STO) equation. By using the extended homogenous balance method, we reduce the STO equation to a linear PDE and obtain Bäcklund transformation of it. Furthermore, the self-transformation of solutions for the STO equation is obtained. By the Bäcklund transformation and various series solutions of the PDE, abundant exact solutions of the STO equation are obtained including the multi-solitary wave solution, trigonometric function series solution, rational series solution and solution consisting of the three types of solutions.

Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations

Abstract Exact solutions for functionally graded thick plates are presented based on the three-dimensional theory of elasticity. The plate is assumed isotropic at any point, while material properties to vary exponentially through the thickness. The system of governing partial differential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into infinite dual series of trigonometric functions. Interactions between the Winkler–Pasternak elastic foundation and the plate are treated as boundary conditions. The problem is finally solved using the state space method. Effects of stiffness of the foundation, loading cases, and gradient index on mechanical responses of the plates are discussed. It is established that elastic foundations affects significantly the mechanical behavior of functionally graded thick plates. Numerical results presented in the paper can serve as benchmarks for future analyses of functionally graded thick plates on elastic foundations.

Vibration of elastically restrained cross-ply laminated plates with variable thickness

The natural frequencies of symmetrically laminated plates of variable thickness are analyzed using the finite strip transition matrix technique. In this paper, the natural frequencies of such plates are determined for edges with being elastically restrained against both rotation and transition or both. A successive conjunction of the classical finite strip method and the transition matrix method is applied to develop a new modification of the finite strip method to reduce the complexity of the problem. The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction. Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration. The mode shapes and the frequency parameters for different combinations of elastic or translational restraint coefficients have been presented and compared with those available from other methods in the literature. Also, the effect of the tapered ratio and the aspect ratio on the natural frequencies and the mode shapes of the plates are presented. The good agreement with other methods demonstrates the validity and the reliability of the proposed method.

A generalized trigonometric series function model for determining ionospheric delay*

Abstract A generalized trigonometric series function (GTSF) model, with an adjustable number of parameters, is proposed and analyzed to study ionosphere by using GPS, especially to provide ionospheric delay correction for single frequency GPS users. The preliminary results show that, in comparison with the trigonometric series function (TSF) model and the polynomial (POLY) model, the GTSF model can more precisely describe the ionospheric variation and more efficiently provide the ionospheric correction when GPS data are used to investigate or extract the earth's ionospheric total electron content. It is also shown that the GTSF model can further improve the precision and accuracy of modeling local ionospheric delays.

A semi-analytical model for global buckling and postbuckling analysis of stiffened panels

A computational model for global buckling and postbuckling analysis of stiffened panels is derived. The loads considered are biaxial in-plane compression or tension, shear, and lateral pressure. Deflections are assumed in the form of trigonometric function series, and the principle of stationary potential energy is used for deriving the equilibrium equations. Lateral pressure is accounted for by taking the deflection as a combination of a clamped and a simply supported deflection mode. The global buckling model is based on Marguerre’s nonlinear plate theory, by deriving a set of anisotropic stiffness coefficients to account for the plate stiffening. Local buckling is treated in a separate local model developed previously. The anisotropic stiffness coefficients used in the global model are derived from the local analysis. Together, the two models provide a tool for buckling assessment of stiffened panels. Implemented in the computer code PULS, developed at Det Norske Veritas, local and global stresses are combined in an incremental procedure. Ultimate limit state estimates for design are obtained by calculating the stresses at certain critical points, and using the onset of yielding due to membrane stress as the limiting criterion.

On the Order of the Mertens Function

We describe a numerical experiment concerning the order of magnitude of q(x) := M (x)/√X, where M(x) is the Mertens function (the summatory function of the Mobius function). It is known that, if the Riemann hypothesis is true and all nontrivial zeros of the Riemann zeta-function are simple, q(x) can be approximated by a series of trigonometric functions of log x. We try to obtain an ω-estimate of the order of q(x) by searching for increasingly large extrema of the sum of the first 102, 104, and 106 terms of this series. Based on the extrema found in the range 104 ≤ x ≤ 101010 we conjecture that q(x) = ω±,().

A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling

A computational model for analysis of local buckling and postbuckling of stiffened panels is derived. The model provides a tool that is more accurate than existing design codes, and more efficient than nonlinear finite element methods. Any combination of biaxial in-plane compression or tension, shear, and lateral pressure may be analysed. Deflections are assumed in the form of trigonometric function series. The deformations are coupled such that continuity of rotation between the plate and the stiffener web is ensured, as well as longitudinal continuity of displacement. The response history is traced using energy principles and perturbation theory. The procedure is semi-analytical in the sense that all energy formulations are derived analytically, while a numerical method is used for solving the resulting set of equations, and for incrementation of the solution. The stress in certain critical points are checked using the von Mises yield criterion, and the onset of yielding is taken as an estimate of ultimate strength for design purposes.

QUASI-PERIODIC VIBRATION OF CRACKED ROTOR ON FLEXIBLE BEARINGS

A cracked rotor on flexible bearings is studied in this paper. The vibration of such a system has many complexities because of the crack and bearing flexibility. However, if the properties of the bearings are known, the system can be simplified by supposing that, the vibration due to weight is dominant. Equations of motion are derived, and a linear system in which the crack has been considered as an external disturbance described by a series of trigonometric functions is obtained. Consequently, the quasi-periodic vibrations of the rotor and bearings are established by harmonic balance method and approximate values of the vibration determined by truncating the higher order terms. It is believed that the simulated results will be useful for crack detection in the case of weight-dominant rotors.

Free Vibration of Orthotropic Skew Plates

The main purpose of this paper is to develop a fast converging semianalytical method for assessing the vibration effect on thin orthotropic skew (or parallelogram/oblique) plates. Since the geometry of the skew plate is not helpful in the mathematical treatments, the analysis is often performed by more complicated and laborious methods. A successive conjunction of the Kantorovich method and the transition matrix is exploited herein to develop a new modification of the finite strip method to reduce the complexity of the problem. The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction. Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration. The influence of the skew angle, the aspect ratio, the properties of orthotropy, and the prescribed boundary conditions are investigated. Convergence of the solution is investigated and the accuracy of the results is compared with that available from other numerical methods. The numerical results show that the convergence is rapidly deduced and the comparisons agree very well with known results. [S0739-3717(00)00202-6]

A new two-dimensional theory for vibrations of piezoelectric crystal plates with electroded faces

A system of two-dimensional (2-D) governing equations for piezoelectric plates with general crystal symmetry and with electroded faces is deduced from the three-dimensional (3-D) equations of linear piezoelectricity by expansion in series of trigonometric functions of thickness coordinate. The essential difference of the present derivation from the earlier studies by trigonometrical series expansion is that the antisymmetric in-plane displacements induced by gradients of the bending deflection (the zero-order component of transverse displacement) are expressed by the linear functions of the thickness coordinate, and the rest of displacements are expanded in cosine series of the thickness coordinate. For the electric potential, a sine-series expansion is used for it is well suited for satisfying the electrical conditions at the faces covered with conductive electrodes. A system of approximate first-order equations is extracted from the infinite system of 2-D equations. Dispersion curves for thickness shear, flexure, and face-shear modes varying along x1 and those for thickness twist and face shear varying along x3 for AT-cut quartz plates are calculated from the present 2-D equations as well as from the 3-D equations, and comparison shows that the agreement is very close without introducing any corrections. Predicted frequency spectra by the present equations are shown to agree closely with the experimental data by Koga and Fukuyo [J. Inst. Elec. Comm. Engrs. of Japan 36, 59 (1953)] and those by Nakazawa, Horiuchi, and Ito [Proceedings of 1990 IEEE Ultrasonics Symposium (IEEE, New York, 1990)].

Thickness-shear and flexural vibrations of contoured crystal strip resonators

A system of two-dimensional equations of motion of successively higher-order approximations for contoured crystal plates are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions of the thickness coordinate of the plates. By removing the first-order thickness-stretch mode and the first-order x2x3 (or fast) thickness-shear mode and all the higher ones, a set of first-order equations of motion is obtained for contoured crystal plates and for frequencies up to and including those of the fundamental x1x2 (or slow) thickness-shear mode. The coupled thickness-shear and flexural vibrations are studied for contoured quartz strip resonators having the shape of (a) a double wedge and (b) a plate with beveled edges. Exact solutions in terms of infinite power series are obtained by Frobenius method for the plates with linearly varying thickness in the x1 direction. Frequency spectra and mode shapes are computed for both types of resonators. The effects of the contour on the frequencies and modes are examined.

Frequency analysis of rectangular isotropic plates carrying a concentrated mass

A free vibration analysis of rectangular isotropic plates carrying a concentrated mass is analysed and presented in this contribution. The deflection of the plate is postulated by a multi-term trigonometric series function. The Ritz approach is applied to rectangular plates with various edge support combinations of clamp and simple support conditions. A thorough comparison with well known published results is presented for the case of free vibration of unloaded plates and good agreement is observed. An extensive comparison of the predicted results with experimental data is presented for a rectangular aluminium plate with S-C-S-C edge supports and carrying a concentrated mass. The effect of different locations of the concentrated mass on the fundamental frequency of the plate is presented in detail.

Vibrations of AT-cut quartz strips of narrow width and finite length

A system of one-dimensional equations of motion for AT-cut quartz strip resonators with narrow width and for frequencies up to and including the fundamental thickness shear is deduced from the two-dimensional, first-order equations for piezoelectric crystal plates by Lee, Syngellakis, and Hou [J. Appl. Phys. 61, 1249 (1987)] by expanding the mechanical displacements and electric potentials in a series of trigonometric functions of the width coordinate. By neglecting the piezoelectric coupling and the weak mechanical coupling through c56, four groups of coupled equations of motion are obtained. For the equations of each group, closed form solutions are obtained and the traction-free conditions at four edges are accommodated. Dispersion curves and frequency spectrum are computed for quartz strips. The predicted frequency as a function of the length-to-thickness ratio and as a function of the width-to-thickness ratio of the quartz strips is compared with experimental data with good agreement.

Buckling of generally laminated composite plates with various edge support conditions

A total potential energy approach was employed in conjunction with the Rayleigh-Ritz method to study the stability behaviour of generally laminated composite plates with various edge support conditions. Classical edge support conditions such as simple support, clamp or a combination of both are imposed on the plate system and their effect on the plate's buckling behaviour are analysed thoroughly. Multi-term trigonometric series functions are used in the postulation of the deflection function to cater for the bending-twisting coupling responses which are dominant in symmetrically laminated composite plates. Detailed comparison of the 144-term solution obtained using the energy approach with the exact solution for some classes of laminated composite plates subjected to uniaxial compression and biaxial compression is presented to verify the theoretical prediction of the present analysis. The effect of the number of terms used in the postulation of the deflection function on the buckling behaviour is also discussed. Lastly, extensive comparison with previously published experimental data for the case of C-C-C-C and C-S-C-S laminated composite plates is carried out and discussed.

Two-dimensional equations for guided electromagnetic waves in dielectric plates surrounded by free space

Two-dimensional governing equations of successively higher-order approximations for guided electromagnetic (EM) waves in an isotropic dielectric plate surrounded by free space are deduced from the three-dimensional Maxwell’s equations by expanding the EM vector potential in a series of trigonometric functions of a thickness coordinate in the plate and in exponentially decaying functions of a thickness coordinate in the upper and lower halves of free space. By further satisfying the continuity conditions of the EM field at the interfaces between the plate and free space, a single system of two-dimensional governing equations is obtained. Solutions and dispersion relations are obtained from the two-dimensional approximate equations. Dispersion curves are computed and compared with the corresponding ones obtained from the solutions of the three-dimensional Maxwell’s equations for the transverse electric (TE) and transverse magnetic (TM) waves of the first four modes and for values of the refractive index n̂=1.5, 5, 15. It is shown that the agreement between the approximate and exact dispersion curves is very close for various order of TE and TM waves and for a broad range of the values of n̂. For bounded plates with edges in contact with free space, a uniqueness theorem for the solutions of the system of two-dimensional equations is derived from which the specification of continuity conditions on the components of the two-dimensional H and E fields at the edges are established.

Strain energy release rate for an interface crack in linearized couple-stress theory

In this paper, the effect of couple-stresses on the strain energy release rate for an interface crack is examined. An internal pressure is applied on surfaces of the crack situated between two dissimilar half-planes. The oscillatory stress singularities and material overlappings usually appear near the crack tips in the so-called exact solutions. To avoid these irregularities the crack surface displacements and the crack surface rotation are expanded in series of trigonometrical functions. A modified version of the Schmidt method is then used to determine those coefficients in the series. Numerical calculations are carried out to reveal the effect of the couple-stresses on the strain energy release rate for the interface crack.

Transient dynamic stresses around a rectangular crack under an impact shear load

In this paper, the three-dimensional impact response of a rectangular crack in an infinite elastic body subjected to an impact shear load is considered. The mixed boundary value equations for the crack are reduced to a set of dual integral equations in the Laplace transform domain with the use of the Fourier transform technique. To solve the equations, the Laplace transformed crack surface displacement is expanded in a double series of trigonometrical functions. The unknown coefficients accompanied in that series are solved with the aid of the Schmidt method. Stress intensity factors are calculated numerically for some shapes of the rectangular crack.