Three-dimensional Equations Of Motion Research Articles

Flight Path Optimization of a Hang-Glider in a Thermal Updraft

Thermal upwinds induced through mountains and valleys allow flights of hang-gliders over a long time or over a large range. In reality these flights include spiralling trajectories in thermal updrafts. We present new numerical results for several new thermal updraft models included into the threedimensional equations of motion.

Geometrically-induced singularities in functionally graded magneto-electro-elastic wedges

Asymptotic solutions are proposed for geometrically-induced magneto-electro-elastic (MEE) singularities at the vertex of a rectilinearly polarized wedge that is made of functionally graded magneto-electro-elastic (FGMEE) materials. The material properties are assumed to vary along the thickness of the wedge, and the direction of polarization may not be parallel or perpendicular to the thickness direction. An eigenfunction expansion approach with the power series solution technique and domain decomposition is adopted to establish the asymptotic solutions by directly solving the three-dimensional equations of motion and Maxwell׳s equations in terms of mechanical displacement components and electric and magnetic potentials. Since the direction of polarization can be arbitrary in space, the in-plane components of displacement, electric and magnetic fields are generally coupled with the out-of-plane components. The solutions are presented here for the first time. The correctness of the proposed solutions is confirmed by comparing the orders of MEE singularities with the published results for wedges under anti-plane deformation and in-plane electric and magnetic fields. The proposed solutions are further employed to investigate the effects of the direction of polarization, vertex angle, boundary conditions and material-property gradient index on the MEE singularities in BaTiO3–CoFe2O4 wedges.

Wave propagation in a rectangular bar, exposed to the impact tangential forces

In modern technology, there are more and more cases of action of impact loads on work items of buildings, machines and constructions, therefore, the strength calculations of these items under the dynamic effects become important. To solve the problems, arising from the dynamic effects, it is necessary to use the continuum mechanics methods and, in particular in many cases, the methods of the dynamic theory of elasticity. Analytical studies allow to find the exact problem solutions, which is very important, because the exact solutions allow to estimate the main features of the solution in general - the nature and the extent of influence of various set parameters on it. On the other hand, exact solutions are always reference and needed, in particular, for developing numerical methods for more complex cases. This work is a continuation of the works of N. Rassoulova, G. Shamilova, dedicated to studying the propagation of unsteady waves in the prisms of rectangular cross section. Approach to solving this problem differs markedly from all previous issues of the dynamics of rectangular prisms, which mainly investigated their dispersion characteristics. This paper deals with studying the process of propagation of unsteady waves in semi-infinite rectangular bars, exposed to impact shear forces on the face platform. System of exact three-dimensional motion equations of an isotropic elastic body is used. Applying a peculiar integration method, previously developed by the authors of this paper, exact analytic solutions to the posed problem for the final time value are found. The results can be implemented in the production in designing special constructions, where there are impulsive effects on them.

Optimization Analysis on Gyroscope’s Three Dimensions Complicated Motion and Lash-Force

The complicated motion of the gyroscope includes the rotation, the precession, the nutation and the sliding motion which are coupled, and it is affected by the lash-force by the whip. Taking the gyroscope which is moving stably as the research subject, according to the motion characteristics of the gyro’s rotation and precession, the gyro is assumed to be the two-force bar, and the coupled relationship of the gyroscope is analyzed quantitatively. The three-dimensional complicated motion in the different categories is analyzed. The nonlinear differential equations in the different categories is derived and linearized by the minor variables method, and then the three-dimensional motion equations on the rotation, precession, nutation and slipping of the gyroscope in the different categories is established. Take the big gym gyroscope as the example, the influence of the force, time and position of the whip on the motion state of the gyroscope is analyzed in detail, and provides the basis for the design of the gyroscope.

Dynamic response of towed line array

Three-dimensional motion equations with consideration of the bending and torsional effects of a marine towed cable system under large elastic deformation conditions are formulated using the lumped parameter model. The lumped mass model is used to simulate a circular maneuver of a towed horizontal array that was first presented by Ablow. The results of this paper are in a good agreement with those obtained by finite difference schemes, and the minimum depth is closer to the experimental result. Although the calculations take more computation time, the lumped mass model is a good method, which can also be used to solve problems of towed line array, especially multi-branched towed line array, because of its flexibility.

Three-dimensional dynamics of long pipes towed underwater. Part 1: The equations of motion

This paper is the first in a two-part study of the dynamics of long pipes towed underwater. In this part, Part 1, the three-dimensional equations of motion are derived for the dynamics of the system. Inviscid forces are modelled by adapting Lighthill's slender-body work to the problem at hand, while the viscous forces are modelled by elaboration of Taylor's expressions. The effect of cross-currents is also considered in the equations of motion. The pipeline is flexibly connected to the towing and trailing vessels via cables at its upstream and downstream ends. The cables are modelled as linear springs and the body is assumed to be of null buoyancy. The linearized equations of motion and boundary conditions are solved in Part 2 by using a finite difference technique, and the results are discussed therein.

Three-dimensional Impact Angle Control Guidance Law for Missiles Using Dual Sliding Surfaces

Guidance law considering terminal impact angle constraints for missiles is proposed based on sliding mode control scheme with dual sliding surfaces. The merits of this method are that the angular constraints are satisfied prior to the impact and is maintained until interception with stable fashion. Therefore, excessive acceleration command at homing phase is not required. Based on the nonlinear three-dimensional equations of motion for the missile and stationary target, two acceleration commands, pitch and yaw channel command, are generated with the proposed guidance law. Dual sliding surface based guidance law without terminal constraints is also developed for the comparison purpose. Consideration on singularities are made and singular avoidance rule is inherited. To verify the effectiveness of the proposed scheme, 6-DOF numerical simulation is performed. Simulation results demonstrate that the guidance law has good performance in spite of the terminal constraints, nonlinearities, and cross-coupling effect of the engagement environment.

A variant of the classical von Karman flow for a Jeffrey fluid

An attempt is made to study the classical Von Karman flow problem for Jeffrey fluid by using a generalized non-similarity transformation. This approach is different from Von Karman's evolution of the flow. Here, physical quantities are allowed to develop non-axisymmetrically. The three-dimensional equations of motion for Jeffrey fluid are treated analytically yielding the derivation of a closed form solution for the velocity field. Exact expressions for the vorticity components, shear stresses and boundary layer thickness are also presented. Mathematics Subject Classification: 76A05

Invariant and approximately invariant solutions of non-linear internal gravity waves forming a column of stratified fluid affected by the Earth's rotation

The exact solutions of three-dimensional equations of motion for internal gravity waves in cylindrical coordinates in unbounded media are found by means of approximate transformation groups of equations with a small parameter. Introduction of the small parameter has been motivated by justifying the analogy of the Kelvin hypothesis on vanishing the component of the velocity ur normal to wall for the rectilinear motion. In the present case of the cylindrical domain, ur is non-zero and achieves its maximum in the interior, which also agrees with analytical predictions in [6]. However, as linear analysis shows, ur can be considered to be small in the limiting case when the aspect ratio σ=H/r0 is small, in which H and r0 are the basin's depth and radius respectively. As a particular applications to the ocean and atmospheric modeling, in terms of linear modeling, the time series of the energy density were visualized as spinning patterns that appear to be rotating in an anticlockwise sense when looking from above the North Pole. Such spinning patterns were compared with the flow around a low-pressure area that is usually being linked with a modeling of hurricanes. In terms of zeroth-order approximate transformations, the invariant solutions were visualized as funnels having something in common with the geometric structure of oceanic whirlpools.

Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method (RPIM)

This paper considers a functionally graded (FG) shell using a meshless radial point interpolation method (RPIM). The material is assumed to be bidirectional FG, where the variation is present in both the radial and the axial directions. Based on the three-dimensional equations of motion, the frequency equations are stated using RPIM. Numerical results are presented for a thick shell for various boundary conditions. These results illustrate the influence from the material variation concerning eigenfrequencies and eigenmodes. In addition, the study shows that the RPIM is an efficient method to solve dynamical shell problems.

Modeling of the dynamic contact in stick-slip microdrives using the method of reduction of dimensionality

In this work we present a new method to describe movements of stick-slip microdrives. On the microscopic scale we model the contact between actuator and slider as a dynamic tangential contact using the method of reduction of dimensionality. On the macroscopic scale simple one- and three-dimensional equations of motion are derived. An algorithm to solve these equations will be introduced. The results of the simulation will be compared qualitatively and quantitatively to experimental investigations. Even for the simplest assumed model it proves that experimental and numerical values correlate excellently.

The 2011 Koiter Lecture: The Simple Logic of Classical Nonlinear Thermodynamic Shell Theory

A classical nonlinear thermodynamic theory of elastic shells is derived by specializing the three-dimensional equations of motion and the second law of thermodynamics to a very general, shell-like body. No assumptions are made on how unknowns vary through the thickness. Extensional and bending strains are derived from the equations of motion via the principle of virtual power. The Coleman-Noll procedure plus the second law applied to an assumed form of the first law leads to constitutive relations plus reduced forms of the first and second laws. To avoid potential ill conditioning, a Legendre-Fenchel transformation is used to define a mixed-energy density, the logical place to impose the constitutive Kirchhoff hypothesis, if desired, because such an energy density rests, ultimately, on experiments. The Ladevèze-Pécastaings treatment of three-dimensional edge effects to obtain accurate two-dimensional solutions is discussed.

RETRACTED: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a porous rotating disk flow with Hall current

Abstract The purpose of the present work is to obtain explicit analytical solutions for the flow of a viscous hydromagnetic fluid due to the rotation of an infinite disk in the presence of an axial uniform steady magnetic field with the inclusion of Hall current effects and porosity. A treatment of three-dimensional equations of motion results in exact formulas for the angular velocity components, current density field as well as the wall shear stresses. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude, approaching their hydrodynamic value in the limit of large Hall numbers. Moreover, the injection-like impact of the Hall current is clarified. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the convection, dissipation function and Joule heating. According to Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though it increases by the presence of magnetic field, the increase is slowed down by the Hall effect eventually reaching its hydrodynamic limit.

Effects of Partial Slip on the Analytic Heat and Mass Transfer for the Incompressible Viscous Fluid of a Porous Rotating Disk Flow

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a porous disk rotating with a constant angular speed about its axis. The recent study (Turkyilmazoglu, 2009, “Exact Solutions for the Incompressible Viscous Fluid of a Porous Rotating Disk Flow,” Int. J. Non-Linear Mech., 44, pp. 352–357) is extended to account for the effects of partial flow slip and temperature jump imposed on the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions for the flow and temperature fields. Explicit expressions representing the flow properties influenced by the slip as well as a uniform suction and injection are extracted, including the velocity, vorticity and temperature fields, shear stresses, flow and thermal layer thicknesses, and Nusselt number. The effects of variation in the slip parameters are better visualized from the formulae obtained.

Three-dimensional simulation of wave-induced circulation: Comparison of three radiation stress formulations

[1] A three-dimensional current-wave modeling system, Curvilinear-grid Hydrodynamics 3D (CH3D)-Simulating Waves Nearshore (SWAN), has been used to simulate wave-induced circulation and compare the performances of three radiation stress (RS) formulations: two depth-dependent formulations (M08 by Mellor (2008) and X04 by Xia et al. (2004)) and one depth-independent formulation (LHS by Longuet-Higgins and Stewart (1964)). While all are based on linear wave theory, LHS uses the vertically integrated equations of motion, and M08 and X04 consider the three-dimensional equations of motion. Results of CH3D-SWAN with three RS formulations are compared with steady state wave setup, observed data in an undertow experiment by Ting and Kirby (1994) (TK94), and observed data in a laboratory fringing reef. All three RS formulations reproduce the analytical solution of wave setup very well. Simulated wave-induced currents and turbulence for TK94 are the best when M08 is used and worst when X04 is used, apparently due to the errors in the X04 formulation. All three RS formulations give good simulation of wave setup in the fringing reef. Wave-induced currents in the fringing reef simulated by the three RS formulations are quite different: M08 produces a single large clockwise gyre in the x-z plane, LHS produces a weaker gyre, and X04 produces a clockwise gyre plus a counterclockwise gyre inside the surf zone. Using the CH3D-Storm Surge Modeling System and M08, storm surge and currents in the Outer Banks and Chesapeake Bay during Hurricane Isabel are simulated. Compared to the earlier simulation obtained with the LHS, M08 produces similar storm surge but slightly improved the wave-induced currents.

RETRACTED: Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating-disk flow with Hall current

The focus of the present study is to obtain exact solutions for the flow of a viscous hydromagnetic fluid due to the rotation of an infinite disk in the presence of an axial uniform steady magnetic field with the inclusion of Hall current effect. In place of the traditional von Karman's axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here, whose governing equations allow an exact solution to develop bounded everywhere in the normal direction to the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman's conducting flow. Making use of this solution, analytical formulas for the angular velocity components, for the current density field as well as for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude, approaching their hydrodynamic value in the limit of large Hall numbers. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation function. According to the Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though it increases by the presence of magnetic field, the increase is slowed down by the Hall effect eventually reaching its hydrodynamic limit.

Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass

The three-dimensional (3-D) dynamics of a fluid-conveying cantilevered pipe fitted with an end-mass and additional intra-span spring-support is investigated in this paper, both theoretically and experimentally. The main objective is to examine how the dynamics of a cantilevered pipe with additional spring-support is modified by the presence of a small mass attached at the free end. In the theoretical study, the nonlinear three-dimensional equations of motion are discretized via Galerkin's method, and the resulting equations are solved by a finite difference method. For the cases studied, the system was found to lose stability by planar flutter; as the flow velocity is increased beyond that point, a sequence of higher order bifurcations ensue, involving 2-D and 3-D periodic and quasiperiodic motions, as well as chaotic ones. In the experiments, performed with elastomer pipes and water flow, similarly 2-D and 3-D periodic, quasiperiodic and chaotic oscillations were observed. Theory and experiments have been shown to be in good qualitative and quantitative agreement. The experimental behaviour is illustrated by video clips (electronic annexes). Moreover, the effects of (i) small stiffness imperfections and (ii) excitation by a point-force are explored theoretically in a preliminary way.

Exact Solutions for the Incompressible Viscous Fluid of a Rotating Disk Flow

The present paper is devoted to a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a disk rotating with a constant angular speed. In place of the traditional von Karman's axisymmetric evolution of the flow, the rotational non-axisymmetric stationary flow is taken into account here. As a consequence, the governing equations allow an exact solution to develop over the disk, which is influenced by a fixed point on the disk and also bounded everywhere in the normal direction to the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman's flow. Making use of this solution, analytical formulas for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. Analytic calculations show that for the specific flow the thicknesses take the same value. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. According to the Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, which is proportional to product of the thermal conduction, Reynolds number and temperature difference. Key Words: Exact Solution; Rotating-Disk Flow; Shear Stresses; Heat Transfer

Natural frequency analysis of 2D-FGM thick hollow cylinder based on three-dimensional elasticity equations

In this paper, natural frequencies characteristics of a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) based on three-dimensional equations of elasticity is considered. The axisymmetric conditions are assumed for the 2D-FGM cylinder. The material properties of the cylinder are varied in the radial and axial directions with power law functions. Effects of volume fraction distribution and FGM configuration on the natural frequencies of a simply supported cylinder are analyzed. Also, the effects of length and thickness of the cylinder are considered for different material distribution profiles. Three-dimensional equations of motion are used and the eigen value problem is developed based on direct variational method. Finite element method with graded material characteristics within each element of the structure is used for solution. The study shows that the 2D-FGM cylinder exhibit interesting frequency characteristics when the constituent volume fractions and its configuration are varied.

Nonlinear hydrodynamic response of marine cable -body system under random dynamic excitations

Three-dimensional motion equations of a marine Tethered Remotely Operated Vehicle (TROV) system with large elastic deformation and snap loads are formulated using the lumped parameter model. This formulation is simple and quite general in solving a wide range of offshore-related cable and pipeline problems using only three degrees of translational freedom for each computational nodes. The model is convenient in dealing with non-uniform features of the cable with low or even zero tension. The snap load that would damaged the sling is predicted by computing the dynamic tension and the instant configuration of the cable. This investigation may help avoid resonance, extend the cable life, reduce the test expense and avoid operating accidents.