Calculating bearing stiffness based on preload

Within this study the preload of the bearings is selected based purely on required stiffness of the bearings. The other aspects of the preload selection are neglected to keep focus on the rotor dynamics. Bearing stiffnesses are calculated using Rotor-Bearing Dynamics (RoBeDyn) toolbox for MATLAB -software. RoBeDyn toolbox is developed in LUT university and is used to simulate rotor and bearing dynamics. Package includes function to calculate bearing stiffnesses and thus is useful in this study. The bearing stiffness function is based on Hertzian contact theory and results are calculated using Newton-Raphson iteration. The stiffness calculation method is described in detail in following paragraphs.

In Figure 6 main dimensions of a ball bearing are presented. Where 𝑑 is ball diameter, 𝑟_𝑜𝑢𝑡 is the outer groove radius, 𝑟_𝑖𝑛 is the inner groove radius, 𝑐_𝑑 is the diametral clearance, 𝐷_ℎ is

the bearing housing diameter, 𝑑_𝑚 is the pitch diameter, 𝑑_𝑠 is the bore diameter, 𝐷_𝑖 is the inner raceway diameter and 𝐷_𝑜 is the outer raceway diameter. (Kurvinen, et al., 2015, p. 13)

Figure 6. Main dimensions of a ball bearing (Kurvinen, et al., 2015, p. 13)

As described in article by E. Kurvinen et al. the elliptical contact conjunction is used to calculate contact stiffness between rolling elements and bearing raceway. Figure 7 the geometry of elliptical contact conjunction is shown with the normal force that acts on the solids. It is noteworthy that radius of contact conjunction is defined negative when surface is concave. (Kurvinen, et al., 2015, p. 11)

Figure 7. Elliptical contact conjunctions (Kurvinen, et al., 2015, p. 11)

Contact deformation are defined with aid of curvature sum, 𝑅 , and curvature difference, 𝑅_𝑑, as shown in equations 13 & 14.

1

In case of angular contact bearing, the contact angle must be considered. Figure 8 shows a cross section of angular contact bearing. In Figure 9 the cross section is taken in plane that is colinear with the contact angle line. In Figure 8 𝜙 is the contact angle of the bearing, 𝑟_𝑏𝑦^𝑖𝑛 is the groove radius of bearing inner ring, 𝑟_𝑏𝑦^𝑜𝑢𝑡 is the groove radius of bearing outer ring, 𝑟_𝑏 is the radius of the bearing ball, and 𝑑_𝑗 is the distance along contact line between the outer and inner raceways. In Figure 9 𝑟_𝑏𝑥^𝑜𝑢𝑡 is the radius of outer raceway and 𝑟_𝑏𝑥^𝑖𝑛 is the radius of inner raceway.

Figure 8. Cross section of angular contact ball bearing geometry.

Figure 9. Contact plane cross section of angular contact ball bearing 𝑟_𝑏𝑦^𝑜𝑢𝑡

𝑟_𝑏𝑦^𝑖𝑛

𝜙

𝑟_𝑏

𝑑_𝑗

𝑟_𝑏𝑥^𝑖𝑛 𝑟_𝑏𝑥^𝑜𝑢𝑡

When contact angle is considered, the radiuses of the inner race are calculated as shown equation 17-19 (Kurvinen, et al., 2015, p. 12)

𝑟_𝑎𝑥^𝑖𝑛 = 𝑟_𝑎𝑦^𝑖𝑛 =𝑑

For outer raceway the corresponding equations are 20-22 (Kurvinen, et al., 2015, p. 12)

𝑟_𝑎𝑥^𝑜𝑢𝑡 = 𝑟_𝑎𝑦^𝑜𝑢𝑡 = 𝑑

When stiffness of angular contact ball bearing is calculated equations 17-22 must be used in formulas 13 & 14 to retrieve curvature sum, 𝑅 , and curvature difference, 𝑅_𝑑, that in turn, will be used in later calculations.

When a load is applied to the solids, contact point is expanded to an ellipse. The ellipticity parameter, 𝑘_𝑒, is described as shown below, in equation 23 (Sopanen & Mikkola, 2003, p.

7).

In turn the ellipticity parameter is defined as shown in equation 24 (Sopanen & Mikkola,

Where elliptical integrals 𝜉 & 𝜁 are as follows

𝜉 = ∫ [1 − (1 − 1 simplify calculation following approximated values can be used (Kurvinen, et al., 2015, p.

12) (Sopanen & Mikkola, 2003, p. 8)

𝑘̅_𝑒 = 1,0339 (𝑅_𝑦

Where

𝐸′ is the effective modulus of elasticity 𝐸_𝑎 is the modulus of elasticity for material a 𝐸_𝑏 is the modulus of elasticity for material b 𝑣_𝑎 is the Poisson’s ratio for material a 𝑣_𝑏 is the Poisson’s ratio for material b

Using equations 27-30 shown above, the contact stiffness coefficient 𝐾_𝑐 can be calculated with equation 31

𝐾_𝑐 = 𝜋𝑘̅_𝑒𝐸^′√ 𝑅𝜉̅

4,5𝜁̅³ (31)

Total stiffness of a single bearing ball can be calculated when contacts with inner and outer race are combined with equation 32 as shown below

𝐾_𝑐^𝑡𝑜𝑡 = 1

𝐾_𝑐^𝑖𝑛 is the contact stiffness between bearing ball and inner raceway 𝐾_𝑐^𝑜𝑢𝑡 is the contact stiffness between bearing ball and outer raceway

The ball bearing forces and moments can be calculated using the relative displacements of the raceways. The displacements of each bearing balls can be evaluated in radial direction using equation 33 and in axial direction with equation 34 (Kurvinen, et al., 2015).

𝑒_𝑗^𝑟= 𝑒_𝑥𝑐𝑜𝑠𝛽_𝑗+ 𝑒_𝑦𝑠𝑖𝑛𝛽_𝑗 (33)

𝑒_𝑗^𝑡 = 𝑒_𝑧− (−𝛾_𝑥𝑠𝑖𝑛𝛽_𝑗+ 𝛾_𝑦𝑐𝑜𝑠𝛽_𝑗)(𝑅_𝑖𝑛+ 𝑟_𝑖𝑛) (34)

Where

𝑒_𝑥 is relative displacement in x direction 𝑒_𝑦 is relative displacement in y direction 𝑒_𝑧 is relative displacement in z direction 𝛽_𝑗 is the attitude angle of ball bearing j

𝛾_𝑥 is the relative misalignment of the inner and outer raceway about x axis 𝛾_𝑦 is the relative misalignment of the inner and outer raceway about y axis

The displacements of raceways and bearing balls are visualised in Figure 10.

Figure 10. Axial and transverse cross-section in the A-A plane of illustrative structure of a ball bearing (Kurvinen, et al., 2015, p. 14)

The contact angle of a single bearing ball, 𝜙_𝑗, is expressed as shown in equation 35

𝜙_𝑗 = tan⁻¹( 𝑒_𝑗^𝑡

𝑅_𝑖𝑛+ 𝑟_𝑖𝑛+ 𝑒_𝑗^𝑟− 𝑅_𝑜𝑢𝑡+ 𝑟_𝑜𝑢𝑡) (35) Where,

𝑅_𝑖𝑛 is the radius of the inner raceway 𝑅_𝑜𝑢𝑡 is the radius of the outer raceway 𝑟_𝑖𝑛 is the radius of the inner raceway groove 𝑟_𝑜𝑢𝑡 is the radius of the outer raceway groove

the distance between raceways along the contact line of the bearing 𝑑_𝑗 is expressed by equation 36. (Kurvinen, et al., 2015, p. 16)

𝑑_𝑗 = 𝑟_𝑜𝑢𝑡+ 𝑟_𝑖𝑛−𝑅_𝑖𝑛+ 𝑟_𝑖𝑛+ 𝑒_𝑗^𝑟− 𝑅_𝑜𝑢𝑡+ 𝑟_𝑜𝑢𝑡

𝑐𝑜𝑠𝜙_𝑗 (36)

The total elastic deformation of a single bearing ball, 𝛿_𝑗^𝑡𝑜𝑡, is calculated with equation 37 (Kurvinen, et al., 2015, p. 16)

𝛿_𝑗^𝑡𝑜𝑡 = 2𝑟_𝐵− 𝑑_𝑗 (37)

Where 𝑟_𝐵 is the radius of the bearing ball.

Then the forces acting on a single bearing ball, 𝐹_𝑗, can be calculated using equation 38 (Kurvinen, et al., 2015, p. 16).

𝐹_𝑗 = 𝐾_𝑐^𝑡𝑜𝑡(𝛿_𝑗^𝑡𝑜𝑡)

3

2 (38)

The forces of each individual bearing ball can be combined to form bearing forces of the whole ball bearing using equations 39 – 43 (Kurvinen, et al., 2015, p. 16). Sum of forces and

moments should include only components where bearing balls are under compression (Sopanen & Mikkola, 2003, p. 12).

𝐹_𝑋 = − ∑ 𝐹_𝑗𝑐𝑜𝑠𝜙_𝑗𝑐𝑜𝑠𝛽_𝑗 𝑧 is the number of bearing balls

In the RoBeDyn toolbox for MATLAB the bearing displacements are calculated using Newton Raphson iteration with equation 44.

𝒆^(𝑛+1)= 𝒆^(𝑛)− (𝑲_𝑇^(𝑛))⁻¹𝑸^(𝑛) (44)

Where

𝒆^(𝑛+1) is matrix containing the displacements of the bearing on iteration step 1+n 𝒆^(𝑛) is matrix containing the displacements of the bearing on iteration step n 𝑲_𝑇^(𝑛) is the tangent stiffness matrix vector on iteration step n

𝑸^(𝑛) is matrix containing the forces acting on bearing on iteration step n

bearing forces 𝑸^(𝑛) include bearing and external forces as described in equation 45

𝑸^(𝑛) = 𝑸_𝑏^(𝑛)− 𝑸_𝑒𝑥𝑡^(𝑛) (45)

Where

𝑸_𝑏^(𝑛) is the matrix of bearing forces 𝑸_𝑒𝑥𝑡^(𝑛) is the matrix of external forces

External forces include bearing preload forces and thus amount of preload is factor when calculating bearing stiffness.

The stiffness matrix for each iteration step, 𝑲_𝑇^(𝑛), is presented in equation 46

𝑲^(𝑛)_𝑇 =𝜕𝑸^(𝑛)

𝜕𝒆^(𝑛) (46)

Newton Raphson iteration is continued until convergence criterion, shown in equation 47, is met

|𝑸| < 𝐾_{𝑐𝑜𝑛𝑣}∙ |𝑸_𝒆𝒙𝒕| (47)

Where

𝐾_{𝑐𝑜𝑛𝑣} is the convergence criterion factor

Bearing stiffness matrix 𝒌_𝑏 is the tangent stiffness matrix from last converged iteration step.

From E. Krämer’s book we can retrieve an estimation for damping factor in rolling element bearings. The damping matrix of the bearing is estimated as shown in equation 48. (Krämer, 1993)

𝒄_𝒃 = (0,25 … 2,5) ∙ 10⁻⁵𝒌_𝒍𝒊𝒏 (48)

where

𝒄_𝒃 is the damping matrix of the bearing

𝒌_𝑙𝑖𝑛 is the linearized stiffness of the bearings in N/mm

A constant is chosen within the limits shown in equation 48. For calculating damping matrices, the equation 49 is used.

𝒄_𝒃 = 2,5 ∙ 10⁻⁵𝒌_𝒃 (49)

Similar calculation method specifically for angular contact bearings has been introduced by Noel, et al. in their study (2013). In their study this method has proven good results with not considerable increase in calculation effort. (Noel, et al., 2013) Since RoBeDyn -toolbox was available to use, the effort of implementing analytical method was considered to be out of scope of this study.

The procedure of bearing stiffness calculation using RoBeDyn MATLAB toolbox is presented in flowchart in Figure 11.

Figure 11. Flowchart bearing stiffness calculation using RoBeDyn-toolbox

In document Bearing system design in outer rotor high speed motor (sivua 28-40)