A Study on Mode Shape and Natural Frequency of Rotating Flexible Cracked Annular Thin Disk

As an important rotating component, the flexible annular thin disk is widely used in mechanical engineering. Cracks may occur in some weak disk parts, which will greatly shorten the equipment service life and even cause equipment failure. Due to the centrosymmetric structure of the flexible annular disk, two typical cracks are studied in this paper; one is radial crack parallel to diameter, including radial closed crack (RC-crack) and radial opening crack (RO-crack); the other one is vertical crack perpendicular to diameter, including circumferential crack (CF-crack) and tangential crack (TG-crack). The effect of crack parameters, such as crack length, direction, and position, on disk vibration characteristics are studied through theoretical simulation and experimental verification. The research shows that the effect of cracks on vibration characteristic gets more obvious with cracks extending in most cases, RO-crack decreases the natural frequency obviously, and vertical cracks would affect mode shapes. In addition, the bigger the nodal diameter is, the more obvious the effect gets. Meanwhile, the most obvious effect appears in the mode of a nodal diameter locating on the crack. The research possesses some guiding significance in industrial production; by comparing with the vibration characteristics of the flawless disk, the integrity of the rotating flexible disk can be judged to prevent possible equipment damage.

On Non-Spherical Models for Spiral Galaxy Gravitation Potential Yielding Flat Rotation Curve

A two component model of gravitation potential for spiral galaxies has been proposed which couples a spherically symmetric component with a second component that observes planar radial symmetry on the galactic plane and vanishes outside an annular disk beyond the edge of galaxy's effective radius. It is shown that such a model for potential satisfying Poisson Equation would produce rotation velocity curve towards the edge of the galaxy which is flat over distance from the galactic centre. This relationship, which is experimentally observed in many spiral galaxies, is shown as a consequence of classical understanding of gravity and specific symmetry of the gravitational potential without any extrinsic requirement of dark matter. It is also demonstrated that this potential directly yields a relationship between inner mass of the galaxy and terminal rotation velocity, which has been empirically observed and known as Baryonic Tully-Fisher relations. Furthermore a direct test has been proposed for experimental verification of the proposed theory.

ROLLERS OF DIFFERENT CONSTRUCTIONS. ENGINEERING ANALYSIS

Comparative field tests of similar in design ring-spur and gear-spur rollers for soil rolling provide their developers and users with information on the understanding of the design and technological features. The Goal of research – engineering analysis of structures of ring-spur and gear-spur rollers according to the criteria of weight, shape of working surfaces, soil pressure, completeness of compaction, penetrating ability in the range of depth of pre-sowing and surface tillage. Methods and Materials. Engineering analysis involved the study of three types of rollers. The research hypothesis assumed that the combination of weight characteristics of rollers and their geometric shapes of working surfaces in contact with the soil, in statics will provide the inherent properties of each cat (the first stage of research), which will potentially focus on choosing the most acceptable criteria for comparing functional features of rollers. necessary for field research in dynamics (the second stage of research). According to the criteria of static research were selected: penetrating ability (1); soil pressure (2); completeness of compaction of the area (3); direction of clamping forces (4); features of working surfaces (5); the nature of the mounting of solid disks on the shaft (6). Methodologically, the first three criteria required special adaptations (capacities) with the ability to change the height of the soil layer in the range of depths of pre-sowing and sowing tillage. Other criteria were supposed to be carried out by organoleptic evaluation. The study was supposed to be carried out on soil characteristic of the Forest-Steppe zone and homogeneous in fractional composition, to exclude the influence of soil inclusions of different sizes. Evaluation of soil characteristics was performed by thermostatic-weight method, according to the generally accepted method of "cutting ring" [Kaczynski, 1947] with a cylinder volume of 100 cm3. To determine the area of one elementary working surface of the annular disk, the application of a marking grid on its imprint on the surface left in the soil layers hmin, hmid, hmax was used. When determining the pressure on the soil, depending on the height of the treated layer, the marking grid was superimposed on the imprint of several areas of elementary working surfaces, which were fixed on the chord of immersion in the soil of the annular segment. Statistical analysis of experimental data was performed by the method of analysis of variance according to [Dospekhov, 1985] and interpreted by standard computer programs Excel and graphs. Results. According to the physical characteristics (weight and geometric parameters) of the roller combines the proximity of the diameter of the disks, the width of the working surfaces on the edge of the generator, the presence of spurs. The difference lies in the dimensional characteristics that fall on the width of the grip, the presence of aggressive protruding shapes (teeth) in the gear-spur structure. Each roller has an individual layout of elementary work surfaces. This allowed the conditional marking of rollers as "classic", "modernized" and "specialized". The working surfaces of the classic and specialized rollers provide vertical and lateral action of the clamping forces, while the modernized forms them in the vertical direction. These forces for each of the rollers are derived from the orientation of the working surfaces of different configurations (flat narrow, flat wide; a combination of point-focused with trapezoidal). In addition, work surfaces can have a one- or two-level sequence of contact with the soil at a variable depth. It is established that the nature of the attachment of adjacent disks on the shaft can be rigid or movable. Definitions of concepts and their estimation on indicators of pressure, completeness of consolidation, penetrating ability are resulted. Conclusions. According to the indicators of engineering analysis, each of the three rollers is predicted to have its inherent dominant feature for the most effective use in milling, pre-sowing and post-sowing tillage operations. It is determined that the most acceptable criteria for further research in the dynamics of variable speed should be the intensity of crushing lumps, the degree of soil compaction, the intensity of soil compaction, sticking of the working surface of the roller in high humidity, self-cleaning ability. The criteria of further research in dynamics are determined.

Elastic Analysis for Rotating Functionally Graded Annular Disk with Exponentially-Varying Profile and Properties

Functionally graded materials have been widely used in engineering and human health applications. The issues about mechanical behavior of functionally graded material have received considerable attention. However, because of the complexity of material property, geometric profile, and mechanical load, there is still lack of proper analytic solutions about deformation and stress in many articles. The principal goal of this research is to study the effect of mechanical load on deformation and stress in rotating thin-walled functionally gradient material annular disk with exponentially-varying profile and properties. The inner and outer surfaces of annular disk are subjected to different pressures simultaneously. For this purpose, the infinitesimal theory of elasticity and axisymmetric plane stress assumptions has been proposed to formulate the governing equation. The governing equation is a generalized confluent hypergeometric differential equation, based on Whittaker’s functions; this is the first time that closed-form solutions of mechanical behaviors are revealed about proposed functionally gradient material model. Besides, another four boundary conditions are also discussed, i.e., the inner and outer surfaces of the annular disk are considered to be the combinations of free and clamped conditions. Numeric examples of two different functionally graded material properties are given to demonstrate displacement and stress solutions. Moreover, uniform disks made of homogeneous material under different boundary conditions are investigated, which are special cases of the proposed rotating functionally gradient material disks. Finally, some conclusions are made at the end of the present paper.

Thermoelastic Response of Functionally Graded Fiber-Reinforced Rotating Disk With Non-Uniform Thickness Profile Under Variable Angular Velocity

Displacement and stress fields in a functionally graded (FG) fiber-reinforced rotating annular disk with a non-uniform thickness profile, subjected to angular deceleration and a temperature profile were investigated. Unidirectional fibers were considered to be circumferentially distributed within the disk with fiber volume fraction changing radially. The governing equations for displacement, stress, and temperature fields were solved using finite difference method. The results indicated that thermal induced stresses were more dominate than the rotational induced stresses. Disks which were fiber rich at the inner radius, the fibers made negligible difference on the displacement and stress fields when compared to a homogenous disk made from the matrix material. In addition, it was found that the deceleration magnitude had no effect on the radial and hoop stresses, nor the temperature on the developed shear stress. The shear stress was only affected by the disk deceleration. Tsai-Wu failure criterion was applied for decelerating disks to ascertain their failure behavior. The results show that Tsai-Wu failure index is dominated by the thermal stresses.