Hybrid SEM-FEA-Experimental Dynamic Models of Frame Structures

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/310587420 Hybrid SEM-FEA-Experimental Dynamic Models of Frame Structures Conference Paper · June 2002 CITATIONS 0 READS 10 2 authors: Khaled Ahmida University of Tripoli 44 PUBLICATIONS148 CITATIONS SEE PROFILE J.R.F. Arruda State University of Campinas (UNICAMP) 242 PUBLICATIONS2,024 CITATIONS SEE PROFILE All content following this page was uploaded by Khaled Ahmida on 16 September 2025. The user has requested enhancement of the downloaded file. Proceedings of the International Conference on Structural Dynamics Modelling Test, Analysis, Correlation and Validation Madeira Island, Portugal 3 - 5 June 2002 Organised by: Institute Superior TScnico, Technical University of Lisbon Universidade da Madeira 1. KEYNOTE LECTURE - ASCI COMPUTING FOR STRUCTURAL DYNAMICS Girrens, S. 13 Session 1A - Analytical Methods I 2. Hybrid Vibration Modelling of Structures Using Analytical Formulations 25 Pavic, G. 3. Hybrid SEM-FEA-Experimental Dynamic Models of Frame Structures 33 Ahmida, K.M., Arruda, J.R.F. 4. A Generalized Formulation for Linear MDOF System Utilizing Complex Operators 43 Viola, E., Zonta, D., Saccone, G. 5. Use of the Continuous Wavelet Transform for the Identification of Damping 53 Boltezar, M., Slavic, J. Session IB - Modelling I 6. Construction of a Finite Element Damping Matrix from Measured Modal Data 63 Willner, K., Gaul, L. 7. Techniques for the Estimation of Angular FRF in Modal Testing 71 Varoto, T.S., Lofrano, M., Oliveira, L.P.R. 8. Measurement of Rotational DOF Frequency Response Functions with Pure Moment Excitation 81 Trethewey, M.W., Sommer, H.J. 9. Identification of "Engine-Attachment-Airframe" System Model Basing on Impedance Test Results 89 Baklanov, V. 10. KEYNOTE LECTURE - MODEL UPDATING OF JOINTS AND CONNECTIONS Mottershead, J., Friswell, M. 95 Session 2A - Modelling II 11. 3-D Finite Element Model Used to Study Effects of Lime-cement Columns Against Train Induced Ground Vibrations 105 Bahrekazemi, M., Bodare, A. 12. Development and Implementation of 2D-Plane Element to Model Dynamic Behaviour of Cracked Reinforced Concrete 115 Maeck, J., Teughels, A., De Roeck, G. 13. Numerical Modelling of Vibration Sources in Hard Disk Drive by Inverse Methods 127 Kim, J.T.; Waters, T.; Nelson, P.A. Session 2B - Damage Detection I 14. Increased Reliability of Reference-Based Damage Identification Techniques by Using Output-Only Data 137 Parloo, E.; Vanlanduit, S.; Guillaume, P.; Verboven, P. 15. Damage Detection Benchmark on a Three-Storey Steel Frame 147 Molina, F.J., Gonzalez, A., Zapico, J.L., Magonette, G. HYBRID SEM-FEA-EXPERIMENTAL DYNAMIC MODELS OF FRAME STRUCTURES Khaled M. Ahmida and José Roberto F. Arruda Departamento de Mecânica Computacional, FEM, UNICAMP, C.P. 6122, Campinas, SP, 13083-970 Brazil Fax: +55-19-2893722 [khaled][arruda]@fem.unicamp.br SUMMARY: The use of a hybrid technique is proposed for modeling frame structures with complex boundary conditions. The hybrid technique consists of combining spectral elements, finite elements, and experimentally obtained impedances. The Spectral Element Method (SEM) is formulated in the frequency domain and is based on exact shape functions, thus providing accurate solutions even at high frequencies without the necessity of mesh refinement, as it is the case with Finite Elements (FE). Nevertheless, the SEM has a relatively poor library of elements. In the proposed hybrid technique, finite elements are used to model the elements that cannot be represented via SEM, and then coupled to the spectral elements. Furthermore, in order to take into account complex boundary conditions, the two elements can be combined with measured impedances. In this paper, the method is illustrated with simple frame structures such as a clamped beam and a beam plunged into a sandbox. KEYWORDS: Spectral elements, finite elements, hybrid modeling, frame structures, impedance. INTRODUCTION Approximate numerical methods, such as Finite Elements (FE) and Boundary Elements (BE), provide a suitable tool for the dynamic analysis of structures and are increasingly being used in the industry. However, the discrepancy between predictions of the dynamical behavior and the real behavior tends to increase with frequency. This discrepancy is usually due to poor discretization of the continuum and/or low of the order of the finite element used, differences in the physical parameters, and poorly represented boundary conditions. Some boundary conditions cannot be easily represented in the numerical models, as it is the case with soil-structure interaction and interaction with the surrounding structures through complex joints. The is currently a search for methods that would overcome the limitations of the FE analysis at higher frequencies without having to go all the way into a statistical analysis such as SEA [1], where only asymptotical, averaged results (both spatially and in frequency) can be obtained. One trend is to try to combine FE with SEA in a hybrid model [2]. Many researchers are currently investigating the possibilities of this approach, originally shown to work for simple bar structures. Another is to try to simplify the wave equations and solve them with a thermal analogy – the so-called Energy Finite Elements [3]. A third line of current research is to use multiple elementary sources with a simple fundamental solution for an infinite structure to impose arbitrary boundary conditions. The latter can be classified as meshless methods [4,5]. Another approach consists of trying to solve the wave equation for the medium within a simple element and then assemble the elements using the direct stiffness method, so popular due to FE, which we call here the Spectral Element Method (SEM) [6]. SEM is a systematic way of solving structural dynamic problems using a wave-based approach (fundamental solution). Although still limited to relatively simple elements – basically homogeneous frames and flat plates with particular boundary conditions [6] –, it is very easy to implement. The SEM is formulated directly in the frequency domain and yields results that are exact within the framework of the strength of materials theory used to derive the element. Thus, for homogeneous elements, in contrast to the FE, SEM provides accurate solutions at arbitrarily high frequencies without the necessity of mesh refinement. In this paper, the formulation of the SEM for frame elements is reviewed and it is shown how some of the limitations of SEM can be overcome if it is combined with FE and experimentally obtained impedances. The use of the proposed methodology is illustrated with the characterization of a clamped beam and a beam plunged into a sandbox. REVIEW OF SEM In the literature, spectrally formulated finite and semi-infinite elements known as spectral elements [6] are presented for Timoshenko beams. In contrast to the conventional FE, this formulation provides exact mass distribution within each structural element. In general, a structure is discretized in a small number of spectral elements (one for each span between two discontinuities) and then these elements are assembled using the direct stiffness approach, in an analogous way to that used in FE analysis. The main difference is that the calculations in SEM are all carried out in the frequency domain. Time-domain responses can be obtained by performing an inverse FFT. SEM can be seen as a combination of the exact wave method and the assembling features of FE. Two different types of beam elements can be used in this method: 2-noded and throw-off, the latter being a semiinfinite element. Due to space restrictions, the equations of the SEM for beams are not shown extensively in this paper. For the beam wave-guide obeying the Timoshenko beam theory, the equation of motion will have a fourcoefficient exact solution given as, )xL(ik2 )xL(ik1 xik2 xik1 )xL(-ik)xL(-ikx-ikx-ik 2121 2121 eReReReR)x(v ˆ eeee)x( ˆ −− −− −−+= +++=φ DCBA DCBA (1) where R1 and R2 are defined as the amplitude ratios, k1, k2 are the wave numbers, and A, B, C, and D are constants obtained through the application of the different boundary conditions. For throw-off elements, the equations of motion can easily be established from Eqn. 1 by putting C=D=0, as waves propagate in only one direction. The solution to these equations can be written in terms of the nodal displacements, and a relation between the global shear forces and moments and the global nodal degrees of freedom can be established as, [ ] { } { } F ˆ U ˆ K ˆgg =(2) where [ ] g K ˆ is the global dynamic stiffness matrix, which is symmetric and generally complex. The individual elements of this matrix can be found in [6]. When frame structures are analyzed in three-dimensional space, longitudinal and torsional waves also propagate. To model these kinds of waves, spectral rod and shaft elements must be used. The spectral solutions for the rod and shaft equations of motion are of the same type as φ ˆ in Eqn. 1. The fact that only one element is needed between any two discontinuities plays the role of making the number of elements needed to model a structure relatively small when compared to the FE. The solution is accurate independently of element’s length, even at higher frequencies. Thus, the response at different nodal degrees of freedom can be recovered with less computational effort, even when solving this system of equations for each frequency line. The use of the frame spectral element for three-dimensional frame structures is shown in detail in [7]. CASE STUDY I: CLAMPED-FREE BEAM This simple case study is frequently used to verify numerical dynamic analysis methods. The clamped condition of beams is ideally modeled by considering an infinite stiffness at the clamped degrees of freedom (DOF). This boundary condition is imposed simply by eliminating the columns and rows of the mass and stiffness matrices that correspond to the clamped DOF. However, even a very rigid clamping cannot be realistically modeled with this assumption. To show how the clamping conditions can be characterized, consider a straight beam clamped at one end with an excitation applied at a point x along its length, as shown in Fig. 1. 23 1 0.3 m 0.2 m v1,φ1v2,φ2v3,φ3 F v1,φ1v2,φ2v3,φ3 F Fig. 1: Scheme of the experimental configuration and SEM model of a clamped-free beam and its DOF. The beam is discretized into two spectral elements with two DOF at each node. Thus, the global dynamic stiffness matrix is a (6x6) matrix given as,                       − − =                                           0 0 0 . 11 1 3 3 2 2 1 1 66 5655 464544 36353433 2625242322 161514131211 F K vK v v v K KKsym KKK KKKK KKKKK KKKKKK v φ φ φ φ φ (3) where the force at node 1 is represented by the translational and rotational dynamic reaction forces 11v vK− and 11 Kφ− φ, respectively, with 1v K and 1 Kφ defined as translational and rotational dynamic stiffness at node 1. The displacement DOF are classified as follows: • v2 and v3 are the DOF easy to measure • v1 and φ1 are the clamped DOF that cannot be measured • φ2 and φ3 are the DOF that are normally hard to measure Equation 3 can be written in a simplified form as, [][] [][] {} {} {} {}       =             2 1 2 1 2221 1211 f f U U MM MM (4) where the different sub-matrices are defined as, [] [] [] [] {} {} {} {}       =       φ− − =               φ φ φ =       =             =             =       =       = φ0 F f, K vK f, v U, v v U KKKK KKKK KKKK KKKK M, KK KK KK KK M KKKK KKKK M, KK KK M 2 11 11v 1 3 2 1 1 2 3 2 1 66646261 56545251 46444241 36343231 22 6563 5553 4543 3533 21 26242221 16141211 12 2523 1513 11 (5) Using the second matrix equation of Eqn. 4, the variables in {U2} can be solved as, {} [] {} [][] {} 121 1 222 1 222 UMMfMU −− −= (6) and using the first equation of Eqn. 4, the dynamic stiffness 1v K and 1 Kφ that characterize the clamped condition can be solved as, {} [] {} [] {} {} {} 1 1 1 1 1 1v 2121111 )2(f K, v )1(f K UMUMf φ −=−= += φ (7) These dynamic stiffness can then be used to model the clamping boundary condition (or other types of boundary conditions) by using them instead of the infinite stiffness that are associated with the ideal clamping. An experiment was performed with aluminum beam with the following properties: L=0.685m, E=7x1010N/m2, ρ=2800Kg/m3, A=25.6x3.3mm2, I=7.666x10-11m4, ν=0.3, and with a hysteretic damping factor of η=10-6. A random excitation force was applied by an electro-dynamic shaker at a point x=0.27m from the clamped end as shown in Fig. 2. A frequency range of 1Hz to 3.2kHz, with a resolution of 3200 frequency lines, was analyzed. The flexural frequency response functions (FRF) were measured at the excitation point and at the free end, by using Laser Doppler Vibrometer (LDV) with a sensitivity of 40V/m/s. The LDV was connected to a data acquisition system. ICP Conditioner Spectral Analyzer Shaker Signal Generator Laser Doppler Vibrometer Fig. 2: Experimental setup of the clamped-free beam experiment. The measured FRFs were then used in Eqns. 4 and 7, and the dynamic stiffness of the clamping condition are determined, as shown in Fig. 3. 0500 1000 1500 2000 2500 3000 3500 10 20 30 40 50 60 70 Frequency, Hz Dynamic stiffness [dB ref. 1 N/m] Kv1 K?1 Fig. 3: Translational and rotational dynamic stiffness of the clamped end obtained using experimental data. The same beam was modeled via SEM as free-free beam. The characterized dynamic stiffness coefficients were then added to the corresponding DOF of the global matrix, and the FRF at the excitation point was calculated. The clamped condition was also modeled in the usual manner, i.e., by eliminating the rows and columns corresponding to the clamped DOF. A comparison between the different FRFs is shown in Fig. 4. 0500 1000 1500 2000 2500 3000 3500 -90 -80 -70 -60 -50 -40 -30 -20 -10 Frequency, Hz FRF [dB ref. 1 m/N] experimental SEM adjusted numerical clamping Fig. 4: FRF (DOF:v2) of the clamped-free beam: measured, computed with ideal clamping, and computed with the experimental impedance. The result of the experimentally adjusted FRF is shown in Fig. 4. As expected, the current methodology yielded FRFs which are nearly identical to the measured FRFs. On the other hand, the FRF computed using the ideal clamping (canceling columns and rows corresponding to the clamped DOF) shows agreement with the measured FRF only for the first three modes. Beyond this mode, the FRF becomes quite different from the exact one, due to the poor modeling of the clamping boundary condition. This demonstrates the importance of effectively modeling the real boundary conditions in numerical simulations. CASE STUDY II: STRUCTURE-SOIL COUPLING Sandboxes are frequently used in structural intensity investigations as quasi-anechoic terminations. Besides, beams plunged into sandboxes can also present a behavior that is similar to that of piles. The same methodology presented and used in the previous section is again used in this case study. For this sake, an aluminum tube plunged into a sandbox with the following properties was used: outer diameter ϕ1= 11.175mm, inner diameter ϕ2= 9.375mm, E=7x1010N/m2, ρ=2800Kg/m3, ν=0.3 and η=10-6. The total length is 1.36m with 1m plunged into the sandbox. The sandbox is a box made of wood and filled with sand, and has the dimensions of 0.3mx0.3mx1m. It is important to note that the sand at the beam entrance is made in a conic shape with foam in order to reduce the impedance discontinuity that causes flexural waves reflections at the entrance of the box. An electro-dynamic shaker was used as an excitation source at a point located x=0.19m away from the free end, as shown in Fig. 5. A frequency range of 1Hz to 3.2kHz, with a resolution of 3200 frequency lines, was analyzed. The FRF was measured at the excitation point using an LDV. The measured FRFs were again used in Eqns. 4 and 7, and the dynamic stiffness representing the sand box are determined and shown in Fig. 6. The FRF of the beam with free-free condition is then calculated via SEM and adjusted using the identified dynamic stiffness of the corresponding DOF that represent the sand box. Fig. 7 shows that the adjusted FRF is very similar to the measured FRF. Fig. 5: Experimental setup of the beam in the sand box. 0500 1000 1500 2000 2500 3000 3500 25 30 35 40 45 50 55 60 65 70 Frequency, Hz Dynamic stiffness [dB ref. 1 N/m] Kv1 K?1 Fig. 6: Translational and rotational dynamic stiffness representing the impedance of the sand box. 0500 1000 1500 2000 2500 3000 3500 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 Frequency, Hz FRF [dB ref. 1 m/N] Experimental SEM adjusted Fig. 7: FRF (DOF:v2), measured and numerically adjusted via SEM.