Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'Innovative concepts for drive train components' taking place on Thursday, 13 March 2014 at 11:15-12:45. The meet-the-authors will take place in the poster area.

**Andreas Heege**LMS-SAMTECH, A Siemens Business, Spain

Co-authors:

Andreas Heege (1)

^{F}

^{P}Loic Bastard (1)

(1) LMS-SAMTECH, A Siemens Business, Barcelona, Spain

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## Abstract

**Non-linear dynamic analysis of a wind turbine direct drive generator by a coupled FEM-MBS approach
****Introduction**

A cost efficient design of wind turbines requires among others an optimization of generator components with respect to static and structural dynamic properties. That requirement leads generally to lighter and more slender wind turbine components which are more susceptible to structural vibrations and resonances.

Central theme of the present work is the non-linear transient analysis of a direct drive generator with particular focus on dynamic amplifications and resonances. Objective of the transient simulations is to support the optimization of the structural dynamic properties of a direct drive generator.

**Approach**

The applied mathematical formulation is based on an implicit non-linear dynamic Finite Element Method/FEM which incorporates simultaneously all common Multi-Body System/MBS functionalities and aero-elastic load procedures (Blade Element Momentum theory) [1][2][3][4].

The generator rotor and stator are modelled initially by detailed FEM models. Prior to integration of the component FEM-models in the global MBS-model of the generator, a condensation technique is applied. This is required because a non-linear analysis in time domain over some hundreds of seconds is to be performed. Computational speed has to be sufficient in order to complete overnight the numerical analysis on a standard Personal Computer.

Accordingly the generator rotor and stator are presented in terms of Super Elements (Craig & Bampton). Non-condensed Super Element nodes are retained on the one hand at each pole of the stator, and, on the other hand, at each permanent magnet of the rotor. Further non-condensed Super Element nodes are retained at the kinematical coupling points like the bearing locations or the elastic couplings of the generator housing.

Special attention is dedicated to the implementation of electro-magnetic excitations that act in between the poles of the stator (see Figure 1) and the rotor respectively (see Figure 2).

The modeling approach of the electro-magnetic forces which act in between poles and pole shoes/permanent magnets is inspired from solution techniques which are frequently applied in the solution of structural contact problems [2][5]. Non-linear electro-magnetic forces are formulated for each combination of pairs of poles and pole shoes (permanent magnets). Analogous to common solution schemes for frictional contact problems in structural analysis, first a contact search algorithm establishes the kinematical relations in between each combination of pole pairs. The kinematical variables of each combination of pole pairs are defined in terms of “normal distances” and in terms of “tangential distances”. The normal distance is directly analogous to the “air gap” of the pole pairs. The normal distance is defined for each combination of poles and permanent magnets. Analogously, the “tangential distances” are defined for each combination of pole pairs through the circumferential distances in between each pole and each magnet.

**Main body of abstract**

Before presenting the transient results of a startup simulation of the generator, some aspects of the implementation of the electro-magnetic loads are commented. The electro-magnetic forces which act in between each pole pair are defined implicitly as a non-linear function of the normal distance (see “air-gap” presented in Figure 3), the tangential distance and the instantaneous generator power. In order to establish a kinematical relation in between each pole and each permanent magnet, there is connected to each retained Super Element node of each pole of the stator model a rigid surface. These rigid surfaces are connected individually to the retained Super Element nodes of each stator pole.

In order to compute the “kinematical pole pair variables” (i.e. the normal and the tangential distances), there is performed a projection of the “slave nodes” of the rotor magnets onto the contact surfaces of the stator poles. That projection is performed for all possible combinations of poles and magnets for each iteration of each time step of the transient analysis. Accordingly, there are computed for each combination of poles and magnets the electro-magnetic loads in tangential and normal direction. The final electro-magnet loads are obtained from the integration of all individual combinations of poles and permanent magnets. The described generator model is based on a non-linear FEM-MBS formalism which is formulated for arbitrary large rotations, a feature which is required for the transient analysis of the operating generator.

Figure 4 presents the electro-magnetic loads which act in the normal direction to the contact surfaces of the pair of poles for quasi-stationary rotation of the rotor. It is emphasized that these electro-magnetic loads present analysis results and not boundary conditions of the simulation run. The normal distance in between each pole pairs is known at the iterative solution of each time step of the numerical integration procedure. At the iterative solution of the associated non-linear equations, the electro-magnetic forces, the inertia forces and the elastic bearing and housing loads keep the generator assembly in dynamic equilibrium.

As follows the dynamic response of the generator is analyzed for a startup from rest to 9[Rpm].

Acceleration transients in axial rotor direction are presented for a startup simulation of a 4[MW] generator in Figure 5.

The computed acceleration transients reveal resonances with important dynamic amplifications for specific instances of the startup (i.e. for specific rotational speeds of the generator rotor). These resonances are clearly visible in the computed accelerations transients and can be related to specific Eigen-modes of the generator rotor, or respectively of the generator stator. Available experimental measurements validate the computer simulation of the startup.

Figure 6 presents the spectrogram of the transient which is presented in Figure 5. The resonances which can be identified in Figure 5 and Figure 6 correspond to specific Eigen-modes of the stator, rotor and generator assembly.

Even though the generator is analyzed in the present application as it is mounted on a test rig, the applied computer program supports the integration of complex FEM or MBS sub-models like generators or drive-trains in one global aero-elastic wind turbine model [1][2][3][4]. As it is further commented in the conclusions, the integration of the presented generator model in a global aero-elastic wind turbine model will permit to obtain further insight on the dynamic generator behavior, but with different boundary conditions.

Figure 1 Definition of “retained Master Nodes” which are located on the poles of the stator model.

Figure 2 Definition of “retained Slave Nodes” which are located on the magnets of the rotor model.

Figure 3 Electro Magnetic Forces as a function of normal distance (air gap).

Figure 4 Transient of Electro Magnetic Forces in between a pole pair for stationary-operation.

Figure 5 : Stator accelerations in axial direction: Identification of resonances in time domain.

Figure 6 : Identification of resonances in time domain by a spectrogram of axial accelerations.

**Conclusion**

The bandwidth of significant low frequent structural Eigen-frequencies of an assembled direct drive generator of the multi-megawatt class is generally included in the domain from 10[Hz] to 100[Hz]. The elastic deformation energy of excited Eigen-modes is generally accumulated by the stator, rotor and supporting structures including the rotor bearings. Even though the excitation frequency of the rotating rotor is much lower than the first generator Eigen-frequencies, a crossing of Eigen-frequencies and excitation frequencies occurs because the latter are proportional to “rotor speed” times the “number of pole-pairs”. No resonance phenomena should occur if structural generator components are dimensioned properly from a dynamic point of view for nominal speed. However, during the startup resonances phenomena are difficult to avoid, because the entire frequency bandwidth up to approximately 50[Hz] is excited.

Conclusions from test-rig measurements and simulations permit to obtain better insight on the activation and de-activation speeds of Eigen-modes. An interesting observation is the possible mutation of an excited Eigen-mode to a different one when rotational speed of the generator is altered. In that case the deformation energy of an excited Eigen-mode might switch “seam-less” to a different Eigen-mode.

It should be noted that conclusions about dynamic characteristics which are obtained from a generator test rig (or more generally power train test-rig) cannot always be transferred to the dynamic properties on an operating wind turbine. The dynamic response of an assembled power train on an operating wind turbine can be substantially different due to the modified inertias, flexibilities and excitations.

Future work concerns the integration of the presented generator model in a pre-existing global aero-elastic wind turbine model. Objective is to compare the dynamics of the generator during operation on a holistic wind turbine with the dynamic behavior on a test-rig.

**Learning objectives**

The detected resonances put in evidence the complexity and challenges of “dynamic dimensioning” of wind turbine components. Future work will focus on methodologies which permit to improve the “dynamic dimensioning” of assembled wind turbines.

**References**

[1] Samcef Wind Turbines User Manual V3.3, http://www.samtech.com, 2013.

[2] Samcef/Mecano ®. User Manual V. 14.1 Samtech SA, http://www.samtech.com

[3] Michel Geradin and Alberto Cardona, Flexible Multibody Dynamics: A Finite Element Approach. John Wiley and Sons Ltd, 2001.

[4] Heege A., Betran J., Radovcic Y. "Fatigue Load Computation of Wind Turbine Gearboxes by Coupled Finite Element, Multi-Body-System and Aerodynamic Analysis". Wind Energy, 2007; vol. 10:395-413, John Wiley & Sons Ltd. (www.interscience.wiley.com: DOI 10/1002/we.226).

[5] A. Heege and P. Alart, “A frictional contact element for strongly curved contact problems”, International Journal for Numerical Methods in Engineering, Vol. 39, 165-184, Wiley & Sons (1996).

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