366 G. Canbalog˘lu and H.N. Özgüven In engineering problems there are usually nonlinearities in structures; therefore it is vital to have model updating techniques for nonlinear structures as well. In literature there are different studies performed to update directly the nonlinear model of a structure [9–11]. In a more recent work, Isasa et al. [12] presented an approach which is based on multi-harmonic balance method and extended constitutive relation error for the updating of nonlinear models. For nonlinear structures, it is possible to employ the model updating techniques developed for the linear systems, provided that the dynamic characteristics of the linear part of the structure are extracted, which may require identification of nonlinearity in the system first. Kerschen et al. [13] presented a literature survey which is one of the most detailed nonlinear system identification literature surveys in which more than 400 papers were cited. Worden et al. [14] applied various time and frequency based nonlinear identification methods to a damper of an automobile. Eriten et al. [15] presented nonlinear system identification (NSI) approach in which experimental measurements are combined with slow-flow dynamic analysis and empirical mode decomposition. Very recently, Doranga and Wu studied [16] the Nonlinear Resonant Decay method for parameter identification of nonlinear dynamic systems. Canbaloglu and Özgüven, in another recent work, developed a method to identify nonlinearity and to obtain linear FRFs of nonlinear structures having multiple nonlinearities including friction type of nonlinearity, by using nonlinear FRF measurements [17], and used this method in the nonlinear model updating approach proposed [18]. The proposed method is experimentally validated by applying it to a real nonlinear T-beam test structure [19]. In this study, the method developed by the authors for nonlinear model updating [17, 18] is experimentally applied to the gun barrel of a battle tank. Dynamic modelling of the gun barrel of a battle tank is studied in different studies to improve the accuracy of the shooting and stabilization performance [20, 21] and it is shown that the fundamental mode of the gun barrel plays a key role in the response of the system. In this perspective, detailed model of the gun barrel at the fundamental mode is studied in this work. An equivalent single degree of freedom nonlinear model of the system is built for the fundamental mode of system. First, using the PRD method, both linear FRFs and the nonlinearities in the system are obtained from experimentally measured nonlinear FRFs. Afterwards, linear FE model of the test structure is built in ANSYS and it is updated by using the linear FRFs obtained through the PRD method. Thus, an updated nonlinear model of the test structure is constructed by using the identified nonlinearity and updated linear FE model of the system. Finally, predicted and measured FRFs of the test structure are compared at different forcing levels in order to demonstrate the accuracy of the updated nonlinear model of the system. 34.2 Theory The model updating method developed by Canbaloglu and Özgüven [17, 18] is employed in this study for updating the FE model of a gun barrel. Only a very brief summary of the method is presented here. The theory of the method is given in detail in Refs. [17] and [18]. For a nonlinear system, it is possible to write the following equation. Œ D f CŒ HF D HNL 1 HL 1 (34.1) where [ ], [ f ], [ HF], [HNL], [HL] are the nonlinearity matrix, nonlinearity matrix due to friction, nonlinearity matrix due to remaining nonlinearities that are dominant at high forcing levels of excitation, response level dependent nonlinear and linear FRF matrices, respectively. Measuring FRFs experimentally several times at the same frequency but at different forcing levels the following set of equations can be written: Œ HF iC1 f 1 D HNL iC1 1 HNL 1 1 i D1;2; : : : ; .n 1/ (34.2) In Eq. (34.2), subscript 1 indicates low forcing case and subscripts 2, 3, : : : n indicate high forcing cases. The nonzero elements in the nonlinearity matrices at the left hand side which can be written as polynomial functions of response amplitudes with unknown coefficients are the describing functions of the corresponding nonlinearities. Applying polynomial fit to (n-1) data points in a least square sense, the equation of the corresponding regression curve can be obtained in order to find the unknown coefficients. By comparing the terms of the regression equation with the corresponding describing functions, nonlinearities can be identified and then linear FRFs can easily be calculated as [17] HL 1 D h HNL 1 1 f 1 i 1 (34.3)
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