scholarly journals TWO MODELS OF NONSMOOTH DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (10) ◽  
pp. 2853-2860 ◽  
Author(s):  
MADELEINE PASCAL
Keyword(s):  
Dry Friction ◽  
Coupled Oscillators ◽  
Degrees Of Freedom ◽  
Analytical Form ◽  
Critical Value ◽  
Form Solution ◽  
Nonsmooth Systems ◽  
The Stability ◽  
Close Form ◽  
Nonsmooth Dynamical Systems

Two examples of nonsmooth systems are considered. The first one is a two degrees of freedom oscillator in the presence of a stop. A discontinuity appears when the system position reaches a critical value. The second example consists of coupled oscillators excited by dry friction. In this case, the discontinuity occurs when the system's velocities take a critical value. For both examples, the dynamical system can be partitioned into different configurations limited by a set of boundaries. Within each configuration, the dynamical model is linear and the close form solution is known. Periodic orbits, including several transitions between the various configurations of the system, are found in analytical form. The stability of these orbits is investigated by using the Poincaré map modeling.

scholarly journals SLIDING BIFURCATIONS: A NOVEL MECHANISM FOR THE SUDDEN ONSET OF CHAOS IN DRY FRICTION OSCILLATORS

2003 ◽  
Vol 13 (10) ◽  
pp. 2935-2948 ◽  
Author(s):  
M. DI BERNARDO ◽  
P. KOWALCZYK ◽  
A. NORDMARK
Keyword(s):  
Dry Friction ◽  
Sudden Onset ◽  
Stick Slip ◽  
Border Collision Bifurcations ◽  
Sliding Bifurcation ◽  
Nonsmooth Systems ◽  
Different Types ◽  
Nonsmooth Dynamical Systems ◽  
Sudden Jump ◽  
Friction Oscillators

Recent investigations of nonsmooth dynamical systems have resulted in the study of a class of novel bifurcations termed as sliding bifurcations. These bifurcations are a characteristic feature of so-called Filippov systems, that is, systems of ordinary differential equations (ODEs) with discontinuous right-hand sides. In this paper we show that sliding bifurcations also play an important role in organizing the dynamics of dry friction oscillators, which are a subclass of nonsmooth systems. After introducing the possible codimension-1 sliding bifurcations of limit cycles, we show that these bifurcations organize different types of "slip to stick-slip" transitions in dry friction oscillators. In particular, we show both numerically and analytically that a sliding bifurcation is an important mechanism causing the sudden jump to chaos previously unexplained in the literature on friction systems. To analyze such bifurcations we make use of a new analytical method based on the study of appropriate normal form maps describing sliding bifurcations. Also, we explain the circumstances under which the theory of so-called border-collision bifurcations can be used in order to explain the onset of complex behavior in stick-slip systems.

scholarly journals Constant Threshold Correction to Electrically Charged Dilatonic Black Holes

1997 ◽  
Vol 12 (22) ◽  
pp. 1597-1603 ◽  
Author(s):  
Kwan-Leung Chan
Keyword(s):  
Black Hole Solution ◽  
Exact Analytical Solution ◽  
Analytical Form ◽  
Form Solution ◽  
Close Form Solution ◽  
Threshold Correction ◽  
Extremal Limit ◽  
Electromagnetic Tensor ◽  
Close Form ◽  
Electrically Charged

We investigate the effect of a constant threshold correction to a general non-extreme, static, spherically symmetric, electrically charged black hole solution of the dilatonic Einstein–Maxwell Lagrangian, with an arbitrary coupling β between the electromagnetic tensor and the dilaton field. For a small β, an exact analytical solution is obtained. For an arbitrary β, a close form solution, up to first-order in the constant threshold correction, of the metric and the dilaton are presented. In the extremal limit, the close form solution is reduced to an exact analytical form.

scholarly journals Numerical Algorithm for Delta of Asian Option

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Boxiang Zhang ◽  
Yang Yu ◽  
Weiguo Wang
Keyword(s):  
Efficient Solution ◽  
Variance Reduction ◽  
Analytical Form ◽  
Form Solution ◽  
Asian Option ◽  
Asian Options ◽  
Hedging Strategy ◽  
Close Form Solution ◽  
Reduction Methods ◽  
Close Form

We study the numerical solution of the Greeks of Asian options. In particular, we derive a close form solution ofΔof Asian geometric option and use this analytical form as a control to numerically calculateΔof Asian arithmetic option, which is known to have no explicit close form solution. We implement our proposed numerical method and compare the standard error with other classical variance reduction methods. Our method provides an efficient solution to the hedging strategy with Asian options.

Waves at liquid surfaces: coupled oscillators and mode mixing

1991 ◽  
Vol 433 (1889) ◽  
pp. 663-678 ◽  
Keyword(s):  
Surface Waves ◽  
Linear Theory ◽  
Coupled Oscillators ◽  
Degrees Of Freedom ◽  
Physical Reality ◽  
Critical Value ◽  
Surface Element ◽  
Surface Viscosity ◽  
Frame Work ◽  
Mode Mixing

The dispersion of surface waves on liquids has been reconsidered in the frame work of currently established linear theory. Mixed excitations, which are neither capillary nor dilational in nature, can occur due to the coupling between the lossy oscillator formed by the vertical and horizontal motions of a surface element. The results emphasize that the capillary and dilational waves are only approximately transverse and longitudinal in nature and that it cannot, in general, be correct to neglect the coupling between these degrees of freedom. In the present case mode mixing only occurs when a particular surface viscosity - that affecting shear normal to the surface - exceeds a critical value. Experimentally accessible tests of the predicted mode mixing are proposed, which would further test the physical reality of the surface viscosity involved.

The Research of the Influence of Damping on Automonile Brake Groan

2011 ◽  
Vol 295-297 ◽  
pp. 2231-2234
Author(s):  
Xian Jie Meng ◽  
Jun Wei Li
Keyword(s):  
Nonlinear Dynamics ◽  
Calculation Result ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Critical Value ◽  
Damping Coefficient ◽  
Dynamics Model ◽  
Brake Disk ◽  
Excited Vibration ◽  
Nonlinear Dynamics Model

A two degrees of freedom nonlinear dynamics model of self-excited vibration induced by dry-friction of brake disk and pads is built firstly, the numerical method is taken to study the impacts of damping on automobile brake groan. The calculation result shows that the variation of damping coefficient and has different influence on automobile brake groan. and all have A critical value, when damping coefficient greater or less than the critical value the brake disk and pad possess the different vibration state respectively.

scholarly journals Using chaos synchronization to estimate the largest lyapunov exponent of nonsmooth systems

2000 ◽  
Vol 4 (3) ◽  
pp. 207-215 ◽  
Author(s):  
Andrzej Stefanski ◽  
Tomasz kapitaniak
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dry Friction ◽  
Chaos Synchronization ◽  
Duffing Oscillator ◽  
Largest Lyapunov Exponent ◽  
Nonsmooth Systems ◽  
The Largest Lyapunov Exponent ◽  
Nonsmooth Dynamical Systems

We describe the method of estimation of the largest Lyapunov exponent of nonsmooth dynamical systems using the properties of chaos synchronization. The method is based on the coupling of two identical dynamical systems and is tested on two examples of Duffing oscillator: (i) with added dry friction, (ii) with impacts.

scholarly journals Analytical Investigation of Periodic Solutions for a Coupled Oscillator With Dry Friction

2009 ◽  
Author(s):  
Madeleine Pascal
Keyword(s):  
Periodic Solutions ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Analytical Investigation ◽  
Contact Forces ◽  
Coulomb’S Friction ◽  
Friction Laws ◽  
The Masses ◽  
Close Form ◽  
Form Stability

In this paper, we present an analytical method to investigate the behavior of a two degrees of freedom oscillator excited by dry friction. The system consists of two masses connected by linear springs. These two masses are in contact with a driving belt moving at a constant velocity. The contact forces between the masses and the belt are obtained from Coulomb’s friction laws. A set of periodic solutions involving a global sticking phase followed by several other phases where one or both masses are slipping, are found in close form. Stability conditions related to these solutions are obtained.

scholarly journals Stability of nonaxisymmetric ferrofluid flow in rotating cylinders with magnetic field

2005 ◽  
Vol 2005 (23) ◽  
pp. 3727-3737 ◽  
Author(s):  
Jitender Singh ◽  
Renu Bajaj
Keyword(s):  
Magnetic Field ◽  
Applied Magnetic Field ◽  
Critical Value ◽  
Annular Space ◽  
Rotating Cylinders ◽  
The Stability

Effect of an axially applied magnetic field on the stability of a ferrofluid flow in an annular space between two coaxially rotating cylinders with nonaxisymmetric disturbances has been investigated numerically. The critical value of the ratioΩ∗of angular speeds of the two cylinders, at the onset of the first nonaxisymmetric mode of disturbance, has been observed to be affected by the applied magnetic field.

scholarly journals Velocity and acceleration level inverse kinematic calculation alternatives for redundant manipulators

Meccanica ◽  
2021 ◽  
Author(s):  
Dóra Patkó ◽  
Ambrus Zelei
Keyword(s):  
Degrees Of Freedom ◽  
Direct Method ◽  
Tracking Error ◽  
Inverse Kinematic ◽  
Linear Feedback ◽  
Joint Position ◽  
Stability Properties ◽  
Acceleration Level ◽  
Error Elimination ◽  
The Stability

AbstractFor both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That is when the desired task possesses lower dimensions than the degrees-of-freedom of a redundant manipulator. The IK calculation can be implemented numerically in several alternative ways not only in case of the redundant but also in the non-redundant case. We study the stability properties and the feasibility of a tracking error feedback and a direct tracking error elimination approach of the numerical implementation of IK calculation both on velocity and acceleration levels. The feedback approach expresses the joint position increment stepwise based on the local velocity or acceleration of the desired TCP trajectory and linear feedback terms. In the direct error elimination concept, the increment of the joint position is directly given by the approximate error between the desired and the realized TCP position, by assuming constant TCP velocity or acceleration. We investigate the possibility of the implementation of the direct method on acceleration level. The investigated IK methods are unified in a framework that utilizes the idea of the auxiliary input. Our closed form results and numerical case study examples show the stability properties, benefits and disadvantages of the assessed IK implementations.

Stability of Ring-Type MEMS Gyroscopes Subjected to Stochastic Angular Speed Fluctuation

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Samuel F. Asokanthan ◽  
Soroush Arghavan ◽  
Mohamed Bognash
Keyword(s):  
Angular Velocity ◽  
Degrees Of Freedom ◽  
Microelectromechanical Systems ◽  
Damping Ratio ◽  
Operating Conditions ◽  
System Stability ◽  
System Response ◽  
Ring Type ◽  
Mems Gyroscopes ◽  
The Stability

Effect of stochastic fluctuations in angular velocity on the stability of two degrees-of-freedom ring-type microelectromechanical systems (MEMS) gyroscopes is investigated. The governing stochastic differential equations (SDEs) are discretized using the higher-order Milstein scheme in order to numerically predict the system response assuming the fluctuations to be white noise. Simulations via Euler scheme as well as a measure of largest Lyapunov exponents (LLEs) are employed for validation purposes due to lack of similar analytical or experimental data. The response of the gyroscope under different noise fluctuation magnitudes has been computed to ascertain the stability behavior of the system. External noise that affect the gyroscope dynamic behavior typically results from environment factors and the nature of the system operation can be exerted on the system at any frequency range depending on the source. Hence, a parametric study is performed to assess the noise intensity stability threshold for a number of damping ratio values. The stability investigation predicts the form of threshold fluctuation intensity dependence on damping ratio. Under typical gyroscope operating conditions, nominal input angular velocity magnitude and mass mismatch appear to have minimal influence on system stability.

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