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Identification of Rational Functions with a forced vibration technique
using random motion histories.
Bartosz Siedziako Department of Structural Engineering, Norwegian University of Science and Technology, Richard Birkelands vei 1A, 7491
Trondheim Norway, [email protected]
Ole Øiseth Department of Structural Engineering, Norwegian University of Science and Technology, Richard Birkelands vei 1A, 7491
Trondheim Norway
ABSTRACT:
Rational Functions are used to describe the self-excited forces acting on the bridge deck in the time domain.
They can be identified indirectly based on aerodynamic derivatives or directly with the free (E2RFC method) or
forced vibration technique, which can significantly decrease the testing time. The approach presented herein
enables the extraction of Rational Function Coefficients by testing the section model at only one wind speed.
This aim is achieved by increased complexity of the forced motion compared to the previous tests, which made it
possible to test a wider range of reduced velocities by adjusting the motion frequency. In this study, motion
histories generated from the assumed flat spectra are used. Wind tunnel tests on a streamlined section model
utilizing simultaneous vertical, horizontal and torsional vibrations were performed to extract Rational Function
Coefficients associated with 3-degree-of-freedom motion. Restrictions and improvements arising from the
proposed methodology are described.
Keywords: Rational Functions; Forced Vibration; Section Model; Arbitrary Motion; Bridge Aeroelasticity.
1. INTRODUCTION
Slender structures such as suspension and cable-stayed bridges are especially vulnerable to
wind-induced phenomena, namely flutter, buffeting and galloping. Scanlan and Tomko (1971)
introduced aerodynamic derivatives (ADs) that characterize the aerodynamic performance of the
bridge deck and enable detailed analysis of the bridge’s in-wind behavior in the frequency domain.
The aerodynamic derivatives that define self-excited forces are most commonly derived
experimentally in a series of wind tunnel tests with a section model of the bridge deck using the free or
forced vibration technique. They can be identified at discrete reduced velocities often within a limited
range, depending on the frequencies and velocities tested during the experiments.
Current technological and engineering advances have made it possible to build increasingly slender
bridges with very light road decks, leading to the construction of possibly highly nonlinear structures.
Moreover, the lower damping of the structure due to the reduced mass emphasizes the significance of
aerodynamic damping. Therefore, time-domain flutter and buffeting analyses, which can incorporate
structural and aerodynamic nonlinearities, have become more common in recent years (Salvatori and
Borri, 2007; Øiseth et al., 2011). Formulated in the Laplace domain by Roger (1977), the Rational
Function Approximation (LS-RFA) using least squares enabled the time-domain modeling of the
frequency dependent self-excited forces. Later, (Karpel, 1981) introduced the Minimum State Rational
Function Approximation (MS-RFA), which improved the accuracy and decreased the computational
time compared to LS-RFA. The main objective of these RFA formulations is to identify the Rational
Function Coefficients (RFCs) that define the motion to self-excited forces continuous transfer
functions. However, this approximation involves experimentally obtained aerodynamic derivatives in
the process of linear and nonlinear optimizations (Neuhaus et al., 2009). This motivated other
researchers to find a more direct method to obtain RFCs from wind-tunnel measurements that would
make it possible to skip the process of extracting aerodynamic derivatives. Chowdhury and Sarkar
(2005) proposed a method to directly extract the RFCs from free vibration tests, while Cao and Sarkar
(2012), to overcome some limitations of the free vibration technique, developed a similar algorithm
for the forced vibration testing technique. In both methods, the RFCs can be extracted directly from
time series recorded during wind tunnel experiments at only a few wind velocities (a minimum of two
wind speeds), which can significantly decrease testing time compared to the standard approach with
aerodynamic derivatives. However, in the method proposed by Cao and Sarkar (2012), simultaneous
pitching and heaving harmonic oscillations of the section model were considered. In this study, a more
general, three-degree-of-freedom random motion generated from flat motion spectra is used to identify
RFCs. It is shown that through this approach, a bridge deck section model needs to be tested at only
one wind speed to extract the full set of RFCs.
2. EXPERIMENTAL SETUP
2.1 Forced vibration rig
The forced vibration setup developed at the Norwegian University of Science and Technology has
been used in this study (Siedziako et al., 2017). This setup was especially designed to be capable of
forcing arbitrary motion histories of the bridge deck section model in heaving, swaying and torsional
directions simultaneously. Fig. 1 shows the segment of the wind tunnel with the main construction of
the setup. The section model of the bridge is attached between the two actuators placed outside on
both sides of the wind tunnel. Inside each of the actuators reside two ball screws for the vertical and
horizontal motion and a planetary gear for the torsional motion. Two high-sensitivity load cells
measure 3 force and 3 moment components acting on the section model during the experiments. The
actuators are supported by the steel frame outside the wind tunnel, while the load cells are mounted
between the section model and actuators in the centers of two circular holes made in the wind tunnel
walls.
Figure 1. Experimental forced vibration setup at NTNU (Siedziako et al. 2017).
The described setup makes it possible to move the section model arbitrarily according to the uploaded
motion histories. Data transfer with the time series of displacement is managed using the LabVIEW
program, which is also responsible for triggering motion, monitoring, controlling algorithms and
acquiring data. In this study, the uploaded motion time series were generated with a time step of 1 ms,
while a sampling rate of 250 Hz was set for the data acquisition.
2.2 Wind tunnel
The wind tunnel tests were conducted in the largest wind tunnel in the Fluid Mechanics Laboratory at
NTNU. It is a closed loop wind tunnel with a test section 11 m long, 2 m height and 2.7 m wide with a
maximum speed of 30 m/s. Temperature inside the wind tunnel was measured with a thermocouple to
account for changes in the air density, while to measure the air velocity static, a pitot probe was placed
6.10 m in front of the section model. All the tests presented in this paper were performed in a smooth
flow.
2.3 Bridge deck section model
The bridge deck of the currently longest suspension bridge in Norway, Hardanger Bridge, was used in
this study. The geometric shape of the bridge deck allows it to be considered as a perfect example of a
streamlined section. The cross-sectional dimensions of the model are shown in Fig. 2 together with the
coordinate system applied. Thanks to additional holes and very light filling material, the model is very
light. With a length of L=2.68 m, it weighs only 5.45 kg. The high aspect ratio L/B=7.32 and the fact
that the model is only 3 cm shorter than the width of the wind tunnel, eliminated the need to use
additional end plates.
Figure 2. The cross-sectional dimensions of the bridge deck used in this study.
3. IDENTIFICATION ALGORITHM
An algorithm used in this study, adapted to the forced vibration technique, has been proposed by Cao
and Sarkar (2012) and is based on the previous work by Roger (1977) and Karpel (1981) in the field of
aeronautics; therefore, the authors refer to those publications for more details on its derivation.
Following Roger (1977), the self-excited forces in the 3-DoF system can be expressed in the Laplace
domain as follows:
2
ˆ ˆ /
ˆ ˆ /
ˆ ˆ
1 0 01
0 1 02
0 0
B
BV B
B
x x
z z
θ θ
q r
q Q r
q r
(1)
Here, ρ is the air density; V denotes the mean wind velocity; B is the bridge deck width, and ‘^’
indicates that the variable is in the Laplace domain. Similarly to the description given by Scanlan and
Tomko (1971), Eq. (1) presents a linear relation between aeroelastic forces (qx – drag, qz – lift, qθ –
pitch) and the horizontal (rx), vertical (rz) and torsional vibrations (rθ) of the bridge deck. The matrix Q
of Rational Functions is the transfer function in the Laplace domain given by:
1311 12
0 1 0 1 0 111 11 12 12 13 13
2321 22
0 1 0 1 0 121 21 22 22 23 23
31 32 33
0 1 0 1 0 131 31 32 32 33 33
F pF p F pA A p A A p A A p
p p p
F pF p F pA A p A A p A A p
p p p
F p F p F pA A p A A p A A p
p p p
Q (2)
Here, A0 and A1 and F are, respectively, the stiffness, damping and lag matrices, all of order 3x3 that
contain unknown RFCs. The value λ denotes an unknown lag term, while p=iK represents the
dimensionless Laplace variable, where K= Bω/V is the reduced frequency, and ω is the circular
frequency of motion. The expression approximating the Rational Function in Eq. (2) can be further
extended by including additional lag terms and lag matrices, but previous studies have shown that the
Rational Function Approximation with one lag term as presented herein is sufficient (Cao and Sarkar,
2010, 2012; Chowdhury, 2004; Chowdhury and Sarkar, 2005; Neuhaus et al., 2009) in the case of
bridge decks. By multiplying Eq. (2) by p+λ and applying the inverse Laplace transform, the following
time-domain equations for the self-excited drag, lift and pitching moment can be obtained:
2
2
2
1 2 3
4 5 6
2
7 8 9
1
2
1
2
1
2
x
z
V V BV B
B B V
V V BV B
B B V
V V BV B
B B V
x x
z z
θ θ
q q ψ r ψ r ψ r
q q ψ r ψ r ψ r
q q ψ r ψ r ψ r
(3)
Here, 1x3 size vectors ψi i=1,2…9 contain the unknown RFCs; r is the vibration matrix consisting of
horizontal vertical and torsional vibrations r=[rx/B ry/B rθ]T; andr andr are, respectively, the first and
second derivatives of the displacements. After a slight modification, Eq. (3) can rewritten into the
following expression:
{x, , }n n n n z A C q (4)
where matrices An and Cn are given by Eq. (5):
33 3
2 22 2
32 2
0.50.5 0.5
0.50.5 0.5
0.50.5 0.5
- -- / B / Bx z
V BV V
V BV B V B
VBVB VB
V V
T TT
1 74
2 85
x z θ x z θ
3 96
θx z
ψ ψψ rr r
ψ ψψ rr rA = A = A = C C C
ψ ψψ rr r
qq q / BV
(5)
To find matrices An that contain RFCs, an algorithm that minimizes the sum of squares can be applied:
1
{x, , }T T
n n n n n n z
A C C Cq (6)
In this study, the derivatives of the drag, lift, pitching moment and displacements were obtained by
applying the finite difference algorithm to the recorded time histories. Since the motion considered
herein is a combination of horizontal, vertical and torsional vibrations, all the RFCs can be identified
using the data from a single forced vibration test at a particular wind speed.
4. RANDOM MOTION HISTORIES
The random motion histories used in this study were generated by Monte Carlo simulations
(Aas-Jakobsen and Strømmen, 2001; Øiseth et al., 2011) from an assumed cross-spectral density
matrix of the response Sr (ω). To achieve the maximum possible randomness of the time series and
prove that the experimental setup can induce arbitrary motion of the section model, flat spectra in the
range of 0.3 to 2.5 Hz have been used to generate histories of displacements for later upload to the
actuators. The amplitudes of the spectra Sr (ω) have been scaled to obtain standard deviations of the
horizontal, vertical and torsional responses, respectively, 6.5 mm, 6.5 mm and 1.4°. The time series for
the degree of freedom m {x, z, θ} were obtained using Eq. (7):
1 1
( ) 2 Re ( )exp ( )m N
m ml k k lkl k
tx t L i
(7)
where Lml (ωk) denotes the elements of the lower triangular matrix obtained by factorizing the
cross-spectral density matrix according to the relation given in Eq. (8).
*( ) ( ) ( )k k k rS L L (8)
Fig. 3 presents part of the time series induced on the section model during wind tunnel testing,
generated using Eq. (7). It can be seen that the created motion histories are very chaotic and simulate a
white-noise stochastic process well. The experimental rig used in this study has been designed to
address much larger amplitudes and motion frequencies than used herein, and therefore the motion
during experiments was very smooth, and the actuators perfectly followed the uploaded motion
history.
Figure 3. Part of the time series of the section model used for wind tunnel testing generated from assumed flat
spectra in the range of 0.3 to 2.5 Hz.
5. EXPERIMENTAL RESULTS
To compare the results obtained in this section with Rational Functions, aerodynamic derivatives of
the Hardanger Bridge section model are needed. The aerodynamic derivatives identified in a standard
forced vibration procedure with that section have been presented in (Siedziako et al. 2016, 2017).
Those two references provide more information about the amplitudes, frequencies and wind speeds
tested and also describe the methodology used for extracting self-excited forces, which requires
measuring forces for the same motion in still-air and in-wind conditions. The same methodology has
been applied herein, considering tests with random motion histories. The duration of each test was
taken to be 100 s. Tests have been performed at three wind speeds, V=4, 8 and 10 m/s.
To evaluate the identification algorithm described in chapter 3 and determine the accuracy of the fit,
the extracted RFCs can be used to predict the self-excited forces. Cao and Sarkar (2012) used for this
task an expression that contains a convolution integral; however, it has been shown that it can be
conveniently replaced with a state-space formulation (Chen et al., 2000; Høgsberg et al., 2000; Mishra
et al., 2008). The second approach has been used in this study – see (Øiseth et al., 2012) for more
details. Example time series of recorded and predicted self-excited forces are shown in Fig. 4. Forces
have been calculated based on the Rational Functions identified using the data from the test conducted
at V=4 m/s. Table 1 presents collected information about the correlation coefficients between
measured and predicted forces together with their standard deviations.
Figure 4. Measured forces vs. forces predicted with RFCs induced during execution of the random motion at
V=4 m/s.
Table 1. Correlation coefficients and standard deviations of measured (σM) and predicted (σP) self-excited forces.
Wind speed
Drag Lift Pitch
ρxy σM
[N/m]
σP
[N/m] ρxy
σM
[N/m]
σP
[N/m] ρxy
σM
[Nm/m]
σP
[Nm/m]
V=4 m/s 0.514 0.087 0.066 0.981 0.395 0.381 0.998 0.033 0.033
V=8 m/s 0.300 0.082 0.043 0.998 1.670 1.550 0.999 0.131 0.131
V=10 m/s 0.531 0.115 0.080 0.996 2.77 2.526 0.998 0.209 0.208
It can be seen that a perfect match between measured and predicted with RFC self-excited forces has
been achieved for the pitch and lift. However, in the case of the self-excited drag, the calculated
correlation between the measured and predicted forces is significantly lower than for the lift and pitch.
Recent studies by Xu et al. (2016) have shown that the self-excited drag is prone to higher-order
contributions that cannot be captured by linear load models and can be especially large when
considering streamlined sections, as in this study. This finding agrees with the results presented herein,
as the drag force is clearly underestimated in all tests when comparing the standard deviations of the
measured and predicted drag.
Knowing that the matrix of Rational Functions Q can also be described by Eq. (9), the relations
between particular aerodynamic derivatives and RFCs can be established to allow the direct
comparison of the results obtained here with the ones presented in (Siedziako et al. 2017).
2 * * 2 * * 2 * *
1 4 5 6 2 3
2 * * 2 * * 2 * *
5 6 1 4 2 3
2 * * 2 * * 2 * *
5 6 1 4 2 3
(P i P ) (P i P ) (P i P )
( i ) ( i ) ( i )
( i ) ( i ) ( i )
K K K
K H H K H H K H H
K A A K A A K A A
Q (9)
Figure 5. Aerodynamic derivatives related to velocities or angular velocities. Comparison of experimentally
obtained ADs (Siedziako et al. 2017) and ADs extracted from Rational Functions identified at one wind speed.
Figure 6. Aerodynamic derivatives related to displacements or rotation. Comparison of experimentally obtained
ADs (Siedziako et al. 2017) and ADs extracted from Rational Functions identified at one wind speed.
Fig. 5 and 6 compare all 18 ADs obtained herein from Rational Functions with the ones identified in
the forced vibration tests using the standard procedure. It can be seen that the ADs match very well,
especially the most important ADs, namely A1*, A2
*, A3*, H3
*, and also H2* as the torsional motion is
responsible for most of the induced self-excited forces. However, the ADs found at the lower reduced
velocities seems to correspond better to the original ones than at the higher reduced velocities, which
is especially visible in the case of the ADs extracted from RFCs identified at V=4 m/s.
6. CONCLUSION
In this paper, a recently developed algorithm for the extraction of Rational Function Coefficients has
been used for the first time with a non-harmonic motion pattern. Motion that simultaneously involves
horizontal, vertical and torsional vibrations generated from flat motion spectra has been used to
measure the self-excited forces induced on the streamlined section model. Preliminary studies showed
that the full set of Rational Function Coefficients can be identified from a single test considering only
one wind speed. The identified Rational Function Coefficients provided an excellent fit to time series
of recorded self-excited lift and pitching moment, demonstrating the high performance of the
algorithm used in this study. However, some discrepancies that require separate studies were observed
in the drag force.
In the experiments performed, aeroelastic forces related to the torsional motion dominated the
measured self-excited drag, lift and pitching moment. Moreover, the motion type used in this study
tends to favor the extraction of ADs at the lower reduced velocities. Therefore, suitable design of the
motion histories for the wind tunnel testing might be of key importance for the method described in
the future. Additionally, testing section models of the bridge decks considering motions that resemble
actual bridge motion would presumably eliminate this problem since the Rational Function
Coefficients would be optimized in the range of reduced velocities that correspond to the bridge’s
natural frequencies. It must be emphasized that the method presented herein assumes that the principle
of superposition between the motion and induced self-excited forces is valid. Although the results
presented herein strongly suggest that it is, there should certainly be further investigations to assess
whether this assumption is well founded.
ACKNOWLEDGEMENTS
This research was conducted with financial support from the Norwegian Public Roads Administration. The
authors gratefully acknowledge this support.
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