Free Vibration of a High-Speed Rotating Truncated Spherical Shell

Imperfections and energy barriers in shell buckling

In order to clarify the turbulence behavior in impeller channels of turbomachinery, effects of the Coriolis and centrifugal forces on the turbulent boundary layer on the concave and convex surfaces were studied experimentally, using a rotating channel of rectangular cross section whose aspect ratio was 2 : 1. Velocity profiles and turbulence intensities were also compared with those for a channel having an aspect ratio of 4 : 1. In the present rotating channel it was found that secondary flows generated by the centrifugal and Coriolis forces have a large effect on the time-averaged and turbulence components in the boundary layer along the concave and convex surfaces, and that the suppression of turbulence expected on the lower-pressure side of a two-dimensional flow is not observed because of the accmulation of lower energy fluids and the interference of the fluids with the primary flow.

In this paper, the effects of shape and arrangement of inclusions on the effective Young's modulus of materials are considered by the application of the finite element method through examining a model, which has two groups of periodically arranged inclusions in a matrix. Here, two groups of inclusions A and B are considered, both having equally shaped equally arranged inclusions, which have the same elastic constants but different from the ones of the matrix. This model includes square and hexagonal arrays of inclusions as its special cases. First, the effect of shape of inclusions on the Young's modulus of composite materials is considered from the comparison between the results of rectangular and elliptical inclusions. Next, when the position of group A is fixed, the effect of location of group B is considered. Then. the effective elastic Young's modulus is almost independent of the location of group B if the projected areas of groups A and B are not overlapped. In conclusion, the volume fraction of inclusion and projected area fraction of inclusions are found to be two major parameters controlling the effective Young's modulus of composites.

Effects of rotation on heat transfer on leading and trailing sides of gas turbine blades has been extensively studied in the past. It has been established for typical two-pass channel that radially outward flow (first pass) has higher heat transfer on trailing side and lower heat transfer on leading side and vice versa for radially inward flow (second pass). Rotation induces three forces on the coolant flow — Coriolis, Centrifugal and Buoyancy forces. The direction of Coriolis force depends on the relative angle between the coolant flow and the rotation direction, because of which the direction of Coriolis force is different in individual passes — which in turn results in non-uniform distribution of high heat transfer regions on leading and trailing walls. The present study is focused on utilizing the Coriolis force favorably in both the passes by rotating the typical arrangement of two-pass channels by 90°. Firstly, smooth two pass duct (Model A-smooth) having typical arrangement of coolant flow and rotation direction is studied. The second configuration is the corresponding ribbed channel (Model A-ribbed) featuring V-shaped ribs on both leading and trailing walls. The rib-height-to-channel hydraulic diameter ratio was 0.125, rib pitch-to-rib height ratio was 8, and channel aspect ratio was unity. Model B was obtained by rotating the Model A by 90° and changing the coolant inlet port as well. Model B had three configurations — (a) smooth duct, (b) single sided ribbed duct, and (c) double sided ribbed duct. Detailed heat transfer coefficients were measured by transient liquid crystal thermography under rotating conditions. In order to match the direction of Buoyancy force as it exists in actual engines, colder air was passed during the transient experiment. The heat transfer experiments were carried out at a Reynolds number of 20000 and Rotation numbers of 0, 0.05 and 0.1. The Nusselt numbers have been reported in two forms, (a) normalized with respect to Dittus-Boelter correlation for developed turbulent flow in circular duct, (b) normalized with corresponding Nusselt number obtained from smooth channel experiments. The effects of Coriolis force and centrifugal force on heat transfer has been discussed in detail. A new model has been proposed based on the understanding and findings of the present study, which has positive effects of rotation on both leading and trailing walls.

In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides. In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the ‘tidal breathing’ in eccentric binary systems are investigated. In Section 3, we consider the angular velocity of rotation about theZ′-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles θ and ϕ, and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form. The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.

The present paper studies the locations and linear stability of the triangular equilibrium points when both primaries are radiating and considered as heterogeneous spheroid with three layers of different densities. Additionally, we include the effects of small perturbations in the Coriolis and centrifugal forces and potential from a belt (circumbinary disc). It is observed that the positions of the triangular equilibrium points are substantially affected by all parameters (except a perturbation in Coriolis force) involved in the system.The stabilty of motion is found only when 0 < mu < mu_{c}, where mu_{c} is the critical mass value which depends on the combined effect of radiation pressures and heterogeneity of the primaries, small perturbations and the potential from a belt.It is also seen that the Coriolis force and the belt have stabilizing effect,while the centrifugal force, radiation and heterogeineity of the primaries have destabilizing behaviour.The net effect is that the size of the region of stability decreases when the value of these parameters increases where mu is the mass ratio and k_{1} ,k_{2} characterize heterogeneity of both primaries. A practical application of this model could be the study of motion of a dust grain near the heterogeneous and luminous binary stars surrounded by a belt.Finally, we carried out and discuss numerical experiments aiming at computing the positions of triangular points and critical masses of three binary systems: Archid, Xi Booties and Kruger 60.

This work is devoted to the study of centrifugation processes, which are one of the most complexprocesses of technology. The use of the action of the centrifugal force field for the separationof heterogeneous liquid systems in a rotor system (centrifuge) is very effective. Separability ofcentrifuged materials, multiphase dispersion, the relationship between phases cause ambiguousprocesses of centrifugation, and therefore it is not always possible to accurately predict and evaluatethe main characteristics of the separation of heterogeneous systems in the centrifugal force field.For a qualitative assessment of the main separation characteristics, such as the angular velocityof rotation, the dependence of the angles of inclination of the glasses on the angular velocity, thesettling time of particles, etc., is created a mathematical model of the vertical rotor system. Therotor represents a round disk on which cups (vial) with a multiphase liquid are symmetricallysuspended. A particular case of a fixed rotor is considered. Nonlinear differential equations ofmotion of the suspension particle, which have no exact solution, are obtained. The research isconducted by analytical and numerical methods. The dependence of the slope angles of the tubesfrom the angular rotational speed of the rotor, sedimentation curves are obtained , which allowestimating the time of particle deposition and the effect of the particle size distribution on theseparation process. The results of this research work allow us to determine with sufficient accuracyall the necessary characteristics of the sedimentation working process, and also, in certain cases,it is possible to exclude the conduct of experimental work. The results of the work confirm thephysical meaning of the process, which can serve as a justification for the use and introduction ofthis mathematical model in industrial production.

The nanoindentation technique was used to measure the hardness and Young's modulus of semiconductor and metal thin films on a Si(100) substrate under cyclic loading. The results showed that in all instances and at a constant cyclic load that the loading curves overlapped the previous unloading curve and had a small displacement after each cyclic nanoindentation. It was observed that the plastic energies of metal materials from the first loading–unloading cycle were much larger than that observed in semiconductor materials. Furthermore, the hardness and Young's modulus of the thin films decreased when the number of cyclic nanoindentations was increased. The effect of the cyclic loading on the hardness and Young's modulus of semiconductor material was much larger than that of the metal material. Young's modulus, the hardness and the contact stiffness of thin films conform to the relationship that Young's modulus was proportional to the contact stiffness and the square root of the thin film's hardness.

Effect of compressive prestress on the correlation between thermal conductivity and Young's modulus of carbon materials, fine-grained isotropic graphite IG-430U and a felt type C/C composite CX-2002U, was examined. Young's modulus and thermal conductivity decreased due to compressive prestress. The change in thermal conductivity of carbon materials was related to that of Young's modulus via sound velocity. The decrease in thermal conductivity of IG-430U graphite due to prestress was mostly correlated with that in Young's modulus. Also, decrease in thermal conductivity of C/C composite showed the similar tendency as the case for the graphite.

The amount of effect of the weight of beam on the measuring values of Young's modulus by latelal vibration of cantilevers may be given by considering only the component of weight acting in the direction of the length of the beam. In this paper the approximate solutions of the first and second frequencies of cantilever, by using the Galerkin method, are given from the equation of motion of the beam subjecting along the length of the beam to a uniformly distributed load. Accordingly the calculating equations of Young's modulus are obtained from these solutions. And the experimental results for the fundamental vibration of cantilevers is presented. It is shown that the Young's modulus can be measured exactly by treating the experimental results with these corrected equations.

Free vibrations across the thickness of a closed spherical shell are studied. A finite solution to the one-dimensional (along the radius) wave problem is derived for the shell, which differs from the solution for radial vibrations of a liquid. The solution is used in constructing a model for joint vibrations of three spherical layers. An approximate version of their vibratory motion is suggested, which is based on a shell model with thin stiff layers enveloping a thick soft core layer. The solution to the dynamic problem of free vibrations of the soft core is obtained by the method of separation of variables. The elastic reactions of the thin face layers were taken as boundary conditions, which allowed us to reduce the dynamics of the three layers to a vibratory motion of the midlayer under mixed boundary conditions. A transcendental equation for eigenvalues of the problem on free vibrations is derived and analyzed. A graphic description of the effect of layer thickness and elastic properties on the spectrum of eigenvalues of the mixed boundary problem is represented. Relationships for the main eigenfrequency versus the stiffness of face layers and characteristic geometric parameters of the sandwich shell wall are obtained.

Young's modulus as a function of temperature was determined by a dynamic method for single‐crystal sapphire and ruby and for polycrystalline aluminum oxide, magnesium oxide, thorium oxide, mullite, spinel, stabilized zirconium oxide, silicon carbide, and nickel‐bonded titanium carbide. For the single crystals, Young's modulus was found to decrease linearly with increasing temperature from 100°C. to the highest temperature of measurement. For all the polycrystalline materials, except silicon carbide, stabilized zirconium oxide, and spinel, Young's modulus was found to decrease approximately linearly with increasing temperature until some temperature range characteristic of the material was reached in which Young's modulus decreased very rapidly and in a nonlinear manner with increasing temperature. This rapid decrease at high temperature is attributed to grain‐boundary slip. Stabilized zirconium oxide and spinel were found to have the same rapid decrease in Young's modulus at high temperature, but they also had a decidedly nonlinear temperature dependence at low temperature.

Nonaxisymmetric free vibrations of a spherically isotropic spherical shell embedded in an elastic medium

Interaction of a plane progressive sound wave with a functionally graded spherical shell

On free vibration of a rotating truncated circular orthotropic conical shell