Motion-amplitude-dependent nonlinear VIV model and maximum response over a full-bridge span

Nonlinear motion-amplitude (yT)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y_{T} )$$\end{document}-dependent energy-trapping properties of a bridge model undergoing vortex-induced vibration (VIV) are investigated. Energy-trapping properties of the model undergoing a full-process from still to a limit cycle oscillation (LCO) state are identified. A van der Pol-type model is adapted to describe the amplitude-dependent aerodynamic properties. Nonlinear parameter-amplitude relations, ε-yT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon {-}y_{T}$$\end{document} and ξε-yT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi_{\varepsilon } {-}y_{T}$$\end{document}, are established. Nonlinear aerodynamic damping is separated into two parts: the initial damping which varies with the reduced wind speed, and the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-related part which varies with both the reduced wind speed and the motion amplitude. The initial aerodynamic damping determines the threshold of VIV, while the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-related part dominates the evolution process and the LCO. The identified nonlinear analytical model is capable of predicting VIV responses at higher mechanical damping ratios. The energy-trapping properties of a section model in time are transformed into nonlinear properties distributed in space along an elongated 3-D elastic bridge span. According to this “time-space” transformation, the convection coefficient, which links the maximum response of a 3-D structure with that of a 2-D (1-DOF) sectional model, can be determined. Compared with a constant-parameter analytical model, an adapted nonlinear one brings to light significantly larger convection coefficients. Finally, parameter overflowing phenomena are revealed and discussed.


Introduction
Vortex-induced vibration (VIV) of long-span bridges has been a focal point in wind engineering. The underlying mechanism of VIV is well known, and a lot of research work aimed at a deeper and broader understanding of it has been conducted over the years. Notwithstanding the revealed mechanism, VIV of bridges in reality has kept taking place, among which a recent case is the 888 m main-span Humen suspension bridge in China [1,2], resulting in about 10 days of traffic interruption. VIVs of bridges have been traditionally investigated via sectional model tests, according to which vibrations of corresponding 3-dimensional (3-D) prototype bridge spans are evaluated. To date, very little work has been done to explore the link between a section model and its 3-D prototype. A few papers appearing in the literature dealing with this issue obtained convection coefficients based on analytical Z. Zhang (&) College of Civil Engineering and Architecture, Hainan University, Haikou, China e-mail: zhangzhitian999@126.com models independent on the motion amplitude [3][4][5], which ignores the most pronounced feature that sets the VIV of a 3-D structure apart from that of a 2-dimensional (2-D) one; that is, aerodynamic loadings or, to be more accurately, energy-trapping properties, are dependent on the motion amplitude and hence differ among sections throughout the bridge span.
The establishment of a linkage between a sectional model and a 3-D elastic structure entails appropriate descriptions of the vortex-induced loads [5][6][7][8]. Among innumerable studies in this regard, it is the type of wake-oscillator models that have attracted the most attention. The major characteristic of a wakeoscillator model is an imaginary fluid oscillator being coupled with body motions. Early models of this type go to Bishop and Hassan [9], and Hartlen and Currie [10], where the lift coefficient is used to represent the wake-oscillator motion. Related to wake-oscillator models, the oscillating system is self-exciting and selflimiting, and the Strouhal relationship is satisfied [11]. A large variety of wake-oscillator models have been explored over the past decades. It has been extended from 1-degree-of-freedom (1-DOF) systems where only across-wind body motion is considered [12,13], to 2-DOF systems where both across-wind and streamwise body motions are included [14][15][16][17][18]. In terms of the specific body motions coupled, they can be divided into models coupled with velocity [10,[19][20][21], acceleration [16], or a mix of velocity and acceleration [22]. Moreover, this type of models has been adapted to forced oscillations [23,24]. The number of parameters that need to be identified varies among models. Most of them involve three or four parameters with one or two depending on the others.
In bridge engineering, it is the structure's motion that matters. In this sense, the description of the imaginary wake oscillator is not only tedious but also unnecessary. Actually, the VIV behaviors of a solid body can be simplified into a single equation if the aerodynamic loading is appropriately described. Reference [25] proposed a model where the aerodynamic loading includes three terms: the forced term due to vortex shedding, the aerodynamic stiffness term, and the aerodynamic damping term. Reference [26] argued that, for structures immersed in a wind flow, the forced term and aerodynamic stiffness can be neglected, and the model is finally simplified into a single motion equation excited by one loading term expressed by a two-parameter Van der Pol oscillator. The model of [25] is convenient for engineering practice and without compromising of precision. Therefore, this kind of single-degree model has gained extensive applications [27][28][29]. More recently, other semiempirical models have been used for VIV of bridge structures. Reference [30] brought forward a Volterra series-based model where the VIV properties are represented by kernel functions; [31] tried a describing function method and demonstrated the equivalence to the model of [26]; reference [32] proposed a van der Pol-type model similar to that of [26].
A limit cycle oscillation (LCO) during VIV suggests an balance between the amount of energy absorbed from the wind and the amount dissipated due to mechanical damping [33]. This mechanism applies to both 2-D (with only 1-DOF) and 3-D structures. However, an obvious dissimilarity of the latter is that its motion amplitudes vary continuously throughout the entire span. Therefore, energy trapping and dissipating might not be balanced in every single section but instead, over the entire span, which means a part of the structure keep trapping energy while the other part keep dissipating. In this situation, linkage between the VIV responses of a 2-D model and its 3-D prototype necessitates knowledge of nonlinear aerodynamic properties with respect to the motion amplitude. Despite nonlinear properties inherent in the traditional van der Pol-type models, they are not able to describe the true motion-amplitude-dependent nonlinearities of a given configuration, due to their fixed model parameters at a given wind speed. To this end, the model parameter in this study is described as a function of both wind speed and structural motion amplitude, and relevant identification procedure from experimentally obtained time-domain signals is addressed. In so doing, the traditional van der Poltype model is adapted to the required motion-amplitude-dependent situations and finally leads to the linkage between VIV responses of a 2-D model and its 3-D prototype.
2 Energy-trapping based VIV model

Basic assumptions
Firstly, added mass effects and aerodynamic stiffness are assumed to be negligible. These effects associated with a vibrating body can sometimes be pretty pronounced when it is immersed in a water flow [34]. For a bridge structure, the combined effects of added mass and aerodynamic stiffness can also be significant when it is exposed to very high wind speeds, exhibiting an obvious drop in torsional frequencies [35,36]. However, these effects can be ignored at low wind speeds over a single VIV lock-in range [37].
Secondly, for structures being released to vibrate from still, the nonlinear aerodynamic damping ratio is assumed to be expressed with an initial term n in (negative in value) plus a motion dependent nonlinear term n e . n in is independent on the motion amplitude but differs among wind speeds. This assumption follows from experimental facts that the VIV responses are self-limiting and sensitive to the mechanical damping (see Fig. 1). Generally, the peak responses and the width of the lock-in ranges decrease as the mechanical damping increases, and the resonance could disappear completely beyond a critical mechanical damping n s ¼ n in j j. That means the VIV phenomenon disappears once the mechanical damping ratio is higher than n in j j.
Thirdly, attention is paid solely to energy-trapping properties that are relevant to an analytical model. It has been known that aerodynamic loadings developed around a 2-D body can be very complex. The load components can be narrow or wide banded, depending on the region of Reynolds number [38], and load fluctuations and energetic scales evolve drastically around the entire section [39,40]. Therefore, the selection of a semiempirical model depends on how many details are targeted, and, as concerned aspects widen, it is believed that all analytical models cannot be satisfactory [14], since all of them in essence do not include any analysis of the flow and hence are not able to replace the Navier-Stokes equations. Their value is at best describing and simulating known experimental results [11]. In bridge engineering, how much the structure responds to the wind is often the unique concern of engineers. Therefore, selection of the analytical model can be evaluated by its capability of reflecting the energy-trapping properties dominating the structure's motion evolution. The motion evolution here includes not only the final LCO but also the intermediate process. Modeling the intermediate process entails observing the evolution of the macroscopic damping ratios (negative n) as shown in Fig. 2, which evolves with the motion amplitude y T t ð Þ, and this should be reflected by the adopted analytical model. Once the evolution of n with y T t ð Þ is obtained, the energy-trapping/dissipating along a 3-D elastic full-bridge span can be determined accordingly.

Amplitude-dependent model parameters
The where m is the mass or mass moment of the model; n s is the mechanical damping ratio; x is the model's natural circular frequency; F a is the vortex-induced aerodynamic load acting on the model; and y, _ y, € y are structural displacement, velocity, and acceleration, respectively.
First, the mathematical form of F a proposed by reference [25] is altered slightly in this work as Fig. 1 Schematic VIV response of a sectional model, where n s1 \n s2 \n s3 are mechanical damping ratios, with n s1 \n s2 \ n in j j and n s3 [ n in j j Fig. 2 Variation of the nominal damping ratio during a VIV process from still to a LCO state where F a is the vortex-induced aerodynamic load; q is the air density; U is wind speed; D is the reference height; L is the model length; U r ¼ U=Df is the reduced wind speed; Y 1 is a function of U r that needs to be identified; y and _ y are the displacement and velocity, respectively; and e is a parameter to be identified that provides self-limiting property for the system. The model has been traditional applied with fixed e values identified from the LCO amplitudes. A LCO indicates a state of positive mechanical damping balanced by the negative aerodynamic damping. However, for given y(t) and _ yðtÞ, the aerodynamic damping is determined by a combination of Y 1 U r ð Þ and e, which means their solution might not be unique if aiming at a specific LCO amplitude. Therefore, the two parameters will be addressed in such a fashion that Y 1 U r ð Þ determines the initial aerodynamic damping (for small structural motions), and e describes the motion-amplitude-dependent nonlinear properties, namely: where y T is the time-variant motion amplitude. The amount of work done by the aerodynamic load F a during a single oscillation period is where W a denotes the amount of work and T is the period of oscillation. The displacement and velocity histories of the structure undergoing decaying or diverging vibration are given as where x is the vibrating circular frequency and k is the exponential coefficient. Note that the sign of k determines whether y T decreases or increases with time.
According to Eqs. (3)-(6), one obtains Introducing the relation between k and n, Generally, macroscopic damping ratio n j j 0:05 holds even under very severe VIV responses. Therefore, Eq. (10) can be approximated as a n ð Þ % 1 The error of b n ð Þ relative to a n ð Þ is plotted in Fig. 3, where it is noticed that errors at n j j 0:05 are always less than 1%. Therefore, the expression of W a can be rewritten in a concise form, as On the other hand, the negative work done by the mechanical damping force is where c is the mechanical damping coefficient and c ¼ 2mn s x. Substituting Eq. (6) in (13), one obtains Equations (12) and (14) relate the amounts of work to y T andk, done, respectively, by the aerodynamic and mechanical damping during a single period. In a LCO state, these two amounts of work counteract each other exactly. During the increasing stage of y T , however, there is a net increment of mechanical energy DW, and it can be expressed in terms of structural stiffness and amplitude increment as where Dy and DW are the amplitude and the energy increments, respectively. The energy increment must result from the summation of W a and W c , namely Substituting Eqs. (12) and (14) in (16) results in A basic relation between the k and Dy can be obtained according to principles in structural dynamics, as Therefore, the exponent k can be determined directly according to Eq. (18) once a time history of structural response is tested.
The initial aerodynamic damping depends on the wind speed only, and it is expressed according to Eq. (2) as Applying Eqs. (19) to (17) results in Equation (20) links e with the motion amplitude y T , and this relation is able to be determined by an experimentally obtained time history.
In a LCO state, where y lco is the LCO amplitude. For a specific bridge deck configuration and wind speed, y lco is a function of mechanical damping. Equation (21) states that e in a LCO state, denoted here by e lco , is jointly determined by the mechanical damping, the initial aerodynamic damping, and the final motion amplitude. The calculation of e by Eqs. (20) or (21) implies a precondition of c in being identified first, which can be determined by the foremost cycles of motion as where n is the number of selected cycles of motion and d n is the logarithmic decrement corresponding to n, and it is calculated according to where y T 0 is the motion amplitude at reference time t = 0 andy T n is that at time t = nT. Once Y 1 is determined, the initial aerodynamic damping ratio can be obtained as where n in is the initial aerodynamic damping ratio. Now we turn to the part of aerodynamic damping related to e. Once the initial aerodynamic damping (or Y 1 ) is determined, e U r ; y T ð Þis able to be calculated by Eq. (20). The part of the work done by the e-related term can be separated from Eq. (12), as According to Eq. (14), W e can also be expressed in terms of an effective damping coefficient c e , as Then c e can be obtained by comparing Eq. (26) with (25), as This part of damping contributes a part of damping ratio as Equation (28) determines the evolution of n e . It indicates that, for a specific time history, the part of erelated damping ratio is determined by the initial aerodynamic coefficient Y 1 and the motion amplitude y T . In the final stage of motion evolution, namely the state of LCO, the sum of the initial and e-related aerodynamic damping is balanced by the mechanical damping, namely where n s ¼ c=2mx is the mechanical damping ratio.

Peak response of a full-bridge span
All structural members take part in the vibration when VIV happens, as shown in Fig. 4, although only the bridge deck is assumed to be responsible for energy trapping. The participation of other members is considered in terms of added mass to the bridge deck [4,5]. Once the modal shape and the effective mass property of the bridge deck are determined, the vertical vibration y x; t ð Þ is related to a modal shape function and a generalized coordinate, as where q is the generalized coordinate and u v x ð Þ is the bridge deck's modal shape function in the vertical direction. The governing equation of motion is given as where M q is the generalized mass and m e is the effective mass per unit length of the bridge deck and It is noted that u v x ð Þ is a part of the modal shape function of the whole structure u x ð Þ, and u x ð Þ in this case is normalized to the mass matrix of the whole structure.
Moving the right-side term of Eq. (31) to the left side and using Eqs. (32) and (33) yields The final solution of Eq. (34), namely the LCO of the generalized coordinate q, is determined by the energy balance principle; that is, the two amounts Thus, the LCO amplitude, q 0 , is obtained as However, Eq. (36) is not able to be solved directly because not all parameters that appear on the right-hand side are determined. The reason for that is e x ð Þ depends on the motion amplitude of the deck at location x, namely q 0 u v x ð Þ j j. However, the motion amplitude at location x is unknown yet since q 0 itself needs to be solved. Therefore, an iteration method is required to obtain the final solution, as shown in Fig. 5.
A parameter overflowing phenomenon might happen during the solution, as shown in Fig. 6. This phenomenon occurs when the energy balance of the 3-D structure entails motion amplitude in some zones being larger than the maximum amplitude experienced by the section model. For motions beyond the LCO amplitude of the sectional model, corresponding e is unknown since it is identified based on known motion amplitudes.
In a similar fashion, the motion equation of a stringmounted sectional model is where e lco is identified based on the LCO state (see Eq. (21)). The LCO amplitude of Eq. (36) can be obtained by a similar procedure, as Introducing b v as the ratio of the maximum response of the 3-D structure to that of the 1-DOF model, it can be determined with Eq. (36), Eq. (38), and the modal shape function u x ð Þ, as The underlying mechanism behind Eq. (39) applies to both vertical and torsional responses. It is noted that the conversion coefficient of Eq. (39) implies a 1:1 geometric scale. If a geometrically scaled-down sectional model is used, then the geometric scale needs to be introduced into Eq. (39), as where k g is the prototype-to-model geometric scale. However, it is noted that k g applies exclusively to transverse VIVs, and its torsional response counterpart would always be irrelevant to the geometric scale. If a linear model (linear in terms of model parameters) is concerned, namely the VIV model being independent on the motion amplitude, then 4 Application

Experimental setup and results
A sectional model as shown in Fig. 7 is used to conduct VIV tests in a wind tunnel. The model is made to a geometric scale of 1:50 based on its prototype bridge deck. The experimental setup is shown in Fig. 8. Major properties of the sectional model are listed in Table 1. Traditionally, the VIV properties of a bridge deck model are tested by increasing the wind speed step by step (or gradually), without external interference to the model's motion. That means the model's motion at a new wind speed, y lco U þ DU ð Þ , would evolve based on its motion state at the previous wind speed y lco U ð Þ. Information revealed in such a fashion is incomplete because the aeroelastic properties of motion states of y T \y lco U ð Þ at U þ DU are missing. To avoid this kind of information missing, the response of the model at every wind speed is tested from still to the LCO state.
The lock-in range and VIV amplitudes are plotted in Fig. 9. A total of six time histories have been recorded during the lock-in range, as shown in Fig. 10. The foremost several motion periods are used to identify the initial aerodynamic damping, as shown in Fig. 11. The exact number of periods used for the identification is hard to determine and is subject to a number of factors. However, a specific time history can be divided largely into three zones: the zone for identification of initial aerodynamic damping, the zone of parameter evolution, and the stable zone with a LCO. Model parameters no longer vary once the structure enters the LCO state.

Initial aerodynamic damping
The initial aerodynamic damping are identified and listed in Table 2. The relation between n in and reduced wind speed U r can be calculated by Eq. (24). It is noted that although the VIV amplitudes vary significantly throughout the lock-in range, the initial aerodynamic damping ratio n in stays almost at a constant value. However, this might not be a universal law.

Nonlinear aerodynamic damping
Once Y 1 or c in is determined, eðU r ; y T Þ is able to be calculated by Eq. (20), as shown in Fig. 12. According to Eq. (2), one knows that a positive e contributes to positive aerodynamic damping. Figure 12 shows that e decreases drastically as the motion amplitude increases. However, this does not mean the aerodynamic stability deteriorates as the motion amplitude increases. According to Eq. (28), while the e-related damping ratio is proportional to e, it is also proportional to the power of y T . Therefore, e and y T combined would result in a stabilization effect. As shown in Fig. 13, the e-related damping ratio increases in general as the motion amplitude increases.
The initial aerodynamic damping is a function of the wind velocity only, namely n in ¼ n in U r ð Þ, while n e is a function of both wind velocity and motion amplitude, namely n e ¼ n e U r ; y T ð Þ. A LCO means n e ¼ Àn s À n in . Therefore, once n e U r ; y T ð Þ is experimentally determined, the motion amplitude corresponding to a higher mechanical damping, namely n 0 s [ n s , is able to be predicted since in this case the aerodynamic damping, n 0 e ¼ Àn 0 s À n in \n e , appears in the known path. For a lower mechanical damping n 0 s \n s , however, the VIV response cannot be predicted because n 0 e [ n e locates somewhere beyond (appears above the highest point shown in Fig. 13). VIV responses of the section model at higher damping ratios are predicted and plotted in Fig. 14, where a general decreasing trend of the motion with respect to mechanical damping is noticed. Also, it is noticed that the VIV phenomenon disappears completely when the mechanical damping reaches 0.55%.

Maximum responses of the 3-D bridge structure
Based on the identified nonlinear e À y T relation, values of b v are able to be determined by Eq. (38), as listed in Table 3. It is noted that, throughout the lock-in range, b v cannot be determined for n s ¼ 0:003 and n s ¼ 0:0035 due to the existence of parameter overflowing zones along the bridge deck. The lengths of   Fig. 15, where it can be seen the overflowing zone for n s ¼ 0:003 covers 9 bridge deck segments (about 108 m long). As n s increases slightly to 0.0035, the overflowing zone shrinks drastically to 3 deck segments (36 m long). The overflowing phenomena indicate that, to predict the VIV response of a full-bridge structure, the mechanical damping ratio n s used by the sectional model must be lower than the targeted value at least to some extent. If very low mechanical damping ratios are targeted, it would be necessary to adopt forced vibration techniques, which has been practiced by [41][42][43]. The overflowing zone does not necessarily mean the region enters an aerodynamic positive damping state, which is demarcated by a motion amplitude larger than the LCO amplitude achieved with n s ¼ 0 (see Fig. 16), namely y lco ðn s ¼ 0Þ. However, if a part of the bridge span does exceed this demarcation, as  Zones of the evolution: I. Zone for identification of initial aerodynamic damping; II. zone of parameter evolution; III. the stable LCO zone may occur in cases of very low mechanical damping, then, to avoid parameter overflowing, a negative mechanical damping or a forced vibration method is required to identify the model parameters.
For comparison, b v calculated from the linear van der Pol models, namely by Eq. (41), is presented in Table 4. The most pronounced findings are given as follows: firstly, while b v from the linear model is independent of both the reduced wind velocity and the mechanical damping ratio, that from the nonlinear model varies significantly with both U r and n s . In fact, the linear model based b v is determined uniquely by the modal shape function; secondly, when compared to the linear model based b v , 1.234, those based on the nonlinear model can be as large as 2.171; Finally, while there is not an obvious trend as to the variation of b v with respect to U r , it increases in general as the mechanical damping increases.
It is worth mentioning that the method proposed in this study is not able to be extended to cases where the model experiences coupled VIV between DOFs. DOFs coupling could change drastically the aeroelastic properties. It has been found that the stream-wise oscillations of a circular cylinder have a substantial effect on the transverse vibrations [44]. Reference [45] found that the participation of torsional oscillation drastically decreases the vortex-induced transverse motion amplitude. They also pointed out that there is not only a phenomenon of energy transformation from transverse to torsional direction, but also a significant drop in the total mechanical energy of the system. It is also worthy of noting that the mechanical damping in    this study is viewed as a fixed value independent of the motion amplitude. Recent studies indicate, however, that the mechanical damping of both a section of a fullbridge model varies with motion amplitude [35,46,47]. Therefore, a refined identification of the model parameters should take into account the variation of mechanical damping with respect to motion amplitude. Finally, it should be pointed out that the mechanism is not exclusive to the van der Pol-type analytical models. Any model capable of the required energy description can be used to replace the current one.

Conclusions
A nonlinear motion-amplitude-dependent VIV model is brought forward in this study, and a ''time-space'' transformation method is established to calculate the VIV of 3-D bridge spans based on sectional model testing results. The core mechanism is dependence of the amount of aerodynamically trapped energy on the structural motion amplitude y T , according to which model parameters are nonlinearized. The y T -dependent property, which exhibits for a 2-D (1-DOF) model in terms of y T increasing in time, exhibits for a 3-D model in terms of a model parameter varying in space along the bridge span. The following conclusions are drawn based on the discussions presented in this study: Fig. 15 Overflowing zone of e at U r = 10.19; a n s = 0.003; b n s = 0.0035  (1) The adapted van der Pol-type model is able to describe motion-amplitude-dependent energytrapping/dissipating properties. The nonlinear e-y T relation is established, which enables prediction of the VIV responses at higher mechanical damping ratios, or calculation of the 3-D VIV response based on 1-DOF results. (2) When the ''time-space'' transformation is concerned, parameter overflowing could occur when energy-balancing entails the maximum response of the 3-D structure being larger than the LCO amplitude of the 1-DOF model. To minimize the range of overflowing zone, and to broaden the application of an identified analytical model, the mechanical damping involved in sectional model tests should be set as small as possible. (3) The convection coefficient, which reflects the ratio of the maximum response of the 3-D elastic bridge span to that of the 1-DOF model, is determined based on the method presented in this study. It is revealed that the coefficient varies with both the mechanical damping and the reduced wind speed. The current example shows that it can be as large as 2.171, which is substantially larger than the constant-parameter-model based value of 1.234.