scholarly journals Designing Limit-Cycle Suppressor Using Dithering and Dual-Input Describing Function Methods

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1978
Author(s):  
Elisabeth Tansiana Mbitu ◽  
Seng-Chi Chen
Keyword(s):  
Nonlinear System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Digital Computer ◽  
Describing Function ◽  
Amplitude Function ◽  
Sustained Oscillations ◽  
Analytical Work ◽  
Simulation Results ◽  
Minimum Amplitude

This paper described a method to design a limit-cycle suppressor. The dithering technique was used to eliminate self-sustained oscillations or limit cycles. Otherwise, the Dual Input Describing Function (DIDF) method was applied to design dither parameters and analyze the existence of limit cycles. This method was done in a nonlinear system with relay nonlinearity using three standard dither signals, namely sine, triangle, and square waves. The aim of choosing varying dithers was to investigate the effect of dither shapes and the minimum amplitude required for the quenching strategy. First, the possibility and amplitude of limit cycles were determined graphically on the DIDF curve. Then, the minimum amplitude of dither was calculated based on the DIDF analysis. Finally, a simulation was built to verify the analytical work using a digital computer. The simulation results were related to the analysis results. It was evident that the dithering technique is a simple way to suppress limit cycles in a nonlinear system. This paper also presented that dither is an amplitude function, and square-wave dither has the minimum amplitude to quench limit cycles.

How to prevent undesired oscillation in NC rotary table

2015 ◽  
Vol 23 (20) ◽  
pp. 3490-3503
Author(s):  
Ali Ghaffari ◽  
Ebrahim Mohammadiasl
Keyword(s):  
Experimental Data ◽  
Limit Cycle ◽  
Operating Conditions ◽  
Rotary Table ◽  
Describing Function ◽  
Rotary Axis ◽  
Milling Operations ◽  
Complete Set ◽  
Simulation Results ◽  
Real Experimental Data

Heavy lathe-mill and turn-mill machine tools with both turning and milling operations are usually equipped with a frictional brake system to mitigate the effect of the mechanical backlash on the gear driven rotary table. In this paper the simultaneous effects of the coupled nonlinear frictions and backlashes on the positioning of the rotary axis have been investigated theoretically and empirically. Using the describing function method, it is shown that the undesired oscillations of the system are due to the existence of a limit cycle in the nonlinear closed-loop trajectory pattern of the rotary axis. Some simple practical rules are proposed for parameters adjustment of the rotary table, to assure that limit cycle is not created, and the multi-function machine does not oscillate improperly. The proposed rules can be used both at the designing stage and also during the maintenance of the machine. In order to verify the simulation results, a complete set of experimental data in a heavy lathe-mill machine has been utilized. It is shown that the deviation between the simulation results and the real experimental data at different operating conditions are quite small.

Describing Function Analysis of Limit Cycles in a Multiple Flame Combustor

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Frédéric Boudy ◽  
Daniel Durox ◽  
Thierry Schuller ◽  
Grunde Jomaas ◽  
Sébastien Candel
Keyword(s):  
Transfer Function ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Function Analysis ◽  
Combustion Instabilities ◽  
Describing Function ◽  
Variable Size ◽  
Theoretical Predictions ◽  
Flame Describing Function ◽  
Oscillation Frequencies

A recently developed nonlinear flame describing function (FDF) is used to analyze combustion instabilities in a system where the feeding manifold has a variable size and where the flame is confined by quartz tubes of variable length. Self-sustained combustion oscillations are observed when the geometry is changed. The regimes of oscillation are characterized at the limit cycle and also during the onset of oscillations. The theoretical predictions of the oscillation frequencies and levels are obtained using the FDF. This generalizes the concept of flame transfer function by including dependence on the frequency and level of oscillation. Predictions are compared with experimental results for two different lengths of the confinement tube. These results are, in turn, used to predict most of the experimentally observed phenomena and in particular, the correct oscillation levels and frequencies at limit cycles.

Describing Function Analysis of Limit Cycles in a Multiple Flame Combustor

2010 ◽  
Author(s):  
Fre´de´ric Boudy ◽  
Daniel Durox ◽  
Thierry Schuller ◽  
Grunde Jomaas ◽  
Se´bastien Candel
Keyword(s):  
Transfer Function ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Function Analysis ◽  
Combustion Instabilities ◽  
Describing Function ◽  
Variable Size ◽  
Theoretical Predictions ◽  
Flame Describing Function ◽  
Oscillation Frequencies

A recently developed nonlinear Flame Describing Function (FDF) is used to analyze combustion instabilities in a system where the feeding manifold has a variable size and where the flame is confined by quartz tubes of variable length. Self-sustained combustion oscillations are observed when the geometry is changed. Regimes of oscillation are characterized at the limit cycle and also during the onset of oscillations. Theoretical predictions of the oscillation frequencies and levels are obtained using the FDF. This generalizes the concept of flame transfer function by including a dependence on the frequency and on the level of oscillation. Predictions are compared with experimental results for two different lengths of the confinement tube. These results are in turn used to predict most of the experimentally observed phenomena and in particular the correct oscillation levels and frequencies at limit cycles.

Nonlinear Flame Describing Function Analysis of Galloping Limit Cycles Featuring Chaotic States in Premixed Combustors

2012 ◽  
Author(s):  
Frédéric Boudy ◽  
Daniel Durox ◽  
Thierry Schuller ◽  
Sébastien Candel
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Function Analysis ◽  
Perforated Plate ◽  
Operating Conditions ◽  
Nonlinear Prediction ◽  
Constant Amplitude ◽  
Describing Function ◽  
Flame Describing Function ◽  
Velocity Pressure

Nonlinear prediction of combustion instabilities in premixed systems is undertaken on a generic configuration featuring an adjustable feeding manifold length, a multipoint injector composed of a perforated plate and a flame confinement tube. By changing the feeding manifold or flame tube lengths, the system exhibits different types of combustion regimes for the same flow operating conditions. Velocity, pressure and heat release rate measurements are used to examine oscillations during unstable operation. For many operating conditions, a limit cycle is reached at an essentially fixed oscillation frequency and quasi-constant amplitude. In another set of cases, the system features other types of oscillations characterized by multiple frequencies, amplitude modulation and irregular bursts which can be designated by “galloping” limit cycles or GLC. These situations are explored in this article. Imaging during GLCs indicates that the flame is globally oscillating but that the cycle is irregular. Prediction of these special oscillation states is tackled within the Flame Describing Function (FDF) framework. It is shown that it is possible to predict with a reasonable degree of agreement the ranges where a quasi-constant amplitude limit cycle will be established and ranges where the oscillation will be less regular and take the form of a galloping limit cycle. It is found that the FDF analysis also provides indications on the bounding levels of the oscillation envelope in the latter case.

BIFURCATION STRUCTURE OF A DRIVEN, MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR (I): THE SUPERHARMONIC RESONANCE STRUCTURE

1991 ◽  
Vol 01 (02) ◽  
pp. 485-491 ◽  
Author(s):  
F. KAISER ◽  
C. EICHWALD
Keyword(s):  
Limit Cycle ◽  
External Force ◽  
Limit Cycles ◽  
The Self ◽  
Van Der Pol Oscillator ◽  
Resonance Structure ◽  
Bifurcation Structure ◽  
Van Der Pol ◽  
Superharmonic Resonance ◽  
Sustained Oscillations

Bifurcations in the superharmonic region of a generalized version of the van der Pol oscillator which exhibits three limit cycles are investigated. An external force causes the subsequent breakdown of the self-sustained oscillations. Beyond these series of bifurcations chaotic solutions also exist. In this first part we concentrate on a discussion of the bifurcation structure of the system.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

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