applied sciences
Article Wind Tunnel Study on Bidirectional Vibration Control of Lattice Towers with Omnidirectional Cantilever-Type Eddy Current TMD
Wenjuan Lou, Zuopeng Wen , Yong Chen * and Mingfeng Huang
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China * Correspondence: [email protected]
Received: 13 June 2019; Accepted: 21 July 2019; Published: 25 July 2019
Abstract: An omnidirectional cantilever-type eddy current tuned mass damper (ECTMD) for lattice towers is introduced to suppress bidirectional vibration of lattice towers, in a form of a cantilever beam with a tip magnet mass. The damping of the ECTMD can be easily changed by tuning the amount of the current. A typhoon-fashion wind environment is simulated for the wind tunnel test. Test results show that there exists an optimal damping of the ECTMD along with an optimal frequency ratio. The scaled aeroelastic model is tested under various wind conditions, and a good effectiveness of ECTMD is observed. The wind directions, perpendicular and parallel to the cross-arm, are the most critical for the design of ECTMD, as the vibration mitigation in either of these two cases is relatively weaker. Finally, a simplified model is established for theoretical analysis in the frequency domain, whereby the variance responses of the tower with and without ECTMD are computed. The numerical results agree well with the experimental results, which corroborates the feasibility of using the proposed omnidirectional cantilever-type ECTMD in suppressing the vibrations of the tower in both along-wind and across-wind directions.
Keywords: omnidirectional cantilever-type ECTMD; lattice tower; wind tunnel test; frequency domain analysis; bidirectional vibration control
1. Introduction Lattice towers are widely used for power transmission, telecommunication, wind turbine, and observation towers, etc. Tall lattice towers usually have relatively low natural frequencies and small damping ratios, which may result in large amplitude vibration under dynamic excitation such as wind loads [1,2]. The excessive vibration may diminish the serviceability of the towers, and cause fatigue damage, joint bolt looseness and even structural failure [3–7]. The lattice transmission tower employed in transmission line systems is an important type of lattice tower. Many collapse accidents happened to transmission towers under strong winds [8–10], as shown in Figure1. The failure of the tower may cause massive blackouts, which results in a lot of economic loss and secondary disasters. With rapid development of power industry, many high-voltage transmission towers with greater height and span were constructed, especially in China. For example, the world’s highest transmission tower currently is located at Zhoushan in China, with a height of 380 m. These tall transmission towers, which are prone to large vibrations, pose new challenges to engineering design. Hence, improving the performance of wind resistance to ensure the structural safety of the tower has become an important issue, and has attracted great attention.
Appl. Sci. 2019, 9, 2978; doi:10.3390/app9152978 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 2978 2 of 23 Appl. Sci. 2019, 9, x 2 of 23
Figure 1. Collapse of a transmission tower in China caused by wind loading.loading.
There areare twotwo mainmain countermeasurescountermeasures to to improve improve the the wind wind resistance resistance of of lattice lattice towers. towers. The The first first is tois to increase increase structural structural sti stiffnessffness by by enlarging enlarging the the size size of of the the members members oror addingadding secondarysecondary members. However, this maymay significantlysignificantly increaseincrease thethe materialmaterial consumptionconsumption and thethe projectproject cost,cost, whichwhich compromises the advantage of of the the low low self-weight self-weight of of lattice lattice towers. towers. The The second second countermeasure countermeasure is to is toinstall install vibration vibration control control devices devices to suppress to suppress the the vibration. vibration. In this In this way, way, the thedynamic dynamic response response of the of theinternal internal forces forces of the of the members members can can be be mitigated. mitigated. Consequently, Consequently, small small size size members members can can meet thethe requirement,requirement, whichwhich meansmeans aa lowerlower cost.cost. Various typestypes of of control control devices devices installed installed on structures on structures were studied,were studied, such as such tuned as mass tuned dampers mass (TMD)dampers [11 (TMD)–19], tuned [11–19] liquid, tuned dampers liquid (TLD) dampers [20], (TLD) liquid [20], column liquid vibration column absorbers vibration (LCVA) absorbers [5], (LCVA) friction dampers[5], friction [21 dampers], etc. Among [21], theseetc. Among control these devices, control using devices, TMD in using tower TMD vibration in tower control vibration seems control to be a promisingseems to be solution a promising due tosolution its efficiency, due to convenienceits efficiency, and convenience low cost [and14]. low A TMD cost typically[14]. A TMD consists typically of a mass,consists a springof a mass, and a a damper. spring Theand controla damper. effectiveness The control of a TMDeffectiveness depends of on a itsTMD parameters, depends such on its as mass,parameters, frequency such and as dampingmass, frequency ratio. Hence, and damping it is of great ratio. significance Hence, it tois of obtain great the significance optimal parameters to obtain forthe theoptimal TMD [parameters15–17]. For TMDsfor the on TMD lattice [15–17]. towers underFor TMDs wind on loads, lattice theoretical towers orunder numerical wind studiesloads, ontheoretical it were conducted.or numerical Pelc studies and Kolator on it were [18] performed conducted. a threePelc and dimensional Kolator (3D)[18] performed finite element a three (FE) analysisdimensional of a truss-type(3D) finite telecommunicationelement (FE) analysis tower of a withtruss-type a TMD te installed,lecommunication and the TMDtower ewithffectiveness a TMD wasinstalled, found and to bethe favorable TMD effectiveness in case of was wind found load. to Tian be favorable and Zeng in [19 case] conducted of wind load. a parametric Tian and studyZeng of[19] tuned conducted mass dampersa parametric fora study long spanof tuned transmission mass dampers tower-line for a long system span under transmission wind load, tower-line and the influencesystem under of TMD wind parameters load, and the on TMDinfluence effectiveness of TMD parameters was presented. on TMD effectiveness was presented. Though the eeffectivenessffectiveness ofof TMD can be evaluated theoretically or numerically, windwind tunnel test isis stillstill anan importantimportant methodmethod forfor physicalphysical verification.verification. For high-rise structures under wind loads, the eeffectivenessffectiveness ofof TMDTMD waswas experimentallyexperimentally examinedexamined [[22–25],22–25], whereaswhereas thethe experimentalexperimental studiesstudies onon latticelattice towerstowers with with TMD TMD installed installed are are scarce scarce [26 [26].]. Xu Xu et al.et [al.23 ][23] studied studied the along-windthe along-wind and across-windand across- performancewind performance of a TMD-installed of a TMD-installed building building under di unfferentder different wind speeds wind and speeds wind and directions, wind directions, and their resultsand their clearly results show clearly the eshowffectiveness the effectiveness of the TMD. of the Wang TMD. et al. Wang [26] performedet al. [26] performed wind tunnel wind tests tunnel for a transmissiontests for a transmission tower-line tower-line system with system TMDs with installed, TMDs and installed, the control and etheffectiveness control effectiveness in either along-wind in either oralong-wind across-wind or direction across-wind was favorable direction under was difavofferentrable wind under speeds. different In these wind experiments speeds. [22 In–26 these], the eexperimentsffectiveness of[22–26], TMD wasthe merelyeffectiveness verified of with TMD a limitedwas merely number verified of TMD with parameters, a limited asnumber the adjustment of TMD ofparameters, TMD parameters as the adjustment was inconvenient. of TMD The paramete optimalrs frequency was inconvenient. and damping The parametersoptimal frequency of the TMDs and aredamping thus hard parameters to obtain of experimentally. the TMDs are thus hard to obtain experimentally. For robustness, the vibration control device installed on a lattice tower needs to be capablecapable of controlling thethe vibration vibration of of the the structure structure under under diff erentdifferent wind wind directions. directions. In addition, In addition, the device the device needs toneeds mitigate to mitigate the along-wind the along-wind response response and across-wind and across-wind response response of the tower of the simultaneously. tower simultaneously. Hence, it isHence, of great it is significance of great significance to design an to omnidirectional design an omnidirectional device that is device able to that accommodate is able to diaccommodatefferent wind directionsdifferent wind and vibrationdirections directions. and vibration Eddy directions current tuned. Eddy mass current damper tuned (ECTMD) mass damper is a newly (ECTMD) developed is a andnewly promising developed device and forpromising vibration device mitigation. for vibratio It hasn beenmitigation. implemented It has b ineen tall implemented buildings [14 in] andtall transmissionbuildings [14] towers and transmission [27], and has towers attracted [27], considerable and has attracted interest considerable [28–30]. For interest a traditional [28–30]. TMD For in a whichtraditional viscous TMD dampers in which are viscous used to dampers provide are damping, used to itprovide is found damping, that there it is exists found a concernthat there of exists fluid a concern of fluid leakage, and the damping adjustment is difficult. The concerns raised can be attenuated by using ECTMD, in which an eddy current is employed to provide the damping. The Appl. Sci. 2019, 9, x 3 of 23 relative movement between the magnetic field and the conductor in the device produces a damping Appl. Sci. 2019, 9, 2978 3 of 23 force approximately proportional to the relative velocity, which can be considered linear damping [31] in the case of an ECTMD mass swaying in small amplitude. The separation of damping and stiffnessleakage, in and an theECTMD damping makes adjustment it potentially is di ffifeasiblecult. The to design concerns an raisedadjustable can beTMD attenuated for performing by using passiveECTMD, [32,33] in which or semi-active an eddy currentvibration is control employed in the to further provide study. the damping. The relative movement betweenIn this the paper, magnetic an fieldomnidirectional and the conductor cantilever-type in the device ECTMD produces is adesigned damping and force fabricated. approximately Its adjustabilityproportional is toexamined the relative via a velocity, free vibration which test. can Wind be considered tunnel tests linear are dampingconducted [31 to] attain in the relatively case of an optimalECTMD parameters mass swaying of the in ECTMD small amplitude. installed on The an separationaeroelastic ofmodel damping of a lattice and sti tower,ffness in the an ECTMDcase of prescribedmakes it potentially design wind feasible loading. to design After an adjustableapplying the TMD parameters for performing to the passive ECTMD, [32,33 the] or dynamical semi-active responsesvibration of control the damped in the further tower under study. different wind speeds and wind directions are then measured to verifyIn this the paper, effectiveness an omnidirectional of the ECTMD. cantilever-type Finally, ECTMDa frequency is designed domain and method fabricated. is presented Its adjustability for evaluationis examined of the via dynamic a free vibration response test. of the Wind tower, tunnel and tests the arenumerical conducted results to attainbased relativelyon the proposed optimal computingparameters method of the ECTMD are compared installed wi onth the an aeroelasticexperimental model results. of a lattice tower, in the case of prescribed design wind loading. After applying the parameters to the ECTMD, the dynamical responses of the 2.damped Cantilever-Type tower under ECTMD different wind speeds and wind directions are then measured to verify the effectiveness of the ECTMD. Finally, a frequency domain method is presented for evaluation of the 2.1.dynamic Configuration response of ECTMD of the tower, and the numerical results based on the proposed computing method are comparedAn omnidirectional with the experimental cantilever-type results. ECTMD was designed for the wind tunnel test of an aeroelastic tower model. As shown in Figure 2, the vertical cantilever is attached with a circular mass 2. Cantilever-Type ECTMD block that moves parallel to the copper plate. The mass block is made of magnet. The copper plate is to2.1. be Configurationfixed on the oftower ECTMD model. The stiffness of the ECTMD is provided by the cantilevered beam with a circular cross section, while the damping is achieved by the relative movement between the magnetAn and omnidirectional the copper plate. cantilever-type The four columns ECTMD of was the designed exterior forwooden the wind frame tunnel penetrate test of anthrough aeroelastic the smalltower holes model. on As the shown copper in Figureplate. 2The, the connectio vertical cantilevern between is attachedthe frame with and a circularthe copper mass plate block was that strengthenedmoves parallel by toremovable the copper hot plate. melt Theadhesive, mass block if the isdistance made of between magnet. the The mass copper and platethe copper is to be plate fixed areon determined. the tower model. Thus, The the sti distanceffness of can the ECTMDbe change isd provided independently by the cantilevered from the cantilever beam with length a circular by removingcross section, the hot while melt the adhesive. damping is achieved by the relative movement between the magnet and the copperThe plate.natural The frequency four columns of ECTMD of the exterior can be woodenindependently frame penetrate tuned by through changing the the small length holes of on the the cantilevercopper plate. while The the connection distance between between the the frame ma andgnet the and copper the platecopper was plate strengthened remains byunchanged. removable Correspondingly,hot melt adhesive, the if damping the distance of ECTMD between can the be mass independently and the copper changed plate by are adjusting determined. the Thus,distance the betweendistance the can magnet be changed and the independently copper plate from while the the cantilever cantilever length length by remains removing unchanged. the hot melt adhesive.
FigureFigure 2. 2. SchematicSchematic of of ECTMD ECTMD with with and and wi withoutthout exterior exterior wooden wooden frame. frame.
2.2. DynamicThe natural Characteristics frequency of ECTMD can be independently tuned by changing the length of the cantilever while the distance between the magnet and the copper plate remains unchanged. The dynamic characteristics of the ECTMD can be estimated theoretically in advance. The Correspondingly, the damping of ECTMD can be independently changed by adjusting the distance estimation results can guide the process of selecting appropriate materials to fabricate the ECTMD. between the magnet and the copper plate while the cantilever length remains unchanged. The force resulting from ECTMD, and acting on the main structure takes a form of [29] + fECTMD =cvdd kx (1) Appl. Sci. 2019, 9, 2978 4 of 23
2.2. Dynamic Characteristics The dynamic characteristics of the ECTMD can be estimated theoretically in advance. The estimation results can guide the process of selecting appropriate materials to fabricate the ECTMD. The force resulting from ECTMD, and acting on the main structure takes a form of [29] Appl. Sci. 2019, 9, x 4 of 23
fECTMD = cdv + kdx (1) where cd and kd are the damping coefficient and stiffness of the ECTMD respectively, and v and wherex are thecd relativeand kd are velocity the damping and displacement coefficient of and the sti massffness to of the the main ECTMD structure respectively, respectively. and v and x are the relativeA simplified velocity method and displacement to predict the of damping the mass ratio to the of main the ECTMD structure is respectively.presented below. Firstly, the magneticA simplified flux density method at the to predictcopper theplate damping is requir ratioed. For of thethe ECTMDcylindrical is presentedmagnet in below. Figure Firstly,3, the themagnetic magnetic flux fluxdensity, density B, at at the the point copper (0, plate0, z) is is [34] required. For the cylindrical magnet in Figure3, the magnetic flux density, B, at the point (0, 0, z) is [34] − zzt− 221/2 2 1/2 rz+ z rzt2 +−()z t 0 0 − ==() 2 2 1/2 2 2 1/2 ≥ BBu 0,0, zB0 [r +z ] − [r +(z t) ] , zt (2) = ( ) = 0 t 0 − Bu B 0, 0, z B0 t , z t (2) 221/2 ≥ rt2+ 2 1/2 [0r0 t ] where BB00 is the magnetic fluxflux densitydensity atat thethe pointpoint (0,(0, 0,0,tt),), r0 andand tt areare the radius and thickness of the cylindrical magnet.
Figure 3. Magnetic flux flux density along cent centerlineerline of cylindrical magnet.
Assume that the center of copper plate is locatedlocated at the point (0, 0, z), and a uniformuniform magnetic fieldfield is assumedassumed toto bebe acrossacross the the copper copper plate. plate. Then Then the the damping damping coe coefficientfficient of of the the eddy eddy current current can can be estimatedbe estimated by by [35 [35]] 2 2 απD Bu t cd =απ 22 (3) DB4ρu t c = (3) d 4ρ where α is a non-dimensional parameter, D is the diameter of the magnet, and ρ is the electrical resistivitywhere α is of a copper.non-dimensional With the parameter, assumption D thatis the the diameter internal of resistance the magnet, of the and copper 𝜌 is the plate electrical at the magneticresistivity fieldof copper. is equal With to the the external assumption resistance, that αthise setinternal as 0.5. resistance The corresponding of the copper damping plate ratio at the of themagnetic ground-mounted field is equal ECTMD to the external can be obtained resistance, by α is set as 0.5. The corresponding damping ratio of the ground-mounted ECTMD can be obtained by 2 2 22πBuD t ζπa BDt= (4) ζ = u 16ρω m (4) a 16ρωm where mmis is the the mass mass of magnet,of magnet, and andω is ω the is natural the natural circular circular frequency frequency of the ground-mounted of the ground-mounted ECTMD. ECTMD.The elastic restoring force of ECTMD is provided by the cantilever, and can be approximately computedThe elastic by restoring force of ECTMD is provided by the cantilever, and can be approximately 3EI computed by fk = x (5) 3EI L3 fx= where E is the Young’s modulus of the cantilever,k 3 I is the second area moment of the cross-section,(5) and L Lwhereis the E length is the ofYoung’s the cantilever. modulus The of the natural cantilever, frequency I is ofthe a second ground-mounted area moment ECTMD, of the designatedcross-section, as theand frequencyL is the length of the of ECTMD,the cantilever. can be The predicted, natural frequency which is based of a ground-mounted on an idealized model ECTMD, of a designated cantilever as the frequency of the ECTMD, can be predicted, which is based on an idealized model of a cantilever beam with a tip mass. Note that the mass of the cantilevered beam is relatively small and negligible. Thus, the corresponding natural circular frequency can be computed by ω = 3EI 3 (6) mL The parameters of the ECTMD are listed in Table 1. Appl. Sci. 2019, 9, 2978 5 of 23 beam with a tip mass. Note that the mass of the cantilevered beam is relatively small and negligible. Thus, the corresponding natural circular frequency can be computed by r 3EI ω = (6) mL3 Appl. Sci. 2019, 9, x 5 of 23 The parameters of the ECTMD are listed in Table1. Table 1. Parameters of ECTMD. Table 1. Parameters of ECTMD. Part Property Symbol Value Part Property Symbol Value Young’s modulus E (N⋅m-2) 1.4×1010 Cantilever 2 10 Moment Young’sof inertia modulus of the cross-section E (NI (mm−4)) 1.02×101.4 -1110 Cantilever · 4 × 11 Moment of inertia of the cross-section I (m ) 1.02 10− Mass m (g) 11.5× Magnet Magnetic fluxMass density at the surface m 𝐵(g) (T) 0.16 11.5 Magnet Magnetic flux densityDiameter at the surface B0D(T) (m) 0.025 0.16 Diameter D (m) 0.025 Electrical resistivity 𝜌 (Ω⋅m) 1.8×10-8 Copper plate 8 Electrical resistivity ρ (Ω m) 1.8 10− Copper plate Thickness t (mm)· 6 × Thickness t (mm) 6 Dynamic characteristics of the ground-mounted ECTMD were identified through free vibration tests.Dynamic The experimental characteristics setup of of the the ground-mounted free vibration tests ECTMD is shown were identified in Figure through 4. To achieve free vibration a free tests.vibration The of experimental the magnet, the setup magnet of the is given free vibration an initial displacement tests is shown and in is Figure then 4released.. To achieve A computer a free vibrationvision-based of the approach magnet, thewith magnet corresponding is given an identifying initial displacement method and[36] isfor then dynamic released. displacement A computer vision-basedresponse is employed approach within the corresponding free vibration identifying test of the methodground-mounted [36] for dynamic ECTMD. displacement A Huawei responseMate 20 isPro employed cellphone in thewas free employed vibration to test record of the the ground-mounted video of motion, ECTMD. and the A Huaweiframe rate Mate is 20235 Pro frames cellphone per wassecond. employed to record the video of motion, and the frame rate is 235 frames per second.
ECTMD
Cellphone recording the motion
Figure 4. Experimental setup of free vibratio vibrationn test of ground-mounted ECTMD.
The vibration responses are firstlyfirstly filteredfiltered with the frequency component near the fundamental frequency left, and then processed byby HilbertHilbert transformtransform [[37]37]
∞ 1 x()τ yt()= Z∞ ( d)τ 1 −xττ (7) y(t) = π −∞ t dτ (7) π t τ − where x(t) is the input signal. Combining x(t) and−∞ y(t) leads to where x(t) is the input signal. Combining=+x(t) and =y(t) leadsitϕ () to =− zt() xt () iyt () ate () ,1 i (8) z(t) = x(t) + iy(t) = a(t)eiϕ(t), i = √ 1 (8) where a(t) and ϕ ()t are the instantaneous amplitude and phase of− x(t) respectively. By using the whereleast squarea(t) and method,ϕ(t) are the the frequency instantaneous and damping amplitude ratio and can phase be identified. of x(t) respectively. By using the least squareFree method, vibration the tests frequency were andperformed damping in two ratio orthog can beonal identified. directions individually. After exploring the resultsFree vibration attained testsvia the were tests performed in these two in twodirectio orthogonalns, it is found directions that individually.either the natural After frequencies exploring theor the results damping attained ratios via of the the tests ECTMD in these in twothe two directions, directions it is foundare very that close. either The the difference natural frequencies between the or results in two directions is less than 2% for natural frequency and 8% for damping ratio, which verifies the direction independent characteristic of the ECTMD used herein. Figure 5 shows the free vibration responses of the ground-mounted ECTMD with and without the copper plate, where the identified damping ratios are 9.2% and 2.4%, respectively. Appl. Sci. 2019, 9, 2978 6 of 23 the damping ratios of the ECTMD in the two directions are very close. The difference between the results in two directions is less than 2% for natural frequency and 8% for damping ratio, which verifies the direction independent characteristic of the ECTMD used herein. Figure5 shows the free vibration responsesAppl. Sci. 2019 of, the 9, x ground-mounted ECTMD with and without the copper plate, where the identified6 of 23 dampingAppl. Sci. 2019 ratios, 9, x are 9.2% and 2.4%, respectively. 6 of 23 1.5 1.5 1.5 1.5 Damping ratio=2.4% Damping ratio=9.2% 1 Damping ratio=2.4% 1 Damping ratio=9.2% 1 1 0.5 0.5 0.5 0.5 0 0 0 0 -0.5 -0.5 -0.5 -0.5 -1 -1 -1 -1 -1.5 -1.5 -1.5 0 0.1 0.2 0.3 -1.5 0 0.1 0.2 0.3 0 0.1Time (s) 0.2 0.3 0 0.1Time (s) 0.2 0.3 Time (s) Time (s) (a) (b) (a) (b) Figure 5. Free vibration of ECTMD. (a) Without copper plate; (b) With copper plate. FigureFigure 5. 5.Free Free vibration vibration of of ECTMD. ECTMD. (a ()a Without) Without copper copper plate; plate; (b ()b With) With copper copper plate. plate. It is found that, given various lengths of the cantilever, the measured ECTMD frequencies agree ItIt is is found found that, that, given given various various lengths lengths of of the the cantilever, cantilever, the the measured measured ECTMD ECTMD frequencies frequencies agree agree well with the predicted values attained via Equation (6), as shown in Figure 6a. In Figure 6b, the wellwell with with the the predicted predicted values values attained attained via via Equation Equationζ (6), (6), as as shown shown in in Figure Figure6a. 6a. In In Figure Figure6b, 6b, the the damping ratio of the ground-mounted ECTMD, ζ t , originates from the intrinsic damping of the dampingdamping ratio ratio of of the the ground-mounted ground-mounted ECTMD,ECTMD, ζt,t originates, originates from from the the intrinsic intrinsic damping damping of of the the cantilever and the additional damping caused by the eddy current. As the magnetic flux density of cantilevercantilever and and the the additional additional damping damping caused caused by by the the eddy eddy current. current. As As the the magnetic magnetic flux flux density density of of the magnetic field increases with the decrease of the distance between the mass and the copper plate, thethe magnetic magnetic field field increases increases with with the the decrease decrease of of the the distance distance between between the the mass mass and and the the copper copper plate, plate, the smaller the distance is, the larger the damping is. The theoretical damping ratio predicted by thethe smaller smaller the the distance distance is, is, the the larger larger the the damping damping is. is. The The theoretical theoretical damping damping ratio ratio predicted predicted by by Equation (4) matches well in general with the measured curve, though there is a slight discrepancy, EquationEquation (4) (4) matches matches well well in in general general with with the the measured measured curve, curve, though though there there is is a a slight slight discrepancy, discrepancy, which may be attributed to the simplified assumption of magnetic flux density. It is also found that whichwhich may may be be attributed attributed to to the the simplified simplified assumption assumption of of magnetic magnetic flux flux density. density. It isIt alsois also found found that that a a wide range of damping ratio can be achieved. The results show that the frequency and damping of widea wide range range of damping of damping ratio ratio can can be achieved. be achieved. The The results results show show that that the frequencythe frequency and dampingand damping of the of the ground-mounted ECTMD are fully adjustable. ground-mountedthe ground-mounted ECTMD ECTMD are fully are adjustable.fully adjustable.
50 30 50 30 Measured value Measured value Measured value 40 Theoretical value 25 MeasuredTheoretical value value 40 Theoretical value 25 Theoretical value 20 30 20 30 15 15 20 20 10 10 10 5 10 5 0 0 060 80 100 120 140 160 0 24681012 60 80Cantilever 100 leng 120th (mm) 140 160 24681012Distance (mm) Cantilever length (mm) Distance (mm) (a) (b) (a) (b) FigureFigure 6. 6.Dynamic Dynamic characteristics characteristics of of ECTMD. ECTMD. (a ()a ECTMD) ECTMD frequency frequency varying varying with with cantilever cantilever length; length; Figure 6. Dynamic characteristics of ECTMD. (a) ECTMD frequency varying with cantilever length; (b()b Damping) Damping ratio ratio varying varying with with distance distance between between copper copper plate plate center center and and magnet magnet surface surface at at ECTMD ECTMD (b) Damping ratio varying with distance between copper plate center and magnet surface at ECTMD frequencyfrequency of of 24 24 Hz. Hz. frequency of 24 Hz. 3. Experimental Setup 3. Experimental Setup 3.1. Finite Element Model of Prototype Transmission Tower 3.1. Finite Element Model of Prototype Transmission Tower The full-scale transmission tower, i.e., the prototype structure, has a height of 43.6 m, and 7.2 × The full-scale transmission tower, i.e., the prototype structure, has a height of 43.6 m, and 7.2 × 7.2 m base. It is recognized that the lower the fundamental frequency is, the more significant the 7.2 m base. It is recognized that the lower the fundamental frequency is, the more significant the resonant dynamic response of a lattice tower will be [2], and the control effectiveness of ECTMD can resonant dynamic response of a lattice tower will be [2], and the control effectiveness of ECTMD can be expected to be better. Consequently, with a relatively high fundamental frequency of 1.88Hz, the be expected to be better. Consequently, with a relatively high fundamental frequency of 1.88Hz, the selected prototype structure is appropriate to examine the lower bound of the ECTMD’s selected prototype structure is appropriate to examine the lower bound of the ECTMD’s effectiveness. effectiveness. Appl. Sci. 2019, 9, 2978 7 of 23
3. Experimental Setup
3.1. Finite Element Model of Prototype Transmission Tower The full-scale transmission tower, i.e., the prototype structure, has a height of 43.6 m, and 7.2 7.2 m base. It is recognized that the lower the fundamental frequency is, the more significant the × resonantAppl. Sci. 2019 dynamic, 9, x response of a lattice tower will be [2], and the control effectiveness of ECTMD7 of can 23 be expected to be better. Consequently, with a relatively high fundamental frequency of 1.88Hz, the selectedA finite prototype element structure model is of appropriate the prototype to examine structur thee is lowerestablished bound by of using the ECTMD’s ANSYS software. effectiveness. The first Asix finite natural element frequencies model of are the listed prototype in Table structure 2, with is establishedcorresponding by using mode ANSYS shapes software. shown in The Figure first six7. natural frequencies are listed in Table2, with corresponding mode shapes shown in Figure7.
TableTable 2. 2.First First sixsix naturalnatural frequenciesfrequencies of of prototype prototype structure. structure.
ModeMode FrequencyFrequency (Hz) (Hz) 11 1.8791.879 22 1.8981.898 33 5.3565.356 44 6.6686.668 55 7.0377.037 66 7.0797.079
z z y x y x y x z (a) (b) (c)
z z z x y x y y x (d) (e) (f)
FigureFigure 7.7. ModeMode shapesshapes ofof prototypeprototype structure.structure. ((aa)) 1st1st mode;mode; ((bb)) 2nd2nd mode;mode; ((cc)) 3rd3rd mode;mode; ((dd)) 4th4th mode;mode; (e(e)) 5th 5th mode; mode; ( f()f) 6th 6th mode. mode.
3.2.3.2. AeroelasticAeroelastic ModelModel forfor WindWind TunnelTunnel Test Test InIn thethe windwind tunneltunnel test,test, similaritiessimilarities ofof somesome importantimportant parametersparameters suchsuch asas StrouhalStrouhal number,number, elasticelastic parameter parameter and and inertial inertial parameter parameter need need to beto be fulfilled. fulfilled. The The similarity similarity of Reynolds of Reynolds number number is not is satisfactorilynot satisfactorily achieved achieved in the in wind the tunnelwind tunnel test. However, test. However, for the anglefor the steel angle tower, steel the tower, Reynolds the Reynolds number doesnumber not significantlydoes not significantly affect the resultsaffect [the38], results and the [38], similitude and the requirement similitude ofrequirement Reynolds numberof Reynolds was thusnumber relaxed. was Froudethus relaxed. number Froude similarity number mainly similarity affects mainly lifting aerodynamicaffects lifting forceaerodynamic of the scaled force model. of the scaled model. It plays an important role in wind tunnel test of free-flying models, such as wings. However, for cantilever-type structures, the deflection due to the vertical gravitational loading is relatively small, compared to that caused by horizontal wind loadings. As only the horizontal vibration is considered herein, the similitude requirement of Froude number was relaxed [39]. An aeroelastic model with a scale factor of 1:30 was designed and fabricated, having a height of 1.45 m, as shown in Figure 8. In the figure, the aeroelastic model consists of eight segments in a vertical line and connected by copper tubes. Each segment consists of angle steel members that meet the similitude requirement of windward area. To simplify the design, the members with a relatively small size are discarded, and the corresponding windward areas are incorporated into the remaining Appl. Sci. 2019, 9, 2978 8 of 23
It plays an important role in wind tunnel test of free-flying models, such as wings. However, for cantilever-type structures, the deflection due to the vertical gravitational loading is relatively small, compared to that caused by horizontal wind loadings. As only the horizontal vibration is considered herein, the similitude requirement of Froude number was relaxed [39]. An aeroelastic model with a scale factor of 1:30 was designed and fabricated, having a height of 1.45 m, as shown in Figure8. In the figure, the aeroelastic model consists of eight segments in a Appl.verticalAppl. Sci. 2019 Sci. line ,2019 9, x and , 9, x connected by copper tubes. Each segment consists of angle steel members that8 of meet 823 of 23 the similitude requirement of windward area. To simplify the design, the members with a relatively members. Generally, the natural frequencies of the steel tower model in which only geometric members.small size Generally, are discarded, the natural and the correspondingfrequencies of windwardthe steel tower areas aremodel incorporated in which intoonly the geometric remaining similarity is considered in scaling, are too high for an aeroelastic model-based wind tunnel study. similaritymembers. is Generally,considered the in natural scaling, frequencies are too high of the for steel an aeroelastic tower model model-based in which only wind geometric tunnel similarity study. Hence, copper tubes are placed between the segments as substitutes of the steel member, to reduce Hence,is considered copper tubes in scaling, are placed are too between high for the an aeroelasticsegments as model-based substitutes wind of the tunnel steel member, study. Hence, to reduce copper the overall lateral stiffness of the aeroelastic model. The details of the connection are depicted in thetubes overall are placedlateral betweenstiffness theof the segments aeroelastic as substitutes model. The of the details steel member,of the connection to reduce are the overalldepicted lateral in Figure 9. Figurestiffness 9. of the aeroelastic model. The details of the connection are depicted in Figure9.
ECTMDECTMD
FigureFigureFigure 8. 8.Diagram 8.Diagram Diagram of ofaeroelastic of aeroelastic aeroelastic tower tower tower model. model. model.
WeldedWelded connectionconnection CopperCopper tube tube
AngleAngle members members
FigureFigureFigure 9. 9.Detailed 9.Detailed Detailed connection connection connection of ofcopper of copper copper tube tube tube and and andangle angle angle members. members. members.
TheThe Theacceleration acceleration acceleration responses responses responses of ofthe of the aeroelasticthe aeroelastic aeroelastic model model model are are measuredare measured measured by by the by the sensorsthe sensors sensors installed installed installed at atthe at the the heightheightheight of of0.60 of 0.60 0.60 m, m, 0.90m, 0.90 0.90 m, m, 1.11 m, 1.11 1.11 m, m, 1.34 m, 1.34 1.34 m mand m and and 1.44 1.44 1.44 m mrespectively, m respectively, respectively, as as shown as shown shown in in Figure in Figure Figure 10.10 .The10. The Thesensors sensors sensors areare aresequentially sequentially sequentially numbered, numbered, numbered, and and and the the the‘X’ ‘X’ and‘X’ and and ‘Y’ ‘Y’ in‘Y’in the in the thesensor sensor sensor names names names denote denote denote the the themeasurements measurements measurements in inX in X X directiondirectiondirection and and and Y Ydirection Y direction direction respectively. respectively. respectively. The The Theacceleration acceleration acceleration sensors sensors sensors were were were produced produced produced by by Kistler by Kistler Kistler Group, Group, Group, and and and the massthe mass of each of each sensor sensor is about is about 4.0 g. 4.0 The g. Thesampling sampling frequency frequency adopted adopted in the in datathe data acquisition acquisition system system is 2048is 2048 Hz. Hz.The Theexperimental experimental setup setup of the of aeroelasthe aeroelastic modeltic model with with an ECTMD an ECTMD installed installed on itson top its topfor for windwind tunnel tunnel test testis shown is shown in Figure in Figure 11. 11. Appl. Sci. 2019, 9, 2978 9 of 23 the mass of each sensor is about 4.0 g. The sampling frequency adopted in the data acquisition system is 2048 Hz. The experimental setup of the aeroelastic model with an ECTMD installed on its top for Appl. Sci. 2019, 9, x 9 of 23 windAppl. Sci. tunnel 2019, test9, x is shown in Figure 11. 9 of 23
ECTMDECTMD
Top point Top point TowerTower top top
Figure 10. Locations of acceleration sensors. FigureFigure 10.10. LocationsLocations ofof accelerationacceleration sensors.sensors.
ECTMDECTMD
Figure 11. Experimental setup. FigureFigure 11. 11.Experimental Experimental setup.setup.
ThroughThrough free free vibrationvibration tests, tests, the the primary primary natu naturalnaturalral frequencies frequencies of of the the aeroelasticaeroelastic model model werewere identifiedidentifiedidentified and and these thesethese are areare listed listed in inin Table Table 33, ,3, andand and thethe the dampingdamping damping ratiosratios ratios ofof of thethe the aeroelasticaeroelastic aeroelastic modelmodel model areare are 1.9%1.9% 1.9% andand 2.0% 2.0%2.0% for for the the the first first first mode mode mode in in inthe the the X-direction X-direction X-direction and and and the the thefirst first first mode mode mode in in the the in Y-direction theY-direction Y-direction respectively. respectively. respectively. The The scaleThescale scale factorsfactors factors forfor the forthe the aeroelasticaeroelastic aeroelastic modelmodel model areare are listedlisted listed inin in TableTable Table 4.4.4 . AsAs As the the scale scalescale factor factorfactor forfor frequency frequencyfrequency is isis determineddetermined experimentally experimentally in in advance, advance, the thethe correspon correspondingcorrespondingding scale scalescale factors factorsfactors for forfor dynamical dynamicaldynamical similarity, similarity, e.g., e.g.,e.g., scalescale factors factors for for accelerationacceleration and and wind wind speed, speed,speed, can cancan th thenthenen be bebe deduced. deduced.deduced. According According to toto similarity similarity principles, principles, thethe non-dimensional non-dimensional parameters parameters of of the the scaled scaled ECTMD ECTMD determined determined in in this this study study are are the the same same as as those thosethose μ forfor a a full-scale full-scale ECTMD, ECTMD, which which benefitsbenefits benefits thethe the designdes designign ofof of full-scalefull-scale full-scale ECTMDs.ECTMDs. ECTMDs. TheThe The massmass mass ratio,ratio, ratio, µ μ= = 6.3%, 6.3%, is is denoteddenoted as as the the ratio ratio of of the the ECTMD ECTMDECTMD ma massmassss to toto the thethe modal modalmodal mass massmass (0.183 (0.183(0.183 kg) kg)kg) of ofof the thethe first firstfirst mode modemode in inin Y Y direction. direction.
Table 3. Natural frequencies of towers. Table 3. Natural frequencies of towers. Prototype Frequency (Hz) Scaled Model Frequency (Hz) Prototype Frequency (Hz) Scaled Model Frequency (Hz) First mode in X direction 1.879 20.9 First mode in X direction 1.879 20.9 First mode in Y direction 1.898 21.2 First mode in Y direction 1.898 21.2 Appl. Sci. 2019, 9, 2978 10 of 23
Table 3. Natural frequencies of towers.
Prototype Frequency (Hz) Scaled Model Frequency (Hz) First mode in X direction 1.879 20.9 First mode in Y direction 1.898 21.2
Table 4. Scale factors of aeroelastic tower model.
Scale Factor Value
Geometry CL 1:30 Density CPS 2.05:1 Mass CM 1:13185 Lateral bending rigidity 1:5.8 106 CEI × Frequency CF 11.2:1 Acceleration CA 2.03:1 Wind speed CV 1:2.69
3.3. Wind Environment Simulation The wind tunnel test was carried out in an atmospheric boundary layer (ABL) wind tunnel, name as ZD-1, in Zhejiang University. ZD-1 has a testing segment with a size of 4 3 18 m, and a testing × × wind speed ranges from 3 m/s to 55 m/s. To simulate the ABL wind, several triangle spires are settled at the inflow entrance for forming vortex. In addition, rows of roughness elements are placed on the ground of the wind tunnel to simulate the turbulence induced by the terrain. An ABL wind flow that matches the characteristics of a typhoon was simulated. The power-law exponent of 0.14 was specified, and the corresponding wind profile is in a form of
z 0.14 v = v ( ) (9) 10 10 where v10 is the mean wind speed at the height of 10 m, and z is the height. The terrain category B in the code of China, namely “Load Code for the Design of Building” (GB 5009-2012) [40], was chosen for the turbulence intensity setup. For a typhoon on terrain category B, the turbulence intensity is amplified by the factor of 1.48 according to Sharma’s recommendation [41]. That is to say
z 0.15 I = 1.48I = 0.207 − (10) T 10 where I is the standard turbulence intensity profile for terrain category B. The measured wind speed profile, turbulence profile and wind power spectrum are shown in Figure 12, and they agree well with the theoretical results. In Figure 12b, Kaimal spectrum is used to generate the theoretical results [42], and has a form of
2 100zσv Sv(z, n) = (11) nz 5/3 3v 1 + 50 v
2 where n is the frequency, and σv is the variance of the fluctuating wind speed. Appl. Sci. 2019, 9, x 10 of 23
Table 4. Scale factors of aeroelastic tower model.
Scale Factor Value
Geometry CL 1:30 Density CΡS 2.05:1 Mass CM 1:13185 6 Lateral bending rigidity CEI 1:5.8 ×10 Frequency CF 11.2:1 Acceleration CA 2.03:1 Wind speed CV 1:2.69
3.3. Wind Environment Simulation The wind tunnel test was carried out in an atmospheric boundary layer (ABL) wind tunnel, name as ZD-1, in Zhejiang University. ZD-1 has a testing segment with a size of 4×3×18 m, and a testing wind speed ranges from 3 m/s to 55 m/s. To simulate the ABL wind, several triangle spires are settled at the inflow entrance for forming vortex. In addition, rows of roughness elements are placed on the ground of the wind tunnel to simulate the turbulence induced by the terrain. An ABL wind flow that matches the characteristics of a typhoon was simulated. The power-law exponent of 0.14 was specified, and the corresponding wind profile is in a form of
= z 0.14 vv10 () (9) 10
where v10 is the mean wind speed at the height of 10 m, and z is the height. The terrain category B in the code of China, namely “Load Code for the Design of Building” (GB 5009-2012) [40], was chosen for the turbulence intensity setup. For a typhoon on terrain category B, the turbulence intensity is amplified by the factor of 1.48 according to Sharma’s recommendation [41]. That is to say
−0.15 ==z IIT 1.48 0.207 (10) 10 where I is the standard turbulence intensity profile for terrain category B. The measured wind speed profile, turbulence profile and wind power spectrum are shown in Figure 12, and they agree well with the theoretical results. In Figure 12b, Kaimal spectrum is used to generate the theoretical results [42], and has a form of 100zσ 2 ()= v Sznv , 5/3 nz (11) 3150v+ v Appl. Sci. 2019, 9, 2978 11 of 23 2 where n is the frequency, and σv is the variance of the fluctuating wind speed.
(a) (b)
Appl. Sci.Figure 2019 12., 12.9, xWind Wind environment environment simulation. simulation. (a) Vertical(a) Vertical profiles profiles of horizontal of horizontal wind speed wind and speed turbulence; and11 of 23 (turbulence;b) Wind power (b) Wind spectrum power at spectrum height of 0.4at height m. of 0.4 m. 3.4. Experimental Procedure 3.4. Experimental Procedure Define the reduction ratio, β, as Define the reduction ratio, β, as RR− β 01× =100% (12) RR R β = 0 −0 1 100% (12) R0 × where R1 and R0 are the RMS acceleration responses of tower top (‘tower top’ is indicated in Figure where10), withR1 and Rwithout0 are the ECTMD RMS acceleration respectively. responses The measur of towerements top (‘towerof the acceleration top’ is indicated sensors in Figureat 1.34 10 m,), withnamely and 3(X) without and ECTMD4(Y), are respectively. employed as The the measurements representative of measurements the acceleration for sensors the aeroelastic at 1.34 m, namely model 3(X)response and 4(Y),in wind are employedtunnel tests. as theThat representative is to say, the measurementsresults in the following for the aeroelastic Sections model4 and response5 are of the in windtower tunnel top. tests. That is to say, the results in the following Sections4 and5 are of the tower top. Since thethe firstfirst natural natural frequency frequenc predominatesy predominates in thein the vibration vibration of transmission of transmission towers towers under under ABL wind,ABL wind, the frequency the frequency ratio, ratio,λ, is defined λ, is defined as as
ffECTMD λ = ECTMD (13) λ f (13) 1f1 where 𝑓 is the natural frequency of the ground-mounted ECTMD, and f is the first natural where fECTMD is the natural frequency of the ground-mounted ECTMD, and f 1 is the first natural frequencyfrequency ofof thethe towertower inin YY direction.direction. NoteNote thatthat ff1 = =21.2 21.2 Hz Hz in in this this paper. paper. Figure 1313 showsshows thethe windwind direction direction angles angles used used in in the the test. test. The The wind wind speed speed in in this this paper paper refers refers to theto the mean mean wind wind speed speed at theat the prototype prototype height height of 10of m.10 m.
Cross-arm
Figure 13. Wind direction angles in wind tunneltunnel test.test.
The experiment procedure procedure can can be be divided divided into into two two phases. phases. In the In first the firstphase, phase, in the in case the of case wind of winddirection direction angle angleof 0° ofand 0◦ andwind wind speed speed of 24.7 of 24.7 m/ ms, /s,the the aeroelastic aeroelastic model model is is tested tested with variousvarious combinations of ECTMDECTMD frequencyfrequency andand dampingdamping ratio.ratio. By surveying the maximum amount of thethe reductionreduction ratio,ratio, thethe favorablefavorable parametersparameters ofof thethe ECTMDECTMD cancan bebe identified.identified. In the second phase, using thethe favorablefavorable parametersparameters of of ECTMD ECTMD determined, determined, the th aeroelastice aeroelastic model model is testedis tested with with and and without without the the ECTMD, under wind speeds of 18.5, 22.6 and 24.7 m/s and wind direction angles in the range of 0° to 90°.
4. Results
4.1. Parametric Study of ECTMD In wind tunnel tests, by changing the length of cantilever beam and the distance between the magnet and the copper plate, the frequency and damping parameter of the ECTMD are thus adjusted to fulfil the experimental requirements. Meanwhile, the values of the frequency and damping ratio of the ECTMD can be approximated conveniently by interpolation, according to the curves shown in Figure 6. Given different frequencies and damping ratios of the ECTMD, corresponding reduction ratios are obtained and shown in Figure 14. Appl. Sci. 2019, 9, 2978 12 of 23
ECTMD, under wind speeds of 18.5, 22.6 and 24.7 m/s and wind direction angles in the range of 0◦ to 90◦.
4. Results
4.1. Parametric Study of ECTMD In wind tunnel tests, by changing the length of cantilever beam and the distance between the magnet and the copper plate, the frequency and damping parameter of the ECTMD are thus adjusted to fulfil the experimental requirements. Meanwhile, the values of the frequency and damping ratio of the ECTMD can be approximated conveniently by interpolation, according to the curves shown in Figure6. Given di fferent frequencies and damping ratios of the ECTMD, corresponding reduction ratiosAppl. Sci. are 2019 obtained, 9, x and shown in Figure 14. 12 of 23
25 25
20 20
15 15
10 10
5 Across-wind 5 Across-wind Along-wind Along-wind 0 0 0.8 0.9 1 1.1 1.2 0 0.05 0.1 0.15 0.2 0.25 Frequency ratio Damping ratio (a) (b)
FigureFigure 14. 14.Variation Variation of of reduction reduction ratios ratios with with ECTMD ECTMD parameters parameters under under wind wind speed speed of of 24.7 24.7 m m/s/s and and
windwind direction direction angle angle of 0of◦.( 0°.a) Variation(a) Variation of reduction of reduction ratio with ratio frequency with frequency ratio of ECTMD; ratio of (bECTMD;) Variation (b) ofVariation reduction of ratio reduction with damping ratio with ratio damping of ECTMD. ratio of ECTMD.
ItIt is is found found that, that, in both in both along-wind along-wind and across-wind and across-wind directions, directions, there exists there a favorable exists a frequency favorable ratiofrequency with a ratio corresponding with a corresponding favorable ECTMD favorable damping ECTMD ratio, damping using ratio, which using a maximum which a reductionmaximum ratioreduction can be ratio achieved. can be Theachieved. favorable The frequencyfavorable ratiofrequency is 0.97 ratio and is the 0.97 favorable and the ECTMDfavorable damping ECTMD ratiodamping is 0.14 ratio for bothis 0.14 along-wind for both along-wind and across-wind and acro directions.ss-wind directions. With the favorable With the frequencyfavorable ratiofrequency and ECTMDratio and damping ECTMD ratio damping obtained ratio in obtained the first phasein the offirst the phase experiment, of the experiment, the corresponding the corresponding reduction ratiosreduction are 23.1% ratios forare the23.1% vibration for the invibration the along-wind in the along-wind direction direction and 22.3% and for 22.3% the vibrationfor the vibration in the across-windin the across-wind direction. direction. FigureFigure 15 15 shows shows the the dynamic dynamic acceleration acceleration responses responses of of the the tower tower top top in in both both time time domain domain and and frequencyfrequency domain, domain, in in the the case case of of the the favorable favorable frequency frequency ratio ratio and and damping damping ratio ratio tuned tuned forfor ECTMD.ECTMD. ItIt is is found found that that both both the the vibration vibration amplitude amplitude in in the the time time history history and and the the resonant resonant component component of of frequencyfrequency response response at at the the fundamental fundamental frequency frequency are are significantly significantly suppressed suppressed by by the the ECTMD. ECTMD. In In addition,addition, only only the the first first mode mode has has a a significant significant contribution contribution to to the the response, response, and and the the modes modes with with higher higher modalmodal frequency frequency areare notnot accessibleaccessible within the freque frequencyncy range. range. Figure Figure 16 16 presents presents a diagram a diagram of ofbi- bi-directionaldirectional responses responses of of the the tower tower top, top, in in which which the along-wind data isis thethe datadata inin FigureFigure 15 15aa within within thethe duration duration of of 10.9 10.9 s tos to 11.0 11.0 s. Its. isIt clearlyis clearly observed observed that that the the ECTMD ECTMD can can mitigate mitigate the vibrationsthe vibrations in the in across-windthe across-wind direction direction and theand along-wind the along-wind direction direction in the in meantime. the meantime.
4 Without ECTMD With ECTMD 2
0
-2
-4 10.7 10.75 10.8 10.85 10.9 10.95 11 Time(s) (a) (b) Appl. Sci. 2019, 9, x 12 of 23
25 25
20 20
15 15
10 10
5 Across-wind 5 Across-wind Along-wind Along-wind 0 0 0.8 0.9 1 1.1 1.2 0 0.05 0.1 0.15 0.2 0.25 Frequency ratio Damping ratio (a) (b)
Figure 14. Variation of reduction ratios with ECTMD parameters under wind speed of 24.7 m/s and wind direction angle of 0°. (a) Variation of reduction ratio with frequency ratio of ECTMD; (b) Variation of reduction ratio with damping ratio of ECTMD.
It is found that, in both along-wind and across-wind directions, there exists a favorable frequency ratio with a corresponding favorable ECTMD damping ratio, using which a maximum reduction ratio can be achieved. The favorable frequency ratio is 0.97 and the favorable ECTMD damping ratio is 0.14 for both along-wind and across-wind directions. With the favorable frequency ratio and ECTMD damping ratio obtained in the first phase of the experiment, the corresponding reduction ratios are 23.1% for the vibration in the along-wind direction and 22.3% for the vibration in the across-wind direction. Figure 15 shows the dynamic acceleration responses of the tower top in both time domain and frequency domain, in the case of the favorable frequency ratio and damping ratio tuned for ECTMD. It is found that both the vibration amplitude in the time history and the resonant component of frequency response at the fundamental frequency are significantly suppressed by the ECTMD. In addition, only the first mode has a significant contribution to the response, and the modes with higher modal frequency are not accessible within the frequency range. Figure 16 presents a diagram of bi- directional responses of the tower top, in which the along-wind data is the data in Figure 15a within the duration of 10.9 s to 11.0 s. It is clearly observed that the ECTMD can mitigate the vibrations in Appl. Sci. 2019, 9, 2978 13 of 23 the across-wind direction and the along-wind direction in the meantime.
4 Without ECTMD With ECTMD 2
0
-2
-4 Appl. Sci.10.7 2019 10.75, 9, x 10.8 10.85 10.9 10.95 11 13 of 23 Appl. Sci. 2019, 9, x Time(s) 13 of 23 (a) (b) 4 0.25 4 0.25 Without ECTMD 0.2 Without ECTMD 2 0.2 With ECTMD 2 With ECTMD 0.15 0 0.15 0 0.1 0.1 -2 -2 0.05 Without ECTMD 0.05 WithoutWith ECTMD ECTMD -4 With ECTMD 0 -410.7 10.75 10.8 10.85 10.9 10.95 11 0 0 50 100 150 10.7 10.75 10.8Time 10.85(s) 10.9 10.95 11 0 50Frequency(Hz 100) 150 Time(c) (s) Freq(uencd) y(Hz) (c) (d) Figure 15. Acceleration response under wind speed of 24.7 m/s and wind direction angle of 0°. (a) Figure 15. Acceleration response under wind speed of 24.7 m/s and wind direction angle of 0°. (a) FigureAlong-wind 15. Acceleration time history; response (b) Along-wind under wind Power-spectral speed of 24.7 density m/s and of acceleration; wind direction (c) Across-wind angle of 0◦ . Along-wind time history; (b) Along-wind Power-spectral density of acceleration; (c) Across-wind (a) Along-windtime history; time (d) Across-wind history; (b) Along-windpower-spectral Power-spectral density of acceleration. density of acceleration; (c) Across-wind timetime history; history; (d) Across-wind(d) Across-wind power-spectral power-spectral density density of of acceleration. acceleration.
3 3 Without ECTMD Without ECTMD 2 With ECTMD 2 With ECTMD 1 1 0 0 -1 -1 -2 -2-2 -1 0 1 2 3 4 -2 -1 0 1 2 32 4 Across-wind acceleration (m/s2) Across-wind acceleration (m/s ) FigureFigure 16. 16.Across-wind Across-wind vibration vibration versus versus along-wind along-wind vibration. vibration. Figure 16. Across-wind vibration versus along-wind vibration. FigureFigure 17 illustrates 17 illustrates the the measured measured RMS RMS accelerations, accelerations, along along with with the the correspondingcorresponding reduction reduction Figure 17 illustrates the measured RMS accelerations, along with the corresponding reduction ratiosratios obtained obtained in the in the wind wind tunnel tunnel test. test. It isIt is found found that that the the tower tower vibrates vibrates inin aa fashionfashion of its first first mode mode ratios obtained in the wind tunnel test. It is found that the tower vibrates in a fashion of its first mode mainly,mainly, whether whether the the ECTMD ECTMD is involvedis involved or or not. not. Without Without the the ECTMD, ECTMD, thethe along-windalong-wind vibration of of mainly, whether the ECTMD is involved or not. Without the ECTMD, the along-wind vibration of the tower is a little smaller than the across-wind vibration, except for the top segment of the tower the towerthe tower is a is little a little smaller smaller than than the the across-wind across-wind vibration, vibration, except except forfor thethe toptop segmentsegment of of the the tower tower where the amplitudes of the vibrations in two directions are very close. This finding accounts for the wherewhere the the amplitudes amplitudes of of the the vibrations vibrations in in twotwo directions are are very very close. close. This This finding finding accounts accounts for the for motivation of using omnidirectional ECTMD which is capable of suppressing both along-wind and the motivationmotivation of of using using omnidirectional omnidirectional ECTMD ECTMD which which is capable is capable of su ofppressing suppressing both bothalong-wind along-wind and across-wind vibrations simultaneously. It is found that, with the presence of the ECTMD, the andacross-wind across-wind vibrations vibrations simultaneously. simultaneously. It Itis isfound found that, that, with with the the presence presence of of the the ECTMD, ECTMD, the the vibrations in the two directions are mitigated, demonstrating the effectiveness of ECTMD. vibrationsvibrations in the in the two two directions directions are are mitigated, mitigated, demonstrating demonstrating the the e ffeffectivenessectiveness ofof ECTMD.ECTMD. Appl.Appl. Sci. Sci.2019 2019, 9,, 9 2978, x 1414 of of 23 23
Appl. Sci. 2019, 9, x 14 of 23
40 40
30 30
20
20Height (m) Height (m) 10 10 Across-wind Across-windAlong-wind 0 Along-wind 0 0 5 10 15 20 25 0 5 10 15 20 25 Reduction ratio (%) Reduction ratio (%) (a) (b) (a) (b) FigureFigure 17. 17. Acceleration responses and reduction ratios along height under wind speed of 24.7 m/s Figure 17.Acceleration Acceleration responses responses and and reduction reduction ratios ratios along along height height under under wind wind speed speed of of 24.7 24.7 m m/s/s and windandand directionwind direction angle angle ofangle 0◦.( of ofa 0°.) 0°. Acceleration ( a()a Acceleration) Acceleration responses; responses; responses; (b) Reduction(b )( Reductionb) Reduction ratios. ratios. ratios.
ItItIt is isis also also found found that thatthat the the the maximummaximum maximum reduction reduction ratio ratio ratio appears appears appears at at theat the the measuring measuring measuring point point point that that thatis located is is located located atatat the thethe tower towertower top top rather ratherratherthan thanthan thethe the toptop top point.point. point. Theoreti Theoretically, Theoretically,cally, if if onlyif only only the the the first first first mode mode mode contributes contributes contributes to tothe to the the vibration,vibration,vibration, the the reduction reduction ratio ratio ratio is is is expectedexpected expected to to be be th the the samee same same along along along the the the height. height. height. Ho However, wever,However, the the contributionsthe contributions contributions ofofof the thethe other otherother modes, modes, which whichwhich maymay may notnot not bebe be well well controlle controlled controlled byd by by the the the ECTMD, ECTMD, ECTMD, can can can also also also affect a ffaffectect the the reductionthe reduction reduction ratio.ratio.ratio. The TheThe frequency frequency response responseresponse ofof of thethe the point point on on the the the to tower towerwer top top top shown shown shown in in Figurein Figure Figure 15b 15 15b indicatesb indicates indicates that that thatexcept except except forforfor the thethe first firstfirst mode, mode, the thethe contributions contributionscontributions ofof of thethe the otherother other modes modes are are are negligible. negligible. negligible. This This This may may may account account account for for thefor the the observedobservedobserved phenomenon. phenomenon.
4.2.4.2.4.2. E EffectffEffectect of of Wind Wind Condition ConditionCondition UsingUsingUsing the the determined determined favorable favorablefavorable parameters parameters of of ofthe the the ECTMD, ECTMD, ECTMD, i.e., i.e., i.e., frequency frequency frequency ratio ratio ratio of of0.97 of 0.97 0.97and and and dampingdampingdamping ratio ratio 0f 0f 0.14, 0.14,0.14, the thethe aeroelastic aeroelasticaeroelastic model model model is tested is istested tested with with andwith and without and without without the ECTMD,the theECTMD, ECTMD, under under wind under wind speeds wind ofspeedsspeeds 18.5, 22.6 ofof 18.5, and 22.6 24.7 and mand/s 24.7 and24.7 m/s windm/s and and direction wind wind direction direction angles inangles angles the rangein inthe the ofrange 0range◦ to of 90 0° of◦ .to 0° 90°. to 90°. The RMS accelerations of the tower with and without the ECTMD are plotted against wind speed TheThe RMS RMS accelerations accelerations of of the the tower tower with with and and without without the the ECTMD ECTMD are are plotted plotted against against wind wind speed speed in Figure 18. Figure 18 shows that under all wind directions, the vibration responses of the tower inin Figure Figure 18 18.. Figure Figure 18 18 shows shows that that under under all all wind wind directions, directions, the the vibration vibration responses responses of of the the tower tower without ECTMD increase continuously with the increasing wind speed, and the vibration responses withoutwithout ECTMD ECTMD increase increase continuously continuously with with the the increasing increasing wind wind speed, speed, and and the the vibration vibration responses responses in in the X and Y directions remain relatively close to each other. It is found that the vibration responses thein Xthe and X and Y directions Y directions remain remain relatively relatively close close to each to each other. other. It is It found is found that that the vibrationthe vibration responses responses of theof tower the tower with with ECTMD ECTMD are significantly are significantly lower lower than than those those of the of tower the tower without without ECTMD. ECTMD. This This proves ofproves the tower that the with ECTMD ECTMD employed are significantly is capable of suppressinglower than vibrationthose of responsesthe tower at without different ECTMD. principal This that the ECTMD employed is capable of suppressing vibration responses at different principal axis provesaxis directions that the ofECTMD the tower employed under various is capable wind of conditions. suppressing vibration responses at different principal directionsaxis directions of the of tower the tower under under various various wind conditions.wind conditions. ) ) ) 2 2 2 ) ) ) 2 2 2 RMS acceleration(m/s RMS acceleration(m/s RMS acceleration(m/s RMS RMS acceleration(m/s RMS acceleration(m/s RMS acceleration(m/s RMS
(a) (b) (c)
(a) Figure 18.(b)Cont . (c) Appl. Sci. 2019, 9, 2978 15 of 23 Appl. Sci. 2019, 9, x 15 of 23 ) ) ) 2 2 2 RMS acceleration(m/s RMS acceleration(m/s RMS acceleration(m/s RMS
(d) (e) (f) ) 2 RMS acceleration(m/s RMS
(g)
Figure 18. Variation of RMS acceleration with wind speed and wind direction. (a) 0°; (b) 15°; (c) 30°; Figure 18. Variation of RMS acceleration with wind speed and wind direction. (a) 0◦;(b) 15◦;(c) 30◦; (d) 45°; (e) 60°; (g) 90°. (d) 45◦;(e) 60◦;(g) 90◦.
UnderUnder the the wind wind speed speed from from 18.5 18.5 toto 24.724.7 mm/s,/s, the RMS RMS accelerations accelerations of of the the tower tower with with and and without without thethe ECTMD ECTMD are are plotted plotted against against wind wind directiondirection angleangle in in Figure Figure 19, 19 ,and and the the corresponding corresponding reduction reduction ratiosratios are are plotted plotted against against wind wind direction direction angle angle in Figure in Figure 20. Figure20. Figure 19 shows 19 shows that, forthat, the for tower the tower without ECTMD,without under ECTMD, all windunder speeds, all wind the speeds, vibration the vibration responses responses maintain maintain a high level a high at level the wind at the direction wind direction angles from 0° to 45°, and the responses drop significantly at the angles from 45o to 90o. For angles from 0◦ to 45◦, and the responses drop significantly at the angles from 45◦ to 90◦. For the tower the tower with ECTMD, the vibration responses decrease continuously in general with the wind with ECTMD, the vibration responses decrease continuously in general with the wind direction angle direction angle increasing from 0° to 90°. Figure 20 shows that under all wind speeds, the reduction increasing from 0 to 90 . Figure 20 shows that under all wind speeds, the reduction ratios in the X and ratios in the X and◦ Y directions◦ are generally higher at the wind direction angle ranging from 30° to Y directions are generally higher at the wind direction angle ranging from 30 to 60 . The reduction 60°. The reduction ratios at 0° and 90° wind directions are generally lower, compared◦ ◦ to other wind ratios at 0 and 90 wind directions are generally lower, compared to other wind directions. The results directions.◦ The results◦ indicate that the control effectiveness of the ECTMD becomes not favorable indicatewhen the that wind the controldirection eff approachesectiveness ofthe the principal ECTMD axis becomes directions not of favorable the tower, when similar the to wind the directionresults approachesshown in the the work principal by Xu axis et al. directions [23] in which of the a tall-building tower, similar structure to the was results studied. shown Consequently, in the work 0° by Xuand et al. 90° [23 wind] in whichdirections a tall-building are the critical structure cases to was be studied.studied preferentially, Consequently, for 0◦ designand 90 ◦ofwind ECTMDs. directions In arethe the case critical of the cases wind to bespeed studied of 24.7 preferentially, m/s, the lower for bound design of of reduction ECTMDs. ratio In the identified case of the(the wind smallest speed ofreduction 24.7 m/s, theratio lower among bound those of in reduction all wind ratio directions), identified is higher (the smallest than that reduction found for ratio the among other thosewind in allspeeds, wind directions), which means is higher that the than ECTMD that found effectiveness for the is other most wind favorable speeds, in this which case. means The corresponding that the ECTMD effreductionectiveness ratio is most ranges favorable from 21.7% in this to case.35.7% The for correspondingthe vibration in reductionthe X direction, ratio rangesand from from 18.3% 21.7% to to 35.7%29.2% for for the the vibration vibration in in the the X Y direction, direction. and from 18.3% to 29.2% for the vibration in the Y direction. Appl.Appl. Sci.Sci.2019 2019,,9 9,, 2978 x 16 ofof 2323 Appl. Sci. 2019, 9, x 16 of 23 ) ) 2 2 ) ) ) 2 2 2 ) 2 RMS acceleration(m/s RMS RMS acceleration(m/s RMS RMS acceleration(m/s RMS RMS acceleration(m/s RMS RMS acceleration(m/s RMS RMS acceleration(m/s RMS
(a) (b) (c) (a) (b) (c) Figure 19. Variation of RMS acceleration with wind direction and wind speed. (a) 18.5 m/s; (b) 22.6 Figure 19. Variation of RMS acceleration with wind direction and wind speed. ( (aa)) 18.5 18.5 m/s; m/s; ( b) 22.6 m/s; (c) 24.7 m/s. m/s;m/s; ( c) 24.7 m/s. m/s. Reduction ratio (%) Reduction ratio (%) Reduction ratio (%) Reduction ratio (%) Reduction ratio (%) Reduction ratio (%)
(a) (b) (c) (a) (b) (c) FigureFigure 20. 20.Variation Variation ofof reductionreduction ratioratio withwith windwind directiondirection andand windwind speed.speed. ((aa)) 18.518.5 mm/s;/s; ((bb)) 22.622.6 mm/s;/s; Figure 20. Variation of reduction ratio with wind direction and wind speed. (a) 18.5 m/s; (b) 22.6 m/s; (c(c)) 24.7 24.7 m m/s./s. (c) 24.7 m/s. 5.5. NumericalNumerical VerificationVerification 5. Numerical Verification 5.1. Methodology 5.1. Methodology 5.1. Methodology The frequency domain approach is introduced herein to attain the numerical dynamic response The frequency domain approach is introduced herein to attain the numerical dynamic response of theThe tower. frequency As shown domain in Figure approach 21, foris introduced a structure he withreinn todegree attain ofthe freedom numerical (DOF), dynamic the governing response of the tower. As shown in Figure 21, for a structure with n degree of freedom (DOF), the governing ofequation the tower. of motion As shown is in Figure 21, for a structure with n degree of freedom (DOF), the governing equation of motion is equation of motion is " T #( .. ) " #( . ) " #( ) ( ) m + LLmLLm+d LT mmmd Lx xxc ck0 00 x k 0 xx f ()t f(t) mLL+ T mmdd L.. xx+ ++ck00 . + x =f ()=t (14) TT dd (14) L md L mmmd v vv++0 00cd ck v 0 kd v =v 0 0 T dddd (14) L mmddvv00ckdd v0 wherewherem m, ,c c,, and andk kare are the the mass, mass, damping damping and and sti stiffnessffness matrices matrices of of the the primary primary structure structure respectively, respectively,x where m, c, and k are the mass, damping and stiffness matrices of the primary structure respectively, is the displacement vector, v =vxx=−d xp, xxp isx the pth nodal displacement at the attaching point of the x is the displacement vector, vx=−−d xp , x p is the pth nodal displacement at the attaching point of xECTMD, is the displacementxd is the displacement vector, ofd the ECTMD,p , p is themd, pcthd and nodalkd displacementare the mass, dampingat the attaching coefficient point and of stitheffness ECTMD, of the ECTMDxd is the respectively, displacementL is of the the location ECTMD, vector md of, thecd ECTMD,and kd are and thef(t) mass, is the vectordamping of the ECTMD, xd is the displacement of the ECTMD, md , cd and kd are the mass, damping windcoefficient loading. and Note stiffness that forof the the ECTMD simplified respectively, model shown L is in the Figure location 21, m vectoris a diagonal of the ECTMD, matrix, i.e.,and f(t) coefficient and stiffness of the ECTMD respectively, L is the location vector of the ECTMD, and f(t) is the vector of wind loading. Note that for the simplified model shown in Figure 21, m is a diagonal is the vector of wind loading. Note that for the simplified model shown in Figure 21, m is a diagonal matrix, i.e., m = diag(m1, m2, , mn) (15) matrix, i.e., ··· m =diag(mm , , ⋅⋅⋅ , m ) where mr (r = 1, ... , n) is the rth lumped= mass. 12 ⋅⋅⋅ n (15) m diag(mm12 , , , mn ) (15) where mr (r = 1,…, n) is the rth lumped mass. where mr (r = 1,…, n) is the rth lumped mass. Appl. Sci. 2019, 9, 2978 17 of 23 Appl. Sci. 2019, 9, x 17 of 23
Figure 21. A multi-degree-of-freedom model with an ECTMD. Figure 21. A multi-degree-of-freedom model with an ECTMD. The vector of the frequency response function is then obtained by [43] The vector of the frequency response function is then obtained by [43] ( ) HHq − 1 T T q =AA−1 {}1,1, ⋅⋅⋅ ,1,, 0 1, 0 (16) Hv { ··· } (16) H v wherewhere ω2 M + ϕTLLTϕm + iωC + K ω2ϕTLm A = − d − d (17) −+ωωω22TT2 Tϕ ++−2 + T + ()MφLLφω L mimddd CKω md φLiω mcd kd A = − − (17) −−++ωωω22T Hq is the vector of frequency response functionLφmmick fordddd the primary structure, Hv is the frequency response function of the ECTMD, M = ϕTmϕ, C = ϕTcϕ, and K = ϕTkϕ are the modal mass, modal damping H andH q modal is thesti vectorffness of matrices frequency of the response primary function structure for respectively, the primaryϕ is structure, the mode shapev is matrix,the frequency and ω isresponse the circular function frequency of the of ECTMD, the excitation. M = ϕTmϕ, C = ϕTcϕ, and K = ϕTkϕ are the modal mass, modal dampingIf m and andϕ modalare available, stiffness then matrices the modal of the mass primary matrix, structureM, can respectively, be obtained inφ the is formthe mode of shape matrix, and ω is the circular frequency of the excitation. M = diag(M , M , , Mn) (18) If m and φ are available, then the modal mass1 2matrix,··· M, can be obtained in the form of Then the modal stiffness matrix, K, can be=⋅⋅⋅ obtained by M diag(M12 ,MM , ,n ) (18)
2 Then the modal stiffnessK = matrix,diag(K K, ,K can, be, Kobtainedn), K rby= ω Mr, r= 1, 2, n (19) 1 2 ··· r ··· = ⋅⋅⋅ =ω2 ⋅⋅⋅ K diag(KK12 , , , Knrrr ), K M , r =1,2, n (19) where ωr is the rth natural circular frequency. Furthermore, the modal damping matrix, C, is in a form of ω where r is the rth natural circular frequency. Furthermore, the modal damping matrix, C, is in a C = diag(C , C2, , Cn), Cr = 2ξrω Mr, r= 1, 2, n (20) form of 1 ··· r ··· where ξr is the modal damping= ratio of the ⋅⋅⋅rth mode. If = theξω two modal damping ⋅⋅⋅ ratios of the prescribed C diag(CC12 , , , Cnrrrr ), C 2 M , r =1,2, n (20) modes are specified, the ξr can be thus computed by using Rayleigh damping [44]. Theξ spectral density matrix for acceleration is therefore where r is the modal damping ratio of the rth mode. If the two modal damping ratios of the ξ prescribed modes are specified, the r canXn beX nthus computed by using Rayleigh damping [44]. 4 T Saa = ω ϕrϕ H∗HsS (21) The spectral density matrix for acceleration is therefores r FrFs r=1 s=1 nn Sφφ= ω 4*T HHS (21) where Hr∗ is the complex conjugate ofaa frequency response r s function r s Frs F of the rth mode, Hs is the frequency rs==11 response function of the sth mode, and SFrFs is the cross-spectral density of the generalized load. * Thewhere variance H r is of the acceleration complex atconjugate the jth point of frequency can be computed response by function of the rth mode, H s is the frequency response function of the sth mode, and S is the cross-spectral density of the Z FrsF 2 generalized load. The variance of accelerationσaj = at theSajj jthdω point can be computed by (22) σω2 = ajS ajjd (22) where Sajj is the power-spectral density of acceleration of the jth point. where S ajj is the power-spectral density of acceleration of the jth point. Appl.Appl. Sci.Sci. 20192019,, 99,, 2978x 1818 of 2323
The cross-spectral density of along-wind fluctuations between the points at height z and z’ is The cross-spectral density of along-wind fluctuations between the points at height z and z0 is Szzn(),,′′′= SznSzn ()() , , Coh,,() zzn (23) vvvq ( ) = ( ) ( ) ( ) Sv z, z0, n Sv z, n Sv z0, n Coh z, z0, n ()(23) where the Kaimal spectrum shown in Equation (11) is adopted herein for computing Sznv , , and ()′ whereCohzz the, , n Kaimal is the spectrum coherence shown function in Equationin a form of (11) [45] is adopted herein for computing Sv(z, n), and Coh(z, z , n) is the coherence function in a form of [45] 0 − ′ ′ 7nz z Coh()zz , , n =− exp ! (24) 7n zv z0 Coh(z, z0, n) = exp | − | (24) − v where v is the mean wind speed between the points at height z and z’ . Thus, the power-spectral wheredensityv isof the the mean wind wind pressure speed is between the points at height z and z0. Thus, the power-spectral density of the wind pressure is ()′′′′= ρμ2 ()() μ () () ( ) Swssv zzn,,2 z zvzvz S zzn ,, (25) Sw(z, z0, n) = ρ µs(z)µs(z0)v(z)v(z0)Sv(z, z0, n) (25) μ wherewhereρ ρand andµ s ares theare airthe densityair density and and shape shape coe fficoefficientcient respectively. respectively. The The cross-spectral cross-spectral density density of the of generalizedthe generalized wind wind loads loads is is
HH H H SDzDzzzSzznzz= Z Z () (′′′′ )ϕϕ () ( ) (,, ) dd FFij i j w (26) SFiFj = 00 D(z)D(z0)ϕi(z)ϕj(z0)Sw(z, z0, n)dzdz0 (26) where D is the nominal width of0 the0 windward area. where D is the nominal width of the windward area. 5.2. Numerical Results 5.2. Numerical Results The tower is divided into 17 segments, as shown in Figure 22, and the relevant parameters are listedThe in towerTable is5. dividedThus, for into numerical 17 segments, analysis, as shown a simplified in Figure model 22, andwith the 17 relevantcorresponding parameters nodes are is listedemployed. in Table The5. location Thus, for of numericaleach node is analysis, set at the a simplifiedcenter of the model corresponding with 17 corresponding segment. Therefore nodes the is employed.rth lumped The mass location is the mass of each of nodethe rth is setsegment. at the centerThe mode of the shapes corresponding used herein segment. are derived Therefore from thethe rfiniteth lumped element mass analysis is the mass of ofthe the prototyperth segment. stru Thecture, mode in shapeswhich usedseventeen herein aremode derived shapes from with the finitecorresponding element analysis natural of frequencies the prototype are structure,obtained. inThe which modal seventeen damping mode ratios shapes of the with first corresponding two modes in naturalthe Y direction frequencies are areset as obtained. 4.4%, which The modal is identified damping from ratios the ofexperimental the first two dynamic modes in response the Y direction results. areThus, set the as 4.4%, modal which mass, is modal identified damping from theand experimental modal stiffness dynamic matrixes response can be results. computed, Thus, according the modal to mass,Equations modal (18)–(20). damping and modal stiffness matrixes can be computed, according to Equations (18)–(20).
FigureFigure 22.22. SegmentsSegments ofof prototypeprototype structure.structure. Appl. Sci. 2019, 9, x 19 of 23
Appl. Sci. 2019, 9, 2978 Table 5. Segment parameters of the prototype tower. 19 of 23
Segment No. Segment Mass (kg) Segment Height (m) Shape Coefficient Windward Area (m𝟐) 1 87.5Table 5. Segment parameters 0.5 of the prototype 2.03 tower. 0.26 2 498.0 2.1 1.63 1.71 2 Segment3 No. Segment858.0 Mass (kg) Segment 0.9 Height (m) Shape Coe2.05ffi cient Windward Area2.38 (m ) 4 1418.4 87.5 1.4 0.5 2.012.03 0.26 1.50 5 2686.3 498.0 3.9 2.1 2.151.63 1.71 1.78 3 858.0 0.9 2.05 2.38 6 4699.9 418.4 1.6 1.4 2.182.01 1.50 2.98 7 51006.3 686.3 4.8 3.9 2.232.15 1.78 2.45 8 6603.9 699.9 1.4 1.6 2.162.18 2.98 2.36 9 7767.9 1006.3 2.5 4.8 2.212.23 2.45 1.63 10 8269.0 603.9 1.3 1.4 2.372.16 2.36 0.70 9 767.9 2.5 2.21 1.63 11 10422.1 269.0 2.5 1.3 2.37 0.70 1.47 12 11454.4 422.1 2.6 2.5 2.432.37 1.47 1.58 13 12524.3 454.4 3 2.6 2.482.43 1.58 1.82 14 13842.7 524.3 4.2 3 2.512.48 1.82 2.62 14 842.7 4.2 2.51 2.62 15 640.5 2.6 2.56 1.58 15 640.5 2.6 2.56 1.58 16 16723.1 723.1 1.4 1.4 2.43 1.48 1.48 17 171612.2 1612.2 6.9 6.9 2.59 4.19 4.19
In the case of the along-wind direction, by using the aforementioned theoretical approach in the In the case of the along-wind direction, by using the aforementioned theoretical approach in the frequency domain analysis, the dynamic responses of the tower with and without the ECTMD are frequency domain analysis, the dynamic responses of the tower with and without the ECTMD are attained. For different mass ratios, the reduction ratios varying with the damping ratio of ECTMD attained. For different mass ratios, the reduction ratios varying with the damping ratio of ECTMD and frequency ratio are illustrated in Figure 23. There exists an optimal frequency ratio and ECTMD and frequency ratio are illustrated in Figure 23. There exists an optimal frequency ratio and ECTMD damping ratio, whereby the maximum reduction ratio can be achieved. The maximum reduction damping ratio, whereby the maximum reduction ratio can be achieved. The maximum reduction ratios ratios for mass ratios of 0.02, 0.04, and 0.063 are 15.8%, 21.2%, and 25.1% respectively. For the mass for mass ratios of 0.02, 0.04, and 0.063 are 15.8%, 21.2%, and 25.1% respectively. For the mass ratio of ratio of 0.063 used in the experiment, it is found that the optimal frequency ratio and ECTMD 0.063 used in the experiment, it is found that the optimal frequency ratio and ECTMD damping ratio in damping ratio in theory are 0.98 and 0.11 respectively. theory are 0.98 and 0.11 respectively.
(a) (b) (c)
Figure 23. 23. ReductionReduction ratio ratio over over damping damping ratio ratio and and frequency frequency ratio. ratio. (a) Mass (a) Mass ratio ratio = 0.02;= 0.02;(b) Mass (b) Massratio =ratio 0.04;= (0.04;c) Mass (c) Massratio = ratio 0.063.= 0.063.
By using the parameters in the experiment, the the theoretical results are compared with the experimental results in the along-wi along-windnd direction, as shown in in Figure Figure 24 and Table 66.. ItIt cancan bebe foundfound fromfrom Figure Figure 2424 thatthat the the theoretical theoretical reduction reduction ratios ratios generally generally match match well well with with the the experimental experimental values values as the frequency ratio and damping ratio of the ECTMD change, though the experimental reduction ratios are aa littlelittle lower.lower. The The theoretical theoretical highest highest reduction reduction ratio ratio corresponds corresponds to theto the frequency frequency ratio ratio of 0.97 of 0.97and theand ECTMD the ECTMD damping damping ratio of ratio 0.11, whereasof 0.11, thewher experimentaleas the experimental highest reduction highest ratio reduction corresponds ratio correspondsto the frequency to the ratio frequency of 0.97 ratio and of the 0.97 ECTMD and the damping ECTMD ratiodamping of 0.14. ratio Under of 0.14. the Under frequency the frequency ratio of ratio0.97 andof 0.97 the and ECTMD the ECTMD damping damping ratio of ratio 0.14, of the0.14, experimental the experimental reduction reduction ratio ratio is 23.1%, is 23.1%, whereas whereas the thetheoretical theoretical reduction reduction ratio ratio is 24.8%. is 24.8%.
Appl. Sci. 2019, 9, x 20 of 23
Table 6. Dynamic responses in case of wind speed of 24.7m/s and wind direction angle of 0°.
Wind RMS Acceleration Reduction Ratio (%) Speed Without ECTMD With ECTMD (m/s) Theoretical Experimental Theoretical Experimental Theoretical Experimental 18.5 0.400 0.447 0.304 0.363 24.1 18.8 22.6 0.658 0.672 0.497 0.513 24.5 23.7 24.7 0.818 0.801 0.615 0.616 24.8 23.1
Table 6 shows that at wind speeds of 24.7 and 22.6 m/s, the theoretical results of acceleration and reduction ratio match well with the experimental results. In case of the wind speed of 18.5 m/s, the theoretical results of acceleration are a litter lower, whereas the reduction ratio computed via the frequency domain approach is slightly higher than the experimental results. Generally speaking, the comparison shows that the theoretical approach in frequency domain introduced herein is capable of Appl. Sci. 2019, 9, 2978 20 of 23 producing a good prediction of the dynamic responses of the tower with or without ECTMD. Reduction ratio (%) ratio Reduction (%) ratio Reduction
(a) (b)
FigureFigure 24. 24.Reduction Reduction ratios ratios in in along-wind along-wind direction direction under under wind wind speed speed of of 24.7 24.7 m m/s/s and and wind wind direction direction
angleangle of of 0 0°.◦.( (aa)) VariationVariation of of reduction reduction ratio ratio with with frequency frequency ratio; ratio; ( b(b)) Variation Variation of of reduction reduction ratio ratio with with dampingdamping ratio ratio of of ECTMD. ECTMD.
6. ConclusionsTable 6. Dynamic responses in case of wind speed of 24.7m/s and wind direction angle of 0◦. An omnidirectional cantilever-typeRMS Acceleration ECTMD is introduced herein for vibration control of lattice Wind Speed Reduction Ratio (%) Without ECTMD With ECTMD towers(m under/s) wind loading. A wind tunnel study of a lattice tower with an ECTMD was conducted to gain insight intoTheoretical the performance Experimental of the ECTMD Theoretical used Experimental in suppressing Theoretical vibration of Experimental lattice towers. Conclusions18.5 can be 0.400drawn as follows. 0.447 0.304 0.363 24.1 18.8 22.6 0.658 0.672 0.497 0.513 24.5 23.7 1) The24.7 wind tunnel 0.818 study on 0.801the lattice tower 0.615 shows that 0.616 there exists 24.8a significant 23.1across-wind vibration, and it deserves efforts made to design and manufacture an omnidirectional ECTMD Tableto suppress6 shows the that bidirectional at wind speeds vibration. of 24.7 andThe 22.6parameter m /s, the adjustability theoretical of results the newly of acceleration designed and reductionECTMD ratioin a wide match range well withis verified the experimental experiment results.ally. Thus, In casethe ofvibration the wind of speedthe tower of 18.5 can m /bes, the theoreticalsuppressed results by tuning of acceleration the frequency are a of litter the lower,ECTMD whereas to a value the reductionclose to the ratio first computed natural frequency via the frequencyof the domain tower. approachThe parametric is slightly study higher of ECTMD than the in experimentalwind tunnel test results. shows Generally that with speaking, a frequency the comparisonratio of shows 0.97 and that an the ECTMD theoretical damping approach ratio in of frequency 0.14, the vibration domain introduced of the tower herein can be is capableeffectively of producingmitigated. a good prediction of the dynamic responses of the tower with or without ECTMD. 2) The wind tunnel study on the vibration of a lattice tower with an ECTMD mounted indicates 6. Conclusionsthat the newly designed ECTMD is capable of effectively reducing the bidirectional response under various wind speed and wind directions. This may be attributed to the fact that the first An omnidirectional cantilever-type ECTMD is introduced herein for vibration control of lattice two natural frequencies of the tower are very close. The ECTMD technique presented in this towers under wind loading. A wind tunnel study of a lattice tower with an ECTMD was conducted paper can be applied to tall structures and long-span bridges that undergo significant vibration to gain insight into the performance of the ECTMD used in suppressing vibration of lattice towers. induced by wind/earthquake loadings. Conclusions can be drawn as follows. 3) It is found that the effectiveness of the ECTMD model is least favorable when the wind direction 1) Theis consistent wind tunnel with studythe principal on the latticeaxis directions tower shows of the thatlattice there tower. exists Therefore, a significant theseacross-wind cases should vibration, and it deserves efforts made to design and manufacture an omnidirectional ECTMD to suppress the bidirectional vibration. The parameter adjustability of the newly designed ECTMD in a wide range is verified experimentally. Thus, the vibration of the tower can be suppressed by tuning the frequency of the ECTMD to a value close to the first natural frequency of the tower. The parametric study of ECTMD in wind tunnel test shows that with a frequency ratio of 0.97 and an ECTMD damping ratio of 0.14, the vibration of the tower can be effectively mitigated. 2) The wind tunnel study on the vibration of a lattice tower with an ECTMD mounted indicates that the newly designed ECTMD is capable of effectively reducing the bidirectional response under various wind speed and wind directions. This may be attributed to the fact that the first two natural frequencies of the tower are very close. The ECTMD technique presented in this paper can be applied to tall structures and long-span bridges that undergo significant vibration induced by wind/earthquake loadings. Appl. Sci. 2019, 9, 2978 21 of 23
3) It is found that the effectiveness of the ECTMD model is least favorable when the wind direction is consistent with the principal axis directions of the lattice tower. Therefore, these cases should be considered in the preliminary design of ECTMDs. That is to say, if the vibration of the tower in these cases can be well mitigated, the vibration under other wind conditions can be also suppressed very well. 4) A governing equation is established for the motion of the simplified model, and the equation is transformed in modal space to obtain the frequency response function of each mode. A frequency domain approach is presented to achieve variance of dynamic response. The comparison of numerical and experimental results indicates that the theoretical approach introduced herein is capable of capturing the optimal settings of the ECTMD, and producing a good prediction of the dynamic responses of the tower with or without the ECTMD.
Author Contributions: Conceptualization, W.L.; Investigation, Z.W. and M.H.; Methodology, W.L. and Y.C.; Software, M.H.; Supervision, W.L.; Writing—original draft, Z.W.; Writing—review & editing, Y.C. Funding: This research was funded by National Natural Science Foundation of China (Grant No.: 51878607, 51838012 and 51678525). Conflicts of Interest: The authors declare no conflict of interest.
References
1. Zhang, P.; Ren, L.; Li, H.; Jia, Z.; Jiang, T. Control of Wind-Induced Vibration of Transmission Tower-Line System by Using a Spring Pendulum. Math. Probl. Eng. 2015, 2015, 1–10. [CrossRef] 2. Harikrishna, P.; Shanmugasundaram, J.; Gomathinayagam, S.; Lakshmanan, N. Analytical and experimental studies on the gust response of a 52 m tall steel lattice tower under wind loading. Comput. Struct. 1999, 70, 149–160. [CrossRef] 3. Mulherin, N. Atmospheric icing and communication tower failure in the United States. Cold Reg. Sci. Technol. 1998, 27, 91–104. [CrossRef] 4. Da Silva, J.; Vellasco, P.D.S.; De Andrade, S.; De Oliveira, M.; Da Silva, J.G.S. Structural assessment of current steel design models for transmission and telecommunication towers. J. Constr. Steel Res. 2005, 61, 1108–1134. [CrossRef] 5. Hitchcock, P.; Glanville, M.; Kwok, K.; Watkins, R.; Samali, B. Damping properties and wind-induced response of a steel frame tower fitted with liquid column vibration absorbers. J. Wind. Eng. Ind. Aerodyn. 1999, 83, 183–196. [CrossRef] 6. Qu, W.; Qin, W.; Liang, Z. The wavelet recognition method of joint bolt looseness damage diagnosis based on the wind-induced responses of transmission towers. J. Earthq. Eng. Eng. Vib. 2008, 4, 146–153. (In Chinese) 7. Shu, X.; Hu, X.; Yu, Z. The Accident Analysis of a Collapsed Tower. J. Hunan Univ. 2004, 31, 56–58. (In Chinese) 8. Park, J.H.; Moon, B.W.; Min, K.W.; Lee, S.K.; Kim, C.K. Cyclic loading test of friction-type reinforcing members upgrading wind-resistant performance of transmission towers. Eng. Struct. 2007, 29, 3185–3196. [CrossRef] 9. Tian, L.; Yu, Q.; Ma, R.; Wang, C. The Collapse Analysis of a Transmission Tower Under Wind Excitation. Open Civ. Eng. J. 2014, 8, 136–142. [CrossRef] 10. Jiang, Y.; Pan, C.; Zhuang, Z.; Li, M. Analysis of collapse accident of transmission line tower causing by typhoon attacking in the coastal district of Guangdong province. South Energy Constr. 2016, 3, 82–87. (In Chinese) 11. Jiang, J.; Zhang, P.; Patil, D.; Li, H.N.; Song, G.; Li, H. Experimental studies on the effectiveness and robustness of a pounding tuned mass damper for vibration suppression of a submerged cylindrical pipe. Struct. Control. Health Monit. 2017, 24, e2027. [CrossRef] 12. Tan, J.; Ho, S.M.; Zhang, P.; Jiang, J. Experimental Study on Vibration Control of Suspended Piping System by Single-Sided Pounding Tuned Mass Damper. Appl. Sci. 2019, 9, 285. [CrossRef] 13. Jiang, J.; Ho, S.C.M.; Markle, N.J.; Wang, N.; Song, G. Design and control performance of a frictional tuned mass damper with bearing–shaft assemblies. J. Vib. Control 2019, 25, 1812–1822. [CrossRef] Appl. Sci. 2019, 9, 2978 22 of 23
14. Lu, X.; Zhang, Q.; Weng, D.; Zhou, Z.; Wang, S.; Mahin, S.A.; Ding, S.; Qian, F. Improving performance of a super tall building using a new eddy-current tuned mass damper. Struct. Control Health. Monit. 2017, 24, e1882. [CrossRef] 15. Warburton, G.B.; Ayorinde, E.O. Optimum absorber parameters for simple systems. Earthq. Eng. Struct. Dyn. 1980, 8, 197–217. [CrossRef] 16. Fujino, Y.; Abe, M. Design formulas for tuned mass dampers based on a perturbation technique. Earthq. Eng. Struct. Dyn. 1993, 22, 833–854. [CrossRef] 17. Rana, R.; Soong, T. Parametric study and simplified design of tuned mass dampers. Eng. Struct. 1998, 20, 193–204. [CrossRef] 18. Pelc, J.; Kolator, B. Passive vibration damping in a truss telecommunication tower. Tech. Sci. Warm. Maz. Olsztyn 2014, 17, 249–258. 19. Tian, L.; Zeng, Y. Parametric Study of Tuned Mass Dampers for Long Span Transmission Tower-Line System under Wind Loads. Shock. Vib. 2016, 2016, 1–11. [CrossRef] 20. Barros, R.; Guimarães, H.; César, M.B. Analysis of the dynamic behavior of a radar tower. In Dynamical Systems Theory and Applications, Proceedings of the Dynamical Systems: Modelling, Łód´z,Poland, 7–10 December 2015; Awrejcewicz, J., Ed.; Springer: Cham, Switzerland, 2015; pp. 39–48. 21. Qu, W.; Chen, Z.; Xu, Y.-L. Dynamic analysis of wind-excited truss tower with friction dampers. Comput. Struct. 2001, 79, 2817–2831. [CrossRef] 22. Tanaka, H.; Mak, C. Effect of tuned mass dampers on wind induced response of tall buildings. J. Wind. Eng. Ind. Aerodyn. 1983, 14, 357–368. [CrossRef] 23. Xu, Y.L.; Kwok, K.; Samali, B. Control of wind-induced tall building vibration by tuned mass dampers. J. Wind. Eng. Ind. Aerodyn. 1992, 40, 1–32. [CrossRef] 24. Isyumov, N.; Holmes, J.D.; Surry, D.; Davenport, A.G. A Study of Wind Effects for the First National City Corporation Project—New York, USA; Boundary Layer Wind Tunnel Laboratory Special Study Report No. BLWT-SSI-75; University of Western Ontario: London, ON, Canada, 1975. 25. Vickery, B.J.; Davenport, A.G. An Investigation of the Behaviour in Wind of the Proposed Centrepoint Tower, in Sydney, Australia; Engineering Science Report No. BLWT-1-70; University of Western Ontario: London, ON, Canada, 1970. 26. Deng, H.; Zhu, S.; Chen, Y.; Wang, Z. Study on wind-induced vibration control of long span transmission line system. J. Build. Struct. 2003, 24, 60–64. (In Chinese) 27. Wang, Z. Development and Applications of the Self-Powered Magnetorheological Damper and TMDs Using Eddy Current Damping; Hunan University: Changsha, China, 2011. (In Chinese) 28. Bae, J.S.; Hwang, J.H.; Kwag, D.G.; Park, J.; Inman, D.J. Vibration Suppression of a Large Beam Structure Using Tuned Mass Damper and Eddy Current Damping. Shock Vib. 2014, 2014, 1–10. [CrossRef] 29. Wang, W.; Dalton, D.; Hua, X.; Wang, X.; Chen, Z.; Song, G. Experimental Study on Vibration Control of a Submerged Pipeline Model by Eddy Current Tuned Mass Damper. Appl. Sci. 2017, 7, 987. [CrossRef] 30. Chen, J.; Lu, G.; Li, Y.; Wang, T.; Wang, W.; Song, G. Experimental Study on Robustness of an Eddy Current-Tuned Mass Damper. Appl. Sci. 2017, 7, 895. [CrossRef] 31. Sodano, H.A.; Bae, J.S.; Inman, D.J.; Belvin, W.K. Improved Concept and Model of Eddy Current Damper. J. Vib. Acoust. 2006, 128, 294–302. [CrossRef] 32. LaRose, G.; Larsen, A.; Svensson, E. Modelling of tuned mass dampers for wind-tunnel tests on a full-bridge aeroelastic model. J. Wind. Eng. Ind. Aerodyn. 1995, 54, 427–437. [CrossRef] 33. Livesey, F.; Søndergaard, T. Verification of the effectiveness of TMDs using wind tunnel section model tests. J. Wind. Eng. Ind. Aerodyn. 1996, 64, 161–170. [CrossRef] 34. Ravaud, R.; Lemarquand, G.; Babic, S.; Lemarquand, V.; Akyel, C. Cylindrical Magnets and Coils: Fields, Forces, and Inductances. IEEE Trans. Magn. 2010, 46, 3585–3590. [CrossRef] 35. Wiederick, H.D. Magnetic braking: Simple theory and experiment. Am. J. Phys. 1987, 55, 500. [CrossRef] 36. Huang, M.; Zhang, B.; Lou, W. A computer vision-based vibration measurement method for wind tunnel tests of high-rise buildings. J. Wind. Eng. Ind. Aerodyn. 2018, 182, 222–234. [CrossRef] 37. Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. A Math Phys. Eng. Sci. 1998, 454, 903–995. [CrossRef] Appl. Sci. 2019, 9, 2978 23 of 23
38. Liang, S.; Zou, L.; Wang, D.; Cao, H. Investigation on wind tunnel tests of a full aeroelastic model of electrical transmission tower-line system. Eng. Struct. 2015, 85, 63–72. [CrossRef] 39. Heeg, J.; Spain, C.; Rivera, J. Wind Tunnel to Atmospheric Mapping for Static Aeroelastic Scaling. In Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, CA, USA, 19–22 April 2004; p. 2044. 40. Chinese Standard. GB5009-2012 Load Code for the Design of Building Structures; China Architecture & Building: Beijing, China, 2012. (In Chinese) 41. Sharma, R.; Richards, P. A re-examination of the characteristics of tropical cyclone winds. J. Wind. Eng. Ind. Aerodyn. 1999, 83, 21–33. [CrossRef] 42. Kaimal, J.; Wyngaard, J.; Izumi, Y.; Cote, O. Spectral characteristics of surface-layer turbulence. Q. J. R. Meteorol. Soc. 1972, 98, 563–589. [CrossRef] 43. Den Hartog, J.P. Mechanical Vibrations; Dover Publication, Inc.: Mineola, NY, USA, 1985; pp. 93–94. 44. Clough, R.W.; Penzien, J. Dynamics of Structures; Computers & Structures Incorporated: Berkeley, CA, USA, 2003; pp. 234–236. 45. Davenport, A.F. The relationship of wind structure to wind loading. Wind effects on buildings and structures. In Proceedings of the International Conference on Wind Effects on Buildings and Structures, Teddington, UK, 26–28 June 1963; HMSO: London, UK, 1965.
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