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.

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.

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.

GENERALIZED MANDELBROT SETS FOR MEROMORPHIC COMPLEX AND QUATERNIONIC MAPS

2002 ◽  
Vol 12 (08) ◽  
pp. 1755-1777 ◽  
Author(s):  
WALTER BUCHANAN ◽  
JAGANNATHAN GOMATAM ◽  
BONNIE STEVES
Keyword(s):  
Parameter Space ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Rational Maps ◽  
Fractal Structures ◽  
Stability Regions ◽  
Theoretical Support ◽  
The Stability ◽  
Definition Of ◽  
Meromorphic Maps

The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z2+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

Postmodern More

Moreana ◽  
2003 ◽  
Vol 40 (Number 153- (1-2) ◽  
pp. 219-239
Author(s):  
Anne Lake Prescott
Keyword(s):  
Human Language ◽  
Thomas More ◽  
The Stability ◽  
Definition Of ◽  
Recent Critic

Thomas More is often called a “humanist,” and rightly so if the word has its usual meaning in scholarship on the Renaissance. “Humanist” has by now acquired so many different and contradictory meanings, however, that it needs to be applied carefully to the likes of More. Many postmodernists tend to use the word, pejoratively, to mean someone who believes in an autonomous self, the stability of words, reason, and the possibility of determinable meanings. Without quite arguing that More was a postmodernist avant la lettre, this essay suggests that he was not a “humanist” who stalks the pages of much recent postmodernist theory and that in fact even while remaining a devout Catholic and sensible lawyer he was quite as aware as any recent critic of the slipperiness of human selves and human language. It is time that literary critics tightened up their definition of “humanist,” especially when writing about the Renaissance.

scholarly journals Influence of study model, baseline catalytic concentrations and analytical system on the stability of serum alanine aminotransferase

2020 ◽  
Vol 1 (2) ◽  
Author(s):  
Josep Miquel Bauça ◽  
Andrea Caballero ◽  
Carolina Gómez ◽  
Débora Martínez-Espartosa ◽  
Isabel García del Pino ◽  
...  
Keyword(s):  
Alanine Aminotransferase ◽  
Experimental Models ◽  
Individual Variability ◽  
Serum Samples ◽  
Multicentric Study ◽  
Different Temperatures ◽  
The Stability ◽  
Analytical System ◽  
Definition Of ◽  
Analytical Platforms

AbstractObjectivesThe stability of the analytes most commonly used in routine clinical practice has been the subject of intensive research, with varying and even conflicting results. Such is the case of alanine aminotransferase (ALT). The purpose of this study was to determine the stability of serum ALT according to different variables.MethodsA multicentric study was conducted in eight laboratories using serum samples with known initial catalytic concentrations of ALT within four different ranges, namely: <50 U/L (<0.83 μkat/L), 50–200 U/L (0.83–3.33 μkat/L), 200–400 U/L (3.33–6.67 μkat/L) and >400 U/L (>6.67 μkat/L). Samples were stored for seven days at two different temperatures using four experimental models and four laboratory analytical platforms. The respective stability equations were calculated by linear regression. A multivariate model was used to assess the influence of different variables.ResultsCatalytic concentrations of ALT decreased gradually over time. Temperature (−4%/day at room temperature vs. −1%/day under refrigeration) and the analytical platform had a significant impact, with Architect (Abbott) showing the greatest instability. Initial catalytic concentrations of ALT only had a slight impact on stability, whereas the experimental model had no impact at all.ConclusionsThe constant decrease in serum ALT is reduced when refrigerated. Scarcely studied variables were found to have a significant impact on ALT stability. This observation, added to a considerable inter-individual variability, makes larger studies necessary for the definition of stability equations.

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