Three-dimensional Linear Equations Research Articles

A Method for Measuring the Inclination of Forgings Based on an Improved Optimization Algorithm for Fitting Ellipses

During a piercing or punching process, the center axis of a forging stock is required to be parallel to the direction of movement of the upsetting rod or piercing needle because any small tilt of the stock will cause uneven upsetting or uneven punching wall thickness. In order to detect the inclination of a stock during forging, a method for measuring the inclination of forgings based on an improved optimization algorithm for fitting ellipses is proposed in the work. Specifically, a laser rangefinder is used to measure the distances to the surface of the workpiece, which are converted into Section profile coordinates through coordinate transformation. Next, the randomly divided area and random sampling are presented for screening data points, making it possible to effectively eliminate the influence of abnormal values in the measurement points. Then, taking the point-to-ellipse orthogonal distance as the geometric model, we propose an improved particle swarm optimization (PSO) algorithm to solve the target ellipse parameters. Subsequently, we compare the fitting effect with four other contemporary advanced methods to verify the superiority of the algorithm in ellipse fitting. Finally, the three-dimensional linear equation of the center axis of the forging stock is obtained according to the center point of the long axis of the ellipse in the vertical direction, and the inclination angle of the forging stock is calculated according to the linear equation. The experimental measurement reveals that the error that occurred using the method is less than 0.05°.

Three-dimensional pulse response of curved composite sandwich panel with compressible core using non-linear higher-order theory

In this paper, the dynamic response of cylindrical sandwich panels with compressible core is obtained using the extended non-linear higher-order sandwich panel theory. It is assumed that the sandwich panel has simply supported boundary at all edges and is consisted of orthotropic face sheets and viscoelastic core layer. To describe the mechanical properties of the viscoelastic foam core, the Kelvin-Voigt linear viscoelastic model was applied. Three-dimensional linear equations of motions were used to describe the sandwich panel deformations. The effects of various parameters including the panel span, core and facing thickness, the viscous damping factor, pulse duration, and maximum pressure on the dynamic response of the sandwich cylindrical panel are studied. The results obtained from present method are compared with finite element solutions and those reported in the literature, and consequently, agreement among the results could be observed. The results shown that applied viscoelastic model has a signification effect on the panel response and reduces the magnitude of vibrations. The presented programming code (DQ) needs less computational time and computer hardware capacity and is faster than finite element solution.

REFINEMENT OF THE PARAMETERS OF A MATHEMATICAL MODEL OF QUADCOPTER DYNAMICS

Errors in the calculation of the parameters of quadcopter control models at design stage significantly change the desired aerodynamic properties of the drone and make it difficult to control its flight along the intended path. Therefore, to calculate the adequate operation modes of the blades, it becomes necessary to refine some parameters of the mathematical model of the drone as accurately as possible. This paper shows the possibility of using control parameters (rotational speed of the blades) and information received from navigation devices of the drone to refine the values of the parameters of the mathematical model of the drone. For this purpose, a mathematical model of a quadcopter is built, and the problem of refining the parameters of its dynamic model is investigated based on the information received from navigation devices and the control parameters in the initial period of its flight. From the results obtained from several consecutive measurements, a system of equations expressing a mathematical model is solved. The mean value of the corresponding solutions of the system of three-dimensional linear equations obtained at different time intervals is the refined value of the parameters.

Propagation of Waves in an Elastic Half-Space Interacting with a Viscous Liquid Layer

The problem of the propagation of acoustic waves in a layer of a compressible viscous fluid that interacts with an elastic half-space is solved using the three-dimensional linear equations of classical elasticity theory for the solid and the three-dimensional linearized Navier–Stokes equations for the compressible viscous fluid. The problem statement and approach based on the application of the general solutions of linear equations for elastic bodies and linearized equations for the fluid are employed. The dispersion equation describing the propagation of quasi-Lamb waves in the hydroelastic system is derived. Dispersion curves for normal waves in a wide frequency range are plotted. The effect of the thickness of the liquid layer on the phase velocities and attenuation coefficients of acoustic waves is analyzed. It is shown that the influence of fluid viscosity on the parameters of the wave process is associated with the properties of wave localization. Using the developed approach and obtained results, the limits of the applicability of the models of wave processes based on the model of an ideal fluid are established. The numerical results are presented in the form of graphs and analyzed.

Dispersion Properties of Lamb Waves in an Elastic Layer–Ideal Liquid Half-Space System

The problem of the propagation of normal waves in an elastic layer interacting with a compressible ideal liquid half-space is stated and solved using the three-dimensional linear equations of classical elasticity for the solid and the three-dimensional linearized Euler equations for the compressible ideal fluid. The dispersion curves are plotted for quasi-Lamb modes over a wide frequency range. The effect of the ideal compressible fluid and the elastic layer thickness on the dispersion of the phase velocities of quasi-Lamb modes in hydroelastic waveguides is analyzed. The localization properties of these modes in hydroelastic waveguides are studied. The numerical results are presented in the form of plots and analyzed.

Three-dimensional free vibration of multi-layered piezoelectric plates through approximate and exact analyses

This article presents two models that have the aim of analysing three-dimensional freely vibrating plates made of an arbitrary combination of structural and/or piezoelectric layers. The first model is derived from a displacement-based variational statement, and it investigates the possibility of approaching exact three-dimensional results at any degree of accuracy. This model has been developed as if the plates were virtually made of a single layer, and it is herein referred to as the approximate analysis model. The second model is based on solving the set of three-dimensional linear equations coupling the relevant mechanical and electric quantities, and it therefore provides exact results. The latter model is derived from the transfer-matrix technique which, having shown numerical instability in the multiphysics problem being dealt with, was then successfully modified to provide exact and reliable results. Excellent agreement has been obtained between the models, and this shows how the exact approach here designed is stably able to overcome ill conditioning problems, while the first model, having been validated by the exact results, could be applied to effectively investigate multiphysics problems for general boundary conditions, and for cross- and/or angle-ply laminates, at any level of required accuracy.

Wave propagation in axially polarized piezoelectric hollow cylinders of sector cross section

In this paper, the problem of electroelastic waves propagating in piezoelectric hollow cylinders of sector cross section is studied for the case when the boundary surfaces of sector cut are covered by non-extensible membranes. The three-dimensional linear equations of motion for the piezoelectric cylinder are analytically integrated and different boundary conditions on the cylindrical surfaces yield frequency equations, which relate the frequencies of elastic waves to their wavenumbers. Numerical results for waveguides with various boundary conditions are presented to illustrate the approach. Analysis of the dispersion spectra is carried out, and cutoff frequencies are obtained and characterized; mode asymptotic behavior and amplitude distributions of wave characteristics are analyzed. The main effects of their transformation by variation of the angular measure and the ratio of inner and outer radii are discussed. The results obtained are in good agreement with the results for the special case of a hollow semicircular cylinder.

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.

Tidal Hydrodynamics in Golfo Nuevo, Argentina, and the Adjacent Continental Shelf

Abstract A numerical, non-linear, barotropic, two-dimensional tidal model was implemented and used to simulate the M2, S2, N2, K1, O1 and M4 constituents in Golfo Nuevo (Argentina) and the adjacent continental shelf. The model includes the calculation of the vertical distribution of flow velocities using the linear three-dimensional equations. The tidal circulation was modeled by forcing at the open boundaries. Field measurements have shown that the hydrodynamics of Golfo Nuevo is basically of a semidiurnal character. Because of the difference in magnitude between M2 and the other constituents the M2 tide induces the strongest currents. These are strong on the shelf, with speeds exceeding 50 cm/s at maximum flood and ebb, but close to the coast become parallel to it and somewhat weaker, except at Golfo Nuevo mouth and near Punta Delgada. The direction of maximum current is almost constant in the vertical at the adjacent continental shelf, whereas in the middle of Golfo Nuevo the axes of the ellipse at the...

Frequency-temperature behavior of spurious vibrations of rectangular AT-cut quartz plates

We evaluate the frequency-temperature behavior of spurious modes in a rectangular AT-cut quartz plate resonator based on three-dimensional linear equations. The elastic constants and three geometrical dimensions of the resonator are defined in terms of cubic polynomials of a temperature change. Assuming that the resonator holds its rectangular plate shape irrespective of temperature, we can determine the relationship between frequency and the dimensions of the resonator for a given temperature using the previous technique. We compare the calculated results with our own experimental data, and show that agreement between the calculated and observed data is excellent.

On transition due to three-dimensional disturbances in plane Poiseuille flow

The purpose of the present study is to characterize the process of laminar–turbulent transition at Reynolds numbers which are subcritical from the two-dimensional linear point of view. The development of a point-like disturbance was studied in an air flow channel with hot-wire anemometry at a Reynolds number of 1600. Localized disturbances were triggered at one of the walls and their development followed downstream by traversing the hot-wire probe in the streamwise direction over a distance of 90 half channel heights, as well as in the normal and spanwise directions. The disturbance evolved into elongated streaky structures with strong spanwise shear (i.e. normal vorticity) which grew in amplitude and streamwise extension and thereafter either decayed or gave rise to a turbulent spot. The results indicate that the mechanism underlying the initial growth is a linear one resulting from the coupling between the normal velocity and the normal vorticity, as described by the three-dimensional linear equations. The nonlinear development of the structure leads to the formation of intense normal shear layers and the appearance of oscillations and ‘spikes’, which multiply and form the rear or a turbulent spot.

Clinical determination of the linear equation for the subtalar joint axis

The authors present a methodology to measure the frontal plane angular and linear displacement and the transverse plane angular displacement of subtalar joint movement. This method is combined with a modification of the Kirby method for determining the transverse plane projection of the subtalar joint axis onto the plantar foot. A mathematical model is then used to construct the subtalar joint axis into a three-dimensional linear equation. Data are obtained from an in vivo series of 62 feet that indicates that within acceptable clinical errors of measurement the subtalar joint is a ginglymus type of joint that moves around a single fixed axis. Results also indicate that the subtalar joint axis is more superior and lateral to the neutral foot than any previous studies on cadaver feet have shown. Finally, the authors show that once the subtalar joint axis can be accurately located, the torque on the joint axis produced by ground reactive forces and muscular forces can be computed.

Clinical determination of the linear equation for the subtalar joint axis.

The authors present a methodology to measure the frontal plane angular and linear displacement and the transverse plane angular displacement of subtalar joint movement. This method is combined with a modification of the Kirby method for determining the transverse plane projection of the subtalar joint axis onto the plantar foot. A mathematical model is then used to construct the subtalar joint axis into a three-dimensional linear equation. Data are obtained from an in vivo series of 62 feet that indicates that within acceptable clinical errors of measurement the subtalar joint is a ginglymus type of joint that moves around a single fixed axis. Results also indicate that the subtalar joint axis is more superior and lateral to the neutral foot than any previous studies on cadaver feet have shown. Finally, the authors show that once the subtalar joint axis can be accurately located, the torque on the joint axis produced by ground reactive forces and muscular forces can be computed.

Reality of chaos in four-dimensional hysteretic circuits

This paper gives rigorous evidence for chaos in a certain class of four-dimensional hysteretic circuits. The circuit dynamics are described by two symmetric three-dimensional linear equations which are connected to each other by hysteresis switchings. The author transforms the circuit equation into the Jordan form and derives the two-dimensional return map T. Then the author proves a sufficient condition for T(D'/sub T/) contained in/implied by D'/sub T/ and mod DT mod > 1 on D'/sub T/, where D'/sub T/ is some subset in the domain of T and DT is the Jacobian. It implies that T exhibits area-expanding chaotic attractors. >

Comparison of 2-D and 3-D models of the steady wind-driven circulation in shallow waters

The classical theory of the steady wind-driven circulation in shallow waters described by the linear three-dimensional equations is generalized to include arbitrary variations of the vertical eddy viscosity. The stream function equation is compared to the corresponding equation of a two-dimensional model. The influence, on the solution, of various eddy viscosity profiles and/or bottom boundary conditions is discussed. From the comparison of the equations, a parameterization of the bottom stress in terms of transport and wind stress is derived such that the two-dimensional model can yield results identical to those of the three-dimensional model.

Frequency-temperature behavior of thickness vibrations of doubly rotated quartz plates affected by plate dimensions and orientations

Three-dimensional linear equations of motion for small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz are obtained. The material properties of quartz, such as the elastic stiffnesses and thermal expansion coefficients, are assumed temperature dependent and expressible by third-degree polynomials in temperature change. From the solutions of third-order perturbations of these equations for the thickness resonances of infinite quartz plates, six values of the effective third temperature derivatives of elastic stiffnesses C̃(3)pq are calculated by the use of the measured temperature coefficients of frequency by Bechmann, Ballato, and Lukaszek [Proc. IRE 50, 1812 (1962)] for various doubly rotated cuts and the values of the first temperature derivatives C(1)pq and the effective second temperature derivatives C̃(2)pq obtained in a previous study. An infinite system of two-dimensional equations of motion is derived by Mindlin’s method of power-series expansion for crystal plates subject to a steady, uniform temperature change. Four equations, governing the coupled thickness-shear, thickness-twist, thickness-stretch, and flexural vibrations, are extracted from the infinite set and employed to study the frequency-temperature behavior of thickness vibrations of finite SC-cut quartz plates with a pair of free edges. Changes in the thickness-shear resonance frequencies as a function of temperature are predicted and plotted for various values of orientation angles θ and φ, and length-to-thickness ratio a/b. Effects on the frequency-temperature behavior of the plates due to changes in the values of θ, φ, and a/b are observed and discussed.

Approximation of the three-dimensional linear water wave equations by a set of two-dimensional equations

It is shown how the linear three-dimensional equations for surface water waves over a bottom of slowly varying depth may be approximated by a set of two-dimensional equations, the first of which is a simple extension of the equation derived by Berkhoff. Although the equation coefficients and the set structure are not simple, however this kind of approach does provide a new tool for the numerical solution of certain three-dimensional problems, for which, at present, other alternative approaches do not exist.

Chromatography of nucleoside phosphates on deaespheron 1000 with ethanol-ammonium formate as eluent

Summary The chromatography of 5′-mono-, di- and triphosphates of ribonucleosides on DEAE-Spheron 1000 was studied using various ammonium formate-ethanol mixtures as eluent with pH from 3.5 to 5. The results of log k′ measurements have been described by a second-degree polynomial with three variables, and the reduced height of a theoretical plate function with a three-dimensional linear equation. The influence of all variables on the chromatographic behaviour of this system is described, particularly using the first derivatives of log k′ with respect to all variables. An example of a preparative separation is presented.

Exact equations for analysing thickness-twist trapped-energy modes in monolithic filters

Thickness-twist vibrations with energy trapping in a monolithic filter consisting of an infinite piezoceramic plate withN infinitesimally thin electrodes evaporated on to each face are analysed. By applying the Fourier transform technique, the linear three-dimensional equations for a piezoceramic plate are reduced to integral equations for the charge distributions on the electrodes. An approach for solving these equations and numerical results for a dual are given.