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.

Maximizing the Natural Frequencies and Transverse Stiffness of Centrally damped, Circular Disks by Thickening the Clamped Part of the Disk

1999 ◽  
Vol 66 (4) ◽  
pp. 1017-1021 ◽  
Author(s):  
A. A. Renshaw
Keyword(s):  
Natural Frequency ◽  
Natural Frequencies ◽  
Vibration Modes ◽  
Rotating Disks ◽  
Design Study ◽  
Nodal Diameter ◽  
Transverse Stiffness ◽  
Vibration Data ◽  
Two Alternatives ◽  
Circular Disks

The natural frequencies and transverse stiffness of centrally damped, circular disks are computed taking into account the flexibility of the central clamp and the thickness of the damped part of the disk. When compared to experimental vibration data, these predictions are more accurate than the traditional, perfect clamping predictions, particularly, for zero and one-nodal-diameter vibration modes. The reduction in natural frequency or transverse stiffness caused by clamping flexibility can be mitigated either by increasing the clamping stiffness or by increasing the hub thickness, defined here as the thickness of the disk sandwiched by the central clamp. A design study of these two alternatives for both stationary and rotating disks shows that increasing the hub thickness is often a more attractive design alternative.

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.

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.

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 Measurement and Monitoring of a Rotating Disk Using Contactless Laser Excitation

2013 ◽  
Author(s):  
Itsuro Kajiwara ◽  
Naoki Hosoya
Keyword(s):  
Laser Doppler Vibrometer ◽  
Rotating Disk ◽  
Disk Drive ◽  
Vibration Characteristics ◽  
Vibration Measurement ◽  
Testing System ◽  
Rotating Disks ◽  
Vibration Testing ◽  
Structural Surface ◽  
Theoretical Analyses

This paper proposes a contactless vibration testing system for rotating disks based on an impulse response excited by a laser ablation. High power YAG pulse laser is used in this system for producing an ideal impulse force on structural surface without contact. The contactless vibration testing system is composed of a YAG laser, laser Doppler vibrometer and spectrum analyzer. This system makes it possible to measure vibration characteristics of structures under operation, such as vibration measurement of a rotating disk. The effectiveness of this system is confirmed by experimental and theoretical analyses. In this paper, a platter of hard disk drive is employed as an experimental object. Vibration characteristics of a rotating and non-rotating platter are measured and compared with the results of theoretical analysis.

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.

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.

Thermo-mechanical analysis and optimization of functionally graded rotating disks

2020 ◽  
Vol 55 (5-6) ◽  
pp. 159-171
Author(s):  
Hassan Mohamed Abdelalim Abdalla ◽  
Daniele Casagrande ◽  
Luciano Moro
Keyword(s):  
Finite Element ◽  
Equivalent Stress ◽  
Rotating Disk ◽  
Mechanical Analysis ◽  
Theory Of Elasticity ◽  
Functionally Graded ◽  
Rotating Disks ◽  
Maximum Equivalent Stress ◽  
Quadratic Programming Method ◽  
Stress Uniformity

The behavior of thermo-mechanical stresses in functionally graded axisymmetric rotating hollow disks with variable thickness is analyzed. The material is assumed to be functionally graded in the radial direction. First, a two-dimensional axisymmetric model of the functionally graded rotating disk is developed using the finite element method. Exact solutions for stresses are then obtained assuming that the plane theory of elasticity holds. These solutions are in accordance with finite element ones, thus showing the validity of the assumption. Finally, in order to reduce the maximum equivalent stress along the radius, the optimization of the material distribution is addressed. To avoid subsequent finite element simulations in the optimization process, which can be computationally demanding, a nonlinear constrained optimization problem is proposed, for which the solution is obtained numerically by the sequential quadratic programming method, showing prominent results in terms of equivalent stress uniformity.

Analysis of High-Speed Rotating Disks With Variable Thickness and Inhomogeneity

1994 ◽  
Vol 61 (1) ◽  
pp. 186-191 ◽  
Author(s):  
Kai-Yuan Yeh ◽  
R. P. S. Han
Keyword(s):  
High Speed ◽  
Elevated Temperatures ◽  
Rotating Disk ◽  
Rotating Disks ◽  
Analytic Expressions ◽  
Inhomogeneous Temperature ◽  
Inhomogeneous Temperature Field ◽  
Thermoelastic Stresses ◽  
Shrink Fit ◽  
Annular Disks

A rotating disk with varying thickness and inhomogeneity, and subjected to a steady, inhomogeneous temperature field is analyzed. To handle the arbitrary profile, the disk is discretized into a series of uniform annular disks possessing constant material properties and then solved by the step-reduction method. Analytic expressions for thermoelastic stresses are given, and based on these results, the formulation is extended to include the calculation of shrink fit, the solving of the inverse problem for equistrength rotating disks, and the computations of plastic stresses and creep at elevated temperatures.

Stresses in Rotating Disks at High Temperatures

1946 ◽  
Vol 13 (1) ◽  
pp. A45-A52
Author(s):  
A. Stanley Thompson
Keyword(s):  
Radial Stress ◽  
Coefficient Of Thermal Expansion ◽  
Variable Temperature ◽  
Rotating Disk ◽  
Allowable Stress ◽  
Tabular Form ◽  
Rotating Disks ◽  
Elasticity Coefficient ◽  
Plastic Disk ◽  
General Method

Abstract A general method was found by which the problem of the rotating disk with any arbitrary profile could be solved, including the effect of plastic flow and of variable temperature, and including the change with temperature of modulus of elasticity, coefficient of thermal expansion, and allowable stress. The solution requires for its application to a specific disk only the elementary arithmetic involved in completion of a tabular form sheet. Two applications of the method are made. For an arbitrary disk profile, an integral equation was found which converges rapidly to the radial stress distribution in a series of successive substitutions. For an arbitrary choice of radial stress, the necessary disk profile can be found in one calculation. Appendix 1 gives an example of the use of the method for the design of a partially plastic disk with a central hole.

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