Identification of Rational Functions with a forced vibration technique using random motion histories.

Transcription

1 Identification of Rational Functions with a forced vibration technique using random motion histories. Bartosz Siedziako Department of Structural Engineering, Norwegian University of Science and echnology, Richard Birkelands vei 1A, 7491 rondheim Norway, bartosz.siedziako@ntnu.no Ole Øiseth Department of Structural Engineering, Norwegian University of Science and echnology, Richard Birkelands vei 1A, 7491 rondheim Norway ABSRAC: Rational Functions are used to describe the self-excited forces acting on the bridge deck in the time domain. hey 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. he approach presented herein enables the extraction of Rational Function Coefficients by testing the section model at only one wind speed. his 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. INRODUCION Slender structures such as suspension and cable-stayed bridges are especially vulnerable to wind-induced phenomena, namely flutter, buffeting and galloping. Scanlan and omko (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. he 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. hey 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. herefore, 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. he 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). his 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

2 (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. EXPERIMENAL SEUP 2.1 Forced vibration rig he forced vibration setup developed at the Norwegian University of Science and echnology has been used in this study (Siedziako et al., 2017). his 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. he 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. wo high-sensitivity load cells measure 3 force and 3 moment components acting on the section model during the experiments. he 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 NNU (Siedziako et al. 2017). he 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 he wind tunnel tests were conducted in the largest wind tunnel in the Fluid Mechanics Laboratory at NNU. It is a closed loop wind tunnel with a test section 11 m long, 2 m height and 2.7 m wide with a

3 maximum speed of 30 m/s. emperature 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 he bridge deck of the currently longest suspension bridge in Norway, Hardanger Bridge, was used in this study. he geometric shape of the bridge deck allows it to be considered as a perfect example of a streamlined section. he cross-sectional dimensions of the model are shown in Fig. 2 together with the coordinate system applied. hanks 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. he 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. he cross-sectional dimensions of the bridge deck used in this study. 3. IDENIFICAION ALGORIHM 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: qˆx rˆ x / B 1 2 ˆ V B q ˆ / B z Q r 2 z qˆ 0 0 B ˆ θ rθ (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 omko (1971), Eq. (1) presents a linear relation between aeroelastic forces (q x drag, q z lift, q θ pitch) and the horizontal (r x), vertical (r z) and torsional vibrations (r θ) of the bridge deck. he matrix Q of Rational Functions is the transfer function in the Laplace domain given by:

4 A A p A A p A A p p p p p p p A A p A A p A A p p p p 23 Q A A p A A p A A p (2) Here, A 0 and A 1 and F are, respectively, the stiffness, damping and lag matrices, all of order 3x3 that contain unknown RFCs. he 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. he 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: V 1 2 V B qx q x x V B B 2 ψ r ψ r ψ r B V V 1 2 V B qz q V B z z B 2 ψ r ψ r ψ r B V V V B qθ q V B θ B 2 ψ r ψ r ψ r B V (3) Here, 1x3 size vectors ψ i i=1,2 9 contain the unknown RFCs; r is the vibration matrix consisting of horizontal vertical and torsional vibrations r=[r x/b r y/b r θ] ; andr andr are, respectively, the first and second derivatives of the displacements. After a slight modification, Eq. (3) can rewritten into the following expression: where matrices A n and C n are given by Eq. (5): AC n n q n n{x, z, } (4) 3 3 ψ ψ ψ V r 0.5V r 2 2 ψ ψ ψ V Br 0.5V Br x z θ x 2 z 2 θ ψ ψ ψ 0.5VB 0.5VB r r A = A = A = C C C V B V B r 3 0.5VB r -x - z - qxv / B q zv / B qθv /B o find matrices A n that contain RFCs, an algorithm that minimizes the sum of squares can be applied: 1 {x,, } An q ncn CnCn n z (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 MOION HISORIES he 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 r (5)

5 matrix of the response S r (ω). o 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. he amplitudes of the spectra S r (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. he time series for the degree of freedom m {x, z, θ} were obtained using Eq. (7): m N xm ( t) 2 Re Lml ( k )exp i( kt lk ) (7) l1 k1 where L ml (ω k) denotes the elements of the lower triangular matrix obtained by factorizing the cross-spectral density matrix according to the relation given in Eq. (8). S L L (8) * r ( k ) ( k ) ( k ) 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. he 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. EXPERIMENAL RESULS o compare the results obtained in this section with Rational Functions, aerodynamic derivatives of the Hardanger Bridge section model are needed. he aerodynamic derivatives identified in a standard forced vibration procedure with that section have been presented in (Siedziako et al. 2016, 2017). hose 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. he same methodology has been applied herein, considering tests with random motion histories. he duration of each test was

6 taken to be 100 s. ests have been performed at three wind speeds, V=4, 8 and 10 m/s. o 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). he 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. able 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. able 1. Correlation coefficients and standard deviations of measured (σ M) and predicted (σ P) self-excited forces. Wind speed ρ xy Drag Lift Pitch σ M [N/m] σ P [N/m] ρ xy σ M [N/m] σ P [N/m] ρ xy σ M [Nm/m] σ P [Nm/m] V=4 m/s V=8 m/s V=10 m/s 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. his 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

7 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 * * K (P1 i P 4 ) K (P5 i P 6 ) K (P2 i P 3 ) 2 * * 2 * * 2 * * Q K ( H5 i H6 ) K ( H1 i H 4 ) K ( H 2 i H3 ) (9) 2 * * 2 * * 2 * * K ( A5 i A6 ) K ( A1 i A4 ) K ( A2 i A3 ) 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.

8 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 A 1*, A 2*, A 3*, H 3*, and also H 2 * 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. he 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. herefore, 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

9 presented herein strongly suggest that it is, there should certainly be further investigations to assess whether this assumption is well founded. ACKNOWLEDGEMENS his research was conducted with financial support from the Norwegian Public Roads Administration. he authors gratefully acknowledge this support. REFERENCES Aas-Jakobsen K, Strømmen E (2001) ime Domain Buffeting Response Calculations of Slender Structures. Journal of Wind Engineering and Industrial Aerodynamics 89(5): Cao B, Sarkar P P (2010) Identification of Rational Functions by Forced Vibration Method for ime-domain Analysis of Flexible Structures. In: Proceedings of he Fifth International Symposium on Computational Wind Engineering. Chapel Hill Cao B, Sarkar P P (2012) Identification of Rational Functions Using wo-degree-of-freedom Model by Forced Vibration Method. Engineering Structures 43:21 30 Chen X, Matsumoto M, Kareem A (2000) Aerodynamic Coupling Effects on Flutter and Buffeting of Bridges. Journal of Engineering Mechanics 126:17 26 Chowdhury A G (2004) Identification of Frequency Domain and ime Domain Aeroelastic Parameters for Flutter Analysis of Flexible Structures, Ph.D hesis, Iowa State University, USA, 778 Chowdhury A G, Sarkar P P (2005) Experimental Identification of Rational Function Coefficients for ime-domain Flutter Analysis. Engineering Structures 27: Høgsberg J R, Krabbenhøft J, Krenk S (2000) State Space Representation of Bridge Deck Aeroelasticity. In: Proceedings of the 13th Nordic Seminar on Computational Mechanics, Oslo, pp Karpel M (1981) Design for Active and Passive Flutter Suppression and Gust Alleviation. NASA contractor report No Mishra S S, Kumar K, Krishna P (2008) Multimode Flutter of Long-Span Cable-Stayed Bridge Based on 18 Experimental Aeroelastic Derivatives. Journal of Wind Engineering and Industrial Aerodynamics 96(1): Neuhaus Ch, Mikkelsen O, Bogunovic Jakobsen J, Höffer R, Zahlten W (2009) ime Domain Representations of Unsteady Aeroelastic Wind Forces by Rational Function Approximations. In: EACWE 5 Florence Roger K L (1977) Airplane Math Modeling and Active Aeroelastic Control Design[C]. AGARD-CP-228 Salvatori L, Borri C (2007) Frequency- and ime-domain Methods for the Numerical Modeling of Full-Bridge Aeroelasticity. Computers and Structures 85(11 14): Scanlan R H, and omko J J (1971) Airfoil and Bride Deck Flutter Derivatives. Journal of Engineering Mechanics Division 97(6): Siedziako B, Øiseth O, Rønnquist A (2017) An Enhanced Forced Vibration Rig for Wind unnel esting of Bridge Deck Section Models in Arbitrary Motion. Journal of Wind Engineering and Industrial Aerodynamics 164: Siedziako B, Øiseth O, Rønnquist A (2016) A New Setup for Section Model ests of Bridge Decks. In: Proceedings of 12th UK Conference on Wind Engineering, Nottingham Xu F Y, Wu, Ying X Y, Kareem A (2016) Higher-Order Self-Excited Drag Forces on Bridge Decks. Journal of Engineering Mechanics 142(3):1 11 Øiseth O, Rönnquist A, Sigbjörnsson R (2011) ime Domain Modeling of Self-Excited Aerodynamic Forces for Cable-Supported Bridges: A Comparative Study. Computers and Structures 89(13 14): Øiseth O, Rönnquist A, Sigbjörnsson R (2012) Finite Element Formulation of the Self-Excited Forces for ime-domain Assessment of Wind-Induced Dynamic Response and Flutter Stability Limit of Cable-Supported Bridges. Finite Elements in Analysis and Design 50:173 83