9th SAMTECH Users Conference 2005 1/4

Quantification of wind turbine gearbox loads by coupled structural and mechanisms analysis

Andreas Heege SAMTECH Iberica: TUhttp://www.samcef.com UT

Summary Non-linear Dynamic Analysis of Wind Turbines, Coupled Structural & Mechanism Analysis, Computation of Detailed Gearbox Loads, Comparison of numerical & experimental results.

1. Introduction In order to obtain detailed information about dynamic mechanical loads, a complete wind turbine model is developed which includes all structural components and in particular, a detailed gearbox model. The applied methodology implemented in the software package SAMCEF Mecano /12/, couples consistently non-linear FEM models and advanced multi-body technologies. In the context of numerical analysis of wind turbines, this feature is of particular importance, because the turbine dynamics depend on the entire electro-mechanical system. On one hand there are flexible structural components like composite blades, tower, bedplate, gearbox housing and shafts and on the other hand there are mechanisms type elements like gears, bearings, elastic couplings, clutches and generator mechatronics. All these structural components, mechanisms and mechatronics are consistently coupled in one analysis model. The methodology enables to predict gearbox details like misalignments of gears and shafts produced by dynamic “Operation Deflection Modes” of the entire wind turbine structure. As well numerical, as well as experimental results indicate that commonly used fatigue load spectrums for wind turbine gearboxes might be incomplete. It is stipulated that excessive loads, like for example occurring during emergency stops, sudden change of wind directions, or due to mechatronics failures are not accounted correctly. Numerical results indicate that dynamic load amplifications in the gearbox can not be deduced simply from transient torques of the rotor and generator shafts. Especially during violent transients with backlashes, the gear stages decouple due to clearances and the dynamic load amplification is not necessarily the same for each gearbox stage and/or component. First conclusions derived from numerical analysis will be presented. Numerical models of different wind turbines had been validated successfully in the low frequency domain for several transient manoeuvres like emergency stops. In that case, the dynamic load amplifications are about 300 % and the experimentally measured and numerically simulated torques and velocity transients at rotor and high speed shafts showed deviations less than 20 %. In the context of noise prediction, first promising results are obtained by an updated gearbox model including “parameter excitation” in terms of gear stiffness harmonics. Presently, the numerical

model is in validation process for frequencies up to 2000 [Hz]. Gearbox shafts torques and accelerations are measured experimentally and transferred via FFT to frequency domain. Objective is to match the experimentally measured amplitudes of fundamental and first harmonics of gear mesh frequencies by the updated numerical model. First comparisons of numerical and experimental “Power Spectrum Density” data of torque transients show encouraging qualitative agreement. It is expected that “Model Updating” of the mentioned gear teeth stiffness harmonics, will enable to match experimental and numerical PSD data with satisfactory precision and to construct a solid basis for numerical noise prediction. 2. Coupling Mechanism and Structural Analysis - Mathematical Background In the context of non-linear dynamics of structures, the discretised equilibrium can be formulated for a given state by with classical non-linear Finite Element Method (FEM): To emphasize the non-linear character of stiffness and damping, it is stated explicitly the dependence of stiffness matrix )]([ qK and damping matrix )]([ q&C on the solution vector qr . In order to include further equations typical for Multi-Body-Systems (MBS), the above equations are extended in terms of an Augmented Lagrangian Approach. The additional constraints issued from further mechanisms in terms MBS equations might be either of “holonomic”, or respectively of “non-holonomic” type. It is mentioned that the final discretised equilibrium state is extended by the “Hilber Hughes & Taylor” form /2,3,4,/: Equation (2) presents the equilibrium state of coupled structures (FEM), mechanism (MBS) and control loops for a given time instance. The global equilibrium equation (2) is solved by a “Newton-Raphson” type iteration scheme. For time

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integration an implicit scheme is applied where the variable time step size is adapted such that the truncation error stays below a given threeshold. This is of particular importance when discontinuities like impacts have to integrated. It is emphasized that in case of wind turbine dynamics, the time intervals to be analysed are frequently larger than 60 [s] and as a consequence, an “implicit” solution scheme is the most efficient choice from a computational point of view. In the case of frictional contact problems, the final system of equation becomes non-symmetric and the type and number of equations might change during the “Newton-Raphson” type iterations /5,9/. 3. Gearbox: MBS-modelling & structural components The gear geometry is defined by the helix-, cone- and pressure angles, as well as normal modulus and teeth number. In order to account for geometric imperfections of gear teeth, a “geometrical transmission error” can be included in the analysis model. Gear teeth flexibility is defined either according to ISO 6336, or respectively by a non-linear gap-function which accounts for the stiffness variation, when passing along one tooth engagement (see figure 1). It is emphasized that the proper modelling of gear clearances and bearings is of crucial importance when evaluating gearbox loads during backlashes. As follows related to the “parameter-excitation”, that non-linear “gear teeth contact functions” can be defined in terms of Fourier series containing as many harmonics as necessary to describe the stiffness variation when passing along one tooth engagement /7,8,12/. That extension of the gear tooth flexibility is necessary in order to reproduce properly the higher frequency content produced by teeth meshing which is crucial for acoustic noise prediction.

Figure 1: Normalized gear teeth contact stiffness as a function of relative rotational configuration. All gearbox, rotor or generator shafts are modelled precisely by non-linear beam elements. Every bearing, including the main rotor shaft, the entire gearbox and the generator, is modelled by non-linear axial and radial gap-functions which acount for the coupling of radial and axial bearing stiffness. 4. Application of Super Elements Structural components which present a linear material behaviour within a convective reference frame and which undergo only small

deformations, are generally condensed by the Super Element Method to reduced stiffness, damping and mass matrices. These reduced matrices include generally further internal degrees of freedom in order to present properly the dynamic properties of the condensed FE-structures in terms of “internal-” and “boundary modes” /1,6,12/. Structural components which are suited for condensation are for example the gearbox housing, the planet carrier, the bedplate and with some restrictions, as well the composite blades. 4.1 Blade modelling If blade modelling is done by means of Super Elements, the anisotropic features of the composite material can be accounted for, but only in a linearized form. It has to be emphasized that Super Element theory allows generally only small deformations, i.e second order effects like geometrically or stress induced stiffening can not be taken into acount in an computational efficient way. Especially in the case of strong unsteady wind conditions which produce large deformations of the blades, the application of small deformation theory in terms of Super Elements to composite materials is questionable. If such events shall be reproduced properly by numerical simulation, it is favourable to switch to anisotropic non-linear FEM shell models. The numerical simulation of a transient dynamic event of about 60 seconds “real time”, consumes about 8 hours CPU on a Pentium IV PC, if a fully non-linear anisotropic 3D-FEM blade structure is included. For the same analysis run, about 30 minutes CPU time are consumed, if the non-linear shell model of the blades is condensed to a non-linear Super Element. A further alternative presents the application of non-linear beam elements. In that case, CPU times comparable to Super Elements are obtained, i.e. 60 seconds of “real time” turbine operation is reproduced in about 30 minutes CPU time. 4.2 Generator and Controllers Control loops for generator-converter control, pitching, or braking, are defined either directly using the controllers implemented in SAMCEF, or imported from Matlab Simulink® controller models. The electro-mechanical generator characteristics are defined in terms of non-linear “torque versus speed/slip” functions. That non-linear “generator function” might be modified during certain operation modes due to control actions. It is emphasized that the resulting transient generator torque is an analysis result and not a boundary condition. 5. Application Example: Coupled Structural & Mecanism Analysis Figures 2a,b show typical “Samcef Wind Turbine Models” including on one hand structural “FEM Components” like blades, rotor- and gearbox

9th SAMTECH Users Conference 2005 2/4

shafts, tower structure, gearbox housing, planet-carrier, bedplate etc. and on the other hand “MBS type components” like gears, bearings, elastic couplings or bushings, over-load clutch, and finally the generator & control loops. As a function of the analysis requirements, the fully non-linear FEM models might be condensed subsequently to Super Elements, or respectively not.

Figure 2ª: Complete “wind turbine model” including structural and mechanism type components.

Figure 2b: Non-linear anisotropic composite shell model coupled to planetary gearbox. Components like Gearbox housing and bedplate are condensed to Super Elements. In order to demonstrate the efficiency of the methodology, an emergency stop with a “megwatt class turbine” at low wind speed is simulated. The time interval to analyse are 50 seconds and the corresponding CPU time was about 25 minutes on a Pentium IV PC. The sudden augmentation of gear mesh forces depicted in figure 3, is due to the activation of the disc brake at about time=6[s]. Taking into account that the machine was running below nominal speed, the turbine is at idle after time>10[s]. The gear tooth forces of the planets show within the time interval [6s, 10s] torsional drive train vibrations which correspond to the operation mode: “drive train vibrates against rotor”. These vibrations are characteristics of the respective drive train and are generally visible during braking or unsteady conditions which produce backlash. Once the turbine at idle, that frequency changes to the system frequency “rotor against drive train”. An interesting issue are axial bearing forces of the three planets as shown in figure 4 (by superposed black, red & green plots), because these forces show important dynamic load amplifications. The gear forces of the planets are statically equilibrated due to opposed helix angles at internal rim and sun gear. However, the non-zero helix angle of the sun shaft wheel at the exit of the planetary stage in combination with further excitations related to the respective drive-train design, produce not neglectible dynamic axial planet bearing forces.

Figure 3: Normal Gear Contact Forces [N]: Internal gear <-> planets

Figure 4: Axial bearing forces [N]: planets 6 Comparison of “Numerical Samcef Results” and “Experimentally Measured Data” Figures 5,6 & 7 show the comparison of numerical results to the experimental data during an emergency stop at low wind conditions. In figure 5, the main shaft torque is shown and it can be seen that the numerical model reproduces with a very satisfactory precision the priorly mentioned system change from “braking” to “turbine at idle”. Figure 6 shows a zoom on one single oscillation at turbine idle. The comparison to experimental data shows that the numerical model reproduces properly the “zero torque instances” produced by radial, axial and gear clearances during load inversion. Figure 7 presents the comparison of numerical and experimental data for the axial vibrations of the parallel helical shaft of the second gear stage.

Figure 5: Numerical & experimental results: Main shaft torque [Nm] during emergency stop.

Figure 6: Numerical & experimental results: Zoom on rotor shaft torque oscillation.

Figure 7: Numerical & experimental results: Axial displacement of shaft of gearbox stage 2.

9th SAMTECH Users Conference 2005 3/4

6. Gear mesh parameter excitation Starting from the computed high speed shaft torque in time domain, figure 7 presents the associated transformation in frequency domain in terms of its Power Spectrum Density. It is mentionned that the computed HSS torque corresponds to turbine operation at nearly constant rotor speed of approximately 0.18 [Hz]. As shown in figure 7, the frequencies associated to the peaks of the PSD-plot correspond exactly to the analytical solutions of the gear mesh frequencies of the different gear stages for the respective rotorspeed.

Figure 7: Analytical & numerical results: PSD of HSS torque for constant rotor speed 7. Extension of Load diagrams The application of precise numerical models to the simulation of highly transient operation modes reveals larger load spectrum as presently applied. As indicated in figure 8, it is expected that load spectrums have to be extended particularly in the domains which are outside of the nominal turbine operation modes, including very high drive train loads, but having associated low cycles numbers.

Figure 8: Load spectrum for 20 years life time indicating the domains where an extension of load cycles is expected. 7. Further investigations & future implementations Future investigations will be focused on the numerical optimization of wind turbine power trains and the evaluation of fatigue load spectra. It is mentioned that the presented “SAMCEF” /12/ wind turbine model is parameterized and can be piloted by the optimization platform “BOSS quattro” /13/. That coupling of optimization procedures and detailed numerical modeling allows a very efficient application of optimization algorithms on design parameters like the generator-converter control loops, the braking procedures, as well as the mechanical characteristics of the gearbox in terms of frequency content, bearing properties and/or gear tooth profile corrections. It is deduced as well from experimental data, as from numerical analysis, that the presently applied fatigue load spectra should be revised and

complemented by “load cases” which might occur during transient events like “emergency shut-down”, “electrical grid faults”, or simply very “unsteady wind conditions”. A comparison of numerical results and experimental data has shown that suited numerical models can predict power train loads even during operation modes which produce dynamic load amplification of more than 250% with respect to nominal loads. Under such unsteady conditions, the error in numerical load prediction with respect to experimental data, was proved to be generally less than 20 %. 8. References [1] R. Craig and M. Bampton. Coupling of substructures for dynamic analysis. AIAA Jnl.,6 no. 7:1313-1319,1968. [2] H.M. Hilber, T.J.R. Hughes, and R.L.Taylor. “Improved numerical dissipation for time integration algorithms in structural dynamics”, Earthquake Engng. Struct. Dyn., Vol. 5, pp. 283-292, 1977. [3] A.Cardona and M. Geradin. Time integration of the equations of motion in mechanism analysis, Computers and Structures, 33, No. 3,pp. 801-82, 1989. [4] A. Cardona, M. Geradin, and D.B. Doan. Rigid and flexible joint modelling in multibody dynamics using finite elements. Comp. Meth. Appl. Mech. Engng., 89, pp. 395-418,1991. [5] A. Heege and P. Alart, “A frictional contact element for strongly curved contact problems”, International Journal for Numerical Methods in Engineering, 39, pages: 165-184 (1996). [6] A. Heege and P. Jetteur. “Evaluation of different approaches for coupling structural and mechanism analysis based on the experience of SAMTECH”. Abstracts: Tagung Haus der Technik e.V. Essen: Nr. E-30-206-056-7 “Verbindung von MKS mit FEM Modellen”, 4-5 Februar 1997. [7] A. Cardona. “Three Dimensional Gears Modeling in Multibody Systems Analysis”. International Journal for Numerical Methods in Engineering, Vol. 40, pgs.357-381, 1997. [8] A. Cardona and D. Granville.” Flexible gear dynamics modelling in multibody analysis”. Abstracts 5 US National Congress on Computational Mechanics, Boulder, USA, pg 60, 4-6 August 1999.

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[9] A. Heege. “Implicit time integration of impacts between deformable bodies undergoing finite deformation”, Eccomas 2000 Conference: European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona. John Wiley & Sons, 11-14 September 2000.[10] A. Heege. “Detallierte Simulation von dynamischen Rückschlägen in Windturbinen, Erneuerbare Energien, pgs. 27-30, Edition 7/2003.[11] A. Heege. “Computation of dynamic loads in wind turbine power trains”. DEWI Magazin Nr. 23, pp. 59-64, August 2003. [12] SAMCEF Mecano ®. User Manual version 10.1. SAMTECH s.a., http://www.samcef.com. [13] BOSS quattro ®. User Manual version 5.0. SAMTECH s.a., http://www.samcef.com

9th SAMTECH Users Conference 2005 4/4