Research on the iterative method for model updating based on the frequency response function

Transcription

1 Acta Mech. Sin. 2012) 282): DOI /s RESEARCH PAPER Research on the iterative method for model updating based on the frequency response function Wei-Ming Li Jia-Zhen Hong Received: 18 June 2010 / Revised: 3 October 2011 / Accepted: 12 March 2012 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012 Abstract Model reduction technique is usually employed in model updating process. In this paper, a new model updating method named as cross-model cross-frequency response function CMCF) method is proposed and a new iterative method associating the model updating method with the model reduction technique is investigated. The new model updating method utilizes the frequency response function to avoid the modal analysis process and it does not need to pair or scale the measured and the analytical frequency response function, which could greatly increase the number of the equations and the updating parameters. Based on the traditional iterative method, a correction term related to the errors resulting from the replacement of the reduction matrix of the experimental model with that of the finite element model is added in the new iterative method. Comparisons between the traditional iterative method and the proposed iterative method are shown by model updating examples of solar panels, and both of these two iterative methods combine the CMCF method and the succession-level approximate reduction technique. Results show the effectiveness of the CMCF method and the proposed iterative method. Keywords Model updating Model reduction Frequency response function Iteration 1 Introduction In engineering field, people usually build theoretical model which is usually the finite element FE) model to master the dynamic characteristics of an engineering structure. The project was supported by the Key Project of the National Natural Science Foundation of China ). W.-M. Li J.-Z. Hong Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai, China jzhong@sjtu.edu.cn Nowadays, the appearances of high precision aircrafts and spacecrafts, large scale bridges and other new engineering structures impose higher demands on the reliability and accuracy of their FE models. However, there is usually a big gap between results of the finite element and those of the experiment data resulting from the modeling errors of joints, boundary conditions and structural damping, etc. So it is necessary to adopt the model updating methods to improve the precision and reliability of the finite element model [1, 2]. Nowadays, there are lots of model updating methods, mainly be categorized into two types according to the experimental data used in the model updating process: methods based on the mode data [3 6] and methods based on the frequency response function data [7 10]. Because the latter methods use the experiment data directly, which avoid the modal analysis process and can avoid introducing extra noises, they have drawn more attention. However, most of the model updating methods based on the frequency response function are eigensensitivity-based methods which limit the number of the updating parameters and need huge amounts of computing time. Modak et al. [11] compared the inverse eigensensitivity method and the response function method on the basis of computer simulated experimental data for the purpose of studying the convergence of the two methods and the accuracy. Steenackers et al. [12] concentrated on an updating method based on measured modal data which extended the conventional FE updating techniques by taking into account the uncertainty on the estimated modal parameters. Lin et al. [13] proposed a new model updating method which employed directly the response function data measured under base excitation. Results showed that this method was effective as applied to the identifications of mass and stiffness modeling errors, and had noise-resisting ability. Kanev et al. [14] gave experimental validation of an FE model updating approach to damage which could simultaneously update all three FE model matrices at the same time.

2 Research on the iterative method for model updating based on the frequency response function 451 In addition, the degrees of freedom DOFs) of the FE model greatly exceed those of the experimental measurement. One critical reason is that the rotational DOFs are difficult to be measured and some FE nodes in the internal structure can not be measured too. Meanwhile, the number of the sensors is restricted by the experimental conditions. Therefore, in order to solve the problem of mismatch between the DOFs of the FE model and those of the experimental model, model reduction [15 18] is the most frequently adopted technique. Nowadays, the structures such as aircrafts and spacecrafts are usually very complex and have huge amounts of DOFs. In order to obtain ideal results, a lot of iterations and computing time are still needed in model updating, though the DOFs of these structures are reduced by the model reduction process. In the traditional iterative method, the reduction matrix of the FE model was used to replace that of the experimental model, and then the iteration was employed to eliminate the errors produced by the replacement. The traditional iterative method just uses the simple replacement, which needs many iterations to obtain the ideal model results. In this paper, a new model updating method named as cross-model cross-frequency response function CMCF) method is proposed. In order to speed up convergence of the model updating, a new iterative method is put forward by adding a correction term to the traditional iterative method. The expression of the correction term is deduced with the CMCF method and a complete derivation process of the improved iterative technique is shown. This paper is organized as follows. In Sects. 1 and 2, the CMCF method and new iterative method are presented, respectively. Numerical examples of solar panels with measurement noise are elaborated in Sect. 3, and conclusions are given in Sect Cross-model cross-frequency response function CMCF) method The displacement frequency response function of an undamped linear analytical dynamic system can be described as H = ω 2 i M + K) 1, 1) where K, M are the stiffness matrix and mass matrix of the FE model, respectively; ω i, H are the i-th frequency and displacement frequency response function of the FE model, respectively. The displacement frequency response function of the experimental model which is assumed to be an undamped dynamic system can also be expressed as H = ω 2 j M + K ) 1, 2) where K, M are, respectively, the unknown stiffness matrix and mass matrix of the experimental model, and ω j, H ω j ) are, respectively, the j-th frequency and displacement frequency response function of the experimental model. From Eqs. 1) and 2), one obtains ω 2 i M + K)H = I, 3) ω 2 j M + K )H = I, 4) where I and I are unit matrixes. Premultiplying Eq. 4) by [Hω i )] T yields H T ω 2 j M + K )H = H T I, 5) where the superscript T is a transpose operator. Premultiplying Eq. 3) by H T yields H T ω 2 i M + K)H = HT I. 6) From Eq. 6), one obtains H T ω 2 i M + K)H = I T H. 7) Subtracting Eq. 5) from Eq. 7), one obtains H T K K + ω 2 i M ω2 j M )H = H T I I T H. 8) Suppose the relationships of the matrices between the FE model and the experimental one are K = K + α n K n, M = M + β n M n, 9) where K n, M n denote the stiffness matrix and mass matrix of the n-th element, respectively; α n and β n are the correction coefficients of the n-th stiffness matrix and mass matrix, respectively; Ne is the total number of the elements. Substituting Eq. 9) into Eq. 8) leads to α n H T K n H β n ω 2 j H T M n H = H T I I T H + H T ω 2 j ω 2 i )M H. 10) Assume the frequency order of the FE model that can be used for model updating is x and the columns number of every frequency response function that could be used for model updating is m, similarly, the frequency order of the experimental test model that can be used for model updating is y and the columns number of every frequency response function that could be used for model updating is n, and thus there are xmyn linear equations in Eq. 10). The correction values α n and β n can be obtained by solving Eq. 10), and the updated FE model can be gained from Eq. 9).

3 452 W.-M. Li, J.-Z. Hong 3 New iterative method for model updating based on model reduction Generally speaking, it is unpractical and even impossible to obtain all the mode information in experiment and the DOFs of the FE model usually greatly exceed those of the experimental measurement. In order to solve the problem of mismatch between the DOFs of the FE model and those of the experimental model, model reduction technique is frequently adopted. For the reduced FE model and reduced experimental test model, the CMCF method could be expressed as H T s K s K s + ω 2 i M s ω j 2M s )H s = H T s I I T H s, 11) where M s, K s, H s are the mass matrix, stiffness matrix and frequency response function of the reduced FE model, respectively; M s, K s, H s are those of the reduced experimental model, respectively. Based on the model reduction theory, the relationships between the FE model and the reduced FE model and the relationships between the experimental model and the reduced experimental model can be expressed as M s = T T f MT f, K s = T T f KT f, 12) M s = T T e M T e, K s = T T e K T e, 13) where T f, T e are the reduction matrix of the FE model and the experimental model, respectively. Substituting Eqs. 12) and 13) into Eq. 11) leads to H T s [T T e K T e T T f KT f ω 2 j T T e M T e ω 2 i T T f MT f )]H s = H s I I T H s. 14) Since assumptions exist in the model reduction theory, it is difficult to obtain ideal model updating results through a straightforward calculation process. Therefore, iterations are necessary. Iterating the mass matrix, stiffness matrix and the reduction matrix of the experimental model leads to H T s [T T e,k K k T e,k T T f KT f ω 2 j T T e,k M k T e,k ω 2 i T T f MT f )]H s = H T s I I T H s, 15) where k is the iteration number. The relationship between T f and T e,k is expressed as T e, k = T f + k, 16) where k is the differences between T f and T e,k. Substituting Eq. 16) into Eq. 15) leads to H T s {T f + k ) T K k T f + k ) T T f KT f [ω 2 j T f + k ) T M k T f + k ) ω 2 i T T f MT f ]}H s = H T s I I T H s, 17) rearranging Eq. 17) yields Hs T T f T K k K ) T f Hs ω 2 j Hs T T f T M k T f Hs ω 2 i HT s T T f MT f H s ) = Hs T I I T Hs +Hs T [T f T ω 2 j Mk ) K k k ]Hs +Hs T [ T k ω 2 j Mk ) K k T f ]Hs +Hs T T k ω 2 j Mk ) K k k Hs. 18) During the k-th iteration, the relationships between the FE model and the experimental model before model reduction are Kk = K + α n, k K n, 19) Mk = M + β n, k M n, 20) where α n, k and β n, k are the correction coefficients of the n- th stiffness matrix and mass matrix during the k-th iteration, respectively. Substituting Eqs. 19) and 20) into Eq. 18) yields α n, k H T s T T f K nt f H s ω 2 j β n, k H T s T T f M nt f H s = Hs T I I T Hs + ω 2 j ω 2 i )HT s T f T MT f Hs +Hs T T f T ω 2 j Mk ) K k k Hs +Hs T T k ω 2 j Mk ) K k T f Hs +Hs T T k ω 2 j Mk ) K k k ]Hs. 21) Supposing that K n s = T T f K nt f, 22) M n s = T T f M nt f, 23) substituting Eqs. 22), 23) and 12) into Eq. 21) yields α n, k H T s K n s H s ω 2 j = [H s ω i )] T I I T H s + ω 2 j +H T s T T f ω2 j M k K k ) kh s +H T s T k ω2 j M k K k )T f H s β n, k H T s M n s H s ω 2 i )HT s M s H s +H T s T k [ω2 j M k K k ) kh s. 24)

4 Research on the iterative method for model updating based on the frequency response function 453 Suppose C i j,n = H T s K n s H s, 25) E i j,n = ω 2 j H T s M n s H s, 26) f i j = H T s I I T H s + ω 2 j ω 2 i )HT s M s H s, 27) G k = H T s T T f ω2 j M k K k ) kh s + H T s T k ω2 j M k K k )T f H s + H T s T k ω2 j M k K k ) kh s. 28) Using a new index r to replace i and j and substituting Eqs. 25) 28) into Eq. 24) leads to α n,k C r,n + β n,k E r,n = f r + G k. 29) At last, the updated stiffness matrix and mass matrix are K k+1 = Kα n, k ), 30) M k+1 = Mβ n, k ), 31) where K and M denote the assembling method of the stiffness matrix and mass matrix when parameters are given. Equation 32) is the updating formulation obtained using the traditional iterative method [19,20]. α n, k H T s,k K n s,kh s ω 2 j β n, k H T s,k M n s,kh s = H T s,k I I T H s + ω 2 j ω 2 i,k )HT s,k M s,kh s. 32) Using the traditional iterative method, the reduced FE model has to be computed repeatedly during the iteration process to get the reduced mass matrix, reduced stiffness matrix, reduced frequency response matrix, reduced elemental mass matrix and reduced elemental stiffness matrix. However, in the proposed iterative method, only the value k of the difference between T f and T e,k is computed repeatedly, which decreases the computational time and makes the calculation converge with less iteration. The superiority of the proposed CMCF method and the new iterative method in the model updating process are shown in the following examples. 4 Numerical examples A structure of solar panels is shown in Fig. 1. It is composed of four beams and two identical plates. The structure is discretized with 20 elements and 31 nodes. Elements are beam elements and the others are plate elements. Node 31 is a fixed end and the other connections are articulated, so the total number of DOFs is 90. The geometric properties are shown in Fig. 1. The other properties of the beams are given as follows: mass density is kg/m 3, cross-section area is 1 cm 2, Young s modulus is 70 GPa. The other properties of the plates are given as follows: mass density is kg/m 3, Young s modulus is 193 GPa, thickness is 6 mm, Poisson s ratio is To get the assumed experimental mode parameters, several disturbances are introduced by changing the stiffness of the FE elements. In the following examples, it is assumed that the modeling errors exist in all 4 beams and 2 plates of the FE model. The Young s modulus of the experimental model of elements 17, 18 is 1.3 times larger than that of the FE model. The Young s modulus of the experimental model of elements 19, 20 is 1.5 times larger than that of the FE model. The Young s modulus of the plate elements is 1.2 times larger than that of the FE model. Since the plate elements of the structure have the same material properties in the simulation experiment, adding the restrictions during the calculation process that Young s moduli of elements 1 8 and elements 9 16 are respectively identical. The perturbed model with the former modeling errors assumptions is treated as the simulation experimental model. Fig. 1 A structure of solar panels

5 454 W.-M. Li, J.-Z. Hong Measurement noise which can obviously influence the test results is unavoidable in real experimental tests. In order to verify the accuracy and efficiency of the present method in practical applications, the measurement noise is considered in the simulation experiment, and it is simulated by adding a series of random errors to the theoretically calculated frequency response functions and frequencies of the experimental model The test frequency response functions and frequencies can be rewritten as H i, j = H i, j + ρ HcH i, j, 33) ω j = ω j + ρ ωcω j, 34) H i, j where Hi, j and are exact value and noisy value of the i-th row and the j-th column of the frequency response function when the frequency is ω; ω j and ω j are exact value and noisy value of the j-th frequency; c denotes a random value between 1 and 1; ρ H and ρ ω represent the level of measurement noises of the frequency response function and the frequency which are 5% and 3%, respectively. In practice, it is difficult to obtain all the mode information in experiment. It is assumed that only the transverse DOFs of nodes 1, 3, 7, 9, 13, 15, 16, 18, 22, 24, 28 and 30 which are treated as the master DOFs can be measured in the simulation experiment and the others are the slave DOFs. In order to make the DOFs of the FE model equal to those of the simulation experiment, the FE model is reduced by succession-level approximate reduction SAR) technique [18]. Table 1 shows the relative errors between the frequencies of the FE model and those of the reduced FE model. As can be seen in the table, the model reduction precision increases with increasing iterations of the reduction level l. In other words, the frequencies of the reduced FE model become closer to those of the FE model. For the purpose of obtaining more accurate model updating results, the 4-th level SAR technique is employed in the following examples. For comparing the convergence rate of the two iterative methods, the FE model is updated using the proposed CMCF method and SAR technique which are, respectively, Table 1. Relative error of the frequencies between the FE model and the reduced FE model Order Relative error/% l = 1 l = 2 l = 3 l = combined with the traditional iterative method and the proposed iterative method. In the calculation process, the first 4 frequencies and corresponding frequency response functions of the experimental model are selected, and the former 8 order frequencies and corresponding frequency response functions of the FE model are employed. A total of 15 iterations are used and the Young s modulus of the FE model is selected as the updating parameters. The relative errors of the frequencies before and after model updating using the traditional method and the proposed method are shown in Table 2. The relationships between the relative errors of the frequencies and the iterations number are shown in Figs. 2a 2f. Despite the existence of measurement noise, the first 6 frequencies updated using the proposed CMCF method and SAR technique which are combined with the traditional iterative method and the proposed iterative method are better than the initial ones, and the maximum relative error is limited to only 3.9%. Not only the first 4 frequencies that are used in the model updating process become better but also the fifth and sixth frequencies which are not employed in the model updating process become more accurate than the initial ones. The relative errors of the frequencies obtained using the proposed iterative method are obviously smaller. More importantly, the frequencies are convergent after about 3 iterations using the new iterative method while the relative errors of frequencies convergent after about 7 iterations using the traditional iterative method. It is because that, in the new iterative method, the correction term G k caused by k on the model updating process eliminating the gap between the FE model and the experimental model are considered. Therefore, the correction term G k is rather necessary which significantly accelerate the convergence. Table 2. Relative errors of the frequencies using the traditional and the proposed methods Order Original error/% The traditional The proposed method method As can be seen from Fig. 2, when T f is taken as T e,1 at the first iteration, the results obtained by using the proposed iterative method have the same solutions as obtained by using the traditional iterative methods. However, when the iteration continues, the effect of the correction term G k emerges. Under identical criteria, more computing time can be saved using the new iterative method.

6 Research on the iterative method for model updating based on the frequency response function 455 Fig. 2 Convergence speed of frequencies with measurement noise. a First natural frequency; b Second natural frequency; c Third natural frequency; d Fourth natural frequency; e Fifth natural frequency; f Sixth natural frequency Extract H1, 12) and H 1, 12) from the frequency response function of the FE model and the experimental model, respectively. Comparisons of the frequency response functions between the initial FE model and experimental model are shown in Fig. 3. As can be seen from Fig. 3, the differences of the frequency response functions are very obvious, which indicate that the initial FE model is not accurate and the model updating should be adopted. Figure 4 shows comparison of the frequency response functions between the updated FE model and the experimental model. The ideal updating results can be obtained by the proposed CMCF method using both the proposed iterative method and the traditional iterative method. More important, not only the frequency response function curve that falls within the frequency range used in the model updating process become much closer to those of the experimental model, but also those outside the frequency range used in the model updating process become more precise. To further illustrate the validity of the present method, Table 3 compares the time cost of the present iterative method and that of the traditional iterative method with measurement noise. The simulation CPU time with measurement noise using the traditional iterative method is s compared with s using the proposed iterative method. Therefore, the proposed iterative method has higher calculation accuracy.

7 456 W.-M. Li, J.-Z. Hong correction term to the formula of the traditional iterative method. Numerical examples of solar panels demonstrate that the CMCF method can update the solar panels successfully and the new iterative method makes the calculation converge with less iterations and computing time than those of the traditional method with measurement noise. This advantage is very useful for model updating of large-scale engineering structures; it can save much computing time and improve efficiency and accuracy. References Fig. 3 Comparison of frequency response curves before model updating Fig. 4 Comparison of frequency response curves after model updating Table 3. Comparison of the time cost with measurement noise) The traditional method/s The proposed method/s Time cost Conclusions In this study, a new model updating method named as crossmodel cross-frequency response function CMCF) method is proposed based on the frequency response function, and a new iterative method associating the model updating method and the model reduction technique is developed based on the traditional iterative method. The CMCF method uses directly the frequency response function in the model updating process, which avoids the modal analysis process and thus won t introduce extra noises. CMCF method does not require to pair or scale the measured and the analytical frequency response function, which greatly increases the number of the equations and the updating parameters. The new iterative method eliminates the bad effect of model reduction process on the model updating process by adding a 1 Mottershead, J. E.: Model updating in structural dynamics: a survey. Journal of Sound and Vibration. 1672), ) 2 Friswell, M. I., Mottershead, J. E.: Finite Element Model Updating in Structural Dynamics. Dordrecht: Kluwer Academic Publisher, 1995) 3 Qin, X. R., Zhang, L. M., Gu, M., et al.: Improved modal parameter based on model updating procedure. Journal of Tongji University. 3011), ) in Chinese) 4 Hu, S. J., Li, H. J., Wang, S. Q.: Cross-model cross-mode method for model updating. Mechanical Systems and Signal Processing. 21: ) 5 Carvalho, J., Datta, B. N., Gupta, A.: A direct method for model updating with incomplete measured data and without spurious modes. Mechanical Systems and Signal Processing. 217), ) 6 Bakir, P. G., Reynders, E.: An improved finite element model updating method by the global optimization technique coupled local minimizes. Computers & Structures ), ) 7 Xu, Z. M., Gao T. M., Shen, R. Y.: Improved finite element model updating method based on frequency response functions. Journal of Vibration and Shock. 213), ) in Chinese) 8 Zhu, D. D., Feng, Y. Q., Xiang S. H.: Research of refinement methods of dynamic model based on frequency response functions. Engineering Science. 78), ) in Chinese) 9 Hwang, H. Y.: Identification techniques of structure connection parameters using frequency response functions. Journal of Sound and Vibration. 2123), ) 10 D Ambrogio, W., Fregolent, A.: Results obtained by minimising natural frequency and antiresonance errors of a beam model. Mechanical Systems and Signal Processing. 171), ) 11 Modak S., Kundra T., Nakra B.: Comparative study of model updating methods using simulated experimental data. Computers and Structures ), ) 12 Steenackers G., Guillaume P.: Finite element model updating taking into account the uncertainty on the modal parameters estimates. Journal of Sound and Vibration ), ) 13 Lin R. M., Zhu J.: Finite element model updating using vibration test data under base excitation. Journal of Sound and Vibration ), ) 14 Kanev S., Weber F., Verhaegen M.: Experimental validation of a finite-element model updating procedure. Journal of Sound and Vibration ), )

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