Boundary for Complete Set of Attractors for Forced–Damped Essentially Nonlinear Systems

2015 ◽  
Vol 82 (5) ◽  
Author(s):  
Itay Grinberg ◽  
Oleg V. Gendelman
Keyword(s):  
State Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Boundary Curve ◽  
Trapping Region ◽  
Analytic Procedure ◽  
Strongly Nonlinear ◽  
Damped System ◽  
Analysis Of Stability ◽  
Dynamic Attractors

Forced–damped essentially nonlinear oscillators can have a multitude of dynamic attractors. Generically, no analytic procedure is available to reveal all such attractors. For many practical and engineering applications, however, it might be not necessary to know all the attractors in detail. Knowledge of the zone in the state space (or the space of initial conditions), in which all the attractors are situated might be sufficient. We demonstrate that this goal can be achieved by relatively simple means—even for systems with multiple and unknown attractors. More specifically, this paper suggests an analytic procedure to determine the zone in the space of initial conditions, which contains all attractors of the essentially nonlinear forced–damped system for a given set of parameters. The suggested procedure is an extension of well-known Lyapunov functions approach; here we use it for analysis of stability of nonautonomous systems with external forcing. Consequently, instead of the complete state space of the problem, we consider a space of initial conditions and define a bounded trapping region in this space, so that for every initial condition outside this region, the dynamic flow will eventually enter it and will never leave it. This approach is used to find a special closed curve on the plane of initial conditions for a forced–damped strongly nonlinear oscillator with single-degree-of-freedom (single-DOF). Solving the equations of motion is not required. The approach is illustrated by the important benchmark example of x2n potential, including the celebrated Ueda oscillator for n = 2. Another example is the well-known model of forced–damped oscillator with double-well potential. We also demonstrate that the boundary curve, obtained by analytic tools, can be efficiently “tightened” numerically, yielding even stricter estimation for the zone of the existing attractors.

Damped Transition of a Strongly Nonlinear System of Coupled Oscillators Into a State of Continuous Resonance Scattering

2011 ◽  
Author(s):  
David Andersen ◽  
Xingyuan Wang ◽  
Yuli Starosvetsky ◽  
Kevin Remick ◽  
Alexander Vakakis ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
Resonance Scattering ◽  
Linear Oscillator ◽  
System Response ◽  
Analytical Treatment ◽  
Strongly Nonlinear ◽  
Damped System ◽  
Initial Excitation ◽  
Strongly Nonlinear System ◽  
Energy Plane

We examine analytically and experimentally a new phenomenon of ‘continuous resonance scattering’ in an impulsively excited, two-mass oscillating system. This system consists of a grounded damped linear oscillator with a light, strongly nonlinear attachment. Previous numerical simulations revealed that for certain levels of initial excitation, the system engages in a special type of response that appears to track a solution branch formed by the so-called ‘impulsive orbits’ of this system. By this term we denote the periodic (under conditions of resonance) or quasi-periodic (under conditions of non-resonance) responses of the system when a single impulse is applied to the linear oscillator with the system being initially at rest. By varying the magnitude of the impulse we obtain a manifold of impulsive orbits in the frequency-energy plane. It appears that the considered damped system is capable of entering into a state of continuous resonance scattering, whereby it tracks the impulsive orbit manifold with decreasing energy. Through analytical treatment of the equations of motion, a direct relationship is established between the frequency of the nonlinear attachment and the amplitude of the linear oscillator response, and a prediction of the system response during continuous scattering resonance is provided. Experimental results confirm the analytical predictions.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Dynamics of nanodust particles emitted from elongated initial orbits

2018 ◽  
Vol 617 ◽  
pp. A43 ◽  
Author(s):  
A. Czechowski ◽  
I. Mann
Keyword(s):  
Solar Wind ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Dust Cloud ◽  
Theoretical Models ◽  
Ecliptic Plane ◽  
Parent Body ◽  
The Sun ◽  
Trapping Region ◽  
Eccentric Orbits

Context. Because of high charge-to-mass ratio, the nanodust dynamics near the Sun is determined by interplay between the gravity and the electromagnetic forces. Depending on the point where it was created, a nanodust particle can either be trapped in a non-Keplerian orbit, or escape away from the Sun, reaching large velocity. The main source of nanodust is collisional fragmentation of larger dust grains, moving in approximately circular orbits inside the circumsolar dust cloud. Nanodust can also be released from cometary bodies, with highly elongated orbits. Aims. We use numerical simulations and theoretical models to study the dynamics of nanodust particles released from the parent bodies moving in elongated orbits around the Sun. We attempt to find out whether these particles can contribute to the trapped nanodust population. Methods. We use two methods: the motion of nanodust is described either by numerical solutions of full equations of motion, or by a two-dimensional (heliocentric distance vs. radial velocity) model based on the guiding-center approximation. Three models of the solar wind are employed, with different velocity profiles. Poynting–Robertson and the ion drag are included. Results. We find that the nanodust emitted from highly eccentric orbits with large aphelium distance, like those of sungrazing comets, is unlikely to be trapped. Some nanodust particles emitted from the inbound branch of such orbits can approach the Sun to within much shorter distances than the perihelium of the parent body. Unless destroyed by sublimation or other processes, these particles ultimately escape away from the Sun. Nanodust from highly eccentric orbits can be trapped if the orbits are contained within the boundary of the trapping region (for orbits close to ecliptic plane, within ~0.16 AU from the Sun). Particles that avoid trapping escape to large distances, gaining velocities comparable to that of the solar wind.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

scholarly journals A Type of Set Membership Estimation Designed for Dynamic Flux Balance Models

Processes ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1762
Author(s):  
Xin Shen ◽  
Hector Budman
Keyword(s):  
State Space ◽  
Initial Conditions ◽  
Variable Structure ◽  
Upper And Lower Bounds ◽  
Structure System ◽  
E Coli ◽  
Flux Balance ◽  
System A ◽  
Variable Structure System ◽  
Set Membership

Dynamic flux balance models (DFBM) are used in this study to infer metabolite concentrations that are difficult to measure online. The concentrations are estimated based on few available measurements. To account for uncertainty in initial conditions the DFBM is converted into a variable structure system based on a multiparametric linear programming (mpLP) where different regions of the state space are described by correspondingly different state space models. Using this variable structure system, a special set membership-based estimation approach is proposed to estimate unmeasured concentrations from few available measurements. For unobservable concentrations, upper and lower bounds are estimated. The proposed set membership estimation was applied to batch fermentation of E. coli based on DFBM.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals Elastic shocks in relativistic rigid rods and balls

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180858 ◽  
Author(s):  
João L. Costa ◽  
José Natário
Keyword(s):  
Free Boundary ◽  
Minkowski Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Space Time ◽  
Spherically Symmetric ◽  
Soft Phase ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We study the free boundary problem for the ‘hard phase’ material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130 , 343–400), both for rods in (1 + 1)-dimensional Minkowski space–time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space–time. Unlike Christodoulou, we do not consider a ‘soft phase’, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

Vibrational Characteristics of Ball Bearings

1977 ◽  
Vol 99 (2) ◽  
pp. 284-287 ◽  
Author(s):  
P. K. Gupta ◽  
L. W. Winn ◽  
D. F. Wilcock
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Dominant Frequency ◽  
Hertzian Contact ◽  
Analytical Simulation ◽  
Contact Frequency ◽  
Ball Contact ◽  
Vibrational Characteristics ◽  
General Motion

The classical differential equations of motion of the ball mass center in an angular contact thrust loaded ball bearing are integrated with prescribed initial conditions in order to simulate the natural high frequency vibrational characteristics of the general motion. Two distinct frequencies are identified in the analytical simulation and their existence is also confirmed experimentally. One of the frequencies is found to be associated with the Hertzian contact spring at the ball race contact and it is therefore defined as the “elastic contact frequency”, Ωe. The other dominant frequency corresponding to oscillatory motion of the ball in the raceway groove appears to be kinematic in nature and it is, therefore, termed as the “bearing kinematic frequency”, Ωk. It is shown that for a given bearing Ωe and Ωk, vary as, respectively, 1/6 and 1/2 powers of the ball contact load and, therefore, for a given load these frequencies correspond to the natural frequencies of the bearing as applied in any vibrational analysis or simulation.

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