Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate

Chapter 37Structural Dynamics Model Calibration and Validationof a Rectangular Steel Plate Structure

Hasan G. Pasha, Karan Kohli, Randall J. Allemang, Allyn W. Phillips, and David L. Brown

Abstract To characterize the dynamics of a structure accurately and to minimize uncertainties, it is important to performmodal testing in various configurations. However, performing such rigorous testing of structures can be a resource intensiveprocess. In addition, simulating certain conditions may not be possible in the lab. Developing a calibrated and a validatedmodel that can predict the dynamic response of a structure accurately can be a key to address this issue. In this paper, a modelcalibration and validation case study performed on a rectangular steel plate structure is presented. The geometric and materialproperties used in the model were updated to calibrate the model. The accuracy of the calibrated model was confirmed byperforming a validation process involving perturbed mass and constrained boundary condition FE modal analysis and modaltesting. The validation criteria were achieved using the calibrated model and thus proved that the model could reliably predictthe 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� Mass per unit area of a plate (kg=m2)

� Density of the material (kg=m3)

� 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

37.1 Introduction

Testing a structure to determine its dynamic characteristics is both time consuming and expensive. This situation has made itnecessary to develop analytical/mathematical models that can predict the response of system in real time without performingactual tests. Such models need to undergo verification, calibration and validation with actual test data before they can berelied upon to extrapolate results for future studies. In this context, verification and validation (V&V) are defined as below:

H.G. Pasha (�) • K. Kohli • R.J. Allemang • A.W. Phillips • D.L. BrownUniversity of Cincinnati – Structural Dynamics Research Lab (UC-SDRL), Cincinnati, OH, USAe-mail: [email protected]

© The Society for Experimental Mechanics, Inc. 2015H.S. Atamturktur et al. (eds.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedingsof the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15224-0_37

351

mailto:[email protected]

352 H.G. Pasha et al.

Fig. 37.1 Verification,calibration and validationprocesses

Verification: refers to process of determining that a model implementation accurately represents the developer’s conceptualdescription of the model and the solution to the model [1]. Frequently, the verification portion of the process answers thequestion “Are the model equations correctly implemented?”

Validation: refers to the process of determining the degree to which a model is an accurate representation of the real worldsystem 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 frequenciesand mode shapes). A step by step approach was taken to verify, calibrate and validate the results generated by the model indifferent test configurations (Fig. 37.1).

37 Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate Structure 353

37.2 Verification

37.2.1 Analytical Modeling and Formulation

Ideally, a plate is considered a two-dimensional layer of elastic material made up of a sheet that lies in a plane. A plateas a structure possesses bending rigidity due to its thickness and elasticity. As a result, when the plate vibrates, it deformsprimarily by flexing perpendicular to its own plane.

The following assumptions were made while performing the analytical formulation of the rectangular plate structure [9]:

• 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

z DX

i

X

j

Aij Qzijsin�2� fijtC�ij

�(37.1)

where Qzij is the mode shape of the mode .i; j/, Aij is the modal amplitude and �ij is the phase. The modal frequency of the

plate is defined as fij D �2ij

2� l2

qEh3

12�.1��2/, where �ij is a dimensionless parameter, which is generally a function of the mode

indices .i; j/. Flexural rigidity of plate is defined as D D Eh3

12�.1��2/.

37.2.2 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 rectangularplate model, with aspect ratio of 1:5 was developed, such that the first mode is a torsion mode roughly at 40 Hz and thesecond mode is a bending mode roughly at 44 Hz. Closed form expressions [9] were used to develop an analytical model(Figs. 37.2 and 37.3).

A comparable rectangular plate FE model was developed using the dimensions and material properties chosen for theanalytical closed form model of the plate. Analytical modal analysis was performed on the plate FE model. The modalfrequencies and mode shapes for the first six deformation modes were retrieved and compared with the analytical modelprediction.

The modes were in the expected sequence and the relative difference in modal frequencies was less than 2%. This verifiedthe FE model (Table 37.1).

37.3 Calibration

Calibration of a model involves comparing the analytical results to equivalent experimental results. Calibration in this caseinvolves correlating the modal parameters obtained from an analytical modal analysis of the rectangular plate FE model andimpact testing of the fabricated rectangular plate structure. The details of the various activities performed in the calibrationprocess are explained in the subsequent sections.


Fig. 37.2 2-D drawing showingplate dimensions

Plate Thickness-0.25in

22.50in

1.00

34.00in.

1.00 2.00in.

2.f0.25in.(1/4-20)

f0.19

4.f0.25in.(1/4-20)

2.f0.25(1/4-20)

Fig. 37.3 Rectangular plate model. (a) CAD model. (b) Mesh

Table 37.1 Preliminaryverification of the FEmodel—modal frequencycomparison

Modal frequency (Hz)

Mode # Description Analytical FE Rel. diff. (%)

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


37.3.1 Analytical Modal Analysis

Analytical modal analysis was performed in ANSYSr Workbench 14:5. The default material model for steel in the ANSYSr

material library was chosen. The modal frequencies and mode shapes were extracted in the frequency range 0–250 Hz.

37.3.2 Experimental Modal Analysis

The plate structure was tested using the multi-reference impact testing (MRIT) technique with 21 accelerometers mounted onthe rectangular plate at locations identified in Fig. 37.4. For simulating the free-free boundary condition, the plate was initiallysuspended using shock cords. However, the support system was dynamically coupled with the third mode of the rectangularplate (second torsion mode) and it acted as a vibration absorber. Subsequently, the plate was supported on racquet balls,as shown in Fig. 37.5. Though the racquet ball supports offered a lesser frequency separation ratio between the rigid bodymodes and the first elastic mode (compared to the shock cords), they were well isolated from the test structure [11].

Fig. 37.4 Rectangular plate—reference locations

Fig. 37.5 Rectangular plate—support system


Fig. 37.6 Complex mode indicator function

On processing the measured data, nine deformation modes in the 0–250 Hz frequency range were identified, as evidentfrom the CMIF plot shown in Fig. 37.6. The first torsion and the first bending mode were at 42:16 Hz and 44:16 Hzrespectively. The mode shapes are shown in Fig. 37.7.

37.3.3 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.

37.3.4 Model Calibration

From the correlation of the modal parameters from the preliminary analytical modal analysis and impact testing, it wasidentified that there were various sources of uncertainty. On the modeling side, the sources of uncertainty were geometricand material properties of the steel plate. In addition, the soft supports had not been modeled. On the testing side, the supportsystem, location of supports, sensor calibration, impact hammer tip and signal processing parameters were considered to bethe sources of uncertainty. The relative difference in modal frequencies was desired to be less than 2 % and was set as thecalibration criteria or calibration metric.


Fig. 37.7 Experimental mode shapes

37.3.4.1 Model Updating

Geometric properties: It was established that the plate thickness, originally intended to be 0:25 in: was not constant. Theplate was weighed (52:5 lbm) and the thickness of the plate model was adjusted (0:243 in:) to match the measured weightof the plate.

Material model: The relative difference for the third mode (second torsion mode) was reasonably high compared to othermodes. A parametric study, in which the effect of varying the Young’s modulus, mass density and Poisson ratio on themodal frequencies, was studied. For the calibrated model, the material properties were established as: Young’s modulusE D 2:05x1011 Pa .2:9734x107 psi/, mass density � D 7; 850 kg=m3 .0:2836 lb=in3/ and Poisson ratio � D 0:29.

Modeling support system: Four racquet balls were used to support the plate during impact testing. These racquet balls actas springs and their stiffness value was estimated to be roughly 40 lbf=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.

37.3.4.2 Testing Related Changes

Support system: Free-free boundary conditions are relatively easier to achieve compared to other boundary conditionssuch as a fixed boundary condition. Support systems add stiffness and can also interact with the test structure. Certainprecautions 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 supportsdynamically interacted with the third deformation mode (second torsion mode), and acted as a vibration absorber. As aresult, 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 isacquired. Sensors generally tend to have variability in the sensitivity in the order of 2–5 %, which when not accounted forcould 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 areadequately 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 impactlocation. As a result, an appropriate hammer tip should be selected. A soft hammer tip was selected as the plate structurewas lightly damped.

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 shouldbe 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 tocondition the signal to avoid leakage and measurement noise. The number of ensembles acquired per average also affectsthe ability to obtain good quality data and minimize measurement noise. In this test, data was acquired in the 0–100 Hzand 0–250 Hz frequency ranges. The frequency resolution (�F) was chosen for each case such that each ensemble wasobserved for 16 s. The force-exponential window was chosen, with a cutoff value of 10 % for the force window and anend value for exponential window as 20 %.

37.3.4.3 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 the corresponding synthesized FRFs as a further calibration metric that involvesdamping and sensor calibration issues. A plot comparing the measured FRF with the synthesized FRF for a driving-pointis shown in Fig. 37.8. It is apparent from the comparison that the peaks of the synthesized FRF are sharp as damping wasnot prescribed in the FE model. However, the peaks of the measured FRF are damped. In reality the modes do not have theamount of damping that is evident in the FRF comparison plot as the rectangular plate is a lightly damped structure. Thedamped peaks in the measured FRF are a result of the artificial damping introduced by the force-exponential window thatwas used to eliminate leakage while performing the impact testing.

Fig. 37.8 Comparison of adriving-point FRF


Table 37.2 Correlation of modal frequencies for free-free case with no perturbed mass andupdated FE models

h D 6:17 mm .0:243”), E D 2:05x1011 Pa .2:9734x107 psi/FE model I: � D 7; 810 kg=m3 .0:2821 lb=in3/, m D 23:8 kg .52:44 lb/

FE model II: � D 7; 850 kg=m3 .0:2836 lb=in3/, m D 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

The results of modal correlation are presented in Table 37.2. The calibration criteria, a relative difference for modalfrequencies less than 2 % was met and a calibrated rectangular plate FE model was obtained.

37.4 Validation

Validation is the process of determining the correctness of a model in its description of the reference system under a set oftest conditions [1, 2]. For validating the calibrated rectangular plate FE model and thereby establishing the robustness of themodel, it was decided to analyze and to perform testing for the following two cases:

1. Perturbed mass analytical modal analysis and testing2. 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 perturbedmass 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 while testing.In addition, modeling real boundary conditions is involved. As a result it was decided to set the validation criteria less than5% relative difference for modal frequencies for the constrained boundary condition case.

37.4.1 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 plateFE 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 and2 lb 1:5 oz perturbed mass cases are listed in Tables 37.3 and 37.4 respectively. It is evident that the validation criteria,relative difference for modal frequencies less than 2 %, was met.

37.4.2 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 toa huge isolated mass. The test setup is shown in Fig. 37.9. Impact testing was performed on the clamped plate.


Table 37.3 Correlation of modal frequencies for free-free case with perturbed mass of0:496 kg (1 lb 1:5 ozand updated FE models)





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 37.4 Correlation of modal frequencies for free-free case with perturbed mass of0:95 kg(2 lb 1:5 oz) and updated FE models,





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

Subsequently, the FE model was analyzed after applying constraints in vertical direction at the location corresponding tothe clamping points. The modal frequencies for the constrained boundary analysis and modal testing are listed in Table 37.5.

The larger relative difference for this case was expected, and can be attributed to the uncertainties involved in modelingthe boundary constraints used in the test. While it is aimed to constrain only the vertical direction translation of the plateat two points, it is practically not possible to achieve this boundary condition. The support system constrains the motion ofthe plate at a small but finite patch near the clamping location. The boundary condition of the constrained plate test setup isdifficult to incorporate in a FE model. As the FE model is not a true representation of the test setup, the relative differencefor this case is higher. However, the validation criteria was met for the constrained boundary condition case as well.

37.5 Conclusions

• Testing is expensive and resource intensive. Therefore, it is necessary to develop a well calibrated and validated FE modelto 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 oncorrelation with experimental results.


Fig. 37.9 Rectangular plate withconstraints. (a) Test setup;(b) supports and clampinglocation

Table 37.5 Comparison of modal frequencies for constrained boundary conditioncase

Modal frequency (Hz)Mode # Description FE Experimental Rel. diff. (%)

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

• A plate structure was fabricated and tested to estimate its dynamic properties. A FE model of the plate was developed toperform modal correlation and validation.

• The results obtained from both the model and the test showed some differences in modal frequencies due to uncertaintiesin geometric parameters, the material model and boundary conditions.

• Once corrections to material model and geometric parameters were applied, the FE model was calibrated. An improvedsupport 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 0–250 Hz.The first nine elastic modes showed good agreement in determining torsion, bending or combination of both torsion andbending modes.

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


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Structural Dynamics Model Calibration and Validation of a Rectangular Steel Plate