Mass and Stiffness Matrix Updating of Damped Coupled Structure Using Normal Response Function Method

摘要

Quite often a structural dynamic finite element model is required to be updated so as to accurately predict the dynamic characteristics like natural frequencies & the mode shapes. Since in many situations undamped natural frequencies and mode shapes need to be predicted, it has generally been the practice in these situations to seek updating of only mass & stiffness matrix so as to obtain a reliable prediction model. Updating using frequency response functions (FRFs) has been one of the widely used approaches for updating, including updating of mass & stiffness matrices. However, the problem with FRF based methods, for updating mass and stiffness matrices, is that these methods are based on the use of complex FRFs. Use of complex FRFs to update mass and stiffness matrices is not theoretically correct as complex FRFs are not only affected by these two properties but also by damping. Therefore, in situations where updating of only mass and stiffness matrices using FRFs is required, the use of complex FRFs based updating formulation is not fully justified and would lead to inaccurate updated models. This difficulty is addressed in this paper through an FRF based finite element model updating procedure that is based on the concept of normal FRFs. The normal FRFs, which represent the FRFs of a structure in the absence of damping, are used thereby developing a formulation that is consistent to update mass and stiffness matrices. This paper addresses two issues. Firstly it considers mass and stiffness matrix updating of structures that have close modes and have high levels of damping. Updating of mass and stiffness matrix using complex FRFs is especially difficult in this situation because the complex FRF data is heavily affected by damping and tends to strongly couple the frequency response around close modes. This probably represents the worst scenario for mass and stiffness matrix updating using complex FRFs. This paper evaluates the effectiveness of the proposed Normal FRF based updating method for updating of such structures through a numerical study on an H-shape frame resembling an automotive chassis. It also compares the performance of the method with an existing FRF based updating method that makes use of complex FRFs. It is found that the normal FRF based method is able to yield accurate estimates of the mass and the stiffness matrix even under the presence of high damping and close modes due to its ability to remove the influence of damping from the updating process via use of normal FRFs. The second issue that is addressed in this paper is to evaluate the effectiveness of the method when used in practice using experimental data. Since the developed normal FRF based method requires estimates of the normal FRFs, which are obtained from the measured complex FRFs, it is necessary to examine the pros and cons of the process of estimation of the normal FRFs itself and subsequently its effect on updating. For this purpose, an experimental study on a representative beam structure is conducted. It is found that the incompleteness of the experimental data is an important issue in the estimation of the normal FRFs that needs to be carefully dealt in the implementation of the method in an experimental framework. In the present study the technique of regeneration of the unmeasured FRFs has been used and is found to be effective in the estimation of normal FRFs and updating.