Unstable Limit Cycle Research Articles

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.

Bifurcation in a 3-DOF Airfoil with Cubic Structural Nonlinearity

Limit cycle oscillations (LCOs) as well as nonlinear aeroelastic analysis of a 3-DOF aeroelastic airfoil motion with cubic restoring moments in the pitch degree of freedom are investigated. Aeroelastic equations of an airfoil with control surface in an incompressible potential flow are presented in the time domain. The harmonic balance (HB) method is utilized to calculate the LCO frequency and amplitude for the airfoil. Also the semi-analytical method has revealed the presence of stable and unstable limit cycles, along with stability reversal in the neighborhood of a Hopf bifurcation. The system response is determined by numerically integrating the governing equations using a standard Runge-Kutta algorithm and the obtained results are compared with the HB method. Also the results by the third order HB (HB3) method for control surface are consistent with the other numerical solution. Finally, by combining the numerical and the HB methods, types of bifurcation, be it supercritical, subcritical, or divergent flutter area are identified.

On forced oscillations of a simple model for a novel wave energy converter

The dynamics of a simple model for an ocean wave energy converter is discussed. The model for the converter is a hybrid system consisting of a pair of harmonically excited mass–spring–dashpot systems and a set of four state-dependent switching rules. Of particular interest is the response of the model to a wide spectrum of harmonic excitations. Partially because of the piecewise-smooth dynamics of the system, the response is far more interesting than the linear components of the model would suggest. As expected with hybrid systems of this type, it is difficult to establish analytical results, and hence, with the assistance of an extensive series of numerical integrations, an atlas of qualitative results on the limit cycles and other forms of bounded oscillations exhibited by the system is presented. In addition, the presence of unstable limit cycles, the stabilization of the unforced system using low-frequency excitation, the peculiar nature of the response of the system to high-frequency excitation, and the implications of these results on the energy harvesting capabilities of the wave energy converter are discussed.

Bifurcation Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube

A bifurcation analysis of the dynamical behavior of a horizontal Rijke tube model is performed in this paper. The method of numerical continuation is used to obtain the bifurcation plots, including the amplitude of the unstable limit cycles. Bifurcation plots for the variation of nondimensional heater power, damping coefficient and the heater location are obtained for different values of time lag in the system. Subcritical bifurcation was observed for variation of parameters and regions of global stability, global instability and bistability are characterized. Linear and nonlinear stability boundaries are obtained for the simultaneous variation of two parameters of the system. The validity of the small time lag assumption in the calculation of linear stability boundary has been shown to fail at typical values of time lag of system. Accurate calculation of the linear stability boundary in systems with explicit time delay models, must therefore, not assume a small time lag assumption. Interesting dynamical behavior such as co-existing multiple attractors, quasiperiodic behavior and period doubling route to chaos have been observed in the analysis of the model. Comparison of the linear stability boundaries and bifurcation behavior from this reduced order model are shown to display trends similar to experimental data.

대향류 확산화염에서 맥동 불안정성의 비선형 거동에 대한 수치해석

대향류 확산화염에서 확산-전도 불안정성에 의한 맥동불안정성의 비선형 거동을 수치 해석적으로 연구하였다. Lewis 수를 1보다 충분히 크게 두고 일차원 준정상상태의 화염의 해로부터 Damkohler 수를 섭동시켜 시간에 따른 화염의 전개를 계산하였다. 맥동 불안정성에 의한 비선형 화염전개는 세 가지 다른 형태, 즉 교란이 점점 감소되는 경우, 교란이 증폭되어 안정된 주기적 진동이 일어나는 경우, 그리고 교란이 계속 증폭되어 화염이 소염되는 경우 등으로 나타났다. 스트레치를 받지 않는 화염의 결과와 달리 대향류 유동장의 화염에서는 안정된 한계순환 맥동 불안정이 존재하였다. 세 가지 다른 형태의 화염 전개를 보이는 임계 Damkohler 수를 계산하여 동적 소염이 일어나는 영역을 표시하였고, 이는 층류소화염의 국소소염 계산에 이용될 수 있다. 불안정성이 나타나는 갈래질의 구조는 초임계 및 임계이하 Hopf 갈래질로 나타났다. 특정한 . Damkohler 수의 영역에서 안정된 한계 순환 갈래질이 나타났으며, 화염온도가 증가함에 따라 영역이 축소되어 안정된 한계순환이 일어나는 영역은 사라지고 불안정한 한계순환 갈래질이 나타났다. 안정된 한계순환 영역이 확장되는 영역이 존재하며, 이는 단순한 한계순환 불안정성이 주기배증에 의한 Rossler 갈래질이 나타나면서 한계 영역이 확장되었다.

Feedback control of vortex shedding from an inclined flat plate

Open- and closed-loop control of vortex shedding in two-dimensional flow over a flat plate at high angle of attack is numerically investigated at a Reynolds number of 300. Unsteady actuation is modeled as a body force near the leading or trailing edge and is directed either upstream or downstream. For moderate angles of attack, sinusoidal forcing at the natural shedding frequency results in phase locking, with a periodic variation of lift at the same frequency, leading to higher unsteady lift than the natural shedding. However, at sufficiently high angles of attack, a subharmonic of the forcing frequency is also excited and the average lift over the forcing period varies from cycle-to-cycle in a complex manner. It is observed that the periods with the highest averaged lift are associated with particular phase differences between the forcing and the lift, but that this highest-lift shedding cycle is not always stably maintained with open-loop forcing. We design a feedback algorithm to lock the forcing with the phase shift associated with the highest period-averaged lift. It is shown that the compensator results in a stable phase-locked limit cycle for a broader range of forcing frequencies than the open-loop control, and that it is able to stabilize otherwise unstable high-lift limit cycles that cannot be obtained with open-loop control. For example, at an angle of attack of 40°, the feedback controller can increase the averaged magnitude of force on the plate by 76% and increase the averaged lift coefficient from 1.33 to 2.43.

High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations

This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed.

Spatial limit cycles around the moon in the TBP

Abstract Stable and unstable limit cycles are important orbits in chaotic systems. So many works are done to find and to quench chaos by stabilizing them. In this paper a new family of limit cycle orbits around the Moon is introduced as a result of restricted three-body problem. The family is completely spatial and can be used as an out-of-plane velocity magnifier. Lyapunov exponent in the Floquet theory has also been checked and stability of the orbits has been measured.

Noise induced escape through an unstable limit cycle in the presence of a fluctuatingbarrier

The escape of Brownian particles through an unstable limit cycle in the presence ofdeterministic barrier fluctuations has been studied. It has been observed that withincreasing non-Gaussian character of the noise, resonant activation gets suppressed andperiodically forced resonant activation becomes robust. Secondly, the ratio of the meanlifetimes in the presence and in the absence of barrier fluctuation increases very rapidlywith the damping constant for Gaussian noise unlike that for the non-Gaussian case. Thisratio decreases sharply for deviation of the noise behavior from Gaussian characteristics.The Arrhenius plot, on the other hand, exhibits deviation from linearity particularly atlarge noise strength for both Gaussian and non-Gaussian cases. The deviation is large fornon-Gaussian noise. We have also shown that additive colored non-Gaussian noise caninduce RA-like phenomena even in a system having harmonic potential energy.

Homoclinic Loop and Degenerate Hopf Bifurcations Introduced by Load Dynamics

This letter presents the occurrence and properties of a homoclinic loop bifurcation in a simple power system with induction motor and load tap changer (LTC). This global bifurcation is accompanied by the emergence of an unstable limit cycle, or the disappearance of a stable one. It is also shown that it may coincide with a Hopf bifurcation condition yielding a codimension-2 degenerate Hopf bifurcation.

Nonlinear dynamic behaviour of a railway wheelset

Nonlinear dynamic systems are well known to contain certain characteristic additional phenomena compared with linear systems. One example is the fact that nonlinear systems can have multiple stable solutions for one set of parameters. In that case, which one of the multiple stable solutions will be realised will depend on the initial conditions. From the domains of initial conditions which are domains of attraction, probabilities of occurrence of the stable solutions can be calculated. The described dynamical behaviour is studied in the present paper using two examples. To introduce the basic phenomena, the well-known academic Duffing oscillator with harmonic excitation is considered. Domains of attraction are shown for the two stable solutions and the probability of occurrence of the two solutions in the case of equally distributed initial conditions is calculated. The main example to be considered in this paper is the railway wheelset, which is known to show (depending on the nonlinearities in the model) a subcritical Hopf bifurcation depending on the vehicle speed. In that case a range of speeds exists, where a stable trivial solution and a stable limit cycle solution coexist in addition to an unstable limit cycle solution. Again domains of attraction of the two stable solutions are calculated and their dependence on the vehicle speed is shown. The probabilities of occurrence of the two stable solutions are calculated, where special interest is given to the probability of occurrence of the undesired limit cycle solution.

An investigation of stability of a control surface with structural nonlinearities in supersonic flow using Zubov's method

It is well known that the presence of nonlinearities may significantly affect the aeroelastic response of an aerospace vehicle structure. In this paper, the aeroelastic behaviour at high Mach numbers of an all-moving control surface with a nonlinearity in the root support is investigated. Very often, under certain flight conditions, a stable equilibrium point, corresponding to zero displacement of the structure, together with an unstable limit cycle arising from a sub-critical Hopf bifurcation results from the presence of the nonlinearity. The dynamic aeroelastic response to external excitation is also of interest, and when sinusoidal forcing is applied, the stable equilibrium point may then be replaced by a periodic attractor, and the limit cycle by an unstable multi-periodic solution. With or without this forcing, there is an attractor which will possess a domain of attraction. In this paper, the problem of estimating these domains of attraction is tackled using Zubov's method. In the absence of forcing, the method is applied directly to the aeroelastic equations, while for the forced system, the method of averaging is applied to approximate the aeroelastic equations by an autonomous system. The behaviour of the system with forcing is also investigated for flight speeds below a threshold which may occur where the unstable limit cycle of the unforced system disappears. In this regime, the nonlinear system may nevertheless still possess multiple attractors, and their domains of attraction are investigated, again using an averaged form of the aeroelastic equations. In this study, the nonlinearity in the root support was assumed to be due to a cubic hardening restoring moment. The Zubov approach, which always yields conservative estimates, was shown to be capable of rapidly giving a good indication of stability domain boundaries under many conditions. Although this investigation focuses on an aeroelastic system, the general form of equations considered arises in many other settings, so that the approach would be relevant to a whole range of engineering applications.

Hybrid dislocated control and general hybrid projective dislocated synchronization for the modified Lü chaotic system

This paper introduces a modified Lü chaotic system, and some basic dynamical properties are studied. Based on these properties, we present hybrid dislocated control method for stabilizing chaos to unstable equilibrium and limit cycle. In addition, based on the Lyapunov stability theorem, general hybrid projective dislocated synchronization (GHPDS) is proposed, which includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. The drive and response systems discussed in this paper can be strictly different dynamical systems (including different dimensional systems). As examples, the modified Lü chaotic system, Chen chaotic system and hyperchaotic Chen system are discussed. Numerical simulations are given to show the effectiveness of these methods.

The dynamics of an omni-mobile vehicle

The motion of an omni-mobile vehicle on a horizontal plane is considered. The wheels are modelled by absolutely rigid discs. Slippage in a certain direction, which makes a constant non- zero angle with the plane of a disc, is possible at the contact point of a wheel and the plane, and the planes of the discs are fixed with respect to the platform of the vehicle. The dynamic equations of motion are obtained for vehicles of this type with an arbitrary number and arrangement of the wheels. A complete qualitative description of the inertial motion of a vehicle is given (there are no control actions and it is assumed that there is no friction in the axes). The result is presented in the form of a phase portrait of the system. The motion of a vehicle is then considered in the case when control moments are applied to the axes of the wheels. The stability and branching of a certain class of steady motions of the vehicle are investigated. The domain of parameters is separated out where Andronov-Hopf bifurcation occurs with the formation of unstable limit cycles.

Chaos through Sliding Bifurcations in a Boost Converter under a SMC Strategy

Chaos in a hybrid model of a DC-DC boost converter operating in Discontinuous Conduction Mode (DCM), when controlled by Sliding Mode Control (SMC), is detected. The chaotic regime coexists with the stable operating point. It is rigorously shown that there is an unstable limit cycle in the sliding dynamics due to a subcritical Hopf bifurcation. The existence of this limit cycle and its evolution with parameters turn out to be of great importance for the stability and robustness control of this type of power converter.

On the Use of Soft Computing Techniques for Helicopter Control in Environment Protection Mission Scenarios

Abstract This paper presents some results related to fuzzy control from the research on autonomous helicopter control conducted in the framework of the COMETS project. Soft-computing techniques have been applied to improve performance of the Multi-Loop PID model based helicopter controller without increasing significantly the computational requirements. Thus, the resulting control system can be implemented by using a simple on-board fixed point microcontroller. Moreover, fuzzy logic has been also used for testing the helicopter model by inducing nonlinear behaviors (including multiple stable and unstable equilibriums and limit cycles). Experimental results with MARVIN helicopter illustrate the methods (on-line videos are available).

A Passive 2-DOF Walker: Hunting for Gaits Using Virtual Holonomic Constraints

A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.

Derivation of ODEs and Bifurcation Analysis of a Two-DOF Airfoil Subjected to Unsteady Incompressible Flow

An airfoil subjected to two-dimensional incompressible inviscid flow is considered. The airfoil is supported via a translational and a torsional springs. The aeroelastic integro-differential equations of motion for the airfoil are reformulated into a system of six first-order autonomous ordinary differential equations. These are the simplest and least number of ODEs that can present this aeroelastic system. The differential equations are then used for the bifurcation analysis of an airfoil with a structural nonlinearity in the pitch direction. Sample bifurcation diagrams showing both stable and unstable limit cycle oscillation are presented. The types of bifurcations are assessed by evaluating the Floquet multipliers. For a specific case, a period doubling route to chaos was detected, and mildly chaotic behavior in a narrow range of velocity was confirmed via the calculation of the Lyapunov exponents.

Nonlinear Stability Analysis of Control Surface Flutter with Freeplay Effects

Results are presented from nonlinear stability analyses of control surface flutter that were conducted on a threedegree-of-freedom typical-section airfoil with free play about the control surface hinge line. The describing function (harmonic-balance) method has been used to predict response amplitudes, and a locally parameterized continuation method has been used to track all flutter modes. Themethodology is directly applicable to systems with unsteady air loads expressed in either the frequency or time domains. Both stable and unstable limit cycles have been identified. Limit cycle stability has been assessed by examining the change in growth rate due to amplitude perturbations at a given velocity. Limit cycle amplitudes predicted by the describing functionmethod,with frequency-domain unsteady aerodynamics, are shown to compare very well with the magnitudes of time-history responses obtained by direct integration of the equations of motion, using rational function approximation time-domain aerodynamics. Results from these nonlinear analyses are compared with analytical and experimental results of similar analyses from the open literature.

Bifurcation analysis for an SMIB power system with series capacitor compensation associated with sub-synchronous resonance

In this paper, an introduction to the bifurcation theory and its applicability to the study of sub-synchronous resonance (SSR) phenomenon in power system are presented. The continuation and bifurcation analysis software AUTO97 is adopted to investigate SSR for a single-machine-infinite-bus power system with series capacitor compensation. The investigation results show that SSR is the result of unstable limit cycle after bifurcation. When the system exhibits SSR, some complex periodical orbit bifurcations, such as torus bifurcation and periodical fold bifurcation, may happen with the variation of limit cycle. Furthermore, the initial operation condition may greatly influence the ultimate state of the system. The time-domain simulation is carried out to verify the effectiveness of the results obtained from the bifurcation analysis.