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.

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.

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.

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.

scholarly journals Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

2020 ◽  
Vol 11 (1) ◽  
pp. 127
Author(s):  
Fuchun Yang ◽  
Dianrui Wang
Keyword(s):  
High Speed ◽  
Differential Operators ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Distribution Area ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Wide Range ◽  
Forced Responses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

Mineral interfacial processes in the method of induced polarization

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 1105-1114 ◽  
Author(s):  
James D. Klein ◽  
Tom Biegler ◽  
M.D. Horne
Keyword(s):  
Frequency Domain ◽  
Diffusion Layer ◽  
Electrode Potential ◽  
Rotation Speed ◽  
Low Frequency ◽  
Rotating Disk ◽  
Active Species ◽  
Hydrodynamic Theory ◽  
Electrode Polarization ◽  
Disk Electrodes

A phenomenological laboratory investigation has been conducted of the IP response of pyrite, chalcopyrite, and chalcocite. The technique that was used is standard in electrochemistry and employs rotating disk electrodes. The effect of rotation is to stir the electrolyte and thus to restrict the maximum distance available for diffusion of electroactive aqueous species. For high rotation speed and low excitation frequencies, the mean diffusion length exceeds the thickness of the diffusion layer. The net effect is to reduce the electrode impedance at low frequency. The thickness of the diffusion layer and thus the impedance at low frequency can be controlled by the rotation speed. Measurements using rotating disk electrodes have been conducted in both the time domain and the frequency domain. For both pyrite and chalcopyrite, the results were the same: no dependence on rotation was observed. For frequency domain measurements with chalcocite, a strong dependence on rotation was observed. The interpreted diffusion layer thickness was found to depend on rotation speed to the [Formula: see text] power, in agreement with results predicted by hydrodynamic theory. The results of this study imply that there are two physical processes responsible for electrode polarization in the IP method. For chalcocite and perhaps other related copper sulfide minerals, the probable mechanism is diffusion of copper ions in the groundwater. In case, the phenomenon is correctly described by the Warburg impedance. Chalcocite’s distinctive response is thought to be related to its forming a reversible oxidation‐reduction couple with cupric ions in solution. No other common sulfide mineral forms a reversible couple with its cations in solution. For the other minerals of this study, the lack of dependence on rotation implies that diffusion of active species in the electrolyte is not the controlling process. Possible alternate mechanisms include surface controlled processes such as surface diffusion or adsorption phenomena. Ancillary data obtained during this study indicate the interface impedance of chalcopyrite is proportional to the electrode potential which in turn can be controlled by rotation speed, electrolyte composition, or application of an external dc current or voltage. This implies that the surface concentration of active species is dependent on electrode potential.

Nonlinear Resonant Motion of Traveling Waves in Rotating Disks

2000 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chin An Tan
Keyword(s):  
Traveling Waves ◽  
Rotation Speed ◽  
Rotating Disk ◽  
Disk Drive ◽  
Computer Memory ◽  
Rotating Disks ◽  
Resonant Wave ◽  
Linear And Nonlinear ◽  
Resonant Spectrum ◽  
Disk Drive Industry

Abstract The resonant conditions for traveling waves in rotating disks are derived. The nonlinear resonant spectrum of a rotating disk is computed from the resonant conditions. Such a resonant spectrum is useful for the disk drive industry to determine the range of operational rotation speed. The resonant wave motions for linear and nonlinear, rotating disks are simulated numerically for a 3.5-inch diameter computer memory disk.

Vibration of High-Speed Compliant Gear Pairs

2016 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
System Structure ◽  
Vibration Modes ◽  
Mesh Stiffness ◽  
Nodal Diameter ◽  
Gear Teeth ◽  
Wide Range ◽  
System Coupling

This study analytically investigates the vibration of high-speed, compliant gear pairs using a model consisting of coupled, spinning, elastic rings. The gears are elastically coupled by a space-fixed, discrete stiffness element that represents the contacting gear teeth. Hamilton’s principle is used to derive the nonlinear governing equations of motion and boundary conditions. These equations are linearized for small vibrations about the steady equilibrium due to rotation. The equations are cast in operator form, which exemplifies their gyroscopic system structure. The eigenvalue problem is discretized using Galerkin’s method. The natural frequencies and vibration modes for an example aerospace gear pair are numerically calculated for a wide-range of rotation speeds. The system coupling leads to multiple eigenvalue veering regions as the gear rotation speed varies. Highly coupled vibration modes that have meaningful deflection in the discrete mesh stiffness occur within a set frequency band. The vibration modes within this band have distinct nodal diameter components that evolve with rotation speed.

Thermal Effects on the Natural Frequency of Nonlinear, Co-Rotating Disks

2005 ◽  
Author(s):  
Albert C. J. Luo ◽  
Nader Saniei ◽  
William Ray Harp
Keyword(s):  
Natural Frequency ◽  
Thermal Effects ◽  
Natural Frequencies ◽  
Computer Memory ◽  
Uniform Temperature ◽  
Rotating Disks ◽  
Nonlinear Free Vibration ◽  
Nodal Diameter ◽  
Disk Rotation ◽  
Asymmetric Responses

Thermal effects on the natural frequency for the nonlinear free vibration of co-rotating disks are investigated for non-uniform temperature distributions relative to airflow induced by disk rotation. The natural frequencies for symmetric and asymmetric responses of a 3.5 inch diameter computer memory disk are calculated. When the disk is heated, its stiffness becomes larger for the two lowest nodal diameter numbers and smaller for the other nodal diameter numbers. It implies that the vibration of heated, rotating disks for the higher nodal diameter numbers may be induced more easily than the cooled one.

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