Characteristic Orthogonal Polynomials Research Articles

Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates With Bilinear Thickness Variation

An analysis and numerical results are presented for buckling and transverse vibration of orthotropic nonhomogeneous rectangular plates of variable thickness using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method on the basis of classical plate theory when uniformly distributed in-plane loading is acting at two opposite edges clamped/simply supported. The Gram–Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of various plate parameters such as nonhomogeneity parameters, aspect ratio together with thickness variation, and in-plane load on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported, and free edges correct to four decimal places. Three dimensional mode shapes for a specified plate for all the four boundary conditions have been plotted. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed correct to six significant digits. A comparison of results with those available in the literature has been presented.

Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation are presented here using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory. Gram-Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of nonhomogeneity parameters, aspect ratio and thickness variation together with foundation parameter on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported and free edges correct to four decimal places. Three dimensional mode shapes for specified plate for all the four boundary conditions have been plotted. A comparison of results in special cases with published one has been presented.

Application of Boundary Characteristic Orthogonal Polynomials on Vibration of Circular Plates with Circular Eccentric Holes

Application of Boundary Characteristic Orthogonal Polynomials on Vibration of Circular Plates with Circular Eccentric Holes

Characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular orthotropic plates of bilinearly varying thickness

Effect of nonhomogeneity on the vibrational characteristics of thin orthotropic rectangular plates of bilinearly varying thickness has been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. The thickness variation is taken as the Cartesian product of linear variations along two concurrent edges of the plate. The orthogonal polynomials in two variables are generated using the Gram-Schmidt process. The nonhomogeneity of the plate material is assumed to arise due to linear variations in Young’s moduli, shear modulus and density of the plate with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of thickness variation together with varying values of aspect ratio and nonhomogeneity on the natural frequencies is illustrated for the first three modes of vibration. Three dimensional mode shapes have been presented. Comparison has been made with the known results.

In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

In-plane freevibrations of an elasticand isotropic annular disk withelasticconstraints at theinner and outer boundaries, whichareappliedeither alongtheentireperipheryof thediskorat apoint areinvestigated. Theboundary characteristicorthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

Numerical computation of BCOPs in two variables for solving the vibration problem of a CF-elliptical plate

Boundary characteristic orthogonal polynomials in xy-coordinates have been built up over an elliptical domain occupied by a thin elastic plate. Half of the plate boundary is taken clamped while the other half is kept free. Coefficients of these polynomials have been computed once and for all so that an orthogonal polynomial sequence is generated from a set of linearly independent functions satisfying the essential boundary conditions of the problem. Use of this sequence in Rayleigh–Ritz method for solving the free vibration problem of the plate makes it faster in convergence and leads to a simplified system whose solution is comparatively easier. Three-dimensional solution surfaces and the associated contour lines have been plotted in some selected cases. Comparison have been made with known results whenever available.

Minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by Rayleigh–Ritz method

The Rayleigh–Ritz method is employed to determine the minimum stiffness location of the elastic point support for raising the fundamental natural frequency of a rectangular plate to the second frequency of the unsupported plate, which usually is the upper limit of the first frequency for a single support. Based on the optimal design of the support position, the minimum stiffness can be obtained numerically by solving a characteristic eigenvalue sub-problem. In the Rayleigh–Ritz procedure the boundary characteristic orthogonal polynomials are used for the admissible functions. Several typical examples of plate structures with the additional point supports are analyzed in detail, and the results prove that the proposed method is very effective in the solution to the optimal design of the supports. It will be shown that elastic supports can be designed like rigid ones for the purpose of increasing the fundamental natural frequency of a rectangular plate.

Vibration of a tapered composite aircraft wing using the Rayleigh–Ritz method with boundary characteristic orthogonal polynomials

Geometric and material factors can cause coupling between bending and torsional deflections in composite beams. The geometric coupling occurs when the elastic axis does not coincide with the mass a...

In-plane free vibration of circular annular disks

This paper presents a generalized formulation for the in-plane modal characteristics of circular annular disks under combinations of all possible classical boundary conditions. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The boundary characteristic orthogonal polynomials are employed in the Rayleigh–Ritz method to obtain the natural frequencies and associated mode shapes. Two approaches have been used to represent a clamped boundary condition. The first approach assumes a polynomial expression that satisfies the clamped conditions, while the second approach uses a disk with free boundary supported on artificial springs with stiffness tending to infinity. The natural frequencies are tabulated and compared with data available in the literature. Mode shapes are presented to illustrate the free vibration behavior of the disk.

Boundary characterization of MEMS structures through electro-mechanical testing

Micro-electro-mechanical-systems (MEMS) are coupled electro-mechanical microsystems used in many different sensing and actuation applications. The mechanical characteristics of the microsystem depend in large part to the choice of material and microfabrication process. In microsystems where there are suspended structures, as in the case of atomic force microscope (AFM) microcantilever probes, the boundary supports are uniquely defined by the microfabrication process implemented. In practice, AFM microcantilever probes suffer from large variations in natural frequencies due to the inherent microfabrication limitations, such as, material non-uniformity, geometrical non-uniformity and non-classical boundary support conditions. Out of these influences, non-classical boundary support conditions are difficult to estimate and are also very important for design and modeling considerations. Hence, this paper presents a testing method in which most of the influences were quantified or made variant while the support condition was kept invariant. This approach will help to identify the influence of microfabrication to the support boundary condition. This paper presents the results on the characterization of non-classical support boundary conditions of microcantilevers through electro-mechanical testing. A laser based non-contact optical testing approach is employed for the dynamic testing of the microstructure. The theoretical formulation is based on the Rayleigh-Ritz energy method using boundary characteristic orthogonal polynomials.

Quantitative Boundary Support Characterization for Cantilever MEMS.

Microfabrication limitations are of concern especially for suspended Micro-Electro-Mechanical-Systems (MEMS) microstructures such as cantilevers. The static anddynamic qualities of such microscale devices are directly related to the invariant and variantproperties of the microsystem. Among the invariant properties, microfabrication limitationscan be quantified only after the fabrication of the device through testing. However, MEMSare batch fabricated in large numbers where individual testing is neither possible nor costeffective. Hence, a suitable test algorithm needs to be developed where the test resultsobtained for a few devices can be applied to the whole fabrication batch, and also to thefoundry process in general. In this regard, this paper proposes a method to test MEMScantilevers under variant electro-thermal influences in order to quantify the effectiveboundary support condition obtained for a foundry process. A non-contact optical sensingapproach is employed for the dynamic testing. The Rayleigh-Ritz energy method usingboundary characteristic orthogonal polynomials is employed for the modeling andtheoretical analysis.

Effect of non-homogeneity on natural frequencies of vibration of elliptic plates

This paper presents the effect of non-homogeneity of the material of plate structures on the vibration frequencies. The non-homogeneity of the plate is characterized by taking a variety of combinations of linear as well as quadratic variations in the Young’s modulus and density of the material. Boundary characteristic orthogonal polynomials using Gram-Schmidt procedure have been employed in the Rayleigh Ritz method. The results have been included for the first few frequencies of the plate element for all the boundary conditions viz. clamped, simply supported and free. Related Tables and graphs are incorporated to show the effect of the non-homogeneity parameter on the frequencies of the vibration of elliptic and circular plates.

Shape optimizations and static/dynamic characterizations of deformable microplate structures with multiple electrostatic actuators

Micromirrors used in many optoelectronic devices can be considered as microplates. The functions and accuracy of the micromirrors depend on the static and dynamic deflection shapes of microplates. The objective of this study is to develop an efficient method for predicting the shapes of the microplates subjected to unsymmetrical electrostatic forces produced by electrostatic actuators such that micromirrors can be effectively optimized and controlled in their real time operation. The non-classical boundary conditions which result from the microfabrication process were modeled with artificial springs at the edges. A classical energy method using boundary characteristic orthogonal polynomials was applied to formulate the equations of motion of the microsystem. Based on this method, influence functions were built and least squares method was used to optimize the desired deflection under electrostatic forces from the electrostatic actuators. Softening effect of the electrostatic stiffness was also evaluated and considered in the simulation. Static deflections and dynamic responses were compared with those from finite element analysis (FEA) using Reduced Order Modeling (ROM) method. This study found that the static and dynamic responses of microplates predicted from the proposed method were highly consistent with those calculated from FEA. However, the proposed method is simpler and more efficient than FEA and can be conveniently used for any non-classical boundary condition situations. These features make the proposed method useful to effectively control and optimize the shape of a microplate with multiple electrostatic actuators.

Shape optimization and dynamic characterization of multiple-electrostatically deformable microbridges

The nonlinear nature of electrostatic fields in micromachined structures, such as, cantilevers, bridges or plates, makes it difficult to achieve desired deflection shapes. An analytical approach to predict the static deflection and dynamic behavior of a microbridge subjected to electrostatic fields of multiple electrostatic actuators is presented in this paper. The boundary support conditions of the micromachined structures are nonhomogeneous in nature and are modeled with artificial translational and rotational springs. The static and dynamic behaviors of the microbridge are investigated by the Rayleigh–Ritz method using boundary characteristic orthogonal polynomials. The deflection of the microbridge and the natural frequencies under certain applied voltages are also presented. Least-squares fitting method is used to optimize the applied voltage of actuators in order to generate the desired static deflection. The proposed method is simple and can be easily extended to complicated configurations which are suitable for adaptive optics applications.

Flexural-torsional coupled vibration of slewing beams using various types of orthogonal polynomials

Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. Performance of various orthogonal polynomials is compared to each other in terms of their efficiency and accuracy in determining the required natural frequencies. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the special trigonometric functions used in conjunction with Hermite cubics, and beam characteristic orthogonal polynomials. A total of 5 cases of beam boundary conditions and rotation are studied for their natural frequencies. The obtained natural frequencies and mode shapes are compared to those available in various references and the results for coupled flexural-torsional vibrations are especially compared to both previously available references and with those obtained using NASTRAN finite element package. Among all the examined orthogonal functions, Legendre orthogonal polynomials are the most efficient in overall CPU time, mainly because of ease in performing the integration required for determining the stiffness and mass matrices.

Vibration of Nonhomogeneous Orthotropic Elliptic and Circular Plates With Variable Thickness

In this paper, study of nonhomogeneity as well as variable thickness in elliptic and circular orthotropic plates is undertaken. Nonhomogeneity of plate material is assumed to be a quadratic variation of Young’s modulii and density whereas shear modulus, is considered to vary linearly along both the axes. The quadratic thickness variation in orthotropic nonhomogeneous plates is also considered. Effect of variation of these parameters on vibrational characteristics are analyzed for various boundary conditions at the edges. Results are obtained using boundary characteristic orthogonal polynomials generated by using Gram-Schmidt orthogonalization procedure in Rayleigh-Ritz method.

Boundary characterization of microstructures through thermo-mechanical testing

A variety of silicon foundry processes available for microsystem implementation are available at the present time. The manufacturing methods and the associated process tolerances employed at a particular foundry will determine the performance of the finished devices. Moreover, micro-electro-mechanical systems (MEMS) often require processes that are difficult to control. Device-to-device variations can occur even in batch microfabricated systems. One particular limitation of MEMS foundry processes, in general, is associated with non-classical boundary support conditions due to over/under etching of silicon. These non-classical support conditions will affect the static and dynamic performance of the microsystem. This condition has important implications in atomic force microscopy applications where the targeted natural frequencies are given a wide tolerance due in large part to microfabrication limitations. This paper presents the boundary characterization of single crystal silicon microcantilevers through thermo-mechanical testing. A non-contact optical sensing approach is used for the experimentation. The Rayleigh–Ritz energy method incorporating boundary characteristic orthogonal polynomials is used for the prediction analysis.