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 Treatment of Damping Forces in Vibration Theory

1955 ◽  
Vol 59 (539) ◽  
pp. 738-742 ◽  
Author(s):  
R. E. D. Bishop
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Forced Oscillations ◽  
Viscous Damping ◽  
Vibration Theory ◽  
Hysteretic Damping ◽  
Damped Oscillation ◽  
The Subject ◽  
Definition Of

SummaryThis paper is the first of a series of three which are concerned with the subject of “ hysteretic damping.” This type of damping, in a simple system with one degree of freedom, is like the familiar “ viscous damping ” in that it implies a resisting force which is in phase with velocity; but it is unlike viscous damping in that the magnitude of the force is not proportional to the velocity but to the displacement. When a system has n degrees of freedom, hysteretic damping implies that damping forces exist which are proportional to relative displacement but which are in phase with relative velocity.From a physical standpoint, hysteretic damping may give a better representation of the facts when the damping arises from the internal friction of solid materials. On the side of theory, it raises considerations which it is the purpose of these three papers to elucidate. It may be said, at the outset, that the notion of hysteretic damping raises no great mathematical difficulty; on the contrary, a main reason for presenting the theory is that it appears to allow of a much simpler discussion (than does viscous damping) of the nature of steady damped oscillation of systems having n degrees of freedom.In the first paper, the purpose is discussed of mathematical theories of damping in vibration theory. It is concluded that the theory of “ hysteretic damping ” is a useful one since it provides an alternative to the fiction of “ viscous ” damping while retaining the mathematical linearity of equations of motion. The word “ hysteretic” is proposed for use in this sense instead of the previously used adjective, namely “ structural.” “ Complex damping ” is related to hysteretic damping in a way which is explained.The theory is given for forced oscillations of a system with one degree of freedom. It is shown that free vibration cannot be treated satisfactorily unless the definition of hysteretic damping is widened in some way to cover non-harmonic motion.

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.

Investigation of manipulator dynamics with a viscous damping model

2007 ◽  
Vol 221 (5) ◽  
pp. 513-517
Author(s):  
P Herman
Keyword(s):  
Degrees Of Freedom ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Viscous Damping ◽  
First Order ◽  
Manipulator Dynamics ◽  
Interesting Insight ◽  
Damping Model ◽  
Insight Into

In this article, some remarks concerning dynamics investigation of a manipulator described using first-order equations of motion with a viscous damping model is conducted. The viscous damping model arises from the Rayleigh dissipation potential and decomposition of the manipulator mass matrix. As a result, it takes into account both kinematic and mechanical parameters of the system. Moreover, the use of first-order equations of motion leads to obtaining some interesting insight into the manipulator dynamics. The proposed approach was tested on a three-degrees-of-freedom, three-dimensional Yasukawa-like robot.

scholarly journals Hydrodynamic Interaction of Ocean Wave Energy Point Absorbers

2016 ◽  
Author(s):  
Lorenzo Banos Hernandez ◽  
Jose Maria Emperador Alzola
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Radiation Force ◽  
Domain Model ◽  
Irregular Waves ◽  
Diffraction Radiation ◽  
Wave Direction ◽  
Ocean Wave Energy ◽  
Near Resonance ◽  
Radiation Boundary

This work condenses various modeling techniques for different Point Absorber configurations. A combined frequency - time domain model will be developed in Matlab-FORTRAN in order to compute the displacement, velocities and the power absorbed in the heave mode. Additionally, a single buoy motion including multiple degrees of freedom will be investigated as well. Therefore, the diffraction-radiation Boundary Element Method solvers NEMOH and BEMIO will be applied in the calculation of the hydrodynamic coefficients, which will determine the solution of Newtons impulse equations of motion. Initially, the Wave to wire model will be validated through comparison with previous experimental results for a submerged cone cylinder shape (Buldra-FO3). A single, generic, vertical floating cylinder will be contemplated then, that responds to the action of the passing waves excitation. Later, two vertical floating cylinders aligned with the incident wave direction will be modeled for a variable distance between the bodies. For both unidirectional regular and irregular waves as an input in deep water, the convolutive radiation force function term will be hereby approximated through the Prony method. By changing the spatial disposition of the axisymmetric buoys, using for instance triangular or diamond shaped arrays of three and four bodies respectively, the study will focus on the interaction effects for regular waves. The results will highlight the most efficient layout for maximizing the energy production whilst providing important insights into their performance, revealing for instance displacement amplification or capture width ratios in near-resonance conditions.

scholarly journals Elucidation of near-resonance vibronic coherence lifetimes by nonadiabatic electronic-vibrational state character mixing

2018 ◽  
Vol 116 (37) ◽  
pp. 18263-18268 ◽  
Author(s):  
Shu-Hao Yeh ◽  
Ross D. Hoehn ◽  
Marco A. Allodi ◽  
Gregory S. Engel ◽  
Sabre Kais
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Underlying Mechanism ◽  
Electronic Energy ◽  
Molecular Aggregates ◽  
Vibrational States ◽  
Near Resonance ◽  
Harvesting Systems ◽  
Coupling Parameters ◽  
Vibronic States

Recent work suggests that the long-lived coherences observed in both natural and artificial light-harvesting systems (such as the Fenna–Matthews–Olson complex) could be attributed to the mixing of the pigments’ electronic and vibrational degrees of freedom. To investigate the underlying mechanism of these long coherence lifetimes, a sophisticated description of interactions between the molecular aggregates and the nonequilibrium fluctuations in the surrounding environment is necessary. This is done by implementing the hierarchical equations of motion approach on model homodimers, a method used in the intermediate coupling regime for many molecular aggregates wherein the nonequilibrium environment phonons play nontrivial roles in exciton dynamics. Here we report a character change in the vibronic states—reflective of property mixing between the electronic and vibrational states—induced by an interplay between system coupling parameters within the exciton-vibrational near-resonance regime. This mixing dictates vital aspects of coherence lifetime; by tracking the degree of mixing, we are able to elucidate the relationship between coherence lifetime and both the electronic energy fluctuation and the vibrational relaxation dephasing pathways.

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.

A Resonance Chart for Damped Vibration

1955 ◽  
Vol 59 (540) ◽  
pp. 850-852 ◽  
Author(s):  
R. E. D. Bishop
Keyword(s):  
Linear System ◽  
Convenient Method ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Viscous Damping ◽  
Similar Form ◽  
Hysteretic Damping ◽  
Resonance Curves ◽  
Phase Angles

A convenient method is pointed out for calculating the response of a damped linear system with one degree of freedom to harmonic excitation. Results of such calculations are usually represented by the familiar “ resonance curves ”—one curve being plotted for each intensity of damping. These curves are not particularly convenient to use and Yates has overcome several of their defects by throwing them into a nomographic form. Yates' nomogram is based upon the concept of viscous damping and it does not give the information of a conventional set of resonance curves in that it relates to the velocity of vibration. By changing over to hysteretic damping, a nomogram of somewhat similar form may be constructed such that it gives amplitudes and phase angles of displacements while retaining the advantages, over resonance curves, of this form of representation.

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

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