Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate Structure
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1 Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate Structure Hasan G. Pasha, Karan Kohli, Randall J. Allemang, Allyn W. Phillips and David L. Brown University of Cincinnati Structural Dynamics Research Lab (UC-SDRL), Cincinnati, Ohio, USA Abstract. To characterize the dynamics of a structure accurately and to minimize uncertainties, it is important to perform modal testing in various configurations. However, performing such rigorous testing of structures can be a resource intensive process. In addition, simulating certain conditions may not be possible in the lab. Developing a calibrated and a validated model that can predict the dynamic response of a structure accurately can be a key to address this issue. In this paper, a model calibration and validation case study performed on a rectangular steel plate structure is presented. The geometric and material properties used in the model were updated to calibrate the model. The accuracy of the calibrated model was confirmed by performing a validation process involving perturbed mass and constrained boundary condition FE modal analysis and modal testing. The validation criteria were achieved using the calibrated model and thus proved that the model could reliably predict the dynamic response of the structure. Keywords. Model Verification, Model Calibration, Model Validation, Modal Correlation, Model Updating, Perturbed Boundary Conditions, Constrained Boundary Condition Testing Notation Symbol Description γ Mass per unit area of a plate (kg/m 2 ) ρ Density of the material (kg/m 3 ) ν Poisson ratio b Width of the plate (m) f ij h i j l Modal frequency (Hz) Thickness of the plate (m) Mode index, number of half waves in mode shape along horizontal direction Mode index, number of half waves in mode shape along vertical direction Length of the plate (m) D Flexural rigidity (N m 2 ) E Young s modulus (Pa) CMIF Complex Mode Indicator Function FEA Finite Element Analysis MRIT Multi-reference Impact Testing 1. Introduction Testing a structure to determine its dynamic characteristics is both time consuming and expensive. This situation has made it necessary to develop analytical/mathematical models that can predict the response of system in real time without performing actual tests. Such models need to undergo verification, calibration and validation with actual test data before they can be relied upon to extrapolate results for future studies. In this context, verification and validation (V&V) are defined as below: Verification: refers to process of determining that a model implementation accurately represents the developer s conceptual description of the model and the solution to the model [1]. Frequently, the verification portion of the process answers the question Are the model equations correctly implemented?
2 Validation: refers to the process of determining the degree to which a model is an accurate representation of the real world system from the perspective of the intended uses of the model [1]. Frequently, the validation process answers the question Are the correct model equations implemented? In this paper, a FE model of the plate structure was developed to determine its dynamic characteristics (modal frequencies and mode shapes). A step by step approach was taken to verify, calibrate and validate the results generated by the model in different test configurations. Figure 1: Verification, calibration and validation processes
3 2. Verification 2.1. Analytical Modeling and Formulation. Ideally, a plate is considered a 2-dimensional layer of elastic material made up of a sheet that lies in a plane. A plate as a structure possesses bending rigidity due to its thickness and elasticity. As a result, when the plate vibrates, it deforms primarily by flexing perpendicular to its own plane. The following assumptions were made while performing the analytical formulation of the rectangular plate structure [?]: The plate is composed of homogeneous, linear, elastic, isotropic material The plate is flat and has constant thickness The plate is thin, with a thickness of less than 1/10 of the minimum lateral plate dimension The in-plane load on the plate is zero and it deforms through flexural deformation The deformation is small in comparison with the thickness of the plate. There is no deformation at normal to the mid surface or nodal lines. In addition, the rotary and shear deformations are ignored. In the case of free vibration, total transverse deformation is defined as the sum of all modal deformations such that (1) z = A ij z ij sin (2πf ij t + φ ij ) i j where z ij is the mode shape of the mode (i, j), A ij is the modal amplitude and φ ij is the phase. The modal frequency of the plate is defined as f ij = λ2 ij 2πl 2 Eh 3 12γ(1 ν 2 ), where λ ij is a dimensionless parameter, which is Eh generally a function of the mode indices (i, j). Flexural rigidity of plate is defined as D = 3 12γ(1 ν 2 ) Verification. Verification is the process of comparing different analytical models to be certain that the analysis results are correct [1, 2]. In this case, a FE model is compared to the analytical, closed form results for a rectangular plate. An analytical rectangular plate model, with aspect ratio of 1.5 was developed, such that the first mode is a torsion mode roughly at 40 Hz and the second mode is a bending mode roughly at 44 Hz. Closed form expressions [9] were used to develop an analytical model. Figure 2: 2-D drawing showing plate dimensions A comparable rectangular plate FE model was developed using the dimensions and material properties chosen for the analytical closed form model of the plate. Analytical modal analysis was performed on the plate FE model.
4 (a) CAD Model (b) Mesh Figure 3: Rectangular plate model The modal frequencies and mode shapes for the first 6 deformation modes were retrieved and compared with the analytical model prediction. The modes were in the expected sequence and the relative difference in modal frequencies was less than 2%. This verified the FE model. Table 1. Preliminary verification of the FE model Modal frequency comparison Mode # Description Modal frequency (Hz) Rel. diff. Analytical FE (%) 1 First torsion First X bending Second torsion First Y bending Second X bending Anti symmetric X bending Calibration Calibration of a model involves comparing the analytical results to equivalent experimental results. Calibration in this case involves correlating the modal parameters obtained from an analytical modal analysis of the rectangular plate FE model and impact testing of the fabricated rectangular plate structure. The details of the various activities performed in the calibration process are explained in the subsequent sections Analytical Modal Analysis. Analytical modal analysis was performed in ANSYS R Workbench The default material model for steel in the ANSYS R material library was chosen. The modal frequencies and mode shapes were extracted in the frequency range Hz Experimental Modal Analysis. The plate structure was tested using the multi-reference impact testing (MRIT) technique with 21 accelerometers mounted on the rectangular plate at locations identified in Fig. 4. For simulating the free-free boundary condition, the plate was initially suspended using shock cords. However, the support system was dynamically coupled with the third mode of the rectangular plate (second torsion mode) and it acted as a vibration absorber. Subsequently, the plate was supported on racquet balls, as shown in Fig. 5. Though the racquet ball supports offered a lesser frequency separation ratio between the rigid body modes and the first elastic mode (compared to the shock cords), they were well isolated from the test structure [11]. On processing the measured data, nine deformation modes in the Hz frequency range were identified, as evident from the CMIF plot shown in Fig. 6. The first torsion and first bending mode were at Hz and Hz respectively. The mode shapes are shown in Fig Modal Correlation. The relative difference between the modal frequencies obtained from FE modal analysis and impact testing is > 2 3%. The sequence of the modes was established to be in the same order for both cases by viewing the mode shape animations.
5 Figure 4: Rectangular plate reference locations Figure 5: Rectangular plate support system
6 Figure 6: Complex mode indicator function Figure 7: Experimental mode shapes
7 3.4. Model Calibration. From the correlation of the modal parameters from the preliminary analytical modal analysis and impact testing, it was identified that there were various sources of uncertainty. On the modeling side, the sources of uncertainty were geometric and material properties of the steel plate. In addition, the soft supports had not been modeled. On the testing side, the support system, location of supports, sensor calibration, impact hammer tip and signal processing parameters were considered to be the sources of uncertainty. The relative difference in modal frequencies was desired to be less than 2% and was set as the calibration criteria or calibration metric Model Updating. Geometric properties: It was established that the plate thickness, originally intended to be 0.25 in was not constant. The plate was weighed (52.5 lb m ) and the thickness of the plate model was adjusted (0.243 in) to match the measured weight of the plate. Material model: The relative difference for the third mode (second torsion mode) was reasonably high compared to other modes. A parametric study, in which the effect of varying the Young s modulus, mass density and Poisson ratio on the modal frequencies, was studied. For the calibrated model, the material properties were established as: Young s modulus E = 2.05x10 11 Pa (2.9734x10 7 psi), mass density ρ = 7850 kg/m 3 ( lb/in 3 ) and Poisson ratio ν =.29. Modeling support system: Four racquet balls were used to support the plate during impact testing. These racquet balls act as springs and their stiffness value was estimated to be roughly 40 lb f /in. At locations corresponding to the supports, linear springs were added in the FE model. This reduced the relative difference for the modal frequencies further Testing Related Changes. Support system: Free-free boundary conditions are relatively easier to achieve compared to other boundary conditions such as a fixed boundary condition. Support systems add stiffness and can also interact with the test structure. Certain precautions need to be followed while selecting support systems [11]. Initially, shock chords were used to support the structure. However, it was observed that the shock chord supports dynamically interacted with the third deformation mode (second torsion mode), and acted as a vibration absorber. As a result, a racquet ball support system, that was reasonably isolated from the test structure, was selected. Sensor calibration: The sensors (accelerometers and impact hammer load cells) need to be calibrated before data is acquired. Sensors generally tend to have variability in the sensitivity in the order of 2 5%, which when not accounted for could skew the measurement magnitudes (but would have no effect on the calibration metric involving modal frequencies). Impact hammer tip: The impact hammer tip should be selected such that the modes in the required frequency range are adequately excited without exciting higher frequency modes. When the focus is to acquire data in lower frequency range, using a harder tip would possibly cause overloads on the data acquisition channels associated with sensors near the impact location. As a result, an appropriate hammer tip should be selected. Signal processing parameters: The rectangular plate is a very lightly damped structure. It was noted that after every impact, it took over 16 s for the response to die out. After selecting the frequency bandwidth, the frequency resolution should be chosen such that the response at all the reference locations is completely observed in the captured time segment. If this condition is not met, it would result in leakage errors. If desired, the force-exponential windows can be applied to condition the signal to avoid leakage and measurement noise. The number of ensembles acquired per average also affects the ability to obtain good quality data and minimize measurement noise. In this test, data was acquired in the Hz and Hz frequency ranges. The frequency resolution ( F) was chosen for each case such that each ensemble was observed for 16 s. The force-exponential window was chosen, with a cutoff value of 10% for the force window and an end value for exponential window as 20% Modal Correlation after Calibrating the FE Model. Modal correlation was performed after updating the FE model and incorporating the changes identified for the test setup. The relative difference for modal frequencies reduced to a value less than 2%. FRFs from the FE model were synthesized at a few locations corresponding to the driving-points in the impact test. The measured FRFs were compared with corresponding synthesized FRFs as a further calibration metric that involves damping and sensor calibration issues. Plots comparing the measured FRF with the synthesized FRF for a driving-point is shown in Fig. 8. It is apparent from the comparison that the peaks in the synthesized FRF does not
8 have any damping as damping was not modeled. However, the measured FRF peaks are damped. The rectangular plate is a lightly damped structure. In reality the modes do not have the amount of damping that is visible in the FRF comparison plot. As impact testing method was used to acquire the FRF data, most of the damping stems from the force-exponential window used to eliminate leakage while acquiring FRF measurements. Figure 8: Comparison of a driving-point FRF The results of modal correlation are presented in Table 2. The calibration criteria, a relative difference for modal frequencies less than 2% was met and a calibrated rectangular plate FE model was obtained. Table 2. Correlation of modal frequencies for free-free case with no perturbed mass and updated FE models h = 6.17 mm (.243 ), E = 2.05x10 11 Pa (2.9734x10 7 psi) FE Model I: ρ = 7810 kg/m 3 ( lb/in 3 ), m = 23.8 kg (52.44 lb) FE Model II: ρ = 7850 kg/m 3 ( lb/in 3 ), m = 23.9 kg (52.69 lb) Mode Modal frequency (Hz) Rel. diff.(%) Experimental FE Model I FE Model II FE Model I FE Model II First torsion First X bending Second torsion First Y bending Second X bending Anti sym. X bending Third torsion X and Y bending Third X bending
9 4. Validation Validation is the process of determining the correctness of a model in its description of the reference system under a set of test conditions [1, 2]. For validating the calibrated rectangular plate FE model and thereby establishing the robustness of the model, it was decided to analyze and to perform testing for the following two cases: (1) Perturbed mass analytical modal analysis and testing (2) Constrained boundary analytical modal analysis and testing The relative difference in modal frequencies was chosen as the validation criteria or the validation metric. For the perturbed mass case, the validation criteria was set to less than 2% relative difference in modal frequencies. Constraining specific points is possible theoretically in a FE model; however, it is impractical to achieve. In addition, modeling real boundary conditions is involved. As a result it was decided to set the validation criteria less than 5% relative difference for modal frequencies for the constrained boundary condition case Perturbed Mass Modal Analysis. Two separate cases were studied by attaching two cylindrical masses, 1 lb 1.5 oz and 2 lb 1.5 oz each, to the calibrated plate FE model. The perturbation masses were also attached to the fabricated plate structure and impact testing was conducted. The results obtained from the modal analysis and impact testing were correlated. The correlated results for 1 lb 1.5 oz and 2 lb 1.5 oz perturbed mass cases are listed in Tables 3 and 4 respectively. It is evident that the validation criteria, relative difference for modal frequencies less than 2%, was met. Table 3. Correlation of modal frequencies for free-free case with perturbed mass of kg (1 lb 1.5 ozand updated FE models) h = 6.17 mm (.243 ), E = 2.05x10 11 Pa (2.9734x10 7 psi) FE Model I: ρ = 7810 kg/m 3 ( lb/in 3 ), m = 23.8 kg (52.44 lb) FE Model II: ρ = 7850 kg/m 3 ( lb/in 3 ), m = 23.9 kg (52.69 lb) Mode Modal frequency (Hz) Rel. diff.(%) Experimental FE Model I FE Model II FE Model I FE Model II First torsion First X bending Second torsion First Y bending Second X bending Anti sym. X bending Third torsion X and Y bending Third X bending
10 Table 4. Correlation of modal frequencies for free-free case with perturbed mass of 0.95 kg(2 lb 1.5 oz) and updated FE models, h = 6.17 mm (.243 ), E = 2.05x10 11 Pa (2.9734x10 7 psi) FE Model I: ρ = 7810 kg/m 3 ( lb/in 3 ), m = 23.8 kg (52.44 lb) FE Model II: ρ = 7850 kg/m 3 ( lb/in 3 ), m = 23.9 kg (52.69 lb) Mode Modal frequency (Hz) Rel. diff.(%) Experimental FE Model I FE Model II FE Model I FE Model II First torsion First X bending Second torsion First Y bending Second X bending Anti sym. X bending Third torsion X and Y bending Third X bending Constrained Boundary Condition Modal Analysis. The plate was clamped to ground at two locations near one of the edges using steel spacers, which in turn were grounded to a huge isolated mass. The test setup is shown in Fig. 9. Impact testing was performed on the clamped plate. Subsequently, the FE model was analyzed after applying constraints in vertical direction at the location corresponding to the clamping points. The modal frequencies for the constrained boundary analysis and modal testing are listed in Table 5. (a) Test setup (b) Supports and clamping location Figure 9: Rectangular plate with constraints The larger relative difference for this case was expected, and can be attributed to the uncertainties involved in modeling the boundary constraints used in the test. While it is aimed to constrain only the vertical direction
11 Table 5. Comparison of modal frequencies for constrained boundary condition case Mode # Description Modal frequency (Hz) Rel. diff. FE Experimental (%) 1 Psuedo-pitch First X bending Second torsion First Y bending Second X bending Anti symmetric X bending Third torsion X and Y bending Third X bending translation of the plate at two points, it is practically not possible to achieve this boundary condition. The support system constrains the motion of the plate at a small but finite patch near the clamping location. The boundary condition of the constrained plate test setup is difficult to incorporate in a FE model. As the FE model is not a true representation of the test setup, the relative difference for this case is higher. However, the validation criteria was met for the constrained boundary condition case as well. 5. Conclusions Testing is expensive and resource intensive. Therefore, it is necessary to develop a well calibrated and validated FE model to assist in predicting system response. This paper presents an example of model calibration and validation of the FE model of a rectangular steel plate based on correlation with experimental results. A plate structure was fabricated and tested to estimate its dynamic properties. A FE model of the plate was developed to perform modal correlation and validation. The results obtained from both the model and the test showed some differences in modal frequencies due to uncertainties in geometric parameters, the material model and boundary conditions. Once corrections to material model and geometric parameters were applied, the FE model was calibrated. An improved support system was used for testing. Subsequently, a good agreement between FE and modal testing results was achieved. The relative error was reduced to 1 2%, which met the validation criteria. Both FE model and modal testing results had similar modal frequencies obtained in the frequency range of Hz. The first 9 elastic modes showed good agreement in determining torsion, bending or combination of both torsion and bending modes. The robustness of the validated model was checked by comparing the results predicted by the FE model with the results of perturbed mass and constrained boundary testing. References [1] AIAA,Guide for the Verification and Validation of Computational Fluid Dynamics Simulations, AIAA G , 1998 [2] ASME, Guide for Verification and Validation in Computational Solid Mechanics, ASME V&V , 2006 [3] W.L. Oberkampf and C.J. Roy, Verification and Validation in Scientific Computing, Cambridge University Press, 2010 [4] R.J. Allemang, Validation of the Digital Twin of a Reusable, Hot Structure, Hypersonic Vehicle, US Air Force Summer Faculty Fellowship Program, 2010 [5] T.L. Paez, Introduction to Model Validation, Proceedings of the 27 th IMAC, SEM, Orlando, FL, 2009 [6] C. Hirsch, Verification and Validation for Modeling and Simulation in Computational Science and Engineering Applications, Foundations for Verification and Validation in the 21 st Century Workshop, Johns Hopkins University/Applied Physics Laboratory, October 22-23, 2002 [7] T.G. Trucano, L.P. Swiler, T. Igusa, W.L. Oberkampf and M. Pilch, Calibration, Validation, and Sensitivity analysis: What s what, Reliability Engineering and System Safety, 2006 [8] D. Mariappan, Simulation and Experimental Validation of Automotive Components, Proceedings of the 27 th IMAC, SEM, Orlando, FL, 2009 [9] R.D. Blevins, Formulas for Natural Frequency and Mode Shape, pp , 1979 [10] B.J. Schwarz and M.H. Richardson, Experimental Modal Analysis, CSI Reliability Week, Orlando, FL, October, 1999 [11] H.G. Pasha, R.J. Allemang, D.L. Brown, A.W. Phillips and K. Kohli, Support Systems for Developing System Models, Proceedings of the International Modal Analysis Conference (IMAC) XXXII A Conference and Exposition on Structural Dynamics, 2014
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