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
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.
Abstract.
Model Verification, Model Calibration, Model Validation, Modal Correlation, Model Updating, Perturbed
Boundary Conditions, Constrained Boundary Condition Testing
Keywords.
Notation
Symbol Description
kg/m2 )
𝛾
Mass per unit area of a plate (
𝜌
Density of the material (
3
𝜈
Poisson ratio
b
Width of the plate (
fij
h
kg/m
)
m)
Hz)
Modal frequency (
m)
Thickness of the plate (
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 (
N · m2 )
D
Flexural rigidity (
E
Young’s modulus (
CMIF
FEA
MRIT
m)
Pa)
Complex Mode Indicator Function
Finite Element Analysis
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 devel-
oper’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?”
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
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 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
z=
(1)
∑︁ ∑︁
i
where
z̃ij
is the mode shape of the mode
frequency of the plate is defined as
fij =
generally a function of the mode indices
2.2.
Verification.
Aij z̃ij sin (2𝜋fij t + 𝜑ij )
j
(i, j), Aij
√︁
𝜆2
ij
2𝜋l2
(i, j).
is the modal amplitude and
Eh3
12𝛾(1−𝜈 2 ) , where
𝜆ij
𝜑ij
is the phase. The modal
is a dimensionless parameter, which is
Flexural rigidity of plate is defined as
D=
Eh3
12𝛾(1−𝜈 2 ) .
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
torsion mode roughly at
40 Hz
1.5
was developed, such that the first mode is a
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.
(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 #
1
2
3
4
5
6
Modal frequency (Hz) Rel. diff.
Analytical
FE
(%)
First torsion
41.55
41.34
0.51
First X bending
44.59
43.95
1.46
Second torsion
96.3
95.12
1.24
First Y bending
103.8
102.99
0.79
Second X bending
121
119.16
1.54
Anti symmetric X bending
139.3
137.39
1.39
Description
3. 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.
3.1.
Analytical Modal Analysis.
Analytical modal analysis was performed in ANSYS
default material model for steel in the ANSYS
shapes were extracted in the frequency range
3.2.
Experimental Modal Analysis.
(MRIT) technique with
21
r
material library was chosen.
r
Workbench
14.5.
The
The modal frequencies and mode
0 − 250 Hz.
The plate structure was tested using the multi-reference impact testing
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
0 − 250 Hz
frequency range were identified, as
evident from the CMIF plot shown in Fig. 6. The first torsion and first bending mode were at
42.16 Hz and 44.16 Hz
respectively. The mode shapes are shown in Fig. 7.
3.3.
Modal Correlation.
and impact testing is
The relative difference between the modal frequencies obtained from FE modal analysis
> 2 − 3%.
The sequence of the modes was established to be in the same order for both cases
by 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 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.
3.4.1. Model Updating.
Geometric properties:
It was established that the plate thickness, originally intended to be
constant. The plate was weighed (52.5
lbm )
0.25 in
was not
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 com-
pared 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.
rial properties were established as:
𝜌 = 7850 kg/m3 (0.2836 lb/in3 )
Modeling support system:
For the calibrated model, the mate-
E = 2.05x1011 Pa (2.9734x107 psi),
𝜈 = .29.
Young’s modulus
and Poisson ratio
mass density
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 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.
3.4.2. 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 fre-
quency 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:
after every impact, it took over
The rectangular plate is a very lightly damped structure. It was noted that
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
0 − 100 Hz
and
0 − 250 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
value for exponential window as
10%
for the force window and an end
20%.
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.
reduced to a value less than
The relative difference for modal frequencies
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
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 = 2.05x1011 Pa (2.9734x107 psi)
3
3
FE Model I: 𝜌 = 7810 kg/m (0.2821 lb/in ), 𝑚 = 23.8 kg (52.44 lb)
3
3
FE Model II: 𝜌 = 7850 kg/m (0.2836 lb/in ), 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
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 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.
4.1.
Perturbed Mass Modal Analysis.
1 lb 1.5 oz
and
2 lb 1.5 oz
Two separate cases were studied by attaching two cylindrical masses,
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
0.496 kg (1 lb 1.5 ozand updated FE models)
ℎ = 6.17 mm (.243” ), E = 2.05x1011 Pa (2.9734x107 psi)
3
3
FE Model I: 𝜌 = 7810 kg/m (0.2821 lb/in ), 𝑚 = 23.8 kg (52.44 lb)
3
3
FE Model II: 𝜌 = 7850 kg/m (0.2836 lb/in ), 𝑚 = 23.9 kg (52.69 lb)
Mode
Modal frequency (Hz)
Experimental
FE Model I
Rel. diff.(%)
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
of
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 = 2.05x1011 Pa (2.9734x107 psi)
3
3
FE Model I: 𝜌 = 7810 kg/m (0.2821 lb/in ), 𝑚 = 23.8 kg (52.44 lb)
3
3
FE Model II: 𝜌 = 7850 kg/m (0.2836 lb/in ), 𝑚 = 23.9 kg (52.69 lb)
Modal frequency (Hz)
Mode
4.2.
Rel. diff.(%)
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
0.39
Second torsion
86.62
86.92
86.28
0.34
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
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
Table 5. Comparison of modal frequencies for constrained boundary condition case
Mode #
Modal frequency (Hz)
Description
Rel. diff.
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 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
0 − 250 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.
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