Investigations have been carried out both numerically and experimentally to settle with a practically feasible set of proportional viscous damping parameters for the accurate prediction of responses of fibre reinforced plastic beams over a chosen frequency range of interest. The methodology needs accurate experimental modal testing, an adequately converged finite element model, a rational basis for correct correlations between these two models, and finally, updating of the finite element model by estimating a pair of global viscous damping coefficients using a gradient-based inverse sensitivity algorithm.
The present approach emphasises that the successful estimate of the damping matrix is related to a-priori estimation of material properties, as well. The responses are somewhat accurately predicted using these updated damping parameters over a large frequency range. In the case of plates, determination of in-plane stiffness parameters becomes easier, whereas for beam specimens, transverse material properties play a comparatively greater role and need to be determined. Moreover, for damping matrix parameter estimation, frequency response functions need to be used instead of frequencies and mode shapes. The proposed method of damping matrix identification is able to reproduce frequency response functions accurately even outside the frequency ranges used for identification.
1. INTRODUCTION
The accurate determination of dynamical responses is very
important from the viewpoint of safety, serviceability, and op-
eration of any structure. The geometrical complexities, mate-
rial property distributions, existing boundary conditions, and
applied loading are the key factors that influence the dynamic
responses. The elastic and inertial properties are somewhat
correctly represented through finite element modelling with
suitable simplifying assumptions, whereas uncertainties in re-
sponse prediction still remain due to imperfect boundary con-
ditions and the presence of damping, which are difficult to deal
with. No generally acceptable modelling techniques for damp-
ing have been proposed in previous research that can be con-
fidently used for complicated structures. The damping mech-
anism may comprise three effects -- material damping result-
ing from micro-structural behaviour, friction damping result-
ing from looseness at boundaries, and environmental damping
effects, such as interaction with the surrounding fluid. Depend-
ing on the practical situation, one or more component may be
less significant than the others, making the modelling effort
easier for the particular structure under consideration.
Although the phenomenon of damping is mostly nonlinear,
the assumption of small damping makes many equivalent lin-
ear models practically acceptable. For example, Dowell and
Schwartz
1
presented a methodology for accounting for dry
friction damping arising from axial sliding of surfaces inside
supports, and concluded from studies on plates and beams that
an equivalent linear viscous damping ratio can be agreed upon,
even if nonlinear Coulomb law for the friction and geometric
nonlinearity of beams are present. Tang and Dowell
2
further
investigated experimentally to verify the theory presented in
the previous paper. It was concluded that the methodology
works well in lower mode ranges, especially with the funda-
mental mode. Sometimes, it will be possible to accept on prac-
tical terms the linear damping models for much more compli-
cated environmental effects, such as interaction of a beam with
the surrounding air, etc. Filipiak, et al.
3
presented such an ap-
proach to determine the effects of air damping on small beams
housing miniature sensors. How far such efforts are applicable
to realistic full scale structures remain an open question.
If linear damping is agreed upon with small damping as-
sumptions, it can be modelled as a multiplier of conveniently
chosen state variables with constant coefficients. The success
of such a model can be judged by its ability to replicate the
actual observable responses over a frequency range of inter-
ests. Then, the entire domain of linear modal testing can be
employed and a damping matrix can finally be put forward in
the model to be treated in a fashion similar to the stiffness and
mass matrices. Mostly, instantaneous velocity is chosen as the
state variable, and the damping can be stated to be viscous.
4
Fibre Reinforced Plastics (FRP) have long been used in
weight-sensitive aerospace, naval, automotive, and high per-
formance sports applications. However, it took some time for
the engineering community to appreciate the other positive as-
pects of FRPs, such as durability, fatigue, and corrosion resis-
tance to pave the way for its infrastructural applications. Re-
cently, many standard structural forms such as various beam
sections, plates, and shells have been routinely manufactured
and employed for structural applications. Pultruded sections
in regular forms such as rectangular, angle, `T', etc., are likely
to replace most of the current infrastructures made of conven-
tional material such as steel. Condition assessment and health
monitoring of such huge infrastructures made of FRPs will
be a challenge in the future, especially if they degrade over
long periods of time, but still remain serviceable. The exist-
2
ing stiffness and damping properties need to be correctly as-
sessed from time to time using a reliable non-destructive in-
verse technique. Unlike the stiffness and inertia parameters,
uncertainties in damping parameters will further increase, as
the mechanism may include one or more effects which were
initially absent. For example, loosening of joints and supports
may result in increased friction component of damping as time
passes.
Literature related to the modelling of damped FRP struc-
tures is very scant. Zhuang and Crocker
5
presented a review on
the damping of composite structures. Gelman, et al.
6
proposed
a methodology of diagonalisation of the damping matrix based
on measured frequency response functions (FRF). Akrout, et
al.
7
conducted numerically simulated investigations of vibroa-
coustic behaviour of two thin film-laminated glass panels in
the presence of a fluid layer. Assaf
8
analysed sandwich com-
posite beams and investigated the effects of ply-stacking se-
quences, core-to-face stiffness ratio, etc. on natural frequen-
cies and modal damping.
The current literature is very rich in information related to
inverse detection of stiffness parameters from measured vibra-
tion responses,
9
whereas only limited attempts have been made
to inversely estimate the damping parameters for FRP type of
structures. The main reason is that for small damping, the
resonant frequencies and mode shapes change very little with
damping coefficients, but the responses change drastically, es-
pecially near resonances. Literature related to damping iden-
tification using a beam type of specimen is very rare. Reix, et
al.
10
used the FRF information of a beam to update the damp-
ing matrix using a nonlinear least square optimization tech-
nique.
Inverse detection of damping using an iterative procedure
demands proper forward simulation of the damped responses
of the FRP structures in the iterative loop. The most popular
forward damping model is due to Rayleigh, in which the damp-
ing matrix is assumed as a weighted linear combination of the
mass and stiffness matrix
C = a
0
M + a
1
K.
(1)
A more generalised viscous proportional damping matrix has
been proposed by Caughey and Kelly,
11
and can be written as
C = M
r-1
n=0
a
n
M
-1
K
n
.
(2)
Woodhouse
12
has given an account of various linear damp-
ing models useful for structural applications. The main diffi-
culty of all such models is that the damping parameters remain
somewhat insensitive to frequency variations. Moreover, stiff-
ness and mass distributions should be exactly determined a-
priori, which is impossible in most practical cases. Adhikari
13
incorporated the frequency variation of damping factors within
the framework of a generalised damping model. As a contin-
uation of the above methodology, Adhikari and Phani
14
pro-
posed a proportional damping matrix obtained from a single
driving point FRF. Minas and Inman
15
used incomplete exper-
imental modal data and reduced mass and stiffness matrices to
identify a non-proportional damping matrix in a weighted least
square sense. Lancaster and Prells
16
used the inverse spectral
method to estimate the damping matrix from complex eigen-
vectors. Pilkey
17
proposed direct and iterative approaches for
damping matrix reconstruction. Friswell, et al.
18
used a direct
approach to identify damping and the stiffness matrix together
using FRF information. Chen and Tsuei
19
distinguished be-
tween the viscous and structural damping components from
the measured complex FRF matrix. Some investigators tried
to estimate mass, stiffness, and damping matrices together.
20
The main drawback of the investigations on damping matrix
identification proposed in a great deal of research is that they
demand availability of accurate information about the stiffness
and mass of the system, as well as the availability of accurate
modal properties. Even in newly built FRP structures, there
are large variations between the predicted stiffness parameters
as compared to those existing, due to the fact that the struc-
tural fabrication and material fabrication are one unified pro-
cess for FRP, and the actual existing material properties vary
greatly from those mentioned in manufacturer's manual or in
standard handbooks. For the FRP type of anisotropic layered
composites, such uncertainties are greater as compared to sim-
ilar constructions made of isotropic and homogeneous materi-
als. It thus appears that a proper inverse regularised technique
augmented by a-priori stiffness estimation procedure will be
appropriate for realistic damping parameter identification. If
the damping is small, which is the case in most practical struc-
tures, a linear model will suffice.
The objective of the present investigation is to apply a
gradient-based model updating technique to estimate viscous
damping parameters along with the stiffness parameters for
pultruded FRP beams using measured FRFs. The efficiency
of the algorithm will be judged by comparing the regenerated
FRFs to the experimentally obtained values to examine if the
responses match accurately. Information related to FRF-based
updating is abundant in current literature,
21
although most of
it is related to the estimation of stiffness parameters as far as
applications to FRP structures are concerned.
The complete process for identification of damping of FRP
beams includes a-priori estimate of stiffness parameters us-
ing measured modal and FRF data, converged finite element
modelling, correlations between them, and finally updating the
global proportional damping parameters using the gradient-
based inverse sensitivity technique in a nonlinear least square
sense.
22
The methodology is first established through a numer-
ically simulated example, followed by real experimental case
studies involving different boundary conditions.
2. MATHEMATICAL FORMULATION
The equation of motion of a multiple-degrees-of-freedom
system in a discretized form can be written as
M ¨
x(t) + C
x(t) + Kx(t) = f (t);
(3)
where M , K, and C are the mass, stiffness, and damping ma-
trix, respectively. Here, equivalent viscous damping has been
considered as the major dissipation mechanism. In modal co-
ordinates, the equation can be written as a set of single-degree-
of-freedom (SDOF) uncoupled equations
[m]¨
x(t) + [c]
x(t) + [k]x(t) = {u}(t);
(4)
3
where
[m] =
T
[M ] = modal mass matrix,
[k] =
T
[K] = modal stiffness matrix,
[c] =
T
[C] = modal damping matrix.
(5)
The free vibration equation can be expressed as
-
2
[m] + i[c] + [k] {u} = 0.
(6)
If damping is neglected, the undamped equation of motion can
be solved from the eigenequation
Ku =
2
M u.
(7)
These undamped eigenvalues and eigenvectors can be used to
form the acceleration FRFs, and can be expressed as a function
of modal damping factor
H
ij
() = -
2
N
k=1
ik
jk
k
- + 2i
k
k
.
(8)
It is to be noted that the Rayleigh damping coefficients can be
related to the modal damping factor, as shown below:
i
=
1
2
a
0
1
+ a
1
i
.
(9)
The disadvantage of Rayleigh damping is that the effects
of the higher modes are usually weighted more than the lower
modes. As only two modes are used at a time for the estimation
of modal damping, the effects of other modes cannot be taken
care of. The present investigation is focused on removing this
difficulty by including the effects of multiple modes through
measured FRFs within a frequency range of interest for a FRP
beam.
The expression for the FRF can be modified to include the
damping coefficients as
H
ij
() = -
2
N
k=1
ik
jk
k
- + i (a
1
2
+ a
0
)
;
(10)
where H
ij
is the acceleration response at point i due to ex-
citation at point j. As it is a common practice to deploy ac-
celerometers for measuring accelerations directly and compute
displacement and velocities as derived quantities as and when
required, all response quantities are expressed in terms of ac-
celerations in this paper. In case of other measurement tech-
niques employing measured displacements as first hand in-
formation, such as in full field measurements using scanning
Laser Doppler Vibrometer types of non-contact devices, the
formulations can be modified to deal with displacements di-
rectly.
Since the order of magnitude of the terms of the damping
matrix is much lower as compared to the stiffness and mass
matrices, it will be efficient to have the stiffness properties up-
dated prior to the updating of the damping parameters. Thus,
a two-stage model updating algorithm is implemented here.
Moreover, the global stiffness properties can be updated more
efficiently with the help of measured natural frequencies and
mode shapes, whereas updating the damping matrix coeffi-
cients a
1
and a
0
requires the information from measured FRFs.
At present, the inertia properties are assumed to be determined
accurately, as this is generally the case in practice.
The objective functions involving the measured and mod-
elled FRFs can be written in a weighted least square sense
E =
q
i=1
w
ii
||H
exp
() - H
nu
()||
2
;
(11)
where w
ii
are the weights and q is the number of FRFs con-
sidered. The sensitivities of these FRFs with respect to the
damping or elastic parameters can be computed using
S
ij
=
H()
r
j
;
(12)
where i = 1 to n and j = 1 to m. Here, the order of the sensi-
tivity matrix is n × m. The linearized first order approximation
of the relationship between changes in measured modal prop-
erties (i.e. frequencies, mode shapes or FRFs) and the changes
in the parameters to be updated can be related through the first
order sensitivity matrix
{f } = [S]{r}.
(13)
In the updating process, changes (r) are made to the initial
guesses of parameters within reasonable bounds, and the finite
element model of the pultruded FRP beam is updated as fol-
lows:
{r}
i+1
= {r}
i
+ {r}
i
.
(14)
The error between the experimental observation and the fi-
nite element modelling is thus minimised through this inverse
sensitivity method. In the present investigation, the param-
eters can be the in-plane elastic material constants, such as
the Young's modulus and the shear modulus, the out-of-plane
shear modulus, and the modal damping coefficients. A Block
Lanczos Algorithm has been implemented for the eigensolu-
tions.
In an inverse problem related to anisotropic materials, mode
sequences need to be properly checked using certain estab-
lished correlation criteria. In the present investigation, Modal
Assurance Criteria (MAC) is used at each step of iteration to
determine how similar or dissimilar the analytical modal vec-
tor is as compared to the experimentally measured modes; a
value close to 1 indicates good correlations:
23
MAC (
nu
,
exp
) =
{
nu
}
T
{
exp
}
2
{
nu
}
T
{
exp
} {
exp
}
T
{
exp
}
. (15)
Here, represents the realised eigenvectors from the mea-
sured complex modes. The analytical and experimental FRFs
are similarly correlated using Signature Assurance Criteria
(SAC),
22
which is basically a global Frequency Response As-
surance Criteria (FRAC):
24
SAC (H
nu
i
, H
exp
i
) =
H
T
exp
i
H
nu
i
2
H
T
exp
i
H
exp
i
H
T
nu
i
H
nu
i
.
(16)
Furthermore, Cross Signature Assurance Criteria (CSAC) is
the correlation function, checking the FRF correlations
22
lo-
4
Figure 1. Flow chart of the model updating algorithm.
cally:
CSAC (
k
) =
H
T
exp
i
(
k
)H
nu
i
(
k
)
2
H
T
exp
i
(
k
)H
exp
i
(
k
)
H
T
nu
i
(
k
)H
nu
i
(
k
)
,
k = 1, 2, . . . , N.
(17)
The amplitude correlations of FRFs are taken care of by Cross
Signature Scale Factor (CSF):
22
CSF (
k
) =
2 H
T
exp
i
(
k
)H
nu
i
(
k
)
H
T
exp
i
(
k
)H
exp
i
(
k
) + H
T
nu
i
(
k
)H
nu
i
(
k
)
,
k = 1, 2, . . . , N.
(18)
First, an isotropic beam is investigated numerically to see
if the damping parameters can be conveniently determined.
The `experimental' FRFs in this simulated study are also de-
termined using the same finite element programming. Subse-
quently, a real experiment is conducted on a Pultruded FRP
beam, both in cantilever and under free boundary conditions.
For the updating process, the initial finite element model com-
putes the FRFs using the undamped frequencies, modal vec-
tors, and initially assumed modal damping factors. Analyti-
cal and experimental FRFs are correlated as explained earlier
to form the objective functions. A first order sensitivity ma-
trix is computed for the selected parameters by a finite differ-
ence approximation of variables. Finally, the inverse sensitiv-
ity method is used to update the stiffness parameters, first us-
ing the modal information, followed by an estimation of modal
damping coefficients using the FRFs. A Bayesian approach is
used to include the variance of the response data. The entire
procedure is explained through a flow chart in Fig. 1.
3. NUMERICALLY SIMULATED EXAMPLE
To check the stability and efficiency of the algorithm de-
scribed above, first a numerically simulated example involving
a rectangular isotropic beam of dimensions 500 mm × 40 mm
and having thickness of 10 mm is considered and is shown in
Fig. 2. A three-noded quadratic beam element (B32)
25
is used
Table 1.
Numerically simulated `experimental' frequencies and assumed
modal damping factors.
Mode
Frequency (Hz)
Modal damping factor (%)
1
24.93
5.0E-2
2
99.28
1.3E-2
3
155.90
1.1E-2
4
435.05
1.25E-2
5
528.57
1.50E-2
6
604.20
0.8E-2
Figure 2. Numerical model of the cantilever beam.
for the finite element modelling of the isotropic beam.
25
The
present investigation deals with the average stiffness properties
and global average damping parameters. A 40-mesh division
along the length was found to be sufficient for convergence of
eigenproperties. The actual existing Young's modulus E and
in-plane Poission's ratio are taken to be 30 GPa and 0.3, re-
spectively. The mass density is assumed as 2012 kgm
-3
. With
the above set of data, the simulated `experimental' modal prop-
erties are computed and presented in Table 1, along with the
assumed modal damping factors.
As explained earlier, the determination of material constants
from the modal data using the inverse sensitivity method is
taken up first. It has been observed that changing the modal
damping coefficients to have a different set of `experimental'
modal data has very little effect on the accuracy of the esti-
mation of these stiffness parameters, and thus is not reported
here. The updated values of the elastic material constants were
used for further updating of the damping coefficients, which
requires the simulated `experimental' FRF data.
It is readily observed that the estimated values of the damp-
ing parameters a
0
and a
1
differ depending upon the modes
considered. The results are shown in Table 2 for a few se-
lected arbitrary combinations of modes using Eq. (9).
26
The
corresponding values of
i
and
i
are taken from Table 1. The
first three sets show the variations of the two estimated damp-
ing parameters due to the incorporation of up to the first five
modes.
While implementing the inverse FRF-based updating algo-
rithm, these values of a
0
and a
1
are chosen as initial values
to see if all trials converge to a unique set of parameters. To
check the robustness of this FRF-based inverse algorithm, two
additional arbitrary sets of values of a
0
and a
1
are also cho-
sen (Trial 4 and Trial 5) that do not immediately correspond
to any combination of modes and may not have any physical
significance.
Figure 3 shows the monotonic convergence curves for both
the parameters, the final values of which are a
0
= 14.95 rad/s
and a
1
= 31.60E-6 s/rad, respectively. The updated mass
proportional damping coefficient converged to a value that is
quite higher, while the stiffness proportional damping coeffi-
cient converged to a somewhat lower value.
A typical set of regenerated FRF curves with different mode
combinations are shown in Fig. 4. It clearly shows that the
most accurate global representation of damping parameters de-
5
Table 2.
Initial values of damping parameters for numerically simulated
example.
Mode considered
Mode considered
for average values
for average values
a
0
a
1
of
1
and
1
of
2
and
2
Trial 1
1
2
2.48
9.55E-6
Trial 2
1, 2
5
3.74
4.33E-5
Trial 3
2
3
2.00
5.87E-5
Trial 4
--
--
40
5.0E-6
Trial 5
--
--
50
1.0E-6
(a) Convergence of a
0
(b) Convergence of a
1
Figure 3. Convergence curves for damping coefficients.
pends upon the participation of modes. If that matches with
the selected modes, then only the SAC value indicating better
correlation between the observation and model will approach
unity. This is difficult to predict in practice, and several tri-
als with various combinations of modes are necessary before
settling with the most appropriate solution.
4. EXPERIMENTAL INVESTIGATION
Equipped with the knowledge gained from the numerically
simulated example, a rectangular FRP composite beam of the
same size is fabricated using the pultrusion process with Wo-
ven Roving (WR) glass fibres and an epoxy matrix. The exact
final average thickness of the beam comes to be 10.12 mm.
The mass density was measured to be exactly 2012 kgm
-3
.
First, the modal testing was carried out with a fixed bound-
ary at one end of the beam, the other end being free. An
(a) Comparison of FRFs before updating
(b) Comparison of FRFs after updating
Figure 4. Comparison of FRFs computed using modal damping to those com-
puted using trial values and updated values of damping parameters.
Figure 5. Measurement grid points of the FRP cantilever beam.
impact hammer fitted with a force transducer (B&K, number
8206-002) was used for exciting the beam at different prede-
fined locations, and the resulting responses were picked up
by an accelerometer (IEPE DeltaTron 4507) at a particular
node. Both the signals were Fourier transformed in a B&K
spectrum analyser, and the FRFs were obtained utilising the
PULSE-LabShop software.
27
The frequencies, modes shapes,
and modal damping factors were extracted from the measured
FRFs using the post-processing software MEScope.
28
Figure 5
shows the position of the accelerometer (point 23) and also the
nodes where forces were imparted through the impact hammer
in turn (33 nodal points altogether). Figure 6 shows the exper-
imental setup for the modal testing. Very heavy steel billets
are used to ensure proper fixity after using a properly bolted
connection at the cantilever end. Accelerometers were placed
near the supports to check whether near zero support mobility
is achieved during testing.
The finite element modelling of the beam was done with
shell element (S8R),
25
implementing an equivalent single layer
6
Figure 6. Experimental modal testing of the FRP cantilever beam.
Figure 7. Experimentally and numerically obtained mode shapes for the FRP
cantilever beam.
theory for layered composites.
29
The parameters selected for
updating are the in-plane stiffness parameters of the beam, out-
of-plane shear modulus, and the damping constants. First, the
material constants are updated, followed by the estimation of
damping coefficients. The initial values to start the iterative
model updating process are selected from standard handbooks
and manufacturer's data. Figure 7 shows the comparison be-
tween properly correlated experimental and numerical mode
shapes.
Frequencies obtained using updated elastic material param-
eters, along with the experimentally measured frequencies, are
shown in Table 3. The experimentally obtained modal damp-
ing factors are also shown.
The final MAC values between the experimental and up-
dated mode shapes indicate an excellent correlation. To have
better insight into the global updating process, the correlation
quantities CSAC and CSF were also computed, and excellent
correlations were achieved, except near some anti-resonant
points.
Table 3. Updated and experimentally observed natural frequencies and damp-
ing factors of the FRP cantilever beam.
Mode
Updated Freq.
Exp. Freq.
Modal Damping
No.
(Hz)
(Hz)
Factors (%)
1
25.63
25.61
15.56
2
160.06
160.85
2.84
3
399.28
399.28
2.43
4
445.54
442.65
1.47
5
865.86
867.96
1.26
Table 4. Experimentally obtained and updated material parameters.
Experimentally obtained
Updated and finally used
Parameters
elastic parameters (GPa)
elastic parameters (GPa)
for damping matrix updating
E
x
33.05
31.66
E
y
31.80
31.80
G
xy
5.73
6.30
G
xz
--
5.37
G
yz
--
5.37
Table 5. Initial values of damping coefficients from experimentally obtained
modal damping factors of the FRP cantilever beam.
Mode considered
Mode considered
for average values
for average values
a
0
a
1
of
1
and
1
of
2
and
2
Trial 1
1
2
7.93
4.71E-5
Trial 2
1, 2
5
17.07
6.42E-6
Trial 3
2
3
7.22
8.24E-5
Trial 4
2
4
8.78
2.19E-5
Trial 5
--
--
60
1.00E-4
Trial 6
--
--
20
2.00E-6
Next, static characterisation tests were carried out using
coupons that were prepared and tested quasi-statically as per
ASTM standard (No.D3039/D3039M),
30
and the results are
presented in Table 4.
The experimentally obtained Poisson's ratio is 0.15. For the
updating of the damping parameters, the initial values are se-
lected again from Eq. (9), in the same way as they were se-
lected in the numerically simulated example. A few such se-
lected sets of damping coefficients are shown in Table 5, along
with two arbitrary values to test the robustness of the algorithm
from distant points in this practical example.
The FRFs using different trial values of damping parameters
are shown in Fig. 8, and it is clear that they still differ from the
experimentally observed values. The regenerated responses
using the updated damping parameters, however, match almost
exactly with the experimentally obtained FRFs, as shown in
Fig. 9. The SAC value also approaches 1, indicating very good
global correlations.
Figure 10 shows comparisons between the experimental
FRFs and the numerically regenerated FRFs that use the ex-
perimentally obtained modal damping factors. The conver-
gence curves of the damping parameters from various initial
values are shown in Fig. 11, and are found to be monotonic
in all cases.
The last updated parameters are found to be
a
0
= 42.25 rad/s and a
1
= 1.10E-5 s/rad, respectively, for this
FRP beam, as shown in the Fig. 11, considering five modes
altogether. The apparent improvement in global response pre-
diction can be attributed to the incorporation of a number of
modes, rather than using only a few selected modes (Eq. (9)).
To test the authenticity of the methodology developed, some
more FRFs are compared that were not used in the updat-
ing process, and the correlations are found to be excellent.
This was observed at most of the anti-resonant points, as well
7
(a) FRFs compared before updating for Trial 1
(b) FRFs compared before updating for Trial 4
Figure 8. Typical comparison of experimental and trial FRFs at two selected
points using different values of viscous damping coefficients.
(as shown in Fig. 12).
Even, the modes beyond the fre-
quency ranges considered also showed improved correlations
(as shown in Fig. 13).
The experimental investigation was then extended to cater
to the free boundary conditions, as well, and a two-step finite
element model updating procedure was performed. The free
boundary conditions were achieved by hanging the beam from
soft rubber threads of sufficient length so that the frequency
of oscillation was much lower as compared to the fundamen-
tal frequency of the free-free beam. The last measured width
and length of the beam were found to be 40 mm and 501 mm,
respectively, whereas the average thickness was measured to
be 10.12 mm. The schematic diagram showing the measure-
ment points, as well as a photograph of modal testing under
free boundary conditions, are shown in Fig. 14.
First the material properties were updated, followed by up-
dating the viscous damping parameters from the FRF data us-
ing the sensitivity-based model updating algorithm.
The updated in-plane Young's moduli (E
x
and E
y
) were
found to be 31.85 GPa and 30.86 GPa, respectively. The in-
plane shear modulus (G
xy
) was updated to a value of 6.31 GPa,
and the out-of-plane shear modulus (G
xz
and G
yz
) were up-
dated to 4.77 GPa and 5.56 GPa, respectively. The Poisson's
ratio was measured to be 0.15. The experimentally obtained
first four frequencies were 165.87 Hz, 453.39 Hz, 800.44 Hz,
and 886.53 Hz, and the modal damping coefficients were mea-
(a) Comparison of FRFs after updating at point 23
due to force at point 14
(b) Comparison of FRFs after updating at point 23
due to force at point 29
Figure 9. Typical comparison of experimental and updated FRFs at two se-
lected points.
sured to be 2.59%, 1.30%, 2.52%, and 1.40%, respectively.
The experimental mode shapes are compared in Fig. 15 to the
finite element mode shapes, indicating very good correlations.
A typical set of FRF curves before and after updating are
also shown in Fig. 16, indicating good correlations in terms of
improved SAC values. The updated damping parameters are
found to be 49.3 rad/s and 5.09E-6 s/rad, which reproduces the
modal damping almost exactly.
5. CONCLUSIONS
A finite element model updating algorithm using measured
frequency response functions has been implemented to esti-
mate proportional damping parameters of a fibre-reinforced
plastic beam with different boundary conditions over a selected
frequency range of interest. It has been observed that the ma-
terial constants need to be updated a-priori before estimating
the damping parameters. The number of frequencies to be in-
cluded is case-specific, and for this example it gives very good
accuracy with only a few modes, with which even the out-of-
range frequency responses were regenerated with acceptable
accuracy. At present, the methodology assumes equivalent vis-
cous damping for all combined effects, such as boundary fric-
tion, etc.
8
(a) Comparison of FRFs after updating at point 23
due to force at point 29
(b) Comparison of FRFs after updating at point 23
due to force at point 14
Figure 10. Typical comparison of experimental and numerically regenerated
(using the modal damping factors) FRFs.
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11
Frequently asked questions
What is the primary objective of this investigation?
The objective of the present investigation is to apply a gradient-based model updating technique to estimate viscous damping parameters along with the stiffness parameters for pultruded FRP beams using measured FRFs.
Why is accurately determining dynamical responses important?
The accurate determination of dynamical responses is important from the viewpoint of safety, serviceability, and operation of any structure.
What factors influence the dynamic responses of a structure?
The geometrical complexities, material property distributions, existing boundary conditions, and applied loading are the key factors that influence the dynamic responses.
What are the challenges in predicting dynamic responses?
Uncertainties in response prediction remain due to imperfect boundary conditions and the presence of damping, which are difficult to deal with.
What does the damping mechanism comprise of?
The damping mechanism may comprise three effects -- material damping resulting from micro-structural behaviour, friction damping resulting from looseness at boundaries, and environmental damping effects, such as interaction with the surrounding fluid.
What is Fibre Reinforced Plastic (FRP) commonly used for?
Fibre Reinforced Plastics (FRP) have long been used in weight-sensitive aerospace, naval, automotive, and high-performance sports applications.
What are the advantages of using FRPs in infrastructures?
Durability, fatigue, and corrosion resistance.
What inverse technique can assess the stiffness and damping properties?
A reliable non-destructive inverse technique.
What makes uncertainties increase in damping parameters?
Uncertainties in damping parameters will further increase, as the mechanism may include one or more effects which were initially absent. For example, loosening of joints and supports may result in increased friction component of damping as time passes.
Why are there limited attempts to estimate inversely the damping parameters for FRP structures?
The main reason is that for small damping, the resonant frequencies and mode shapes change very little with damping coefficients, but the responses change drastically, especially near resonances.
What is the most popular forward damping model?
The most popular forward damping model is due to Rayleigh, in which the damping matrix is assumed as a weighted linear combination of the mass and stiffness matrix.
What does a successful model rely on?
A model's ability to replicate the actual observable responses over a frequency range of interests.
Why is the gradient-based model updating technique used?
To estimate viscous damping parameters along with the stiffness parameters for pultruded FRP beams using measured FRFs.
What is the process for identification of damping of FRP beams?
The complete process for identification of damping of FRP beams includes a-priori estimate of stiffness parameters using measured modal and FRF data, converged finite element modelling, correlations between them, and finally updating the global proportional damping parameters using the gradient-based inverse sensitivity technique in a nonlinear least square sense.
What is Modal Assurance Criteria (MAC) used for?
To determine how similar or dissimilar the analytical modal vector is as compared to the experimentally measured modes; a value close to 1 indicates good correlations.
What are Signature Assurance Criteria (SAC) and Cross Signature Assurance Criteria (CSAC)?
SAC is used to correlate analytical and experimental FRFs, while CSAC is a correlation function used to check the FRF correlations locally.
What does the inverse detection of damping using an iterative procedure demand?
Proper forward simulation of the damped responses of the FRP structures in the iterative loop.
What is Cross Signature Scale Factor (CSF)?
Cross Signature Scale Factor takes care of the amplitude correlations of FRFs.
What equation of motion is used?
The equation of motion of a multiple-degrees-of-freedom system in a discretized form is: M x''(t) + C x'(t) + Kx(t) = f(t).
What are the sensitivities of these FRFs computed?
Using Sij = H(ω)/δrj.
What is the relation between changes in measured modal properties and changes in parameters to be updated?
{δf} = [S]{δr}.
How is the stiffness more efficiently updated?
With the help of measured natural frequencies and mode shapes.
What is required to update damping matrix coefficients?
The information from measured FRFs.
What is the two-stage updating algorithm?
1 - Stiffness properties are updated prior to the updating of the damping parameters
2 - a-priori estimate of stiffness and damping is achieved from measure FRF data
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- Subhajit Mondal (Autor), Sushanta Chakraborty (Autor), 2014, An Inverse Approach for the Determination of a Viscous Damping Model of Fibre Reinforced Plastic Beams using Finite Element Model Updating, Múnich, GRIN Verlag, https://www.grin.com/document/318609
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