DYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES


 Asher Price
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1 DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation and maintenance. An undestanding of the loadings on wind tubines and thei oigins, as well the tubines esponse to them is cucial to avoid thei catastophic failue. Fig. 1: ollapsed wind tubine in Januay 2007 in Gemany. Souce: AP. TYPES OF LOADS The types of loads a wind tubine is subject to duing sevice can be ckassified as: STATI LOADING This loading is constant in time and the esulting deflection of the stuctue is constant and popotional to its stiffness. YLI LOADING Two types of cycling loadings pesent themselves. In quasi static cycling, the loading vaies slowly enough wheeas the deflection of the stuctue is popotional to the loading. In dynamic cycling the loading esults in a deflection elated to the damping
2 foces of the stuctue, paticulaly when the load application fequency is close to the natual vibation fequency of the stuctue. STOHASTI LOADING This type of loading vaies in a andom manne. It esults pedominantly fom wind tubulence and is elevant to the fatigue esponse of the wind tubine stuctue. AERODYNAMI LOADING This is loading deived fom the foce of the wind. The oto of a wind tubine convets the wind's kinetic enegy into useful mechanical wok though aeodynamic effects. The basic undelying concept is the consevation of momentum. Most of the momentum exchange takes place in the wind flow diection, although the useful powe is poduced by foces in the oto plane, which is pependicula to the oncoming steam. A significant slowing down of the oncoming wind steam is necessay to poduce useful mechanical powe. Lage steady loads ae geneally not poblematic in tems of design, and they can be educed by blade coning, to be descibed late. The constant thust loads imposed by the wind may not be dangeous as long as they ae aptly accounted fo at the design stage. The cyclic and stochastic tubulence deived loads ae the ones that can cause most stuctual failues, paticulaly those due to fatigue. The ate of change of thust T on a tubine is popotional to the squae of the appaent wind velocity W: dt c = = ρ ( cosφ+ sin ) dt 2 φ (1) 2 T V L D whee: T V φ ρ c L D is the axial wind thust[ knewton] is the appaent wind speed [ m / s] is the inflow angle is the lift coefficient is the dag coefficient 3 [ / ] is theai density kg m is a constant Gusting involving apid wind change can be hazadous paticulaly if the natual fequency of the tubine stuctue lies within the fequency ange of the wind gusting.
3 Fig. 2: Lift and dag on the oto of a wind tubine. Wind shea coesponding to the incease of wind velocity with height can exacebate the effects of the wind speed due to gusting. A simple fom of the basic elation govening wind shea as a function of height is: H V( H) = Vef H ef α (2) whee: V ( H ) is the wind speed at the nacelle height H [ m / s] Vef isthe wind speed at a efeenceheight[ m / s] H is the elevation above gound [ m] α is the shea coefficient Fo a oto with a adius of 20 metes, the tip to tip distance is 40 metes and the wind speed at the top and bottom of the blade otation can diffe by as much as 30 pecent. If the shea effect coincides with sevee gusting the consequences on the oto can be seious possibly leading to failue. The aeodynamic loading is also affected fom the oientation of the tubine stuctue if it comes out of alignment with the diection of the wind. Yaw misalignment is a situation aising fom a discepancy between the wind diection and the oientation of the oto axis. Shaft tilt is a design featue used to incease the cleaance of the oto blades fom the towe in high winds. Howeve, it can cause sinusoidal cyclic loadings to be imposed on the oto and the dive tain.
4 These aeodynamic loadings can become significant in apidly veeing wind conditions whee a significant lag exists as the yaw mechanism of the tubine attempts to follow the wind diection. MEHANIAL LOADING Loadings that esult fom the mass o the momentum of the wind tubine's stuctue ae classified as mechanical loads. These include: 1. Gavity Loading Gavity can impose lage fatigue stesses on the moving oto, paticulaly at the egion of the oot and at any lamina joints. In the cases of coned and tilted otos, the load can also be out of the oto plane, leading to a flap wise bending aound the blade's chod. The bending moment at the oot of the blade can be expessed as: M = g m( ) sinψ d (3) b whee: m g ψ is theeffective massof the oto blade[ kg] is the distance fom the oto ' s oot[ m] is the gavity acceleation constant is the angleof the blade with the vetyical diection 2. oning Effect oning which is the bending of the oto blades in high winds intoduces centifugal foce loads which act against the aeodynamic steady thust loads, thus educing the mean blade loading, and is a desiable featue intoduced into the design of the oto blade. Howeve, the emoval of steady thust loads may also cause oscillations away fom the mean stess level, which could become vey damaging. 3. Yaw Foces The yaw o gyoscopic foces ae significant in fee yaw machines, whee instantaneous yaw velocities can cause sevee flapping of the oto blades. 4. Tansient Loads Tansient loads such as occu at the stat up and shut down times, ae dependent on the machine and contol system chaacteistics. Inheent flaws in cetain components such as a micoscopic void in a slow speed shaft may esult in fatigue damage unde tansient loading.
5 TURBINE DYNAMI RESPONSE The dynamic esponse of a wind tubine stuctue to the imposed loads affects the oto, the powe tain and the stuctual towe. Undestanding the behavio of these components unde both static and vaying loads is cucial to avoiding potentially dangeous esponses. It is impeative that the otational fequency of the tubine oto is diffeent fom any hamonics of the stuctue's natual vibation esonance fequency. The excitation of the esonant conditions in any dynamic stuctual component must be avoided, o passed though quickly at statup o shut down to avoid catastophic failue. The dynamic esponse of wind tubines encompasses diffeent situations. 1. Static Loads In the case of static loads the extent of deflection unde a load depends upon the stiffness of the stuctue and the size of the load. This can descibed as: whee: F k x F is the estoation foce is the stiffness facto is the deflection magnitude = kx (4) The estoing foce F acts in the opposite diection to the deflection x. This is the simplest model of the static esponse of a stuctue and it elies upon the assumptions that the esponse of the stuctue is linea and that the stuctue is allowed to each equilibium o its static condition balancing the applied loads and the intenal eactions in the stuctue.
6 Fig. 3: Static esponse of a wind tubine stuctue to an applied constant axial load. The static esponse of a wind tubine to an applied axial load F can be due to the thust of a steady wind on the powe tain. Fo a nacelle height h, the eaction at the foundation is given by a bending moment: M b = Fh. (5) An opposite eaction foce of magnitude F also aises at the foundation. Equilibium is eached fo both the extenal applied foce and the esulting moments at the foundation. The intenal eactions in the stuctue due to the elastic effects ae equal to the applied foces and moments. The esponse of most mateials is assumed to be linea so long as stesses in the mateial do not appoach the yield stess of the mateial. The lowe end of the stessstain cuve is a staight line fo most common stuctual mateials. 2. yclic Loads To estimate the dynamic esponse of the stuctue, seveal simplifications ae common: 1. The existence of a linea quasi static esponse 2. A linea modal esponse 3. A non linea quasi static esponse
7 Quasi static hee means that although the stuctue is moving, the motion may be subdivided into small time intevals so that at each inteval the system is teated as if it had eached equilibium. This is not totally accuate since the stiffness modulus of the stuctue will incease at high speeds; a situation designated as stain ate stiffening. Anothe eason is that the damping within the mateials of the stuctue geneates exta loads that ae popotional to the ate of defomation. In addition some load aise due to ai fiction caused by the motion These exta effects ae usually small and so it is easonable to neglect them in most cases. If a stuctue is teated as linea, the natual modes of vibation may be found using the technique of Modal Analysis. The mode shape of a natual mode is the shape of the stuctue at a point in a single peiod of the vibation when the defomation is lagest in magnitude. The fequency is the ecipocal of the time fo one peiod of vibation. Pactically, damping in the stuctue changes the theoetical mode shape and the fequency of vibation. The measued fequencies ae often vey close to those pedicted assuming a linea behavio model. Fig. 4: Mode shape of a natual vibation mode. The mode shape of a natual vibation mode looks like the static defomation of
8 the stuctue with the pactically simila. Howeve, the stuctue's defomation oscillates duing the vibation cycle fom its static shape to its maximum mode shape. Mathematical fomulae fo the natual modes and fequencies of simple systems can be analytically deived. 3. Dynamic Loading Thee possible esponses of a tubine towe to applied loads can occu: tosional, longitudinal and tansvese. In pactice, the actual esponse is a composite of the thee esponses. If the applied load vaies with a fequency appoaching that of the stuctue s natual fequency, then a mechanism exists wheeby enegy may be continuously added to the system. This leads to a pogessive incease in the vibation amplitude, which continues until stuctual failue occus and is called dynamic loading. Resonant wind conditions tansfeed enegy fom the wind and waves unde the Tacoma Naows Bidge in the State of Washington in the USA leading to its collapse, and tuning it into an example of esonant fequency failue. Moe infomation about the Tacoma Naows Bidge is coveed in the Appendices. 4. Vibation Damping The damping chaacteistics of the stuctue dictate how fast the system can estoe itself to the equilibium condition when the stess foces ae emoved. A stuctue is citically damped when the enegy dissipating damping foces acting on it ae just enough to soak up all the enegy impated to it by the applied load befoe the neutal position is eached. Tubine blades ae patially filled with a foam mateial, which helps to dampen the vibation of the blade unde tubulent wind conditions. This pevents the blade fom geneating a esonant esponse. At the citical damping, the citical damping coefficient is given by: = 4km (6) The citical damping atio is defined as: cit = (7) It is usually expessed as a pecentage. Thus a one pecent citical damping coesponds to = 0.01 o 1/100 of the value of the citical damping coefficient. Instability due to aeodynamic and mechanical loads becomes moe of a poblem as the stuctues become softe as thei stuctual modulus deceases. The designe must ensue that the esonant fequencies ae not excited excessively. Special attention must be given to the following situations and factos:
9 a. The stat up and shut down tansients. b. The stuctual towe shadow effects, paticulaly fo downwind machines. c. Excessive toque loadings on the hub and the dive tain. d. Roto speed vaiations paticulaly fo vaiable speed tubines designs. 5. Towe and blade vibations It is impotant to undestand the esponse of tubines with nacelles on slende wieguyed towes. Fom the pespective of the stuctual towe and oto blades, esonances may cause unwelcome noise emissions and fatigue. The towe esonances may feed back to the electical output to the gid, causing powe suges. Stuctual towe and blade vibation may be studied by using the example of a simple cantileve beam. Fo the towe, the eal system includes othe factos such as inetial effects fom the nacelle and esonant effects fom the guy cables and the gin pole. These can be accounted fo by using a patial diffeential modeling technique. In this case we ae mainly concened with the simple case. In the moe ealistic analysis of the dynamic behavio of multiple degee of feedom cantileve systems, assumptions can be made to simplify the analysis: a) The stain enegy o potential enegy in the exteme position is consideed as equal to the kinetic enegy in the neutal position implying that thee ae no damping losses. b) The natual fequency is elated to the deflected shape of the cantileve implying that thee could be esonant vibation. c) The cantileve has seveal natual fequencies at which tue dynamic esponse is possible, whee the applied load vaies at the system's natual fequency. d) Each fequency is associated with a paticula deflected shape o the mode shape. Fo a oto blade o towe, and fo moe complex mode shapes, the stain enegy is highe fo a paticula deflection in compaison to simple mode shapes. This leads to a coesponding highe natual fequency. These mode shapes and natual fequencies ae fundamental to the system, and ae independent of the loading chaacteistics in a given vibation situation. 6. Dynamic esponse to stochastic andom load The stochastic andom loads occu pedominantly fom wind tubulence. In addition to fatigue, tubulence affects many othe design paametes such as the maximum load pediction, the stuctual excitation, and the contol and powe quality. The non linea natue of fatigue, thee esults that a doubling of the load amplitude has a vey stong effect on the fatigue life. The occuence of high load amplitudes esulting fom tubulent stochastic wind gusts educes a stuctue's fatigue esistance.
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