A Semi-Analytical Method for Evaluation of the Dynamic Responses of an

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ASemi-Analytical Method for Evaluation of the Dynamic Responses of anActual Wind Turbine Blade under Different Loadings

ASemi-Analytical Method for Evaluation of the Dynamic Responses of anActual Wind Turbine Blade under Different Loadings


Theaim of this paper is to provide a discussion of a semi-analyticsolution for the determination of the vibration equation for a windturbine blade. It is quite difficult to determine the dynamicbehavior of the wind turbine blade using the current analyticalsolutions. Some of the limitations of the current analyticalsolutions include the level of complexity of dynamic equations andmultiple calculations. In the present study, wind turbine blades areconsidered as beams where the achieved forces and movements aretransferred with the state-of-the-art functions, includinggeometrical and vibration characteristic. The objective oftransferring them to the forces and movements of the actual blades isto find realistic results. The wind turbine blade modeled by ANSYSwas loaded with the same masses. The results were then compared witheach other. This comparison was intended to verify the results thatwere obtained using the mapping method and transfer function.

Keywords:Semi-analytical solution, vibration and deformation equation, windturbine blade, force, transfer function, Euler-Bernoulli beam,piezoelectric actuator and sensor.


Anincrease in the size of wind turbines around the blades has a directeffect on their dynamic behaviors. It will be possible to design andmanufacture bigger wind turbines by taking account of space and otherenvironmental elements. However, critical vibration-related problemsaffecting the wind turbines require more considerations. Thevibration caused by the flow of the wind is the most significantparameter considered during the process of designing turbines. Windturbines with 5 MW outcome power generation, 80 m height, and 120 mrotor diameter were manufactured by increasing the size of differentcomponents.

Thereare several factors (such as variation in the size of components)that limit the process of designing wind turbines. A chance in thesize of the wind turbine blades adds to the flexibility of the solidsthat make the understanding of dynamic behaviors more important [1].Sutterland conducted research on the solid flexibility and found thatit caused a dramatic drop in the amount of power that is generated bythe wind turbines [1]. Currently, researchers are interested in thedetermination of the dynamic interaction between the blades and thetower.

Thewind turbine blades can be changed in two ways. The first one is anedge-wise vibration that happens out of the beam’s rotating plate.The second one is the flap-wise vibration that occurs at the rotatingplate. Flap-wise vibrations are similar to the shaking of planes’wings. They cause accidents between blades and tower in wind turbinesin the critical mode. Lavsen [2] conducted a research on the failureof a wind turbine blade in the flap-wise mode. This occurs in thenormal working conditions of the wind turbine. Matagad [3] studiedvertical deformations of the wind turbine blades, including theinteraction between the blade and the tower. The researcher foundthat the folding blade-tower interaction can cause a dramaticincrease in maximum displacement at the tip of the blade. Researchersaim to reduce the flaws in the vibration. The dynamic behaviors(including the normal frequencies, mode shapes, and maximum verticaldisplacement of these blades) should be obtained first. Research ondynamic behavior of the actual blades under different loadings hasnot been possible when using the analytical solutions method.

Thecomplexity of dynamic equations and the huge amount of calculationsare the most significant challenges for semi-analytical solutions.They affect the special complex shapes of solids, such as windturbine blades, helicopter rotors, and robot arms. Twister slight andheavy components have been used widely in engineering applicationsduring recent years. Rigid movements and elastic vibration of twistercomponents in machines are called solid-flexible couple. The simplestexample of this kind of dynamic movement is a cantilever beam.Twister turbines, helicopter, robot arms, vehicle drive system, andmicro electro-mechanic are some of the applications of the twisterbeam. Generally, three methods (including analytical solution,numerical resolution, and the definition of the system via theexperimental method) are used in modeling of different structures.

Vibrationof the twister beams is widely studied using different models of beam(such as Euler-Bernoulli, Timoshenko) and semi-analytical solutions,numerical or finite element (FEM) methods. In finite elementmodeling, Leshman [4] and Avakava [5] used CFD (Computational FluidDynamic) to study the wind flow around the blades and tower of theturbine. Xi [6] studied loads caused by wind flow in stormysituations. Most of the previous studies on twister or under loadingbeams were based on the Euler-Bernoulli model. Most of these studieshave not investigated the interaction between fluid and structure[7].

TheFSI method can provide more precise results in comparison with otherapproaches [7] that will be used in this paper. Dasilva and Hajez [8]used the Hamilton method to determine the dynamic equations of atwister beam regarding initial conical and twisting angles that arevariable along the beam. Hong [9] obtained natural frequencies ofEuler-Bernoulli beam at a high speed using the exponential seriessolution. Arvin and Bakhtiarinejad [10] studied free nonlinearvibration or rotating beam using the Fen-Karman equation. Hoblet andBroks [11] formed deformation equations of Euler-Bernoulli cantileverbeam in flexural-flexural-torsional vibration movements. Halaner andLiv [12] and Feriberg [13] derived an accurate dynamic stiffnessmatrix that disregards the wrap stiffness. Dasilva [14] identifiedthe complete non-linear partial equations of motion of rotating beam.Dasilva determined vibration solution of the system and specialvectors of the turbulent systems through aerodynamic forces.

Yijit[15] investigated the significance of vibration couple in therotational Euler-Bernoulli beam. The researcher determined theequation of motion caused by vibrations couple and the rigid body.Hunagud and Sarcar [16], Baruh and Tadikonda [17] and Choura [18]investigated the dynamic model of a rotating part without regard tocentripetal forces on changes associated with the bending of theEuler-Bernoulli beam. Lee [19] studied flap-wise vibration of acomposite rotational Euler-Bernoulli beam and determined theinteraction between rotational speed and natural frequency.

2.Theory and Modeling


Thefact that piezoelectric material can be used both as a sensor and anactuator is attributed to its electro-mechanical features. Thesedual features in piezoelectric materials make them appropriate foruse in vibration control. They can also be used in the application ofthe outer impulse and force in different kinds of structures, such asbeams, wind turbine, and helicopter blades. Moreover, piezoelectricmaterials can be installed in every structure. In the first part, aninnovative method was used to transfer the actual blade to theEuler-Bernoulli Beam. The transfer function in the actual blade andsimilar sections of the Euler-Bernoulli beam were used to derive anaccurate model. This model made it possible to match the beams.

Secondly,the strain-stress relations for Euler-Bernoulli beam were computed.The kinetic and potential types of energy were used to reach thevibration equations for the beam. Many parameters should beconsidered in order to model the beam with piezoelectric patches. Theimpact of surplus mass on the beams’ surface should be consideredas part of these parameters. This surplus impact is caused by settingpiezoelectric patches and electro-mechanical effect between them andthe beam.

Lastly,the derived equations were assessed using the Rayleigh-Ritz method.Different parts of the beam that were transferred to the actual windturbine blade were related to displacement by applying equationsdetermined in the first part.

2-2Transfer from the actual blade to the Euler-Bernoulli beam

Velocityis only related to frequency variant and not to the time. However,the control system is dependent on time. Therefore, whenever theforces are applied to the structure, there would be no changes insignal domain. The linear features of the system can be used in therecognition of the blade’s vibration properties on any loading thatis made to the structure.

Thesemi-analytical modeling is quite complex, frustrating, and needs along duration compared to the actual wind turbine blade. The lineartime invariant features of the system will be used to represent anefficient and precise model that is feasible at the same time. Thegeometry model of the actual blade was divided into (n) points thatwere set at an equal distance to one another. The geometry of actualblade was transferred to an Euler-Bernoulli beam using a point-wisetransfer function.

Theunit step pulse was applied to every point and then n-1 outcomepluses and n-points’ signal were reached. The estimated figure ofthis system is shown below (Figure 2). This was assumed to be animpulse function in order to obtain the frequency response of thestructure. Instead of sending pulses in the LTI system, the stepfunction can be used to determine the structure frequency response.The structure frequency response is achieved by deriving the stepoutput reaction. Noises caused this action to take place in thedomain. In addition, high frequencies can be eliminated in thefrequency domain. Then per any section i, (N) output will beobtained. By using these outputs, (N) transfer function will becreated in the form of gij.Byrepeating this procedure for every (N) sections, A [n*n] matrix oftransfer function Gswillbe derived.

Theresponse of impact function equals to:

Theoutputs are transferred to the time domain using the inverse Fouriertransfer function. An overvoltage will be required in order toincrease the ratio of signal to noise in the section. Then, everysection’s transfer function will be equal to:

Thesame transfer function process will be derived for theEuler-Bernoulli beam. The transfer function matrix (Gb)in the same sections with main blade will be derived from theincoming dynamic load (Us).

Therefore,vibration and tension outcomes resulting from the semi-analyticalsolution of the Euler-Bernoulli beam under different dynamic loadingscan be mapped to the actual blade.

2-3Derivation of Euler-Bernoulli equation with piezoelectric patches

Thestudy beam in this paper is assumed to be the Euler-Bernoulli withthe length (L), width (b), and height (h). According to theEuler-Bernoulli theory, shearing deformations are neglected byassuming that the beam’s middle plates will be parallel. Therefore,the shearing tension and strain were removed. In this theory, thesheer force is only derived by this equation.

Theboundary conditions were assumed to be the cantilever beam, whichimplies that the edge-wise vibrations were discarded. Therefore,piezoelectric patches were assumed to be continuous in the widthdirection.

Regardingthe acquisition of the beam’s equations, energy (constrain) andassumed mode methods were used. The total beam strain energy withpiezoelectric patches was expressed as:

InTimoshenko’s model and the lateral displacement, the slop of thetangent for every point should be considered in the equations.According to the Timoshenko, equations derive more from deformationthan the Euler-Bernoulli’s beam. This is accomplished under anassumption that the thickness and relative displacement between upperand below the surfaces of the beam are necessary. In this case study,even if the ratio of beam’s stickiness to its length isinsignificant, Timoshenko’s equation is useful because the beam isactuated in high frequencies.

Assuminglinear behavior of piezoelectric patches, strain-stress equation willbe expressed as:

Therelation between electric field and applied voltage to thepiezoelectric patches are:

Asit was shown in upper equations, piezoelectric patches’ strainenergy equation consists of two parts:

a)Piezoelectric mass behavior.

b)Piezoelectric electro-mechanical behavior of the patches.

Piezoelectricpatch was considered as an Euler-Bernoulli beam that constituted thefirst section of the equation. The strain energy was brought about bypiezoelectric patches. The electro-mechanical correlation wasformulated in part b. The kinetic energy equation for theEuler-Bernoulli beam was formulated without considering the effectsof piezoelectric patches.

Thetotal kinetic energy is expressed as:

Solvingequation by assuming the mode method:

Byusing this method, lateral bending of the beam under the actuatingcaused by piezoelectric patches was connected to the beam surface,which is expressed as below

Theequation defines a known function that satisfies the specificboundary conditions. It also defines the unknown function, which willbe determined by solving equations.

Byreplacing strain and kinetic energy equations in the Lagrange model,the time discrete formula of systems is derived as:


Thefindings of vibration modeling that resulted from the wind loads,gravity, and rotating of the rotor are discussed in this section. Theresults were obtained from the control system in the wind turbineblades. The geometrical and vibration characteristic transferfunctions were used to achieve forces and movements of actual windturbine blades. The blade of the considered wind turbine was dividedinto 45 equal pieces. The division was performed on Euler-Bernoullibeam too. It was assumed that every piezoelectric patch would act asactuators and sensors that were set ideally on every piece of thebeam. To verify this transfer function, the wind turbine blade wasmodeled in the ANSYS with every 45 actuators/sensors. Next, the 45*45transfer function of the wind turbine matrix was established. TheANSYS model was then put on three different loadings and themovements of all 45 points obtained, just like what was applied tothe Euler-Bernoulli beam.

Theloadings were performed on Euler-Bernoulli beam in a discrete way.The loadings caused a continuous force on the Euler-Bernoulli beam,which should be represented to the number of divided pieces that are45 in this study. In this paper, the Reilly-Ritz method is used forthe approximation forces. In the first place, forces and theresultant movements of the points had been reached. The movements ofthe points of the actual wind turbine blade were achieved by thetransfer function.

Investigatingthe results of performing a unit step to actuators/sensors on theEuler-Bernoulli beam and the wind turbine blade

Firstly,the movements that are felt by sensors in order to perform the unitstep on the beam and the blade will be investigated. The 4thactuator was selected, which is placed 4 meters away from the base.The 20thsensor was set 20 meters from the base. The upper diagrams presentthe outputs of the Euler-Bernoulli beam and the wind turbine blade.Then, dividing the derivations of every signal to first step incomesresulted in the element of transfer function which is g4,20, inthis case. The total transfer function was achieved by repeating theabove-mentioned action for all actuators and sensors.


Theapplication of the theoretical method will be discussed in thissection. In the first part, the wind turbine that is used in thispaper will be introduced. The piezoelectric patches as well as theirelectro-mechanical properties and the topology of patches’deployment on the beam will also be presented. Next, the windturbine, which will be modeled by the ANSYS, will be discussed.Lastly, the proposed loading in this research will be studied inorder to assess the main approach.

Thewind turbine that was used in this project

Thewind turbine has three blades, and each one of them measures 45meters in length. In this simulation, material of the blade wasassumed to be steel. Steel’s mechanical properties are shown in thetable below:

Thevibrational effects of the tower and the impact caused by interactionbetween rotors and gears are neglected in the present research. Infigure 3, the wind turbine, which is used in this study, is shown.

Oneof the key ceramics that have piezoelectric features is thePlumbum-Titania-Zirconia. The Piezoelectric patches that are used inthis project are of this type and their properties are shown in thetable below. The table shows mechanical and electro-mechanicalproperties of these patches. The coefficients α and β shows dampingcoefficient related to the structure of piezoelectric patches. Theprecise determination of those coefficients has a significant impacton the accuracy of results in finite element modeling in ANSYS.

3-3How to set piezoelectric patches on the blade and the beam: Patchesshould be set at equal distribution on the blade and the beam. Theyshould be set 1m away from each other. To be more precise, the firstpatch should be set at 1m from the base. The last one should beinstalled on the tip of the beam and of course on the blade too.

3-4The Finite Element Modeling in ANSYS:The ANSYS software was used to model the wind turbine blade andaerodynamic force load. The actuator as well as sensor piezoelectricand loading under different conditions will be included. Variousparts of finite element simulation:

1.Wind turbine blade modeling

2.Piezoelectric patches modeling

3.Aerodynamic loads modeling

Hexahedralelements are used in order to mess the model. The wind turbine bladehas 7763 elements and 7566 nodes on it. Figure 4 shows thedistribution of masses on the blade. In this part, the outcomes ofvibration modeling, which resulted from wind loads, gravity, androtation of the rotor, will be discussed. Therefore, the results thatwere obtained from performing control system on the wind turbineblades will be shown. To simplify the derivation of semi analyticalequations as a defining mode space of the wind turbine blade, thestate-of-the-art transfer function was used to achieve forces andmovements of actual wind turbine blades.

Theblade of the considered wind turbine was divided into 45 equalpieces. The division was performed on Euler-Bernoulli beam too. Itwas assumed that every piezoelectric patch acted as actuators andsensors that were set ideally on every piece of the beam. To verifythis transfer function, the wind turbine blade was modeled using theANSYS software in every 45 actuators/sensors. Next, the 45*45transfer function of the wind turbine matrix was achieved. Performingon those steps for Euler-Bernoulli beam caused 45*45 of its transferfunction matrix too. The ANSYS model was then placed on threedifferent loading and the movements of all 45 points obtained. Theseloadings were performed on Euler-Bernoulli beam in discrete form. Inthe first place, movements of the points had been noticed and themovements of these points of the actual wind turbine blade achievedby the transfer function. The comparison of both movements resultedin the verification of the proposed method.

Investigatingthe results of performing of a unit step to actuators/sensors on theEuler-Bernoulli beam and the wind turbine blade

Movementsthat are detected by sensors in order to perform of unit step on thebeam and blade will be investigated. The 4thactuator was selected and placed 4 meters from the base. The 20thsensor was selected and set 20 meters from the base.

3-5:Comparing the wind turbine blades’ results with the outcome fromthe Euler-Bernoulli beam:Movements of wind turbine blade that were caused by three differentloadings were obtained using the ANSYS. They will be compared withthe results of the semi-analytical solution of Euler-Bernoulli beamunder the same loading. In this regard, the figure of the selectedsensor’s outcome signal will be studied. The 20thsensor is placed 20 m away from the base. According to the results ofdiagram 3 and 4, there are satisfying matches between outcome signalsthe result of Euler-Bernoulli beam and the actual wind turbine blade.

Diagram5, 6, and 7 illustrates the error existing between the outcomesignals of the turbine blade and the results of corresponding sensorson the beam after the derivation. No steady behavior for errors wasreported with respect to the placement of the sensors as shown in theresults that are presented on the diagram 5 to 7. These features werecaused by different behaviors in some places between turbine bladesthat were modeled using the ANSYS software and the mapped beam.Taking the distance from the base made it possible to assess theaccuracy of the semi-analytic method. The increase in the bladevibration’s domain with respect to the distance from the base isone of the significant reasons for the occurrence of the error.


Thefirst part of the results focus on functions used to transmit thebeam’s forces and movements to the blade. This transfer functioncan be considered as a solution for semi-analytic equations. Theapplication of the transfer function made it possible to use controlrules to the piezoelectric actuator in a more precise way. TheEuler-Bernoulli beam and the wind turbine blade were placed with thesame loadings in order to assess the proposed approach. The resultsof the Euler-Bernoulli part of the beam were mapped to thecorresponding blade of the turbine. The mapping was accomplishedusing three flow speeds. An increase in the distance between theblades made it possible to lower the signals of the real and transfermethods. This trend can be attributed to an increase in the vibrationof the blades at the tips. From the results, mapping between theactual blade and the beam is satisfying. These results support thesemi-analytic method.

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