Oblique Coordinate System Research Articles

Interpolation-based plane stress anisotropic yield models

In this paper, the interpolation-type plane stress anisotropic yield models are proposed. The models are built in a specific three-dimensional oblique coordinate system by sequentially constructing three specific ``longitude lines" and two sets of quasi-orthogonal ``latitude lines" on the tricomponent yield surface. The three ``longitude lines" act as the trajectories of the endpoints of the ``latitude lines", and the semi-yield surface is formed by the continuous variation of the ``latitude lines" along the trajectories. These curves are constructed by parametric Bézier curves, and different order schemes are adopted for “latitude lines” and “longitude lines”. Some coefficients in the expression of “latitude lines” are represented by specific analytic functions, which are related to the distortion of yield loci with the increase of τxy-component. According to whether the asymmetry of tension and compression when yielding is considered, two corresponding models are proposed. We also obtain corresponding simplified forms by omitting the parameter determination of the "latitude lines". The yield loci, flow stresses and r-valus of five kinds of materials with three lattice types were predicted by full-parameter models and simplified models, which showed the accuracy of the proposed models for predicting yield behavior of various materials.

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogeniza-tion schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.

The velocity updating rule according to an oblique coordinate system with mutation and dynamic scaling for particle swarm optimization

Particle swarm optimization (PSO) has been showing powerful search performance especially in separable and unimodal problems. However, the performance is deteriorated in non-separable problems such as rotated problems. In this study, a new velocity updating rule according to an oblique coordinate system, instead of an orthogonal coordinate system, is proposed to solve non-separable problems. Two mutation operations for the best particle and the worst particle are proposed to improve the diversity of particles and to decrease the degradation of moving speed of particles. In addition, the vectors generated according to the oblique coordinate system are dynamically scaled to improve the robustness and efficiency of the search. The advantage of the proposed method is shown by solving various problems including separable, non-separable, unimodal, and multimodal problems, and their rotated problems and by comparing the results of the proposed method with those of standard PSO.

Oblate spheroidal droplet evaporation in an acoustic levitator

The modelling of evaporation in which droplet geometry deviates from sphericity, i.e., oblate spheroid, when the droplet experiences high dynamic stresses or a high Weber number, is important in many applications. The validation of such theoretical models is often difficult to achieve experimentally. The acoustic levitation technique was used to investigate the evaporation of an oblate spheroid for different liquids. Evaporation of oblate droplet at constant aspect ratio is realized through the course of evaporation in the acoustic levitator by continuously adjusting the applied acoustic force on the droplet. A Two-dimensional axisymmetric computational model in oblate coordinate system is presented to predict droplet evaporation driven by the acoustic boundary layer, the model calculates the vapor flux at each grid point on droplet surface. The evaporation follows the d2-law and a good agreement between model prediction and experiments is demonstrated. The acoustic levitator allows for the study of the evaporation of freely suspended deformed droplets and validates the theoretical model of oblate droplet evaporation.

Free vibration of FG-CNT reinforced composite skew cylindrical shells using the Chebyshev-Ritz formulation

Abstract In the present research, the free vibration characteristics of a skew cylindrical panel made of functionally graded carbon nanotube reinforced composites (FG-CNTRCs) is investigated. A dual distribution of CNTs is considered across the panel thickness, namely a uniform and a nonuniform distribution. A refined rule of mixtures approach is applied to estimate the mechanical properties of the composite body, by means of the introduction of some efficiency parameters. A first order shear deformation shell theory (FSDT) is also combined with the Donnell's kinematic assumptions to determine the basic governing equations of the problem for thin-to-moderately thick shells. The governing equations are here referred to an oblique coordinate system, in order to handle any kind of boundary conditions. With the aid of the Ritz method, a system of homogeneous equations governs the eigenvalue problem, whose shape functions are built on Chebyshev polynomials. This system allows to compute the natural frequencies of the shell. A comparative evaluation of the formulation is performed to demonstrate its accuracy and efficiency. Further parametric studies are aimed at exploring the sensitivity of the response to some reinforcement parameters, as the volume fraction or the distribution of CNTs within the matrix.

Numerical and Experimental Analyses of Dynamic Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the dynamic characteristics of liquid annular seals with helical grooves in the seal stator. In the numerical analysis, the governing equations were the momentum equations with turbulent coefficients and the continuity equation, all averaged across the film thickness and expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. These governing equations were solved numerically to obtain the dynamic characteristics, such as the dynamic fluid-film forces, dynamic coefficients, and whirl-frequency ratio (WFR). The numerical analysis included the effect of both fluid inertia and energy loss at the steps between the helical groove and the land sections. In the experiments, the dynamic fluid-film pressure distributions, which were induced by a small whirling motion of the rotor about the seal center, were measured to obtain the dynamic characteristics. The equivalent numerical results reasonably agree with the experimental results, demonstrating the validity of the numerical analysis. The value of the tangential dynamic fluid force induced by the rotor whirling motion decreased with increasing the helix angle γ. Consequently, the values of the cross-coupled stiffness coefficient and WFR decreased with increasing γ and became negative for large γ. In general, pump rotors rotate with a forward whirling motion under normal operating conditions. Hence, the negative value of WFR for helically grooved seals contributes to rotor stability by suppressing the forward whirling motion of the rotor.

On the use of convected coordinate systems in the mechanics of continuous media derived from a QR factorization of F

An oblique, Cartesian, coordinate system arises from the geometry affiliated with a Gram–Schmidt (QR) factorization of the deformation gradient F, wherein Q is a proper orthogonal matrix and R is an upper-triangular matrix. Here a cube deforms into a parallelepiped whose edges are oblique and serve as the base vectors for a convected coordinate system. Components for the metric tensor, its dual, and their rates, evaluated in this convected coordinate system, are established for any state of deformation. Strains and strain rates are defined and quantified in terms of these metrics and their rates. Quotient laws and their affiliated Jacobians are constructed that govern how vector and tensor fields transform between this oblique coordinate system, where constitutive equations are ideally cast, and the reference, rectangular, Cartesian, coordinate system described in terms of Lagrangian variables, where boundary value problems are solved.

Method of the Determination of Exterior Orientation of Sensors in Hilbert Type Space.

The following article presents a new isometric transformation algorithm based on the transformation in the newly normed Hilbert type space. The presented method is based on so-called virtual translations, already known in advance, of two relative oblique orthogonal coordinate systems—interior and exterior orientation of sensors—to a common, known in both systems, point. Each of the systems is translated along its axis (the systems have common origins) and at the same time the angular relative orientation of both coordinate systems is constant. The translation of both coordinate systems is defined by the spatial norm determining the length of vectors in the new Hilbert type space. As such, the displacement of two relative oblique orthogonal systems is reduced to zero. This makes it possible to directly calculate the rotation matrix of the sensor. The next and final step is the return translation of the system along an already known track. The method can be used for big rotation angles. The method was verified in laboratory conditions for the test data set and measurement data (field data). The accuracy of the results in the laboratory test is on the level of 10−6 of the input data. This confirmed the correctness of the assumed calculation method. The method is a further development of the author’s 2017 Total Free Station (TFS) transformation to several centroids in Hilbert type space. This is the reason why the method is called Multi-Centroid Isometric Transformation—MCIT. MCIT is very fast and enables, by reducing to zero the translation of two relative oblique orthogonal coordinate systems, direct calculation of the exterior orientation of the sensors.

New approach to isometric transformations in oblique local coordinate systems of reference

AbstractThe research article describes a method of isometric transformation and determining an exterior orientation of a measurement instrument. The method is based on a designation of a “virtual” translation of two relative oblique orthogonal systems to a common, known in the both systems, point. The relative angle orientation of the systems does not change as each of the systems is moved along its axis. The next step is the designation of the three rotation angles (e.g. Tait-Bryan or Euler angles), transformation of the system convoluted at the calculated angles and moving the system to the initial position where the primary coordinate system was. This way eliminates movements of the systems from the calculations and makes it possible to calculate angles of mutual rotation angles of two orthogonal systems primarily involved in the movement. The research article covers laboratory calculations for simulated data. The accuracy of the results is 10-6 m (10-3 regarding the accuracy of the input data). This confi rmed the correctness of the assumed calculation method. In the following step the method was verifi ed under fi eld conditions, where the accuracy of the method raised to 0.003 m. The proposed method enabled to make the measurements with the oblique and uncentered instrument, e.g. total station instrument set over an unknown point. This is the reason why the method was named by the authors as Total Free Station - TFS. The method may be also used for isometric transformations for photogrammetric purposes.

Buckling analysis of isotropic skew plates under general in-plane loads by the modified differential quadrature method

• Buckling analysis of thin skew plates using modified DQM is presented. • A complete set of differential equations in oblique coordinate system is established. • Accurate buckling loads are presented and verified by the FEM data. • Effects of boundary conditions, loadings, skew angles and aspect ratios are discussed. Perhaps due to the bending moment singularity at the obtuse plate corners, the existing buckling loads vary appreciably even for simply supported rhombic plate under uniaxial loading. In this paper, the buckling analysis of skew plates under various loadings is presented using the modified differential quadrature method (DQM). A complete set of equations of equilibrium, bending and twisting moments, equivalent shear forces and corner force in oblique coordinate system is given. Eight combinations of simply supported, clamped and free boundary conditions, four skew angles, and five aspect ratios are considered. Convergence studies are made for rhombic plates with the largest skew angle subjected to uniaxial and equal biaxial compressive loads. The solution accuracy is further verified by comparing the results with data obtained by the finite element method using very fine meshes. New data are tabulated and plotted for skew plates with different edge conditions, aspect ratios and skew angles, subjected to uniaxial, biaxial, as well as combined shear with uniaxial loads. Most of the data, not available thus far, may be useful for other researches in testing the validity of their numerical techniques for thin plate analysis.

Numerical and Experimental Analyses of Static Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the static characteristics of liquid annular seals with helical grooves in a seal stator. In the numerical analysis, the momentum equations with turbulent coefficients and the continuity equation, which were averaged across the film thickness, were numerically solved to obtain the leakage flow rate and the pressure distributions in the seal clearance. To accurately define the location of the step between the groove and the land regions in the calculation domain, these governing equations were expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. The numerical analysis included the effects of both fluid inertia and energy loss due to expansion during the passage of fluid from the land region to the helical groove region and that due to contraction from the groove region to the land region. In the experimental analysis, the leakage flow rate and the fluid-film pressure distributions in the seal clearance were measured for the helically grooved seals with different helix angles of the helical groove. The numerical results of leakage flow rate and pressure distributions agree reasonably with the experimental results, which demonstrates the validity of the numerical analysis. The leakage flow rate of the helically grooved seals was influenced by two factors: fluid energy loss during passage through the step between the groove and the land, and the pumping effect by which the spinning motion of the rotor pushes the flow back upstream along the helical grooves. Under a low range of rotor spinning velocity, the leakage flow rate decreased with helix angle because the effect of fluid energy loss in the steps was significant. By contrast, under a high range of spinning velocity, the quantitative difference in the leakage flow rate due to the helix angle decreased compared to that under a low range because the reduction in the leakage flow rate due to the pumping effect was pronounced for a larger helix angle. The effects of helix angle and rotor spinning velocity on the leakage flow rate are explained qualitatively using a simplified model.

Thermal buckling of temperature-dependent FG-CNT-reinforced composite skew plates

ABSTRACTCritical buckling temperatures of skew plates made from a polymeric matrix reinforced by single-walled carbon nanotubes (CNTs) are obtained in the present research. Reinforcements are distributed across the thickness of the plate uniformly or according to a prescribed nonuniform function. All of the thermomechanical properties are assumed to be temperature dependent. First-order shear deformation plate theory is used as the basic assumption to obtain the total strain and potential energies of the plate due to the thermally induced prebuckling loads. A transformation is proposed to express the components of the displacement field in an oblique coordinate system. A Ritz-based solution is implemented to obtain the matrix representation of the stability equations associated to the onset of buckling. Gram–Schmidt process is used to obtain a set of orthogonal shape functions as the basis polynomials of the Ritz method. The obtained eigenvalue problem is solved successively to extract the critical buckling temperature of the skew plate. Convergence and comparison studies are provided to assure the accuracy and correctness of the proposed formulation. Afterward, parametric studies are given to explore the influences of boundary conditions, CNT volume fraction, CNT dispersion profile, aspect ratio, skew angle, and side to thickness ratio.

Free vibration of FG-CNT reinforced composite skew plates

Present study deals with the free vibration analysis of skew plates made from functionally graded carbon nanotube reinforced composites. Carbon nanotubes as reinforcements are distributed across the thickness of the plate. Distribution pattern may be uniform or functionally graded. The developed formulation from a Cartesian coordinate system is transformed to an oblique coordinate system to satisfy the boundary conditions. The virtual strain and kinetic energies of the plate are obtained using the first order shear deformation plate theory. Ritz method whose shape functions are developed according to the Gram–Schmidt process is implemented to construct an eigenvalue problem associated to the natural frequencies of the plate. The developed solution method is general and may be used for arbitrary boundary conditions of the plate. Results are compared for isotropic homogeneous and composite laminated plates in skew shape with the available data in the open literature. Afterwards numerical results are provided for skew plates reinforced with carbon nanotubes. It is shown that volume fraction of carbon nanotubes and their distribution pattern are both influential of natural frequencies of the carbon nanotube reinforced plates. Generally, the higher the volume fraction of carbon nanotubes, the higher the natural frequencies of the skew plate.

Transverse vibrations of moving skew plates made of functionally graded material

The present paper investigates the transverse vibrations and stability of a moving skew thin plate made of functionally graded ceramic–metallic material. The material properties are assumed to vary continuously through the thickness according to a power-law distribution of the volume fractions of the constituents. The Voigt's rule is used to estimate the effective material properties from the volume fractions and the properties of the constituent materials. By the coordinate transformation, the differential equations of motion of the moving functionally graded material (FGM) skew plate are obtained in oblique coordinate system. The boundary conditions with simply supported and clamped edges are obtained in oblique coordinate system. The vibration frequencies are obtained from the solution of a generalized eigenvalue problem. The entire computational work is carried out in a normalized square domain obtained through an appropriate domain mapping technique. Results of the reduced problem revealed excellent agreement with other studies. The dimensionless complex frequencies of the moving FGM skew plate are calculated by the differential quadrature method. The effects of gradient index, aspect ratio, and dimensionless moving speed on the transverse vibration and stability of the moving FGM skew plate are analyzed. Results are furnished in dimensionless amplitude–frequency curves for different dimensionless moving speed and representations of some vibration mode shapes are shown.

Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

When studying axisymmetric particle fluid flows, a scalar function,ψ, is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely,E4ψ=0, whereE2is the Stokes irrotational operator andE4=E2∘E2is the Stokes bistream operator. As it is already known,E2ψ=0in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation acceptsR-separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operatorE4does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, whileE4ψ=0semiseparates variables in the spheroidal coordinate systems and itR-semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equationE′2ψ=0alsoR-separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equationE′4ψ=0 R-semiseparates variables. Since the generalized eigenfunctions ofE′2cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions ofE′2in the modified inverted oblate spheroidal coordinate system.

Constructing optimal transition matrix for Markov chain Monte Carlo

The notion of asymptotic variance has been used as a means for gauging the performance of Markov chain Monte Carlo (MCMC) methods. For an effective MCMC simulation, it is imperative to first construct a Markov model with minimal asymptotic variance. The construction of such a stochastic matrix with prescribed stationary distribution as well as optimal asymptotic variance amounts to an interesting variationally constrained inverse eigenvector problem. Cast against a specially defined oblique coordinate system, the worst-case analysis of the asymptotic variance can be formulated as a problem of minimizing the logarithmic 2-norm of a restricted resolvent matrix over a convex and compact monoid. Based on this framework, this paper proposes employing global optimization techniques as a general instrument for numerical construction of optimal transition matrices. Numerical experiments manifest the complexity of the underlying problem. First, new global solutions different from the conventional structure characterized in the literature are found across the board for reversible problems. Second, local solutions with high frequencies of occurrence appear widespread for nonreversible problems. In all, the approach via the global optimization techniques is a feasible and practical means for numerical construction of the optimal Markov chain.

Fourier method for modeling slanted lamellar gratings of arbitrary end-surface shapes in conical mounting.

An efficient modal method for numerically modeling slanted lamellar gratings of isotropic dielectric or metallic media in conical mounting is presented. No restrictions are imposed on the slant angle and the length of the lamellae. The end surface of the lamellae can be arbitrary, subject to certain restrictions. An oblique coordinate system that is adapted to the slanted lamella sidewalls allows the most efficient way of representing and manipulating the electromagnetic fields. A translational coordinate system that is based on the oblique Cartesian coordinate system adapts to the end-surface profile of the lamellae, so that the latter can be handled simply and easily. Moreover, two matrix eigenvalue problems of size 2N × 2N, one for each fundamental polarization of the electromagnetic fields in the periodic lamellar structure, where N is the matrix truncation number, are derived to replace the 4N × 4N eigenvalue problem that has been used in the literature. The core idea leading to this success is the polarization decomposition of the electromagnetic fields inside the periodic lamellar region when the fields are expressed in the oblique translational coordinate system.

Design of Spherical Spiral Groove Bearings for a High-Speed Air-Lubricated Gyroscope

A spherical spiral groove gas bearing has the ability to not only support radial and axial loads simultaneously but also tolerate extensive misalignment. Thus, this bearing type is considered suitable as a supporting and lubrication component of inertial components such as gyroscopes in the fields of aerospace and navigation. In this study, we propose a numerical method for predicting static and dynamic characteristics and conduct a parametric analysis of an aerodynamic spherical bearing with rotating spiral grooves. The finite difference method and the perturbation method are used to calculate the Reynolds equation to obtain the force coefficients. Given the complicated groove distribution, as well as film discontinuity and compressibility, solving the equations numerically in the spherical coordinates system is difficult. Parameter transformation and oblique coordinate transformation are thus applied in this study to modify the Reynolds equation into the planar oblique coordinate system. An eight-point method is also utilized to deal with film thickness discontinuity. The predictions of this proposed method show good agreement with the available experimental data. Parametric studies on nominal clearance, eccentricity ratio, rotating speed, groove depth, groove angle, and perturbation frequency are then conducted to determine optimum design parameters. The results show that these factors significantly affect bearing characteristics in both the radial and axial directions.

An addressing and routing scheme based on modified Euclidean space for hexagonal networks

The addressing and routing algorithm on hexagonal networks is still an open problem so far. Although many related works have been done to resolve this problem to some extent, the properties of hexagonal networks are still not explored adequately. In this paper, we first create an oblique coordinate system and redefine the Euclidean space to address the hexagonal nodes. Then an optimal routing algorithm using vectors and angles of the redefined Euclidean space is developed. Compared with the traditional 3-directions scheme and the Cayley graph method, the proposed routing algorithm is more efficient and totally independent of the scale of networks with two-tuples addresses. We also prove that the path(s) obtained by this algorithm is always the shortest one(s).

A dynamical basis for slip in centrifugal impellers

The polar angle of a variable point defining the trajectory of a fluid particle in a centrifugal impeller is obtained using mechanics of particles in a rotating coordinate system. The expression for the angle then yielded the impeller outlet relative flow angle, which was used to calculate slip factor. An attempt to explain the phenomenon of slip using an oblique coordinate system with axes in the direction of the tangent to a vane and in the tangential direction then enabled a new expression for slip factor to be obtained. The computed values of slip factor using available pump data compare reasonably well with reference values.