Two-dimensional Polynomial Function Research Articles

Modified lysmer’s analog model for two dimensional mat settlements under vertically uniform load

A two dimensional model of linearly elastic soil spring used for the settlement analysis of the flexible mat foundation is suggested in this study. The spring constants of the soils underneath the foundation were modeled assuming uniformly vertical load applied onto the foundation. The soil spring constants were back calculated using the three-dimensional finite element analysis with Midas GTS NX program. Variation of the soil spring constants was modeled as a two-dimensional polynomial function in terms of the normalized spatial distances between the center of foundation and the analytical points. The Lysmer's analog spring for soils underneath the rigid foundation was adopted and calibrated for the flexible foundation. For validations, the newly proposed soil spring model was incorporated into a two dimensional finite difference analysis for a square mat foundation at the surface of an elastic half-space consisting of soft clays. Comparative study was made for elastic soils where the shear wave velocity is 120~180 m/s and the Poisson's ratio varies at 0.3~0.5. The resulting foundation settlements from the two dimensional finite difference analysis with the proposed soil springs were found in good agreement with those obtained directly from three dimensional finite element analyses. Details of the applications and limitations of the modified Lysmer's analog springs were discussed in this study.

Manipulating the geometry of architectured beams for maximum toughness and strength

Dense architectured materials are made of blocks that can slide, rotate, interlock and jam in powerful mechanisms that can generate simultaneous strength and toughness. Nature abounds of examples of such architectured materials, for example in the segmented structure of vertebrate spines. In this study we consider segmented beams made of stiff blocks and submitted to a transverse force. We start with simple cubes as a geometrical reference, which we then enrich by using two-dimensional polynomial functions. The flexural response of the beam is simulated using finite element modeling (FE-model) to predict strength, toughness and maximum local stresses. Using this procedure we identified the most efficient interface geometries and interlocking mechanisms within a set of polynomial functions and for a given strength of the individual blocks. To illustrate these results, we fabricated segmented beams of ceramic glass using a laser engraver. Experiments on these architectured glass revealed how enriched blocks turned the catastrophic brittle failure of monolithic glass into graceful progressive deformation. Resulting in a tougher response than the monolithic by 370 times and preserved 40% of strength of that of the monolithic.⁎Corresponding author.

Actual Tooth Contact Analysis of Straight Bevel Gears

Theoretically, spherical involutes are used as one of the base topographies for straight bevel gears. Actual bevel gears, however, have deviations from their intended topographies due to manufacturing errors, heat treatment deviations, and finishing processes. Measuring the physical parts with coordinate measuring machines (CMMs), this study proposes a new approach to capture such deviations. The measured deviations from spherical involute are expressed in form of a third-order two-dimensional (2D) polynomial function and added to the base topography to duplicate the geometry of the actual part; tooth thickness deviation is also accounted for and corrected through changing the theoretical tooth thickness. The resultant surfaces are then used to construct ease-off and surface of roll angle topographies and to perform tooth contact analysis (TCA) and calculate motion transmission error (TE). At the end a sample straight bevel gear set is measured and utilizing the proposed approach its predicted TCA is compared to the experimental TCA obtained from roll tester. The results show very good correlation between the predicted and actual TCA of the parts. Utilizing the proposed methodology, the other bevel gear base profile geometries (such as octoids) can also be analyzed. In the proposed approach, the difference between other base geometries and spherical involutes can be treated as deviations from spherical involutes and can be taken into account to perform TCA.

A novel approach to fixture layout optimization on maximizing dynamic machinability

In the aerospace industry, the reasonable layout of fixture can efficiently suppresses machining vibration of thin-walled aerospace structure during machining. Based on the analysis of typical structural components encountered in the aerospace industry, a general frame-structure workpiece with fixture constraints can be equivalent as Mindlin plates with simultaneous elastic edges and internal supports. On basis of the equivalent models, the powerful pb-2 Ritz method defined by the product of a two-dimensional polynomial and basic functions can be introduced to be taken as trial functions. Substituting displacement functions into energy functional and minimizing total energy by differentiation leads to eigenfrequency equations of the workpiece–fixture system. Consequently, a novel nonlinear programming problem based on the frequency sensitivity can be built to optimize the layout of fixture supports to maximize the fundamental nature frequency of the workpiece–fixture system. The feasibility of the proposed approach is validated by a machining case.

An improved analytical method to design CMUTs with square diaphragms

A highly accurate mathematical method has been presented for analytical characterization of capacitive micromachined ultrasonic transducers (CMUTs) built with square diaphragms. The method uses a new two-dimensional polynomial function to more accurately predict the deflection curve of a multilayer square diaphragm subject to both mechanical and electrostatic pressure and a new capacitance model that takes into account the contribution of the fringing field capacitances. Both of the models have been experimentally verified by comparing the models predicted values with measurement results and are found to be in excellent agreement with a maximum deviation of less than 2% for experimentally measured capacitance values. 3-D electromechanical finite element analysis (FEA) for a wide range of material properties, geometric specifications, and loading conditions show that the presented method is highly consistent in accuracy over the typical square-diaphragm CMUT design space.

Square Diaphragm CMUT Capacitance Calculation Using a New Deflection Shape Function

A new highly accurate closed-form capacitance calculation model has been developed to calculate the capacitance of capacitive micromachined ultrasonic transducers (CMUTs) built with square diaphragms. The model has been developed by using a two-dimensional polynomial function that more accurately predicts the deflection curve of a square diaphragm deformed under the influence of a uniform external pressure and also takes account of the fringing field capacitances. The model has been verified by comparing the model-predicted deflection profiles and capacitance values with experimental results published elsewhere and finite element analysis (FEA) carried out by the authors for different material properties, geometric specifications, and loading conditions. New model-calculated capacitance values are found to be in excellent agreement with published experimental results with a maximum deviation of 1.7%, and a maximum deviation of 1.5% has been observed when compared with FEA results. The model can help in improving the accuracy of the design methodology of CMUT devices and other MEMS-based capacitive type sensors built with square diaphragms.

DEVELOPMENT OF A NOVEL DUAL TIME-OF-FLIGHT IMAGING MASS SPECTROMETER: PRINCIPAL, REALIZATION, AND OPTIMAL PERFORMANCE

A novel dual time-of-flight imaging analyzer has been developed for studies of gas phase reactions and the scattering or desorption of ions and molecules from surfaces. The analyzer is capable of experimentally selecting a two-dimensional slice of particles from a three-dimensional flux without the necessity for deconvolution of the resulting velocity images by the Abel transform. The analyzer operates through ionization of the scattered species and their subsequent flight through a field-free region. This initial flight allows a dispersion according to the species natural velocity distribution. The second time of flight deflects the ions through a right angle and through a flight tube allowing dispersion according to mass or charge before detection. The analyzer offers two modes of operation — the first of these produces a mass spectrum of the desorbing species, the second produces a two-dimensional velocity map of the desorbing species. Trial results using an effusive beam source and acetone as a test gas have demonstrated the operation of the analyzer. The operation of the analyzer has been simulated and optimized to reduce ion flight aberrations. A set of orthogonal two-dimensional polynomial functions have been derived to reduce residual aberrations across a broad range of operating conditions. An upper limit to the temporal resolution of the analyzer has been established and a set of working parameters for low distortion electron beam ionization have been presented.

Plastic buckling analysis of thick plates using p-Ritz method

This paper is concerned with the application of the p-Ritz method for the plastic buckling analysis of thick plates. In order to allow for the effect of transverse shear deformation in thick plates, the Mindlin plate theory is adopted. The plastic buckling behaviour of the plate is captured by using the incremental and deformation theories of plasticity. The material property of the plate is assumed to obey the Ramberg–Osgood stress–strain relation. The p-Ritz method will be applied to obtain the governing eigenvalue equation for the plastic buckling analysis of uniformly stressed plates with edges defined by polynomial functions. In the p-Ritz method, the displacement functions of the plate are approximately represented by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. The validity, convergence and accuracy of the method were demonstrated for various plate shapes such as rectangular, triangular and elliptical shapes. A parametric study was also undertaken to study the plastic buckling behaviour and the effect of transverse shear deformation.

Improved Computation of Stress Resultants in the p -Ritz Method

This paper highlights the problems encountered in bending and vibration analysis of thin plates when using the p-Ritz method for computing the stress resultants, especially the twisting moments and shear forces. In the p-Ritz method, products of mathematically complete two-dimensional polynomial functions and boundary polynomial equations raised to appropriate powers are used to approximate the displacements. This ensures the satisfaction of the geometric boundary conditions. However, it is found that the distributions of the twisting moment and shear forces contain unacceptable oscillations, and for plates with free edges these stress resultants do not satisfy the natural boundary conditions. A method to overcome the aforementioned problems is proposed here. The penalty function method is used to satisfy the natural boundary conditions and postprocessing of the stress resultants to eliminate the oscillations. Some examples of static and vibration problems of plates are presented to illustrate the effectiveness of the proposed strategy for overcoming the shortcomings inherent in the p-Ritz method.

Determination of cantilever plate shapes using wavelength division multiplexed fiber Bragg grating sensors and a least-squares strain-fitting algorithm

An algorithm has been developed to determine the full deformation field of a cantilever honeycomb plate under arbitrary loading conditions. The algorithm utilizes strain information from a set of sixteen fiber Bragg grating sensors mounted on the plate so that all sensors measure strains along the clamped-free direction. The sensors were interrogated using a wavelength division multiplexing scheme. A two-dimensional polynomial function which represents the strain field was created using a least-squares surface-fitting algorithm. This function was integrated twice with the known boundary conditions applied to yield the deformation field for the plate. Finite-element comparisons were performed to test the accuracy of the strain-displacement algorithm. Single-point loading tests were also experimentally performed to further verify the accuracy of the algorithm.

Comparative Accuracy of Shallow and Deep Shell Theories for Vibration of Cylindrical Shells

This paper investigates the limits and range of application of the shallow shell theory for vibration of singly curved cylindrical panels. The significance of neglecting tangential displacements in bending and twist in the Donnell-Mushtari-Vlasov shallow shell theory is examined. The Ritz stationary energy principle, with a class of geometrically oriented two-dimensional polynomial functions (p-Ritz method), is employed to determine the frequency parameters. A comparison of numerical results from the shallow and deep shell theories is presented. The paper concludes that the limits of the shallow shell theory are dependent on the aspect ratio and boundary condition. The conclusion regarding limitation for a cantilevered shallow shell available in the literature cannot be directly extended to other types of boundary conditions. For shells with unit aspect ratio, shallow shell solutions are practically accurate even for relatively large subtended angles, but the results could be otherwise for long shells. In general, the shallow shell theory is relatively accurate for shells with a subtended angle of not more than 40°.

Three-dimensional vibratory characteristics of solid cylinders and some remarks on simplified beam theories

A set of linear three-dimensional frequency equations that describes the vibratory characteristics of elastic solid cylinders of different end supports is derived. Starting from the linear, small-strain, three-dimensional elasticity theory, integral expressions for strain and kinetic energies are formulated. The three-dimensional mode shapes are expressed in terms of sets of one- and two-dimensional orthogonal polynomial functions which are to approximate the longitudinal and lateral surface variations of solid cylinders. From the resulting displacement-based energy expressions, the variational form of the three-dimensional energy functional is minimized to yield the linear eigen-value equation. Frequency solutions for elastic solid cylinders of different lengths and end support conditions are determined. The accuracy of solutions is validated through convergence tests and comparisons with the existing three-dimensional analytical solutions and empirical data. From the present three-dimensional elasticity solutions, some remarks on the existing simplified beam theories are made. Particular attention is drawn to Timoshenko's shear deformable beam theory for transverse bending modes. The validity of the one-dimensional wave equation for longitudinal and torsional modes is also discussed.

Vibratory characteristics of general laminates

Abstract This paper presents Part I of investigations of free flexural vibration of multi-layered symmetric and unsymmetric composite laminates with symmetric trapezoidal planform of arbitrary combinations of edge conditions. The Ritz procedure withpb−2shape functions as the admissible functions is used to arrive at the governing eivenvalue equation. The set ofpb−2shape functions is intrinsically a product of a class of orthogonally generated two-dimensional polynomial functions and a basic function which ensures the satisfaction of geometric boundary conditions at the outset. Different boundary conditions, with various combinations of in-plane constraints, are considered. The effects of wide ranges of chord ratios, angles of fibre orientation and stacking sequences on the vibration response are investigated. With the aim of increasing the existing database, sets of previously unavailable vibration frequencies and mode shapes are presented as benchmarks for future reference.

A higher order theory for vibration of shear deformable cylindrical shallow shells

The vibratory characteristics of thick cylindrical shallow shells of rȩctangular planform is investigated in this paper. A higher order shear deformation theory is proposed to analyse the effects of various shell geometries and boundary conditions on the vibration responses. The present higher order theory results in cubic expressions for the transverse shear distribution through shell thickness. The strain and kinetic integral energy expressions involving higher order shear deformation are derived in terms of Cartesian coordinates. An energy functional based on the principle of extremum energy is employed which yields a governing eigen-matrix equation. A set of orthogonally generated two-dimensional polynomial functions is adopted to approximate the deflections and rotations of the shells. The results, where possible, are compared with the available experimental and existing numerical results. Sets of new results for the hard simply supported and fully clamped cylindrical shallow shells are presented. Selected modes are illustrated in three-dimensional displacement plots.

Type I collagen packing, conformation of the triclinic unit cell

The X-ray diffraction pattern of tendon collagen can contain a number ofsharp Bragg peaks indicating three-dimensional crystallinity of the sample. Optimal diffraction images have been obtained with a high flux synchrotron X-ray source and a carefully maintained sample environment and staining techniques. The Bragg peaks are always superimposed on a diffuse background. This makes interpretation of data difficult and a number of conflicting models of collagen packing have been proposed. The removal of the diffuse scatter from the images allows the Bragg peaks to be seen on a relatively flat background. This was conducted by modelling the background points as a series of two-dimensional polynomial functions. The resultant set of observed Bragg reflections serves as an excellent basis to test the validity of two contradictory packing modes; (1) the triclinic model, Fraser et al., (2) the microfibril model, Kajava. From this it can easily be seen that the model proposed by Kajava is inappropriate, since there is limited agreement between predicted positions of reflections and the positions of observable reflections on film. The packing of collagen molecules on a triclinic lattice is favoured by this criterion.

Vibration of perforated doubly-curved shallow shells with rounded corners

This study examines the natural frequency and vibratory characteristics of doubly-curved shallow shells having an outer super-elliptical periphery and an inner super-elliptical cutout. A superelliptical boundary in this context is defined as (2x/a)2n + (2y/b)2n = 1, where n = 1, 2, 3, …, ∞. This class of shells with rounded outer and inner corners has a great advantage over shells with a rectangular planform as stress concentration at the corners is greatly diffused. As a result, the high stress durability of such shells has a great potential for use in practical engineering applications, especially in aerospace, mechanical and marine structures. The doubly-curved shells investigated possess variable positive (spherical), zero (cylindrical) and negative (hyperbolic paraboloidal) Gaussian curvatures. A global energy approach is proposed to the study of such shell problems. The Ritz minimization procedure with a set of orthogonally generated two-dimensional polynomial functions is employed in the current formulation. This method is shown to yield better versatility, efficiency and less computational execution than the discretization methods.

Vibration of Clamped Circular Symmetric Laminates

Treated in this paper is the free-flexural vibration analysis of symmetrically laminated thin circular plates. The total energy functional for the laminated plates is formulated where the pb-2 Ritz method is applied for the solution. The assumed displacement is defined as the product of (1) a two-dimensional complete polynomial function and (2) a basic boundary function. The simplicity and accuracy of the numerical procedure will be demonstrated by solving some plate examples. In the present study, the effects of material properties, number of layers and fiber stacking sequences upon the vibration frequency parameters are investigated. Selected mode shapes by means of contour plots for several 16-ply laminated plates with different fiber stacking sequences and composite materials are presented. This study may provide valuable information for researchers and engineers in design applications. In addition, the present solution plays an important role in increasing the existing data base for future references.

Buckling of triangular Mindlin plates under isotropic inplane compression

This paper considers the elastic buckling of triangular Mindlin plates under isotropic inplane compression. The total potential energy functional for the considered buckling problem is derived and the newly developedpb-2 Rayleigh-Ritz method is applied for the analysis. The method features thepb-2 Ritz function defined by the product of a two-dimensional polynomial function (p-2) and a basic function (b) formed from taking the product of the boundary equations with each equation raised to appropriate powers. New buckling results for isosceles and right-angled triangular Mindlin plates with various apex angles, thickness to width ratios and different combinations of edge conditions are presented. Such tabulated results are not only valuable to designers but also important to researchers as reference values for validating their numerical techniques and software packages.

Buckling of triangular plates under uniform compression

This paper concerns the elastic buckling of triangular plates under uniform compression. Different edge conditions involving free, simple support and clamped have been considered. Moreover, the plates may have internal line supports at prescribed locations. The buckling analysis is performed using the so-called pb-2 Rayleigh-Ritz method where the Ritz function has been defined as the product of: firstly, a two-dimensional complete polynomial function; secondly, equations of the internal line supports and; thirdly, the equations of the three edges with each equation raised to power of either 0, 1 or 2 corresponding to a free, simply supported or clamped edge. Buckling factors are obtained for right-angled and isosceles triangular plates of various height-to-base ratios. These very comprehensive results should be of considerable value and practical significance in design applications.

Vibration studies on skew plates: Treatment of internal line supports

A comprehensive literature survey on the vibration of thin skew plates is presented and a few virgin areas on this subject are identified. As an initial part of a research plan to fill these gaps, the paper focuses on vibrating skew plates with internal line supports. For analysis, the pb-2 Rayleigh-Ritz method is used. The Ritz function is defined by the product of (1) a two-dimensional polynomial function, (2) the equations of the boundaries with each equation raised to the power of 0, 1, or 2 corresponding to a free, simply supported or clamped edge and (3) the equations of the internal line supports. Since the pb-2 Ritz function satisfies the kinematic boundary conditions at the outset, the analyst need not be inconvenient by having to search for the appropriate function; especially when dealing with any arbitrary shaped plate of various combinations of supporting edge conditions. Based on this simple and accurate pb-2 Rayleigh-Ritz method, tabulated vibration solutions are presented for skew plates with different edge conditions, skew angles, aspect ratios and internal line support positions.