Oblique Coordinate System Research Articles

Modeling a triclinic lattice elastic body based on the linear couple stress theory

Triclinic lattice structures with generalized parallelepiped unit cells are crucial targets in understanding the influence of lattice structure on the mechanical behaviors of lattice elastic bodies. This study models a three-dimensional elastic body with a triclinic lattice microstructure comprising three uniform elastic members rigidly joined at a single point. Based on the linear couple stress theory, a specialized case of the Cosserat theory, this study derives elasticity tensors in the constitutive equations of the triclinic lattice elastic body that depend on the crossing angles between the members. To derive the angular-dependent elasticity tensors, coordinate transformations between oblique and orthogonal coordinate systems are effectively employed in formulating the substitution of deformation and force fields in the continuum for the fields in the lattice structure. Evaluating the elasticity tensors reveals the influences of the crossing angles on the anisotropic tensile and torsional properties, the existence of the angle-covariant lattice number manipulation that cancels out the size effect, and several types of geometrical shape dependence. This study could contribute to the theoretical evaluation of elastic behaviors of both artificially engineered and naturally formed materials with lattice microstructures.

Free vibration behaviour of composite laminated skew cylindrical shells reinforced with graphene platelets

Current Investigation deals with the free vibration response of composite laminated shells which are reinforced with graphene platelet. Shell is in the form of skew cylinder where rectangular and skew plates and cylindrical shell are considered as especial cases. Each layer of the shell may be enriched with different amount of graphene platelets which results in a functionally graded pattern of reinforcements. The basic governing equations of the shell are obtained by means of the first order shear deformation shell theory. To implement the boundary conditions, it is more preferred to use an oblique coordinate system. Consequently, definition of strains and stresses are provided in an oblique coordinate system. After that, total kinetic and strain energies of the shell are established. Following the idea of Ritz method where the basic shape functions are Chebyshev polynomials, the motion equations of the shell associated to free vibrations are established. Results of this research are compared with the available data in the open literature and after that novel numerical results from the present study are provided to explore the effects of skew angle, opening angle, shallowness, side to thickness ratio, boundary conditions, graphene platelet weight fraction and profile. It is shown that all of these factors are important for the free vibration response of such structure.

Simulation of the hexagonal 3D braiding process for stent preforms

Due to the complex braiding process and long development cycle of the hexagonal three-dimensional braided stent, a MatLab based computer-aided braiding method for a stent is proposed to speed up the development process. First, an oblique coordinate system for the chassis and a polar coordinate system for the chassis unit are constructed, respectively, precisely to coordinate the carrier’s movements on the chassis. Subsequently, an iterative formula delineating the trajectory of the carrier is introduced. The formula effectively translates the entire braiding process into the positional coordinates of the carrier on the chassis and the yarn heights on the mandrel during different stages. Based on the specific characteristics of the braiding process and the stent’s structure, the stent is divided into pressing and twisting sections. The interwoven pattern for both the pressing and twisting sections is determined by establishing the basic tiling form, solving the yarn interwoven sequence, judging the interwoven type and computing the number of kinks. Finally, while considering the stent's dimensional parameters and the interwoven pattern of the yarn, the spatial curve equations for the yarns in both the pressing and the twisting sections are formulated. By concatenating these equations for each section, the three-dimensional trajectory equations and a comprehensive solid model of the stent are successfully derived. Through a rigorous comparative analysis of dimensions and the braiding pattern between the three-dimensional solid model and the physical stent preform, the accuracy and fidelity of the model generated through the implementation of computer-aided braiding technology are verified.

Vibration characteristics of skew sandwich plates with functionally graded metal foam core

Current study deals with the free vibration analysis of skew sandwich plates. The sandwich plate is composed of three layers where the core layer is made from a metal foam while the faces are made from pure metal. Different types of functionally graded patterns are assumed for the distribution of pores in the core. The governing equations of the plate are obtained by means of the first order shear deformation theory. An oblique coordinate system is also defined and the basic governing equations are transformed from the orthogonal system to the oblique one. With the general idea of Ritz method where the shape functions are constructed by means of the Chebyshev polynomials, the matrix representation of the governing equations is established and an eigenvalue problem is extracted. Results of this study are given to discuss the effects of patterns and size of pores, boundary conditions, skew angle, host to face thickness ratio and aspect ratio of the plate. It is shown that all of these parameters are important on natural frequencies of the plate.

Nonlinear Vibration of Moving Orthotopic Films Under Oblique Support

To reduce product damage caused by the vibration instability phenomenon associated with the commutation of new material films during long-path transportation, the nonlinear vibration of orthotropic films transported under oblique support is investigated. First, on the basis of von Karman’s large deflection theory and D’Alembert’s principle, then through coordinate system transformation, the vibration partial differential equations of the system in an oblique coordinate system are established, taking into account the influence of damping. Next, the equations are discretized by the Galerkin method to obtain the system’s nonlinear vibration ordinary differential equations. Finally, the fourth-order Runge-Kutta method is used to carry out numerical calculations of the system to analyze the effects of the oblique support angle, orthotopic coefficient, and damping on the nonlinear vibration of films in transportation. The research results show that the selection of these three important parameters has an impact on the vibration characteristics of moving orthotopic films, and reasonable suggestions are made based on the parameter range for the stable motion of the system, which will help in structural adjustment and stability control during actual printing and manufacturing.

ПРИМЕНЕНИЕ МЕТОДА КОРРЕКЦИИ БАЗИСА ДЛЯ УСТРАНЕНИЯ НЕОПРЕДЕЛЕННОСТИ ПРИ СИНТЕЗЕ ИНВЕРСНОГО ФИЛЬТРА

The process of linear filtering is considered as a process of representing a signal in an oblique coordinate system. An approach to the elimination of uncertainty of the "divide by zero" type in the synthesis of an inverse filter, based on the basis correction, is proposed.

A Hybrid Multivector Model Predictive Control for an Inner-Interleaved Hybrid Multilevel Converter

A hybrid multivector model predictive control strategy for an inner-interleaved hybrid multilevel converter is brought forward in this article, which can enable the separation of the low- and high-frequency stages. It initiates with the use of sign patterns of the reference vector in the stationary reference frame to determine switching states of the low-frequency stage. Then, the reference vector is converted to the inner virtual space vector diagram, which is further transformed into a 120° oblique coordinate system, where the adjacent vectors are selected. Further, the current tracking is realized through the duty cycle optimization of the chosen vectors. Finally, a symmetric switching sequence (SS), which serves as the best one for dc capacitors voltages balancing and circulating current suppression, is selected among all the SSs that belong to the three chosen vectors with optimal duty cycles. The proposed method reduces both current ripples and computational burden while achieving a constant equivalent switching frequency. Comprehensive experimental results performed on an all silicon-carbide prototype verify the effectiveness of the proposed control.

A Hybrid Model Predictive Control for a Seven-Level Hybrid Multilevel Converter With Independent Low-Frequency and High-Frequency Stages

This article puts forward a hybrid model predictive control (H-MPC) that enables the decoupling of low- and high-frequency stages in a seven-level hybrid multilevel converter (7L-HMC). It first uses the sign pattern of the reference voltage vector to determine switching states of the low-frequency stage. Next, the reference vector is shifted to the internal hexagon of the original space vector diagram, which is thereby converted to a 120° oblique coordinate system, where the adjacent vectors can be rapidly selected. Then, the optimal current tracking is primarily safeguarded by the duty cycle optimization of the three adjacent vectors. Finally, an optimized symmetric switching sequence minimizing the dc voltage deviation over one control iteration is selected through the assessment of possible switching sequences that are subject to the chosen vectors with optimal duty cycles. The proposed H-MPC method can reduce both the current ripple and computational burden while achieving a constant switching frequency. Comprehensive simulation and experimental comparisons on an all-silicon-carbide 7L-HMC prototype verify the effectiveness of the proposed H-MPC.

Structural Design and Optimization of Herringbone Grooved Journal Bearings Considering Turbulent

In order to improve the performance of the traditional constant-width herringbone grooved journal bearing in a computed tomography tube under a high-temperature environment, the present study designed a convergent herringbone grooved journal bearing (HGJB) structure lubricated by liquid metal. The bearing oil film thickness and the Reynolds equation considering the influence of turbulence are established and solved by using the finite difference method in the oblique coordinate system. The performance of the two bearings was compared, and the static and dynamic performance change trends of the two bearing structures under different eccentricities were systematically studied. The results show that the convergent herringbone grooved journal bearings are superior to the constant-width herringbone grooved journal bearings in terms of bearing capacity and stiffness coefficient. At the same time, the influence of structural parameters, such as the number of grooves, helix angle, groove to ridge ratio, groove depth on the performance of the constant-width herringbone grooved journal bearings, and the convergent herringbone grooved journal bearings was studied. Finally, we conclude that the performance of the convergent herringbone grooved journal bearings is optimal when the number of grooves is 15–20, the helix angle is 30°–45°, the ratio of the groove to ridge is 1, and the groove depth is 0.02 mm –0.024 mm. This research has provided the thinking and reference basis for the design of liquid metal bearings for high-performance CT equipment.

Contravariant tensor algebra for anisotropic hyperelasticity

Analysis of tensors in oblique Cartesian coordinate systems always requires the definition of a set of orthogonal covariant basis vectors called the Reciprocal basis. This increases the complexity of the analysis and hence makes the method cumbersome. In this work a novel method is presented to effectively carry out the various transformations of tensors to and between oblique coordinate system/s without the need to create the covariant reciprocal basis. This will simplify the procedure of transformations involving problems where tensors are required to be defined in the oblique coordinate system. This work also demonstrates how the analysis of contravariant tensors can be applied to hyperelasticity. Continuum material and damage models can integrate this approach to model anisotropy and non linearity using a much simpler approach. The accuracy of the models was illustrated by matching the predictions to experimental results. A finite element analysis of material and damage model based on contravariant tensors was also carried out on a simple geometry with a re-entrant corner.

The Use of Contravariant Tensors to Model Anisotropic Soft Tissues

Biological tissues have been shown to behave isotropically at lower strain values, while at higher strains the fibers embedded in the tissue straighten and tend to take up the load. Thus, the anisotropy induced at higher loads can be mathematically modeled by incorporating the strains experienced by the fibers. From histological studies on soft tissues it is evident that for a wide range of tissues the fibers have an oblique mean orientation about the physiological loading directions. Thus, we require a mathematical framework of tensors defined in nonorthogonal basis to capture the direction-dependent response of fibers under high induced loads. In this work, we propose a novel approach to determine the fiber strains with the aid of the contravariant tensors defined in an oblique coordinate system. To determine the fiber strains, we introduce a fourth-order contravariant fiber orientation transformation tensor. The approach helps us successfully in determining the fiber strains, for a family of symmetrically and asymmetrically oriented fibers, with the aid of a single anisotropic invariant. The proposed model was fitted with the experimental data from literature to determine the corresponding material parameters.

Prediction of shear thickening of particle suspensions in viscoelastic fluids by direct numerical simulation

Abstract

The Relations between the Various Critical Temperatures of Thin ‎FGM Plates

This work investigates the relations between the critical temperature of the thin FGM plates under various temperature distributions through the thickness resting on the Pasternak elastic foundation. Both rectangular and skew plates are investigated. The uniform, linear, and nonlinear temperature distributions through the plate’s thickness are considered. Formulations are derived based on the classical plate theory (CPT) considering the von Karman geometrical nonlinearity taking the physical neutral plane as the reference plane. The partial differential formulation is separated into two sets of ordinary differential equations using the extended Kantorovich method (EKM). The stability equations and boundary conditions terms are derived according to Trefftz criteria using the variational calculus expressed in an oblique coordinate system. Novel multi-scale plots are presented to show the linear relations between the critical temperatures under various temperature distributions. The critical temperature of plates with different materials are also found linearly related. Resulting relations should be a huge time saver in the analysis process, as by knowing one critical temperature of the one FGM plate under one temperature distribution many other critical temperatures of many other FGM plates under any temperature distributions can be obtained instantly.

Bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation

This paper analyzes bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation. Through coordinate transformation, the Von Karman nonlinear plate theory was employed to derive the non-linear governing equations of membrane on oblique supports in terms of axial movement in the oblique coordinate system. The boundary conditions were also given in the oblique coordinate system. The approximate solution for the governing equations was found via the Galerkin method as well as the fourth-order Runge-Kutta numerical computing method. The nonlinear dynamic techniques including Lyapunov exponents diagrams, bifurcation plots, Poincare maps, phase trajectories, and time histories were introduced to analyze the impacts of the angles of oblique supports, aspect ratios and traveling velocities on dynamics responses and the various forms of vibration regarding the membrane system.

Vibration characteristics and damping properties of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures under hygrothermal conditions

The present article deals with the vibration and damping characteristics of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structure in different hygrothermal conditions. Carbon nanotube reinforced polymer as a matrix phase and carbon fibre as a reinforcing phase are used, and carbon fibre is graded with uniform distribution along the thickness direction for the shell panel according to the power law distribution. The Mori–Tanaka scheme and strength of materials are used to determine the mechanical properties of such functionally graded carbon nanotubes reinforced hybrid composite materials. Finite element modelling has been done by considering an eight-noded shell element with the transverse shear effect according to Mindlin’s hypothesis, and an oblique coordinate system is used for the functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures. Damping is incorporated into such carbon nanotube–based hybrid skewed shell structure based on the Rayleigh damping model. A MATLAB-based in-house computer code has been developed for the proposed formulation and verified with published research work before using for the present dynamic analysis of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structure under hygrothermal conditions. The effect of the carbon nanotube, carbon fibre, material distribution as per power law index and hygrothermal conditions on the damping behaviour of such functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures have been studied. Furthermore, parametric studies are carried out for the first resonant frequency, absolute amplitude, settling time and carbon nanotube impact on the vibrational behaviour of different functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures under different hygrothermal conditions.

Process modelling of 3D hexagonal braids

The hexagonal braiding is a novel 3D braiding technique that can generate a large family of complex fiber architectures for composite preforms. In order to fully exploit this technology, a fiber architecture based model was constructed connecting the braiding process parameters with the resulting fabric structures through machine emulation. First, an oblique coordinate system is established according to the loom arrangement, which is the foundation to digitalize the braiding process. Then, an algorithm is proposed to trace the carrier path through the calculation on digitalized braiding process parameters. Finally, with a knowledge of the relationship between the carrier path and yarn trajectory, a simulation tool was developed to generate fabric structures from the input braiding process parameters. The simulation results were verified by experiments and discussions were made on the effect of process parameters on the fiber architecture.

An efficient algorithm for finding all solutions of nonlinear equations using parallelogram LP test

This paper presents an efficient algorithm for finding all solutions of nonlinear equations using linear programming. This algorithm is based on a simple test (called the LP test) for nonexistence of a solution to a system of nonlinear equations in a given region. In the conventional LP test, a system of nonlinear equations is formulated as a linear programming problem by surrounding component nonlinear functions by rectangles. In the proposed algorithm, we first use rectangles, and when the nonlinearity of functions becomes weak, we switch to parallelograms. It is shown that we can use the dual simplex method throughout the algorithm by applying the variable transformation to the oblique coordinate system, by which the LP test becomes more efficient. Moreover, since polygons with proper sizes are used, the LP test becomes more powerful. By numerical examples, it is shown that the proposed algorithm is more efficient than the conventional algorithms using rectangles only or parallelograms only. We also consider the special case where component nonlinear functions are locally convex and monotone, and propose an efficient LP test algorithm using rectangles and triangles.

Buckling of thin skew isotropic plate resting on Pasternak elastic foundation using extended Kantorovich method

The extended Kantorovich method (EKM) is implemented to numerically solve the elastic buckling problem of thin skew (parallelogram) isotropic plate under in-plane loading resting on the Pasternak elastic foundation. EKM has never been applied to this problem before. Investigation of the EKM accuracy and convergence is conducted. Formulations are based on classical plate theory (CPT). Stability equations and boundary conditions terms are derived from the principle of the minimum total potential energy using the variational calculus expressed in an oblique coordinate system. The resulting two sets of ordinary differential equations are solved numerically using the Chebfun package in MATLAB software. In-plane compression and shear loads are considered along with various boundary conditions and aspect ratios. Results are compared to the analytical and numerical solutions found in the literature, and to the finite element solutions obtained using ANSYS software. The effects of the skew angle, stiffness of elastic foundation, and aspect ratio on the buckling load are also investigated. For plates with zero skew angle, i.e. rectangular plates, with various boundary conditions and aspect ratios under uniaxial and biaxial loading resting on elastic foundation, the single-term EKM is found accurate. However, more terms are needed as the skew angle gets bigger. The multi-term EKM is found accurate in the analysis of rectangular and skew plates with various boundary conditions and aspect ratios under uniaxial, biaxial, and shear loading resting on elastic foundation. Using EKM in buckling analysis of thin skew plates is found simple, accurate, and rapid to converge.

On the surface elastic-based shear buckling characteristics of functionally graded composite skew nanoplates

Abstract The prime objective of the present investigation is to predict the shear buckling characteristics of skew nanoplates made of a functionally graded material (FGM) in the presence of surface stress effect. For this purpose, the Gurtin-Murdoch surface theory of elasticity is applied to the higher-order shear deformation plate theory within the framework of the oblique coordinate system. Different types of the homogenization scheme including Reuss model, Voigt model, Mori-Tanaka model, and Hashin-Shtrikman bounds model are taken into consideration in order to extract the effective mechanical properties of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is utilized to obtain the surface elastic-based shear buckling loads of FGM skew nanoplates. It is indicated that by increasing the value of the index associated with the material property gradient, the significance of the surface stress type of size effect on the shear buckling behavior of a FGM skew nanoplate improves. Moreover, by changing the boundary conditions from simply supported ones to clamped ones, the influence of the skew angle on the surface elastic-based shear buckling load of a FGM skew nanoplate increases. Also, it is illustrated that by increasing the width to thickness ratio of a skew nanoplate, the free surface area increases which results in to enhance the effect of surface residual stress on its shear buckling characteristics.

Constructing a method for the geometrical modeling of the lame superellipses in the oblique coordinate systems

The elliptic curves possess a certain disadvantage related to that at the point of intersection with the coordinate axes the ellipses have tangents perpendicular to these axes. However, such a situation is undesirable for some practical applications of ellipses. It can be prevented by modeling the specified curves in oblique coordinates, which, in turn, are related to a certain original Cartesian coordinate system. The Lame superellipses are understood to be the curves whose equations include the exponents that differ from those inherent in regular ellipses. Variating these exponents can produce a wide range of different curves. This paper has proposed a method for the geometric modeling of superellipses in the oblique coordinate systems. The source data for modeling are the coordinates of the two points with the known angles of the tangent slope. The accepted axes of the oblique coordinate system are the straight lines drawn as follows. Through the first point, a line parallel to the tangent at the second point is built, and at the second point, a line parallel to the tangent at the first point is constructed. It has been shown that these operations could yield the desired values of tangent angles at intersection points of the superellipse with axial lines. It has been proven that the superellipse arc could be drawn through a third given point with the required angle of the tangent; that, however, would require determining the exponents in the superellipse equation by a numerical method. Such a situation occurs, for example, when designing the projected profiles of axial turbine blades. Based on the proposed method of modeling the superellipse curves, a computer code has been developed that could be used in describing the contours of components applied in the technologically complex industries