Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD

Tianyou Tao; Hao Wang; Chengyuan Yao; Xuhui He

doi:10.3390/app7040395

Tao, Tianyou;Wang, Hao;Yao, Chengyuan;He, Xuhui

2017-04-14 00:00:00

applied sciences Article Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD 1 , 2 1 , 2 , 2 3 Tianyou Tao , Hao Wang *, Chengyuan Yao and Xuhui He Key Laboratory of C&PC Structures of Ministry of Education, Southeast University, Nanjing 210096, China; taotianyou@seu.edu.cn School of Civil Engineering, Southeast University, Nanjing 210096, China; yaochengyuan@seu.edu.cn School of Civil Engineering, Central South University, Changsha 410075, China; xuhuihe@csu.edu.cn * Correspondence: wanghao1980@seu.edu.cn Academic Editor: Alkiviadis Paipetis Received: 7 February 2017; Accepted: 12 April 2017; Published: 14 April 2017 Abstract: The long-span triple-tower suspension bridge is a brand-new type of structural form with typical wind-sensitive features. Inevitable wind-induced vibrations will heavily inﬂuence the driving comfort during strong winds and shorten the fatigue life of the bridge structure, which highlights the demand for the mitigation of wind-induced vibrations of long-span bridges. In this study, a parametric analysis is performed to investigate the control effect of multiple tuned mass dampers (MTMD) on the buffeting responses of a long-span triple-tower suspension bridge. Taking Taizhou Bridge as an example, the buffeting analysis is conducted via the ﬁnite-element-based approach. A step-updating framework is presented to guide the parametric analysis and determine the preferred values of the mechanical parameters of the MTMD. Speciﬁcally, four parameters—including the number of TMDs, the mass ratio, the damping ratio, and the frequency bandwidth ratio—are included in the parametric analysis. The robustness of the MTMD is also evaluated by changing another parameter called the frequency ratio. It is found that the performance of the MTMD is sensitive to the variation of the four parameters and the robustness can be adjusted by increasing the frequency bandwidth ratio. Utilizing the MTMD with reasonable mechanical parameters, the buffeting responses of the long-span triple-tower suspension bridge under strong winds can be effectively mitigated. Keywords: triple-tower suspension bridge; buffeting; vibration control; multiple tuned mass dampers (MTMD); ﬁnite element analysis 1. Introduction In recent decades, wind disasters have become more and more violent across the world. Damages to property with attendant economic losses and casualties are frequently reported by the public media [1,2]. Among the disasters, extreme wind events—e.g., typhoons/hurricanes, downbursts, tornadoes, etc.—have become the most severe cases that will cause destructive damages or signiﬁcant performance deteriorations to engineering structures [2,3]. With the worsening of global climatic environment, many places have been known as typical wind-prone areas. For example, every year the coastal area of East China will be attacked many times by the severe typhoons from the Paciﬁc Ocean. However, in the wind-prone areas above, there are a great number of long-span bridges constructed or still under construction over rivers or even seas, which is catering to the developments of local transportation and economics [4]. With the rapid increase of the bridge span, the bridge will be a typical slender structure which is sensitive to the wind effects [5]. There are four main Appl. Sci. 2017, 7, 395; doi:10.3390/app7040395 www.mdpi.com/journal/applsci Appl. Sci. 2017, 7, 395 2 of 22 types of wind-induced vibrations occurring on bridges known as ﬂutter, buffeting, galloping, and vortex-induced vibration. Among them, buffeting is a typical forced vibration induced by the turbulence existing inherently in natural wind. Different from the other three vibrations, buffeting is daily experienced by long-span bridges and becomes more prominent with the increase of the bridge span. Although the buffeting will not cause direct catastrophic failure to structures, it will result in instability of the vehicles travelling on the deck as well as discomfort of pedestrians, it will also severely shorten the fatigue life of structural components [6–10]. Many studies have been conducted on the buffeting performance of cable-supported bridges. For example, Xu and Zhu investigated the buffeting responses of Tsing Ma Bridge under skew winds during Typhoon Sam [11]. Wang et al. conducted a comparative study on the buffeting performance of Sutong Cable-Stayed Bridge based on design and measured spectrum [10]. In recent years, a typical bridge with a brand new type of structural form, which is called the triple-tower suspension bridge, has been put into engineering practices for long-span bridges, e.g., Taizhou Yangtze River Bridge and Ma’anshan Yangtze River Bridge in China. The triple-tower suspension bridge has not only two towers at the ends of two main spans, but also a dominant mid-tower between two main spans to alleviate the strain of main cables. Compared to the double-tower suspension bridge, structural characteristics of a triple-tower suspension bridge are strikingly different because of the existence of the middle tower and an additional main span [12,13]. Most existing research focuses on the static analysis of long-span triple-tower suspension bridges [14–16]. Several studies also made efforts to the ﬂutter stability [17–19]. However, few researches are concerned about the buffeting problem of the long-span triple-tower suspension bridge. With the increase of the bridge span, buffeting may perform an important role in the structural design of long-span triple-tower suspension bridges, especially for the ones located in the typhoon/hurricane-prone areas. In order to improve the travelling comfort and keep sufﬁcient structural fatigue life, effective measures need to be adopted to mitigate the buffeting responses of long-span triple-tower suspension bridges during strong winds. There are a lot of dampers available in the vibration control of engineering structures—e.g., viscous damper, magnetorheological damper, tuned mass damper (TMD) [20–24]. Buffeting responses of long-span bridges are prominent in the main girder, which is a linear and ﬂexible structure. Obviously, the viscous damper and magnetorheological damper are not easy to be applied in the vibration control since they are usually installed between two adjacent joints to alleviate the comparative motions. However, the TMD is proved to be an effective device for the vibration control of long-span structures as the tacit mechanism is inertia. TMD is a classical device with a mass, a spring and a damper attached to attenuate undesirable vibrations of engineering structures [25]. However, it is known that the performance of the TMD is sensitive to the frequency variation of the main structure [4,26,27]. A solving scheme is realized by the MTMD, which is equipped with a series of TMDs with distributed frequencies. Hence, there is a frequency band formed in the MTMD to cope with any target frequency lying in this band, which means the robustness of the TMD has been signiﬁcantly improved. A lot of pioneering work has been done on the performance of the MTMD [26,28,29]. Also, a few studies have been conducted on the buffeting control of the double-tower cable-stayed bridge with MTMD. For example, Gu et al. conducted a parametric study on the MTMD for buffeting control of Yangpu Bridge, which is a cable-stayed bridge with a main span of 602 m [27]; Wang et al. presented a simulation study on the optimal control of a long-span cable-stayed bridge with MTMD via the ﬁrst-order optimization method [4]; However, few studies are reported on the buffeting control of the novel long-span triple-tower suspension bridge. In this study, a parametric sensitivity analysis is conducted on the buffeting control of a long-span triple-tower suspension bridge. The Taizhou Yangtze River Bridge, “Taizhou Bridge” for short, is taken as the research object. The buffeting performance of the long-span triple-tower suspension bridge is ﬁrstly analyzed with the unconditionally simulated wind ﬁeld. Then, a step-updating framework is presented to guide the parametric analysis and determine the preferred values of the mechanical parameters of the MTMD. Four parameters, which are the number of TMDs, the mass ratio, the Appl. Sci. 2017, 7, 395 3 of 22 Appl. Sci. 2017, 7, 395 3 of 21 analysis. Meanwhile, the robustness of the MTMD is also evaluated by changing another parameter of damping ratio and the frequency bandwidth ratio, are speciﬁcally included in the parametric analysis. MTMD which is called the frequency ratio. The analytical results are expected to provide references Meanwhile, the robustness of the MTMD is also evaluated by changing another parameter of MTMD for the buffeting control and wind-resistant design of long-span triple-tower suspension bridges. which is called the frequency ratio. The analytical results are expected to provide references for the buffeting control and wind-resistant design of long-span triple-tower suspension bridges. 2. Structural Description and Finite Element Model 2. Structural Description and Finite Element Model 2.1. Description of the Taizhou Bridge 2.1. Description of the Taizhou Bridge The Taizhou Yangtze River Bridge, ‘Taizhou Bridge’ for short, is located in the lower reach of The Taizhou Yangtze River Bridge, ‘Taizhou Bridge’ for short, is located in the lower reach of the the Yangtze River in China, and first establishes the connection of land transportation between Yangtze River in China, and ﬁrst establishes the connection of land transportation between Taizhou Taizhou and Zhenjiang, as shown in Figure 1. The bridge has been a landmark project across the and Zhenjiang, as shown in Figure 1. The bridge has been a landmark project across the Yangtze Yangtze River and the longest triple-tower suspension bridge in the world since it was open to River and the longest triple-tower suspension bridge in the world since it was open to trafﬁc in 2012. traffiW c i ith n 201 respect 2. Wi to th respect to the geographi the geographical location, T c aizhou al locaBridge tion, Ta isiz located hou Brin idge the ieastern s locatepart d in t of he e theaAsian stern part continent, which belongs to a wind-prone area that is vulnerable to the typhoons from the west Paciﬁc of the Asian continent, which belongs to a wind-prone area that is vulnerable to the typhoons from Ocean in the summer and the strong northern wind from Siberia in the winter. the west Pacific Ocean in the summer and the strong northern wind from Siberia in the winter. Figure 1. Geographical location of Taizhou Bridge. Figure 1. Geographical location of Taizhou Bridge. As shown in Figure 2a, Taizhou Bridge is equipped with two main spans. Each main span is 1080 m long. The two main spans are continuously connected at the mid-tower while they are separated 390 1080 1080 390 from side spans with two expansion joints installed at the side towers, respectively. The streamline ﬂat-steel-box girder is employed as the bridge deck, as shown in Figure 2b. The total width of the bridge deck is 39.1 m with the wind mouth included. Two steel-wired cables are installed in parallel to support the static and dynamic loads on the main girder, and the sag-to-span ratio of the main cable is 1/9. The two side towers are reinforced concrete structures with a height of 178 m, and the middle (a) tower is a steel tower with a height of 192 m. To constrain the lateral and vertical displacements of the main girder, lateral and vertical bearings are installed on lower cross beams of side towers, while 2% 2% only lateral bearings are installed on the mid-tower. Elastic restraints made of steel-stranded wires are installed in the longitudinal direction to control the relative displacement between the main girder and mid-tower. Since Taizhou Bridge is such a long-span bridge with a novel structural type, the buffeting performance attracts extensive attention. (b) Figure 2. Schematic description of Taizhou Bridge: (a) Elevation (unit: m); (b) Section of the main girder (unit: mm). As shown in Figure 2a, Taizhou Bridge is equipped with two main spans. Each main span is 1080 m long. The two main spans are continuously connected at the mid-tower while they are separated from side spans with two expansion joints installed at the side towers, respectively. The streamline flat-steel-box girder is employed as the bridge deck, as shown in Figure 2b. The total width of the bridge deck is 39.1 m with the wind mouth included. Two steel-wired cables are installed in parallel to support the static and dynamic loads on the main girder, and the sag-to-span ratio of the main cable is 1/9. The two side towers are reinforced concrete structures with a height of 3500 Appl. Sci. 2017, 7, 395 3 of 21 analysis. Meanwhile, the robustness of the MTMD is also evaluated by changing another parameter of MTMD which is called the frequency ratio. The analytical results are expected to provide references for the buffeting control and wind-resistant design of long-span triple-tower suspension bridges. 2. Structural Description and Finite Element Model 2.1. Description of the Taizhou Bridge The Taizhou Yangtze River Bridge, ‘Taizhou Bridge’ for short, is located in the lower reach of the Yangtze River in China, and first establishes the connection of land transportation between Taizhou and Zhenjiang, as shown in Figure 1. The bridge has been a landmark project across the Yangtze River and the longest triple-tower suspension bridge in the world since it was open to traffic in 2012. With respect to the geographical location, Taizhou Bridge is located in the eastern part of the Asian continent, which belongs to a wind-prone area that is vulnerable to the typhoons from the west Pacific Ocean in the summer and the strong northern wind from Siberia in the winter. Appl. Sci. 2017, 7, 395 4 of 22 Figure 1. Geographical location of Taizhou Bridge. 390 1080 1080 390 Appl. Sci. 2017, 7, 395 4 of 21 178 m, and the middle tower is a steel tower with a height of 192 m. To constrain the lateral and (a) vertical displacements of the main girder, lateral and vertical bearings are installed on lower cross beams of side towers, while only lateral bearings are installed on the mid-tower. Elastic restraints 2% 2% made of steel-stranded wires are installed in the longitudinal direction to control the relative displacement between the main girder and mid-tower. Since Taizhou Bridge is such a long-span bridge with a novel structural type, the buffeting performance attracts extensive attention. (b) Figure 2. Schematic description of Taizhou Bridge: (a) Elevation (unit: m); (b) Section of the main 2.2. Finite Element Figure M 2. Schematic odel description of Taizhou Bridge: (a) Elevation (unit: m); (b) Section of the main girder (unit: m girder (unit: mmm). ). According to the ultimate design scheme of Taizhou Bridge, the finite element (FE) model is established As shown in based on Fig ANSY ure 2a S , Ta (ANS izho YS, u Bri Inc. d,ge Canon is equ sb ipped wit urg, PA, h t US wo mai A), as n spans. shown E in acFigur h main e 3. span In th is e FE 2.2. Finite Element Model 1080 m long. The two main spans are continuously connected at the mid-tower while they are model, the BEAM4 element is utilized to simulate the main girder and the towers, and the LINK10 According to the ultimate design scheme of Taizhou Bridge, the ﬁnite element (FE) model is separated from side spans with two expansion joints installed at the side towers, respectively. element is selected for the simulation of the main cable and suspenders. The main girder and established based on ANSYS (ANSYS, Inc., Canonsburg, PA, USA), as shown in Figure 3. In the The streamline flat-steel-box girder is employed as the bridge deck, as shown in Figure 2b. The total FE model, the BEAM4 element is utilized to simulate the main girder and the towers, and the suspenders are connected via the zero-mass rigid beams. To model the real restrictions between the width of the bridge deck is 39.1 m with the wind mouth included. Two steel-wired cables are LINK10 element is selected for the simulation of the main cable and suspenders. The main girder and main girder and the towers, the lateral, vertical, and torsional degree of freedoms (DOF) of the main suspenders are connected via the zero-mass rigid beams. To model the real restrictions between the installed in parallel to support the static and dynamic loads on the main girder, and the sag-to-span girder are coupled with the lower beams of the side towers, while only the lateral DOF of the main main girder and the towers, the lateral, vertical, and torsional degree of freedoms (DOF) of the main ratio of the main cable is 1/9. The two side towers are reinforced concrete structures with a height of girder is coupled with the mid tower. The COMBIN14 element with only stiffness coefficient is applied girder are coupled with the lower beams of the side towers, while only the lateral DOF of the main to model the elastic cable between the main girder and the mid-tower in a longitudinal direction. girder is coupled with the mid tower. The COMBIN14 element with only stiffness coefﬁcient is applied to model the elastic cable between the main girder and the mid-tower in a longitudinal direction. Figure 3. Finite element model of Taizhou Bridge. Figure 3. Finite element model of Taizhou Bridge. 2.3. Modal Analysis of Taizhou Bridge 2.3. Modal Analysis of Taizhou Bridge Modal analysis of Taizhou Bridge is conducted with the subspace iteration method based on Modal analysis of Taizhou Bridge is conducted with the subspace iteration method based on ANSYS (ANSYS, Inc., Canonsburg, PA, USA), and 200 modes of the structure are extracted. Among them, the 13 typical modes with natural frequencies and the corresponding mode shapes are presented ANSYS (ANSYS, Inc., Canonsburg, PA, USA), and 200 modes of the structure are extracted. Among in Table 1, and six typical mode shapes of Taizhou Bridge are shown in Figure 4. them, the 13 typical modes with natural frequencies and the corresponding mode shapes are presented in Table 1, and six typical mode shapes of Taizhou Bridge are shown in Figure 4. Table 1. Typical modes of Taizhou Bridge. No. Frequency/Hz Modal Shape 1 0.07163 First antisymmetric lateral vibration of the main girder (AS-L-1) 2 0.08023 First antisymmetric vertical vibration of the main girder (AS-V-1) 3 0.09512 First symmetric lateral vibration of the main girder (S-L-1) 4 0.11489 Second antisymmetric vertical vibration of the main girder (AS-V-2) 5 0.11758 First symmetric vertical vibration of the main girder (S-V-1) 6 0.13709 Second symmetric vertical vibration of the main girder (S-V-2) 7 0.17089 Third antisymmetric vertical vibration of the main girder (AS-V-3) 8 0.18518 Third symmetric vertical vibration of the main girder (S-V-3) 9 0.23058 Second antisymmetric lateral vibration of the main girder (AS-L-2) 10 0.23786 Fourth antisymmetric vertical vibration of the main girder (AS-V-4) 11 0.23983 Fourth symmetric vertical vibration of the main girder (S-V-4) 13 0.27290 First antisymmetric torsional vibration of the main girder (AS-T-1) 25 0.36613 First symmetric torsional vibration of the main girder (S-T-1) AS—Antisymmetric; S—Symmetric; V—Vertical; L—Lateral; T—Torsional. The number represents the order. 3500 Appl. Sci. 2017, 7, 395 5 of 22 Table 1. Typical modes of Taizhou Bridge. No. Frequency/Hz Modal Shape 1 0.07163 First antisymmetric lateral vibration of the main girder (AS-L-1) 2 0.08023 First antisymmetric vertical vibration of the main girder (AS-V-1) 3 0.09512 First symmetric lateral vibration of the main girder (S-L-1) 4 0.11489 Second antisymmetric vertical vibration of the main girder (AS-V-2) 5 0.11758 First symmetric vertical vibration of the main girder (S-V-1) 6 0.13709 Second symmetric vertical vibration of the main girder (S-V-2) 7 0.17089 Third antisymmetric vertical vibration of the main girder (AS-V-3) 8 0.18518 Third symmetric vertical vibration of the main girder (S-V-3) 9 0.23058 Second antisymmetric lateral vibration of the main girder (AS-L-2) 10 0.23786 Fourth antisymmetric vertical vibration of the main girder (AS-V-4) 11 0.23983 Fourth symmetric vertical vibration of the main girder (S-V-4) 13 0.27290 First antisymmetric torsional vibration of the main girder (AS-T-1) 25 0.36613 First symmetric torsional vibration of the main girder (S-T-1) AS—Antisymmetric; S—Symmetric; V—Vertical; L—Lateral; T—Torsional. The number represents the order. Appl. Sci. 2017, 7, 395 5 of 21 (a) (b) (c) (d) (e) (f) Figure 4. Typical fundamental modal shapes of Taizhou Bridge: (a) AS-L-1; (b) AS-V-1; (c) S-L-1; Figure 4. Typical fundamental modal shapes of Taizhou Bridge: (a) AS-L-1; (b) AS-V-1; (c) S-L-1; (d) (d) S-V-1; (e) AS-T-1; (f) S-T-1. AS—Antisymmetric; S—Symmetric; V—Vertical; L—Lateral; S-V-1; (e) AS-T-1; (f) S-T-1. AS—Antisymmetric; S—Symmetric; V—Vertical; L—Lateral; T—Torsional. T—Torsional. The number represents the order. The number represents the order. 3. Wind Loads on Structures 3. Wind Loads on Structures To compute the buffeting responses of Taizhou Bridge in time domain, an important step is to To compute the buffeting responses of Taizhou Bridge in time domain, an important step is integrate the wind loads on structures. In this study, the Alan G. Davenport Wind Loading Chain [30,31] to integrate the wind loads on structures. In this study, the Alan G. Davenport Wind Loading is utilized for analysis, in which the aerostatic forces, buffeting forces, and the self-excited forces Chain [30,31] is utilized for analysis, in which the aerostatic forces, buffeting forces, and the self-excited are included. forces are included. Under the framework of finite element approach, the governing equation for the buffeting Under the framework of ﬁnite element approach, the governing equation for the buffeting analysis analysis of Taizhou Bridge under turbulent winds can be expressed as of Taizhou Bridge under turbulent winds can be expressed as MX(t) CX(t) KX(t) F  F  F (t) F (t) (1) .. . G st b se MX(t) + CX(t) + KX(t) = F + F + F (t) + F (t) (1) G st se where M, C, and K are structural mass, damping and stiffness matrices, respectively; X , X , and X .. . wher are ac ecele M, ra C,ti and on, K vear locity e str,uctural and displ mass, acemen damping t vectand ors; stif FG fness = structur matrices, al gravi respectively; ty vector; X Fst , X is , and the X static are wind force vector; Fb is the buffeting force vector; Fse is the self-excited force vector. For such a acceleration, velocity, and displacement vectors; F = structural gravity vector; F is the static wind st for long ce-span vector; cab Fleis -suppo the buf rted b feting ridfor ge,ce the vector; actions F of is gr the avity self-excited and aero-for static ce vector forces. will he For such avily a long-span influence b se the initial geometry and stiffness of the structure. Hence, the static forces must be firstly applied to cable-supported bridge, the actions of gravity and aero-static forces will heavily inﬂuence the initial the bridge structure to establish an initial structural state for the buffeting analysis. 3.1. Wind Forces on the Bridge The quasi-steady equations accompanied by the aerodynamic admittance functions are utilized to simulate the buffeting forces [30,32]. The buffeting forces, which include lift force, drag force, and pitching moment, acting on the bridge deck per unit length are given as Equations (2)–(4). 1 u(t) w(t)  2' L (t) U B 2C   (C  C ) (2) b  L Lu L D Lw 2 UU  1 u(t) w(t)  2' D (t) U B 2C   (C C ) (3) b  D Du D L Dw 2 UU  1 u(t) w(t)  2 2 ' M (t) U B 2C  C  (4) b  M Mu M Mw 2 UU  where  is the air density; U is the average wind velocity; B is the width of the bridge deck; ' ' ' C C C , , and are steady-state coefficients; C  dC / d ; C  dC / d ; C  dC / d ; α is L D M LL DD MM the attack angle;  ,  ,  ,  ,  , and  are the aerodynamic admittance functions; Lu Lw Du Dw Mu Mw ut () and wt () are fluctuating wind velocities in along-wind and vertical direction, respectively. Appl. Sci. 2017, 7, 395 6 of 22 geometry and stiffness of the structure. Hence, the static forces must be ﬁrstly applied to the bridge structure to establish an initial structural state for the buffeting analysis. 3.1. Wind Forces on the Bridge The quasi-steady equations accompanied by the aerodynamic admittance functions are utilized to simulate the buffeting forces [30,32]. The buffeting forces, which include lift force, drag force, and pitching moment, acting on the bridge deck per unit length are given as Equations (2)–(4). 1 u(t) w(t) L (t) = rU B 2C c + (C + C )c (2) b L Lu D Lw 2 U U 1 u(t) w(t) D (t) = rU B 2C c + (C C )c (3) D Du L Dw 2 U U 1 u(t) w(t) 2 2 M (t) = rU B 2C c + C c (4) b M Mu Mw 2 U U where r is the air density; U is the average wind velocity; B is the width of the bridge deck; C , C , L D 0 0 0 and C are steady-state coefﬁcients; C = dC /da; C = dC /da; C = dC /da; is the attack M L D M L D M angle; c , c , c , c , c , and c are the aerodynamic admittance functions; u(t) and w(t) are Lu Lw Du Dw Mu Mw ﬂuctuating wind velocities in along-wind and vertical direction, respectively. Self-excited forces are included in the buffeting analysis to describe the coupled and feedback relationships between structural motions and wind ﬁeld [32]. The self-excited forces acting on the bridge deck per unit length can be expressed as linear functions of nodal displacement and nodal velocity as Equations (5)–(7). " # . . 1 h(t) Ba(t) h(t) p(t) p(t) 2 2 2 2 L (t) = rU B KH + KH + K H a(t) + K H + KH + K H (5) ae 1 2 3 4 5 6 2 U U B U B " # . . 1 p(t) Ba(t) p(t) h(t) h(t) 2 2 2 2 D (t) = rU B KP + KP + K P a(t) + K P + KP + K P (6) ae 2 3 5 6 1 4 2 U U B U B " # . . 1 h(t) Ba(t) h(t) p(t) p(t) 2 2 2 2 2 M (t) = rU B KA + KA + K A a(t) + K A + KA + K A (7) ae 2 3 5 6 1 4 2 U U B U B where K = wB/U is the reduced circular frequency; h(t), p(t), and a(t)are vertical, lateral, and . . torsional displacements, respectively; h(t), p(t), and a(t) are vertical, lateral, and torsional velocities, respectively; A , H , and P (i = 1, 2, . . . , 6) are ﬂutter derivatives, which are expressed in terms i i i of reduced wind velocity U = U/ f B and determined by geometric conﬁguration of the deck’s cross-section; f is the natural frequency. Equations (5)–(7) is expressed in a frequency manner for complex sinusoidal structural motions [33], but the self-excited forces can be reshaped as aerodynamic stiffness and aerodynamic damping [6,9,34,35]. Hence, the governing equation by Equation (1) can be further expressed as .. . MX(t) + (C C )X(t) + (K K )X(t) = F + F + F (t) (8) se se st G b where C and K are the aerodynamic damping matrix and the aerodynamic stiffness se se matrix, respectively. In such a case, the self-excited forces can be treated as parts of structural components instead of direct loadings. In this study, self-excited forces are realized in ANSYS with the user-deﬁned element MATRIX27 [19,35]. Then, the ultimate FE model for the buffeting analysis of Taizhou Bridge is shown in Figure 5. Appl. Sci. 2017, 7, 395 6 of 21 Self-excited forces are included in the buffeting analysis to describe the coupled and feedback relationships between structural motions and wind field [32]. The self-excited forces acting on the bridge deck per unit length can be expressed as linear functions of nodal displacement and nodal velocity as Equations (5)–(7). 1( ht) Bα(t) h(t) p(t) p(t) 2* * 2 * 2 * * 2 * Lt ()=+ρα U B KH KH +K H ()t+K H +KH +K H  (5) ae 12 3 4 5 6 2 UU B U B     1(p t) Bα(t) p(t) ht() ht() 2* * 2 * 2 * * 2 * D () t=+ρα U B KP KP + K P (t)+ K P + KP + K P (6)  ae 12 3 4 5 6 2 UU B U B     1(ht) Bα(t) ht() p(t) p(t) 2 2 * * 2* 2* * 2 * Mt ()=+ρα U B KA KA +K A ()t+K A +KA +K A (7) ae 12 3 4 5 6 2 UU B U B  where K = ωBU / is the reduced circular frequency; ht () , pt () , and α() t are vertical, lateral, and torsional displacements, respectively; ht () , pt () , and α() t are vertical, lateral, and torsional * * * A H P velocities, respectively; , , and (i = 1, 2, …, 6) are flutter derivatives, which are expressed i i i in terms of reduced wind velocity UU = /fB and determined by geometric configuration of the deck’s cross-section; f is the natural frequency. Equations (5)–(7) is expressed in a frequency manner for complex sinusoidal structural motions [33], but the self-excited forces can be reshaped as aerodynamic stiffness and aerodynamic damping [6,9,34,35]. Hence, the governing equation by Equation (1) can be further expressed as   MX()tt++ () C-C X( ) ( K-K ) X()t= F+ F+ F ()t (8) se se G st b where C and K are the aerodynamic damping matrix and the aerodynamic stiffness se se matrix, respectively. In such a case, the self-excited forces can be treated as parts of structural components instead of direct loadings. In this study, self-excited forces are realized in ANSYS with the user-defined element MATRIX27 [19,35]. Then, the ultimate FE model for the buffeting analysis of Taizhou Bridge Appl. Sci. 2017, 7, 395 7 of 22 is shown in Figure 5. Figure 5. FE model of Taizhou Bridge with MATRIX27 element included. Figure 5. FE model of Taizhou Bridge with MATRIX27 element included. In the buffeting analysis of Taizhou Bridge, the steady-state coefficients and flutter derivatives Inwere obt the bufa feting ined from the sect analysis of ion- Taizhou model wind t Bridge, unnel the tests conducted in the St steady-state coefﬁcients ate Key Lab for Disaster and ﬂutter derivatives Reduction of Civil Engineering at Tongji University [17,19]. As there is no available measured admittance were obtained from the section-model wind tunnel tests conducted in the State Key Lab for Disaster function, it is supposed to be one in the analysis. The wind is assumed to be perpendicular to the main Reduction of Civil Engineering at Tongji University [17,19]. As there is no available measured girder, which means the yaw wind issue is not considered in this study. From Equations (2)–(4), it is easy admittance function, it is supposed to be one in the analysis. The wind is assumed to be perpendicular to the main girder, which means the yaw wind issue is not considered in this study. From Equations (2)–(4), it is easy to ﬁnd that the turbulence is another vital factor that must be included in the time-domain buffeting analysis. Hence, the simulation of the ﬂuctuating wind ﬁeld is conducted Appl. Sci. 2017, 7, 395 7 of 21 in the following section. to find that the turbulence is another vital factor that must be included in the time-domain buffeting 3.2. Simulation of Wind Field analysis. Hence, the simulation of the fluctuating wind field is conducted in the following section. The spectral representation method proposed by Deodatis [36] is utilized to unconditionally 3.2. Simulation of Wind Field simulate the spatial ﬂuctuating wind ﬁeld around the triple-tower suspension bridge. To enhance The spectral representation method proposed by Deodatis [36] is utilized to unconditionally the computational efﬁciency, the ﬂuctuating wind inputs are exerted on the uniformly distributed simulate the spatial fluctuating wind field around the triple-tower suspension bridge. To enhance locations [34the computational effic ], as shown in Figur iene cy, the 6. Ther fluctua e ar tin eg wi 55 nd i simulated nputs are points exerted on the uni in total on form the ly di main stributed girder. Due to locations [34], as shown in Figure 6. There are 55 simulated points in total on the main girder. Due to their insigniﬁcant contributions to the buffeting response, the wind effects on the main towers, the their insignificant contributions to the buffeting response, the wind effects on the main towers, main cables, and suspenders are neglected for the sake of brevity [37]. the main cables, and suspenders are neglected for the sake of brevity [37]. Figure 6. Layout of simulated points for the fluctuating wind field. Figure 6. Layout of simulated points for the ﬂuctuating wind ﬁeld. Kaimal spectrum [38] and Panofsky spectrum [39] are utilized to simulate the along-wind and vertical fluctuating wind velocities with a sampling frequency of 1 Hz, respectively. Davenport Kaimal spectrum [38] and Panofsky spectrum [39] are utilized to simulate the along-wind and coherence function [40] is adopted to present the spatial correlation of fluctuating winds and the vertical ﬂuctuating wind velocities with a sampling frequency of 1 Hz, respectively. Davenport coherence coefficient is conservatively taken as 7. With the Deodatis method, the wind field for coherence function [40] is adopted to present the spatial correlation of ﬂuctuating winds and the Taizhou Bridge is successfully simulated. A typical set of generated along-wind and vertical fluctuating winds at Point 15 is presented in Figure 7. coherence coefﬁcient is conservatively taken as 7. With the Deodatis method, the wind ﬁeld for Taizhou Bridge is successfully simulated. 14 A typical set of generated along-wind and vertical ﬂuctuating winds Point 15 at Point 15 is presented in Figure 7. To verify the effectiveness of the simulated wind ﬁeld, the power spectral densities (PSDs) and -7 cross-correlation functions of the simulated wind samples are compared with theoretical target values. -14 0 600 1200 1800 2400 3000 3600 Some cases of them are selected as examples, which are shown in Figures 8 and 9. The targets refer to t / s Kaimal and Panofsky spectrum in Figure 8, while the target cross-correlation functions in Figure 9 are (a) acquired by the inverse Fourier transform to the corresponding cross-PSDs. In Figure 9, R denotes Point 15 i ,j the cross-correlation function of ﬂuctuating winds at Point i and Point j. It can be seen that both the PSDs and cross-correlation functions conform well to the targets, indicating the high ﬁdelity of the -4 simulated ﬂuctuating wind ﬁeld. -8 0 600 1200 1800 2400 3000 3600 t / s (b) Figure 7. Typical samples from the simulated fluctuating wind field: (a) Along-wind case; (b) Vertical case. To verify the effectiveness of the simulated wind field, the power spectral densities (PSDs) and cross-correlation functions of the simulated wind samples are compared with theoretical target values. Some cases of them are selected as examples, which are shown in Figures 8 and 9. The targets refer to Kaimal and Panofsky spectrum in Figure 8, while the target cross-correlation functions in Figure 9 are acquired by the inverse Fourier transform to the corresponding cross-PSDs. In Figure 9, Ri,j denotes the cross-correlation function of fluctuating winds at Point i and Point j. It can be seen that both the PSDs and cross-correlation functions conform well to the targets, indicating the high fidelity of the simulated fluctuating wind field. -1 -1 Wind velocity / m·s Wind velocity / m·s Appl. Sci. 2017, 7, 395 7 of 21 to find that the turbulence is another vital factor that must be included in the time-domain buffeting analysis. Hence, the simulation of the fluctuating wind field is conducted in the following section. 3.2. Simulation of Wind Field The spectral representation method proposed by Deodatis [36] is utilized to unconditionally simulate the spatial fluctuating wind field around the triple-tower suspension bridge. To enhance the computational efficiency, the fluctuating wind inputs are exerted on the uniformly distributed locations [34], as shown in Figure 6. There are 55 simulated points in total on the main girder. Due to their insignificant contributions to the buffeting response, the wind effects on the main towers, the main cables, and suspenders are neglected for the sake of brevity [37]. Figure 6. Layout of simulated points for the fluctuating wind field. Kaimal spectrum [38] and Panofsky spectrum [39] are utilized to simulate the along-wind and vertical fluctuating wind velocities with a sampling frequency of 1 Hz, respectively. Davenport coherence function [40] is adopted to present the spatial correlation of fluctuating winds and the coherence coefficient is conservatively taken as 7. With the Deodatis method, the wind field for Taizhou Bridge is successfully simulated. A typical set of generated along-wind and vertical Appl. Sci. 2017, 7, 395 8 of 22 fluctuating winds at Point 15 is presented in Figure 7. Point 15 -7 -14 0 600 1200 1800 2400 3000 3600 t / s (a) Point 15 -4 -8 0 600 1200 1800 2400 3000 3600 t / s (b) Figure 7. Typical samples from the simulated fluctuating wind field: (a) Along-wind case; Figure 7. Typical samples from the simulated ﬂuctuating wind ﬁeld: (a) Along-wind case; (b) Vertical case. (b) Vertical case. Appl. Sci. 2017, 7, 395 8 of 21 Appl. Sci. 2017, 7, 395 8 of 21 To verify the effectiveness of the simulated wind field, the power spectral densities (PSDs) and Point 12 10 Point 6 cross-correlation functions of the simulated wind samples are compared with theoretical target Point 12 Point 6 10 Point 25 Point 18 Point 25 Point 18 values. Some cases of them are selected as Targ exampl et es, which are shown in Figures 8 and 9. The Target targets Target Target refer to Kaimal and Panofsky spectrum in Figure 8, while the target cross-correlation functions in Figure 9 are acquired by the inverse Fourier transform to the corresponding cross-PSDs. In Figure 9, Ri,j denotes the cross-correlation function of fluctuating winds at Point i and Point j. It can be seen that both the PSDs and cross-correlation functions conform well to the targets, indicating the high fidelity of the si 10 mulated fluctuating wind field. -1 -1 -3 -2 -1 -3 -2 -1 10 10 10 10 10 10 10 -3 -2 -1 -3 -2 -1 10 10 10 10 10 10 n / Hz n / Hz n / Hz n / Hz (a) (b) (a) (b) Figure 8. Comparison of the PSD of the simulated wind velocity and the target function: (a) Along-wind Figure 8. Comparison of the PSD of the simulated wind velocity and the target function: (a) Along-wind Figure 8. Comparison of the PSD of the simulated wind velocity and the target function: (a) Along-wind case; (b) Vertical case. case; (b) Vertical case. case; (b) Vertical case. 8 0.20 8 0.20 Target Target Target Target R R 10,12 12,20 0.16 R R 10,12 12,20 6 0.16 0.12 0.12 0.08 0.08 2 0.04 0.04 0 0.00 -2000 0 -1000 0 1000 2000 0.00 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000 t / s t / s t / s t / s (a) (b) (a) (b) Figure 9. Cross-correlation functions of the simulated wind samples with comparison to target Figure 9. Cross-correlation functions of the simulated wind samples with comparison to target Figure 9. Cross-correlation functions of the simulated wind samples with comparison to target values: values: (a) Along-wind case; (b) Vertical case. values: (a) Along-wind case; (b) Vertical case. (a) Along-wind case; (b) Vertical case. 4. Buffeting Analysis of Taizhou Bridge 4. Buffeting Analysis of Taizhou Bridge 4.1. Buffeting Responses 4.1. Buffeting Responses Based on the simulated wind field and the FE model shown in Figure 5, the buffeting responses Based on the simulated wind field and the FE model shown in Figure 5, the buffeting responses of Taizhou Bridge are calculated via the governing equation by Equation (8) in time domain. of Taizhou Bridge are calculated via the governing equation by Equation (8) in time domain. The RMS distribution of buffeting displacements of the main girder is presented in Figure 10. The RMS distribution of buffeting displacements of the main girder is presented in Figure 10. To unify the unit, the torsional buffeting displacement has been multiplied by half of the width of To unify the unit, the torsional buffeting displacement has been multiplied by half of the width of the main girder. the main girder. 0.12 0.12 0.09 0.09 0.06 0.06 0.03 0.03 0.00 0.00 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 Bridge Axis / m Bridge Axis / m (a) (a) 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 Bridge Axis / m Bridge Axis / m (b) (b) 2 -1 2 -1 2 -2 S (n) / m ·s 2 -2 S (n) / m ·s Cross-correlation function / m ·s Cross-correlation function / m ·s -1 -1 Wind velocity / m·s Wind velocity / m·s Displacement / m Displacement / m Displacement / m Displacement / m 2 -2 2 -2 Cross-correlation function / m ·s 2 -1 Cross-correlation function / m ·s 2 -1 S (n) / m ·s S (n) / m ·s Appl. Sci. 2017, 7, 395 8 of 21 Point 12 Point 6 Point 25 Point 18 Target Target -1 -3 -2 -1 -3 -2 -1 10 10 10 10 10 10 n / Hz n / Hz (a) (b) Figure 8. Comparison of the PSD of the simulated wind velocity and the target function: (a) Along-wind case; (b) Vertical case. 8 0.20 Target Target R R 10,12 12,20 0.16 0.12 0.08 0.04 0 0.00 -2000 -1000 0 1000 2000 -2000 -1000 0 1000 2000 t / s t / s (a) (b) Figure 9. Cross-correlation functions of the simulated wind samples with comparison to target values: (a) Along-wind case; (b) Vertical case. Appl. Sci. 2017, 7, 395 9 of 22 4. Buffeting Analysis of Taizhou Bridge 4. Buffeting Analysis of Taizhou Bridge 4.1. Buffeting Responses 4.1. Buffeting Responses Based on the simulated wind field and the FE model shown in Figure 5, the buffeting responses Based on the simulated wind ﬁeld and the FE model shown in Figure 5, the buffeting responses of Taizhou Bridge are calculated via the governing equation by Equation (8) in time domain. of Taizhou Bridge are calculated via the governing equation by Equation (8) in time domain. The RMS The RMS distribution of buffeting displacements of the main girder is presented in Figure 10. distribution of buffeting displacements of the main girder is presented in Figure 10. To unify the unit, To unify the unit, the torsional buffeting displacement has been multiplied by half of the width of the torsional buffeting displacement has been multiplied by half of the width of the main girder. the main girder. 0.12 0.09 0.06 0.03 0.00 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 Bridge Axis / m (a) 0.4 0.3 0.2 0.1 0.0 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 Bridge Axis / m Appl. Sci. 2017, 7, 395 9 of 21 (b) 0.025 0.020 0.015 0.010 0.005 0.000 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 Bridge Axis / m (c) Figure 10. RMS distribution of the buffeting displacements of the main girder: (a) Lateral case; Figure 10. RMS distribution of the buffeting displacements of the main girder: (a) Lateral case; (b) Vertica (b) Vertical l case; ( case; c) Tor (c) Torsional sional case case. . As shown in Figure 10, the buffeting displacements of Taizhou Bridge exhibit the most prominent As shown in Figure 10, the buffeting displacements of Taizhou Bridge exhibit the most prominent in th in e c the ent central ral regriegion on of e ofaeach ch m main ain sspan. pan. D Dif ifffer ere ent nt from the other two cases, t from the other two cases, the h maximum e maximu ofm the of the vertical buffeting displacement shows a leaning trend to the mid-tower. The primary reason is that the vertical buffeting displacement shows a leaning trend to the mid-tower. The primary reason is that vertical stiffness of the main girder is mainly dominated by the longitudinal stiffness of the towers. the vertical stiffness of the main girder is mainly dominated by the longitudinal stiffness of the Obviously, the longitudinal stiffness of the mid-tower is much smaller than side towers, which are towers. Obviously, the longitudinal stiffness of the mid-tower is much smaller than side towers, tightly restricted by the anchorages. which are tightly restricted by the anchorages. As a typical sample, the buffeting displacements in the middle of one span are presented as shown As a typical sample, the buffeting displacements in the middle of one span are presented as in Figure 11. It can be seen that the lateral and vertical buffeting displacements are very prominent, shown in Figure 11. It can be seen that the lateral and vertical buffeting displacements are very while the torsional displacement is generally small. Hence, the torsional vibration of the main girder prominent, while the torsional displacement is generally small. Hence, the torsional vibration of the will not be considered in the following vibration control stage of buffeting responses. main girder will not be considered in the following vibration control stage of buffeting responses. 0.4 0.2 0.0 -0.2 -0.4 0 600 1200 1800 2400 3000 3600 Time / s (a) 1.2 0.6 0.0 -0.6 -1.2 0 600 1200 1800 2400 3000 3600 Time / s (b) 0.08 0.04 0.00 -0.04 -0.08 0 600 1200 1800 2400 3000 3600 Time / s (c) Figure 11. Buffeting displacements in the middle of the main girder in one span: (a) Lateral case; (b) Vertical case; (c) Torsional case. 4.2. Time-Frequency Analysis Before conducting the buffeting control of the long-span triple-tower suspension bridge, a prior step is to find the dominant modes in charge of the buffeting responses. These modes are the basis for the design of the MTMD system. Hence, the wavelet transform is selected to characterize the energy distribution of vibrations in frequency domain. In addition, the energy variation along the 2 -1 2 -2 S (n) / m ·s Cross-correlation function / m ·s Displacement / m Displacement / m Displacement / m Displacement / m Displacement / m Displacement / m 2 -2 2 -1 Cross-correlation function / m ·s S (n) / m ·s w Appl. Sci. 2017, 7, 395 9 of 21 0.025 0.020 0.015 0.010 0.005 0.000 -1080 -900 -720 -540 -360 -180 0 180 360 540 720 900 1080 Bridge Axis / m (c) Figure 10. RMS distribution of the buffeting displacements of the main girder: (a) Lateral case; (b) Vertical case; (c) Torsional case. As shown in Figure 10, the buffeting displacements of Taizhou Bridge exhibit the most prominent in the central region of each main span. Different from the other two cases, the maximum of the vertical buffeting displacement shows a leaning trend to the mid-tower. The primary reason is that the vertical stiffness of the main girder is mainly dominated by the longitudinal stiffness of the towers. Obviously, the longitudinal stiffness of the mid-tower is much smaller than side towers, which are tightly restricted by the anchorages. As a typical sample, the buffeting displacements in the middle of one span are presented as shown in Figure 11. It can be seen that the lateral and vertical buffeting displacements are very prominent, while the torsional displacement is generally small. Hence, the torsional vibration of the Appl. Sci. 2017, 7, 395 10 of 22 main girder will not be considered in the following vibration control stage of buffeting responses. 0.4 0.2 0.0 -0.2 -0.4 0 600 1200 1800 2400 3000 3600 Time / s (a) 1.2 0.6 0.0 -0.6 -1.2 0 600 1200 1800 2400 3000 3600 Time / s (b) 0.08 0.04 0.00 -0.04 -0.08 0 600 1200 1800 2400 3000 3600 Time / s (c) Figure 11. Buffeting displacements in the middle of the main girder in one span: (a) Lateral case; Figure 11. Buffeting displacements in the middle of the main girder in one span: (a) Lateral case; (b) Vertical case; (c) Torsional case. (b) Vertical case; (c) Torsional case. 4.2. Time-Frequency Analysis 4.2. Time-Frequency Analysis Before conducting the buffeting control of the long-span triple-tower suspension bridge, a prior Before conducting the buffeting control of the long-span triple-tower suspension bridge, a prior step is to find the dominant modes in charge of the buffeting responses. These modes are the basis step is to ﬁnd the dominant modes in charge of the buffeting responses. These modes are the basis for for the design of the MTMD system. Hence, the wavelet transform is selected to characterize the the design of the MTMD system. Hence, the wavelet transform is selected to characterize the energy energy distribution of vibrations in frequency domain. In addition, the energy variation along the distribution of vibrations in frequency domain. In addition, the energy variation along the time will also be clearly presented. With the Moret wavelet as the mother wavelet, the wavelet transform is applied to the lateral and vertical buffeting displacements. The distribution of the modules of the wavelet coefﬁcients in time-frequency domain are shown in Figure 12. It is found that the energy distribution at each time instant is presented, and the dominant frequency can gather to form a bright frequency band along the time. In such a case, the energy variation is easily evaluated via the primary dominant frequency band. For example, some intermittence can be found in each bright frequency band as the input energy varies with the turbulence. As can be seen in Figure 12, the energy of the lateral buffeting displacement mainly concentrates on two frequencies (0.072 Hz and 0.095 Hz), which correspond to the AS-L-1 mode and the S-L-1 mode. It means that the lateral buffeting displacement is mainly dominated by these two modes. For the vertical buffeting displacement, 0.080 Hz is the most prominent frequency that plays a dominant role in frequency domain. Although there are some vague high-frequency bands captured in the time-frequency spectrum, they are so minor compared to the bright band of 0.080 Hz that they can be neglected. With the aid of Table 1, the vertical buffeting displacement of Taizhou Bridge is proved to be dominated by the AS-V-1 mode. Hence, AS-L-1 and S-L-1 modes are determined as the target modes for the design of vertical MTMD system while AS-V-1 for the lateral MTMD system. Displacement / m Displacement / m Displacement / m Displacement / m Appl. Sci. 2017, 7, 395 10 of 21 time will also be clearly presented. With the Moret wavelet as the mother wavelet, the wavelet transform is applied to the lateral and vertical buffeting displacements. The distribution of the modules of the wavelet coefficients in time-frequency domain are shown in Figure 12. It is found that the energy distribution at each time instant is presented, and the dominant frequency can gather to form a bright frequency band along the time. In such a case, the energy variation is easily evaluated via the primary dominant frequency band. For example, some intermittence can be found Appl. Sci. 2017, 7, 395 11 of 22 in each bright frequency band as the input energy varies with the turbulence. (a) (b) Figure 12. Time-frequency spectrum of the buffeting displacements: (a) Lateral; (b) Vertical. Figure 12. Time-frequency spectrum of the buffeting displacements: (a) Lateral; (b) Vertical. As can be seen in Figure 12, the energy of the lateral buffeting displacement mainly concentrates 5. MTMD Applications on Taizhou Bridge on two frequencies (0.072 Hz and 0.095 Hz), which correspond to the AS-L-1 mode and the S-L-1 mode. It means that the lateral buffeting displacement is mainly dominated by these two modes. For 5.1. Layout of MTMD on the Main Girder the vertical buffeting displacement, 0.080 Hz is the most prominent frequency that plays a dominant role in frequency domain. Although there are some vague high-frequency bands captured in the To exert the effect of MTMD to the maximum extent, the MTMD system should be deposited near time-frequency spectrum, they are so minor compared to the bright band of 0.080 Hz that they can the maximum amplitude of the dominant mode. As the lateral and vertical buffeting displacements be neglected. With the aid of Table 1, the vertical buffeting displacement of Taizhou Bridge is proved are both dominated by the mode whose maximum amplitude appears in the middle of the span, the to be dominated by the AS-V-1 mode. Hence, AS-L-1 and S-L-1 modes are determined as the target MTMD modes fo systems r the are design o attached f ver to tthe ical MTMD center r system egion while of Taizhou AS-V-1 for Bridge the lateral in eachMTMD sy span, as shown stem. in Figure 13. Appl. Sci. 2017, 7, 395 11 of 21 5. MTMD Applications on Taizhou Bridge 5.1. Layout of MTMD on the Main Girder To exert the effect of MTMD to the maximum extent, the MTMD system should be deposited near the maximum amplitude of the dominant mode. As the lateral and vertical buffeting displacements are both dominated by the mode whose maximum amplitude appears in the middle of the span, the MTMD systems are attached to the center region of Taizhou Bridge in each span, as shown in Figure 13. (a) Vertical TMD Vertical TMD Steel box girder Lateral TMD (b) Figure 13. Distribution of the MTMD system in the main girder: (a) Top view; (b) in the main girder. Figure 13. Distribution of the MTMD system in the main girder: (a) Top view; (b) in the main girder. In this study, there are two independent MTMD systems, a lateral MTMD system and a vertical MTMD system, for the buffeting control of Taizhou Bridge. The layouts of TMDs in the two main spans are entirely the same considering the structural symmetry. In the longitudinal direction of each span, the TMDs of two systems are both symmetric about the central line of the main girder. For the lateral MTMD system in each span, there is only one MTMD included. As shown in Figure 13a, the TMDs are distributed on the axial line of the main girder with equivalent distance between each other. The two TMDs symmetrical about the central line are equipped with the same mechanical parameters. For the vertical MTMD system in each span, there are two MTMDs with the same configuration included. Identical to the TMDs in the lateral MTMD system, the TMDs in one vertical MTMD are distributed in parallel to the axial line with equivalent distance between each other. The two TMDs symmetrical about the central line are also equipped with the same mechanical parameters. For the two corresponding TMDs from the two vertical MTMD, they are symmetric about the axial line with the same mechanical parameter. As there is a distance between the vertical MTMD and the axial line, the MTMD can also to some extent participate in the mitigation of torsional buffeting responses. The section view of TMDs of lateral and vertical MTMD systems is shown in Figure 13b. For the simulation of TMDs, there is no element directly available in ANSYS. It is simulated with the combination of two basic elements, the MASS21 element and COMBIN14 element. The MASS21 element is utilized to simulate the mass block of TMD, and the COMBIN14 element is applied to simulate the spring and the damper with the stiffness coefficient and the damping coefficient included. Hence, TMD can be represented via the connection of the MASS21 element and COMBIN14 element. 5.2. Design Parameters of the MTMD The MTMD is consisted of groups of TMDs, whose frequencies are uniformly distributed around the frequency of the dominated mode. The MTMD in Figure 13a is equipped with two groups of TMDs with a shared TMD on the central line. For the TMDs from the central one to the last one on each side in one MTMD, they are named from TMD1 to TMDn using the subscript 1, 2, …, n. For convenience, the mass of the block as well as the damping ratio of each TMD in the same MTMD is assumed to be the same. The basic frequencies of TMDs from TMD1 to TMDn, f1, f2, …, fn, Appl. Sci. 2017, 7, 395 12 of 22 In this study, there are two independent MTMD systems, a lateral MTMD system and a vertical MTMD system, for the buffeting control of Taizhou Bridge. The layouts of TMDs in the two main spans are entirely the same considering the structural symmetry. In the longitudinal direction of each span, the TMDs of two systems are both symmetric about the central line of the main girder. For the lateral MTMD system in each span, there is only one MTMD included. As shown in Figure 13a, the TMDs are distributed on the axial line of the main girder with equivalent distance between each other. The two TMDs symmetrical about the central line are equipped with the same mechanical parameters. For the vertical MTMD system in each span, there are two MTMDs with the same conﬁguration included. Identical to the TMDs in the lateral MTMD system, the TMDs in one vertical MTMD are distributed in parallel to the axial line with equivalent distance between each other. The two TMDs symmetrical about the central line are also equipped with the same mechanical parameters. For the two corresponding TMDs from the two vertical MTMD, they are symmetric about the axial line with the same mechanical parameter. As there is a distance between the vertical MTMD and the axial line, the MTMD can also to some extent participate in the mitigation of torsional buffeting responses. The section view of TMDs of lateral and vertical MTMD systems is shown in Figure 13b. For the simulation of TMDs, there is no element directly available in ANSYS. It is simulated with the combination of two basic elements, the MASS21 element and COMBIN14 element. The MASS21 element is utilized to simulate the mass block of TMD, and the COMBIN14 element is applied to simulate the spring and the damper with the stiffness coefﬁcient and the damping coefﬁcient included. Hence, TMD can be represented via the connection of the MASS21 element and COMBIN14 element. 5.2. Design Parameters of the MTMD The MTMD is consisted of groups of TMDs, whose frequencies are uniformly distributed around the frequency of the dominated mode. The MTMD in Figure 13a is equipped with two groups of TMDs with a shared TMD on the central line. For the TMDs from the central one to the last one on each side in one MTMD, they are named from TMD to TMD using the subscript 1, 2, . . . , n. For convenience, 1 n the mass of the block as well as the damping ratio of each TMD in the same MTMD is assumed to be the same. The basic frequencies of TMDs from TMD to TMD , f , f , . . . , f , are assumed to increase 1 n 1 2 n or decrease with a constant difference. The frequency of the TMD is adjusted by changing the stiffness of the spring component. In the application of MTMD, the mechanical parameter of each TMD, including the mass of the block, the stiffness of the spring, and the damping ratio, must ﬁrst be known. This raises the determination of limited key parameters of MTMD to facilitate the damper design. In this study, ﬁve key parameters are utilized to characterize the structure of the MTMD. They are the number of TMDs N, the mass ratio m, the damping ratio z, the frequency bandwidth ratio d, and the frequency ratio l, respectively. Once the ﬁve parameters are given, the MTMD will naturally be determined. The number of TMDs N refers to the number of TMDs included in one MTMD. In this study, this number will be odd considering the distribution in Figure 13a. The relationship between N and subscript n can be expressed as Equation (9). When the other parameters are given, more TMDs will lead to more intensive natural frequencies. This means the frequency difference between two adjacent TMDs will be smaller with the increment of N. N = 2n 1 (9) The mass ratio m is a key parameter reported to heavily inﬂuence the control effectiveness of the MTMD system. m denotes the ratio of the total mass of TMDs in a MTMD to the equivalent mass of the dominated structural mode, which is deﬁned as (2n 1)m m = (10) equ Appl. Sci. 2017, 7, 395 13 of 22 where m is the mass of the block for each TMD; M is the equivalent mass of the dominated mode. equ To facilitate the design of the MTMD system, the damping ratios for all the TMDs in the same MTMD are assumed to be identical. z is deﬁned as Equation (11) in detail. z = (11) 4pm f where c and f are the damping coefﬁcient and the natural frequency of the ith TMD, respectively. i i Frequency bandwidth ratio d is a measurement of the range of frequencies that can be controlled by the MTMD. It is a vital parameter indicating the robustness of the MTMD and makes the MTMD insensitive to frequency variations of the main structure. It is deﬁned as the ratio of the maximum frequency difference in the TMDs to the central frequency, which is detailed as j f f j d = (12) where f and f are natural frequencies of TMD and TMD , respectively. f is the central frequency n n c 1 1 which is the average of frequencies from TMD to TMD . 1 n The frequency ratio l denotes the ratio of central frequency to the dominated frequency of the main structure, as expressed in Equation (13). In engineering applications, this ratio is often determined as 1 for the MTMD case. Moreover, the modal frequencies of such a long-span bridge will not vary too much after the installation of the MTMD with a not very large mass ratio. When the natural frequency of the main structure varies with the performance deteriorations or maintenance, this parameter is utilized to evaluate the frequency shift of the installed MTMD system and for further performance evaluation of the damper. l = (13) Once the ﬁve parameters above are given, the mechanical parameter for each TMD can be determined. The procedure is detailed as follows, (1) The frequency difference D between two adjacent TMDs is j f f j l f n 1 s D = = (14) n 1 n 1 (2) The frequency from TMD to TMD can be either ascending or descending. It is set as descending 1 n in this study. The natural frequency of the ith TMD is f = 1.5l f (i 1)D (15) i s (3) The mass of the block for each TMD is mM equ m = (16) 2n 1 (4) The damping coefﬁcient of the ith TMD is c = 4pm f z (17) i i (5) The stiffness of the spring for the ith TMD is 2 2 k = 4p m f (18) i Appl. Sci. 2017, 7, 395 14 of 22 5.3. Step-Updating Framework for the Determination of Parameters In order to quantify the control efﬁciency of the MTMD, the attenuation factor is deﬁned as b a h = 100% (19) where a and b are the RMS (Root Mean Square) of the buffeting displacements with and without MTMD, respectively. This attenuation factor is a performance-oriented index that directly presents the effect of the MTMD system [27]. As shown in Figure 14, a step-updating framework is presented to analyze the parameter sensitivity of the control efﬁciency for each MTMD, and determine the preferred values for the main parameters with the expected attenuation factor. The frequency ratio is set as 1 in the analysis, but it is also changed within 10% to investigate the robustness of the MTMD system. The step-updating framework gets the main parameters updated one by one via the parametric analysis. A feedback Appl. Sci. 2017, 7, 395 14 of 21 in each step will be included for the next round of updating, which will save time for the parameter Step 7 determination. The final values of the By the experiencefour main obtained in par testing ameters the framework, are determined the pr aescribed s the ones attenuation in Step 6 factor and are can be satisﬁed renamed within as Nd 2–3 , μdr , ounds. ζd, and ηd, respectively. N ζ 0 0 0 η >η N μ ζ d d Figure 14. Step-updating framework for the determination of the main parameters. Figure 14. Step-updating framework for the determination of the main parameters. 5.4. Parametric Analysis on the MTMD Application There are seven steps included in the whole procedure, which are detailed as Before the parametric analysis on the MTMD application, the dominant frequency and Step 1 Set the initial values for the four main parameters, including the number of TMDs N , the corresponding equivalent mass should be firstly determined. Since the vertical buffeting response of mass ratio m , the damping ratio z , and the frequency bandwidth ratio d , by experience. 0 0 0 Taizhou Bridge is dominated by the AS-V-1 mode, the modal frequency and equivalent modal mass for the vertical MTMD design is definite, as shown in Table 2. However, the lateral buffeting response is mainly dominated by two modes (AS-L-1 and S-L-1). Hence, a weighting calculation is performed on the two modes. The weight coefficients are given according to the average energy represented by the two modes over the time. For example, if the averaged energies on two frequencies are E1 and E2, respectively, the corresponding weight coefficients will be E1/(E1 + E2) and E2/(E1 + E2). Then, the dominant modal mass and frequency are expressed as f=+ wf w f (20) L 1 AS-L-1 2 S-L-1 Mw=+ M wM (21) L 1 AS-L-1 2 S-L-1 where fL and ML are the dominant modal mass and frequency, respectively. fAS-L-1 and fS-L-1 are the modal frequencies of Mode AS-L-1 and Mode S-L-1, respectively. MAS-L-1 and MS-L-1 are the equivalent modal masses of Mode AS-L-1 and Mode S-L-1, respectively. w1 and w2 are weight coefficients, and in this study correspond to 0.7 and 0.3, respectively. In such a case, the dominant modal mass for the lateral MTMD is set as 25,318.6 t and the dominant modal frequency is 0.07868 Hz. The modal mass for vertical MTMD is 24,200.4 t and the modal frequency is 0.08023 Hz. Then, a parametric analysis is conducted in the following section. Appl. Sci. 2017, 7, 395 15 of 22 Step 2 Conduct the parametric analysis on the number of TMDs N with the other parameters unchanged, and determine a preferred value for N in this round. Then, include the positive range between N and the attenuation factor in the next round. Step 3 Conduct the parametric analysis on the mass ratio m with the updated N and two initial other parameters, and determine a preferred value for m in this round. Then, include the positive range between m and the attenuation factor in the next round. Step 4 Conduct the parametric analysis on the damping ratio z with the updated N and m as well as d , and determine a preferred value for z in this round. Then, include the positive range between z and the attenuation factor in the next round. Step 5 Conduct the parametric analysis on the frequency bandwidth ratio d with the updated N, m, and z , and determine a preferred value for d in this round. Then, include the positive range between d and the attenuation factor in the next round. Step 6 Compare the damping ratio h of the MTMD with the updated parameters to the prescribed threshold h . If h h , go to Step 7; Otherwise, go to Step 2. 0 0 Step 7 The ﬁnal values of the four main parameters are determined as the ones in Step 6 and are renamed as N , m , z , and h , respectively. d d d d 5.4. Parametric Analysis on the MTMD Application Before the parametric analysis on the MTMD application, the dominant frequency and corresponding equivalent mass should be ﬁrstly determined. Since the vertical buffeting response of Taizhou Bridge is dominated by the AS-V-1 mode, the modal frequency and equivalent modal mass for the vertical MTMD design is deﬁnite, as shown in Table 2. However, the lateral buffeting response is mainly dominated by two modes (AS-L-1 and S-L-1). Hence, a weighting calculation is performed on the two modes. The weight coefﬁcients are given according to the average energy represented by the two modes over the time. For example, if the averaged energies on two frequencies are E and E , 1 2 respectively, the corresponding weight coefﬁcients will be E /(E + E ) and E /(E + E ). Then, the 1 1 2 2 1 2 dominant modal mass and frequency are expressed as f = w f + w f (20) L 1 AS-L-1 2 S-L-1 M = w M + w M (21) L 1 AS-L-1 2 S-L-1 where f and M are the dominant modal mass and frequency, respectively. f and f are the L L AS-L-1 S-L-1 modal frequencies of Mode AS-L-1 and Mode S-L-1, respectively. M and M are the equivalent AS-L-1 S-L-1 modal masses of Mode AS-L-1 and Mode S-L-1, respectively. w and w are weight coefﬁcients, and in 1 2 this study correspond to 0.7 and 0.3, respectively. In such a case, the dominant modal mass for the lateral MTMD is set as 25,318.6 t and the dominant modal frequency is 0.07868 Hz. The modal mass for vertical MTMD is 24,200.4 t and the modal frequency is 0.08023 Hz. Then, a parametric analysis is conducted in the following section. Table 2. Dominant frequency and corresponding equivalent mass for MTMD design. Lateral Vertical Dominant Mode AS-L-1 S-L-1 AS-V-1 Equivalent modal mass/t 25,988.8 23,754.8 24,200.4 Modal frequency/Hz 0.07163 0.09512 0.08023 Weight coefﬁcient 0.7 0.3 1 Dominant modal mass/t 25,318.6 24,200.4 Dominant modal frequency/Hz 0.07868 0.08023 Appl. Sci. 2017, 7, 395 15 of 21 Table 2. Dominant frequency and corresponding equivalent mass for MTMD design. Lateral Vertical Dominant Mode AS-L-1 S-L-1 AS-V-1 Equivalent modal mass/t 25,988.8 23,754.8 24,200.4 Modal frequency/Hz 0.07163 0.09512 0.08023 Weight coefficient 0.7 0.3 1 Dominant modal mass/t 25,318.6 24,200.4 Appl. Sci. 2017, 7, 395 16 of 22 Dominant modal frequency/Hz 0.07868 0.08023 5.4.1. Number of TMDs 5.4.1. Number of TMDs The control e The control ef fﬁciency ficiency o of f the later the lateral al and and vertic vertical al MT MTMD MD systems is systems is investigated investigated w with ith dif differ ferent ent numbers of TMDs, as shown in Figure 15. During the process, the other three parameters are constant. numbers of TMDs, as shown in Figure 15. During the process, the other three parameters are These constant. The three parameters se three parameters fo for lateral MTMD r late ar ral MT e m = MD 0.02,are z =μ2%, 0 = 0. d 02 =, 0.5, ζ0 = 2 while %, δ those 0 = 0.for 5, whi vertical le thMTMD ose for 0 0 0 are m = 0.02, z = 2%, d = 0.2. The lateral buffeting response is dominated by two modes, so the vertical MTMD are μ0 = 0.02, ζ0 = 2%, δ0 = 0.2. The lateral buffeting response is dominated by two 0 0 0 modes, selection so the of frequency selection bandwidth of frequency ratio ban needs dwidth r to cover atio both needs to cover both of th of the two modes. Ine t such wo modes. In a consideration, such the frequency bandwidth for lateral MTMD is wider than the vertical one. a consideration, the frequency bandwidth for lateral MTMD is wider than the vertical one. 18% 30% 16% 28% λ=0.90 λ=0.90 14% 26% λ=0.95 λ=0.95 λ=1.00 λ=1.00 λ=1.05 12% 24% λ=1.05 λ=1.10 λ=1.10 10% 22% 13579 11 13 15 17 19 21 23 1 3 5 7 9 111315 171921 23 Number of TMDs N Number of TMDs N (a) (b) Figure 15. Influence of the number of TMDs on the control efficiency of MTMD: (a) Lateral case; Figure 15. Inﬂuence of the number of TMDs on the control efﬁciency of MTMD: (a) Lateral case; (b) Vertical case. (b) Vertical case. According to Figure 15, the control efficiency firstly increases with the augment of the number of According to Figure 15, the control efﬁciency ﬁrstly increases with the augment of the number TMDs, but the rising trend gradually levels off and finally becomes stable. This phenomenon is of TMDs, but the rising trend gradually levels off and ﬁnally becomes stable. This phenomenon consistent in both lateral and vertical MTMDs. When the number of TMDs is small (e.g., it is taken as is consistent in both lateral and vertical MTMDs. When the number of TMDs is small (e.g., it is two), the MTMD system will mainly dedicate itself to mitigating the vibration components taken as two), the MTMD system will mainly dedicate itself to mitigating the vibration components characterized by the two frequencies, even though the MTMD system is equipped with a wide characterized by the two frequencies, even though the MTMD system is equipped with a wide frequency bandwidth. However, as the number of TMDs increase, the effective frequencies in the frequency bandwidth. However, as the number of TMDs increase, the effective frequencies in the given frequency bandwidth will be densely distributed so that the effect of the MTMD can be fully given frequency bandwidth will be densely distributed so that the effect of the MTMD can be fully exerted. The maximum control efficiencies of the lateral and vertical MTMDs in this round can be exerted. The maximum control efﬁciencies of the lateral and vertical MTMDs in this round can be 17.9% and 30.4%, respectively. 17.9% and 30.4%, respectively. Except the control efficiency, the robustness is another consideration in the performance of the Except the control efﬁciency, the robustness is another consideration in the performance of the MTMD. The parameter sensitivity of the number of TMDs is also investigated with the frequency MTMD. The parameter sensitivity of the number of TMDs is also investigated with the frequency ratio ratio ranging from 0.90 to 1.10. As is apparent from Figure 15, the difference of the control efficiency ranging from 0.90 to 1.10. As is apparent from Figure 15, the difference of the control efﬁciency for for different frequency ratios will be nearly constant when the number of TMDs is large enough. different frequency ratios will be nearly constant when the number of TMDs is large enough. This This means the attenuation factor will not deviate too much from the prediction even if there is a big means the attenuation factor will not deviate too much from the prediction even if there is a big shift shift captured in the modal frequency of the main structure. captured in the modal frequency of the main structure. Hence, to keep the MTMD with a stable performance, the number of TMDs should lie in the Hence, to keep the MTMD with a stable performance, the number of TMDs should lie in the aforementioned steady range. However, considering the increasing construction difficulty of more aforementioned steady range. However, considering the increasing construction difﬁculty of more TMDs, the lower limit of the range is a preferable selection of this number. In this round, the TMDs, the lower limit of the range is a preferable selection of this number. In this round, the numbers numbers of TMDs in the lateral and vertical MTMD system are set as 13 and 11, respectively. of TMDs in the lateral and vertical MTMD system are set as 13 and 11, respectively. 5.4.2. Mass Ratio The control efﬁciency of the lateral and vertical MTMD systems is also investigated with different mass ratios, as shown in Figure 16. During the process, the damping ratio and frequency bandwidth ratio are kept the same as the parametric analysis on the number of TMDs above, while the number of TMDs is updated with 13 for the lateral MTMD and 11 for the vertical MTMD. Control efficiency Control efficiency Appl. Sci. 2017, 7, 395 16 of 21 Appl. Sci. 2017, 7, 395 16 of 21 5.4.2. Mass Ratio 5.4.2. Mass Ratio The control efficiency of the lateral and vertical MTMD systems is also investigated with different mass ratios, as shown in Figure 16. During the process, the damping ratio and frequency The control efficiency of the lateral and vertical MTMD systems is also investigated with bandwidth ratio are kept the same as the parametric analysis on the number of TMDs above, while different mass ratios, as shown in Figure 16. During the process, the damping ratio and frequency Appl. Sci. 2017, 7, 395 17 of 22 the number of TMDs is updated with 13 for the lateral MTMD and 11 for the vertical MTMD. bandwidth ratio are kept the same as the parametric analysis on the number of TMDs above, while the number of TMDs is updated with 13 for the lateral MTMD and 11 for the vertical MTMD. 32% 40% 32% 28% 40% 36% 24 28% % 36% 32% 20 24% % 32% 28% 20% 16% 28% 24% 12 16% % 24% 20% 12% 8% 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 20% Mass ratio μ 8% Mass ratio μ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 (a) (b) Mass ratio μ Mass ratio μ (a) (b) Figure 16. Influence of the mass ratio on the control efficiency of MTMD: (a) Lateral case; (b) Vertical case. Figure 16. Inﬂuence of the mass ratio on the control efﬁciency of MTMD: (a) Lateral case; (b) Figure 16. Influence of the mass ratio on the control efficiency of MTMD: (a) Lateral case; (b) Vertical case. Vertical case. As shown in Figure 16, the control efficiency of the lateral MTMD increases with the augment of the mass ratio, but the ascending gradient is gradually decreasing. For example, the control As shown in Figure 16, the control efficiency of the lateral MTMD increases with the augment of As shown in Figure 16, the control efﬁciency of the lateral MTMD increases with the augment of efficiency goes up by 8.99% when the mass ratio varies from 0.01 to 0.03, however, it increases by the mass ratio, but the ascending gradient is gradually decreasing. For example, the control the mass ratio, but the ascending gradient is gradually decreasing. For example, the control efﬁciency only 7.18% when the mass ratio changes from 0.03 to 0.10. It is indicated that the improvement of the efficiency goes up by 8.99% when the mass ratio varies from 0.01 to 0.03, however, it increases by goes up by 8.99% when the mass ratio varies from 0.01 to 0.03, however, it increases by only 7.18% control efficiency is rather limited when the mass ratio is larger than 0.03. Hence, the mass ratio is only 7.18% when the mass ratio changes from 0.03 to 0.10. It is indicated that the improvement of the when the mass ratio changes from 0.03 to 0.10. It is indicated that the improvement of the control updated as 0.03 for the lateral MTMD. control efficiency is rather limited when the mass ratio is larger than 0.03. Hence, the mass ratio is efﬁciency is rather limited when the mass ratio is larger than 0.03. Hence, the mass ratio is updated as For the vertical MTMD, there is also a rising trend between the control efficiency and the mass updated as 0.03 for the lateral MTMD. 0.03 for the lateral MTMD. ratio. But the control efficiency gradually decreases when the mass ratio is larger than 0.07. This is For the vertical MTMD, there is also a rising trend between the control efficiency and the mass For the vertical MTMD, there is also a rising trend between the control efﬁciency and the mass mainly because the mass distribution of the bridge structure will be changed when the mass ratio of ratio. But the control efficiency gradually decreases when the mass ratio is larger than 0.07. This is ratio. But the control efﬁciency gradually decreases when the mass ratio is larger than 0.07. This is the MTMD is too large. Then, the natural frequency of the vertical dominant mode will have a big mainly because the mass distribution of the bridge structure will be changed when the mass ratio of mainly because the mass distribution of the bridge structure will be changed when the mass ratio shift from the design value, which accounts for the decreasing of the attenuation factor of the the MTMD is too large. Then, the natural frequency of the vertical dominant mode will have a big of the MTMD is too large. Then, the natural frequency of the vertical dominant mode will have a MTMD. This phenomenon is not captured in the lateral case, which means the lateral mode is not shift from the design value, which accounts for the decreasing of the attenuation factor of the big shift from the design value, which accounts for the decreasing of the attenuation factor of the obviously influenced by the investigated mass ratios. Since the control efficiency is improved the MTMD. This phenomenon is not captured in the lateral case, which means the lateral mode is not MTMD. This phenomenon is not captured in the lateral case, which means the lateral mode is not most dramatically from 0.01 to 0.02 and the value when mass ratio is 0.02 has reached 30.33%, obviously influenced by the investigated mass ratios. Since the control efficiency is improved the obviously inﬂuenced by the investigated mass ratios. Since the control efﬁciency is improved the most the preferred mass ratio of the vertical MTMD is determined as 0.02 in this round. most dramatically from 0.01 to 0.02 and the value when mass ratio is 0.02 has reached 30.33%, dramatically from 0.01 to 0.02 and the value when mass ratio is 0.02 has reached 30.33%, the preferred the preferred mass ratio of the vertical MTMD is determined as 0.02 in this round. mass ratio of the vertical MTMD is determined as 0.02 in this round. 5.4.3. Damping Ratio 5.4.3. Damping Ratio 5.4.3. Damping Ratio The control efficiency of the lateral and vertical MTMD systems is also investigated with different damping ratios, as shown in Figure 17. During the process, the frequency bandwidth ratio The control efficiency of the lateral and vertical MTMD systems is also investigated with The control efﬁciency of the lateral and vertical MTMD systems is also investigated with different are kept the same as the one in the parametric analysis above, while the other parameters are different damping ratios, as shown in Figure 17. During the process, the frequency bandwidth ratio damping ratios, as shown in Figure 17. During the process, the frequency bandwidth ratio are kept the updated as N = 13, μ = 0.03 for lateral MTMD and N = 11, μ = 0.02 for vertical MTMD. are kept the same as the one in the parametric analysis above, while the other parameters are same as the one in the parametric analysis above, while the other parameters are updated as N = 13, updated as N = 13, μ = 0.03 for lateral MTMD and N = 11, μ = 0.02 for vertical MTMD. m = 0.03 for lateral MTMD and N = 11, m = 0.02 for vertical MTMD. 22% 32% 21% 22% 31% 32% 20% 21% 30% 31% 19% 20% 29% 30% 18% 19% 28% 29% 17% 18% 27% 28% 16% 26% 17% 27% 0% 2% 4% 6% 8% 10% 0% 2% 4% 6% 8% 10% 16% Damping Ratio ζ 26% Damping Ratio ζ 0% 2% 4% 6% 8% 10% 0% 2% 4% 6% 8% 10% Dampin (a) ( g Ratio ζ Damping Ratio b) ζ (a) (b) Figure 17. Influence of the damping ratio on the control efficiency of MTMD: (a) Lateral case; (b) Vertical case. Figure 17. Influence of the damping ratio on the control efficiency of MTMD: (a) Lateral case; Figure 17. Inﬂuence of the damping ratio on the control efﬁciency of MTMD: (a) Lateral case; (b) (b) Vertical case. Vertical case. As illustrated in Figure 17, when the damping ratio changes from 0% to 10%, the control efﬁciencies for lateral and vertical MTMD vary only 1.41% and 3.44%, respectively, indicating that ContC roo l effi ntrocien l effi cy ciency Control efficiency Control efficiency Contro Con l etffi rocien l effi cy ciency Contro Co l eff ntro icie l eff nc icie y ncy Appl. Sci. 2017, 7, 395 17 of 21 Appl. Sci. 2017, 7, 395 18 of 22 As illustrated in Figure 17, when the damping ratio changes from 0% to 10%, the control efficiencies for lateral and vertical MTMD vary only 1.41% and 3.44%, respectively, indicating that the damping ratio the has less in damping fluenc ratio e on has th less e coinﬂuence ntrol efficion ency. theFor contr thol e lateral MTMD syst efﬁciency. For theem, wi lateralth the augm MTMD system, ent of with the damping ratio the control efficiency first increases and then gradually declines, which peaks at 20.27% the augment of the damping ratio the control efﬁciency ﬁrst increases and then gradually declines, which with the dam peaks at pin 20.27% g ratio of with 2%. T the h damping is is mainly ratio because i of 2%. n This creasi is ng mainly the da because mping is incr beneficial easing the for damping the energy is dissipation of the structure. However, when the damping ratio goes up to a certain level, the movement beneﬁcial for the energy dissipation of the structure. However, when the damping ratio goes up to a certain of the level, mass the bloc movement k will be s of upp theremass ssed, block leadin will g to be th suppr e dec essed, line of leading the MT toMD the decline control of effthe ect. MTMD Hence, the preferred value of the damping ratio for the lateral MTMD is set as 2% in this round. control effect. Hence, the preferred value of the damping ratio for the lateral MTMD is set as 2% in For the vertical MTMD system, the control efficiency decreases with the augment of the this round. damping ratio, which means only the suppression effect of damping is presented in the movement For the vertical MTMD system, the control efﬁciency decreases with the augment of the damping of the mass block. When the damping ratio is between 0% and 2%, the reduction of the control ratio, which means only the suppression effect of damping is presented in the movement of the mass efficiency is unapparent, but the subsequent reduction is nearly in a big constant gradient. Since the block. When the damping ratio is between 0% and 2%, the reduction of the control efﬁciency is un-damped TMD is very sensitive to the variation of the natural frequency of the main structure [27], unapparent, but the subsequent reduction is nearly in a big constant gradient. Since the un-damped the preferred value of the damping ratio for the vertical MTMD is also set as 2%. TMD is very sensitive to the variation of the natural frequency of the main structure [27], the preferred value of the damping ratio for the vertical MTMD is also set as 2%. 5.4.4. Frequency Band Width Ratio 5.4.4. Frequency Band Width Ratio The control efficiency of the lateral and vertical MTMD systems is investigated with different The control efﬁciency of the lateral and vertical MTMD systems is investigated with different frequency bandwidth ratios, as shown in Figure 18. During the process, the other updated frequency bandwidth ratios, as shown in Figure 18. During the process, the other updated parameters parameters are N = 13, μ = 0.03, ζ0 = 2% for the lateral MTMD and N = 11, μ = 0.02, ζ0 = 2% for the are N = 13, m = 0.03, z = 2% for the lateral MTMD and N = 11, m = 0.02, z = 2% for the vertical MTMD. vertical MTMD. 0 0 22% 32% λ=0.90 λ=0.95 20% 28% λ=1.00 λ=1.05 18% λ=1.10 24% λ=0.90 16% λ=0.95 λ=1.00 20% 14% λ=1.05 λ=1.10 12% 16% 0.00.2 0.40.6 0.81.0 0.0 0.2 0.4 0.6 0.8 1.0 Frequency bandwidth ratio δ Frequency bandwidth ratio δ (a) (b) Figure 18. Influence of the frequency bandwidth ratio on the control efficiency of MTMD: (a) Lateral Figure 18. Inﬂuence of the frequency bandwidth ratio on the control efﬁciency of MTMD: (a) Lateral caese; (b) Vertical case. caese; (b) Vertical case. According to Figure 18, a typical feature in the parametric analysis is that the control efficiency According to Figure 18, a typical feature in the parametric analysis is that the control efﬁciency first increases to a peak and then decreases with the augment of the frequency bandwidth ratio. ﬁrst increases to a peak and then decreases with the augment of the frequency bandwidth ratio. When When the frequency bandwidth ratio is small, the effective frequencies that can be controlled are the frequency bandwidth ratio is small, the effective frequencies that can be controlled are limited. limited. An extreme case is that, when the frequency bandwidth ratio is zero, the MTMD is reduced An extreme case is that, when the frequency bandwidth ratio is zero, the MTMD is reduced to the to the TMD, which means only the central frequency is determined as the controlled target. Hence, TMD, which means only the central frequency is determined as the controlled target. Hence, with the with the increase of the frequency bandwidth ratio, the control efficiency of the MTMD will have a increase of the frequency bandwidth ratio, the control efﬁciency of the MTMD will have a prominent prominent improvement, but when the frequency bandwidth ratio becomes larger, the frequency improvement, but when the frequency bandwidth ratio becomes larger, the frequency difference difference between two adjacent TMDs is increasing. This will lead to the same problem resulting between two adjacent TMDs is increasing. This will lead to the same problem resulting from the case from the case with small number of TMDs. In such a status, only limited discrete frequencies can be with small number of TMDs. In such a status, only limited discrete frequencies can be controlled controlled by the MTMD system, which accounts for the decreasing attenuation factor in the high by the MTMD system, which accounts for the decreasing attenuation factor in the high frequency frequency bandwidth range. bandwidth range. The robustness of the MTMD system is also investigated via changing the frequency ratio, The robustness of the MTMD system is also investigated via changing the frequency ratio, which which is also shown in Figure 18. In this study, there is a 10% maximum frequency shifting to the is also shown in Figure 18. In this study, there is a 10% maximum frequency shifting to the central central frequency considered. For the lateral MTMD, the control efficiencies in the five cases do not frequency considered. For the lateral MTMD, the control efﬁciencies in the ﬁve cases do not deviate deviate too much from each other. However, for the vertical MTMD, the performance is sensitive to too much from each other. However, for the vertical MTMD, the performance is sensitive to the the frequency variation in the range of small frequency bandwidth ratios. This phenomenon can be frequency variation in the range of small frequency bandwidth ratios. This phenomenon can be obviously alleviated utilizing high frequency bandwidth ratios. The lateral buffeting responses are obviously alleviated utilizing high frequency bandwidth ratios. The lateral buffeting responses are dominated by two modes, which means the buffeting responses cannot be well mitigated in the dominated by two modes, which means the buffeting responses cannot be well mitigated in the cases with low frequency bandwidth ratios as the central frequency is between the frequencies of the Control efficiency Control efficiency Appl. Sci. 2017, 7, 395 19 of 22 Appl. Sci. 2017, 7, 395 18 of 21 cases with low frequency bandwidth ratios as the central frequency is between the frequencies of the two modes. That is why the performance of the MTMD sensitive to frequency variations in low two modes. That is why the performance of the MTMD sensitive to frequency variations in low frequency-bandwidth-ratio cases is not captured in the lateral MTMD. frequency-bandwidth-ratio cases is not captured in the lateral MTMD. For the lateral MTMD, the frequency bandwidth ratio should cover the two dominant modes. For the lateral MTMD, the frequency bandwidth ratio should cover the two dominant modes. To‘make the control efﬁciency of the MTMD sufﬁciently performed, the preferred value of the frequency To make the control efficiency of the MTMD sufficiently performed, the preferred value of the bandwidth ratio for the lateral MTMD is determined as 0.5. When the frequency bandwidth ratio is frequency bandwidth ratio for the lateral MTMD is determined as 0.5. When the frequency 0.3, the control efﬁciency of the vertical MTMD is only a little smaller than the peak value, but the bandwidth ratio is 0.3, the control efficiency of the vertical MTMD is only a little smaller than the robustness is obviously improved and can be accepted for application. Hence, the preferred value of peak value, but the robustness is obviously improved and can be accepted for application. Hence, the frequency bandwidth ratio is updated as 0.3 for the vertical MTMD. the preferred value of the frequency bandwidth ratio is updated as 0.3 for the vertical MTMD. In this study, the prescribed thresholds for the control efﬁciencies of the lateral and vertical MTMD In this study, the prescribed thresholds for the control efficiencies of the lateral and vertical systems are both set as 20%. Thus, with only one round, the main parameters of the MTMD systems MTMD systems are both set as 20%. Thus, with only one round, the main parameters of the MTMD for the buffeting control of Taizhou Bridge are determined via the step-updating framework. Of course, systems for the buffeting control of Taizhou Bridge are determined via the step-updating a better attenuation factor can be obtained after another step or two, but the increment would not be as framework. Of course, a better attenuation factor can be obtained after another step or two, but the big as the ﬁrst step due to the small gradient observed in the parametric analysis. The preferred values increment would not be as big as the first step due to the small gradient observed in the parametric for the design parameters of the MTMDs are presented in Table 3. analysis. The preferred values for the design parameters of the MTMDs are presented in Table 3. Table 3. Preferred values for the design parameters of MTMD. Table 3. Preferred values for the design parameters of MTMD. Cases Number of TMDs N Mass Ratio m Damping Ratio z (%) Frequency Band-Width d d d d d Cases Number of TMDs Nd Mass Ratio μd Damping Ratio ζd (%) Frequency Band-Width δd Lateral MTMD 13 0.03 2 0.5 Lateral MTMD 13 0.03 2 0.5 Vertical MTMD 11 0.02 2 0.3 Vertical MTMD 11 0.02 2 0.3 5.5. Performance Evaluation of the MTMD Application 5.5. Performance Evaluation of the MTMD Application With the parameters for the MTMD systems above, the buffeting displacement of Taizhou Bridge With the parameters for the MTMD systems above, the buffeting displacement of Taizhou in the middle of the main girder is compared to the non-MTMD case to make a performance evaluation Bridge in the middle of the main girder is compared to the non-MTMD case to make a performance of the MTMD application. The buffeting displacements of Taizhou Bridge with and without MTMDs evaluation of the MTMD application. The buffeting displacements of Taizhou Bridge with and are shown in Figure 19. without MTMDs are shown in Figure 19. 0.4 without MTMD with MTMD 0.2 0.0 -0.2 -0.4 0 900 1800 2700 3600 Time / s (a) 1.2 without MTMD with MTMD 0.6 0.0 -0.6 -1.2 0 900 1800 2700 3600 Time / s (b) Figure 19. Comparison of the buffeting displacements at the midspan with and without MTMD: (a) Figure 19. Comparison of the buffeting displacements at the midspan with and without MTMD: Lateral; (b) Vertical. (a) Lateral; (b) Vertical. As shown in Figure 19, the lateral and vertical buffeting displacements of the main girder of As shown in Figure 19, the lateral and vertical buffeting displacements of the main girder of Taizhou Bridge are both successfully suppressed. The RMS of the lateral buffeting displacement has Taizhou Bridge are both successfully suppressed. The RMS of the lateral buffeting displacement has been reduced to 0.083 m from 0.105 m with a 21.0% reduction, while the RMS in the vertical case is decreased to 0.255 m from 0.361 m with a 29.4% reduction. To better present the performance of the MTMD system, a spectral analysis is conducted on the buffeting displacements in Figure 19. The PSDs of the displacements with and without MTMD are presented in Figure 20. It is obvious that the Displacement / m Displacement / m Appl. Sci. 2017, 7, 395 20 of 22 been reduced to 0.083 m from 0.105 m with a 21.0% reduction, while the RMS in the vertical case is decreased to 0.255 m from 0.361 m with a 29.4% reduction. To better present the performance of the MTMD system, a spectral analysis is conducted on the buffeting displacements in Figure 19. The PSDs Appl. Sci. 2017, 7, 395 19 of 21 of the displacements with and without MTMD are presented in Figure 20. It is obvious that the energies energies contained in the dominant frequencie contained in the dominant frequencies are remarkably s are rema mitigated rkably mi bytithe gated MTMD, by the MTMD, which means whi the ch means the damper system has fully exerted its function. In conclusion, with the preferred mechanical damper system has fully exerted its function. In conclusion, with the preferred mechanical parameters parameters determined by the step-updating framework for MTMD application, the buffeting responses determined by the step-updating framework for MTMD application, the buffeting responses of the of the long-span triple-tower suspension bridge under strong winds can be effectively mitigated. long-span triple-tower suspension bridge under strong winds can be effectively mitigated. 1.0 20 without MTMD without MTMD with MTMD with MTMD 0.8 0.6 0.4 0.2 0.0 0 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 n / Hz n / Hz (a) (b) Figure 20. PSD comparison of the buffeting displacements with and without MTMD: (a) Lateral; Figure 20. PSD comparison of the buffeting displacements with and without MTMD: (a) Lateral; (b) Vertical. (b) Vertical. 6. Conclusions 6. Conclusions In this study, a parameter sensitivity analysis is conducted on the buffeting control of a long-span In this study, a parameter sensitivity analysis is conducted on the buffeting control of a long-span triple-tower suspension bridge with MTMD. The following conclusions can be drawn accordingly: triple-tower suspension bridge with MTMD. The following conclusions can be drawn accordingly: • The buffeting responses of Taizhou Bridge are the most prominent in the central region of each The buffeting responses of Taizhou Bridge are the most prominent in the central region of each main span. The lateral buffeting displacement is dominated by the AS-L-1 mode and the S-L-1 main span. The lateral buffeting displacement is dominated by the AS-L-1 mode and the S-L-1 mode while the vertical displacement is dominated by the AS-V-1 mode. mode while the vertical displacement is dominated by the AS-V-1 mode. • The control efficiency of MTMD firstly increases with the augment of the number of TMDs, but The control efﬁciency of MTMD ﬁrstly increases with the augment of the number of TMDs, but the rising trend gradually levels off and finally becomes stable. The effect of MTMD can be fully the rising trend gradually levels off and ﬁnally becomes stable. The effect of MTMD can be fully exerted when the amount of TMDs is big enough with effective frequencies densely distributed exerted when the amount of TMDs is big enough with effective frequencies densely distributed in in the given frequency bandwidth. the given frequency bandwidth. • The augment of the mass ratio has a positive contribution to the control efficiency of the The augment of the mass ratio has a positive contribution to the control efﬁciency of the MTMD. MTMD. However, the structural modal frequency will have a big shift from the design value However, the structural modal frequency will have a big shift from the design value when the when the mass ratio is too large, which even leads to the decreasing of the attenuation factor of mass ratio is too large, which even leads to the decreasing of the attenuation factor of the MTMD. the MTMD. The damping ratio has less inﬂuence on the control efﬁciency of the MTMD. Increasing the • The damping ratio has less influence on the control efficiency of the MTMD. Increasing the damping damping is b is beneﬁcial eneficial for for the the energ energy y dissipation dissipatioof n of the the struct structure, ure, but when the but when the damping dampi ratio ng ra goes tio up to a certain level, the movement of the mass block will be suppressed to cause the decline of goes up to a certain level, the movement of the mass block will be suppressed to cause the declin the MTMD e of thcontr e MTMD cont ol effect. rol effect. • When When the f the fr reequency quency ba bandwidth ndwidth ra ratio tio is is sma small, ll, the the effect effective ive frequencies t frequencies hathat t can can be con be t contr rolled ar olled e rather limited. Hence, the control efficiency of the MTMD firstly increases to a peak with the are rather limited. Hence, the control efﬁciency of the MTMD ﬁrstly increases to a peak with augment of the frequency the augment of the frequency bandwidth ratio. Ho bandwidth ratio. However wever, when the , when the frequency bandwidth ra frequency bandwidth ratio tio increases, the frequency difference between two adjacent TMDs increases. Then, only some increases, the frequency difference between two adjacent TMDs increases. Then, only some discret discrete e frequencies can frequencies can be becont contr rolled, which accou olled, which account nts sfor the for thedecrea decreasing sing of the of thecontr contr ool l efficienc efﬁciency y. . • The performance of the MTMD with small frequency bandwidth ratios is sensitive to the frequency variation, but it can be obviously alleviated when utilizing high frequency bandwidth ratios. That means increasing the frequency bandwidth ratio can improve the robustness of the MTMD. • The preferred mechanical parameters of the MTMD can be easily determined by the presented step-updating framework. With the MTMD application, the buffeting responses of the long-span triple-tower suspension bridge under strong winds can be effectively mitigated. 2 -3 PSD / m ·s 2 -3 PSD / m ·s Appl. Sci. 2017, 7, 395 21 of 22 The performance of the MTMD with small frequency bandwidth ratios is sensitive to the frequency variation, but it can be obviously alleviated when utilizing high frequency bandwidth ratios. That means increasing the frequency bandwidth ratio can improve the robustness of the MTMD. The preferred mechanical parameters of the MTMD can be easily determined by the presented step-updating framework. With the MTMD application, the buffeting responses of the long-span triple-tower suspension bridge under strong winds can be effectively mitigated. Acknowledgments: The authors would like to gratefully acknowledge the supports from the National Basic Research Program of China (973 Program) (Grant No. 2015CB060000), the National Science Foundation of China (Grant No. 51378111), the Fundamental Research Funds for the Central Universities (Grant No. 2242015K42028), the Funding of Jiangsu Innovation Program for Graduate Education (Grant No. KYLX16_0258), the Scientiﬁc Research Foundation of Graduate School of Southeast University (Grant No. YBJJ1659), the National Project Funded by the China Scholarship Council (Grant No. 201606090054), and the Project of Innovation-Driven Planning in Central South University (2015CX006). Author Contributions: All authors discussed and agreed upon the idea and made scientiﬁc contributions. Tianyou Tao did the numerical simulation and parametric analyses. Hao Wang and Tianyou Tao wrote the paper and completed the revisions. 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Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD

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Publisher: Multidisciplinary Digital Publishing Institute
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DOI: 10.3390/app7040395
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Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD

Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD

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Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD

Parametric Sensitivity Analysis on the Buffeting Control of a Long-Span Triple-Tower Suspension Bridge with MTMD

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