Bifurcations of Eigenvalues of Gyroscopic Systems With Parameters Near Stability Boundaries

2000 ◽  
Vol 68 (2) ◽  
pp. 199-205 ◽  
Author(s):  
A. P. Seyranian ◽  
W. Kliem
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundaries ◽  
Stability And Instability ◽  
Generalized Eigenvectors ◽  
Double Eigenvalues ◽  
Normal Vectors ◽  
Axially Moving ◽  
The Stability ◽  
Derivatives Of ◽  
Gyroscopic Systems

This paper deals with stability problems of linear gyroscopic systems Mx¨+Gx˙+Kx=0 with finite or infinite degrees-of-freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulas for the behavior of eigenvalues under a change of parameters are obtained. It is shown that the bifurcation (splitting) of double eigenvalues is closely related to the stability, flutter, and divergence boundaries in the parameter space. Normal vectors to these boundaries are derived using only information at a boundary point: eigenvalues, eigenvectors, and generalized eigenvectors, as well as first derivatives of the system matrices (or operators) with respect to parameters. These results provide simple and constructive stability and instability criteria. The presented theory is exemplified by two mechanical problems: a rotating elastic shaft carrying a disk, and an axially moving tensioned beam.

On Instability Pockets and Influence of Damping in Parametrically Excited Systems

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Degrees Of Freedom ◽  
State Transition ◽  
Degree Of Freedom ◽  
Transition Matrices ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Parametric Space ◽  
Stability Diagrams ◽  
Wide Range ◽  
The Stability

In most parametrically excited systems, stability boundaries cross each other at several points to form closed unstable subregions commonly known as “instability pockets.” The first aspect of this study explores some general characteristics of these instability pockets and their structural modifications in the parametric space as damping is induced in the system. Second, the possible destabilization of undamped systems due to addition of damping in parametrically excited systems has been investigated. The study is restricted to single degree-of-freedom systems that can be modeled by Hill and quasi-periodic (QP) Hill equations. Three typical cases of Hill equation, e.g., Mathieu, Meissner, and three-frequency Hill equations, are analyzed. State transition matrices of these equations are computed symbolically/analytically over a wide range of system parameters and instability pockets are observed in the stability diagrams of Meissner, three-frequency Hill, and QP Hill equations. Locations of the intersections of stability boundaries (commonly known as coexistence points) are determined using the property that two linearly independent solutions coexist at these intersections. For Meissner equation, with a square wave coefficient, analytical expressions are constructed to compute the number and locations of the instability pockets. In the second part of the study, the symbolic/analytic forms of state transition matrices are used to compute the minimum values of damping coefficients required for instability pockets to vanish from the parametric space. The phenomenon of destabilization due to damping, previously observed in systems with two degrees-of-freedom or higher, is also demonstrated in systems with one degree-of-freedom.

Stability and Dynamics of Axially Moving Unidirectional Plates Partially Immersed in a Liquid

2014 ◽  
Vol 14 (04) ◽  
pp. 1450010 ◽  
Author(s):  
Yan Qing Wang ◽  
Xing Hui Guo ◽  
Zhen Sun ◽  
Jian Li
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Instability ◽  
Plate Theory ◽  
Fluid Pressure ◽  
Added Mass ◽  
Chaotic Motions ◽  
Bernoulli’S Equation ◽  
Singular Functions ◽  
Axially Moving ◽  
The Stability

The stability and dynamics of an axially moving unidirectional plate partially immersed in a liquid and subjected to a nonlinear aerodynamic excitation are investigated. The method of singular functions is adopted to study the dynamic characteristics of the unidirectional plates with discontinuous characteristics. Nonlinearities due to large-amplitude plate motions are considered by using the classical nonlinear thin plate theory, with allowance for the effect of viscous structural damping. The velocity potential and Bernoulli's equation are used to describe the fluid pressure acting on the unidirectional plate. The effect of fluid on the vibrations of the plate may be equivalent to added mass of the plate. The formulation of added mass is obtained from kinematic boundary conditions of the plate–fluid interfaces. The system is discretized by Galerkin's method while a model involving two degrees of freedom, is adopted. Attention is focused on the behavior of the system in the region of dynamic instability, and several motions are found by numerical simulations. The effects of the moving speed and some other parameters on the dynamics of the system are also investigated. It is shown that chaotic motions can occur in this system in several certain regions of parameter space.

Response and Discretization Methods for Axially Moving Materials

1991 ◽  
Vol 44 (11S) ◽  
pp. S279-S284 ◽  
Author(s):  
J. A. Wickert ◽  
C. D. Mote
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
The State ◽  
Space Formulation ◽  
Axially Moving ◽  
Effective Use ◽  
Gyroscopic Systems ◽  
Acceleration Component

Through a convective acceleration component, the equations of motion for axially-moving materials are skew-symmetric in the state space formulation, so that the response problem is best analyzed within the broader context of continuous gyroscopic systems. With particular application to the prototypical traveling string and beam models, a modal analysis that associates degrees of freedom with the complex state eigenfunctions and their conjugates is presented. This procedure is well-suited for harmonic excitation sources, and in some instances, it is more convenient than previous methods which decompose the modal coordinates, eigenfunctions, and generalized forces into real and imaginary components. Also from the state space perspective, Rayleigh’s quotient for gyroscopic systems provides a variational method for determining the eigensolutions of axially-moving materials. Ritz discretization of the quotient can make effective use of the speed-adapting modes of the traveling string and beam models as they are rich in phase, as well as amplitude, content.

Stability of Linear Conservative Gyroscopic Systems

1991 ◽  
Vol 58 (1) ◽  
pp. 229-232 ◽  
Author(s):  
J. A. Walker
Keyword(s):  
Sufficient Conditions ◽  
Design Constraints ◽  
Stability And Instability ◽  
The Stability ◽  
Gyroscopic Systems

Sufficient conditions are obtained for the stability and instability of linear conservative gyroscopic systems. The conditions are nonspectral, involve only the definiteness of certain combinations of the coefficient matrices, and may yield useful design constraints. An example is employed to compare these results with earlier results of the same type.

Determining Stability Boundaries Using Gyroscopic Eigenfunctions

2003 ◽  
Vol 125 (3) ◽  
pp. 405-407 ◽  
Author(s):  
Anthony A. Renshaw
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
The Stability ◽  
Simplified Procedure ◽  
Gyroscopic Systems

By taking advantage of modal decoupling and reduction of order, we derive a simplified procedure for applying the method of multiple scales to determine the stability boundaries of parametrically excited, gyroscopic systems. The analytic advantages of the procedure are illustrated with three examples.

Modeling and Stability Analysis of an Axially Moving Beam With Frictional Contact

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
Gottfried Spelsberg-Korspeter ◽  
Oleg N. Kirillov ◽  
Peter Hagedorn
Keyword(s):  
Stability Analysis ◽  
Frictional Contact ◽  
Equations Of Motion ◽  
Axially Moving Beam ◽  
Perturbation Techniques ◽  
Stability Boundaries ◽  
Analytical Approximations ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

This paper considers a moving beam in frictional contact with pads, making the system susceptible for self-excited vibrations. The equations of motion are derived and a stability analysis is performed using perturbation techniques yielding analytical approximations to the stability boundaries. Special attention is given to the interaction of the beam and the rod equations. The mechanism yielding self-excited vibrations does not only occur in moving beams, but also in other moving continua such as rotating plates, for example.

scholarly journals DETERMINATION OF THE STABILITY BOUNDARIES FOR THE HAMILTONIAN SYSTEMS WITH PERIODIC COEFFICIENTS

2005 ◽  
Vol 10 (2) ◽  
pp. 191-204 ◽  
Author(s):  
A. N. Prokopenya
Keyword(s):  
Hamiltonian Systems ◽  
Computer Algebra System ◽  
Many Body ◽  
Stability Boundaries ◽  
Periodic Coefficients ◽  
Stability And Instability ◽  
Existence Of Periodic Solutions ◽  
Linear Differential ◽  
The Stability

We consider the hamiltonian system of linear differential equations with periodic coefficients. Using the infinite determinant method based on the existence of periodic solutions on the boundaries between the domains of stability and instability in the parameter space we have developed the algorithm for analytical computation of the stability boundaries. The algorithm has been realized for the second and the fourth order hamiltonian systems arising in the restricted many-body problems. The stability boundaries have been found in the form of powers series, accurate to the sixth order in a small parameter. All the computations are done with the computer algebra system Mathematica. Nagrinejama Hamiltono tiesiniu diferencialiniu lygčiu su periodiniais koeficientais sistema. Remiantis tuo, kad parametru erdveje stabilumo ir nestabilumo sritis skiriančioje sienoje egzistuoja periodinis sprendinys, sukurtas analitinis minetos sienos apskaičiavimo algoritmas. Algoritmas realizuotas antros ir ketvirtos eiles Hamiltono sistemoms, kylančioms nagrinejant apribotu keleto kūnu uždavinius. Stabilumo srities siena randama laipsnines eilutes pavidalu mažojo parametro šešto laipsnio tikslumu. Skaičiavimai atlikti skaičiavimo algebros paketo Mathematica pagalba.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

The identity policy and its effect in the stability of Middle East after 2010

2019 ◽  
pp. 434
Author(s):  
Ali Hussein Kadhim Alesammi
Keyword(s):  
Middle East ◽  
International Relations ◽  
Social Construct ◽  
Stability And Instability ◽  
The Social ◽  
The Stability ◽  
International Interactions ◽  
Regional Interests ◽  
Political Groups ◽  
Identity Policy

Since 2010 Middle East have many events or what they call "Arab spring events" which it result of overthrow governments and the rise of new political groups, all of this elements was resulting of many international and regional activities and making new regional and international axles, as well as the intersections of the different regional interests, therefore this research will try to study the stability and instability in the region as an independent variable not according to the neorealism or neoliberalism theories, but according to the constructivism theory which it base their assumptions on:  "In the international relations the non-physical structures of international interactions are determined by the identities of the players, which in turn determine the interests that determine the behavior of international players." So the research questions are: 1-What is the identity policy and haw affect in international relations? 2-How the social construct affect in international relations? 3-How the elite's identities for the main actors in the Middle East affect in the regional axles?  

Binding of calcium and lead ions to carboxystarch prepared by action of nitrogen dioxide on native starch and its 2,3-dialdehyde derivatives

1986 ◽  
Vol 51 (6) ◽  
pp. 1340-1351 ◽  
Author(s):  
Rudolf Kohn ◽  
Karol Tihlárik
Keyword(s):  
Nitrogen Dioxide ◽  
Alkaline Medium ◽  
Binding Capacity ◽  
Lead Ions ◽  
Carbonyl Groups ◽  
Weakly Alkaline ◽  
Exchange Cations ◽  
Counter Ions ◽  
The Stability ◽  
Derivatives Of

The binding of calcium and lead ions to carboxy derivatives of starch prepared by allowing nitrogen dioxide to act on native maize starch (procedure A) and on starch 2,3-dialdehyde derivatives of degrees of oxidation DO(d.a.) ≥ 0.94 (procedure B) was studied. The carboxy group content of the samples in the H+ form was 4.6 - 12.1 mmol g-1. The effect of alkaline medium on the stability of the carboxy derivatives and on their ability to bind and exchange cations was examined. The Ca2+ → 2K+ exchange was evaluated in terms of the decrease in the electrostatic free enthalpy Δ(Gel/N)KCa, determined by alkalimetric potentiometric titrations, and the binding of Pb2+ ions was evaluated in terms of the activity of the Pb2+ counter-ions determined in suspensions of Pb salts of the carboxy derivatives by means of an ion specific electrode. The IR and CD spectra revealed that the carboxystarch preparations obtained by procedure A contained, in addition to free carboxy groups, a considerable amount of carbonyl groups. During the conversion of the latter groups to the former, even in a weakly alkaline medium, the carboxy derivatives undergo an appreciable degradation and lose, to a great extent, their ability to bind and exchange cations. Procedure B, on the other hand, leads to highly selective starch and amylose carboxy derivatives, exhibiting a small amount of carbonyl groups and featuring a relative stability towards alkaline medium; their binding capacity is as high as 12 milliequivalents of cations per g of sample.

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