scholarly journals Stability Analysis of a Spinning Elastic Disk Under a Stationary Concentrated Edge Load

1994 ◽  
Vol 61 (4) ◽  
pp. 788-792 ◽  
Author(s):  
Jen-San Chen
Keyword(s):  
Stability Analysis ◽  
Traveling Waves ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Transverse Component ◽  
Membrane Stress ◽  
Reflected Wave ◽  
Spinning Disk ◽  
Elastic Disk ◽  
Stationary Type

The natural frequencies and stability of a spinning elastic disk subjected to a stationary concentrated edge load are investigated both numerically and analytically. It is found that a stationary, conservative, compressive edge load decreases the natural frequencies of the forward and backward traveling waves, but increases the natural frequency of the so-called reflected wave. It is also found that the significance of the effect of the conservative edge load on the natural frequencies and stability of the spinning disk is solely through the transverse component of the edge load on the boundary, and not through the membrane stress field it produces inside the disk. In addition, the compressive edge load induces a stationary-type instability before the critical speed, and induces a merged-type instability after the critical speed when a reflected wave meets a forward or a backward wave. The expression for the derivative of the eigenvalues of the spinning disk with respect to the edge load is derived to verify the numerical results.

scholarly journals Vibration and Stability of a Spinning Disk in Contact With Evenly-Spaced Stationary Load Systems

1998 ◽  
Vol 120 (1) ◽  
pp. 301-302
Author(s):  
Jen-San Chen ◽  
Cheng-Chou Wong
Keyword(s):  
Finite Element ◽  
Natural Frequency ◽  
Critical Speed ◽  
Complex Conjugate ◽  
Element Calculation ◽  
Modal Interactions ◽  
Reflected Wave ◽  
Spinning Disk ◽  
Divergence Instability ◽  
The Difference

The titled problem is studied numerically by finite element calculation. Attention is focused on the behavior of modal interactions when two modes are almost degenerate. In the case when the difference between the numbers of nodal diameters of these two modes is equal to a multiple of the number of the stationary load systems, the frequency loci may merge together (when one of these two modes is a reflected wave) or veer away (when both modes are non-reflected). Otherwise, the natural frequency loci simply cross each other and no instability is induced. When a backward wave meets its complex conjugate at the critical speed, it is found that divergence instability is induced when two times the number of nodal diameters is equal to a multiple of the number of stationary springs.

Vibration and Stability of a Spinning Disk Under Stationary Distributed Edge Loads

1996 ◽  
Vol 63 (2) ◽  
pp. 439-444 ◽  
Author(s):  
Jen-San Chen
Keyword(s):  
Natural Frequency ◽  
Natural Frequencies ◽  
Modal Interaction ◽  
Reflected Waves ◽  
Reflected Wave ◽  
Nonzero Integer ◽  
Spinning Disk ◽  
The Difference

The vibration and stability of a spinning disk under conservative distributed edge tractions are studied both numerically and analytically. The edge traction is circumferentially stationary in the space. When the compressive traction is uniform, it is found that no modal interaction occurs and the natural frequencies of all nonreflected waves decrease, while the natural frequencies of the reflected waves increase. When the spinning disk is under distributed traction in the form of cos kθ, where k is a nonzero integer, it is found that the eigenvalue only changes slightly under the edge traction if the natural frequency of interest is well separated from others. When two modes are almost degenerate, however, modal interaction may or may not occur. It is observed that when the difference between the number of nodal diameters of these two modes is equal to ±k, frequency veering occurs when both modes are nonreflected, and merging occurs when one of these two modes is a reflected wave. In applying this rule, the number of nodal diameters of the forward and the reflected wave is considered as negative.

In-Plane Parametric Vibrations of Curved Bellows Subjected to Oscillating Internal Fluid Pressure Excitation

2004 ◽  
Author(s):  
Masahiro Watanabe ◽  
Eiji Tachibana ◽  
Nobuyuki Kobayashi
Keyword(s):  
Stability Analysis ◽  
Natural Frequencies ◽  
Parametric Instability ◽  
Fluid Pressure ◽  
Parametric Vibrations ◽  
Internal Fluid ◽  
Mathieu’S Equation ◽  
Stability Maps ◽  
Plane Direction ◽  
Pressure Excitation

This paper deals with the theoretical stability analysis of in-plane parametric vibrations of a curved bellows subjected to periodic internal fluid pressure excitation. The curved bellows studied in this paper are fixed at both ends rigidly, and are excited by the periodic internal fluid pressure. In the theoretical stability analysis, the governing equation of the curved bellows subjected to periodic internal fluid pressure excitation is derived as a Mathieu’s equation by using finite element method (FEM). Natural frequencies of the curved bellows are examined and stability maps are presented for in-plane parametric instability. It is found that the natural frequencies of the curved bellows decrease with increasing the static internal fluid pressure and buckling occurs due to high internal fluid pressure. It is also found that two types of parametric vibrations, longitudinal and transverse vibrations, occur to the curved bellows in-plane direction due to the periodic internal fluid pressure excitation. Moreover, effects of axis curvature on the parametric instability regions are examined theoretically.

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.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2020 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems, a general algorithm has been developed to compute natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam’s cross section.

scholarly journals Linear flexural natural frequencies and stability analysis of spinning Rayleigh beams: application to clamped-clamped beams.

2018 ◽  
Vol 241 ◽  
pp. 01002
Author(s):  
Mohamed Amine Aouadi ◽  
Faouzi Lakrad
Keyword(s):  
Stability Analysis ◽  
Boundary Conditions ◽  
Linear Models ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Analytical Computation ◽  
Rayleigh Beam ◽  
Clamped Beams

In the present paper 3D bending linear free vibrations of spinning Rayleigh beams are investigated. Four linear models, that differ in the linearization process, are studied. A focus on analytical computation of natural frequencies for a broad range of boundary conditions is highlighted. Then, the conditions of occurrence of divergence and flutter instabilities are determined. Finally, a case study consisting of a clamped-clamped Rayleigh beam is studied. It is found that the free vibrations destabilization process depends on the used linearization approach.

scholarly journals Mathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System

1992 ◽  
Vol 59 (2) ◽  
pp. 390-397 ◽  
Author(s):  
Jen-San Chen ◽  
D. B. Bogy
Keyword(s):  
Friction Force ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Mathematical Structure ◽  
Transverse Mass ◽  
Modal Interactions ◽  
Spinning Disk ◽  
Physical Insight ◽  
Insight Into

In a previous paper (Chen and Bogy, 1992) we studied the effects of various load parameters, such as friction force, transverse mass, damping, stiffness and the analogous pitching parameters, of a stationary load system in contact with the spinning disk on the natural frequencies and stability of the system when the original eigenvalues of interest are well separated. This paper is a follow-up investigation to deal with the situations in which two eigenvalues of the freely spinning disk are almost equal (degenerate) and strong modal interactions occur when the load parameters are introduced. After comparing an eigenfunction expansion with the finite element numerical results, we find that for each of the transverse and pitching load parameters, a properly chosen two-mode approximation can exhibit all the important features of the eigenvalue changes. Based on this two-mode approximation we study the mathematical structure of the eigenvalues in the neighborhood of degenerate points in the natural frequency-rotation speed plane. In the case of friction force, however, it is found that at least a four-mode approximation is required to reproduce the eigenvalue structure. The observations and analyses presented provide physical insight into the modal interactions induced by various load parameters in a spinning disk-stationary load system.

Critical Speeds of Turbomachinery: Computer Predictions vs. Experimental Measurements—Part I: The Rotor Mass—Elastic Model

1987 ◽  
Vol 109 (1) ◽  
pp. 1-7 ◽  
Author(s):  
J. M. Vance ◽  
B. T. Murphy ◽  
H. A. Tripp
Keyword(s):  
Research Program ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Computer Programs ◽  
Elastic Model ◽  
Experimental Measurements ◽  
Critical Speeds ◽  
Improve Accuracy ◽  
Speed Computer ◽  
Rotor Mass

This is the first part (Part I) of two papers describing results of a research program directed at verifying computer programs used to calculate critical speeds of turbomachinery. This research program was undertaken since questions existed about the accuracy of calculations for the second and higher critical speeds. Part I describes improvements in computer programs and data modeling that resulted from comparing measured and calculated “free-free” natural frequencies of several shafts and rotors. Program modifications to improve accuracy include consideration of the effect of disk/shaft attachment stiffness, revised treatment of the end masses, and an improved convergence. Modifications resulting from the study are applicable to many other damped and undamped critical speed computer programs.

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