The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 Excitation mechanism of

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1 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 xcitation mechanism of rain-wind induced vibration of cables: unsteady and nonlinear aspects Teng Wu a, Ahsan Kareem a, Shouying Li b a Nathaz Modeling Laboratory, University of Notre Dame, Notre Dame, IN, USA b Wind ngineering Research Center, Hunan University, Changsha, China ABSTRACT: Two aerodynamic effects, i.e., unsteady and hysteretic behavior on the water rivulet along a stay cable are considered in the proposed analytical models for rain-wind induced vibration (RWIV). Neither the unsteadiness nor hysteresis can be considered using conventional quasi-steady (QS) model. The new analysis framework is also convenient to effectively take into account the turbulence effects in RWIV. Besides, the higher-order spectrum is utilized to detect the potential nonlinear interactions in RWIV. Numerical examples for unsteady and hysteretic considerations are illustrated based on the wind-tunnel data. The results are compared with the QS results. KYWORDS: Rain-wind induced vibration; Unsteadiness; Nonlinearity; Higher-order spectrum 1 INTRODUCTION Among various wind-induced vibrations of cables of cable-stayed bridges (such as ice galloping, Karman vortex shedding induced vibration, wake induced vibration, vortex-induced oscillation at high wind velocity, rain-wind induced vibration (RWIV) etc.), RWIV is significant due to its large amplitude, frequent occurrence and low critical wind speed. Since the first observation of this phenomenon during the construction of a cable-stayed bridge (Hikami and Shiraishi 1988), focus has been on this topic to better understand and model these vibrations. Though considerable research effort has been made, no consensus exists regarding the underlying mechanism due in part to the extremely complex nature of the gas-liquid-solid interaction. On the other hand, agreement exists among researchers regarding the formation of rivulet on the cable surface due to the rain and the attendant axial flow formation (Matsumoto et al. 2005) as two main factors responsible for RWIV. Predictive models have been established based on the quasi-steady (QS) assumption (e.g., Gu et al. 2009), where the rivulet formation on the cable is regarded as a dominant excitation factor. The QS assumption obviously leaves room for improvement due to extremely complicated flow field around the rivulet. For example, observation in wind tunnel has pointed out that the rain-wind induced vibrations have the self-excited characteristics (Verwiebe and Ruscheweyh 1998). In this paper, the unsteady effect is considered with a scheme parallel to Scanlan s analysis framework of self-excited forces on bridge decks (Scanlan and Tomko 1971). The aerodynamic coefficients in the proposed unsteady model are first identified in the wind tunnel based on the measurement of the oscillatory pressure. On the other hand, RWIV is well recognized as a nonlinear interaction between the solid, liquid and air. However, neither the QS model, which is able to simulate static nonlinearity, nor the developed unsteady model, which is able to simulate the linear fluid memory effect, can take into account the hysteretic (dynamic) nonlinearity with the higher-order memory effect in the RWIV. As a result, a new analytical model based on the Moore-Penrose pseudoinverse identification scheme is developed to capture the nonlinear hysteretic aerodynamics of RWIV in this paper. A detailed discussion is given in the end of this paper to investigate the excitation mechanism of RWIV. 654

2 2 ANALYTICAL MODLS This paper focuses on the analysis of the RWIV in plane, the consideration of the out-of-plane coupling effect is referred to the research work by Li et al. (2011). 2.1 Conventional Quasi-Steady Model The basic assumption of the QS theory is that the wind velocity should be high enough where the unsteady aerodynamic forces acting on a oscillating structure could be modeled by utilizing the steady-state situation without the fluid memory consideration. Actually, an underlying assumption is that it is convenient to define the steady-state situation of the aerodynamic system (Van Oudheusden 1995). The main reason that the simulation fidelity is usually degraded as the QS model is applied to the torsional oscillation case is bacause it is intractable to define a steadystate situation for the torsional case as each point of the cross section has a different local relative angle of attack. Though it is never explicitly stated in the literature, the basis of the applicability of the QS assumption in the RWIV, where the torsional motion is critical, is that: (1) for the cable only one local relative angle of attack is sufficient to define the steady state of the cable since the rotational motion is uniquely determined by the rivulet position and (2) for the rivulet the rotational center is far from the geometric center where the situation approaches the translational case. The QS based scheme recently developed treats the rivulet-cable system as a coupled one. A two dimensional model is applied to simulate the motions of cable and rivulet based on QS assumption. The coupled equations of motion are given as (Gu et al. 2009) 2 y2 y y y y y F y M (1a) D D m F0 sign( ) c r Fr my cos( ) mg cos( )cos( ) (1b) 2 2 where M is the mass per unit length of cable; y is the vertical displacement of cable; and y and y are circular frequency and damping ratio of the cable, respectively; m is the mass per unit length of upper rivulet; is the position of the rivulet; c r and F 0 are the linear damping coefficient and the Coulomb damping force between the rivulet and cable surface; and is the cable inclined angle; the aerostatic forces F y and F r are obtained based on QS assumption, which could be represented as 1 2 Fy DUrelCL( )cos( ) CD( )sin( ) (2a) Fr RUrelcl( )cos( ) cd( )sin( ) (2b) 2 where D and R are the characteristic sizes of the cable and upper rivulet, respectively; is the air density; C L and C D are lift and drag force coefficients of the cable, respectively; c l and c d are lift and drag force coefficients of the upper rivulet; represents the angle between relative wind velocity and x axis; and ' the angle between relative wind velocity and position of upper rivulet. 2.2 Unsteady model of RWIV The aerodynamic forces on the cable may be modeled well based on QS assumption since the RWIV usually happens at a high reduced wind velocity. However, the QS assumption is not appropriate for the upper rivulet since the flow around it usually becomes very unsteady, which could be observed form the intense changes of the pressure coefficients around the rivulet with 655

3 1 A a b B c C d D e 6 f F 7 G g h 8 H 9 I 10J i j k l m R 18 s 19 S t T 20 u U 21 v 22 V w W 23 x 24X y Y 25 z 26Z aa AA 27 ab 28 AB ac AC 29 ad 30 AD ae A 31 af 32 AF ag 33 ah aq 43 AP 42 ANAOap ao 41 an AQ as AS 45 AR ar 44 AT 46 at AU au 47 awaw 49 AV av ba AZ53 BA az ayay AX ax 1 A a 2 B b 3 C c 4 D d 5 e F f G g 8 H h 9 I i 10 J j The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 various locations, as shown in Fig. 1, where the data is obtained from the wind tunnel tests to be discussed in the following section. As indicated in the figure of wind pressure coefficients on cable, in the range of 25 o to 58 o for the angles of attack (also rivulet position) there is large difference between the pressures of the points before and after the rivulet, which may indicate the flow separates around the rivulet. After this range, the difference is gradually decreasing with respect to the increasing of angle of attack, which indicates the flow separates before the rivulet. The similar observation is also reported in recent literature (Gu et al. 2009). The wind pressure coefficient on rivulet further consolidates this observation. As indicated in the figure of wind pressure coefficients on rivulet, in the range of 25 o to 58 o for the angles of attack (also rivulet position) there is large difference between the pressures of the two portions of points on rivulet which are on windward side and on leeward side. The CFD results obtained from Direct Numerical Simulation (DNS) also show the similar behavior of flow around the rivulet (Li and Gu 2006). As indicated in the CFD results, the flow separates at the point of rivulet for both large and small angles of attack. However, when the angle of attack is small, the separation will reattach soon. This phenomenon is difficult to observe in the wind pressure coefficient results K 12 L 13 M 14 N O P Q n o 17 p q r AG AH AI AJ AK AL AM ai aj ak 39 al 40 am Phase angle of pressure taps with respect to the local coordinate of cable ( o ) ang.0 ang.5 ang.10 ang.15 ang.20 ang.25 ang.30 ang.35 ang.40 ang.45 ang.50 ang.52 ang.54 ang.56 ang.58 ang.60 1 ang.62 A ang.64 a ang.66 ang.68 ang.70 ang.74 ang.78 ang.82 ang.86 ang (a) Wind pressure coefficients on cable (b) Wind pressure coefficients on rivulet Figure 1. Wind pressure coefficients on cable and rivulet with different angles of attack. These observations represent the extremely complicated interaction of the wind-rivulet-cable system especially around the rivulet. However, the averaged aerostatic force cannot take into account the unsteady effect since there is little discrepancy between lift and drag coefficients of the static and oscillatory situations. As a result, a more advanced unsteady model for RWIV is proposed in this paper. The unsteady aerodynamic load on the rivulet is represented as u 1 2 Fr RU cd( 0)sin( 0) cl( 0)cos( 0) Lcos( 0) Dsin( 0) (3) * y * R 2 * 2 * y L U (2 R) KyHR1( Ky) K HR2( K ) K HR3( K ) KyHR4( Ky) (4a) 2 U U R 1 2 * R 2 * * y 2 * y D U (2 R) K PR2( K ) K PR3( K ) KyPR5( Ky) KyPR6( Ky) (4b) 2 U U R u where F r represents the wind induced force on the rivulet using unsteady theory; 0 means the equilibrium potion of rivulet and is the dynamic circumferential displacement of 0 rivulet on the cable; * * * * * * * H R1, H R2, H R3, H R4, P R2, P R3, P R5 and P * R6 are aerodynamic coefficients Phase angle of pressure taps with respect to the local coordinate of rivulet ( o ) 6 7 ang.0 ang.5 ang.10 ang.15 ang.20 ang.25 ang.30 ang.35 ang.40 ang.45 ang.50 ang.52 ang.54 ang.56 ang.58 ang.60 1 ang.62 A ang.64 a ang.66 ang.68 ang.70 ang.74 ang.78 ang.82 ang.86 ang

4 (flutter derivatives) with respect to the reduced frequency Ky yr U or K R U. These aerodynamic coefficients play transfer-function roles in this unsteady model, e.g., H * R1 takes into account the lift force exerting on the rivulet induced by the velocity of vertical vibration. In the case of bridge deck, usually some aerodynamic coefficients are negligible, e.g. the value of aerodynamic coefficient H * 4 of the vertical displacement is very small comparing to other aerodynamic coefficients because the aerodynamic stiffness is extremely small comparing to the bridge deck stiffness. However, this conclusion cannot apply to the RWIV system since the stiffness of rivulet is negligible as mentioned in the proceeding content. 2.3 Nonlinear hysteretic model Recently, the nonlinear hysteretic behavior of bridge aerodynamics is observed and analyzed (Diana et al. 2010). Similar to the bridge deck, intense hysteretic behavior is observed in the wind-rivulet-cable interaction, which indicate that the higher-order memory in the wind-rivuletcable system is another indispensable effect. Typically, the steady-state aerodynamic coefficients are modeled by a nonlinear polynomial in terms of the angle of attack. However, the hysteretic behavior can be best described by higher-order (nonlinear) polynomial involving the dynamic angle of attack (Resulting from turbulence components and deck motions) and its derivative, e.g. (Wu and Kareem 2012), c (5) n hyst l (, ) j, k j k jk, where subscript "l" represents "L", "D" or "M", which denotes coefficients related to lift, drag and pitch, respectively; jk, is the coefficient corresponding to (j+k) th -order term; 2n is the highest order of the polynomial possible to parsimoniously model hysteretic behavior. The actual order of the model depends on the data used for fitting and the identification scheme used. Besides, the contributions of some of the terms in the polynomial may be negligible which leads to simplified expression. For example, for the lift coefficient of a stay cable of a cable-stayed bridge, the following model can represent the hysteretic behavior: hyst c (, ) c ( ) (6) L L 0 0,0 1,0 0,1 2,0 1,1 3,0 4,0 where the steady-state coefficient c ( ) L 0 corresponding to the equilibrium position is extracted from the constant term in order to emphasize the difference between the steady-state and the hysteretic cases. 3 PARAMTR IDNTIFICATION 3.1 Wind tunnel test set-up The artificial rivulet is applied in the wind tunnel test set-up since the pressure on the rivulet will be measured. Only the upper rivulet is simulated. The end plates are installed at each end of the two dimensional rigid cable to control the end flow condition. The rigid cable is mounted inside the wind tunnel to form a closed type test condition, which is presented in Fig. 2. Negligible turbulence is generated in the incoming laminar flow. The overall steady wind force on the cable-rivulet system and the pressures on cable and upper rivulet surfaces are measured under the forced pitching and plunging vibrations. The frequency of both pitching and plunging vibrations is 1 Hz. The double-amplitudes are 4 mm for pitching and 10 degrees for plunging. There are total four sections of the rigid cable in which the pressure taps are installed, and 63 pressure taps for each section. The position of these pressure taps is defined by the angle, which is deter- 657

5 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 mined from the stagnation point to the center of taps. Since the pressure on the rivulet will be recorded, the sizes of rivulet and the corresponding cable are amplified comparing to the ones in the real world. The rigid cable is made of acryl glass with the diameter of 350 mm and the length of 1540 mm while the rigid rivulet is made of polyethylene. The shape of the rivulet is one section of arc of 42 mm which belongs to a circle with a 25 mm radius. The schematic illustrations of the test section coordinates, the rivulet and pressure holes positions, and the rivulet and cable sizes are presented in Fig. 3. The angle represents the position of upper rivulet and the relative angle of attack induced by vertical motion of cable. Figure 2. Photo of wind tunnel setup and model. Figure 3. Coordinate system and arrangement of pressure taps. 3.2 Steady-state coefficients utilized in quasi-steady model The steady-state coefficients for cable and rivulet are measured through wind tunnel experiment and shown in Fig. 4. In order to conveniently obtain their derivatives in terms of the angle of attack, which will be used in the calculation of RWIV, both of them are fitted using nonlinear least square technique. xperimental Drag Coefficient xperimental Lift Coefficient 1.2 Numerical Drag Coefficient Numerical Lift Coefficient Angle of attack ( o ) (a) Steady-state coefficients of cable (b) Steady-state coefficients of rivulet Figure 4. xperimental and numerical steady-state coefficients xperimental Drag Coefficient xperimental Lift Coefficient Numerical Drag Coefficient Numerical Lift Coefficient Angle of attack ( o ) 3.3 Aerodynamic coefficients utilized in unsteady model The scheme based on the measurement of the oscillatory pressure is applied to extract the aerodynamic coefficients of the rivulet. A detailed validation procedure for the identified aerodynamic coefficients are carried out. For example, in order to validate the identification results based on pressure measurement, the forced oscillation scheme is also utilized for the cable. Fig. 5 shows 658

6 the identified aerodynamic coefficients (only aerodynamic coefficients for the lift force are presented here) of the upper rivulet. As shown in Fig. 5, the aerodynamic coefficients vary tremendously with respect to the wind angle of attack (e.g., the position of rivulet). The values of these aerodynamic coefficients become very large in the range of 58 o -62 o, which indicates the intense unsteady load in this range. This range is critical for RWIV since the existing of large negative slope of the life coefficient of cable which induced the large amplitude oscillation based on the * galloping theory. Besides, the aerodynamic coefficient H 4 of the vertical displacement is not negligible comparing to other aerodynamic coefficients, which indicates the large difference with bridge system. If the aerodynamic coefficients are presented utilizing Zasso's form (Zasso 1996), the identified aerodynamic coefficients per se straightforwardly show the intensive unsteadiness of the aerodynamic forces on the rivulet in the wind velocity range of interest. H * R A1 a B 2b C3 Dd c 4 e5 F6 f g 7 G 0 o 30 o 35 o 40 o 45 o 50 o 52 o 54 o 56 o 58 o 60 o 62 o 64 o 66 o 68 o 70 o 1 74 o A 78 o a 82 o 86 o 90 o 180 o H 2R A1 a B 2b C3 Dd 4 e c 5 F6 f g 7 G 0 o 30 o 35 o 40 o 45 o 50 o 52 o 54 o 56 o 58 o 60 o 62 o 64 o 66 o 68 o 70 o 1 74 o A 78 o a 82 o 86 o 90 o 180 o Reduced wind velocity (U/fR) Reduced wind velocity (U/fR) H * R o 30 o 35 o 40 o 45 o 50 o 52 o 54 o 56 o 58 o G 7 60 o 62 o 6 64 o f 66 o Dd B 1 2 C F 3 4 Aa b c e 5 68 o g 70 o H * R o 30 o 35 o 40 o 45 o 50 o 52 o 54 o 56 o 58 o G 60 o 7 62 o 6 64 o f 66 o Dd B2 C F A1 a b 3c e o g 70 o 1 74 o 1 74 o A 78 o A 78 o a 82 o a 82 o 86 o 86 o o o 180 o o Reduced wind velocity (U/fR) Reduced wind velocity (U/fR) Figure 5. Identified aerodynamic coefficients for upper rivulet. 3.4 Hysteretic loop utilized in nonlinear hysteretic model The Moore-Penrose pseudoinverse scheme is applied to identify the coefficients of the hysteretic numerical model. The hysteretic behavior in terms of steady-state coefficients and the numerical simulation results are shown in Fig. 6 (here only the lift coefficient are presented) at some certain reduced wind velocity. Because the facility in the wind tunnel can only oscillate with the maximum double-amplitude 10 o, the results shown in these figures are limited consideration of nonlinear hysteretic phenomenon of the RWIV. 659

7 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, xperimental lift coefficient Numerical lift coefficient 0.58 xperimental lift coefficient Numerical lift coefficient Angle of attack ( o ) Angle of attack ( o ) Figure 6. Hysteretic simulation of (a) cable and (b) rivulet (0=66o). 4 NUMRICAL XAMPL The rivulet-cable model used in the numerical example is described in the proceeding section. Besides, the natural frequency of the cable is set the same with the frequency of forced vibration for aerodynamic coefficients identification. The mass and damping ratio of cable are 6 kg per unit length and 0.001, respectively. The linear damping coefficient and the Coulomb damping force between rivulet and cable are set as N s/m 2 and 20% of the weight of rivulet, respectively. The mass of rivulet is set kg per unit length according to the density of water and the assumed area of the upper rivulet on the cable. Moreover, the reasonable range of the circumferential motion of rivulet is assumed between 0 to /2. For the unsteady model, the equilibrium position needs to be determined firstly. The equilibrium position of the rivulet on the cable based on the QS model is utilized in this example. Although the equilibrium position based on QS assumption is somehow different with the observation in the wind tunnel, fortunately the results of unsteady model proposed in this paper are not sensitive to the equilibrium position of rivulet. As long as this equilibrium position in a reasonable range, the calculation of this unsteady model will converge to the same result. The equilibrium position of the rivulet on the cable based on the QS model could be replaced in the proposed unsteady model if reliable data based on the wind tunnel results are available. Some reliable records using an ultrasonic transmission thickness measurement system (UTTMS) are given in recent literature (e.g., Li et al. 2010). A range of wind velocities are calculated for the RWIV utilizing these three models. The calculation responses of the cable based on the QS, unsteady and nonlinear hysteretic models at the wind velocity 9.5 m/s are shown in Fig. 7 while the responses of the rivulet are presented in Fig. 8. Amplitude (m) QS result of cable Amplitude (m) 0.10 Unsteady result of cable Nonlinear hysteretic result of cable (i) QS model (ii) Unsteady model (iii) Nonlinear hysteretic model Amplitude (m) Figure 7. Comparison of calculation results of the cable based on various models. 660

8 1.10 QS result of rivulet 1.10 Unsteady result of rivulet 1.10 Nonlinear hysteretic result of rivulet Amplitude (rad) 1.00 Amplitude (rad) 1.00 Amplitude (rad) (i) QS model (ii) Unsteady model (iii) Nonlinear hysteretic model Figure 8. Comparison of calculation results of the rivulet based on various models As shown in this figure, the vibration behaviors of the cable and rivulet obtained from QS, unsteady and nonlinear hysteretic models are similar while the vibration amplitude of unsteady results is smaller and the vibration amplitude of nonlinear hysteretic results is larger compared with the calculation based on the QS model. This comparison indicates that the linear unsteadiness will alleviate the RWIV while the nonlinear hysteretic behavior will intensify the responses of the rivulet-cable system. Fig. 9 shows the comparison of the amplitudes of cable and the wind velocity range in which the large limited amplitude oscillation appears between numerical results based on QS and unsteady models (here the comparison with nonlinear hysteretic results are not presented due to the large number of computational work). As indicated in the figure, both of these models could obtain wind velocity limited and amplitude limited results. The limited wind velocity is almost the same for these two models while the limited amplitude obtained from unsteady model is just half of the one obtained based on QS model QS model Unsteady model 0.08 Cable amplitude (m) Wind velocity U (m/s) Figure 9. Comparison of cable responses based on QS and unsteady models. It should be noticed that the oscillation frequency of the rivulet is set to be 1 Hz during the calculation process. In other words, the reduced frequency K y and K are not changed under a certain wind velocity. A more sophisticated procedure involving iterative calculation may give the unsteady results with higher accuracy, in which the oscillation frequency of the rivulet is treated as an unknown. However, based on the experience of the flutter analysis on bridge deck, this simplification will change the result negligibly. This unsteady model mainly focuses on the investigation of the wind induced force on the rivulet, while other parameters of the motion of rivulet especially the Coulomb damping force is very important and needed to be investigated in the future research. Besides, although the turbulence has significant effect on the motion of the rivulet-cable system, this effect has not been well understood to date. It is challenging to take into account the 661

9 The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 2012 turbulence effect reasonably and conveniently based on the conventional QS model. On the other hand, this proposed unsteady model could effectively calculate the turbulence effect based on the hybrid scheme proposed by Chen and Kareem (2003). As the aerodynamic coefficients identified in the turbulent condition are available the effect of turbulence could be taken into account immediately with this proposed unsteady model. 5 DISCUSSION One of the main features of RWIV is that the vibration is wind velocity and amplitude limited. Based on the observation of the lift coefficient curve, the mechanism of galloping is a reasonable explanation of RWIV phenomenon. Under this assumption, the effective cross-section of cable or the circumferential motion of rivulet on the cable is critical to the RWIV. Current coupling model based on the QS assumption successfully reproduce the wind velocity and amplitude limited characteristics of RWIV (e.g., Gu et al. 2009). However, it is not difficult to reproduce this limited property of RWIV regardless of the motion of rivulet. For example, with the assumption of the rivulet vibrating as a harmonic or a narrow banded stochastic process, the obtained results also present the limited feature in a similar range of wind velocity and with the same order of the magnitude of vibrating amplitude. Since the amplitude of RWIV is very sensitive to the test conditions (Gu and Du 2005), there is no strong evidence to support the superiority of any of these mentioned models with respect to reproducing the RWIV phenomenon. Furthermore, even if the rivulet rests on the cable at the equilibrium position, which belongs to the classical galloping class, the calculation results of the example in this paper show that at the same range of wind velocity, the magnitude of the limited amplitude is in the same order as the harmonic/stochastic/qs results. Since the equilibrium position of the rivulet is changed as the wind velocity increasing, only in a certain range of wind velocity the position of rivulet is in the negative slope range of the lift coefficient where the large amplitude vibration occurs. The relatively small magnitude of the vibration amplitude for this "classical" galloping may result from the extremely narrow range of the negative slope of the lift coefficient. Assuming the cable vibrating harmonically, the amplitude of velocity of the cable, which contributes to the relative angle of attack, will be the amplitude of the displacement multiplied by the vibrating frequency. As a result, as the amplitude of the vibration increases, the total angle of attack is very easy to escape out of the negative slope range, especially for the large vibrating frequencies or high modes. The higher-order spectrum could be utilized to analyze the data in RWIV (Wu and Kareem 2012). It is shown that there are intense nonlinear coupling and interactions that exist in RWIV. The harmonic distortion and super-harmonic (or sub-harmonic) behaviors could be a potentially convincing illustration of the appearance of the lower or higher frequency components in the aerodynamic forces on the cable or rivulet, where usually the axial flow theory is utilize (Matsumoto et al. 2005). The current state of the art for the RWIV cannot predict the wind velocity range in which the large amplitude oscillation appears and the amplitude of the limited cycle oscillation with high accuracy. The main reason may be the inaccurate simulation of the forces exerting on the rivulet, which is the primary motivation of this research. The main objective of this paper is to reveal the mechanism of RWIV, especially the aerodynamic forces exerting on the rivulet. The effort of simulating the forces exerting on the rivulet with high fidelity is a promising venue of research to improve predictive capability of RWIV. The proposed unsteady model and nonlinear hysteretic model are more advanced and necessary. Other forces exerting on the rivulet especially the Coulomb damping force is also very important and needs to be investigated in the future research. 662

10 6 CONCLUDING RMARKS It is shown that the flow around the rivulet in the rivulet-cable system is extremely complicated. Current model to calculate the rain-wind induced vibration (RWIV) is mainly based on the quasisteady (QS) assumption, which is probably not appropriate for the circumferential motion of the rivulet on the cable. The unsteady model which parallels Scanlan s analysis framework for the bridge aerodynamics and the nonlinear hysteretic model which utilizing the Moore-Penrose pseudoinverse identification scheme are proposed in this paper. Both of these new models could take into account the unsteady effect of the aerodynamic forces exerting on the rivulet. Besides, the turbulence effect could be reasonably and conveniently considered with the unsteady model. The aerodynamic coefficients are identified based on the measured oscillatory pressure data. To the authors' knowledge, this is the first time that the aerodynamic force coefficients of rivulet are identified in the literature. The results of the QS, unsteady and nonlinear hysteretic models are significant different with each other. This comparison indicates that the linear unsteadiness will alleviate the RWIV while the nonlinear hysteretic behavior will intensify the responses of the rivulet-cable system. The unsteady aspects of the aerodynamic load on the cable are not considered in this paper for the sake of simplicity, however, it is straightforward to incorporate this effect in the proposed models. 7 ACKNOWLDGMNTS The support for this project provided by the NSF Grant # CMMI which is gratefully acknowledged. The authors are also thankful to Mr. Wenfeng Sun, Hunan University for his help on a series of wind tunnel tests. 8 RFRNCS 1 Chen, X. and Kareem, A., Aeroelastic analysis of bridges: effects of turbulence and aerodynamic nonlinearities. J. ng. Mech., 129 (8), Diana, G., Rocchi, D., Argentini, T. and Muggiasca, S., Aerodynamic instability of a bridge deck section model: Linear and nonlinear approach to force modeling. Journal of Wind ngineering and Industrial Aerodynamics, v 98, n 6-7, Gu, M. and Du, X., xperimental investigation of rain-wind-induced vibration of cables in cable-stayed bridges and its mitigation. Journal of Wind ngineering and Industrial Aerodynamics, 93 (1), Gu, M., Du, X. Q. and Li, S. Y., xperimental and theoretical simulations on wind-rain-induced vibration of 3-D rigid stay cables. Journal of Sound and Vibration, 320 (1-2), Hikami, Y. and Shiraishi, N., Rain-wind induced vibrations of cables in cable stayed bridges. Journal of Wind ngineering and Industrial Aerodynamics, 29 (1-3), Li, H., Chen, W. L., Xu, F., Li, F. C. And O, J. P., A numerical and experimental hybrid approach for the investigation of aerodynamic forces on stay cables suffering from rain-wind induced vibration. Journal of Fluids and Structures, 26 (7-8), Li, S. and Gu, M., Numerical simulations of flow around stay cables with and without fixed artificial rivulets. The Fourth International Symposium on Computational Wind ngineering (CW2006), Yokohama, Japan, pp Li, S. Y, Chen, Z. Q., Wu, T. and Kareem, A., On the Rain Induced Vibration of Cables. Journal of ngineering Mechanics, ASC. Submitted. 9 Matsumoto, M., Yagi, T., Sakai, S., Ohya, J. and Okada, T., Steady wind force coefficients of inclined stay cables with water rivulet and their application to aerodynamics. Wind and Structures, 8 (2), Scanlan, R. H. and Tomko, J. J., Airfoil and bridge deck flutter derivatives. J.ngrg. Mech.Div.ASC, 97(M6), van Oudheusden, B. W., On the quasi-steady analysis of one-degree-of-freedom galloping with combined translational and rotational effects. Nonlinear Dynamics, 8, Verwiebe, C. and Ruscheweyh, H., Recent research results concerning the exciting mechanisms of rainwind-induced vibrations. Journal of Wind ngineering and Industrial Aerodynamics, 74-76, Wu, T. and Kareem, A., Aerodynamics and Aeroelasticity of Cable-Supported Bridges: Identification of Nonlinear Features. Journal of ngineering Mechanics, ASC. Submitted. 14 Zasso, A., Flutter derivatives: advantages of a new representation convention. Journal of Wind ngineering and Industrial Aerodynamics, 60,