Elsevier

Journal of Sound and Vibration

Volume 260, Issue 3, 20 February 2003, Pages 499-518
Journal of Sound and Vibration

Quantification of non-viscous damping in discrete linear systems

https://doi.org/10.1016/S0022-460X(02)00952-5Get rights and content

Abstract

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.

Introduction

The true nature of the damping forces in a dynamic system is often not known with any great accuracy. The most common approach in vibration modelling is to assume a model in which it is supposed that the instantaneous generalized velocities are the only relevant state variables that determine damping. This approach was first introduced by Rayleigh [1] via his famous ‘dissipation function’, a quadratic expression for the energy dissipation rate with a symmetric matrix of coefficients, the ‘damping matrix’. The equations of motion describing the free vibration of such a linear system with N degrees of freedom can be expressed asMq̈(t)+Cq̇(t)+Kq(t)=0.Here M, C and K are the N×N mass, damping and stiffness matrices respectively, and q(t) is the vector of the generalized co-ordinates (A list of nomenclature is given). The dynamics of such systems have been extensively studied, and are quite well understood. This damping model, which we will call ‘viscous damping’, is generally the only damping model allowed in commercial finite element (FE) codes [2], and it is also the only damping model usually taken into account in experimental modal analysis (EMA) [3]. In fact, in most EMA and FE methods a further idealization of viscous damping is used, known as ‘proportional damping’ or ‘classical damping’. This simplification, also pointed out by Rayleigh, allows the damping matrix to be diagonalized simultaneously with the mass and stiffness matrices, preserving the simplicity of real normal modes as in the undamped case.

It is well recognized that proportional damping is very rarely a physically realistic model because practical experience in modal testing shows that most real-life structures possess complex modes instead of real normal modes. Complex modes can arise from viscous damping, provided it is non-proportional. However, the physical justification for viscous damping is hardly more convincing than that for proportional damping. Any causal model which makes the energy dissipation functional non-negative is a possible candidate for a damping model. Such damping models, in which the damping force depends on anything other than the instantaneous generalized velocities, will be called in this paper ‘non-viscous’ damping models. Recently in a series of papers Adhikari and Woodhouse [4], [5], [6], [7] have considered the problem of identification of viscous and non-viscous damping from vibration measurements. In these studies, attention was focused on the following questions:

  • (1)

    From experimentally determined complex modes can one identify the underlying damping mechanism? Is it viscous or non-viscous? Can the correct model parameters be found experimentally?

  • (2)

    Is it possible to establish experimentally the spatial distribution of damping?

  • (3)

    Is it possible that more than one damping model with corresponding ‘correct’ sets of parameters may represent the system response equally well, so that the identified model becomes non-unique?

  • (4)

    Does the selection of damping model matter from an engineering point of view? Which aspects of behaviour are wrongly predicted by an incorrect damping model?

These questions highlight the distinction between viscous and non-viscous damping models, simply because most vibration analysis textbooks and computer packages only allow viscous damping, and the aim here is to address the question of whether this restriction matters in practice.

Section snippets

Linear models of non-viscous damping

Damping models in which the dissipative forces depend on any quantity other than the instantaneous generalized velocities, then will be called non-viscous damping models. Clearly a wide range of choice is possible. The discussion in this paper is confined to linear systems only, and the most general way to model damping within the linear range is through the class of damping models which depend on the past history of motion via convolution integrals over suitable kernel functions or Green

Analytical background

Woodhouse [19] and Adhikari [20], [21] have shown that conventional modal analysis can be extended to non-viscously damped systems of form (2). The eigenvalue problem associated with this equation can be defined by taking the Laplace transformλk2MzkkGk)zk+Kzk=0,where G(s) is the Laplace transform of G(t). The eigenvalue problem of form (7) has been discussed by Adhikari [20], [21]. The eigenvalues, λk, associated with Eq. (7) are roots of the characteristic equationdetD(s)=0.Because G(t) is

Indices based on the non-viscous damping matrix

In this section two indices of non-viscosity will be developed. It is assumed that the non-viscous damping matrix G(t) is available beforehand. Thus, the indices to be developed here are best suited for analytical applications.

Example 1: A four-d.o.f. system

A four-d.o.f. system with non-viscous damping is considered to illustrate the use of the four non-viscosity indices suggested above. The mass and stiffness matrices of the system are taken to beM=diag[1,2,2,1]andK=5−300−37−400−47−300−35.The matrix of damping functions is assumed to be of the formG(t)=diagδ(t)+μ1e−μ1t,δ(t)+μ2e−μ2t5,δ(t)+μ2e−μ2t5,δ(t)+μ3e−μ3t10.This implies that the damping mechanism is a linear combination of viscous and exponential damping models. It may be verified that none

Error analysis

The numerical values of the non-viscosity indices proposed here are unbounded except that γi⩾0,∀i. The lack of an upper bound may be regarded as a possible drawback because from a single value of γi it is not in general possible to comprehend the degree of non-viscosity of damping. One useful way to interpret the non-viscosity indices is to analyze the errors that arise if one makes the assumption of viscous damping for a non-viscously damped system. There are various possible choices of a

Conclusions

Quantification of the amount of non-viscosity of damping in linear multiple-degree-of-freedom dynamic systems has been considered. Four indices, based on (1) moments of the non-viscous damping matrix, (2) the Laplace transform of the non-viscous damping matrix, (3) transfer function residues and (4) complex modes, have been proposed. The first and the second indices are suitable for analytical studies, while the other two are aimed at using experimental data. The relative merits and demerits of

References (28)

  • D.J. Ewins

    Modal Testing: Theory and Practice

    (1984)
  • D.F. Golla et al.

    Dynamics of viscoelastic structures—a time domain finite element formulation

    Transactions of American Society of Mechanical Engineers, Journal of Applied Mechanics

    (1985)
  • D.J. McTavish et al.

    Modeling of linear viscoelastic space structures

    Transactions of American Society of Mechanical Engineers, Journal of Vibration and Acoustics

    (1993)
  • M.A. Biot

    Variational principles in irreversible thermodynamics with application to viscoelasticity

    Physical Review

    (1995)
  • Cited by (62)

    • A model order reduction technique for systems with nonlinear frequency dependent damping

      2020, Applied Mathematical Modelling
      Citation Excerpt :

      Frequency dependent damping is a commonly occurring phenomenon in many engineering applications. In structural dynamics, for instance, unless idealised by viscous or proportional damping, material damping is typically frequency dependent or non-viscous [1,2]. Similarly, in acoustics, damping due to impedance walls, like those present in turbo fan aircraft engine inlets [3] or those used in duct acoustics for approximating acoustic boundary layers [4], is often frequency dependent.

    • Microstructural topology optimization of viscoelastic materials of damped structures subjected to dynamic loads

      2018, International Journal of Solids and Structures
      Citation Excerpt :

      The microstructural topology optimization based on ESL (Equivalent Static Loads) method was proposed by Xu et al. (2016). However, these research considered only viscous damping, which is rarely present in realistic damping materials (Adhikari and Woodhouse, 2003). Therefore, the viscoelastic damping (also referred to as non-viscos damping) should be considered for the damping material.

    View all citing articles on Scopus
    View full text