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.

Instability Mechanisms of a Spinning Disk Under Rotating Damping Forces

1995 ◽  
Vol 48 (11S) ◽  
pp. S127-S131
Author(s):  
F. Y. Huang ◽  
C. D. Mote
Keyword(s):  
Periodic Solution ◽  
Parameter Space ◽  
Stability Boundary ◽  
Rotating Disk ◽  
Positive Definite ◽  
Stable Region ◽  
Damping Force ◽  
Galerkin Solution ◽  
The Stability ◽  
Definition Of

Stability of a rotating disk under rotating positive-definite damping forces is investigated analytically. The stability boundary is located exactly in parameter space by the criterion that at least one non-trivial periodic solution is necessary at every boundary point. The stable region of parameter space is identified through perturbation of a Galerkin solution. A non-trivial periodic solution is shown to exist only when damping forces are not generated with respect to that solution. Instability in the disk-damping system occurs when the wave speed of any mode in the undamped disk, when observed on the disk, is less that the speed of the damping force relative the disk. The instability occurs independent of the magnitude of the force and the definition of the positive-definite damping operator.

scholarly journals Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol

2019 ◽  
Vol 3 (2) ◽  
pp. 156
Author(s):  
Yuni Yulida ◽  
Muhammad Ahsar Karim
Keyword(s):  
Periodic Solution ◽  
Limit Cycle ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Van Der Pol Equation ◽  
Damping Force ◽  
Van Der Pol ◽  
Scale Method ◽  
Existence And Stability ◽  
The Stability

Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.

Super-Harmonic Resonance Analysis on Torsional Vibration of Misaligned Rotor Driven by Universal Joint

2010 ◽  
Vol 26-28 ◽  
pp. 1226-1231 ◽  
Author(s):  
Chang Lin Feng ◽  
Yong Yong Zhu ◽  
De Shi Wang
Keyword(s):  
Periodic Solution ◽  
Torsional Vibration ◽  
Stable Region ◽  
Universal Joint ◽  
Weakly Nonlinear ◽  
Resonance Analysis ◽  
Actual Error ◽  
The Stability ◽  
Harmonic Resonance ◽  
Super Harmonic Resonance

The super-harmonic resonance of the nonlinear torsional vibration of misaligned rotor system driven by universal joint was studied considering both natural structure misalignment and actual error misalignment. Utilizing multi-scale method, the periodic solution of weakly nonlinear torsional vibration equation was obtained corresponding to super-harmonic resonance, including amplitude-frequency and phase-frequency characteristic expressions of steady periodic solution. The stability of equilibrium point was investigated using Lyapunov first approximate stability theory, then the stable region and unstable region on the amplitude of the super-harmonic resonance periodic solution, which varied with the detuning parameter. At last, the driving shaft’s steady periodic motion of the first approximation and its calculation simulation were carried out according to the kinetic relation about driven shaft and driving shaft. It is found that jumping phenomenon and dynamic bifurcation occur when the rotating angular velocity of driving shaft is half of the natural frequency of the deriving system. The results above indicate the fundamental characteristic of the nonlinear dynamic on the misaligned rotor, also applying the foundation for advanced bifurcation and singularities analysis.

scholarly journals Mapping the stability of stellar rotating spheres via linear response theory

2019 ◽  
Vol 487 (1) ◽  
pp. 711-728 ◽  
Author(s):  
S Rozier ◽  
J-B Fouvry ◽  
P G Breen ◽  
A L Varri ◽  
C Pichon ◽  
...  
Keyword(s):  
Parameter Space ◽  
Rotation Rate ◽  
Linear Response ◽  
Globular Clusters ◽  
Stability Boundary ◽  
Linear Response Theory ◽  
Marginal Stability ◽  
Anisotropic Spheres ◽  
Pattern Speeds ◽  
The Stability

Abstract Rotation is ubiquitous in the Universe, and recent kinematic surveys have shown that early-type galaxies and globular clusters are no exception. Yet the linear response of spheroidal rotating stellar systems has seldom been studied. This paper takes a step in this direction by considering the behaviour of spherically symmetric systems with differential rotation. Specifically, the stability of several sequences of Plummer spheres is investigated, in which the total angular momentum, as well as the degree and flavour of anisotropy in the velocity space are varied. To that end, the response matrix method is customized to spherical rotating equilibria. The shapes, pattern speeds and growth rates of the systems’ unstable modes are computed. Detailed comparisons to appropriate N-body measurements are also presented. The marginal stability boundary is charted in the parameter space of velocity anisotropy and rotation rate. When rotation is introduced, two sequences of growing modes are identified corresponding to radially and tangentially biased anisotropic spheres, respectively. For radially anisotropic spheres, growing modes occur on two intersecting surfaces (in the parameter space of anisotropy and rotation), which correspond to fast and slow modes, depending on the net rotation rate. Generalized, approximate stability criteria are finally presented.

Blood Pressure Increases, Birth Weight-Dependent Stability Boundary, and Intraventricular Hemorrhage

PEDIATRICS ◽  
1990 ◽  
Vol 85 (5) ◽  
pp. 727-732
Author(s):  
Edward H. Perry ◽  
Henrietta S. Bada ◽  
John D. Ray ◽  
Sheldon B. Korones ◽  
Kris Arheart ◽  
...  
Keyword(s):  
Blood Pressure ◽  
Intensive Care ◽  
Birth Weight ◽  
Intraventricular Hemorrhage ◽  
Stability Boundary ◽  
Stable Region ◽  
Low Birth Weight Infant ◽  
Motor Activities ◽  
The Stability ◽  
Transcutaneous Po2

The blood pressure (BP) and transcutaneous Po2, (TcPo2) changes associated with intensive care procedures were evaluated to determine whether responses differ between babies with and without periventricular-intraventricular hemorrhage (PV-IVH). Fifty-three inborn babies ≤1500 g were studied using a microcomputer-based monitoring system. With almost any procedure including a seemingly benign one such as a diaper change, peak systolic BP increased and TcPo2 decreased. However, responses to interventions did not differ between babies with PV-IVH and those without PV-IVH. Neither did these responses differ between those with birth weight ≤1000 g and >1000 g. When each baby's record was scanned for the highest peak systolic BP before diagnosis of PV-IVH or within 48 hours in those with no PV-IVH and their BP points plotted against birth weight, a stable region was evident wherein PV-IVH occurred at a lower incidence (13%). When peak systolic BP was beyond this stable region, the incidence of PV-IVH was significantly higher, 70% (P < .0001). The stability boundary for the maximum systolic BP is birth weight-dependent; the limit for the highest tolerable peak systolic BP is lower for the low-birth-weight infant. In over 70% of instances the highest peak systolic BP was associated with motor activities either induced by nursery procedures or spontaneous. We speculate that decreasing the frequency of intensive care interventions may decrease episodic BP increases to levels beyond the birth weight-dependent stability boundary where PV-IVH is likely to occur.

scholarly journals A New Design Method for PI-PD Control of Unstable Fractional-Order System with Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Min Zheng ◽  
Tao Huang ◽  
Guangfeng Zhang
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Graphical Method ◽  
Stability Boundary ◽  
Fractional Order System ◽  
Stable Region ◽  
Order System ◽  
Pd Controller ◽  
The Stability ◽  
System With Time Delay

In this paper, a practical PI-PD controller parameter tuning method is proposed, which uses the incenter of the triangle and the Fermat point of the convex polygon to optimize the PI-PD controller. Combined with the stability boundary locus method, the PI-PD controller parameters that can ensure stability for the unstable fractional-order system with time delay are obtained. Firstly, the parameters of the inner-loop PD controller are determined by the centre coordinates of the CSR in the kd−kf plane. Secondly, a new graphical method is used to calculate the parameters of the PI controller, in which Fermat points in the CSR of (kp−ki) plane are selected. Furthermore, the method is extended to uncertain systems, and the PI-PD controller parameters are obtained by using the proposed method through common stable region of all stable regions. The proposed graphical method not only ensures the stability of the closed-loop system but also avoids the complicated optimization calculations. The superior control performance of this method is illustrated by simulation.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals Stability and Instability in the Lysogenic State of Phage Lambda

2010 ◽  
Vol 192 (22) ◽  
pp. 6064-6076 ◽  
Author(s):  
John W. Little ◽  
Christine B. Michalowski
Keyword(s):  
Intrinsic Rate ◽  
Growth Conditions ◽  
Emergent Properties ◽  
Regulatory Circuits ◽  
Stability And Instability ◽  
Natural Range ◽  
Analysis Of Stability ◽  
Stable Association ◽  
The Stability ◽  
Gene Regulatory Circuits

ABSTRACT Complex gene regulatory circuits exhibit emergent properties that are difficult to predict from the behavior of the components. One such property is the stability of regulatory states. Here we analyze the stability of the lysogenic state of phage λ. In this state, the virus maintains a stable association with the host, and the lytic functions of the virus are repressed by the viral CI repressor. This state readily switches to the lytic pathway when the host SOS system is induced. A low level of SOS-dependent switching occurs without an overt stimulus. We found that the intrinsic rate of switching to the lytic pathway, measured in a host lacking the SOS response, was almost undetectably low, probably less than 10−8/generation. We surmise that this low rate has not been selected directly during evolution but results from optimizing the rate of switching in a wild-type host over the natural range of SOS-inducing conditions. We also analyzed a mutant, λprm240, in which the promoter controlling CI expression was weakened, rendering lysogens unstable. Strikingly, the intrinsic stability of λprm240 lysogens depended markedly on the growth conditions; lysogens grown in minimal medium were nearly stable but switched at high rates when grown in rich medium. These effects on stability likely reflect corresponding effects on the strength of the prm240 promoter, measured in an uncoupled assay system. Several derivatives of λprm240 with altered stabilities were characterized. This mutant and its derivatives afford a model system for further analysis of stability.

Sign in / Sign up

Export Citation Format

Share Document