scholarly journals Stabilizing and destabilizing perturbations of PT -symmetric indefinitely damped systems

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120051 ◽  
Author(s):  
O. N. Kirillov
Keyword(s):  
Tangent Cone ◽  
Symmetric Matrix ◽  
Stability Boundary ◽  
Stability Domain ◽  
Exceptional Point ◽  
Damped System ◽  
Whitney Umbrella ◽  
The Stability ◽  
Damped Systems ◽  
Pure Imaginary Eigenvalues

Eigenvalues of a potential dynamical system with damping forces that are described by an indefinite real symmetric matrix can behave as those of a Hamiltonian system when gain and loss are in a perfect balance. This happens when the indefinitely damped system obeys parity–time ( ) symmetry. How do pure imaginary eigenvalues of a stable -symmetric indefinitely damped system behave when variation in the damping and potential forces destroys the symmetry? We establish that it is essentially the tangent cone to the stability domain at the exceptional point corresponding to the Whitney umbrella singularity on the stability boundary that manages transfer of instability between modes.

Domains and Types of Aeroelastic Instability of Slender Beam

2007 ◽  
Author(s):  
Jirˇi´ Na´prstek
Keyword(s):  
Stability Boundary ◽  
Stability Loss ◽  
Stability Domain ◽  
Theoretical Explanation ◽  
Point Of View ◽  
System Response ◽  
Physical Point ◽  
Model Response ◽  
Gyroscopic Forces ◽  
The Stability

Slender structures exposed to a cross air flow are prone to vibrations of several types resulting from aeroelastic interaction of a flowing medium and a moving structure. Aeroelastic forces are the origin of nonconservative and gyroscopic forces influencing the stability of a system response. Conditions of a dynamic stability loss and a detailed analysis of a stability domain has been done using a linear mathematical model. Response properties of a system located on a stability boundary together with tendencies in its neighborhood are presented and interpreted from physical point of view. Results can be used for an explanation of several effects observed experimentally but remaining without theoretical explanation until now.

scholarly journals Singularities in Structural Optimization of the Ziegler Pendulum

10.14311/1400 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
O. N. Kirillov
Keyword(s):  
Structural Optimization ◽  
Degrees Of Freedom ◽  
Optimization Problems ◽  
Stability Boundary ◽  
Research Area ◽  
Stability Domain ◽  
Conservative Systems ◽  
The Stability ◽  
Flutter Load ◽  
The Masses

Structural optimization of non-conservative systems with respect to stability criteria is a research area with important applications in fluid-structure interactions, friction-induced instabilities, and civil engineering. In contrast to optimization of conservative systems where rigorously proven optimal solutions in buckling problems have been found, for nonconservative optimization problems only numerically optimized designs have been reported. The proof of optimality in non-conservative optimization problems is a mathematical challenge related to multiple eigenvalues, singularities in the stability domain, and non-convexity of the merit functional. We present here a study of optimal mass distribution in a classical Ziegler pendulum where local and global extrema can be found explicitly. In particular, for the undamped case, the two maxima of the critical flutter load correspond to a vanishing mass either in a joint or at the free end of the pendulum; in the minimum, the ratio of the masses is equal to the ratio of the stiffness coefficients. The role of the singularities on the stability boundary in the optimization is highlighted, and an extension to the damped case as well as to the case of higher degrees of freedom is discussed.

Sensitivity Analysis of Dissipative Reversible and Hamiltonian Systems: A Survey

2009 ◽  
Author(s):  
Oleg N. Kirillov ◽  
Ferdinand Verhulst
Keyword(s):  
Sensitivity Analysis ◽  
Hopf Bifurcation ◽  
Structural Stability ◽  
Hamiltonian Systems ◽  
Perturbation Analysis ◽  
Stability Boundary ◽  
Conservative System ◽  
Small Dissipation ◽  
Whitney Umbrella ◽  
The Stability

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler’s paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation and related bifurcations.

Numerical Research on Machining Stability Subject to Delayed PID Control

2018 ◽  
Author(s):  
Mingjie Li ◽  
Xiaojian Zhang ◽  
Yakun Xie
Keyword(s):  
Pid Control ◽  
Production Efficiency ◽  
Stability Boundary ◽  
Stability Domain ◽  
Integral Parameter ◽  
Controlled System ◽  
Critical Boundary ◽  
Numerical Research ◽  
The Stability ◽  
Delay Differential

Machining stability analysis is important for chatter avoiding and production efficiency improvement. This paper conducts the numerical research on machining stability subject to delayed PID active control which is used to avoid chatter in machine milling. This control strategy is introduced into a two-degree-of-freedom milling system for illustration, the resulting hybrid system with both regenerative and feedback delays is represented as a delay-differential equation with time-periodic coefficients. From the comparison of stability region, it is found that the delayed PID control with proper parameters can lift the stability boundary largely compared with the case without control. To evaluate the stabilizability of the controlled system in cutting, the sensitivity of the stability boundary with respect to the PID parameters is analyzed. The numerical simulation of critical axial depth to PID parameters indicate that the milling stability critical boundary varies drastically with the derivative parameter. It also demonstrates that the stability critical boundary is strongly influenced by the proportional parameter, but it is less effected by the integral parameter. Hence, the stability domain can be expanded drastically with appropriate PID parameters based on the analysis above.

Dynamics and Stability of Pipes Conveying Fluid

1994 ◽  
Vol 116 (1) ◽  
pp. 57-66 ◽  
Author(s):  
C. O. Chang ◽  
K. C. Chen
Keyword(s):  
Equations Of Motion ◽  
Stability Boundary ◽  
Harmonic Component ◽  
Mean Value ◽  
Transverse Motion ◽  
Perturbation Techniques ◽  
Damped System ◽  
Pipes Conveying Fluid ◽  
The Stability ◽  
Conveying Fluid

This paper deals with the dynamics and stability of simply supported pipes conveying fluid, where the fluid has a small harmonic component of flow velocity superposed on a constant mean value. The perturbation techniques and the method of averaging are used to convert the nonautonomous system into an autonomous one and determine the stability boundaries. Post-bifurcation analysis is performed for the parametric points in the resonant regions where the axial force, which is induced by the transverse motion of the pipe due to the fixed-span ends and contributes nonlinearities to the equations of motion, is included. For the undamped system, linear analysis is inconclusive about stability and there does not exist nontrivial solution in the resonant regions. For the damped system, it is found that the original stable system remains stable when the pulsating frequency increases cross the stability boundary and becomes unstable when the pulsating frequency decreases cross the stability boundary.

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 ANALYSIS AND SIMULATION OF N-CLASS RETRIAL SYSTEM WITH CONSTANT RETRIAL RATES AND POISSON INPUTS

2014 ◽  
Vol 31 (02) ◽  
pp. 1440002 ◽  
Author(s):  
K. AVRACHENKOV ◽  
E. MOROZOV ◽  
R. NEKRASOVA ◽  
B. STEYAERT
Keyword(s):  
Queueing System ◽  
Stability Domain ◽  
Single Server ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Single Orbit ◽  
Transmission Control Protocols ◽  
Regenerative Approach ◽  
The Stability ◽  
Multi Access

In this paper, we study a new retrial queueing system with N classes of customers, where a class-i blocked customer joins orbit i. Orbit i works like a single-server queueing system with (exponential) constant retrial time (with rate [Formula: see text]) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, we present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, we deduce a set of necessary stability conditions for such a system. We will show that these conditions have a very clear probabilistic interpretation. We also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that we focus on.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

Analytical and Experimental Flutter Analysis of a Typical Wing Section Carrying a Flexibly Mounted Unbalanced Engine

2019 ◽  
Vol 19 (02) ◽  
pp. 1950013 ◽  
Author(s):  
A. S. Mirabbashi ◽  
A. Mazidi ◽  
M. M. Jalili
Keyword(s):  
Stability Domain ◽  
Governing Equations ◽  
Lagrange Equations ◽  
Wing Section ◽  
Unbalanced Force ◽  
Engine Thrust ◽  
Finite State Model ◽  
Flutter Analysis ◽  
Finite State ◽  
The Stability

In this paper, both experimental and analytical flutter analyses are conducted for a typical 5-degree of freedon (5DOF) wing section carrying a flexibly mounted unbalanced engine. The wing flexibility is simulated by two torsional and longitudinal springs at the wing elastic axis. One flap is attached to the wing section by a torsion spring. Also, the engine is connected to the wing by two elastic joints. Each joint is simulated by a spring and damper unit to bring the model close to reality. Both the torsional and longitudinal motions of the engine are considered in the aeroelastic governing equations derived from the Lagrange equations. Also, Peter’s finite state model is used to simulate the aerodynamic loads on the wing. Effects of various engine parameters such as position, connection stiffness, mass, thrust and unbalanced force on the flutter of the wing are investigated. The results show that the aeroelastic stability region is limited by increasing the engine mass, pylon length, engine thrust and unbalanced force. Furthermore, increasing the damping and stiffness coefficients of the engine connection enlarges the stability domain.

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