In-Plane Vibration Mode Shapes for Rotating Disks: Exact Solution

An analytical method is developed for the determination of modal vibration characteristics of high speed rotating annular disks. A systematic approach based on established solutions for the linear in-plane free vibrations of the disks which satisfy the displacement and stresses compatibilities is developed. The disk is considered to be a homogeneous, thin and elastic isotropic, and it is rotating at constant angular speed. The developed analytical solution was obtained by implementing the two-dimensional plane stress theory. In this research, fixed-free and free-free boundary conditions for the annular disks are considered, and natural frequencies, as well as mode shapes of the rotating disks, are computed. The mode shapes are represented by eight functions in polar coordinates. The dimensionless natural frequency parameters are depicted for free vibration of the system for a range of dimensionless rotation speed and radius ratios. Also, the results provide several non-dimensional critical speeds.

Modeling Gas–Liquid Flow Between Rotating and Nonrotating Annular Disks

When a liquid is forced to flow radially outward in the gap between two coaxial, parallel annular disks—one rotating and one stationary—the liquid occupies the entire gap until the speed of the rotating disk reaches a critical value. Beyond that critical speed, gas from the outer radius starts to enter into the gap, a process referred to as aeration. The higher the rotational speed, the greater is the extent of penetration by the gas into the gap. The extent of gas penetration strongly affects the torque exerted between the two disks because of the large difference in the gas and liquid viscosities. In this study, a reduced-order model is developed to predict the onset of aeration, extent of gas penetration into the gap, and drag torque as a function of the disk's rotational speed, gap between disks, properties of the liquid, and mass flow rate of the liquid forced through the gap. The model developed was validated by comparing predictions with experimental data.

Elastic and Viscoelastic Stresses of Nonlinear Rotating Functionally Graded Solid and Annular Disks with Gradually Varying Thickness

Analytical and numerical nonlinear solutions for rotating variable-thickness functionally graded solid and annular disks with viscoelastic orthotropic material properties are presented by using the method of successive approximations.Variable material properties such as Young’s moduli, density and thickness of the disk, are first introduced to obtain the governing equation. As a second step, the method of successive approximations is proposed to get the nonlinear solution of the problem. In the third step, the method of effective moduli is deduced to reduce the problem to the corresponding one of a homogeneous but anisotropic material. The results of viscoelastic stresses and radial displacement are obtained for annular and solid disks of different profiles and graphically illustrated. The calculated results are compared and the effects due to many parameters are discussed.