Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture

2000 ◽  
Vol 68 (1) ◽  
pp. 42-48 ◽  
Author(s):  
A. F. Vakakis ◽  
O. Gendelman
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Analytical Techniques ◽  
Resonance Capture ◽  
Damped System ◽  
Energy Pumping ◽  
Analytical Approximations ◽  
Nonlinear Mechanical ◽  
Nonlinear Transient ◽  
Mechanical Oscillators

We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging can be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an Oε neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed.

Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems

2000 ◽  
Vol 68 (1) ◽  
pp. 34-41 ◽  
Author(s):  
O. Gendelman ◽  
L. I. Manevitch ◽  
A. F. Vakakis ◽  
R. M’Closkey
Keyword(s):  
Hamiltonian System ◽  
Equations Of Motion ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Part ◽  
Energy Pumping ◽  
Weakly Coupled ◽  
Nonlinear Mechanical ◽  
Low Energies ◽  
Dynamical Flow ◽  
Mechanical Oscillators

The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) “channeling” of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

scholarly journals Speed Oscillations of a Vehicle Rolling on a Wavy Road

2021 ◽  
Vol 11 (21) ◽  
pp. 10431
Author(s):  
Walter V. Wedig
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Travel Speed ◽  
Differential Algebraic Equations ◽  
Polar Coordinates ◽  
Algebraic Equations ◽  
Traffic Lights ◽  
Analytical Approximations ◽  
Vertical Vibrations ◽  
Nonlinear Equations Of Motion

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential–algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

2014 ◽  
Author(s):  
Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rational Functions ◽  
Special Treatment ◽  
Air Bearings ◽  
Fluid Forces ◽  
Stability Diagrams ◽  
Nonlinear Transient

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed in [1]. The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by second order rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace Transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor lead to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However the method [1] requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors [2] showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in [2] was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach enhances also the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.

Reduced Order Modelling in Nonlinear Mechanical Oscillators With Energy Pumping

2005 ◽  
Author(s):  
Xianghong Ma ◽  
Alexander F. Vakakis ◽  
Lawrence A. Bergman
Keyword(s):  
Dimensional Model ◽  
Reduced Order ◽  
Energy Pumping ◽  
Weakly Coupled ◽  
Nonlinear Mechanical ◽  
Dimensional Models ◽  
Mechanical Oscillators ◽  
Low Dimensional ◽  
Nonlinear Components ◽  
And Control

Energy pumping in nonlinear mechanical oscillators has been discovered and studied in mechanical systems consisting of weakly coupled, linear and nonlinear components [1–3]. In this paper this phenomenon is further studied and numerically verified on an 11 degree of freedom system. It also presents a technique to create low dimensional models for energy pumping systems using the Karhunen-Loeve (K-L) decomposition method. It is shown that energy pumping can be identified from the dominant K-L modes. The low dimensional models are used to reconstruct the system responses. From the comparisons between the reconstructed and simulated response, we can see that the K-L mode-based low-dimensional model can represent the system responses; it can be used for monitoring, diagnosis and control purposes.

Nonlinear Modes of a 2-DOF Inverted Pendulum

2011 ◽  
Author(s):  
Elvidio Gavassoni ◽  
Paulo Batista Gonc¸alves ◽  
Deane M. Roehl
Keyword(s):  
Degrees Of Freedom ◽  
Inverted Pendulum ◽  
Asymptotic Methods ◽  
Equations Of Motion ◽  
Conservative System ◽  
Nonlinear Modes ◽  
Fitting Procedure ◽  
Analytical Approximations ◽  
Complex Features ◽  
Analytical Expressions

A number of practical structures such as compliant offshore towers can be physically modeled, as a first approximation, as an inverted pendulum. The nonlinear dynamics of such model can present some complex features due to the nonlinear coupling among its degrees-of-freedom. In this paper a model of a spatial inverted pendulum restrained by three inclined extensional springs is adopted. Geometric imperfections are considered, and the solutions for both perfect and imperfect systems are presented herein. Especial attention is given to the determination of the nonlinear vibration modes. Non-similar and similar NNMs are obtained analytically by direct application of asymptotic methods and the results show important NNM features such as instability and multiplicity of modes. Poincare´ maps of the conservative system are used to identify the existence of modes that do not have a linear counterpart. Analytical approximations are derived for these nonlinear modes by the use of a two-dimensional polynomial fitting procedure obtained by the application of the Levenberg-Marquardt method. The points coordinates used in the fitting procedure were obtained through numerical integration of the governing equations of motion. Such analytical expressions can then be used in modal reduction, parametric analysis and in the derivation of important features of the system such as its frequency-amplitude relations and resonance curves.

The Behaviour of Damped Linear Systems in Steady Oscillation

1956 ◽  
Vol 7 (2) ◽  
pp. 156-168 ◽  
Author(s):  
R. E. D. Bishop
Keyword(s):  
Linear Systems ◽  
Classical Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Viscous Damping ◽  
Steady Oscillation ◽  
Harmonic Vibrations ◽  
Damped System ◽  
Near Resonance ◽  
Hysteretic Damping

SummaryThe classical theory of small harmonic vibrations of a linear damped system embodies the notion of “ viscous damping.” The equations of motion which result are somewhat complicated and, when there are more than two degrees of freedom, they are usually too unwieldy to be of much practical value. When the damping is small, however, approximating assumptions may be made which permit the treatment of systems which are near resonance as if they possess but one degree of freedom. But the effects of making these assumptions are by no means easily assessed, and even their justification is tedious.It is shown that these difficulties may be greatly diminished by postulating hysteretic damping instead of viscous damping; the concept of hysteretic damping has been dealt with in two previous papers. The equations then take a much simpler form and the justification for, and validity of, the foregoing approximations are more easily seen. Moreover, the effects of damping upon the principal modes which the system possesses in the absence of its damping may be elucidated in this way.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

scholarly journals Spin-Mechanics with Nitrogen-Vacancy Centers and Trapped Particles

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 651
Author(s):  
Maxime Perdriat ◽  
Clément Pellet-Mary ◽  
Paul Huillery ◽  
Loïc Rondin ◽  
Gabriel Hétet
Keyword(s):  
Degrees Of Freedom ◽  
Theoretical Background ◽  
Nitrogen Vacancy ◽  
Full Potential ◽  
Strongly Coupled ◽  
Trapped Particles ◽  
Quantum Regime ◽  
Atomic Spins ◽  
Nitrogen Vacancy Centers ◽  
Mechanical Oscillators

Controlling the motion of macroscopic oscillators in the quantum regime has been the subject of intense research in recent decades. In this direction, opto-mechanical systems, where the motion of micro-objects is strongly coupled with laser light radiation pressure, have had tremendous success. In particular, the motion of levitating objects can be manipulated at the quantum level thanks to their very high isolation from the environment under ultra-low vacuum conditions. To enter the quantum regime, schemes using single long-lived atomic spins, such as the electronic spin of nitrogen-vacancy (NV) centers in diamond, coupled with levitating mechanical oscillators have been proposed. At the single spin level, they offer the formidable prospect of transferring the spins’ inherent quantum nature to the oscillators, with foreseeable far-reaching implications in quantum sensing and tests of quantum mechanics. Adding the spin degrees of freedom to the experimentalists’ toolbox would enable access to a very rich playground at the crossroads between condensed matter and atomic physics. We review recent experimental work in the field of spin-mechanics that employ the interaction between trapped particles and electronic spins in the solid state and discuss the challenges ahead. Our focus is on the theoretical background close to the current experiments, as well as on the experimental limits, that, once overcome, will enable these systems to unleash their full potential.

scholarly journals BIOMIMETIC DESIGN OF LOW STIFFNESS ACTUATOR SYSTEMS FOR PROSTHETIC LIMBS

2011 ◽  
Author(s):  
D. L. Russell ◽  
M. McTavish
Keyword(s):  
Mechanical Properties ◽  
Degrees Of Freedom ◽  
Analytical Techniques ◽  
Energetic Costs ◽  
Multiple Degrees Of Freedom ◽  
Prosthetic Limbs ◽  
Limb Stiffness ◽  
Human Arm ◽  
Biomimetic Approach ◽  
Low Stiffness

The various relationships that are possible between the mechanical properties of single actuators and the overall mechanism (in this case a human arm with or without a prosthetic elbow) are discussed. Graphical and analytical techniques for describing the range of overall limb stiffnesses that are achievable and for characterizing the overall limb stiffness have been developed. Using a biomimetic approach and, considering energetic costs, stability and complexity, the implications of choosing passive or active implementations of stiffness are discussed. These techniques and approaches are particularly applicable with redundant (agonist - antagonist) actuators and multiple degrees of freedom. Finally, a novel biomimetic approach for control is proposed.

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