Analysis of wind turbine blade vibration and drivetrain loads

The reliability of wind turbines is a major issue for the industry. Drivetrain and blade failures are common, costly and not fully understood. Designers must thus examine and understand the key parameters that influence reliability. As wind turbines increase in size, the blades are designed to be more lightweight and flexible, increasing the potential for large-displacement oscillations during operation. This necessitates the incorporation of nonlinearity in the formulation of the blade model to better understand the dynamics and stability characteristics. Also, oscillations in the blade impart dynamic loading onto the gearbox. Understanding these dynamic loads is essential for the design of reliable gears and bearings, and hence economically viable wind turbines. Traditional studies of wind turbines have focused on the aerodynamic performance of the blades, the reliability of gearbox and its components, grid connectivity and improvements in power distribution. The aspect of blade vibration from a dynamics point of view has garnered interest but not been fully developed and understood. In this work, the partial and ordinary differential equations that govern the in-plane and out-of-plane motion of a wind turbine blade subject to gravitational and aerodynamic loading are developed using Hamilton's principle and Lagrange formulations respectively. These differential equations include nonlinear terms due to nonlinear curvature and nonlinear foreshortening, as well as parametric and direct excitation at the frequency of rotation. The equations are reduced using an assumed uniform cantilevered beam mode to produce single second-order ordinary differential equations (ODE) to approximate the blade model for the case of constant rotation rate. Embedded in the ODE's are terms of a forced Mathieu equation with cubic nonlinearity. Different variations of the forced Mathieu equation are analyzed for resonances by using the method of multiple scales. The forced Mathieu equation has instabilities and resonances at multiple superharmonic and subharmonic frequencies. Second-order expansions are used to unfold the expressions that govern the amplitude of response at these critical resonances. The equations of motion (EOM's) also have regions of instability and we employ perturbation analysis to identify the stability transition curves of the system. These calculations compare well with numerical simulations for simple systems under study. The formulation is then extended to wind turbine blades. The effect of various parameters on the amount of blade oscillation is demonstrated using the amplitude-frequency curve. Aerodynamic forces on the wind turbine blades are calculated using the Blade Element Momentum (BEM) theory and its extensions. Commercial software, such as FAST, have also been used to simulate responses for specific blades to understand influence of various blade parameters. For current production wind turbine blades, the parameters are such that the superharmonic resonances are not excited significantly. From parametric studies of the blade EOM we can understand the parameter values at which these resonances become dominant. It is shown that as wind turbine blades become larger they are prone to superharmonic resonances, whose existence may not be within the scope of current design strategies. The amplitude of response at all resonances tend to become amplified for much larger blades. Both in-plane and out-of-plane responses will increase the loading at the rotor hub and consequently, increase the loads and moments on the wind turbine drivetrain. To capture the effect of increased loading on the wind turbine drivetrain, we follow two approaches. First, using RomaxWind, we model the 750 kW gearbox used as a part of the Gearbox Reliability Collaborative (GRC) headed by National Renewable Energy Labs (NREL). For this, we partnered with Romax Technology Ltd. to analyze the sensitivities of the load on the elements of the gearbox to variations in the input loads. Using the Romax gearbox model, we suggest methods to optimize the gear geometry to improve reliability of the drivetrain by minimizing influence of manufacturing and assembly tolerances and misalignments. We also designed novel approaches to predict gearbox vibration using the models and suggested changes that are required to improve the overall design of the gearbox (these have been implemented while manufacturing newer generations of the NREL GRC gearbox). Second, for the case of in-plane blade vibration, we use a simple torsional model of the wind turbine gearbox to study the influence of a time varying load on the torsional response of the drivetrain. The effect of increased loading on larger wind turbine systems is shown by scaling values of blade and gearbox properties. The fundamental work formulated in this thesis can be extended to more complex models to understand other system level dynamics of interest (multi-mode interaction, multi-blade resonance, etc.). More detailed formulation of aerodynamic loads (for example by using ONERA semi-empirical approach) would also improve model fidelity for predicting the influence of aerodynamic loads on blade vibration.