The rotating elastic-plastic hollow shaft heated at its inner surface

Based on Tresca's yield criterion and the flow rule associated with it, the behaviour of a rotating linearly strain-hardening elastic-plastic hollow shaft heated at its inner surface is studied. Moreover, the residual stresses at stand-still are depicted. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic-plastic stress distribution in a rotating solid shaft

Based on Tresca's yield condition and the associated flow rule, the stress distribution in a rotating solid cylinder of elastic-perfectly plastic material under plane strain is discussed. It is shown that the plastic core consists of two parts with different regimes of the yield condition. With increasing angular velocity, a second plastic zone forms at the boundary and spreads inwards.

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Zusammenfassung Untersucht wird die Spannungsverteilung in einem rotierenden Zylinder aus elastisch-ideal-plastischem Material unter der Voraussetzung des ebenen Verzerrungszustandes. Zugrundegelegt ist die Trescasche Fließbedingung und die zugeordnete Fließregel. Der plastische Kern besteht aus zwei Teilen mit unterschiedlichen Regimes der Fließbedingung. Bei Steigerung der Winkelgeschwindigkeit bildet sich am Rand eine zweite Fließzone aus, die nach innen wandert.

     

Transient heating of a rotating elastic-plastic solid shaft

Based on Tresca's yield criterion and the flow rule associated with it, the transient plasticization in a rotating elastic–plastic solid shaft with temperature dependent yield stress subject to a temperature cycle is studied. Special attention is paid to the residual stresses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stresses in a hollow rotating cylindrically orthotropic tube

Explicit stress distributions for a hollow cylindrically orthotropic tube due to spinning are presented for immediate use in everyday engineering work. The solutions are obtained directly without using the stress function concept. Various boundary condition combinations present in practical applications are considered. Examples are given in order to indicate how the degree of anisotropy influences the stress distributions.Department of Mechanical Engineering, University of Oulu, P.O. Box 444, 90571 Oulu, Finland. Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp 835–841, November–December, 1996.     

Nonlinear reynolds equation for lubrication of a rapidly rotating shaft

Using the homogenization theory, we derive the nonlinear Reynolds equation governing the process of lubrication of a slipper bearing with rapidly rotating shaft. We prove that this nonliner lubrication law is an approximation of the full Navier-Stokes equations in a thin cylinder with periodic roughness. The analyticity of the nonlinear function giving the relation between the velocity and the pressure drop is proved. The first term in its Taylor's expansion is the classical linear Reynolds law. Boundary layer correctors are computed.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

Elastic-plastic states of two-layer curved bar under pure bending

Analytical solution of an elastic-partially plastic two-layer curved bar subjected to couples at both ends is obtained. The solution is based on Tresca's yield criterion and its associated flow rule. Numerical results for real engineering materials in partially plastic stress state are presented in graphical form. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of plane parallel shear flow in a rotating layer heated from below

The discussion of stability of plane parallel shear flow in an infinite rotating layer heated from below requires a mathematical analysis of this problem in dependence on four parameters. These are the Reynolds- and Rayleigh-number, controlling the strength of the shear flow and the heating power, respectively, the Prandtl-number, which measures the relative influence of viscosity and thermal conductivity, and the rotation rate of the layer. After discussing some physical background, possible applications and laboratory experiments two major problems are addressed: i) To find out the cases where unconditional (global) stability up to criticality takes place. In these situations theory makes the clearest predictions and coincidence between experiments and mathematical theory can be expected. ii) To prove that the (monotonic) energy-stability limit is assumed by 2-dimensional (with respect to the spatial variables) perturbations. The solution of this variational problem shows that in certain situations the critical perturbations are 2-dimensional. In these situations, at least, the stability problem is completely solved. Received May 21, 1996 / Accepted December 17, 1996     

Non-stationary nonlinear analysis of a composite rotating shaft passing through critical speed

In this paper, nonlinear non-stationary dynamics of a nonlinear composite shaft passing through critical speed is studied. The nonlinearity is due to the large amplitude of shaft vibration. The equations of motion are obtained by three-dimensional constitutive relationships of composite materials. The gyroscopic effect, rotary inertia and coupling caused by material anisotropy are considered but shear deformation is neglected. Without any simplification, axial-flexural-flexural-torsional equations of motion (EOM) for the elastic composite shaft with variable rotational speed are obtained. The approximate analytical method namely asymptotic method is applied to analyze the nonstationary behavior of the composite shaft with constant acceleration. First, the EOMs are discretized using one and two-term Galerkin method. Then, the resulted equations are transformed to normal coordinates. Finally, the asymptotic method is applied to equations described in normal coordinates. Analytical expressions governing the amplitude and phase of motion during passage through critical speeds are obtained. By comparing the results obtained from analytical solutions, it is shown that discretization by one mode is not enough due to the existence of coupling in the equations and at least two modes are necessary for this purpose. Effects of damping, eccentricity, initial angular velocity and fiber angle on response amplitude are investigated. For verification, the results of perturbation theory are compared with numerical simulations and it is shown that there is good agreement between both methods.     

The effect of bearing cage run-out on the nonlinear dynamics of a rotating shaft

This paper presents an analytical model to investigate the nonlinear dynamic behavior of an unbalanced rotor-bearing system due to cage run-out. Due to run-out of the cage, the rolling elements no longer stay equally spaced. The mathematical model takes into account the sources of nonlinearity such as Hertzian contact forces and cage run-out, and the resulting transition from a state of no contact to contact between the rolling elements and the races. The contact between the rolling elements and races is treated as nonlinear springs and the system is analyzed for varying numbers of balls. The results are presented in the form of fast Fourier transformations and Poincaré maps. The results show that the ball passage frequency is modulated with the rotational frequency. The response falls into three regimes: periodic motion, quasi-periodic oscillations, and chaotic response.     

Non-stationary analysis of a rotating shaft with geometrical nonlinearity during passage through critical speeds

In this paper, analysis of a rotating shaft with stretching nonlinearity during passage through critical speeds is investigated. In the model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. First, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with non-constant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the non-stationary vibration of the rotating shaft, the asymptotic method is applied to the equations expressed in normal coordinates. Two analytical expressions, as function of system parameters that describe the amplitude and phase of motion during passage through critical speeds are derived. The effects of angular acceleration, stretching nonlinearity, eccentricity and external damping on maximum amplitude of the shaft are investigated. It is shown that the nonlinearity has important effect on maximum amplitude when the rotating shaft passing through critical speeds, especially in low angular acceleration. To validate the results of the perturbation method, numerical simulation is applied.     

Critical states of thin-wall conical gyroscopes elastically attached to a rotating platform

The analysis of effects of emergence of critical states of rigid and elastic thin-wall conical shell frustums hinged or elastically attached by their smaller edges to a rotating platform is performed on the basis of hybrid differential equations. The cases of simple and compound rotations of the carrying platform are considered. It is established that in the case of simple rotation the peculiarities of the critical state onset for the rigid and elastic shells are determined by relationships between their axial inertia moments, whereas in compound rotation the first frequencies of precession resonant vibrations of elastic shells do not depend on elastic pliability of the ties between the shell and the platform.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, T_A1T_{A_1}, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, T_A1T_{A_1} , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.     

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, medium permeability, D _a , and rotation, , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M ₃, and Darcy number, D _a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.      

Semi-exact solution for thermo-mechanical analysis of functionally graded elastic-strain hardening rotating disks

In this paper, distributions of stress and strain components of rotating disks with non-uniform thickness and material properties subjected to thermo-elasto-plastic loading are obtained by semi-exact method of Liao’s homotopy analysis method (HAM) and finite element method (FEM). The materials are assumed to be elastic-linear strain hardening and isotropic. The analysis of rotating disk is based on Von Mises’ yield criterion. A two dimensional plane stress analysis is used. The distribution of temperature is assumed to have power forms with the hotter point located at the outer surface of the disk. A mathematical technique of transformation has been proposed to solve the homotopy equations which are originally hard to be handled. The domain of the solution has been substituted by a new domain through which the unknown variable has been taken out from the argument of the function. This makes the solution much easier. A numerical solution of the governing differential equations is also presented based on the Runge–Kutta’s method. The results of three methods are presented and compared which shows good agreements. This verifies the implementation of the HAM and demonstrates its applicability to provide accurate solution for a very complicated case of strongly high nonlinear differential equations with no exact solution. It is important to notice that compared with other methods, HAM needs significant more computation time and computer hardware requirements which limit its application for those problems that other methods can easily handle them.     

Transient entropy generation analysis of vortex liquid isolated by hollow heated cylinder

The total transient entropy generation of a system that consists of a liquid vortex within a hollow cylinder as a heat source is investigated in this article. The hollow cylinder insulates the liquid vortex, and generates an air vacuum above the vortex which raises its level within the cylinder. The liquid vortex, at a volume of 20% and 60%, partially fills the hollow cylinder. In both cases, the heat transfer was partially established between the inner surface of the hollow cylinder and the vortex liquid. This analysis focused on the transient exchange of entropy generation between the cylinder and fluid. The heat exchange between the hollow heated cylinder and the fluids takes 15 s. The analysis of entropy generated includes only thermal irreversibility of this system; hydraulic irreversibility is neglected.