Linear and Nonlinear Stability Analysis of a Fixed Caliper Brake During Forward and Backward Driving

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Oliver Stump ◽  
Ronaldo Nunes ◽  
Karl Häsler ◽  
Wolfgang Seemann
Keyword(s):  
Limit Cycle ◽  
Initial Conditions ◽  
Robustness Analysis ◽  
Stable Limit Cycle ◽  
Brake System ◽  
Complex Eigenvalue ◽  
Stable Limit ◽  
Nonlinear Stability Analysis ◽  
Linear And Nonlinear ◽  
Small Models

The parking maneuver of a passenger car is known by bench and vehicle testers to sometimes produce brake squeal, even though the brake system is otherwise quiet. This phenomenon is examined in this work. Pressure foil measurements at the pad caliper contact and acceleration measurements are done on a real break system in order to better understand the mechanisms of the forward–backward driving maneuver. The contact area at the caliper is under a large change during a forward and backward driving maneuver. The measurements motivate linear and nonlinear simulations. A proposal has been made to include the linear effects of parking into the standard robustness analysis with the complex eigenvalues calculation. A time integration of the full nonlinear system shows a possible stable limit cycle, when the brake pad moves from the leading to the trailing side, like in a parking maneuver. This growth of amplitude is not anticipated from the complex eigenvalue analysis (CEA), because no instable eigenvalue is found in the linearized equation of motion at that working point. This subcritical flutter-type behavior is known for small models in the literature and is examined in this paper with a more realistic brake system. It is found that the resulting error of the linearization cannot be neglected. Furthermore, different initial conditions are analyzed to narrow the zone of attraction of the stable limit cycle and the decrease of the critical friction value due to this kind of bifurcation behavior.

Stability of a relay dynamic system with non-linear speed sensor and delay under constant disturbance

2020 ◽  
Author(s):  
R.P. Simonyants ◽  
B.R. Khudaybergenov
Keyword(s):  
Dynamic System ◽  
Limit Cycle ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Approximate Determination ◽  
Cycle Phase ◽  
Stable Limit ◽  
Speed Sensor ◽  
Unstable Limit Cycle ◽  
The Stability

The paper considers the joint effect of the control delay and speed sensor output signal limiting on the stability of the relay dynamic system under the constant disturbance. It is shown that in this case a new property is detected in the system – the appearance of the unstable limit cycle. Phase trajectories are drawn to a stable limit cycle only from the area of initial conditions where their boundaries are determined by the trajectory of an unstable limit cycle. Using the method of Poincare mappings, the parameters of fixed points defining the unstable limit cycle as the boundary of the stability region are found. A simplified method for approximate determination of simple limit cycles and stability in the “large” is proposed based on the property of dynamic symmetry of the system. The method allows the study of the problem under consideration to be limited to applying shift and symmetry mappings to the switching lines.

Coexistence of Multiple Attractors, Metastable Chaos and Bursting Oscillations in a Multiscroll Memristive Chaotic Circuit

2017 ◽  
Vol 27 (05) ◽  
pp. 1750067 ◽  
Author(s):  
N. Henry Alombah ◽  
Hilaire Fotsin ◽  
Kengne Romanic
Keyword(s):  
Limit Cycle ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Stable Limit Cycle ◽  
Phase Portraits ◽  
Stable Limit ◽  
Nonlinear Phenomenon ◽  
Chaotic Circuit ◽  
Bursting Oscillations ◽  
Wide Range

In this paper, some complex nonlinear behaviors in a four-dimensional multiscroll autonomous memristor based chaotic system are investigated. This system is derived from the three-dimensional autonomous charge-controlled Muthuswamy–Chua simplest chaotic circuit. The system can generate four different coexisting attractors for a fixed set of parameters and different initial conditions. This phenomenon is relatively rare given that we have four different attractors namely: an equilibrium point, a stable limit cycle, a 16-peak limit cycle and a strange attractor that coexist in the system within a wide range of parameters. The nonlinear phenomenon of transient chaos is studied and revealed numerically in Matlab and Pspice environments. The complex transient dynamics of this memristive system under different initial states shows that the transient time depends strongly on the initial conditions. Moreover, this model displays spiking and bursting oscillations. The bursting behavior is classified according to the dynamics of separated slow and fast subsystems. It is shown to be of the fold-Hopf type. These complex dynamical behaviors of this system are investigated by means of numerical simulations and via Pspice circuit simulations. The use of bifurcation diagrams, Lyapunov exponents diagrams, power spectrums, phase portraits, time series, isospike diagram, basin of attraction, clearly shows these complex phenomena.

Structural instability and sustained oscillations in an extended Brusselator model

1990 ◽  
Vol 68 (9) ◽  
pp. 743-750 ◽  
Author(s):  
M. Otwinowski ◽  
W. G. Laidlaw ◽  
R. Paul
Keyword(s):  
Limit Cycle ◽  
Initial Conditions ◽  
Stable Limit Cycle ◽  
Kinetic Scheme ◽  
Structural Instability ◽  
Stable Limit ◽  
Simple Modification ◽  
Unbounded Solutions ◽  
Brusselator Model ◽  
Multiple Focus

When all reactions in the "Brusselator" kinetic scheme are allowed to be reversible one can demonstrate, by analyzing the focal values, that, for a range of rate constants, a unique limit cycle is created from a multiple focus. A simple modification of the kinetic scheme leads to a model that has two saddle nodes in addition to the well-known unstable focus and a stable limit cycle. In the modified system, for some parameter values, the limit cycle disappears after evolving into a saddle-node connection. For some initial conditions the modified system has unbounded solutions that describe a possible explosion. The analysis that yields these results is based on a general, constructive procedure, which can be applied to higher order physical and chemical systems.

scholarly journals STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

2020 ◽  
Vol 2 (1) ◽  
pp. 27-31
Author(s):  
Abdulghafoor Jasim Salim ◽  
Kais Ismail Ebrahem ◽  
Suhirman
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Autoregressive Model ◽  
Stable Limit Cycle ◽  
Linearization Method ◽  
Stable Limit ◽  
Local Linearization ◽  
Non Linear ◽  
The Stability ◽  
Local Linearization Method

Abstract: In this paper we study the stability of one of a non linear autoregressive model with trigonometric term  by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude  that the proposed model under certain conditions have a non-zero singular point which is  a asymptotically salable ( when  0 ) and have an  orbitaly stable limit cycle . Also we give some examples in order to explain the method. Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2016 ◽  
Vol 26 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Jicai Huang ◽  
Xiaojing Xia ◽  
Xinan Zhang ◽  
Shigui Ruan
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Positive Equilibrium ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Stable Equilibria

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

scholarly journals Height control for a two-mass hopping robot based on stable limit cycle

2016 ◽  
Vol 13 (6) ◽  
pp. 172988141665774
Author(s):  
Taihui Zhang ◽  
Honglei An ◽  
Qing Wei ◽  
Wenqi Hou ◽  
Hongxu Ma
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Inverted Pendulum Model ◽  
Hopping Robot ◽  
Control Objective ◽  
Upper Level ◽  
Mass Spring ◽  
And Control

Differing from the commonly used spring loaded inverted pendulum model, this paper makes use of a two-mass spring model considering impact between the foot and ground which is closer to the real hopping robot. The height of upper mass which includes the upper leg and body is the main control objective. Then we develop a new kind of control algorithm acting on two levels: The upper level aims to achieve the desired velocity of the upper mass based on a stable limit cycle, where three different controllers are used to regulate the limit cycle; the target of the lower level is to drive the system to converge to the desired state and control the contact force between the foot and ground within an appropriate range based on the inner force control at the same time. Simulation results presented in this paper confirm the efficiency of this control algorithm.

On the Poincaré–Hill cycle map of rotational random walk: locating the stochastic limit cycle in a reversible Schnakenberg model

2009 ◽  
Vol 466 (2115) ◽  
pp. 771-788 ◽  
Author(s):  
Melissa Vellela ◽  
Hong Qian
Keyword(s):  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Quantitative Measure ◽  
Power Spectral Analysis ◽  
Chemical System ◽  
Deterministic System ◽  
Stable Limit ◽  
Novel Device ◽  
Schnakenberg Model ◽  
Cycle Map

Recent studies on stochastic oscillations mostly focus on the power spectral analysis. However, the power spectrum yields information only on the frequency of oscillation and cannot differentiate between a stable limit cycle and a stable focus. The cycle flux, introduced by Hill (Hill 1989 Free energy transduction and biochemical cycle kinetics ), is a quantitative measure of the net movement over a closed path, but it is impractical to compute for all possible cycles in systems with a large state space. Through simple examples, we introduce concepts used to quantify stochastic oscillation, such as the cycle flux, the Hill–Qian stochastic circulation and rotation number. We introduce a novel device, the Poincaré–Hill cycle map (PHCM), which combines the concept of Hill’s cycle flux with the Poincaré map from nonlinear dynamics. Applying the PHCM to a reversible extension of an oscillatory chemical system, the Schnakenberg model, reveals stable oscillations outside the Hopf bifurcation region in which the deterministic system contains a limit cycle. Bistable behaviour is found on the small volume scale with high probabilities around both the fixed point and the limit cycle. Convergence to the deterministic system is found in the thermodynamic limit.

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