586 S¸. Tol and H.N. O¨ zgu¨ven [2–10]. For example, Tsai and Chou utilized the substructure FRF synthesis method in the formulation of the joint parameter identification [2]. Wang and Liou [3] improved the work of Tsai and Chou [2]. They avoided inversion of matrices in their algorithm; hence, tried to reduce noise effect in the identification. Hwang [4] used the same formulation and improved the results using an averaging process to exclude highly sensitive regions. Ren and Beards [6] developed a generalized coupling method taking into account the physical restrictions of the real structures, and identified joint parameters with this new method. However, they avoided stiff joints in order to avoid ill-conditioned matrices, and used weighting techniques for better accuracy [1]. Celic and Boltezar [7] improved the method developed by Ren and Beards [1,6] by including the effects of rotational degrees of freedom (RDOFs). Another approach similar to the one presented in [6] is proposed by Maia et al. [8]. They reformulated the impedance uncoupling technique, and identified joints without using joint related FRFs. However, this method has not been validated with an experimental study. Yang et al. [9] derived identification equations employing substructure synthesis. They modeled a joint in terms of translational and rotational stiffness values and used singular value decomposition to avoid noise effect. However, joint damping was not included in their work. In this study an experimental identification method based on FRF decoupling and optimization algorithm is proposed for modeling structural joints. The method developed is an extension of the method proposed by the authors in an earlier work [11]. In the method proposed in the earlier work FRFs of two substructures connected with a bolted joint are measured, while the FRFs of the substructures are obtained theoretically or experimentally. Then the joint properties are calculated in terms of translational, rotational and cross-coupling stiffness and damping values by using FRF decoupling. In this present work, an optimization algorithm is proposed to update the values obtained from FRF decoupling. The validity and application of the proposed method are demonstrated with several experimental studies by using beams connected with hexagonal bolts. 57.2 Theoretical Formulation 57.2.1 Identification of Dynamic Properties of Joints Using FRF Decoupling Frequency response function coupling is one of the most widely used methods in the literature in analyzing two structures coupled elastically. Consider substructures A, B and their assembly (structure C) obtained by coupling them with a flexible element as shown in Fig. 57.1. The coordinates j and k represent joint degrees of freedoms (DOFs), while r and s are the ones that belong to the selected points of substructures A and B, respectively, excluding joint DOFs. By using the elastic coupling equations (explained in more detail in [11]), it is possible to decouple and thus to calculate the complex stiffness matrix representing joint stiffness and damping as shown below: ŒK DŒŒHjr :ŒŒHrr ŒHC rr 1:ŒHrj ŒHjj ŒHkk 1 (57.1a) ŒK DŒŒHks :ŒHC rs 1:ŒHrj ŒHjj ŒHkk 1 (57.1b) ŒK DŒŒHjr :ŒHC sr 1:ŒHsk ŒHjj ŒHkk 1 (57.1c) ŒK DŒŒHks :ŒŒHss ŒHC ss 1:ŒHsk ŒHjj ŒHkk 1 (57.1d) As it was discussed in the earlier work of the authors [11] the most accurate results are obtained with Eq. (57.1a). However, from the experimental applicability point of view among the decoupling equations the most practical one is Eq. (57.1d). Equation (57.1a) requires the measurements of FRFs which belong to substructure A at both joint and non-joint DOFs, which k j s r Substructure B Substructure A Structure C r s [K*] Fig. 57.1 Elastic coupling of two substructures
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