Control of Limit Cycle Amplitude Through Nonsmooth Bifurcations

2006 ◽  
Author(s):  
F. Angulo ◽  
M. di Bernardo ◽  
G. Olivar
Keyword(s):  
Limit Cycle ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Reduction of Limit Cycle Amplitude in the Presence of Backlash

1999 ◽  
Vol 121 (2) ◽  
pp. 278-284 ◽  
Author(s):  
Ronen Boneh ◽  
Oded Yaniv
Keyword(s):  
Nonlinear Control ◽  
Limit Cycle ◽  
Closed Loop ◽  
Controller Design ◽  
Linear Time ◽  
Design Technique ◽  
Feedback Systems ◽  
Mass System ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

The majority of feedback systems driven by an electric motor can be represented by a two-mass system connected via a flexible drive element. Owing to the presence of backlash, the closed-loop performance such as precision speed, position and force control that can be achieved using a linear time invariant controller is limited, and it is expected that a nonlinear control would be superior. In this paper a nonlinear control structure is proposed and a systematic design technique presented. The advantages of the proposed design technique are: (i) It is robust to plant and backlash uncertainty; (ii) it is quantitative to specifications on the maximum limit cycle amplitude; and (iii) the closed loop is superior to a linear controller design both in lower bandwidth and in lower limit cycle amplitude. A design example is included.

Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

2012 ◽  
Author(s):  
Karthik Kashinath ◽  
Santosh Hemchandra ◽  
Matthew P. Juniper
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Release Rate ◽  
Time Domain ◽  
Limit Cycles ◽  
Integral Relation ◽  
Single Mode ◽  
Rate Of Change ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies, amplitudes and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, but can be used to calculate the amplitudes of limit cycles, as well as their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modelled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearised momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state, for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, as well as twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact resulting in a larger limit cycle amplitude. Multi-mode simulations show that in some situations the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

Limit Cycle Behavior of a Flexible Truck

1987 ◽  
Vol 54 (4) ◽  
pp. 930-934 ◽  
Author(s):  
A. M. Whitman ◽  
J. E. Molyneux
Keyword(s):  
Limit Cycle ◽  
Critical Speed ◽  
Perturbation Procedure ◽  
Cycle Amplitude ◽  
Creep Force ◽  
Limit Cycle Amplitude ◽  
Limit Cycle Behavior

We calculate the variation in critical speed of a flexible truck as a function of limit cycle amplitude and truck parameters (i.e., shear and bending stiffnesses, and truck geometry), by means of a perturbation procedure. We find that the creep force nonlinearity is dominant, and that it can cause the nonlinear critical speed to be either lower or higher than the linear critical speed, depending on the values of the two stiffnesses.

Sensitivity Enhancement of Cantilever-Based Sensors Using Feedback Delays

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Calvin Bradley ◽  
Mohammed F. Daqaq ◽  
Amin Bibo ◽  
Nader Jalili
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Sensitivity Enhancement ◽  
Design Parameters ◽  
Feedback Signal ◽  
Stable Limit ◽  
Theoretical Understanding ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

This paper entails a novel sensitivity-enhancement mechanism for cantilever-based sensors. The enhancement scheme is based on exciting the sensor at the clamped end using a delayed-feedback signal obtained by measuring the tip deflection of the sensor. The gain and delay of the feedback signal are chosen such that the base excitations set the beam into stable limit-cycle oscillations as a result of a supercritical Hopf bifurcation of the trivial fixed points. The amplitude of these limit-cycles is shown to be ultrasensitive to parameter variations and, hence, can be utilized for the detection of minute changes in the resonant frequency of the sensor. The first part of the manuscript delves into the theoretical understanding of the proposed mechanism and the operation concept. Using the method of multiple scales, an approximate analytical solution for the steady-state limit-cycle amplitude near the stability boundaries is obtained. This solution is then utilized to provide a comprehensive understanding of the effect of small frequency variations on the limit-cycle amplitude and the sensitivity of these limit-cycles to different design parameters. Once a deep theoretical understanding is established, the manuscript provides an experimental study to investigate the proposed concept. Experimental results demonstrate orders of magnitude sensitivity enhancement over the traditional frequency-shift method.

Algorithm for Controlling Spillback from Ramp Meters

1996 ◽  
Vol 1554 (1) ◽  
pp. 162-171 ◽  
Author(s):  
Robert L. Gordon
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Arrival Rate ◽  
Sampling Period ◽  
High Amplitude ◽  
Ramp Metering ◽  
Control Algorithms ◽  
Considerable Distance ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

The queue control algorithms that are commonly used by ramp-metering controllers often permit the queue to extend a considerable distance upstream of the queue detector on the ramp. This detector senses the presence of a queue with the intent of preventing queue spillback onto the surface street or intersection that is located upstream of the ramp. Queue control is usually obtained by switching between the desired metering rate and a higher metering rate. In some cases metering is terminated. This type of limit cycle control often results in considerable variation in the length of the queue, even for a constant average vehicle arrival rate at the ramp. This high-amplitude limit cycle operation is indicated, by simulation, to result from the relatively long detector sampling and computation period (typically 1 min) that is commonly used. A control algorithm using a 10-sec sampling period is indicated to substantially reduce the limit cycle amplitude. Performance is further improved when the control algorithm anticipates the buildup of the queue over the spillback detector.

Flutter of variable stiffness composite laminates in supersonic flow with temperature effects

2021 ◽  
pp. 002199832110075
Author(s):  
Xiaosui Ouyang ◽  
Yi Liu
Keyword(s):  
Supersonic Flow ◽  
Limit Cycle ◽  
Composite Laminates ◽  
Dynamic Pressure ◽  
Variable Stiffness ◽  
First Order ◽  
Cycle Amplitude ◽  
Critical Dynamic ◽  
Limit Cycle Amplitude ◽  
Variable Stiffness Composite Laminates

The nonlinear thermal flutter behavior of variable stiffness composite laminates (VSCL) with curvilinear fibers in high supersonic flow is investigated. The first order shear deformation theory (FSDT) combining von Karman large-deflection strain-displacement relations, quasi-steady first-order piston theory aerodynamics and quasi-steady thermal stress theory are used to formulate the nonlinear panel flutter finite element equations of motion. The fiber orientation within a layer is assumed to vary linearly from [Formula: see text] at the center to [Formula: see text] at the vertical edges of the rectangular lamina. The flutter characteristics of variable stiffness composite laminates with different temperature distributions are then studied. The results show that the critical dynamic pressure decreases as [Formula: see text] or [Formula: see text] increases, whereas the limit cycle amplitude increases as [Formula: see text] or [Formula: see text] increases for the same dynamic pressure. The critical dynamic pressure and limit cycle amplitude both increase when the temperature gradient along panel thickness increases. Simple harmonic motions, unharmonic but periodic motions, and chaotic motions can be observed on VSCL under different temperatures. It also turns out that temperature distribution has similar influence on both the critical dynamic pressure and limit cycle amplitude of VSCL.

Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Karthik Kashinath ◽  
Santosh Hemchandra ◽  
Matthew P. Juniper
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Release Rate ◽  
Time Domain ◽  
Limit Cycles ◽  
Integral Relation ◽  
Single Mode ◽  
Rate Of Change ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Nonlinear analysis of thermoacoustic instability is essential for the prediction of the frequencies, amplitudes, and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper, an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, however, it can be used to calculate the amplitudes of limit cycles and their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modeled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearized momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, along with twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact, resulting in a larger limit cycle amplitude. Multimode simulations show that, in some situations, the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

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