Short CommunicationDirect identification and expansion of damping matrix for experimental–analytical hybrid modeling
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: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 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
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