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, bartosz.siedziako@ntnu.no

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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