Nonviscous Damping Research Articles

On the interaction of exponential non-viscous damping with symmetric nonlinearities

This paper studies the interaction between non-viscous damping and nonlinearities for nonlinear oscillators with reflection symmetry. The non-viscous damping function is an exponential damping model which adds a decaying memory property to the damping term of the oscillator. We consider the case of softening and hardening behaviour in the frequency response of the system. Numerical simulations of the Duffing oscillator show a significant enhancement of the resonance peaks for increasing levels of non-viscous damping parameter in the hardening case, but not in the softening case. This can be explained in the general context by an energy balance analysis of the undamped unforced oscillator, which shows that for hardening nonlinearities the growth of damping with the energy level is an order of magnitude smaller in the exponential case than in the viscous case.

Dynamic Response Characteristics of a Nonviscously Damped Oscillator

The characteristics of the frequency response function of a nonviscously damped linear oscillator are considered in this paper. It is assumed that the nonviscous damping force depends on the past history of velocity via a convolution integral over an exponentially decaying kernel function. The classical dynamic response properties, known for viscously damped oscillators, have been generalized to such nonviscously damped oscillators. The following questions of fundamental interest have been addressed: (a) Under what conditions can the amplitude of the frequency response function reach a maximum value?, (b) At what frequency will it occur?, and (c) What will be the value of the maximum amplitude of the frequency response function? Introducing two nondimensional factors, namely, the viscous damping factor and the nonviscous damping factor, we have provided exact answers to these questions. Wherever possible, attempts have been made to relate the new results with equivalent classical results for a viscously damped oscillator. It is shown that the classical concepts based on viscously damped systems can be extended to a nonviscously damped system only under certain conditions.

Calculation of Eigensolution Derivatives for Nonviscously Damped Systems

A method to calculate the derivatives of the eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters is presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over given kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy the classical orthogonality relationship. The proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned. Numerical examples are provided to illustrate the derived results. Nomenclature c j = constant associated with the derivative of u j D(s) = dynamic stiffness matrix d j = constant associated with the derivative of v j G(s) = Laplace transform of G(t) G(t) = nonviscous damping matrix K = stiffness matrix L[ ] = Laplace transform of [ ] M = mass matrix m = total number of eigenvectors N = number of degrees of freedom of the system n = number of relaxation parameters p = design parameter rank () = rank of a matrix s = Laplace domain parameter u j = jth eigenvector (t) = displacement vector v j = jth adjoint (left) eigenvector x j = vector associated with the derivative of u j yj = vector associated with the derivative of v j θ j = normalization constant for the jth eigenvector λ j = jth eigenvalue μk = relaxation parameters Subscripts () T = matrix transpose () � = derivative with respect to s

A Galerkin method for distributed systems with non-local damping

In this paper, a non-local damping model including time and spatial hysteresis effects is used for the dynamic analysis of structures consisting of Euler–Bernoulli beams and Kirchoff plates. Unlike ordinary local damping models, the damping force in a non-local model is obtained as a weighted average of the velocity field over the spatial domain, determined by a kernel function based on distance measures. The resulting equation of motion for the beam or plate structures is an integro-partial-differential equation, rather than the partial-differential equation obtained for a local damping model. Approximate solutions for the complex eigenvalues and modes with non-local damping are obtained using the Galerkin method. Numerical examples demonstrate the efficiency of the proposed method for beam and plate structures with simple boundary conditions, for non-local and non-viscous damping models, and different kernel functions.

Qualitative dynamic characteristics of a non-viscously damped oscillator

This paper considers the linear dynamics of a single-degree-of-freedom non-viscously damped oscillator. It is assumed that the non-viscous damping force depends on the history of velocity via a convolution integral over an exponentially decaying kernel function. Classical qualitative dynamic properties known for viscously damped oscillators have been generalized to such non-viscously damped oscillators. The following questions of fundamental interest have been addressed: (i) under what conditions can a non-viscously damped oscillator sustain oscillatory motions? (ii) how does the natural frequency of a non-viscously damped oscillator compare with that of an equivalent undamped oscillator? and (iii) how does the decay rate compare with that of an equivalent viscously damped oscillator? Introducing two non-dimensional factors, namely, the viscous damping factor and the non-viscous damping factor, we provide answers to these questions. Wherever possible, attempts are made to relate the new results with equivalent classical results for a viscously damped oscillator.

Discrete element modelling of railway ballast

The discrete element method has been used to simulate the behaviour of railway ballast under different test conditions. Single particle crushing tests have been simulated using agglomerates of bonded balls, and the distribution of strengths correctly follows the Weibull distribution, and the size effect on average strength is also consistent with that measured in the laboratory. Realistic fast fracture can be obtained if non-viscous damping is reduced. Oedometer tests on aggregates of crushable ballast particles have also been simulated and compared with the results from laboratory tests. Finally, box tests which simulate traffic loading have been simulated using both spherical balls and 8-ball clumps. It is found that the 8-ball clumps give much more realistic behaviour due to particle interlocking.

302 補助空気室付きローリングシール型空気ばねの減衰特性

Fig. A2 Comparison of theoretical and experimental results Fig.A1 Analytical model of air spring Table A1 Dimensions of the air spring shown in Fig.A1In this paper, we analyzed a rolling seal type air spring with an orifice. Two types of damping (viscous damping and non-viscous damping) through the orifice are taken in our analysis. The experimental result shows that the air spiring with circular orifice reveals the non-viscous damping characteristics.

Direct time-domain integration method for exponentially damped linear systems

Time-domain analysis of multiple-degree-of-freedom linear non-viscously damped systems is considered. It is assumed that the non-viscous damping forces depend on the past history of velocities via convolution integrals over exponentially decaying kernel functions. A direct time-domain integration method is proposed. The proposed approach is based on an extended state-space representation of the equations of motion. The state-space method, in turn, is based on introduction of a set of internal variables. The numerical method for the calculation of the displacements eliminates the need for explicit calculation of the velocities and usually large number of internal variables at each time step. This fact particularly makes this method numerically efficient. The proposed method is illustrated by two numerical examples.

Analysis of Asymmetric Nonviscously Damped Linear Dynamic Systems

Multiple-degree-of-freedom linear asymmetric nonviscously damped systems are considered. It is assumed that the nonviscous damping forces depend on the past history of velocities via convolution integrals over exponentially decaying kernel functions. An extended state-space approach involving a single asymmetric matrix is proposed. The nature of the eigensolutions in the extended state space has been explored. Some useful results relating the modal matrix in the extended state space and the modal matrix in the original space has been derived. Numerical examples are provided to illustrate the results.

Symmetric State-Space Method for a Class of Nonviscously Damped Systems

Multiple-degree-of-freedom linear nonviscously damped systems are considered. It is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions. The traditional state-space approach, well known for viscously damped systems, is extended to such nonviscously damped systems using a set of internal variables. Suitable numerical examples are provided to illustrate the proposed approach. I. Introduction V ISCOUSdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems. This model, e rst introduced by Rayleigh, 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping. Viscous damping models are used widely for their simplicity and mathematical convenience, even though the behavior of real structural materials is, at best, poorly mimicked by simple viscous models.Forthisreason,itiswellrecognizedthat,ingeneral,aphysically realistic model of damping will not be viscous. Damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models. Mathematically, any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model. Clearly, a wide range of choice is possible, either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments. Here, we will use a particular type of damping model that is not viscous, and throughout the paper the terminology nonviscous damping will refer to this specie c model only. Possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions. 2 The equations of motion of an N-degree-of-freedom linear system with such damping can be expressed by

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.

Derivative of Eigensolutions of Nonviscously Damped Linear Systems

Derivatives of eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters are presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy any orthogonality relationship. It is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems. Numerical examples are provided to illustrate the derived results.

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.

IDENTIFICATION OF DAMPING: PART 3, SYMMETRY-PRESERVING METHODS

In two recent papers (Adhikari and Woodhouse 2001 Journal of Sound and Vibration243, 43–61; 63–88), methods were proposed to identify viscous and non-viscous damping models for vibrating systems using measured complex frequencies and mode shapes. In many cases, the identified damping matrix becomes asymmetric, a non-physical result. Methods are presented here to identify damping models which preserve symmetry of the system. Both viscous and non-viscous models are considered. The procedure is based on a constrained error minimization approach and uses only experimentally identified complex modes and complex natural frequencies together with, for the non-viscous model, the mass matrix of the system. The methods are illustrated by numerical examples.

Eigenrelations for Nonviscously Damped Systems

Multiple-degree-of-freedom linear dynamic systems with nonviscous damping is considered. The nonviscous damping is such that the damping forces depend on the past history of the velocities via convolution integrals over some kernel functions. For the sake of generality, the system matrices are allowed to be asymmetric. The mode-orthogonality relationships, known for undamped or viscously damped systems, have been generalized to such nonviscously damped systems. Some expressions are suggested for the normalization of the eigenvectors. A number of useful results relating the system matrices and the eigensolutions have been established. Numerical examples are provided to illustrate the derived results.

IDENTIFICATION OF DAMPING: PART 2, NON-VISCOUS DAMPING

In a companion paper (see pp. 43–61 of this issue), it was shown that when a system is non-viscously damped, an identified equivalent viscous damping model does not accurately represent the damping behaviour. This demands new methodologies to identify non-viscous damping models. This paper takes a first step, by outlining a procedure for identifying a damping model involving an exponentially decaying relaxation function. The method uses experimentally identified complex modes and complex natural frequencies, together with the knowledge of the mass matrix for the system. The proposed method and several related issues are discussed by considering numerical examples of a linear array of damped spring-mass oscillators. It is shown that good estimates can be obtained for the exponential time constant and the spatial distribution of the damping.

Dynamics of structures: theory and applications to earthquake engineering

Dynamics of structures: theory and applications to earthquake engineering

A model for the time evolution of the internal friction observed in some fcc metals under irradiation: The peaking effect

The time evolution of the internal friction that has been observed in certain fcc metals (Cu, Ag, Al) under irradiation, and known generally as the ’’peaking effect,’’ has been shown by a recent set of experiments1–2 to be inadequately explained by the existing models which describe the energy dissipation through dislocation motion by the well-known analogy to a vibrating string with viscous damping. A phenomenological model is thus proposed here that introduces a source of energy dissipation through dislocation motion which is independent of the dislocation velocity, viz., nonviscous damping. With dislocation pinning by thermally diffusing point defects and strain- dependent depinning also included in the model, the observed internal-friction dependence upon irradiation time as well as upon the applied strain amplitude and frequency are well described.

Finite Element Model with Nonviscous Damping

A constitutive law by which structural damping is modeled as a relationship between stress, strain, and strain rate in a material is used in conjunction with the finite element method to develop general integral expressions for viscous and nonviscous damping matrices. To solve the set of nonlinear equations resulting from the presence of nonviscous damping, a solution technique is developed by modifying the Newmark method to accommodate an iterative solution and treat the nonviscous damping as a pseudo-force. The technique is then checked for accuracy and convergence in single- and multi-degree-of-freedom problems, and is found to be accurate and efficient for initial-condition problems with small nonviscous damping.

On the linear theory of density perturbations in a neutrino+baryon universe

Various aspects of the linear theory of density perturbations in a universe containing a significant population of massive neutrinos are calculated. Because linear perturbations in the neutrino density are subject to nonviscous damping on length scales smaller than the effective neutrino Jeans length, the fluctuation spectrum of the neutrino density perturbations just after photon decoupling is expected to peak near the maximum neutrino Jeans mass. The gravitational effects of nonneutrino species are included in calculating the maximum neutrino Jeans mass, which is found to be (M/sub J/(t))/sub max/approx.10/sup 17/M/sub sun//(m/sub ..nu../(eV))/sup 2/, about an order of magnitude smaller than is obtained when nonneutrino species are ignored. An explicit expression for the nonviscous damping of neutrino density perturbations less massive than the maximum neutrino Jeans mass is derived. The linear evolution of density perturbations after photon decoupling is discussed. Of particular interest is the possibility that fluctuations in the neutrino density induce baryon density perturbations after photon decoupling and that the maximum neutrino Jeans determines the characteristic bound mass of galaxy clusters.