Stability Analysis of the Rocking Block: Numerical Investigations on Analytical Stability Boundaries

2003 ◽  
Author(s):  
Alessio Ageno ◽  
Anna Sinopoli
Keyword(s):  
Initial Conditions ◽  
General Procedure ◽  
Dynamic Responses ◽  
Stability Range ◽  
Stability Boundaries ◽  
Strong Discontinuities ◽  
Rocking Block ◽  
Analytical Stability ◽  
The Stability ◽  
Characteristic Multipliers

The present paper illustrates some recently developed techniques to evaluate stability features, such as characteristic multipliers and Lyapunov’s exponents, for a problem with strong discontinuities. The problem analyzed concerns the plane dynamics of a rigid block simply supported on a harmonically moving rigid ground. In previous papers [1–3] many aspects have been investigated and the following results have been carried out: 1. the general procedure to approach numerically the problem has been outlined; 2. a rigorous method has been established to calculate characteristic multipliers and Lyapunov’s exponents at the instants of discontinuities; 3. the stability boundaries of symmetric sub-harmonic responses have been drafted by means of closed-form analytical methods based on various hypotheses of linearization. In this paper the new capacities allowed by the techniques and methods pointed out at the previous points 1., 2. and 3. are exploited with the aims: • to investigate numerically in the occurrence of dissipative impacts the features of dynamic responses across the upper boundary of a stability range (i.e. that of the (1,3) sub-harmonic response) included in a larger one (i.e. that of the (1,1) response); • to analyze the responses attainable with a value of the restitution coefficient equal to 1 to describe the impulsive phases (namely for non-dissipative impacts); in previous works, these responses have been classified as quasi-periodic or chaotic [4]. By means of the new techniques proposed and implemented, it has been possible to classify and analyze more deeply such presumed quasi-periodic or chaotic responses and at the same time to clarify the role played by the initial conditions.

Parallel-Process (Simultaneous) Machining and its Stability

1999 ◽  
Author(s):  
O. Burak Ozdoganlar ◽  
William J. Endres
Keyword(s):  
Initial Conditions ◽  
General System ◽  
Parallel Process ◽  
Analytical Stability ◽  
Multiple Processes ◽  
Or Practice ◽  
Machining System ◽  
Eigenvector Analysis ◽  
The Stability ◽  
Machining Operations

Abstract This paper presents a mathematical perspective, to complement the intuitive or practice-oriented perspective, to classifying machining operations as parallel-process (simultaneous) or single-process in nature. Illustrative scenarios are provided to demonstrate how these two perspectives may lead in different situations to the same or different conclusions regarding process parallelism. A model representation of a general parallel-process machining system is presented, based on which the general parallel-process stability eigenvalue problem is formulated. For a special simplified case of the general system, analytical methods are employed to derive a fully analytical stability solution. Thorough study of this solution through eigenvector analysis sheds light on some fundamental phenomena of parallel-process machining stability, such as dependence of the stability solution on phasing of the initial conditions (disturbances). This establishes the importance, when employing numerical time-domain simulation for such analyses, of specifying initial conditions for the multiple processes to be arbitrarily phased so that correct results are achieved across all spindle speeds.

Modern design of belt conveyors in the context of stability boundaries and chaos

1992 ◽  
Vol 338 (1651) ◽  
pp. 491-502 ◽  
Keyword(s):  
Initial Conditions ◽  
Nonlinear Resonance ◽  
Stability Boundaries ◽  
Belt Speed ◽  
Bearing Life ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration ◽  
Belt Conveyors ◽  
The Stability ◽  
Two Parameters

Belt conveying of bulk materials has evolved to the point where the demands of the modern mine to increase capacity is limited by the ability of engineers to design dynamically stable conveyors. Belt speed and width are two parameters that may be varied in the design to provide the required material flow rate. For certain values of belt speed, width and tension, unstable transverse belt vibration has been observed. Large-amplitude vibration may be so severe that the life of the supporting idler bearings is reduced significantly due to dynamic loads. Monitoring idlers for bearing failure in modern conveyors with lengths up to 20 km is practically difficult since there may be as many as 20000 idler sets. Chaotic transverse belt vibrations occur for certain levels of excitation, further complicating the prediction of bearing life. Before conveyor installation, an estimate of the stability boundaries for resonance-free operation is an essential precursor to failure-free conveyor operation. Nonlinear resonance phenomena such as belt flap is sensitive to initial conditions. The effects of chaotic vibrations on the predictability of design stability is reviewed using some examples of forced vibrations.

Dynamic Stability of a Rotor Filled or Partially Filled With Liquid

1996 ◽  
Vol 63 (1) ◽  
pp. 101-105 ◽  
Author(s):  
Wen Zhang ◽  
Jiong Tang ◽  
Mingde Tao
Keyword(s):  
Dynamic Stability ◽  
Dynamic Pressure ◽  
Flow Analysis ◽  
Previous Literature ◽  
Stability Criteria ◽  
Stability Boundaries ◽  
Planar Flow ◽  
Harmonic Motion ◽  
Analytical Stability ◽  
The Stability

The dynamic stability of a high-spinning liquid-filled rotor with both internal and external damping effects involved in is investigated in this paper. First, in the case of the rotor subjected to a transverse harmonic motion, the dynamic pressure of the liquid acting on the rotor is extracted through a planar flow analysis. Then the equation of perturbed motion for the liquid-filled rotor is derived. The analytical stability criteria as well as the stability boundaries are given. The results are extensions of those given by previous literature.

Shimmy model for electric vehicle with independent suspensions

2017 ◽  
Vol 232 (3) ◽  
pp. 330-340 ◽  
Author(s):  
Tian Mi ◽  
Gabor Stepan ◽  
Denes Takacs ◽  
Nan Chen
Keyword(s):  
Electric Vehicle ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Structural Parameters ◽  
Contact Point ◽  
Dynamic Responses ◽  
Simulation Results ◽  
The Stability ◽  
Linearized Model ◽  
Wheel Shimmy

A 5-degrees-of-freedom shimmy model is established to analyse the dynamic responses of an electric vehicle with independent suspensions. Tyre elasticity is considered by means of Pacejka’s magic formula. Under the nonslip assumption for the leading contact point, tyre–road constraint equations are derived. Numerical simulation is conducted with different structural parameters and initial conditions to observe the shimmy phenomenon. Simulation results indicate that Hopf bifurcation occurs at a certain vehicle forward speed. Moreover, suspension structural parameters, such as caster angle, affect wheel shimmy. The linearized model of the system presents the stability boundaries, which agree with the simulation results. The results of this study not only provide a theoretical reference for shimmy attenuation, but also validate the effectiveness of the provided model, which can be used in further dynamic analysis of vehicle shimmy.

Lyapunov’s Exponents for Non-Smooth Dynamical Systems: Behavior Identification Across Stability Boundaries With Bifurcations

2005 ◽  
Author(s):  
Alessio Ageno ◽  
Anna Sinopoli
Keyword(s):  
Closed Form ◽  
Stability Boundary ◽  
Period Doubling ◽  
Numerical Algorithms ◽  
Theoretical Method ◽  
Smooth Functions ◽  
Stability Boundaries ◽  
Strong Discontinuities ◽  
Starting Point ◽  
The Stability

In this paper some recently developed techniques to evaluate both analytically and numerically stability features of a non-smooth dynamical system, are used to investigate in detail the stability boundaries of regions corresponding to given stable periodic responses. The problem analyzed is that of a rocking block simply supported on a harmonically moving rigid ground; in this case, if the block is assumed to be a rigid body, the strong discontinuities characterizing the dynamic evolution are due to the impacts occurring each time the block crosses the initial equilibrium configuration. Therefore some special tools, specific for non-smooth functions, must be introduced to perform the stability analysis. In the present study, the theory due to Mu¨ller [19] is used to handle the evaluation of Lyapunov’s exponents upon discontinuities, by introducing the treatment, both analytical and numerical, of “saltation matrices”. Such a general theoretical method on one hand has been adapted to the numerical algorithms needed for the solution of the complete, non-linearized, problem and on the other hand, it allowed the development of the closed-form analytical reference solutions, obtained by linearizing assumptions less restrictive than those used by Hogan [8, 9]. The approximated stability boundaries obtained by the linearized closed-form solutions have been the starting point to guide the choice of the system parameters values to locate the responses in regions where bifurcations can arise. Inside these ranges several examples can be presented to illustrate the trends exhibited by the numerically evaluated Lyapunov’s exponents when the values of the forcing amplitude increase over the stability boundary of the symmetric responses while the value of the forcing frequency is fixed. Among these, investigations on sequences of responses composing period doubling cascades toward chaos, can provide a good and interesting test to appreciate the indications offered by the numerically derived Lyapunov’s exponents.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

scholarly journals Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quintic Nonlinearity

2021 ◽  
Vol 11 (11) ◽  
pp. 4833
Author(s):  
Afroja Akter ◽  
Md. Jahedul Islam ◽  
Javid Atai
Keyword(s):  
Numerical Stability ◽  
Bragg Gratings ◽  
Stability Boundaries ◽  
Quintic Nonlinearity ◽  
Numerical Stability Analysis ◽  
Gap Solitons ◽  
The Stability ◽  
Dual Core

We study the stability characteristics of zero-velocity gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity. The model supports two disjointed families of gap solitons (Type 1 and Type 2). Additionally, asymmetric and symmetric solitons exist in both Type 1 and Type 2 families. A comprehensive numerical stability analysis is performed to analyze the stability of solitons. It is found that dispersive reflectivity improves the stability of both types of solitons. Nontrivial stability boundaries have been identified within the bandgap for each family of solitons. The effects and interplay of dispersive reflectivity and the coupling coefficient on the stability regions are also analyzed.

Dynamic Response of the Spectral Element Model by Using the FFT

2007 ◽  
Vol 345-346 ◽  
pp. 845-848
Author(s):  
Joo Yong Cho ◽  
Han Suk Go ◽  
Usik Lee
Keyword(s):  
Fourier Transforms ◽  
Initial Conditions ◽  
Element Model ◽  
Spectral Element ◽  
Dynamic Responses ◽  
Analysis Method ◽  
Euler Beam ◽  
Exact Analytical Solutions ◽  
Simply Supported ◽  
Spectral Analysis Method

In this paper, a fast Fourier transforms (FFT)-based spectral analysis method (SAM) is proposed for the dynamic analysis of spectral element models subjected to the non-zero initial conditions. To evaluate the proposed SAM, the spectral element model for the simply supported Bernoulli-Euler beam is considered as an example problem. The accuracy of the proposed SAM is evaluated by comparing the dynamic responses obtained by SAM with the exact analytical solutions.

DRY TURBULENCE FROM A TIME-DELAYED CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 645-668 ◽  
Author(s):  
A. N. SHARKOVSKY ◽  
YU. MAISTRENKO ◽  
PH. DEREGEL ◽  
L. O. CHUA
Keyword(s):  
Difference Equation ◽  
Stability Region ◽  
Nonlinear Difference Equation ◽  
Chua’S Circuit ◽  
Stability Boundaries ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Chaotic Region ◽  
The Stability ◽  
Left Portion

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

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