Limitations of the describing function for limit cycle prediction

2002 ◽  
Vol 47 (11) ◽  
pp. 1887-1890 ◽  
Author(s):  
S. Engelberg
Keyword(s):  
Limit Cycle ◽  
Describing Function

Bang-Bang Versus Linear Control of a Second-Order Rate-Type Servomotor

1960 ◽  
Vol 82 (1) ◽  
pp. 66-72 ◽  
Author(s):  
Philip F. Meyfarth
Keyword(s):  
Limit Cycle ◽  
Function Approximation ◽  
Second Order ◽  
Describing Function ◽  
Response Characteristics ◽  
Order Rate ◽  
Maximum Value ◽  
Bang Bang Control ◽  
Low Amplitude ◽  
Rate Type

A simple nonlinear scheme for controlling a second-order rate-type servomotor is described. This scheme is referred to as “bang-bang” control since the input to the servomotor is made to “bang” from its maximum value in one direction to its maximum value in the other direction depending only on the sign of an error signal. This bang-bang system oscillates in a continuous high-frequency, low-amplitude limit cycle. The nature of this limit cycle is studied by the describing function approximation and by an exact method. The step and frequency response characteristics of the bang-bang system are discussed and compared with the characteristics of a simple linear system. It is shown that many aspects of the behavior of the bang-bang system can be predicted from rather simple considerations.

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.

Stability of a Proportional Valve Control System With Backlash and Saturation

2013 ◽  
Author(s):  
Nader Moustafa ◽  
Roger Fales
Keyword(s):  
Limit Cycle ◽  
Function Analysis ◽  
Linear Part ◽  
Open Loop ◽  
Function Technique ◽  
Main Stage ◽  
Describing Function ◽  
Nonlinear Part ◽  
The Stability ◽  
Limit Cycle Behavior

In this work, the describing function technique is used to study the stability of a nonlinear system. All of dynamic systems in industrial and fluid power systems are nonlinear and include uncertainties to some degree. Thus, unexpected changes in the stability can be exhibited and can lead these systems to become unstable or exhibit oscillatory behavior. Engineers have developed nonlinear mathematical models to be able to predict whether or not a designed system will be exposed to such an oscillation before considering building and implementing the system. The focus of this study is to predict the existence of nonlinear oscillation behavior in a dynamic system using a simplified approach. A nonlinear model validation of a solenoid operated proportional control valve was performed using open loop testes. The type of two-stage hydraulic valve considered in this research is used to control the velocity of hydraulic cylinders. The pilot valve, which is the focus of this research, is a pressure control 3-way valve. A number of 30 replications of this pilot spool valve were studied and tested experimentally along with a single main stage valve. The model consists of linear and nonlinear parts. The linear part of the model was developed by linearizing the nonlinear governing equations at nominal conditions. The nonlinear part was constructed by analyzing open loop experimental test data. The data showed that two major nonlinearities are found that are key to describing the behavior of the system: saturation of the current input and backlash hysteresis behavior. These nonlinearities were considered to be the cause of limit cycle behavior. Each one of these nonlinearities was represented by its describing function and limit cycles were predicted using the describing function analysis method. In using the describing function method, the complexities of working with the nonlinear physics based model to determine limit cycle behavior were avoided.

A Weakly Nonlinear Approach Based on a Distributed Flame Describing Function to Study the Combustion Dynamics of a Full-Scale Lean-Premixed Swirled Burner

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Davide Laera ◽  
Sergio M. Camporeale
Keyword(s):  
Heat Release ◽  
Limit Cycle ◽  
Heat Release Rate ◽  
Release Rate ◽  
Gas Turbines ◽  
Full Scale ◽  
Describing Function ◽  
Weakly Nonlinear ◽  
Lean Premixed ◽  
Flame Describing Function

Modern combustion chambers of gas turbines for power generation and aero-engines suffer of thermo-acoustic combustion instabilities generated by the coupling of heat release rate fluctuations with pressure oscillations. The present article reports a numerical analysis of limit cycles arising in a longitudinal combustor. This corresponds to experiments carried out on the longitudinal rig for instability analysis (LRIA) test facility equipped with a full-scale lean-premixed burner. Heat release rate fluctuations are modeled considering a distributed flame describing function (DFDF), since the flame under analysis is not compact with respect to the wavelengths of the unstable modes recorded experimentally. For each point of the flame, a saturation model is assumed for the gain and the phase of the DFDF with increasing amplitude of velocity fluctuations. A weakly nonlinear stability analysis is performed by combining the DFDF with a Helmholtz solver to determine the limit cycle condition. The numerical approach is used to study two configurations of the rig characterized by different lengths of the combustion chamber. In each configuration, a good match has been found between numerical predictions and experiments in terms of frequency and wave shape of the unstable mode. Time-resolved pressure fluctuations in the system plenum and chamber are reconstructed and compared with measurements. A suitable estimate of the limit cycle oscillation is found.

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