Short Communication
Direct identification and expansion of damping matrix for experimental–analytical hybrid modeling

https://doi.org/10.1016/j.jsv.2007.07.044Get rights and content

Abstract

The theory of direct experimental identification of damping matrix based on the dynamic stiffness matrix (DSM) method is further developed in this work. Based on the relationship between the DSMs of the smaller experimental model and larger analytical model, the mathematical relationship between the damping matrices of the two models is established. Examining the relationship, two methods are developed that can be used to expand the experimental damping matrix to the size of the analytical model. Validity of the expansion methods is demonstrated with numerical examples. The expanded damping matrix is intended to be combined with analytically formulated stiffness and mass matrices to build an experimental–analytical hybrid model. To find the frequency range, in which such a hybrid modeling is valid, a simple but effective method is developed.

Introduction

A damping model should represent both the mechanism and spatial distribution of the energy loss in the system. In contrast to the mass and stiffness matrices, formulation of the damping matrix still stands as a big challenge in modeling a linear dynamic system. Commonly used simple models such as the proportional damping or structural damping model are used for no reason but mathematical convenience. Various models such as viscoelastic [1], [2], friction [3], micro-slip [4], and air damping [5], [6] have been used to describe damping mechanism, while much less efforts have been made to represent the spatial distribution of damping.

Motivated by the desire for accurate simulation of dynamic systems, a substantial amount of research effort has been made to develop experimental damping identification methods. In those methods a proportional or structural damping matrix is frequently used, which is not finding but assuming the forms for the mechanism and the spatial distribution of damping. Furthermore, the damping matrix is formulated often by utilizing modal parameters such as natural frequencies, natural modes, and modal damping ratios extracted from measured frequency response functions (FRFs) [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. There are some methods developed to obtain the damping matrix directly from measured FRFs, thus eliminating the need to identify modal parameters [18], [19], [20], [21], [22].

The damping identification method developed by Lee and Kim [23] which identifies damping matrices directly from measured frequency response functions is unique as it formulates the damping matrix from the dynamic stiffness matrix (DSM). The DSM is obtained by inverting the measured FRF matrix, which is utilizing the fact that the imaginary part of the DSM is the damping matrix.

The DSM-based direct damping identification method is very attractive because it can be applied to any linear dynamic systems and does not use any arbitrary assumptions. However two fundamental issues were encountered while implementing the method [24], [25], [26], which are the difficulty of expanding the identified matrix to the size of the analytical model of the system and the high sensitivity of the accuracy of the result to measurement errors. While these issues are being addressed in our ongoing research, this paper deals with the first one, expansion of the identified damping matrix to the size of the analytical model. Major outcomes of the study presented in this paper are: (1) explicit relationship between the smaller, identified damping matrix and the larger damping matrix of the analytical model, (2) error analysis to understand the limitation of the DSM-based damping identification method, and (3) the method to expand the experimentally formulated damping matrix to the size of the analytical model to formulate the hybrid system equation.

Section snippets

Dynamic stiffness matrix-based damping matrix identification [23]

Direct damping identification method developed by Lee and Kim [23] can be summarized in three steps. In the first step, measured frequency response functions are put into a square matrix form as follows:Hexp(ω)=[H11(ω)H21(ω)H1M(ω)H12(ω)H22(ω)HM1(ω)HMM(ω)],where Hexp(ω) is the experimental frequency response function matrix, ω is the circular frequency (rad/s) and M is the number of measurement points. In the second step, the dynamic stiffness matrix SDexp(ω) defined at the experimental

Errors induced by the dynamic stiffness matrix approach itself

The dynamic stiffness matrix method is a frequency domain method; therefore can be applied only to linearly behaving systems. While implementing the dynamic stiffness matrix approach described in Eqs. (2), (3), both heavily and lightly damped systems will potentially pose challenges. Generally, a system is considered lightly damped when the average modal damping ratio is smaller than 1%. The challenge in lightly damped systems stems from the fact that the method tries to find the spatial

Expanding experimental damping matrix for hybrid modeling

Enabling the analytical–experimental hybrid modeling was the original motivation in developing the DSM-based damping identification method [23]. The idea is to build a simulation model of a dynamic system by combining analytically formulated mass and stiffness matrices with an experimentally identified damping matrix. The approach makes a perfect sense because the damping matrix cannot be formulated analytically like the mass and stiffness matrices. However, the damping matrix obtained from the

Conclusion

Application of the DSM-based damping matrix identification method is discussed in this paper. This method offers several distinct advantages over other damping matrix identification methodologies that use the inherent assumption of proportional damping. Firstly, the damping matrix is obtained directly from experimental frequency response functions, which eliminates the need for experimental modal analysis to extract modal parameters. Secondly, the method offers a more accurate spatial

References (37)

  • M. Link, Identification of physical system matrices using incomplete vibration test data, Proceedings of the fourth...
  • M. Link, M. Weiland, J.M. Barragán, Direct physical matrix identification as compared to phase resonance testing: an...
  • Y.W. Luk, Identification of physical mass, stiffness and damping matrices using pseudo-inverse, Proceedings of the...
  • K. Shye, M. Richardson, Mass, stiffness, and damping matrix estimates from structural measurements, Proceedings of the...
  • N.G. Creamer et al.

    Identification method for lightly damped structures

    Journal of Guidance

    (1988)
  • W.F. Tsang, E. Rider, The technique of extraction of structural parameters from experimental forced vibration data,...
  • J.L. Jensen, R. Brincker, A. Rytter, Identification of light damping in structures, Proceedings of the Eighth...
  • D.F. Pilkey, D.J. Inman, An iterative approach to viscous damping matrix identification, Proceedings of the 15th...
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