On the equilibrium configurations of an elastically constrained rotating disk: An analytical approach

According to the linear theory of vibration for spinning disks, the backward traveling waves of some of the modes may have zero natural frequency at what are called the critical speeds. At these speeds, the linear equations of motion cannot properly predict the amplitude response of the spinning disk, and nonlinear equations of motion must be used. In this paper, geometrical nonlinear equations of motion based on Von Karman plate theory are employed to study the dynamics of an elastically constrained disk near its critical speeds. A one-mode approximation is used to examine the effect of elastic constraint on the amplitude response. Presenting the equations in a space-fixed coordinate system, this study aims to find closed-form solutions for some of the equilibrium configurations of an elastically constrained spinning disk. Also, the stability of these configurations is studied using analytical techniques. It is shown that below the critical speed, one neutrally stable equilibrium solution exists, while above it, a bifurcation occurs. In this situation, two more branches of equilibrium configurations emerge, one of which is neutrally stable and the other unstable. Closed-form expressions for the bifurcation points are obtained. Due to the effect of an elastic constraint, a bifurcation occurs and the previously neutrally stable equilibrium configuration turns unstable. Also at this bifurcation point, two more branches of equilibrium solutions emerge.     

On the unsymmetrical deformation and reverse snapping of a spinning non-flat disk

In this paper we study the steady state deflection and reverse snapping phenomenon of a spinning non-flat disk. Both the initial and the deformed shapes of the disk are allowed to have axisymmetrical and unsymmetrical components. For the analysis of a rotating axisymmetrical disk, we conclude that there is no need to include the unsymmetrical assumed modes in the solution because all the unsymmetrical steady state deformations are unstable. In addition, the stability of the axisymmetrical positions will not be affected by the inclusion of the unsymmetrical assumed modes. In the case when the initial shape of the disk contains a dominant axisymmetrical component and a smaller unsymmetrical component, attention is focused on the effect of this unsymmetrical component on the overall deformation and stability of the spinning disk. It is found that the unsymmetrical component with one nodal diameter tends to slightly defer the occurrence of the reverse snapping phenomenon. On the other hand, all other unsymmetrical components with more than one nodal diameter tend to reduce dramatically the reverse snapping speed. Experiment is conducted on a non-flat brass disk containing an axisymmetrical component and an unsymmetrical component with two nodal diameters. The experimental measurement confirms the theoretical prediction.     

Numerical and statistical approach for Casson-Maxwell nanofluid flow with Cattaneo-Christov theory

The rheological features of an incompressible axi-symmetric Casson-Maxwell nanofluid flow between two stationary disks are examined. The lower permeable disk is located at z =-a, while the upper disk is placed at z = a. Both the disks are porous and subjected to uniform injection. The fluid properties such as thermal conductivity vary with temperature. The Cattaneo-Christov thermal expression is implemented along with the Buongiorno nanofluid theory. By operating the similarity functions, the reduced form of the fluid model in terms of ordinary differential equations is obtained and solved by the bvp4 c numerical technique. The physical quantities are demonstrated graphically on the velocity and temperature fields. Three-dimensional flow arrangements and twodimensional contour patterns against several dimensionless variables are also sketched.The numerical values of the local Nusselt and Sherwood numbers for various quantities are presented in tabular set-up. The intensity of the linear relationship between the Nusselt and Sherwood numbers is assessed through Pearson's product-moment correlation technique. The statistical implication of the linear association between variables is also examined by the t-test statistic approach.     

On rubber disks under rotation or axisymmetric stretching

The governing equations for a class of axisymmetric problems under large elastic deformation, concerned with a circular rubber disk with body force as well as non-uniform initial thickness, are formulated in terms of two coupled first-order ordinary differential equations with explicit derivatives. The following two problems subjected to different boundary conditions are solved: (a) Rotating disks with uniform initial thickness (b) Circular disks with non-uniform initial thickness under axisymmetric stretching at the outer boundary. In problem (b), the rubber disk whose initial thickness contour is h₀ = crⁿ (where c and n are any constants), or whose final thickness is a constant, is considered. Highly elastic materials with a Mooney strain-energy function are used for numerical calculations.     

Thermally Induced,Nonlinear Vibrations of Rotating Disks

The natural frequency and responses for the nonlinear free vibration ofheated rotating disks are presented analytically when nonuniformtemperature distributions pertaining to the laminar and turbulentairflow induced by disk rotation are considered. The nonuniformtemperature distributions on the disk are highly dependent on itsrotation speed. The natural frequencies for symmetric and asymmetricresponses of a 3.5 inch diameter computer memory disk are calculated.When the disk is heated, its stiffness becomes larger for the two lowestnodal diameter numbers and smaller for the other nodal diameter numbers.It implies that the vibration of heated, rotating disks for the highernodal diameter numbers may be induced more easily than the cooled one.The results for the nonlinear vibration can reduce to those for thelinear vibration when the nonlinear effects vanish. To furtherinvestigate of the interaction of thermal and nonlinearity of rotatingdisks, the temperature distribution for such a rotating disk needs to bedeveloped.     

Mode identification of a rotating disk

A modal testing method permitting identification of the natural frequencies, the number of nodal diameters and wave motions in a rotating disk is presented in this paper. This method is applicable at arbitrary rotation speed without requiring a priori information about the vibration modes of the stationary disk. The influence of disk rotation speed on the prediction of mode shapes with this method is shown, and experimental predictions of modal parameters are presented for both axisymmetric and asymmetric disks.     

MHD viscous flow between torsionally oscillating disks

The unsteady motion of an incompressible, viscous, stratified fluid between two parallel infinite disks maintained at different temperatures is studied under the influence of a uniform transverse magnetic field. The whole system is under rigid rotation in the initial state and perturbations are created by the small amplitude torsional oscillations of the disks. The time required for the transient velocity and temperature to decay is found for various ranges of the values of the forcing frequency of the disks. The steady state velocity and temperature distributions represent boundary layers on the disks and an interior flow. The interplay between the Hartmann number and the Ekman number in determining the boundary layers on the disks is discussed.     

A system for measuring the surface profile of a spinning disk

An autofocusing, laser-based system for measuring the surface topography of a spinning optical-quality disk is described. The essential feature of the system is high resolution of absolute surface profiles with large peak to peak variations. Illustrative experimental results are presented for glass and plastic disks. Applications of the instrument to surface accelerations and thickness measurements are also discussed.     

Remarks on the non-linear vibration of an axisymmetric circular disk near critical speed

We re-examine the large amplitude transverse oscillations of axisymmetric disks spinning near a critical speed resonance as considered by Raman and Mote (Int. J. Non-linear Mech. 34 (1) (1999) 139). Averaged equations in traveling wave based coordinates are shown to provide a transparent explanation for certain bifurcations presented in Raman and Mote (Int. J. Non-linear Mech. 34 (1) (1999) 139). The theoretical results in Raman and Mote (Int. J. Non-linear Mech. 34 (1) (1999) 139) are proved analytically, and an enhanced interpretation of their results is presented.     

Resonant and Stationary Waves in Rotating Disks

The analytical conditions for resonant and stationary waves inrotating disks are presented. These conditions are derived from anonlinear plate theory pertaining to initial configurations and areapplicable to rotating disks with initial waviness and/or undergoinglarge-amplitude displacements. The rotational speeds at which theresonant and stationary waves occur for a 3.5-inch diameter computermemory disk are computed. The resonant waves for linear and nonlinear,rotating disks are simulated numerically. It is found that some diskmodes exhibit a hardening effect under which the rotational speeds forthe resonant and stationary waves increase with increasing waveamplitude, while other modes experience a softening effect with thoserotational speeds decreasing with increasing wave amplitude. Therotating-disk resonant spectrum presented in this paper is relevant tothe disk drive industry for determining the range of operationalrotation speed.     

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation. The project supported by the National Natural Science Foundation of China (10472066, 10372057). The English text was polished by Yunming Chen.     

Axisymmetric Thermoelastoplastic Stress State of Isotropic Solids of Revolution Under Impulsive Loading

Consideration is given to the solution of a dynamic problem for a solid of revolution with an arbitrary meridional section under impulsive thermomechanical loading inducing elastoplastic strains. The theory of small elastoplastic deformations is used. The constitutive equations are linearized by the variable-parameter method. The unloading process is described by a linear law. The solution technique involves the finite-element approximation in spatial coordinates and the finite-difference representation of time derivatives. Based on the principle of linear summation, recurrent relations are derived for successive evaluation of nodal displacements by an explicit scheme in time. Solution for cylinders and disks are presented to illustrate the influence of elastoplastic deformations on wave processes     

Limit angular velocities of variable thickness rotating disks

Elastic and plastic limit angular velocities are calculated for rotating disks of variable thickness in power function form. Analytical solution is obtained and used to calculate elastic limit angular velocities of variable thickness rotating annular disks and annular disks with rigid inclusion. An efficient numerical solution procedure is designed and used to obtain the elastic limit angular velocities of variable thickness rotating solid disks. Von Mises yield criterion and its flow rule is used to estimate plastic limit angular velocities. Both linear and nonlinear hardening material behaviors are treated numerically. The results are verified by comparing with those of uniform thickness rotating solid disks available in the literature. Elastic and plastic limit angular velocities are found to increase with the reduction of the disk thickness at the edge as well as the reduction in the disk mass due to the shape of the profile.     

Transient squeeze flow of viscoplastic materials

The transient, axisymmetric squeezing of viscoplastic materials under creeping flow conditions is examined. The flow of the material even outside the disks is followed. Both cases of the disks moving with constant velocity or under constant force are studied. This time-dependent simulation of squeeze flow is performed for such materials in order to determine very accurately the evolution of the force or the velocity, respectively, and the distinct differences between these two experiments, the highly deforming shape and position of all the interfaces, the effect of possible slip on the disk surface, especially when the slip coefficient is not constant, and the effect of gravity. All these are impossible under the quasi-steady state condition used up to now. The exponential constitutive model, suggested by Papanastasiou, is employed. The governing equations are solved numerically by coupling the mixed finite element method with a quasi-elliptic mesh generation scheme in order to follow the large deformations of the free surface of the fluid. As the Bingham number increases, large departures from the corresponding Newtonian solution are found. When the disks are moving with constant velocity, unyielded material arises only around the two centers of the disks verifying previous works in which quasi-steady state conditions were assumed. The size of the unyielded region increases with the Bingham number, but decreases as time passes and the two disks approach each other. Their size also decreases as the slip velocity or the slip length along the disk wall increase. The force that must be applied on the disks in order to maintain their constant velocity increases significantly with the Bingham number and time and provides a first method to calculate the yield stress. On the other hand, when a constant force is applied on the disks, they slow down until they finally stop, because all the material between them becomes unyielded. The final location of the disk and the time when it stops provide another, probably easier, method to deduce the yield stress of the fluid.     

Optimal Design of Rotating Disks in Creep

Abstract

Minimum-weight design of axially-symmetric rotating disks in a state of stationary creep is determined for a prescribed value of the creep velocity at the outer edge of the disk. The constitutive equations used for stationary creep are Norton's law generalized to multiaxial states of stress based on von Mises' criterion and associated flow rule. The resulting nonlinear optimization problem is solved iteratively using a series expansion to approximate the thickness variation of the disk, In each iteration step the nonlinear creep equations are solved for the stresses and a linearized perturbation problem is solved for the stress gradients. The optimization procedure is used to determine the optimal shape of a solid rotating disk carrying a uniform traction at the outer edge, and this result is compared with the corresponding disk of uniform strength. Variations in the optimal shape due to a central hole and due to temperature distributions are illustrated by some examples.     

Thermally induced natural-frequency variations in a thin disk

A number of studies has shown that natural frequencies and, accordingly, the minimum critical speed for the formation of a standing wave in thin, rotating, circular disks can be beneficially altered by purposely induced initial membrane stresses. The possibility of controlling natural frequencies by induced thermal membrane stresses, rather than initial stresses, has received some previous theoretical attention and is experimentally examined here for a stationary, constant-thickness, centrally clamped, circular disk. The primary advantages of thermal membrane stresses are manifested in the inherent flexibility in adjustment of the thermal as opposed to the initial stresses. Increases in the minimum critical speed, which is proportional here to the zero nodal circle—two nodal-diameter natural frequency, of 20 percent were determined with moderate heating. This can be considered a relatively small critical-speed increase when compared with variations expected in many common rotating disk environments. A thermal model, which utilizes as input the peripheral disk heat flux and the controlled disk temperature at some known radius, is shown to predict the temperature distribution and natural frequencies with reasonable accuracy. The applicability of this model enhances the potential practicality of the induced thermal-membrane-stress method of natural frequency and/or critical speed control.     

A simple spinning laminated composite shaft model

A simple spinning composite shaft model is presented in this paper. The composite shaft contains discrete isotropic rigid disks and is supported by bearings that are modeled as springs and viscous dampers. Based on a first-order shear deformable beam theory, the strain energy of the shaft are found by adopting the three-dimensional constitutive relations of material with the help of the coordinates transformation, while the kinetic energy of the shaft system is obtained via utilizing the moving rotating coordinate systems adhered to the cross-sections of shaft. The extended Hamilton’s principle is employed to derive the governing equations. In the model the transverse shear deformation, rotary inertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. To verify the present model, the critical speeds of composite shaft systems are compared with those available in the literature. A numerical example is also given to illustrate the frequencies, mode shapes, and transient response of a particular composite shaft system.