scholarly journals The second approximation in a small parameter to the solution of the problem of loss of the stability of a rotating disk in the refined formulation

2018 ◽  
pp. 33-39 ◽  
Author(s):  
D.M. Lila ◽  
Keyword(s):  
Small Parameter ◽  
Rotating Disk ◽  
Refined Formulation ◽  
The Stability

Zeros of a Class of Transcendental Equation with Application to Bifurcation of DDE

2016 ◽  
Vol 26 (04) ◽  
pp. 1650062 ◽  
Author(s):  
Kit Ian Kou ◽  
Yijun Lou ◽  
Yong-Hui Xia
Keyword(s):  
Small Parameter ◽  
Climate Changes ◽  
Transcendental Equation ◽  
Bifurcation Parameter ◽  
Parameter Perturbation ◽  
Distribution Of Zeros ◽  
World Model ◽  
The Real ◽  
Stability And Bifurcations ◽  
The Stability

Zeros of a class of transcendental equation with small parameter [Formula: see text] are considered in this paper. There have been many works in the literature considering the distribution of zeros of the transcendental equation by choosing the delay [Formula: see text] as bifurcation parameter. Different from standard consideration, we choose [Formula: see text] as bifurcation parameter (not the delay [Formula: see text]) to discuss the distribution of zeros of such transcendental equation. After mathematical analysis, the obtained results are successfully applied to the bifurcation analysis in a biological model in the real word phenomenon. In the real world model, the effect of climate changes can be seen as the small parameter perturbation, which can induce bifurcations and instability. We present two methods to analyze the stability and bifurcations.

Stability Increase of Aerodynamically Unstable Rotors Using Intentional Mistuning

2006 ◽  
Author(s):  
Carlos Martel ◽  
Roque Corral ◽  
Jose´ Miguel Llorens
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Structural Dynamics ◽  
Low Pressure ◽  
Vibration Characteristics ◽  
Stability Properties ◽  
First Order ◽  
Low Pressure Turbine ◽  
Turbine Rotors ◽  
The Stability

A new simple asymptotic mistuning model (AMM), which constitutes an extension of the well known Fundamental Mistuning Model for groups of modes belonging to a modal family exhibiting a large variation of the tuned vibration characteristics, is used to analyze the effect of mistuning on the stability properties of aerodynamically unstable rotors. The model assumes that both, the aerodynamics and the structural dynamics of the assembly are linear, and retains the first order terms of a fully consistent asymptotic expansion of the tuned system where the small parameter is the blade mistuning. The simplicity of the model allows the optimization of the blade mistuning pattern to achieve maximum rotor stability. The results of the application of this technique to realistic welded-in-pair and interlock low-pressure-turbine rotors are also presented.

scholarly journals Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Small Parameter ◽  
Initial Conditions ◽  
Geometric Interpretation ◽  
Rigid Bodies ◽  
Parameter Method ◽  
Large Parameter ◽  
Small Parameter Method ◽  
Rotational Motions ◽  
The Stability

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.

scholarly journals The centrifugal instability of the boundary-layer flow over slender rotating cones

2014 ◽  
Vol 755 ◽  
pp. 274-293 ◽  
Author(s):  
Z. Hussain ◽  
S. J. Garrett ◽  
S. O. Stephen
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Rotating Disk ◽  
Alternative Formulation ◽  
Centrifugal Instability ◽  
Half Angle ◽  
Layer Flow ◽  
The Stability ◽  
Experimental And Theoretical Studies ◽  
Numerical Approaches

AbstractExisting experimental and theoretical studies are discussed which lead to the clear hypothesis of a hitherto unidentified convective instability mode that dominates within the boundary-layer flow over slender rotating cones. The mode manifests as Görtler-type counter-rotating spiral vortices, indicative of a centrifugal mechanism. Although a formulation consistent with the classic rotating-disk problem has been successful in predicting the stability characteristics over broad cones, it is unable to identify such a centrifugal mode as the half-angle is reduced. An alternative formulation is developed and the governing equations solved using both short-wavelength asymptotic and numerical approaches to independently identify the centrifugal mode.

Mathematical Analysis of Stability of a Spinning Disk Under Rotating, Arbitrarily Large Damping Forces

1996 ◽  
Vol 118 (4) ◽  
pp. 657-662 ◽  
Author(s):  
F. Y. Huang ◽  
C. D. Mote
Keyword(s):  
Periodic Solution ◽  
Parameter Space ◽  
Perturbation Technique ◽  
Stability Boundary ◽  
Rotating Disk ◽  
Stable Region ◽  
Damping Force ◽  
Analysis Of Stability ◽  
The Stability ◽  
Nontrivial Periodic Solution

Stability of a rotating disk under rotating, arbitrarily large damping forces is investigated analytically. Points possibly residing on the stability boundary are located exactly in parameter space based on the criterion that at least one nontrivial periodic solution is necessary at every boundary point. A perturbation technique and the Galerkin method are used to predict whether these points of periodic solution reside on the stability boundary, and to identify the stable region in parameter space. A nontrivial periodic solution is shown to exist only when the damping does not generate forces with respect to that solution. Instability occurs when the wave speed of a mode in the uncoupled disk, when observed on the disk, is exceeded by the rotation speed of the damping force relative to the disk. The instability is independent of the magnitude of the force and the type of positive-definite damping operator in the applied region. For a single dashpot, nontrivial periodic solutions exist at the points where the uncoupled disk has repeated eigenfrequencies on a frame rotating with the dashpot and the dashpot neither damps nor energizes these modes substantially around these points.

Self-Excited Vibration in Flexible Rotating Disks Subjected to Various Transverse Interactive Forces: A General Approach

1999 ◽  
Vol 66 (3) ◽  
pp. 800-805 ◽  
Author(s):  
J. Tian ◽  
S. G. Hutton
Keyword(s):  
Time Lag ◽  
Rotating Disk ◽  
Flux Analysis ◽  
Clear Understanding ◽  
Rotating Disks ◽  
Generalized Fourier Series ◽  
Excited Vibration ◽  
The Stability ◽  
Rotating Flexible Disk ◽  
Saw Blade

A generalized approach to predict the physical instability mechanisms that are involved in the interaction between a rotating flexible disk and a stationary constraining system is developed. Based upon equations derived for an energy flux analysis, unified instability conditions for various lateral interactive forces are presented. These developments lead to a clear understanding of the physical mechanisms involved in the development of vibrational instabilities. New developments also involve the stability analysis of a rotating disk subjected to multiple moving concentrated regenerative and follower interactive forces that act over a space-fixed sector. The lateral regenerative interactive forces that are responsible for self-excited vibrations in saw-blade cutting are identified and modeled. The generalized Fourier series method is proposed to develop a characteristic equation for time-varying dynamic systems with or without time lag. The resulting equation can be solved efficiently by using Mu¨ller’s algorithm with deflation.

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