Brachistochrone for a rolling cylinder

The motion of a heavy homogeneous cylinder is considered as a no-slip rolling along the desired curve. We obtain a functional in the form of the total time of the cylinder rolling and solve the corresponding variational problem of minimizing this functional. We obtain an algebraic equation for the directional line of steepest descent, brachistochrone, in parametric form. We use the equation of motion of the cylinder with constraint reaction to determine the conditions of implementation of its pure rolling without separation and slip with respect to the brachistochrone.     

Cycloidal pendulum with a rolling cylinder

Free vibrations of a heavy homogeneous cylinder rolling in a cylindrical cavity whose directing curve is a brachistochrone are considered. The equation of motion of the cylinder is derived and the circular frequency of free vibrations of the cylinder center of mass is determined. An analogy between the cycloidal pendulum with a rolling cylinder and the classical cycloidal pendulum in the form of a material point is obtained.     

Brachistochrone with Coulomb friction as the solution of an isoperimetrical variational problem

The problem of finding the plane supporting curve (brachistochrone) along which a heavy particle released from rest at the given starting point slides with dry (Coulomb) friction and reaches the given destination point in least time, is stated as a variational isoperimetrical problem. The finite parametric equations of the extremal curve are obtained. This curve is the sought-for brachistochrone if the solution of the problem exists. Several numerical examples are given.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

A qualitative analysis of the brachistochrone problem with dry friction and maximizing the horizontal range

The problem of maximizing the horizontal coordinate of a point moving in a vertical plane under the action of gravity and dry friction and the corresponding brachistochrone problem are considered. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. A qualitative analysis of the trajectories of this system is carried out, their typical features are found and illustrated by numerical solving of the boundary value problem. It is shown that the normal component of the support reaction should be positive when moving along the optimal curve. The optimality of the found extremals is discussed.     

Vibrations of a Semiinfinite,Orthotropic, Cylindrical Shell of Open Profile

Natural vibrations localized at the free edge of a semiinfinite, elastic, orthotropic, circular cylindrical shell of open profile are studied. The cylinder is hinged along the bounding generatrices. Dispersion equations are derived from the classical equations describing the dynamic equilibrium for orthotropic cylindrical shells. It is established that these dispersion equations and the dispersion equations for a semiinfinite orthotropic plate strip are in an asymptotic relationship. A procedure for analysis of the possible types of vibrations at the free edge of the cylinder is described. Approximate values of the dimensionless natural frequency and damping factor are determined for shells of different radii     

Dynamics of vibration isolation system with a quasi-isochronous roller shock absorber

The low-frequency vibrations of a vibration isolation system of rigid bodies (roller shock absorber and carrying body) under external harmonic loading are considered. The working surface of the absorber has the form of a brachistochrone. The equations describing the slip-free motion of the absorber over the hinged roller and the motion of the carrying body are derived. A graphical method for optimizing the parameters of the roller absorber as a component of the vibration isolation system is proposed     

Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid

The wave propagation in an infinite, transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid is studied using the Fourier expansion collocation method, within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equation of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion along the radial, circumferential and axial directions. The frequency equations of longitudinal and flexural (symmetric and antisymmetric) modes are analyzed numerically for an elliptic and cardioidal cross-sectional transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid. The computed non-dimensional wavenumbers are presented in the form of dispersion curves for the material zinc. The general theory can be used to study any kind of cylinder with proper geometric relations.     

Efficiency of a vibroprotection system with an isochronous roller damper

Low-frequency vibrations of a vibroprotection “roller damper-movable bearing body” system of rigid bodies under the action of an external harmonic excitation are considered. The working surface of the damper working body is formed by a brachistochrone. The dynamic equations of common no-slip motion of the damper working body on a hinged roller and of the bearing body are formulated. The roller damper tuning parameters are determined.     

Conjugate free convection over a vertical slender hollow cylinder embedded in a porous medium

A numerical study of the steady conjugate free convection over a vertical slender, hollow circular cylinder with the inner surface at a constant temperature and embedded in a porous medium is reported. The governing boundary layer equations for the fluid-saturated porous medium over the cylinder along with the one-dimensional heat conduction equation for the cylinder are cast into dimensionless form, by using a non-similarity transformation. The resulting non-similarity equations with their corresponding boundary conditions are solved by using the Keller box method. Emphasis is placed on the effects caused by the wall conduction parameter, p, and calculations have covered a wide range of this parameter. Heat transfer results including the temperature profiles, the interface temperature profiles and the local Nusselt number are presented. Received on 17 November 1997     

Elastic field of a composite cylinder with a spatially varying dynamic eigenstrain

The analytical solutions of displacements and stresses for an eigenstrain problem in a composite bi-layered coaxial cylinder are presented in this article. The inner cylinder is assumed to undergo a dynamic, spatially varying eigenstrain. The spatial distribution of the eigenstrian is taken to be a quadratic polynomial with arbitrary coefficients along the radial direction. Furthermore, the eigenstrain is assumed to be harmonically time-dependent. Elasticity equations are constructed to directly solve the problem. The effect of spatial distribution, as well as the angular frequency of the eigenstrain on the elastic response of the composite cylinder has been illustrated graphically.     

An improved HSFR method for natural vibration analysis of an immersed cylinder pile with a tip mass

Immersed cylinder piles are usually modelled as immersed cantilever cylinder columns carrying a tip mass and rotary moment of inertia. In this paper, the equations of motion of an immersed cylinder pile along transversal modes of vibration are developed. Compressibility of water and structural damping are included in the formulation. Natural frequencies of the immersed pile are obtained from the developed equations using harmonic sweep frequency response analyses. The proposed method is applied to numerical examples, and the results obtained are shown satisfactory when compared to other numerical solutions in the literature, or to finite element solutions and experimental data.     

具有黏滞阻力时的最速降线

针对实际应用中存在黏滞阻力的最速降线的问题, 首先推导出适于此类问题的解除约束的 广义变分原理, 它适用于具有摩擦阻尼和多自由度系统优化的问题. 得到描述有黏滞阻力情 况下最速降线相关函数的微分方程, 它在黏滞阻力为零时即退化为滚轮线. 利用MATLAB数值 计算给出了最速降线受黏滞阻力的影响: 在黏滞阻力系数较小时最速降线趋于变凹, 当阻力 系数增大到一定值之后最速降线趋于平缓, 当阻力系数很大时最速降线趋于直线.     

Vortex shedding from a square cylinder in presence of a moving wall

A numerical study on the flow past a square cylinder placed parallel to a wall, which is moving at the speed of the far field has been made. Flow has been investigated in the laminar Reynolds number (based on the cylinder length) range. We have studied the flow field for different values of the cylinder to wall separation length. The governing unsteady Navier–Stokes equations are discretized through the finite volume method on a staggered grid system. A SIMPLE type of algorithm has been used to compute the discretized equations iteratively. A shear layer of negative vortex generates along the surface of the wall, which influences the vortex shedding behind the cylinder. The flow‐field is distinct from the flow in presence of a stationary wall. An alternate vortex shedding occurs for all values of gap height in the unsteady regime of the flow. The strong positive vortex pushes the negative vortex upwards in the wake. The gap flow in the undersurface of the cylinder is strong and the velocity profile overshoots. The cylinder experiences a downward force for certain values of the Reynolds number and gap height. The drag and lift are higher at lower values of the Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic extension and twist of a non-linear elastic tube

The propagation of waves in a non-linear cylindrical elastic membrane is considered when one end is fixed and the other is subjected to a dynamic extension and twist. The governing equations are derived for a hyperelastic material with a general strain energy function. In order to obtain specific results the equations are specialised to deal with neo-Hookian materials and in this case we show that there are three real wave speeds in each direction along the cylinder. Numerical results are given and a limiting case considered which provides a check on these results.     

Heat transfer on a cylinder in accelerated slip flow

The axisymmetric laminar boundary layer flow along the entire length of a semi-infinite stationary cylinder under an accelerated free-stream is investigated. Considering flow at reduced dimensions, the boundary layer equations are developed with the conventional no-slip boundary condition for tangential velocity and temperature replaced by a linear slip-jump boundary condition. Asymptotic series solutions are obtained for the heat transfer coefficient in terms of the Nusselt number. These solutions correspond to prescribed values of the momentum and temperature slip coefficients and the index of acceleration. Heat transfer at both small and large axial distances is determined in the form of series solutions; whereas at intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the heat transfer along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid

The free vibration of an arbitrarily thick orthotropic piezoelectric hollow cylinder with a functionally graded property along the thickness direction and filled with a non-viscous compressible fluid medium is investigated. The analysis is directly based on the three-dimensional exact equations of piezoelasticity using the so-called state space formulations. The original functionally graded shell is approximated by a laminate model, of which the solution will gradually approach the exact one when the number of layers increases. The effect of internal fluid can be taken into consideration by imposing a relation between the fluid pressure and the radial displacement at the interface. Analytical frequency equations are derived for different electrical boundary conditions at two cylindrical surfaces. As particular cases, free vibration of multi-layered piezoelectric hollow cylinder and wave propagation in infinite homogeneous cylinder are studied. Numerical comparison with available results is made and dispersion curves predicted from the present three-dimensional analysis are given. Numerical examples are further performed to investigate the effects of various parameters on the natural frequencies.     

Detached flows past a cylinder with a flat end

In the framework of the Navier-Stokes equations, a numerical solution is found to the problem of longitudinal (axisymmetric) flow of a viscous perfect heat conducting gas past a cylinder with a flat front end at subsonic and transonic velocities of the oncoming flow. The flow in the neighborhood of the corner of the cylinder is investigated and a study made of the occurrence and development of detached flow along the side of the cylinder. Tie heat fluxes to the surface of the cylinder are determined, and the influence of the temperature of the wall of the cylinder on the development of the separation is found. The investigations yield the critical Reynolds number Re₀ for the occurrence of a detached flow as a function of the Mach number of the oncoming flow for the investigated bodies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–10, March–April, 1979.I am grateful to Yu. A. Dem'yanov for discussing the work and for a number of valuable comments.     

Numerical study of evaporation by mixed convection of a binary liquid film flowing down the wall of two coaxial cylinders

Evaporation by mixed convection of a binary liquid film flowing down the external wall of a vertical cylinder has been investigated numerically. Two cases were considered: one where the cylinder wall is soaked with a liquid, and another where a liquid film flows along this wall. Heat, mass and momentum transfer in the liquid film and the vapor phase are modelled by mixed convection equations. In order to locate the liquid–vapor interface, a suitable coordinate transformation is carried out with suitable variables. The discretization of the dimensionless equations by an implicit difference scheme leads to a system of algebraic equations, which are solved by using Gauss algorithm for the momentum conservation equations and Thomas algorithm for the energy and diffusion conservation equations. The film thickness is calculated by the Newtons method. Results show, in particular, that the film thickness cannot be neglected and that the latent heat transfers are increasingly significant as the liquid film components become more volatile.     

Three-dimensional instability of an oscillating viscous flow past a circular cylinder

IntroductionTheunsteadyflowpastacircularcylinderhasreceivedagreatdealofattentionowingmainlytoitstheoreticalandpracticalsignificance .Theflowgeneratedbytheoscillationofthecylinder,oroscillatingflowsaroundthecylinder,canbecharacterizedbytwoparameters.OneistheKeulegan_Carpenternumber,definedasKC =UmT/D ,andtheotheristheReynoldsnumberRe=UmD/ν,orafrequencyparameter,definedasβ=D2 / (νT) =Re/KC) ,whichisoftenusedtoreplacetheReynoldsnumberasthesecondparameter.Here,Umisthemaximumvelocityofth…