2.3 Wind turbines
3.1.2 Power losses in rolling element bearings
Power losses in bearings will be discussed from the point of view of rolling element bear-ings since the tested bearbear-ings are of this category, as will be discussed in section 3.2.2.
Furthermore, it should be noted that multiple dierent models for rolling bearing power losses have been presented [21, pp. 59]. Generally simpler models do not consider load independent losses while advanced models take these into account. Load independent losses manifest mainly from the drag caused by the lubrication oil.
In the scope of this work only the models from bearing manufacturers SKF and Schaef-er are presented because SKF models are used in Moventas bearing design calculations, and Amesim uses the Schaeer model among others for simulation. It should be also noted that bearing dimensions are by convention expressed in millimetres and conse-quently the frictional torques are conventionally given in (Nmm) units. This convention is retained here.
3.1.2.1 Simple models Simple forms of frictional power losses typically assume a constant friction coecient µand do not take load independent losses into account. The frictional moment is then simply
M =µF r = µP dm
2 , (Nmm) (1)
where P is the equivalent dynamic load from combined axial and radial forces and dm is the bearing mean diameter
dm = d+D 2 ,
where d is the bearing bore diameter and Dthe outer diameter as shown in gure 6b.
As recount by Fernandes [21, pp. 63], the so called Coulomb model assumes vectorially added loads for the equivalent dynamic load
P =p
Fr2+Fa2, (2)
where Fr and Fa are the radial and axial loads, respectively. A slightly more complex model is the so called simple SKF model [19]. According to this model a quick estimate of bearing power losses can be calculated from
M = µP d
2 (Nmm).
It should be noted that here the bearing bore diameter d is used in stead of the mean diameter dm. The SKF simple model uses the general form of the equivalent dynamic load
P =XFr+Y Fa. (3)
The coecients X and Y depend on the bearing type and ratio of forces Fa/Fr. This ratio is usually compared to characteristic ratio e which is a bearing parameter
Since power losses in rotating system can be calculated from the product of angular velocity ω and frictional moment, the general form for frictional power losses NR is
NR=ωM = 2πn
60 M =µP dmπn
60 , (W) (4)
where the angular velocity is replaced by rotational speednand the frictional torque must be converted to SI units of (Nm).
The SKF model expresses this equivalently as
NR= 1.05×10−4M n = 5.24×10−5µP dn , (W) (5) where the factor
1.05×10−4 ≈ 2π 60·1000
arises from the fact that the frictional torque is given in (Nmm).
The friction coecient µ depends on the bearing type and ranges from 0.001 to 0.005.
The above estimate is only valid in normal operating conditions with good lubrication and at loads corresponding to 10 % of the bearing's rated capacity. [19]
(a) Due to deformation of elastic sur-faces the contact area is elliptical and the motion pure rolling only at specic points.
(b) Inlet shear heating reduction factor takes into account the situation where part of the lubrication oil is owing back towards inlet. Figure from [22].
Figure 8
3.1.2.2 New SKF model The so called new SKF model [22] considers frictional losses from rolling and sliding motions as well as losses from bearing seals and drag.
Accordingly, the total frictional moment is expressed as
M =Mrr+Msl+Mseal+Mdrag (Nmm).
Even though rolling element bearings aim to reduce friction by utilizing rolling friction in place of sliding friction, the contact between rollers and the bearing raceway is pure rolling only at specic points. This is because applied loads deform the surfaces, resulting in elliptical contact areas as described in gure 8a.
Rolling frictional moment is given by
Mrr =ϕishϕrsGrr(νn)0.6, (Nmm) (6) where
• ϕish is the inlet shear heating reduction factor describing the fact than not all of the lubrication oil is fed to the contact area, rather causing an reverse ow as in gure 8b.
• ϕrs is the kinematic replenishment/starvation factor which accounts for situations where not enough lubricant is injected to replenish the lubricant displaced by the rollers
• Grr is a variable depending on the bearing type, dimensions, and applied loads
• ν is the kinematic viscosity of the lubricant and n the rotational speed.
Due to complexity of the full equations and factors bearing calculations are done with ded-icated programs and in this context the preceding and following factors are not presented in detail. Details can be found in the SKF frictional model theory [22].
Sliding frictional moment is
Msl=Gslµsl (Nmm). (7)
Here Gsl is analogous to coecient Grr for sliding motion and µsl is the sliding friction coecient. Seal losses are expressed as
Mseal=KS1dβs +KS2 (Nmm).
Here ds is the seal counter-face diameter and all the other constants dependent on the seal and bearing type.
Drag losses are calculated depending on oil inlet conditions, bearing type, dimensions and operating conditions. The full empirical equation for roller element bearings in oil bath lubrication is Factors considered before retain their usual meanings. Other variables included in the drag loss calculation are
• VM is the drag loss factor, which depends on the height of the oil level inside the bearing
• Kroll is a factor depending on bearing geometry
• CW is a polynomial factor depending on bearing dimensions
• B is the bearing width
• ft is a geometrical term depending on the bearing mean diameter and oil level
• RS is a factor depending on bearing geometry and oil level
Finally, the value calculated for drag losses in oil bath is multiplied by a factor of 2 in the case of oil jet lubrication, which is in use in the current application. Power losses are again calculated from equation (5).
3.1.2.3 Schaeer model The Schaeer model [23] is similar to the SKF as it con-siders rolling and sliding friction losses as well as uid friction losses. Schaeer model is also relevant because the simulation software Amesim uses the same equations. The total frictional torque in this model is
M =M0+M1, (Nmm)
where M0 depends on rotation speed and consists in principle of losses due to uid ow resistance, and M1 accounts for the load dependent losses. Speed dependent frictional torque is given by
M0 =f0·(νn)2/3d3m·10−7 (Nmm). (9) Here f0 is a friction factor dependent on bearing type and dimensions, ν oil kinematic viscosity in (mm2/s−1), n and dm are the rotational speed and bearing mean diameter as before. If the product of kinematic viscosity and rotation speed is below 2000, it is replaced by this constant value.
Load dependent frictional torque is calculated from
M1 =f1P1dm (Nmm). (10)
Again, f1 is a friction factor dependent of the bearing and the mean diameter. The equivalent dynamic loadP1has the same form as in the SKF model shown in equation (3).
Since cylindrical roller bearings are normally designed to support only radial loads they have X = 1 and Y = 0 in equation (3). And thus the load dependent frictional moment takes the form
M1 =f1Frdm, (Nmm) for cylindrical roller bearings.
Power losses in the Schaeer model are given as NR= (M0+M1)· n
9550, (W) (11)
where yet another approximation for2π/60is used and correction for millimetres applied.
If axial load is applied to cylindrical roller bearings additional frictional moment of the form
M2 =f2Fadm (12)
ensues. Since CRBs should not be loaded axially in normal operation, this moment is not considered here. The frictional factorsf0 andf1 can be read from bearing catalogues [23], and are given when relevant calculations are executed in section 5.
3.1.2.4 Amesim model Amesim is a simulation software from Siemens PLM Sofware company incorporating for example electrical, uid, control logic, and mechanical libraries in a so called multi-physics simulation software [24]. The software was developed by LMS International which was bought in 2012 by Siemens [25]. Development was started over 20 years ago by Imagine S.A. which had been previously bought by LMS. Previous developers are noted in the full name: LMS Imagine.Lab Amesim.
Unlike the other discussed models, Amesim considers the torque values in SI units of (Nm). Also, the relevant diameter for calculations is denoted Dm, diameter of the axis to the center of rolls. This is however equivalent to the bearing mean diameter dm discussed in the previous section.
The total frictional torque is presented as
M =Moil+Mradial L+Maxial L,
which is equivalent to the Schaeer formation. The oil losses have a form
Moil =f0·(νn)2/3d3m·9.78×10−2 (Nm). (13) This is otherwise equivalent to equation (9) except for the dierent coecient due to use of (m) instead of (mm), and again applies for νn >2000.
The frictional torque due to equivalent radial load is
Mradial L=C1f1(P0+P1)dm, (Nm) (14)
which is again similar to equation (10). Coecient C1 can be used to adjust the losses empirically, and P0 is the equivalent static preload. If the bearings are not preloaded P0 = 0. The axial load equation in Amesim is identical to the Schaeer form in equation (12). Since the frictional torques are given (Nm), Amesim calculates the power losses from (4)
NR=ωM = πnM
30 (W). (15)