three dimensional vibrations of thick circular and annular plates abstract the ritz method is applied in a three dimensional 3 d analysis to obtain accurate frequencies for thick circular and annular plates the method is formulated in a manner which allows one to have any combination of free or fixed plate boundaries admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co ordinate and algebraic polynomials in the radial and axial co ordinates upper bound convergence of the non dimensional frequencies to at least four significant figures is demonstrated comparisons of results are made with ones obtained by others using 2 d mindlin thick plate theory and with other 3 d solutions extensive and accurate four significant figure frequencies are presented for completely free circular plates having thickness to diameter ratios of 0 2 0 3 0 4 and 0 5 for poisson s ratios 0 0 3 and 0 499 frequencies are also given for thick annular plates having a thickness to outer diameter of 0 2 inside to outside diameter ratios of 0 1 0 5 and 0 9 and 0 3 all 3 d modes are included in the analyses e g flexural thickness shear in plane stretching and torsional the circular and annular plate frequency data given is exact to at least four digits thus being benchmark data against which results from 2 d thick plate theories or other approximate methods e g finite elements may be compared

three dimensional vibration analysis of circular and annular plates via the chebyshev ritz method abstract a three dimensional free vibration analysis of circular and annular plates is presented via the chebyshev ritz method the solution procedure is based on the linear small strain three dimensional elasticity theory selecting chebyshev polynomials which can be expressed in terms of cosine functions as the admissible functions a convenient governing eigenvalue equation can be derived through the ritz method according to the geometric properties of circular and annular plates the vibration is divided into three distinct categories axisymmetric vibration torsional vibration and circumferential vibration each vibration category can be further subdivided into the antisymmetric and symmetric ones in the thickness direction convergence and comparison study demonstrated the high accuracy and efficiency of the present method the present approach shows a distinct advantage over some other ritz solutions in that stable numerical operation can be guaranteed even when a large number of admissible functions is employed therefore not only lower order but also higher order eigenfrequencies can be obtained by using sufficient terms of the chebyshev polynomials finally some valuable results for annular plates with one or both edges clamped are given and discussed in detail

three dimensional vibration analysis of cantilevered and completely free isosceles triangular plates the free vibration of cantilevered and completely free isosceles triangular plates based on exact three dimensional elasticity theory is investigated the actual plate domain is first mapped onto a basic cubic domain then the ritz method is applied to derive the eigenfrequency equation from the strain energy and the kinetic energy of the plate a set of chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometry boundary conditions of the plate is developed as the admissible functions of each displacement component the convergence and comparison studies show that rather accurate results can be obtained by using this approach the effects of thickness to width ratio and apex angle on eigenfrequencies of the plates are studied in detail sets of valuable results are presented which may serve as the benchmark values for future numerical techniques in thick plate vibration analysis data for the completely free isosceles triangular plates are presented for the first time

3 d free vibration analysis of annular plates on pasternak elastic foundation via p ritz method in this study a three dimensional 3 d free vibration analysis of thick annular plates resting on elastic foundation with different combinations of free soft simply supported hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate is presented on the basis of the polynomials ritz method the elastic foundation is considered as a pasternak model with adding a shear layer to the winkler model the analysis procedure is based on the linear small strain and 3 d elasticity theory in this analysis method a set of orthogonal polynomial series in cylindrical polar coordinate is used to extract eigenvalue equation yielding the natural frequencies and mode shapes for the annular plates the accuracy of these results is verified by appropriate convergence studies and checked with the available literature finite element method fem analysis and the mindlin theory furthermore the effect of the foundation stiffness parameters thickness radius ratio inner outer radius ratio and different combinations of boundary conditions on the ill conditioning of the mass matrix as well as on the vibration behavior of the annular plates is investigated finally the validity and the range of applicability of the results obtained on the basis of the mindlin and classical plate theories for a thin and moderately thick annular plate with different values of the winkler foundation stiffness are graphically presented through comparing them with those obtained by the present 3 d p ritz solution

exact closed form frequency equations for thick circular plates using a third order shear deformation theory abstract this paper presents for the first time exact closed form frequency equations and transverse displacement for thick circular plates with free soft simply supported hard simply supported and clamped boundary conditions based on reddy s third order shear deformation theory hamiltonian and minimum potential energy principles are used to extract the equations of dynamic equilibrium and natural boundary conditions of the plate the new formulation is verified by comparing the results with their counterparts reported in open literature natural frequencies of circular plates with different boundary conditions are tabulated in dimensionless form for various values of thickness radius ratios the results presented on the basis of exact closed form frequency equations are expected to serve as reliable benchmarks

three dimensional vibration analysis of a torus with circular cross section the free vibration characteristics of a torus with a circular cross section are studied by using the three dimensional small strain elasticity theory a set of three dimensional orthogonal coordinates system comprising the polar coordinate r at each circular cross section and the circumferential coordinate around the ring is developed each of the displacement components ur and w in the r and directions respectively is taken as a product of the chebyshev polynomials in the r direction and the trigonometric functions in the and directions eigenfrequencies and vibration mode shapes have been obtained via a three dimensional displacement based extremum energy principle upper bound convergence of the first seven eigenfrequencies accurate to at least six significant figures is obtained by using only a few terms of the admissible functions the eigenfrequency responses due to variation of the ratio of the radius of the ring centroidal axis to the cross sectional radius are investig

axisymmetric straining modes in the vibration of circular plates axisymmetric straining modes which are different from the axisymmetric flexural modes of vibrating circular plates and cannot be revealed by conventional plate theory based approaches are investigated in the present study mode shapes of an example problem with clamped and four different simply supported boundary conditions are presented to show what straining modes may look like the straining modes are classified into three types and a criterion is set up to predict the appearance of straining modes some observations about straining modes are obtained and discussions are made results for different poisson ratios are also shown

two sided hyperbolic svd abstract in this paper we propose the two sided hyperbolic svd 2hsvd for square matrices i e a u v where u and v are j unitary j diag 1 and is a real diagonal matrix of double hyperbolic singular values we show that with some natural conditions such decomposition exists without the use of hyperexchange matrices in other words u and v are really j unitary with regard to j and not some matrix j which is permutationally similar to matrix j we provide full characterization of 2hsvd and completely relate it to the semidefinite j polar decomposition

impact of incomplete and inaccurate data models on high resolution parameter estimation in multidimensional channel sounding multidimensional channel sounding aims to estimate the geometrical structure of multi path wave propagation in terms of directions of arrival departure doppler shift time delay and complex polarimetric path weights maximum likelihood parameter estimation based upon an underlying data model is used to achieve high resolution of the path parameters and thus renders possible an antenna independent channel characterization however any mismatch of the underlying data model to physical reality imposes limits to accuracy and reliability of the estimation to cope with the limited resolution capability of the setup we are using a propagation data model that does not only contain discrete deterministic components but also a non resolvable stochastic part joint estimation of both components considerably enhances the estimation quality and finally allows the interpretation as specular and diffuse contribution of multi path propagation respectively however besides of noise influence the achievable resolution is further limited by the accuracy of the data model that describes the measurement setup since the antenna characteristics are very susceptible to calibration and modeling errors the directional estimates are most error prone we refer to the antenna array calibration procedure and discuss common pitfalls in high resolution multi path direction estimation that are related to inaccurate and or incomplete device data model depending on the type of the antenna array linear circular this will inherently produce biased and artificially spread angular estimates only with precise knowledge of the model errors the stochastic part can be identified as diffuse propagation component vs modeling error

three dimensional vibration analysis of thick rectangular plates using chebyshev polynomial and ritz method this paper describes a method for free vibration analysis of rectangular plates with any thicknesses which range from thin moderately thick to very thick plates it utilises admissible functions comprising the chebyshev polynomials multiplied by a boundary function the analysis is based on a linear small strain three dimensional elasticity theory the proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions a very simple and general programme has been compiled for the purpose for a plate with geometric symmetry the vibration modes can be classified into symmetric and antisymmetric ones in that direction in such a case the computational cost can be greatly reduced while maintaining the same level of accuracy convergence studies and comparison have been carried out taking square plates with four simply supported edges as examples it is shown that the present method enables rapid convergence stable numerical operation and very high computational accuracy parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail with respect to different thickness side ratios aspect ratios and poissons ratios these results may serve as benchmark solutions for validating approximate two dimensional theories and new computational techniques in future 2002 elsevier science ltd all rights reserved

on the asymptotic performance analysis of subspace doa estimation in the presence of modeling errors case of music this paper provides a new analytic expression of the bias and rms error root mean square error of the estimated direction of arrival doa in the presence of modeling errors in first order approximations of the rms error are derived which are accurate for small enough perturbations however the previously available expressions are not able to capture the behavior of the estimation algorithm into the threshold region in order to fill this gap we provide a second order performance analysis which is valid in a larger interval of modeling errors to this end it is shown that the doa estimation error for each signal source can be expressed as a ratio of hermitian forms with a stochastic vector containing the modeling error then an analytic expression for the moments of such a hermitian forms ratio is provided finally a closed form expression for the performance bias and rms error is derived simulation results indicate that the new result is accurate into the region where the algorithm breaks down

three dimensional elasticity solutions for free vibrations of circular plates a polynomials ritz analysis this paper presents accurate three dimensional elasticity solutions for free vibration of circular plates the derivation of a linear frequency equation based on an exact three dimensional small strain linearly elastic principle is detailed the solution to this problem is made possible by using the ritz method with a set of orthogonal polynomial series to approximate the spatial displacements of the circular plate in cylindrical polar coordinates the perturbation of frequency responses due to the variations of boundary conditions and thickness is investigated first known frequency parameters and three dimensional deformed mode shapes are presented in vivid graphical forms the accuracy of these results are verified by appropriate convergence studies and when possible are checked with existing solutions

vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment abstract the free vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment is carried out by the 3d elasticity theory and the chebyshev ritz method the material properties are assumed to be temperature dependent and graded in the thickness direction the mixed boundary conditions which include upper and lower surfaces partially fixed inner side partially fixed and outer side partially fixed are considered respectively the accuracy of the present approach for solving the free vibration of the plates with different boundary conditions is validated by comparing the present numerical results with the results available the effects of the different mixed boundary conditions the temperature rise the material graded index and the geometrical parameters on the eigen frequencies are studied

rapid analysis of variable stiffness beams and plates legendre polynomial triple product formulation copyright 2017 john wiley sons ltd numerical integration techniques are commonly employed to formulate the system matrices encountered in the analysis of variable stiffness beams and plates using a ritz based approach computing these integrals accurately is often computationally costly herein a novel alternative is presented the recursive analytical polynomial integral definition rapid formulation the rapid formulation offers a significant improvement in the speed of analysis achieved by reducing the number of numerical integrations that are performed by an order of magnitude a common legendre polynomial basis is employed for both trial functions and stiffness load variations leading to a common form for the integrals encountered the legendre polynomial basis possesses algebraic recursion relations that allow these integrals to be reformulated as triple products with known analytical solutions defined compactly using the wigner 3j coefficient the satisfaction of boundary conditions calculation of derivatives and transformation to other bases is achieved through combinations of matrix multiplication with each matrix representing a unique boundary condition or physical effect therefore permitting application of the rapid approach to a variety of problems indicative performance studies demonstrate the advantage of the rapid formulation when compared to direct analysis using matlab s integral and integral2 copyright 2017 john wiley sons ltd

performance prediction of maximum likelihood direction of arrival estimation in the presence of modeling errors this paper provides new analytic expressions for the root mean square rms error and bias of the maximum likelihood ml direction of arrival doa estimator in the presence of steering vector modeling errors previous work has provided a first order approximation of these performance measures which is valid for small modeling errors in order to take into account larger errors and provide tools for designing an experimental setup a more accurate but still easy to use performance analysis is necessary for such an investigation the doa estimation errors are expressed as a hermitian form with a stochastic vector composed of the modeling errors closed form expressions relating the bias and rms errors to the statistical moments of the modeling error are then deduced from the statistics of this hermitian form simulations are provided to illustrate the usefulness of the theoretical results

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

elasticity solutions for free vibrations of annular plates from three dimensional analysis abstract this article examines the vibrational characteristics of annular plates by using the three dimensional elasticity theory it aims to raise the quality of the investigation beyond that provided by the two dimensional plate theories by resorting to a full three dimensional analysis a polynomials ritz model based on sets of orthogonally generated polynomial functions to approximate the spatial displacements of the plates in cylindrical polar coordinates is presented the model is then used to extract the full vibration spectrum of natural frequencies and mode shapes the vibration responses due to the variations of boundary conditions and thickness are investigated frequency parameters and three dimensional deformed mode shapes are presented in vivid graphical forms the accuracy of the method is validated through appropriate convergence and comparison studies

natural vibration of circular and annular thin plates by hamiltonian approach abstract the present paper deals with the natural vibration of thin circular and annular plates using hamiltonian approach it is based on the conservation principle of mixed energy and is constructed in a new symplectic space a set of hamiltonian dual equations with derivatives with respect to the radial coordinate on one side of the equations and to the angular coordinate on the other side are obtained by using the variational principle of mixed energy the separation of variables is employed to solve hamiltonian dual equations of eigenvalue problem analytical frequency equations are obtained based on different cases of boundary conditions the natural frequencies are the roots of the frequency equations and corresponding mode functions are in terms of the dual variables q1 r three basic edge constraint cases for circular plates and nine edge constraint cases for annular plates are calculated and the results are compared well with existing ones

a higher order shear deformation theory for the vibration of thick plates abstract several first order shear deformation theories for thick plates have been formulated during the past half century e g by reissner hencky mindlin and ambartsumyan the mindlin theory has been widely used for vibration analysis it is well known that consideration of shear flexibility corrects the flexural frequencies by reducing them however it has been found that first order shear deformation theories result in over correction in the present work a higher order theory is presented which provides additional freedom to the displacements through the plate thickness and essentially eliminates the over correction a complete theory is developed including energy functionals equations of motion and boundary conditions comparisons of frequencies arising from classical thin plate theory mindlin theory and the present theory are made for completely free rectangular plates of moderate and large thickness completely free boundaries rather than simply supported ones are chosen because they are more consistent with physical models in the theories used

extension and generalization properties of the weighted minkowski inverse in a minkowski space for an arbitrary matrix the weighted minkowski inverse a m n m n m related to the positive definite matrices m m m and n m n of an arbitrary matrix a m m n including singular and rectangular is one of the important generalized inverses for solving matrix equations in minkowski space in this paper the results are introduced in the following three ways first we establish some new and attractive properties of the weighted minkowski inverse a m n in a minkowski space second new representations and conditions for the continuity of the weighted minkowski inverse a m n in a minkowski space are discussed finally some illustrated counterexamples are also studied to show that some well known properties of the weighted moore penrose inverse a m n in a hilbert space h are not valid in the minkowski inverse a m n in a minkowski space

a performance analysis of subspace based methods in the presence of model error ii multidimensional algorithms for pt i see ibid vol 40 no 7 p 1758 74 1992 in pt i the performance of the music algorithms for narrowband direction of arrival doa estimation when the array manifold and noise covariance are not correctly modeled was investigated this analysis is extended to multidimensional subspace based algorithms including deterministic or conditional maximum likelihood md music weighted subspace fitting wsf mode and esprit a general expression for the variance of the doa estimates that can be applied to any of the above algorithms and to any of a wide variety of scenarios is presented optimally weighted subspace fitting algorithms are presented for special cases involving random unstructured errors of the array manifold and noise covariance it is shown that one dimensional music outperforms all of the above multidimensional algorithms for random angle independent array perturbations

three dimensional free vibration of thick circular plates on pasternak foundation abstract this paper describes a study of the three dimensional vibration characteristics of thick circular plates resting on elastic foundation the analysis is based on the three dimensional small strain linear and exact elasticity theory the foundation is described by the pasternak model the ritz method is used to derive the frequency equation of the plate foundation system by augmenting the strain energy of the plate with the elastic potential energy of the foundation a set of chebyshev polynomials multiplied by a boundary function is adopted as the admissible functions of the displacement components in each direction for plates with free edges the effect of foundation medium beyond the edge of the plate has been considered by introducing the generalized shearing force concept in the analysis the convergence and comparison studies demonstrate the correctness and accuracy of the present method it is shown that the present method has rapid convergent rate stable numerical operation and very high accuracy parametric investigations on the dynamic behavior of thick circular plates resting on elastic foundation have been carried out with respect to various thickness radius ratios foundation stiffness parameters and boundary conditions results known for the first time have been reported and discussed in detail some significant conclusions have been drawn

closed form solution for free vibration of piezoelectric coupled annular plates using levinson plate theory abstract free vibration analysis of annular moderately thick plates integrated with piezoelectric layers is investigated in this study for different combinations of soft simply supported hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the levinson plate theory lpt the distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the maxwell static electricity equation is approximately satisfied the differential equations of motion are solved analytically for various boundary conditions of the plate in this study the closed form solution for characteristic equations displacement components of the plate and electric potential are derived for the first time in the literature to demonstrate the accuracy of the present solution comparison studies is first carried out with the available data in the literature and then natural frequencies of the piezoelectric coupled annular plate are presented for different thickness radius ratios inner outer radius ratios thickness of piezoelectric material of piezoelectric and boundary conditions present analytical model provides design reference for piezoelectric material application such as sensors actuators and ultrasonic motors

buckling and vibration of polar orthotropic circular plate resting on winkler foundation buckling and vibrational behavior of polar orthotropic circular plates of linearly varying thickness is presented on the basis of classical plate theory the plate is resting on winkler type foundation an approximate solution has been obtained by ritz method which employs basis functions based upon static deflection of polar orthotropic plates the effect of elastic foundation and that of orthotropy on the natural frequency of plate has been illustrated for different values of taper parameter flexibility parameter in plane force and nodal diameter the critical buckling load for clamped and simply supported plates have been obtained a comparison of results with those available in literature has been presented

vibration of electro elastic versus magneto elastic circular annular plates using the chebyshev ritz method abstract the aim of this paper is to analyze three dimensional free vibration of magneto elastic electro elastic circular annular plates with different boundary conditions using the chebyshev ritz method in which a set of duplicate chebyshev polynomial series multiplied by the boundary function satisfying the boundary conditions are chosen as the trial functions of the displacement components the electric potential and the magnetic potential convergence of the method is checked using various chebyshev polynomial terms the effect of geometrical parameters and material properties of magneto elastic electro elastic circular annular plates on the eigenfrequencies of free vibration is considered

singular value decomposition in minkowski space abstract the well known singular value decomposition theorem states that every matrix m can be written in the form m qdr where d is diagonal and q and r are orthogonal we explore the analogous decomposition where q and r are required to be orthogonal with respect to the minkowski rather than the euclidean metric

effect of built in edges on 3 d vibrational characteristics of thick circular plates the vibrational characteristics of thick circular plates built into a rigid medium in the vicinity of the circumference are studied the analysis procedure is based on the exact small strain three dimensional linear elasticity theory in the analysis the plate is considered as comprising an inner circular disc with free upper and lower surfaces connected to an outer annulus that is built into the rigid medium the chebyshev polynomials multiplied by suitable boundary characteristic functions are selected as the admissible functions of the displacement functions and the three dimensional displacement fields in each part can be expressed accordingly through the ritz method the eigenvalue equations can be established for the inner disc and the outer annulus respectively utilizing the displacement continuity conditions at the interface between the two parts the eigenvalue equation for the whole plate is derived convergence and comparison studies demonstrate the correctness and accuracy of the present method the effect of structural parameters such as plate thickness built in length and the end condition on eigenfrequencies of circular plate is investigated in detail it is shown that with the increase of plate thickness the effect of the built in annulus on eigenfrequencies significantly increases therefore for a thick plate the flexibility of the built in part should be considered in the dynamic analysis