Absence of One Nodal Diameter Critical Speed Modes in an Axisymmetric Rotating Disk

1992 ◽  
Vol 59 (3) ◽  
pp. 687-688 ◽  
Author(s):  
Anthony A. Renshaw ◽  
C. D. Mote
Keyword(s):  
Critical Speed ◽  
Rotating Disk ◽  
Nodal Diameter ◽  
Diameter Critical

Study of In-Plane Motions in a Thin Rotating Disk

2000 ◽  
Author(s):  
Moreshwar Deshpande ◽  
C. D. Mote
Keyword(s):  
Critical Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Rotating Disk ◽  
Travelling Wave ◽  
Linear Strain ◽  
Strain Measure ◽  
Nodal Diameter ◽  
Plane Oscillations ◽  
Disk Energy

Abstract A model for the in-plane oscillations of a thin rotating disk has been derived using a nonlinear strain measure to calculate the disk energy. This accounts for the stiffening of the disk due the radial expansion resulting from its rotation. The corresponding non-dimensionalized natural frequencies are seen to depend only on rotation speed and have been calculated. The radially expanded disk configuration is linearly stable over the range of rotation speeds studied here. The sine and cosine modes for all nodal diameters couple to each other at all nonzero rotation speeds and the strength of this coupling increases with rotation speed. This coupling causes the reported frequencies of the stationary disk to split. The zero, one and two nodal diameter in-plane modes do not have a critical speed corresponding to the vanishing of the backward travelling wave frequency. The use of a linear strain measure in earlier work incorrectly predicts instability of the rotating equilibrium and the existence of critical speeds in these modes.

In-Plane Vibrations of a Thin Rotating Disk

2003 ◽  
Vol 125 (1) ◽  
pp. 68-72 ◽  
Author(s):  
Moreshwar Deshpande ◽  
C. D. Mote,
Keyword(s):  
Critical Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Rotating Disk ◽  
Travelling Wave ◽  
Linear Strain ◽  
Strain Measure ◽  
Nodal Diameter ◽  
Plane Oscillations ◽  
Disk Energy

A model for the in-plane oscillations of a thin rotating disk has been derived using a nonlinear strain measure to calculate the disk energy. This accounts for the stiffening of the disk due to the radial expansion resulting from its rotation. The corresponding nondimensionalized natural frequencies are seen to depend only on the nondimensionalized rotation speed and have been calculated. The radially expanded disk configuration is linearly stable over the range of rotation speeds studied here. The sine and cosine modes for all nodal diameters couple to each other at all non-zero rotation speeds and the strength of this coupling increases with rotation speed. This coupling causes the reported frequencies of the stationary disk to split. The zero, one and two nodal diameter in-plane modes do not have a critical speed corresponding to the vanishing of the backward travelling wave frequency. The use of a linear strain measure in earlier work incorrectly predicts instability of the rotating equilibrium and the existence of critical speeds in these modes.

Analysis of In-Plane Vibration and Critical Speeds of the Functionally Graded Rotating Disks

2019 ◽  
Vol 11 (02) ◽  
pp. 1950020 ◽  
Author(s):  
Emadoddin Bagheri ◽  
Mostafa Jahangiri
Keyword(s):  
Material Properties ◽  
Critical Speed ◽  
Rotational Velocity ◽  
Free Vibration Analysis ◽  
Rotating Disk ◽  
Functionally Graded ◽  
Radial Coordinate ◽  
Rotating Disks ◽  
Nodal Diameter ◽  
Graded Index

In this paper, the in-plane free vibration analysis of the functionally graded rotating disks with variable thickness is presented utilizing DQM. It is assumed that the rotational velocity of the disk is constant and the thickness and material properties including modulus of elasticity and density vary along the radial coordinate. The distribution of the forward and backward traveling waves versus the angular velocity is demonstrated for several modal circles and nodal diameters with respect to the fixed and rotating coordinate systems. After presenting the accuracy and convergence of the numerical method, the derived formulation and the solution method are validated by comparing the results with those obtained in the literature for simple rotating disks. Furthermore, the critical speed of the rotating disk is introduced and obtained for different modes. Finally, the effects of the functionally graded index (describes the distribution of material properties) and geometric shape of the disks (thickness profile and radius ratio) on the natural frequencies and critical speed of the disk are presented. It is observed that as the number of nodal diameter increases, the critical speed of the disk consequently decreases and reaches to an asymptotic value. This value is independent of the geometric characteristics of the disk.

Vibration Response of Functionally Graded Rotating Disk With a Circumferential Crack

2010 ◽  
Author(s):  
Rui Liu ◽  
Hamid Nayeb-Hashemi
Keyword(s):  
Critical Speed ◽  
Crack Depth ◽  
Rotating Disk ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Campbell Diagram ◽  
Circumferential Crack ◽  
Inner Radius ◽  
The Galerkin Method

In this study, the vibration characteristics of a functionally graded rotating hollow disk with the circumferential surface crack are investigated. In order to simplify the problem, the circumferential crack of the rotating hollow disk is modeled as circumferential step indentation. The Galerkin Method is used to obtain the radial and hoop stresses for disks with clamped edge at the inner radius. Finite Difference scheme is adopted to solve the partial differential equation of motion of the rotating hollow disk to obtain the mode shapes and the Campbell Diagram. The first critical speed, which is one of the important parameters limiting the performance of the rotating disk, was obtained from the Campbell Diagram. The results show that the crack will reduce the stiffness and the critical speed of the rotating disk. Critical speed increases with decreasing the distance from inner radius to the crack and decreases with increasing crack depth. Furthermore, considering the functionally graded disk, the distribution of elastic modulus does not change significantly the effects of circumferential cracks on the vibration characteristics of the rotating.

Modeling Gas–Liquid Flow Between Rotating and Nonrotating Annular Disks

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Irsha Pardeshi ◽  
Tom I-P. Shih
Keyword(s):  
Rotational Speed ◽  
Critical Speed ◽  
Rotating Disk ◽  
Critical Value ◽  
Outer Radius ◽  
Order Model ◽  
Reduced Order ◽  
Gas Liquid Flow ◽  
Gas Penetration ◽  
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.

On Transverse Vibrations of Functionally Graded Rotating Hollow Disk

2010 ◽  
Author(s):  
Rui Liu ◽  
Hamid Nayeb-Hashemi ◽  
Masoud Olia ◽  
Ashkan Vaziri
Keyword(s):  
Elastic Modulus ◽  
Critical Speed ◽  
Volume Fraction ◽  
Rotating Disk ◽  
Outer Radius ◽  
Functionally Graded ◽  
Distribution Functions ◽  
Rotating Disks ◽  
Disk Radius ◽  
Power Law Function

We studied the stress field and vibration characteristics of functionally graded rotating disks by solving the governing equation of motion using the finite difference scheme. The material was assumed to have a constant Poisson’s ratio with the elastic modulus varying as a power law function of the disk radius. Such a material could be developed by using particle reinforced composites with various reinforcements or reinforcement volume fraction. The results show that the first critical speed of the rotating disk could be increased by using FGMs. The first critical speed is greater for disks having higher elastic modulus at the outer radius. However, the disk may be unstable for certain distribution functions.

A Nonclassical Vibration Analysis of a Multiple Rotating Disk and Spindle Assembly

1997 ◽  
Vol 64 (1) ◽  
pp. 165-174 ◽  
Author(s):  
I. Y. Shen ◽  
C.-P. R. Ku
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Spindle Assembly ◽  
Rotating Disk ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Vibration Modes ◽  
Nodal Diameter

This paper studies natural frequencies and mode shapes of a spinning disk/spindle assembly consisting of multiple elastic circular plates mounted on a rigid spindle that undergoes infinitesimal rigid-body translation and rotation. Through use of Lagrangian mechanics, linearized equations of motion are derived in terms of Euler angles, rigid-body translation, and elastic vibration modes of each disk. Compared with a single rotating disk whose spindle is fixed in space, the free vibration of multiple disks with rigid-body motion is significantly different in the following ways. First of all, lateral translation of the spindle, rigid-body rotation (or rocking) of the spindle, and one-nodal diameter modes of each disk are coupled together. When all the disks (say N disks) are identical, the coupled disk/spindle vibration splits into N − 1 groups of “balanced modes” and a group of “unbalanced modes.” For each group of the balanced modes, two adjacent disks vibrate entirely out of phase, while other disks undergo no deformation. Because the out-of-phase vibration does not change the angular momentum, the natural frequencies of the balanced modes are identical to those of the one-nodal-diameter modes of each disk. For the group of the unbalanced modes, all disks undergo the same out-of-plane vibration resulting in a change of angular momentum and a steady precession of the spindle. As a result, the frequencies of the unbalanced modes are significantly lower than those of one-nodal-diameter modes of each disk. Secondly, axial translation of the spindle and the axisymmetric modes of each disk are couple together. Similarly, the coupled motion split into N − 1 groups of “balanced modes” and one group of “unbalanced modes,” where the frequencies of the balanced and unbalanced modes are identical to and smaller than those of the axisymmetric modes of each disk, respectively. Thirdly, the rigid-body motion of the spindle does not affect disk vibration modes with two or more nodal diameters. Response of those modes can be determined through the classical vibration analysis of rotating disks. Moreover, vibration response of the disk/spindle assembly from a ground-based observer is derived. Finally, a calibrated experiment is conducted to validate the theoretical predictions.

Aerodynamically and Structurally Coupled Vibration of Multiple Co-Rotating Disks

2004 ◽  
Vol 126 (2) ◽  
pp. 220-228 ◽  
Author(s):  
Jung Seo Park ◽  
I. Y. Shen
Keyword(s):  
Travelling Waves ◽  
Rotating Disk ◽  
Structural Flexibility ◽  
Vibration Modes ◽  
Coupled Vibration ◽  
Rotating Disks ◽  
Structural Coupling ◽  
Nodal Diameter ◽  
Frequency Splitting ◽  
Parametric Studies

This paper studies vibration of multiple, co-rotating, identical disks coupled by air flow and structural flexibility. In particular, the study focuses on coupled vibration of disk modes with two or more nodal diameters. First, frequency response functions of multiple co-rotating disks are measured in air and in vacuum to study the effects of aerodynamic coupling. In vacuum, vibration modes from each rotating disk are aerodynamically uncoupled; therefore, corresponding travelling waves from each disk have the same natural frequency. When the air is present, the air couples the corresponding travelling waves and rearranges them into a group of N traveling waves with distinct frequencies, where N is the number of the disks. A perturbation analysis is developed to prove the phenomenon of frequency splitting. Aside from the air coupling, finite element analyses and experimental measurements indicate that the flexibility of the clamp and spacers between the disks can also couple the disk vibration in the same manner. Moreover, the aerodynamic coupling is more significant for disk modes with high number of nodal diameters (e.g., 4-nodal-diameter modes). In contrast, structural coupling through spacer flexibility is more pronounced for disk modes with low number of nodal diameters (e.g., 2-nodal-diameter modes). Also, parametric studies using FEA indicate that frequency splitting from structural coupling will remain significant over a wide parameter range.

Rocking Vibration of Rotating Disk and Spindle Systems With Asymmetric Bearings

2003 ◽  
Vol 70 (2) ◽  
pp. 299-302 ◽  
Author(s):  
J. S. Park ◽  
I. Y. Shen ◽  
C.-P. R. Ku
Keyword(s):  
Rigid Body ◽  
Perturbation Analysis ◽  
Natural Frequencies ◽  
Rotating Disk ◽  
Mode Shapes ◽  
Nodal Diameter ◽  
Spindle System ◽  
Rocking Motion ◽  
Rocking Vibration

This note presents how bearing asymmetry affects natural frequencies and mode shapes of a rotating disk/spindle system through a perturbation analysis. The analysis will focus on rocking motion of the disk/spindle system that consists of rigid-body rocking of the spindle, one-nodal-diameter modes of each disk, and deformation of spindle bearings.

Intermodal Coupling in Flexible Spinning Disk-Spindle Systems

1999 ◽  
Author(s):  
Moreshwar Deshpande ◽  
C. D. Mote
Keyword(s):  
Rotating Disk ◽  
Free Vibrations ◽  
Characteristic Matrix ◽  
Vibration Modes ◽  
Nodal Diameter ◽  
Spindle System ◽  
Kirchhoff Plates ◽  
Spindle Vibration ◽  
The One ◽  
Plane Disk

Abstract The coupling between the disk and spindle vibration modes of a rotating disk-spindle system is analyzed through the free vibrations of a rotating, flexible spindle with N attached flexible disks. The spindle is modeled as an extensible Kirchhoff-Love rod and the disks as Kirchhoff plates. Couplings between the longitudinal, torsional and flexural deformations of the spindle and the transverse and in-plane motions of the disk are studied analytically. A kinematically rich model captures couplings that have not been predicted previously. Discretization of these modes as a series of orthonormal functions allows for the construction of the characteristic matrix. The structure of this matrix is exploited to partition the eigenvalue problem into six natural classes and to provide simple, exact rules governing the coupling between the modes of the disk-spindle system. The longitudinal spindle vibration modes and the zero nodal diameter transverse disk modes are coupled inertially at all rotation speeds. The torsional spindle modes couple to the zero nodal diameter in-plane disk modes at all non-zero rotation speeds. This coupling is absent in a stationary disk-spindle system. For non-zero rotation speeds, the flexural modes of the spindle in the two orthogonal planes containing the undeformed spindle centerline and the one nodal diameter transverse and in-plane disk modes couple. The one nodal diameter transverse disk modes couple to the one nodal diameter in-plane disk modes through the flexural compliance of the spindle; this coupling cannot be observed through study of the disk alone.

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