On the destabilizing effect of damping on discrete and continuous circulatory systems

The ‘Ziegler paradox’, concerning the destabilizing effect of damping on elastic systems loaded by nonconservative positional forces, is addressed. The paper aims to look at the phenomenon in a new perspective, according to which no surprising discontinuities in the critical load exist between undamped and damped systems. To show that the actual critical load is found as an (infinitesimal) perturbation of one of the infinitely many sub-critically loaded undamped systems. A series expansion of the damped eigenvalues around the distinct purely imaginary undamped eigenvalues is performed, with the load kept as a fixed, although unknown, parameter. The first sensitivity of the eigenvalues, which is found to be real, is zeroed, so that an implicit expression for the critical load multiplier is found, which only depends on the ‘shape’ of damping, being independent of its magnitude. An interpretation is given of the destabilization paradox, by referring to the concept of ‘modal damping’, according to which the sign of the projection of the damping force on the eigenvector of the dual basis, and not on the eigenvector itself, is the true responsible for stability. The whole procedure is explained in detail for discrete systems, and successively extended to continuous systems. Two sample structures are studied for illustrative purposes: the classical reverse double-pendulum under a follower force and a linear visco-elastic beam under a follower force and a dead load.     

Driven linear modes: Analytical solutions for finite discrete systems

We have obtained exact analytical expressions in closed form, for the linear modes excited in finite and discrete systems that are driven by a spatially homogeneous alternating field. Those modes are extended for frequencies within the linear frequency band while they are either end-localized or end-avoided for frequencies outside the linear frequency band. The analytical solutions are resonant at particular frequencies, which compose the frequency dispersion relation of the finite system.     

Dynamics of linear discrete systems connected to local, essentially non-linear attachments

The dynamics of a linear periodic substructure, weakly coupled to an essentially non-linear attachment are studied. The essential (non-linearizable) non-linearity of the attachment enables it to resonate with any of the linearized modes of the subtructure leading to energy pumping phenomena, e.g., passive, one-way, irreversible transfer of energy from the substructure to the attachment. As a specific application the dynamics of a finite linear chain of coupled oscillators with a non-linear end attachment is examined. In the absence of damping, it is found that the dynamical effect of the non-linear attachment is predominant in neighborhoods of internal resonances between the attachment and the chain. When damping exists energy pumping phenomena are realized in the system. It is shown that energy pumping strongly depends on the topological structure of the non-linear normal modes (NNMs) of the underlying undamped system. This is due to the fact that energy pumping is caused by the excitation of certain damped invariant NNM manifolds that are analytic continuations for weak damping of NNMs of the underlying undamped system. The bifurcations of the NNMs of the undamped system help explain resonance capture cascades in the damped system. This is a series of energy pumping phenomena occurring at different frequencies, with sudden lower frequency transitions between sequential events. The observed multi-frequency energy pumping cascades are particularly interesting from a practical point of view, since they indicate that non-linear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure, in compatibility with the design objectives.     

Collisional damping of Bernstein linear echoes

A Fokker Planck collisional model is used to study the time evolution of Bernstein linear wave echoes. It is shown that a very small amount of collision destroys efficiently the upper echoes.     

Experimental identification of viscous damping in linear vibration

This paper is concerned with the experimental evaluation of the performance of viscous damping identification methods in linear vibration theory. Both existing and some new methods proposed by the present authors [A.S. Phani, J. Woodhouse, Viscous damping identification in linear vibration, Journal of Sound and Vibration 303 (3–5) (2007) 475–500] are applied to experimental data measured on two test structures: a coupled three cantilever beam with moderate modal overlap and a free–free beam with low modal overlap. The performance of each method is quantified and compared based on three norms and the best methods are identified. The role of complex modes in damping identification from vibration measurements is critically assessed.     

Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems

A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damping ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsystem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude.     

Dynamic damping in magnetic Sine-Gordon systems

We study the influence of the elastic degrees of freedom of a one-dimensional spin-chain with ferromagnetic Heisenberg coupling in situations, where the magnetic dynamics can be described by a Sine-Gordon model. Compatibility of the Sine-Gordon approach with spin-lattice coupling is shown, and CsNiF₃ and domain wall motion are discussed as concrete physical realizations. As a consequence of the spin-lattice interaction a soliton-phonon coupling arises, which destroys the formal Lorentz invariance of the Sine-Gordon system. The influence of this coupling on the soliton dynamics is investigated. It results in a dynamic damping of the soliton motion, which in general is not of viscous type. Moreover, the influences of the soliton-phonon coupling on the phonon dispersion are estimated and possible experimental consequences are discussed.Project of the Sonderforschungsbereich 65 Festkörperspektroskopie Darmstadt-Frankfurt financed by special funds of the Deutsche Forschungsgemeinschaft     

Transverse damping systems in modern synchrotrons

Transverse feedback systems for suppression of transverse coherent beam oscillations are used in modern synchrotrons for preventing the development of transverse instabilities and damping residual beam oscillations after injection. Information on damper systems for the Large Hadron Collider (LHC; CERN, Geneva) and the accelerator complex FAIR (GSI, Darmstadt) is presented. The project for the LHC is being performed at the Laboratory of Particle Physics of the Joint Institute for Nuclear Research in collaboration with CERN. The information concerning the state of the project and the plans of its completion at the LHC is given. The results of the first design activity on transverse damping systems at the SIS100 and SIS300 synchrotrons, to be created in the framework of the new international project FAIR, are presented.     

On linear modeling of interface damping in vibrating structures

Dissipation of mechanical vibration energy at contact interfaces in a structure, commonly referred to as interface damping, is an important source of vibration damping in built-up structures and its modeling is the focus of the present study. The approach taken uses interface forces which are linearly dependent on the relative vibration displacements at the contact interfaces.The main objective is to demonstrate a straightforward technique for simulation of interface damping in built-up structures using FE modeling and simple, distributed, damping forces localized to interfaces where the damping occurs.As an illustration of the resulting damping the dissipated power is used for evaluation purposes. This is calculated from surface integrals over the contact interfaces and allows for explicit assessment of the effect of simulated interface forces for different cases and frequencies. The resulting loss factor at resonance is explicitly evaluated and, using linear simulations, it is demonstrated that high damping levels may arise even though the displacement differences between contacting surfaces at damped interfaces may be very small.     

The role of linear damping for explosive instabilities

It is shown that conservation laws of energy and momentum contain often sufficient information on various weakly nonlinear interactions. The changes in the temporal development of an explosive instability or its stabilization due to the linear damping of the interacting waves are discussed in more detail.Nademlýnská 600, Praha 9, Czechoslovakia.     

Detection of directional energy damping in vibrating systems

The transmission efficiency, frequency and amplitude alteration have been measured by a simple technique of coupled oscillators with a frequency gradient and in a system of non-Newtonian fluid in the form of corn-flour slime. The system of coupled oscillators was found to exhibit preferential energy transfer towards the low frequency end with the reverse propagation severely damped. Energy transfer in all directions was damped in the non-Newtonian fluid in comparison with water. Also the damping in non-Newtonian fluids works only after a lower limit for input amplitude. While most of the previous studies focussed on dissipation of energy within shock-absorbing systems, we demonstrate the contribution of re-distribution of energy reaching the output end to achieve shock absorbing.      

Quantification of correlations in quantum many-particle systems

We introduce a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems which is based on the relative von Neumann entropy computed from the density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.     

Structural stability in discrete singular systems

In this paper,we study the structural stability of singular discrete systems with perturbations in coefficient matrices,Some sufficient and necessary conditions of structurally stable singular discrete systems are given,Two kinds of structurally stable normal compensators are discussed.     

Response of shock isolators with linear and quadratic damping

Performance of two types of shock isolators is analyzed (a) with linear damping and (b) with quadratic law damping. Input to the systems is a base acceleration pulse of rectangular shape. It is found that with quadratic damping the optimum damping ratio varies for different pulse durations.     

Vibrational resonance in Duffing systems with fractional-order damping

The phenomenon of vibrational resonance (VR) is investigated in over- and under-damped Duffing systems with fractional-order damping. It is found that the factional-order damping can induce change in the number of the steady stable states and then lead to single- or double-resonance behavior. Compared with vibrational resonance in the ordinary systems, the following new results are found in the fractional-order systems. (1) In the overdamped system with double-well potential and ordinary damping, there is only one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for these new resonance behaviors is the value of the fractional-order satisfies α?>?1. (2) In the overdamped system with single-well potential and ordinary damping, there is no resonance, whereas there is a single-resonance for the case of fractional-order damping. The necessary condition for the new result is α?>?1. (3) In the underdamped system with double-well potential and ordinary damping, there are double-resonance and one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for the new single-resonance is α?相似文献   

Driving-dependent damping of Rabi oscillations in two-level semiconductor systems

We propose a mechanism to explain the nature of the damping of Rabi oscillations with an increasing driving-pulse area in localized semiconductor systems and have suggested a general approach which describes a coherently driven two-level system interacting with a dephasing reservoir. Present calculations show that the non-Markovian character of the reservoir leads to the dependence of the dephasing rate on the driving-field intensity, as observed experimentally. Moreover, we have shown that the damping of Rabi oscillations might occur as a result of different dephasing mechanisms for both stationary and nonstationary effects due to coupling to the environment. Present calculated results are found in quite good agreement with available experimental measurements.