A comparison study between the Craig - Bampton model reduction method and traditional finite element method for analyzing the dynamic behavior of vibrating structures

th Many studies have shown that the method of dynamic substructuring of Received 15 Jun 2021 Craig-Bampton (CB) which showed the effectiveness, will be used to study Accepted 16th Aug 2021 the dynamic response of the structure with respect to the excitation Available online 19th Dec 2021 frequency [Dennis Klerk et al, 2008; Duc Thinh Kieu et al, 2019; J. Wijker, 2008; Mapa et al 2021]. Its capacity to effectively reduce the number of Keywords: Craig-Bampton, dynamic degrees of freedom (DOFs) and also the computational costs will be evaluated in comparison with computations carried out with a complete substructuring, model finite element (FE) model. The CB method is one of the most popular reduction method, vibrating substructuring methods and is based on a formalization which will be structure, finite element presented in this paper. In this method, the internal DOFs are separated model from the boundary DOFs, and decomposed onto a basis of static modes, and a basis of fixed interface modes [Craig et Bampton, 1968]. The use of a reduced basis for the fixed interface modes makes it possible to reduce the size of the system to be solved, and therefore to save the computational time compared to a classical finite element computation involving the complete system. We will apply the Craig-Bampton method to an academic structure composed of three plates connected by springs and we will be interested in the frequency response functions of the deformation energies of the different plates. To evaluate the influence of the number of modes selected on the results, we will consider two bases of fixed interface modes of different sizes. Copyright © 2021 Hanoi University of Mining and Geology. All rights reserved. 1. Introduction necessary to reduce the size of the models to solve. One of the efficient model reduction The FE method is a traditional method used to strategies is the well-established Craig-Bampton perform the dynamic analysis of complex method that is a substructuring technique. This industrial structures. However, it usually involves method reduces the number of DOF of models characterized by large numbers of DOFs substructures by approximations, using a limited and leads to large computational costs. It may be 160 HỘI NGHỊ KHOA HỌC TOÀN QUỐC VỀ CƠ KHÍ – ĐIỆN – TỰ ĐỘNG HÓA (MEAE2021) number of fixed interface modes. It is useful for In the classical FE method, the unknown study the dynamic response of the structure with DOFs are obtained by inverting the above respect to the excitation frequency with many system, if the number of DOFs involved is DOFs [Craig et Bampton, 1968; J. Wijker, 2008; important the computational time can be Dennis Klerk et al, 2008]. cumbersome. The Craig-Bampton The aim of this paper is to apply the Craig - transformation consists of expressing the 푇 Bampton method in order to reduce the physical amplitudes {푞퐵 푞퐼} from generalized 푇 computational costs in the analysis of the dynamic coordinates {푞퐵 훼} as follows: behavior of a vibrating structure with many DOFs. 푞 퐼 0 푞 Section 2 presents the Craig - Bampton method. In { 퐵} = [ ] { 퐵}, (4) Sections 3, this method is used to analyze the 푞퐼 푋푠푡 푋푒푙 훼 dynamic response of a system composed of three where 푋푠푡 is the matrix of static modes, plates connected two by two by springs. The −1 which are computed as − 퐾퐼퐼 퐾퐼퐵, 푋푒푙 is the obtained results will be compared with the matrix of fixed interface modes, i.e. the matrix of reference solution that is constituted by the full FE the eigenvectors of (퐾퐼퐼, 푀퐼퐼), and 훼 is the vector model. of the modal amplitudes. 2. The Craig – Bampton method Reducing the size of the problem by Using the Craig-Bampton method [Craig et retaining only a limited number of fixed Bampton, 1968] helps to reduce the size of the FE interface modes of amplitudes 훼̃. The internal models involving large numbers of DOFs. In the DOFs are therefore approximated by: framework of finite element modeling, the degrees of freedom (DOFs) of an undamped {푞̃퐼} ≈ [푋푠푡]{푞퐵} + 푋̃푒푙{훼̃} (5) structure, contained in the vector q, the equation ̃ of motion of a structure is: where 푋푒푙 is a matrix of reduced size. [푀]{푞̈} + [퐾]{푞} = {퐹}, (1) Inserting Eqs. (Error! Reference source not found.) and (Error! Reference source not where [M] and [K] denote respectively the found.) into Eq. (Error! Reference source not mass and stiffness matrices of the structure, {q} found.) leads to a system of reduced size easier the vector of displacements and {F} the vector of to invert. external forces. To apply the Craig-Bampton method, it will 3. Numerical results: case of three plates be necessary to partition the vector of DDLs {푞} connected by springs into the boundary DOFs, 푞퐵, and the internal 3.1. Presentation of the model DDLs, 푞 : 퐼 In this section, we will illustrate the results 푞 obtained by applying the Craig-Bampton 푅푞 = { 퐵}. (2) 푞퐼 method presented previously to a system composed of three plates connected two by two Considering harmonic forces, the above by springs, visible in Figure 1. equation may be rewritten as: The model used to illustrate the results is a 푀 푀 푞퐵 set of three plates of constant 5 푚푚 thickness. −휔2 [ 퐵퐵 퐵퐼] { } 푀퐼퐵 푀퐼퐼 푞퐼 Plates 1 and 3, of dimensions 1푚 × 1푚, are 퐾 퐾 푞 + [ 퐵퐵 퐵퐼] { 퐵} (3) identical and made of steel, density 휌 = 퐾퐼퐵 퐾퐼퐼 푞퐼 7850 푘푔/푚3, Young's modulus 퐸 = 2 × 퐹 = { 퐵}. 1011 푃푎 and Poisson's ratio 휈 = 0.3. Plate 2, 퐹퐼 with dimensions 0.2 푚 × 1 푚, represents a 161 HỘI NGHỊ KHOA HỌC TOÀN QUỐC VỀ CƠ KHÍ – ĐIỆN – TỰ ĐỘNG HÓA (MEAE2021) flexible rubber junction, with the following three DOFs per node, namely displacement 푢푧 3 properties: 휌 = 950 푘푔/푚 , 퐸 = 15 × and two rotations 휃푥 and 휃푦. The meshes of 107 푁/푚2 and 휈 = 0.48. To take into account plates 1 and 3 therefore each comprise 1600 a structural damping of the plates, a loss factor elements and 5043 DOFs, while 320 elements 휂 = 0, 005 is taken into account for each of the and 1107 DOFs are used for plate 2. The total three plates. number of DOFs of the system is therefore In the framework of finite element 11193. modeling, the three plates are meshed using square plate elements of length 0.025m having Figure 1. Model of 3 plates connected by springs. The springs that couple the plates are the system in the interval [0, 50] 퐻푧 using a located at the interfaces between plates 1 and 2 frequency step of 10−3 퐻푧. on the one hand, and plates 2 and 3 on the other 3.2. Reference solution hand, as shown in Figure 1. To each node of the mesh are connected three springs, two torsion The reference solution is constituted by springs of identical stiffness 푘푡 = 20 푁푚/푟푎푑 direct computation finite elements. The matrix around the axes x and y, and a linear spring in to be inverted here is of size 11193 × 11193, the direction z, of stiffness 푘푧 = 150 푁/푚. and the inversion is repeated for each of the 50001 frequencies in the interval [0, 50] Hz. The The whole structure is clamped at both simulations were carried out using Matlab extremities (i.e., left edge of plate 1 and right software on a computer comprising two Intel edge of plate 3). The system is excited by a point (R) Xeon (R) CPU E5-2623 v3 @ 3.00 GHz and harmonic force of amplitude 40N in the z- 32 GB of ram processors. The total simulation direction, located at the node of coordinates time for this case is 5h and 24min. (0.25 푚, 0.25 푚) if the origin is chosen as the lower left corner of plate 1, as shown in Figure 1. We study the frequency response function of 162 HỘI NGHỊ KHOA HỌC TOÀN QUỐC VỀ CƠ KHÍ – ĐIỆN – TỰ ĐỘNG HÓA (MEAE2021) denoted CB3080 in the following. The corresponding calculation times are now 1h 48min for the model CB50130 and 1h 27min for the model CB3080, that is to say, reductions in calculation cost of 66.86% and 73.10% respectively. Each of figures 3 (a), (b) and (c) represents the deformation energies of a given plate obtained by the three methods (complete finite element model and the two reduced models). The three curves are not dissociable in these figures, thus validating the results obtained by the Craig-Bampton method. Slight deviations are nevertheless visible Figure 32. Deformation energies of plates 1, 2 by enlarging the views, in particular around the and 3 as a function of frequency. resonance peaks. The four peaks 2, 5, 7 and 9 of The results are shown for each plate 푖 in the deformation energy of plate 1 can thus be terms of deformation energy: observed in more detail in figure 4. It thus clearly appears that the differences increase 1 with the natural frequency of the peak: peak 2 is 퐸 = 푞푇퐾 푞 (6) 푑푒푓 푖 2 푖 푖 푖 thus faithfully reproduced with the two reduced models compared to the reference solution, Figure 2 represents the frequency while slight amplitude deviations are visible for evolutions of the amplitudes of the three peak 5 with the smaller model CB3080. For deformation energies thus obtained. All three peaks 7 and 9, differences are visible in terms of curves look similar with multiple peaks at the amplitude and frequency of the peak, the resonant frequencies of the plates, and an overall higher energy level for plate 1 which is smaller model CB3080 logically giving larger deviations than the model CB50130. being excited. In the following, we will focus on four particular peaks (peaks 2, 5, 7 and 9 highlighted in Figure 2) in order to estimate the impact of model reduction on the precision of the results. 3.4. Model reduction by the Craig-Bampton method A reduced model resulting from a Craig- Bampton decomposition in which a limited number of fixed interface modes is retained; two CB models will be proposed to study. For the first model, which will be denoted CB50130 in the following, 50 fixed interface modes are (a) used for plates 1 and 3 (out of 4563 modes) and 130 modes for plate 2 (out of 819). For the second model, smaller, 30 modes are then retained for the plates, the plates 1 and 3 and 80 for the plate 2. This second model will thus be 163 HỘI NGHỊ KHOA HỌC TOÀN QUỐC VỀ CƠ KHÍ – ĐIỆN – TỰ ĐỘNG HÓA (MEAE2021) (b) (c) Figure 33. Deformation energies (a) 푬풅풆풇ퟏ, (b) 푬풅풆풇ퟐ and (c) 푬풅풆풇ퟑ according to the frequency (a) (b) (c) (d) Figure 34. (a) Peak 2, (b) peak 5, (c) peak 7 and (d) peak 9 of the deformation energy Edef1 of plate 1 according to the frequency 164 HỘI NGHỊ KHOA HỌC TOÀN QUỐC VỀ CƠ KHÍ – ĐIỆN – TỰ ĐỘNG HÓA (MEAE2021) Table 6. Mean and maximum of the relative errors on the amplitude and frequency of the deformation energies of the three plates obtained by a reduced model compared to the reference solution 퐸푑푒푓1 퐸푑푒푓2 퐸푑푒푓3 Amplitude Frequency Amplitude Frequency Amplitude Frequency Method Mean of relative errors (%) CB3080 7.49 × 10−2 5.08 × 10−3 8.96 × 10−2 5.16 × 10−3 1.06 × 10−1 4.79 × 10−3 CB50130 2.90 × 10−2 1.40 × 10−3 2.33 × 10−2 2.24 × 10−3 2.40 × 10−2 1.79 × 10−3 Maximum of relative errors (%) CB3080 2.27 × 10−1 1.02 × 10−2 4.45 × 10−1 1.28 × 10−2 4.79 × 10−1 1.18 × 10−2 CB50130 1.10 × 10−1 4.89 × 10−3 8.47 × 10−2 5.77 × 10−3 9.43 × 10−2 7.33 × 10−3 165 HỘI NGHỊ KHOA HỌC TOÀN QUỐC VỀ CƠ KHÍ – ĐIỆN – TỰ ĐỘNG HÓA (MEAE2021) To further analyze the accuracy of the dynamic response of a structure having a large results, the relative errors are calculated for each number of DOFs in the FE framework. Two peak of each of the three plates between the reduced models comprising a variable number of deformation energy obtained by a reduced fixed interface modes have been proposed and model and the reference solution involving the validated by comparison with the reference complete finite element model, in terms of solution involving the complete finite element amplitude: model of the system. It has been shown that the Craig-Bampton method allows a substantial |퐸푑푒푓 푗,푟푒푓(푝푖푐푖) − 퐸푑푒푓 푗,퐶퐵(푝푖푐푖)| reduction in the computational cost while (7) guaranteeing a negligible loss of precision. |퐸푑푒푓 푗,푟푒푓(푝푖푐푖)| References And of frequency: Paper published in journal |푓푑푒푓 푗,푟푒푓(푝푖푐푖) − 푓푑푒푓 푗,퐶퐵(푝푖푐푖)| (8) Dennis Klerk, Daniel Rixen et Sven Voormeeren. A |푓푑푒푓 푗,푟푒푓(푝푖푐푖)| general framework for dynamic substructuring. Where 퐸 (푝푖푐 ), 퐸 (푝푖푐 ) history, review and classifcation of techniques. 푑푒푓 푗,푟푒푓 푖 푑푒푓 푗,퐶퐵 푖 AIAA Journal, 46:1169–1181, 01, 2008. denote the amplitudes of the ith peaks of the deformation energies of the plate 푗 (푗 = Mapa, L., das Neves, F. & Guimarães, G.P. Dynamic 1, 2 표푟 3) obtained respectively by the reference Substructuring by the Craig–Bampton Method solution and a reduced model CB; 푓푑푒푓 푗,푟푒푓(푝푖푐푖) Applied to Frames. J. Vib. Eng. Technol. 9, 257– and 푓푑푒푓 푗,퐶퐵(푝푖푐푖) are the natural frequencies of 266 (2021). these peaks. R.R. Craig and M.C.C. Bampton. Coupling of The mean and maximum of these relative Substructures for Dynamic Analyses. AIAA errors, calculated over the first fourteen peaks, Journal, 6(7):1313–1319, 1968. are given for each plate in Table 1. With a Presentation at conferences maximum relative error of less than 0.1%, the accuracy of the results obtained with the model Duc Thinh KIEU, Marie-Laure GOBERT, Sébastien CB50130 is excellent; it remains perfectly BERGER et Jean-Mathieu MENCIK. A model satisfactory using the smallest CB3080 model, reduction method to analyze the dynamic the maximum error this time being less than behavior of vibrating structures with uncertain 0.5%. parameters. SURVISHNO, Lyon, 2019. 4. Conclusion Book This article was presented the model Jacob Job Wijker. Dynamic Model Reduction reduction method of Craig-Bampton, based on a Methods, pages 265–280. Springer projection of the internal DOFs of a structure on Berlin Heidelberg, Berlin, Heidelberg, a basis of static modes and modes of fixed 2008. interface. The method was applied to analyze the 166