IDENTIFICATION OF DAMPING: PART 4, ERROR ANALYSIS

In previous papers (S. ADHIKARI and J. WOODHOUSE 2001 Journal of Sound and Vibration243, 43-61; 63-88; S. ADHIKARI and J. WOODHOUSE 2002 Journal of Sound and Vibration251, 477-490) methods were proposed to obtain the coefficient matrix for a viscous damping model or a non-viscous damping model with an exponential relaxation function, from measured complex natural frequencies and modes. In all these works, it has been assumed that exact complex natural frequencies and complex modes are known. In reality, this will not be the case. The purpose of this paper is to analyze the sensitivity of the identified damping matrices to measurement errors. By using numerical and analytical studies it is shown that the proposed methods can indeed be expected to give useful results from moderately noisy data provided a correct damping model is selected for fitting. Indications are also given of what level of noise in the measured modal properties is needed to mask the true physical behaviour.     

Pole assignment of friction-induced vibration for stabilisation through state-feedback control

Friction-induced self-excited linear vibration is often governed by a second-order matrix differential equation of motion with an asymmetric stiffness matrix. The asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface. When the friction coefficient becomes high enough, the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues (poles) coalesce (when viscous damping is low).This short paper presents a receptance-based inverse method for assigning complex poles to second-order asymmetric systems through (active) state-feedback control of a combination of active stiffness, active damping and active mass, which is capable of assigning negative real parts to stabilise an unstable system.     

Quantification of non-viscous damping in discrete linear systems

The damping forces in a multiple-degree-of-freedom engineering dynamic system may not be accurately described by the familiar ‘viscous damping model’. The purpose of this paper is to develop indices to quantify the extent of any departures from this model, in other words the amount of ‘non-viscosity’ of damping in discrete linear systems. Four indices are proposed. Two of these indices are based on the non-viscous damping matrix of the system. A third index is based on the residue matrices of the system transfer functions and the fourth is based on the (measured) complex modes of the system. The performance of the proposed indices is examined by considering numerical examples.     

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.     

Decay of capillary turbulence on the surface of a viscous liquid

Decay of the turbulence of capillary waves on the surface of a real liquid is studied in the presence of the viscous damping of the waves at all frequencies after stepwise removal of external pumping. The investigation is performed using two different models: the weak turbulence approximation and the local turbulence model in which the energy redistribution over frequencies is described by the polynomial expression in the wave-occupation number. It is shown that the decay of turbulence in the viscous liquid proceeds self-similarly and begins at high frequencies. In the decay process, the frequency distribution of the energy of waves is close to the stationary form E _ω ～ ω^?3/2 in a wide frequency range below the boundary frequency of the inertial range during a relatively long time after removal of the external force. The calculation results agree qualitatively with the results of the experiments on capillary turbulence on the charged surface of liquid hydrogen.     

Design and optimization of periodic structure mechanical filter in suppression of foundation resonances

The paper describes the development of periodic structure mechanical filter (PSMF) that has the potential to reduce vibration transmission and sound radiation at resonances of the foundation in a two-degree-of-freedom (2dof) vibration isolation system by using the band gaps of the periodic structure. The transmission matrix method is used to model vibration transmission of the 2dof system and an analytical expression of sound radiation from the foundation plate is derived. The multi-layer PSMF composed of rigid plates and curved beams is represented by an equivalent m-k-c (viscous damping) model. The propagation/attenuation zones and attenuation ability of PSMF are expressed in the propagation scenario and the iso-attenuation curves by exploiting the unit cell transfer matrix invariant. Influence of the number of unit cells, viscous damping on the mobility of PSMF and vibro-acoustic behavior of the 2dof system is extensively studied. And under the constraints of installation space and stability of the whole system, the more the number of the unit cells, the better attenuation ability in the band gap can be obtained. The interaction between PSMF and the 2dof system is analyzed by the substructure method and contribution of frequency component from different substructures is identified by setting different level of damping for each substructure. Factors influencing the first mounting frequency of the 2dof system with PSMF are discussed and three styles of installing PSMF are studied. The performance of piecewise periodic PSMF and quasi-periodic PSMF is also studied in an attempt to eliminate new-born resonances by PSMF. An optimization scheme involving sensitivity analysis is applied to obtain the optimal values of m and k. And the optimization is effective. The experiment of detecting the band gap of PSMF and the comparative trial of a 2dof system with a flexible plate as the foundation with/without PSMF are carried out. Both numerical and experimental simulation results have demonstrated that by use of PSMF, the vibration transmission at resonances is reduced and the radiation of the foundation at resonances is suppressed.     

Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkin's technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v₁. For v>v₁, there is a value of the internal viscous damping coefficient, C_iv, which depends on the rotational speed and torque, above which destabilization occurs.The value of C_iv (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of C_iv critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%.For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using C_iv critical, parametric instabilities are still found in the practical range of operation.     

Improving the accuracy of the n-dB method for determining damping of non-lightly damped systems

Damping measurements using the spectral magnitude remain popular and are studied here for non-lightly damped systems using the variable bandwidth n-dB method, which is advantageous for non-lightly damped systems. The most commonly used estimator (based on normalised bandwidth) provides significant errors for non-lightly damped systems. An existing more accurate method (using the squares of the frequencies used in the former method) is exact for hysteretic damping, but still provides significant error for viscous damping. Improved estimators are developed in order to correct either exactly, or to insignificant errors, measurements taken with existing estimators. Neither further data nor the individual frequencies are required; the previously calculated damping values are corrected. The application of the improved estimators is dependent upon the existing estimator used and the damping type; however a strategy is suggested to reduce errors if the latter is unknown.     

A general model for power dissipation by domain wall movement in thin ferromagnetic films

In this work we present a model for magnetic domain wall (DW) dynamics in metallic ferromagnets with uniaxial anisotropy. The model allows us to consider both viscous and eddy current dampings. While the latter mechanism involves only one degree of freedom — the coordinate of the DW, the former involves all degrees of freedom of the spins in magnetic domains and in DW’s. The structure of the DW itself must then be considered, the conjecture used allowing one to describe the spin precessions which may appear in low quality factor Q materials. The model built is nonlinear, allowing one to deal with the set of complex susceptibilities characterizing the response to an ac drive field in strongly nonlinear circumstances, and has a general character, due to the normalized quantities used. In this work we analyze the susceptibility and the dissipated power spectra as functions of a dc transversal field, which can be treated only by this kind of models. We show that the viscous damping can become comparable to eddy current damping when precessions are excited.     

A new analytical technique for vibration analysis of non-proportionally damped beams

Vibrating linear mechanical systems, in particular continuous systems, are often modelled considering proportional damping distributions only, although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy. In this paper an analytical method is given to take into account the effects of a more general viscous damping model, referred to as non-proportional damping, on a class of vibrating continuous systems. A state-form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form, i.e., complex modes corresponding to non-synchronous motions. Numerical examples are included in order to show the efficiency of the proposed method; non-proportional damping distributions of different type, such as internal and external lumped or distributed viscous damping, are tested on non-homogeneous Euler-Bernoulli beams in bending vibration with different boundary conditions. Finally, a discussion on root locus diagrams behaviour and on modal damping ratio significance for non-proportionally damped systems is presented.     

Behavior of nonlinear fluid viscous dampers for control of shock vibrations

Fluid viscous dampers have been widely used for suppression of high velocity shocks. While linear fluid viscous dampers have been used for a long time, nonlinear fluid viscous dampers show considerable promise due to their superior energy dissipation characteristics and significant reduction in the damper force compared to a linear fluid viscous damper for the same peak displacement. This paper presents results from experimental study to characterize fluid viscous dampers when subjected to half-cycle sine shock excitation. The mathematical formulation and a numerical study to evaluate the relative performance of structures with fluid viscous dampers subjected to short-duration shock (impulse) loading are also discussed. The influence of damper nonlinearity (α) and the supplemental damping ratio (ξ_sd) on response has been investigated. The supplemental damping ratio of nonlinear fluid viscous dampers when subjected to shock excitation is found by equivalent linearization using the concept of equal energy dissipation. The paper also presents some design charts, which can be used for preliminary decisions on parameters of nonlinear dampers to be used in design.     

Hysteretic damping and causality

It is proven that linear oscillatory systems with hysteretic damping in the form of complex stiffness and/or complex elastic moduli satisfy the causality principle: the response of such a system to an arbitrary external force cannot appear earlier than the onset of the force. The proof, based on a rigorous solution to the problem of forced oscillations, is presented in detail for an oscillator with a complex stiffness, as well as in a brief explanation for a system with N mass. It is also shown that these systems are Lyapunov-unstable. A comparison is made to other linear hysteretic damping models.     

APPROXIMATING THE HYSTERETIC DAMPING MATRIX BY A VISCOUS MATRIX FOR MODELLING IN THE TIME DOMAIN

The paper is concerned with modelling the dynamic behaviour of a structure with damping. Hysteretic damping is commonly accepted to be reasonably accurate in some circumstances, but can only be applied directly in the frequency domain. Dynamic (time) behaviour, however, is most conveniently predicted by a viscous model. A damping matrix is constructed for use in the viscous equation which gives a dissipation of energy approximating to the hysteretic model. The approximation is justified by comparing results in the frequency and time domains and against measured data from a loudspeaker diaphragm. The ability of the matrix to reflect different damping in various components of the structure is considered, together with predicted natural frequencies and modes.     

Ab initio theory and calculations of X-ray spectra

There has been dramatic progress in recent years both in the calculation and interpretation of various x-ray spectroscopies. However, current theoretical calculations often use a number of simplified models to account for many-body effects, in lieu of first principles calculations. In an effort to overcome these limitations we describe in this article a number of recent advances in theory and in theoretical codes which offer the prospect of parameter free calculations that include the dominant many-body effects. These advances are based on ab initio calculations of the dielectric and vibrational response of a system. Calculations of the dielectric function over a broad spectrum yield system dependent self-energies and mean-free paths, as well as intrinsic losses due to multi-electron excitations. Calculations of the dynamical matrix yield vibrational damping in terms of multiple-scattering Debye–Waller factors. Our ab initio methods for determining these many-body effects have led to new, improved, and broadly applicable x-ray and electron spectroscopy codes. To cite this article: J.J. Rehr et al., C. R. Physique 10 (2009).     

Effects of band structure and matrix element on RKKY interaction

The effects of band structure and matrix elements on the RKKY interaction J(R) are separately investigated. When the Fermi surface has planes perpendicular to R, effects appear on the period of oscillation, the phase shift and the amplitude of J(R). The applicable region of the asymptotic form for large R and the validity of the free electron approximation are also examined. If there are no tangential planes perpendicular to R, it is found that: 1) when two interacting localized spins are on lattice points in the crystal, exponential damping appears even for the constant matrix element model and the matrix element effects introduce competing terms causing a sign change; 2) when one of the spins is at an interstitial position, the constant matrix model gives a weaker J(R) ∝ R^-2 damping, but the character of this term changes into the exponential damping by taking into account matrix elements.     

Frequency modulation of waveguides in acoustic band-gap materials consisting of solid cylinders in viscous liquid

We investigated waveguides in acoustic band-gap materials consisting of steel cylinders in a liquid with viscous damping. Numerical results show that when the viscous penetration depth is comparable to the structural length scale, linear defect states fall in complete acoustic band gaps forming waveguides. It is also found that the magnitude of the viscosity in the liquid has an influence on the frequency of waveguides, that large viscous damping can make the defect modes ascend. An expected frequency of waveguides can be obtained by modulating the viscous damping parameter θ.     

Dynamic characterization of high damping viscoelastic materials from vibration test data

The numerical analysis and design of structural systems involving viscoelastic damping materials require knowledge of material properties and proper mathematical models. A new inverse method for the dynamic characterization of high damping and strong frequency-dependent viscoelastic materials from vibration test data measured by forced vibration tests with resonance is presented. Classical material parameter extraction methods are reviewed; their accuracy for characterizing high damping materials is discussed; and the bases of the new analysis method are detailed. The proposed inverse method minimizes the residue between the experimental and theoretical dynamic response at certain discrete frequencies selected by the user in order to identify the parameters of the material constitutive model. Thus, the material properties are identified in the whole bandwidth under study and not just at resonances. Moreover, the use of control frequencies makes the method insensitive to experimental noise and the efficiency is notably enhanced. Therefore, the number of tests required is drastically reduced and the overall process is carried out faster and more accurately. The effectiveness of the proposed method is demonstrated with the characterization of a CLD (constrained layer damping) cantilever beam. First, the elastic properties of the constraining layers are identified from the dynamic response of a metallic cantilever beam. Then, the viscoelastic properties of the core, represented by a four-parameter fractional derivative model, are identified from the dynamic response of a CLD cantilever beam.     

Chains of oscillators with negative stiffness elements

Negative stiffness is not allowed by thermodynamics and hence materials and systems whose global behaviour exhibits negative stiffness are unstable. However the stability is possible when these materials/systems are elements of a larger system sufficiently stiff to stabilise the negative stiffness elements. In order to investigate the effect of stabilisation we analyse oscillations in a chain of n linear oscillators (masses and springs connected in series) when some of the springs? stiffnesses can assume negative values. The ends of the chain are fixed. We formulated the necessary stability condition: only one spring in the chain can have negative stiffness. Furthermore, the value of negative stiffness cannot exceed a certain critical value that depends upon the (positive) stiffnesses of other springs. At the critical negative stiffness the system develops an eigenmode with vanishing frequency. In systems with viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes even with light damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while for the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. A special element with controllable negative stiffness is suggested for designing hybrid materials whose stiffness and hence the dynamic behaviour is controlled by the magnitude of applied compressive force.     

Influence of thermal photons on the fundamental model of quantum optics with a coherent pump

A quantum system consisting of a two-level atom interacting with a single field mode of a high-Qcavity under influence of a coherent pump is considered. The analytical solutions for the P and Q distribution functions are obtained in the limit of large Rabi frequencies. In the presence of thermal photons, the P distribution function loses its property of restriction by the range on the complex plane and becomes an analytical function. When the ratio of the atomic decay rate to the cavity mode damping rate is smaller than 4, the effect of phase bistability appears. Absorptive optical bistability is absent in this case. On the basis of the system of Fokker-Planck equations for the quasi-probabilities corresponding to the atom being on the upper and lower atomic levels, computer simulation of the stochastic trajectory of motion for the system is presented.