Limit Cycle Frequency Research Articles

Describing Function Analysis in the Presence of Uncertainty

A framework is proposed that generalizes describing function (DF) analysis to uncertain systems. By e tting a rational approximation to the DF of nonlinear elements, DF analysis is incorporated into a generalized π framework of robustness analysis. This allows us to consider uncertainty in both the linear and nonlinear components. Information on the size, frequency, and stability of unforced limit cycles is obtained as in the graphical test. The response to sinusoidal driving inputs can also be analyzed in this framework.

Dynamics and chaos stabilization of semiconductor lasers with optical feedback from an interferometer

This paper studies dynamical behaviors and noise properties of semiconductor lasers with optical feedback from an interferometer, i.e., two external cavities. Bifurcation diagrams versus external-cavity length and reflectivity reveal the existence of stable regions for fixed states and limit cycles. The steady-state analysis for a laser diode with two external feedbacks is performed, which shows that oscillation modes are locked at minimum excess gain modes. Oscillation frequencies of limit cycles show discrete transitions with the variation of ratios between two external-cavity lengths and reflectivities. It is proposed that such a model might be applied to stabilization of the feedback-induced chaos in external-cavity semiconductor lasers since the stable region is quite robust for wide parameter ranges. Intensity and phase noises are examined and the results indicate that the low-frequency relative intensity noise level and the linewidth can be greatly improved when the laser output is a fixed state or a limit cycle.

Pilot-Induced Oscillation Analysis and Prediction with Actuator Rate Limiting

Akeyfeatureofall of therecentwell-recordedseverepilot-induced oscillations (PIOs)isthepresenceof actuator rate limiting. As part of attempts to develop a comprehensive understanding of PIO phenomena, the results of an effort focused on improved understanding and analysis of actuator rate limiting are presented. For this effort describing functions were used in concert with modern computer simulation techniques to quantify the added phase lag, magnitude reduction, and bandwidth reduction of a rate-limited actuator in terms of the input and actuator design parameters. The results and inverse describing function techniques are then employed to analyze thelimit cyclepotential of a system that isessentially linearexceptfortheseriesnonlinearity. Thewell-documented X-15 landing e are PIO is used to exemplify these analysis techniques. From this example it is shown that these techniques can be used to provide a prediction of the limit cycle or PIO frequency.

Poly-phase sigma-delta modulation

A new circuit is described for the implementation of analog sigma-delta modulation. It is based upon the use of a poly-phase sampler with the aim of obtaining high over-sampling ratios at low clock frequencies. A detailed analysis of second-order systems is made to predict its performance. The basic modeling comprises the incorporation of sampling noise within limit cycle oscillations of the modulator. It is shown that the polyphase sigma-delta modulator can be conceived as an internally sampled asynchronous sigma-delta modulator. The theoretical model presupposes a low limit cycle frequency as compared with the effective sampling frequency. In spite of the fact that this condition is not met in the extreme of conventional single-phase sigma-delta modulation, the theoretical result obtained for this special case is close to the result obtained from the conventional sampled data model. Poly-phase sigma-delta modulation may therefore be regarded as a generalization of conventional single-phase sigma-delta modulation. The results of the theory are illustrated by simulation results of a practical design example.

Nonlinear interaction of DC cables and magnetically controlled ship-steering autopilots

This paper presents new results on the interaction of DC cables and magnetically controlled ship-steering autopilots. Simulations with a nonlinear model show that a ship using a steering autopilot with a magnetic compass can be captured by a DC cable. That is, the ship tracks the cable, perhaps in an oscillatory fashion. A full-scale experimental study has been performed at the Kontiskan high voltage direct current link to verify calculations and simulation results. Good agreement was obtained between measurements and computer simulations. This paper presents the simulation study, the dynamic model used, and the test results. Simulation and field test show that if the ship is captured by the cable, it can, under certain conditions, track the cable with limit-cycle oscillation. Steady-state stability analysis explains why the ship does not find a stable capture course. Describing function analysis is used to find the limit-cycle amplitude and frequency.

Necessary conditions for limit cycles in multiloop relay systems

The paper examines the behaviour of multivariable systems under multiloop relay feedback control. The analysis is a generalisation of the Tsypkin method (1984) for the prediction of forced oscillations in single variable systems. It is exact in the sense that a complete harmonic balance is considered in all the loops. The results are valid for systems with characteristic loci with phase lags more than 180 degrees . It is shown that such multivariable systems under multiloop relay feedback may exhibit limit cycle oscillations in three possible modes. The first mode consists of identical relay outputs which are square waves with precisely one fundamental frequency. The second mode is characterised by relay outputs which are square waves of different fundamental frequencies in each loop. In this mode, each loop behaves like a single variable system oscillating at a unique limit cycle frequency. The third mode is one of periodic complex oscillations consisting of multiple relay switches within one fundamental period. The necessary conditions derived show that the modes are related to the strength of the interactions in the respective loops. The authors derive a graphical technique to determine when unique oscillations (with distinct frequencies) at the output of each relay may occur and when single frequency or complex oscillations may exist instead. Simulation results are given to illustrate the possible scenarios.

BIFURCATION STRUCTURE OF THE DRIVEN VAN DER POL OSCILLATOR

The bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurcation curves. The two regions in which the driving frequency ω is greater or less than the limit cycle frequency ω0 of the nondriven oscillator are considered separately. For the case ω > ω0, the subharmonic region, the extent and location of the largest Arnol'd tongues are shown, as well as the period-doubling cascades and chaotic attractors that appear within most of them. Special attention is paid to the pattern of the bifurcation curves in the transitional region between low and large dampings that is difficult to approach analytically. In the case ω < ω0, the ultraharmonic region, a recurrent pattern of the bifurcation curves is found for small values of the damping d. At medium damping the structure of the bifurcation curves becomes involved. Period-doubling sequences and chaotic attractors occur.

Active Control Of Flow-induced Vibration

The active control of flow-induced oscillations, specifically, vortex-induced oscillations of circular cylinders and galloping oscillations of circular cylinders and galloping oscillations of a square prism, are considered. In the case of vortex-induced oscillations, the vibrating cylinder is modeled as a single-degree-of-freedom (sdof) linear damped oscillator and the fluid motion as a Rayleigh oscillator. Galloping oscillations are modeled as a sdof limit-cycle system with cubic and higher order non-linearities in the velocity term. In formulating the control method it is assumed that the fluid interaction is an external excitation/disturbance on the structure. A suitable disturbance model is chosen based on the spectral content of the disturbance. For the problems considered here, the disturbance model assumes the form of a harmonic oscillator the frequency of which, in the case of vortex-induced oscillations, is equal to the natural frequency of the structure or the "lock-in" frequency, and in the case of galloping oscillations, is equal to the limit-cycle frequency. A disturbance counteracting control law is synthesized by using a Luenberger observer incorporating the structure and disturbance dynamics. It is shown that using this approach one is able to completely suppress the fluid-induced oscillations of the structure.

Faraday waves in 2:1 internal resonance

Experiments are reported on Faraday waves in a circular cylinder, which are internally resonant with either the subharmonic mode (with frequency ¼ that of the forcing) or the superharmonic mode (with frequency equal to that of the forcing). For subharmonic resonance both modes achieved comparable amplitudes that were steady, or were modulated with one or two periods, or exhibited quasi-periodic or chaotic motions. A stability map of these responses is presented. Theoretical predictions of linear stability and growth rates are tested. Measurements of steady amplitudes, limit-cycle frequencies and wave slopes at breaking are presented. Some of the measured phase-plane trajectories are shown to have theoretical counterparts. For superharmonic resonance the amplitude of the superharmonic was never comparable with that of the Faraday wave. For low modes an energy exchange occurred during the initial period of growth, and a precession instability sometimes developed. For high modes for which both frequencies and wavenumbers are in a 2:1 ratio, superharmonic resonance occurred irreproducibly; it appeared to be overwhelmed by 1:1 interactions among the possible Faraday-wave modes.

Narrowband random excitation of a limit cycle system

The response of the Van der Pol oscillator to stationary narrowband Gaussian excitation is considered. The central frequency of excitation is taken to be in the neighborhood of the system limit cycle frequency. The solution is obtained using a non-Gaussian closure approximation on the probability density function of the response. The validity of the solution is examined with the help of a stochastic stability analysis. Solution based on Stratonovich's quasistatic averaging technique is also obtained. The comparison of the theoretical solutions with the digital simulations shows that the theoretical estimates are reasonably good.

Entrainment in van der Pol's oscillator in the presence of noise

The response of Van der Pol's oscillator to a combination of harmonic and white noise excitations is considered. The harmonic excitation frequency is taken to be in the neighbourhood of the system limit cycle frequency. The effect of addition of noise on the entrainment behaviour is investigated using a combination of methods of stochastic averaging and equivalent nonlinearization. Results based on the gaussian closure technique are also obtained and the theoretical solutions are compared with digital simulations.

Analytical study of the subsonic wing-rock phenomenon for slender delta wings

An analytic expression describing the aerodynamic roll moment has been obtained from the numerical simulation of wing rock. This expression is used in the equation governing the rolling motion of a delta wing around its midspan axis. The result is used to construct phase planes, which reveal the general global nature of wing rock—stable limit cycles, unstable foci, saddle points, and domains of initial conditions leading to oscillatory motion and divergence. An asymptotic approximation to the solution of the governing equation is obtained; this result provides expressions for the amplitudes and frequencies of limit cycles. The present analysis provides a penetrating global view of the wing-rock phenomenon.

Low speed hunting of the pneumatically governed compression ignition engine. (8th Report, Effects of various parameters on the amplitude and frequency of limit cycle: Part II).

For a better understanding of hunting, it is desirable to be able to obtain the amplitude and frequency of hunting analytically. In the author's previous paper as a first step toward nonlinear approximate analysis, the effects of three parameters besides a subventuri pressure lag, i.e. moment of inertia of the crankshaft system, and mass and damping of the governing system, that have a marked influence each on the limit cycle characteristic are estimated as a step to analytically explaining the mechanism of limit cycle evolution. The present report estimates the effects of the other parameters on the limit cycle behavior: (1) the limit cycle lessens with decreasing stiffness of the control rack spring and (2) also with decreasing the slope of engine torque vs. rack displacement, but (3) scarcely depends on the slope of engine torque vs. engine speed, while (4) the calculated amplitude and frequency without considering a component of a higher frequency caused by the suction stroke of each piston do not agree with the experimental.

Low-speed hunting of the pneumatically governed compression ignition engine. Effects of various parameters on the amplitude and frequency of limit cycle.

For a better understanding of hunting, it is desirable to obtain the amplitude and frequency of limit cycle analytically. In the author's last two reports, as a first step toward nonlinear approximate analysis, the effects of individual parameters of a closed-engine-governor system on the instability of the equilibrium state, especially on the frequency and increment of amplitude, were linearly estimated. In the present report, results show that the limit cycle disappears over the whole speed range if the moment of inertia of the crankshaft system exceeds 1.8 times that in the actual engine; this phenomenon resembles the disappearance due to minimized subventuri pressure lag. Results also show that the amplitude decreases, except at extremely low speeds, with decreasing mass of the governing system, although mass reduction is difficult in reality. Further, it is found that the limit cycle does not disappear no matter what value the damping of the governor may take.

Low speed hunting of the pneumatically governed compression ignition engine. (7th report. Effects of various parameters on the amplitude and frequency of limit cycle. Part I).

For a better understanding of hunting, it is desirable to be able to obtain the amplitude and frequency of hunting analytically. In the author's last two reports as a first step toward nonlinear approximate analysis, the effects of individual parameters of a closed-engine-governor system on the instability of equilibrium state are linearly estimated, especially on the frequency and increment of amplitude. In the present report, the effects of three parameters that have a marked influence on the limit cycle characteristic are estimated as a step to analytically explaining the mechanism of limit cycle evolution. Results show that the limit cycle disappears over all speed ranges with increasing moment of inertia of crankshaft system 1.8 times that in the actual engine as with minimizing subventuri pressure lag. Results also show that the amplitude decreases except at extremely low speeds with decreasing mass of the governing system, but it is difficult to decrease the mass in reality. Further, it is found that the limit cycle does not disappear no matter what value the damping of the governor may take.

A nonlinear pilot model for hover

Tx Nomenclature = properties of the ideal relay with dead zone (Af=0.25 in., >1 =0.125 in.) = configuration C of X-22A flight 217 = error in X position, —XC—X (ft) = amplitude of sinusoidal input to nonlinear element N(E) = pilot position error gain (in./ft) = ratio of steady state x translational velocity per inch of pilot's stick position (ft/s/in.) . = minimum pilot position error gain to maintain a stable limit cycle (in./ft) = describing function gain which is a nonlinear function of sinusoidal input amplitude E = Cooper-Harper pilot rating (Ref. 1) = quantization nonlinearity, Fig.9 = Laplace operator for differentiation = equivalent path mode time constant, s = fore and aft position, ft = commanded position, ft = pilot's stick position (positive stick aft, in.) —modified stick position (positive stick forward, in.) = ds = body-axis pitch attitude, deg = limit-cycle frequency

A STABLE OPERATION REGION WITH NO LIMIT CYCLES USING THE METHOD OF DOMAINS SEPERATION

In determining the region of stable operation in the plane of two parameters, the issue of limit cycles need be studied carefully. The resonance problem will be raised if the mechanical resonance frequency of power systems'is equal to the limit cycle frequency. Despite the stability of each point in the stable region, it is crucial to determine the boundaries in which one can choose the Automatic voltage Regulator (AVR) parameters in such a way that the global stability with no limit cycle is guaranteed. Seperation of such boundaries to dangrous and safety borders have been investigated in the last few years. Many techniques have been devoted to enhance the safe region such that one can choose the AVR parameters in a better manner. In this paper, the stable region will be separated to a non-limit cycle region and a limit cycle one. The non-limit cycle region is the region in which the adjustable coefficients can be chosen with guarentee that the resonance will never happen. Two different techniques are used here to solve such a problem. Firstly, by scanning of the whole region; secondly by analyzing the describing function. In spite of using the digital computer in both cases, the second technique is much easier and is a time saver than the first one.

Analysis of limit-cycle conditions in intercoupled relay oscillators with reference to gastrointestinal modelling.

This paper presents an exact analysis of a mutually coupled relay oscillator based on a method orignated by Tsypkin. Limit-cycle frequencies and phases can be determined exactly using this method, unlike other approximate methods based on describing functions and harmonic balance techniques. A new method of exact determination of limit-cycle stability is also shown to give excellent agreement with simulation studies. Different types of intercoupling are shown to give different stability conditions, and these are discussed in relation to gastrointestinal (GI) smooth muscle modelling. GI tract electrical activity has previously been modelled using bidirectionally coupled nonlinear oscillators, and the results of the present analysis of relay oscillators is compared with other studies using van der Pol dynamics.

Limit cycles in relay systems

A method using the z transform is presented for evaluating the frequency of limit cycles in relay systems. The technique as described is only applicable when the relay has no dead zone and the plant transfer function no time delay. An extension of the approach to these systems is possible but requires use of the modified z transform; this makes the calculations more cumbersome and the method loses its simplicity.

The complex Lorenz equations

We have undertaken a study of the complex Lorenz equations x ̇ = −σx + σy . y ̇ = (r − z)x − ay . z ̇ = −bz + 1 2 (x ∗y + xy ∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r 1 + ir 2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r 2 = 0. We have been able to determine analytically two critical values of r 1, namely r1 c and r1 c . The origin is a stable fixed point for 0 < r 1 < r 1 c , but for r 1 > r 1 c , a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ < b + 1. If σ > + 1 then this limit is only stable in the region r 1 c < r 1 < r lc . When r 1 > r lc , a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.