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.

Rocking Vibration of Rotating Disk and Spindle Systems With Asymmetric Bearings

2001 ◽  
Author(s):  
Jung Seo Park ◽  
I. Y. Shen ◽  
C.-P. Roger Ku
Keyword(s):  
Numerical Simulation ◽  
Rotational Speed ◽  
Natural Frequencies ◽  
Rotating Disk ◽  
Mode Shapes ◽  
Unperturbed System ◽  
High Rotational Speed ◽  
Spindle System ◽  
Rocking Motion ◽  
Gyroscopic Effects

Abstract This paper studies how bearing asymmetry affects natural frequencies and mode shapes of a rotating disk/spindle system through a numerical simulation and a perturbation analysis. Existing literature has shown that rocking motion of a rotating disk/spindle system with symmetric bearings consists of rigid body rocking of the spindle, one-nodal-diameter modes of each disk, and deformation of spindle bearings. The rocking motion, characterized by (0,1) unbalanced modes, has repeated natural frequencies when the spindle is stationary, because the disk/spindle system is axisymmetric. For a rotating spindle, (0,1) unbalanced modes evolve into forward and backward precession with circular orbits. In this paper, the numerical simulation shows that bearing asymmetry splits a pair of repeated (0,1) unbalanced modes into two modes with distinct frequencies when the spindle is stationary. Moreover, when the rotational speed increases from zero, the (0,1) unbalanced mode with lower frequency evolves into backward precession and the (0,1) unbalanced mode with higher frequency evolves into forward precession. The precession orbits are elliptical because of the bearing asymmetry. Two perturbation schemes are developed to prove the phenomena observed in the numerical simulation. For low rotational speed, a stationary disk/spindle system with symmetric bearings serves as the unperturbed system. Both the bearing asymmetry and gyroscopic effects from rotation form the perturbation. A contraction iteration predicts the effects of bearing asymmetry on natural frequencies and mode shapes. For high rotational speed, a rotating (gyroscopic) disk/spindle system with symmetric bearings serves as the unperturbed system. The bearing asymmetry forms the perturbation. To obtain a perturbation solution, the solvability condition is first derived for the unperturbed gyroscopic system. Lindsted-Poincaré approach then predicts the effects of bearing asymmetry on natural frequencies and mode shapes of the rotating disk/spindle system.

Effects of Elevated Temperatures on Rocking Vibration of Rotating Disk and Spindle Systems

2002 ◽  
Vol 124 (4) ◽  
pp. 794-800 ◽  
Author(s):  
Chaw-Wu Tseng ◽  
Jr-Yi Shen ◽  
C.-P. R. Ku ◽  
I. Y. Shen
Keyword(s):  
Mathematical Model ◽  
Natural Frequency ◽  
Elevated Temperatures ◽  
Natural Frequencies ◽  
Rotating Disk ◽  
Temperature Variations ◽  
Spindle System ◽  
Radial Bearing ◽  
Bearing Stiffness ◽  
Rocking Vibration

This paper studies how temperature variations affect natural frequencies of rocking vibration of a rotating disk and spindle system through mathematical modeling and experimental measurements. Existing literature has shown that both radial bearing stiffness krr and natural frequency ω01B of one-nodal-diameter disk modes could substantially affect natural frequencies ω01U of rocking vibration. In this paper, a preliminary experiment first identifies that relaxation of bearing stiffness krr is the dominating factor to shift the natural frequency ω01U at elevated temperatures. In addition, the bearing relaxation primarily results from thermal mismatch between the bearing raceways and the rotating hub. Guided by the experimental results, a mathematical model is developed to determine how temperature variations affect bearing contact angles, bearing preloads, and subsequently the radial bearing stiffness krr. Based on the bearing stiffness krr and disk frequency ω01B at elevated temperatures, one can predict natural frequency ω01U of rocking vibration through the mathematical model by Shen and Ku (1997). Finally, ω01U of a rotating disk and spindle system are measured in a thermal chamber to validate the theoretical predictions.

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.

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.

scholarly journals On the detection of natural frequencies and mode shapes of submerged rotating disk-like structures from the casing

2015 ◽  
Vol 60-61 ◽  
pp. 547-570 ◽  
Author(s):  
Alexandre Presas ◽  
David Valentin ◽  
Eduard Egusquiza ◽  
Carme Valero ◽  
Ulrich Seidel
Keyword(s):  
Natural Frequencies ◽  
Rotating Disk ◽  
Mode Shapes

In-Plane Inertial Coupling in Tuned and Severely Mistuned Bladed Disks

1982 ◽  
Author(s):  
E. F. Crawley
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Analytic Model ◽  
Angular Rotation ◽  
Structural Coupling ◽  
Nodal Diameter ◽  
Rigid Disk ◽  
Compressor Rotor ◽  
Rigid Body Modes ◽  
Bending Modes

A model has been developed and verified for blade-disk-shaft coupling in rotors due to the in-plane rigid body modes of the disk. An analytic model has been developed which couples the in-plane rigid body modes of the disk on an elastic shaft with the blade bending modes. Bench resonance tests were carried out on the M.I.T. Compressor Rotor, typical of research rotors with flexible blades and a thick rigid disk. When the rotor was carefully tuned, the structural coupling of the blades by the disks was confined to zero and one nodal diameter modes, whose modal frequencies were greater than the blade cantilever frequency. In the case of the tuned rotor, and in two cases where severe mistuning was intentionally introduced, agreement between the predicted and observed natural frequencies is excellent. The analytic model was then extended to include the effects of constant angular rotation of the disk.

Effect of a Circumferential Arc Crack on the Vibration Characteristics of a Flexible Spinning Disk

2007 ◽  
Author(s):  
Nikhit N. Nair ◽  
Hamid N. Hashemi ◽  
Grant M. Warner ◽  
M. Olia
Keyword(s):  
Natural Frequencies ◽  
Crack Opening ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Rotating Disk ◽  
Vibration Characteristics ◽  
Coupled Systems ◽  
Mode Shapes ◽  
Crack Location ◽  
Inner Region

The vibration characteristics of a circumferentially cracked rotating disk are investigated. The disk is assumed to be axisymmetric, flexible and clamped at the center. The crack increases the local flexibility of the disk at the crack location and is modeled as linear and torsional springs, connecting the two segments of the disk. The spring constants are evaluated by considering crack opening displacements due to bending moment and shear force at the crack location. The equations of motion of two segments of the disk, for disk operating in vacuum as well as subjected to shear fluid flow are developed. Using the Finite Difference Technique, the coupled systems of equations are solved and the natural frequencies and mode shapes are obtained. The mode shapes are seen to be comparatively flattened in the inner region of the crack and heightened towards the periphery of the disk. Shear fluid loading reduces the natural frequencies and results in a quicker onset of instability. It is observed that the effect of the crack on the vibration characteristics of the disk is mainly a function of the crack location.

A Normalized Modal Eigenvalue Approach for Resolving Modal Interaction

1997 ◽  
Vol 119 (3) ◽  
pp. 647-650 ◽  
Author(s):  
M.-T. Yang ◽  
J. H. Griffin
Keyword(s):  
Monte Carlo ◽  
Eigenvalue Problem ◽  
Perturbation Analysis ◽  
Material Properties ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Modal Interaction ◽  
Eigenvalue Approach ◽  
The Difference

Modal interaction refers to the way that the modes of a structure interact when its geometry and material properties are perturbed. The amount of interaction between the neighboring modes depends on the closeness of the natural frequencies, the mode shapes, and the magnitude and distribution of the perturbation. By formulating the structural eigenvalue problem as a normalized modal eigenvalue problem, it is shown that the amount of interaction in two modes can be simply characterized by six normalized modal parameters and the difference between the normalized frequencies. In this paper, the statistical behaviors of the normalized frequencies and modes are investigated based on a perturbation analysis. The results are independently verified by Monte Carlo simulations.

scholarly journals A Normalized Modal Eigenvalue Approach for Resolving Modal Interaction

1996 ◽  
Author(s):  
Ming-Ta Yang ◽  
Jerry H. Griffin
Keyword(s):  
Monte Carlo ◽  
Eigenvalue Problem ◽  
Perturbation Analysis ◽  
Material Properties ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Modal Interaction ◽  
Eigenvalue Approach ◽  
The Difference

Modal interaction refers to the way that the modes of a structure interact when its geometry and material properties are perturbed. The amount of interaction between the neighboring modes depends on the closeness of the natural frequencies, the mode shapes, and the magnitude and distribution of the perturbation. By formulating the structural eigenvalue problem as a normalized modal eigenvalue problem, it is shown that the amount of interaction in two modes can be simply characterized by six normalized modal parameters and the difference between the normalized frequencies. In this paper, the statistical behaviors of the normalized frequencies and modes are investigated based on a perturbation analysis. The results are independently verified by Monte Carlo simulations.

Free Dry and Wet Vibration of 2-Way Tapered Hollow Marine Rudder With Non-Classical Pivot: Theoretical Study

2015 ◽  
Author(s):  
Rahul Jindal ◽  
Nabanita Datta
Keyword(s):  
Rigid Body ◽  
Natural Frequencies ◽  
Added Mass ◽  
Chord Length ◽  
Mode Shapes ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Uniform Beam ◽  
Aspect Ratios ◽  
Beam Mode

A theoretical analysis of free dry and wet vibration of a trapezoidal, 2-way tapered, marine spade rudder, is presented. The rudder is considered as a hollow Kirchhoff’s plate, with the chord section as a NACA profile. The chord length and the thickness taper from the top to the bottom, over the vertical span. The rudder is pivoted at the top, with the pivot behind the leading edge. The pivot is modeled as a combination of a translational and a rotational spring, in order to include the rigid body modes of the rudder vibration. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, in order to establish the non-uniform beam mode shapes. The span-wise beam is a linearly tapered vertical cantilever, with non-classical edge at the top and free at the bottom. The chord-wise section is a 2-span beam with the ends free, and four continuity conditions at the pivot. The non-uniform mode shapes, in either direction, are a weighted summation of the uniform beam mode shapes, which also satisfy the boundary/continuity conditions. They now act as admissible spatial functions to the plate vibration, which is analyzed by the Galerkin’s method. Eigenvalue analysis generates the plate natural frequencies. A weighted superposition, of the product of the beam mode shapes, in either direction, generates the plate mode shapes. Alternately, uniform beam mode shapes are used as admissible functions into the Galerkin’s method for the plate natural frequencies and mode shapes. The natural frequencies are generated for various positions of the rudder stock along the chord length. The pivot conditions (in both translational and rotational rigid body degree of freedom) influence the prominence of the rigid body mode shapes. The natural frequencies are analyzed for various pivot fixities, taper ratios, and aspect ratios of the plate. This is followed by the wet vibration analysis of the rudder. First, 2D strip theory is used to generate the added mass of each chord section. Constant strength source distribution technique is used to generate the added mass in sway and yaw of a 2D aerofoil. Each flexural and torsional mode is associated with its own added mass. Various empirical corrections are done to account for the 3D flow. Finally, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, and aspect ratios of the plate.

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