New strategy for testing new high nitrogen bearing steel for offshore wind turbines

Effect of sliding at the interface under viscous-elastic conditions

The kinematic analysis can be performed, in the first approximation, by assuming that a pure rolling motion occurs between the rollers and the rings and that the rollers and the rings are rigid bodies. Therefore, the centers of the relative rotation can be defined for the three pairs: roller–inner ring, roller–outer ring, and inner–outer rings. However, the rigidity should be conveniently relaxed to admit the hypothesis of elasticity, under which much more useful information can be achieved. For example, assuming an elastic contact between the roller and the inner ring and applying the relative motion theorem, it is possible to obtain an approximation of the sliding velocity w at the interface at the beginning F and at the end E of the contact line. With reference to Figure 1(a), points F and E can be considered as belonging to the roller or to the inner ring, and so, the classic velocity triangle can be easily drawn out for the detection of the sliding (relative) velocity. A short MATLAB Script has been encoded to obtain the theoretical sliding along the contact profile without taking into account the stick regions of the contact. Results are presented in Figure 1(b).

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Figure 1.

(a) Sliding velocity triangle for points F and E under the assumption of the existence of a pseudo-rigid part of the ring and (b) sliding along the tangential interface roller–ring in the contact arc.

In this simplified approach, the obtained sliding velocity is only representative of a nonholonomic constraint, which is active on the contact profiles and induces an adhesive traction stress τ_k, that is, in the first approximation, proportional to sliding w, according to a viscous-elastic model for which the element dynamic balance is described by the following equation

τ k = G γ = η w

(1)

where γ is the deformation angle, G is the elastic tangential module, and η is the viscosity coefficient.

Loads on rollers

Considering the working conditions, each roller can be considered as radially loaded. However, during rolling, the real material shows dissipative phenomena, such as hysteresis and elasticity delay. These two occurrences are responsible for rolling friction. The situation is qualitatively depicted as in Figure 2(a), where the pressure diagram becomes asymmetrical because in the increasing stress regions, the pressure is greater than in those with decreasing stress. Since symmetry is lost, the action (as well as the reaction) force Q is tilted with respect to the case of ideal material. Furthermore, force Q can be reduced to a normal N _e plus a tangential T _e component, which leads to the conclusion that tangential stresses appear on the ring and roller surfaces. Hence, even though the rollers are not designed to theoretically transmit any power from a moving ring to the other ring, a small amount of power is dissipated to ground as heat. In order to guarantee the dynamic balance of the roller, neglecting its weight and the accelerations, any normal force N _e has to be concentrated in the pressure center of the normal stress distribution, which is no more symmetrical because of the elasticity delay. The two normal forces, therefore, are equivalent to a clockwise moment, which must be balanced only by another moment. The balancing moment is given by the tangential adhesion of the contact region which yields, on the inner and outer rings, a traction force T _e.

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Figure 2.

(a) Introduction of a tangential component in the roller loads under stationary conditions and (b) resultant R _e, tangential T _e, and normal N _e forces which act on the inner and outer rings due to roller dynamic balance in stationary regime.

Overall effects on the inner ring

With reference to Figure 3(a), two tangential tensions can be combined. The first one is representative of the tangential stress τ_k due to the viscous-elastic action of the roller surface as described by equation (1). This stress, generated by the roller on the ring, can be regarded as the product of a viscous coefficient multiplied by the sliding velocity, directed as the inlet rolling velocity U. The second one consists in a tangential tension τ_E that must act between the surfaces in order to guarantee the roller dynamic balance. This tension is proportional to the normal stress by a virtual adhesive coefficient µ_0E which must be greater than the actual friction coefficient µ₀. In fact, the integral of all the tensions extended to the contact profile must have a net resultant which is equal to T _e, as shown in Figure 2(b), and so, the constant µ_0E is determined in such a way that the overall traction is equal to T _e.

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Figure 3.

Qualitative distribution of (a) the tangential stress τ_E and τ_k and evaluation of their difference and (b) the net tangential stress τ_E − τ_k and adhesion tangential distribution σ_µ0. Determination of (c) the intersection points and (d) the slip stick regions.

The difference between the two tensions, τ_k − τ_E, is characterized by two external regions where the overall tension is positive (same sense as U) and a central region where this tension is negative (see Figure 3(b)). By comparing this difference function with the adhesive surface capability, equal to σµ₀, five regions can be identified, as shown in Figure 3(c): two external regions with positive slip, one central region with negative slip, and two middle regions where adhesion prevails on the tangential actions exerted by the combination of the external constraints and the roller dynamic balance (see Figure 3(d)). Although the proposed approach has been developed by assuming several approximations, more complex methods give similar results, as those presented in Bentall and Johnson (1967).

Influence of load direction

The waves reported in Figure 4 refer to a specific zone of the inner ring. However, the load conditions change along the contact circle, and so, the wave amplitudes are different in different points of the inner ring. For an externally applied radial load Q, on a bearing having z of rollers, Stribeck’s approximated formula

P 0 = 5 Q z

(2)

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Figure 4.

Qualitative representation of the normal and tangential wave trains, which affect the bearing races. Tangential tensions change their direction sense with a double frequency with respect to the roller passages. Furthermore, normal and tangential amplitudes are modulated by the vibrations generated by the wind turbine, both flexural and torsional.

predicts the maximum load P₀ on a the bottom roller. However, after the cage rotates by a quarter of revolution, only the preload will affect this roller, and after half revolution of the cage, the roller will be in its upper position, that is, in opposition to the highest load configuration. In this case, the overall load is even less than the preload because there is a small displacement of the inner ring downward, and the overall roller–ring slip occurrence probability increases.

Each roller sustains a load cycle, which is reasonably near to a pulsing cycle, namely, variable from about 0 to P₀. However, on a specific point of the inner ring, the amplitude of the load waves, reported in Figure 4, will have a peculiar amplitude depending on the position of the point of interest. The frequency f_S of the exciting wave is proportional to the angular speed n (r/min) of the main shaft, namely

f S = R i 2 ( R i + r ) 60 n

(3)

where r and R_i are the radii of the roller and the inner ring, respectively.