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 importantto perform modal testing in various configurations. However, performing such rigorous testing of structures can bea resource intensive process. In addition, simulating certain conditions may not be possible in the lab. Developinga calibrated and a validated model that can predict the dynamic response of a structure accurately can be a keyto address this issue. In this paper, a model calibration and validation case study performed on a rectangular steelplate structure is presented. The geometric and material properties used in the model were updated to calibrate themodel. The accuracy of the calibrated model was confirmed by performing a validation process involving perturbedmass and constrained boundary condition FE modal analysis and modal testing. The validation criteria were achievedusing 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, PerturbedBoundary Conditions, Constrained Boundary Condition Testing

Notation

Symbol Description

𝛾 Mass per unit area of a plate (kg/m2)

𝜌 Density of the material (kg/m3)

𝜈 Poisson ratio

b Width of the plate (m)

fij Modal frequency (Hz)

h Thickness of the plate (m)

i Mode index, number of half waves in mode shape along horizontal direction

j Mode index, number of half waves in mode shape along vertical direction

l Length of the plate (m)

D Flexural rigidity (N ·m2)

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 situationhas made it necessary to develop analytical/mathematical models that can predict the response of system in realtime without performing actual tests. Such models need to undergo verification, calibration and validation withactual test data before they can be relied upon to extrapolate results for future studies. In this context, verificationand validation (V&V) are defined as below:

Verification: refers to process of determining that a model implementation accurately represents the devel-oper’s conceptual description of the model and the solution to the model [1]. Frequently, the verificationportion of the process answers the question “Are the model equations correctly implemented?”

Validation: refers to the process of determining the degree to which a model is an accurate representation ofthe real world system from the perspective of the intended uses of the model [1]. Frequently, the validationprocess 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 (modalfrequencies and mode shapes). A step by step approach was taken to verify, calibrate and validate the resultsgenerated by the model in different test configurations.

Figure 1: Verification, calibration and validation processes

2. Verification

2.1. Analytical Modeling and Formulation. Ideally, a plate is considered a 2-dimensional layer of elastic mate-rial made up of a sheet that lies in a plane. A plate as a structure possesses bending rigidity due to its thickness andelasticity. 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 normalto 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 deformationssuch that

(1) z =∑︁i

∑︁j

Aijz̃ij sin (2𝜋fijt + 𝜑ij)

where z̃ij is the mode shape of the mode (i, j), Aij is the modal amplitude and 𝜑ij is the phase. The modal

frequency of the plate is defined as fij =𝜆2ij

2𝜋l2

√︁Eh3

12𝛾(1−𝜈2) , where 𝜆ij is a dimensionless parameter, which is

generally a function of the mode indices (i, j). Flexural rigidity of plate is defined as D = Eh3

12𝛾(1−𝜈2) .

2.2. Verification. Verification is the process of comparing different analytical models to be certain that the analysisresults are correct [1, 2]. In this case, a FE model is compared to the analytical, closed form results for a rectangularplate. An analytical rectangular plate model, with aspect ratio of 1.5 was developed, such that the first mode is atorsion 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 forthe analytical closed form model of the plate. Analytical modal analysis was performed on the plate FE model.

(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 theanalytical model prediction.

The modes were in the expected sequence and the relative difference in modal frequencies was less than 2%. Thisverified the FE model.

Table 1. Preliminary verification of the FE model – Modal frequency comparison

Modal frequency (Hz) Rel. diff.Mode # Description

Analytical FE (%)

1 First torsion 41.55 41.34 0.51

2 First X bending 44.59 43.95 1.46

3 Second torsion 96.3 95.12 1.24

4 First Y bending 103.8 102.99 0.79

5 Second X bending 121 119.16 1.54

6 Anti symmetric X bending 139.3 137.39 1.39

3. Calibration

Calibration of a model involves comparing the analytical results to equivalent experimental results. Calibration inthis case involves correlating the modal parameters obtained from an analytical modal analysis of the rectangularplate FE model and impact testing of the fabricated rectangular plate structure. The details of the various activitiesperformed in the calibration process are explained in the subsequent sections.

3.1. Analytical Modal Analysis. Analytical modal analysis was performed in ANSYSr Workbench 14.5. Thedefault material model for steel in the ANSYSr material library was chosen. The modal frequencies and modeshapes were extracted in the frequency range 0 − 250 Hz.

3.2. 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. Forsimulating the free-free boundary condition, the plate was initially suspended using shock cords. However, thesupport system was dynamically coupled with the third mode of the rectangular plate (second torsion mode) and itacted as a vibration absorber. Subsequently, the plate was supported on racquet balls, as shown in Fig. 5. Thoughthe racquet ball supports offered a lesser frequency separation ratio between the rigid body modes and the firstelastic 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 0 − 250 Hz frequency range were identified, asevident from the CMIF plot shown in Fig. 6. The first torsion and first bending mode were at 42.16 Hz and 44.16 Hzrespectively. The mode shapes are shown in Fig. 7.

3.3. Modal Correlation. The relative difference between the modal frequencies obtained from FE modal analysisand impact testing is > 2 − 3%. The sequence of the modes was established to be in the same order for both casesby viewing the mode shape animations.

Figure 4: Rectangular plate – reference locations

Figure 5: Rectangular plate – support system

Figure 6: Complex mode indicator function

Figure 7: Experimental mode shapes

3.4. Model Calibration. From the correlation of the modal parameters from the preliminary analytical modalanalysis 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 supportshad not been modeled. On the testing side, the support system, location of supports, sensor calibration, impacthammer tip and signal processing parameters were considered to be the sources of uncertainty. The relative differencein modal frequencies was desired to be less than 2% and was set as the calibration criteria or calibration metric.

3.4.1. Model Updating.

Geometric properties: It was established that the plate thickness, originally intended to be 0.25 in was notconstant. The plate was weighed (52.5 lbm) and the thickness of the plate model was adjusted (0.243 in) tomatch the measured weight of the plate.

Material model: The relative difference for the third mode (second torsion mode) was reasonably high com-pared to other modes. A parametric study, in which the effect of varying the Young’s modulus, massdensity and Poisson ratio on the modal frequencies, was studied. For the calibrated model, the mate-rial properties were established as: Young’s modulus E = 2.05x1011 Pa (2.9734x107 psi), mass density𝜌 = 7850 kg/m3 (0.2836 lb/in3) and Poisson ratio 𝜈 = .29.

Modeling support system: Four racquet balls were used to support the plate during impact testing. Theseracquet balls act as springs and their stiffness value was estimated to be roughly 40 lbf/in. At locationscorresponding to the supports, linear springs were added in the FE model. This reduced the relative differencefor the modal frequencies further.

3.4.2. Testing Related Changes.

Support system: Free-free boundary conditions are relatively easier to achieve compared to other boundaryconditions such as a fixed boundary condition. Support systems add stiffness and can also interact with thetest 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 chordsupports dynamically interacted with the third deformation mode (second torsion mode), and acted as avibration absorber. As a result, a racquet ball support system, that was reasonably isolated from the teststructure, was selected.

Sensor calibration: The sensors (accelerometers and impact hammer load cells) need to be calibrated beforedata is acquired. Sensors generally tend to have variability in the sensitivity in the order of 2 − 5%, whichwhen not accounted for could skew the measurement magnitudes (but would have no effect on the calibrationmetric involving modal frequencies).

Impact hammer tip: The impact hammer tip should be selected such that the modes in the required fre-quency range are adequately excited without exciting higher frequency modes. When the focus is to acquiredata in lower frequency range, using a harder tip would possibly cause overloads on the data acquisitionchannels associated with sensors near the impact location. As a result, an appropriate hammer tip shouldbe selected.

Signal processing parameters: The rectangular plate is a very lightly damped structure. It was noted thatafter 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 completelyobserved 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 measurementnoise. The number of ensembles acquired per average also affects the ability to obtain good quality data andminimize measurement noise. In this test, data was acquired in the 0 − 100 Hz and 0 − 250 Hz frequencyranges. The frequency resolution (∆F) was chosen for each case such that each ensemble was observed for16 s. The force-exponential window was chosen, with a cutoff value of 10% for the force window and an endvalue for exponential window as 20%.

3.4.3. Modal Correlation after Calibrating the FE Model. Modal correlation was performed after updating the FEmodel and incorporating the changes identified for the test setup. The relative difference for modal frequenciesreduced to a value less than 2%.

FRFs from the FE model were synthesized at a few locations corresponding to the driving-points in the impacttest. The measured FRFs were compared with corresponding synthesized FRFs as a further calibration metric thatinvolves damping and sensor calibration issues. Plots comparing the measured FRF with the synthesized FRF for adriving-point is shown in Fig. 8. It is apparent from the comparison that the peaks in the synthesized FRF does not

have any damping as damping was not modeled. However, the measured FRF peaks are damped. The rectangularplate is a lightly damped structure. In reality the modes do not have the amount of damping that is visible in theFRF comparison plot. As impact testing method was used to acquire the FRF data, most of the damping stemsfrom 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 modalfrequencies 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 updatedFE modelsℎ = 6.17 mm (.243”), E = 2.05x1011 Pa (2.9734x107 psi)FE Model I: 𝜌 = 7810 kg/m3 (0.2821 lb/in3), 𝑚 = 23.8 kg (52.44 lb)FE Model II: 𝜌 = 7850 kg/m3 (0.2836 lb/in3), 𝑚 = 23.9 kg (52.69 lb)

Modal frequency (Hz) Rel. diff.(%)Mode

Experimental FE Model I FE Model II FE Model I FE Model II

First torsion 42.16 41.33 41.34 2 1.98

First X bending 44.16 43.71 43.65 1.02 1.16

Second torsion 95.58 95.04 95.12 0.57 0.48

First Y bending 104.6 103.1 102.99 1.45 1.56

Second X bending 119.06 118.19 118.16 0.74 0.76

Anti sym. X bending 137.98 137.38 137.39 0.44 0.43

Third torsion 176.38 175.52 175.81 0.49 0.32

X and Y bending 203.72 202.86 203.25 0.42 0.23

Third X bending 244.98 243.65 244.1 0.54 0.36

4. Validation

Validation is the process of determining the correctness of a model in its description of the reference system undera set of test conditions [1, 2]. For validating the calibrated rectangular plate FE model and thereby establishing therobustness 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 theperturbed 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.

4.1. 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 tothe fabricated plate structure and impact testing was conducted. The results obtained from the modal analysis andimpact testing were correlated. The correlated results for 1 lb 1.5 oz and 2 lb 1.5 oz perturbed mass cases are listedin Tables 3 and 4 respectively. It is evident that the validation criteria, relative difference for modal frequencies lessthan 2%, was met.

Table 3. Correlation of modal frequencies for free-free case with perturbed mass of0.496 kg (1 lb 1.5 ozand updated FE models)ℎ = 6.17 mm (.243”), E = 2.05x1011 Pa (2.9734x107 psi)FE Model I: 𝜌 = 7810 kg/m3 (0.2821 lb/in3), 𝑚 = 23.8 kg (52.44 lb)FE Model II: 𝜌 = 7850 kg/m3 (0.2836 lb/in3), 𝑚 = 23.9 kg (52.69 lb)

Modal frequency (Hz) Rel. diff.(%)Mode

Experimental FE Model I FE Model II FE Model I FE Model II

First torsion 38.95 38.4 38 1.43 1.06

First X bending 42.42 42.15 41.84 0.63 0.76

Second torsion 89.53 89.73 88.86 0.23 0.98

First Y bending 99.96 98.93 97.79 1.04 1.17

Second X bending 117.39 117.09 116.11 0.26 0.84

Anti sym. X bending 128.95 128.94 127.56 0.31 1.08

Third torsion 167.56 168.34 166.33 0.46 1.21

X and Y bending 197.19 198.36 196.76 0.59 0.81

Third X bending 235.5 237.82 233.8 0.98 0.71

Table 4. Correlation of modal frequencies for free-free case with perturbed mass of0.95 kg(2 lb 1.5 oz) and updated FE models,ℎ = 6.17 mm (.243”), E = 2.05x1011 Pa (2.9734x107 psi)FE Model I: 𝜌 = 7810 kg/m3 (0.2821 lb/in3), 𝑚 = 23.8 kg (52.44 lb)FE Model II: 𝜌 = 7850 kg/m3 (0.2836 lb/in3), 𝑚 = 23.9 kg (52.69 lb)

Modal frequency (Hz) Rel. diff.(%)Mode

Experimental FE Model I FE Model II FE Model I FE Model II

First torsion 36.86 36.47 36.17 1.07 1.92

First X bending 41.22 41.09 40.82 0.33 0.97

Second torsion 86.62 86.92 86.28 0.34 0.39

First Y bending 96.26 95.49 94.5 0.8 1.86

Second X bending 115.98 115.85 114.99 0.11 0.86

Anti sym. X bending 125.37 125.29 124.39 0.26 0.79

Third torsion 163.3 164.29 162.74 0.6 0.34

X and Y bending 195.1 196.22 195.05 0.57 0.23

Third X bending 229.84 231.37 230.8 0.67 0.42

4.2. Constrained Boundary Condition Modal Analysis. The plate was clamped to ground at two locationsnear one of the edges using steel spacers, which in turn were grounded to a huge isolated mass. The test setup isshown 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 correspondingto the clamping points. The modal frequencies for the constrained boundary analysis and modal testing are listedin 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 inmodeling the boundary constraints used in the test. While it is aimed to constrain only the vertical direction

Table 5. Comparison of modal frequencies for constrained boundary condition case

Modal frequency (Hz) Rel. diff.Mode # Description

FE Experimental (%)

1 Psuedo-pitch 10.76 11.02 2.42

2 First X bending 33.34 34.1 2.28

3 Second torsion 53.99 52.71 2.37

4 First Y bending 82.14 83.56 1.75

5 Second X bending 94.92 91.75 3.34

6 Anti symmetric X bending 114.7 118.59 3.39

7 Third torsion 124.4 128.43 3.24

8 X and Y bending 151.9 154.53 1.73

9 Third X bending 205.4 208.28 1.4

translation of the plate at two points, it is practically not possible to achieve this boundary condition. The supportsystem constrains the motion of the plate at a small but finite patch near the clamping location. The boundarycondition of the constrained plate test setup is difficult to incorporate in a FE model. As the FE model is not a truerepresentation of the test setup, the relative difference for this case is higher. However, the validation criteria wasmet 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 andvalidated 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 steelplate based on correlation with experimental results.

∙ A plate structure was fabricated and tested to estimate its dynamic properties. A FE model of the plate wasdeveloped to perform modal correlation and validation.

∙ The results obtained from both the model and the test showed some differences in modal frequencies due touncertainties 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 modaltesting 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 of0−250 Hz. The first 9 elastic modes showed good agreement in determining torsion, bending or combinationof both torsion and bending modes.

∙ The robustness of the validated model was checked by comparing the results predicted by the FE model withthe results of perturbed mass and constrained boundary testing.

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