Recent data shows that 90% of large wind turbines include a gearbox, and industry forecasts expect this fi gure to remain relatively stable. With global annual volumes (2009) of around 18,600 units, the quality, cost and performance of gearboxes is of paramount importance to the wind sector. The industry has been focusing some attention on gearbox reliability, as demonstrated by a growth in the number of specifi c seminars and collaborative programs on this topic. One aspect that needs to be brought to an industry-wide forum is the understanding of the complexity of bearing design in the gearbox and the careful attention that needs to be paid to ensure a reliable gearbox design. This paper seeks to address this issue by clear demonstration of design issues using a model of the gearbox from the National Renewable Energy Lab's Gearbox Reliability Collaborative. Detailed models are presented with focus on determining the quality of the function of the planetary gear stages. Key design drivers are discussed such as the quality of alignment at the gears and bearings and the loads and stresses seen on these components. Under a design load case with a signifi cant rotor off-axis moment the stresses in the planet gears and bearings are investigated. It is shown how the misalignment of the planet pins varies with the rotation of the planetary set and how subsequently time-varying contact stresses and load distributions occur in the planet gears and bearings. These factors strongly infl uence the fatigue life of the gearbox components as well as the level of vibration. Design tools are then used to demonstrate how small variations in the clearances of the planet carrier bearings can have a big effect on the quality of the design. Numerical studies show where optimal clearance settings lie and how the misalignment of the planetary set can be improved. Furthermore, a demonstration is made of how redesign of the bearing arrangement and subsequent optimization of the planet tooth geometry further improves the misalignment and results in signifi cantly reduced time-varying contact stresses, better load distribution and reduced vibration. It is illustrated that small clearances, such as in the carrier bearings, can have a large effect on the performance of the design and a study shows how to identify and reduce time-varying misalignment and contact stresses resulting in lower vibration, lower fatigue and a more reliable product.

A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation with cubic nonlinearity to model its dynamic behavior. This equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. Order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and they are similar to those of the Duffing equation.

The INTRODUCTIONThis paper introduces a model of the lead-lag (in-plane) vibration motion of an operating wind turbine subjected to gravitational loading and aerodynamic loading, and provides initial analyses of resonances by using a simplification of the singlemode reduced-order model.

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The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.