MOTORS, DRIVES & CONTROLS
SECTION TITLE
FIG 2. Influence factors on bearing and spindle stiffness
Note that for the main diagonal in
y-direction for example, the stiffness value of 232.54 N/µm is very different from the inverse of the compliance value 1/0.081688=12.2 N/µm. Te first value is the radial force variation over radial displacement for constant displacements ux, uz and rotations ry, rz while the second ratio is for constant forces Fx, Fz and constant moments My, Mz. Te three curves in Fig. 1 are generated using cyy=∂F (ux, uy, uz, ry, rz)/ ∂uy; cyy2=1/(∂uy(Fx, Fy, Fz, My, Mz)/ ∂Fy) ; cyy3=1/(∂uy(Fx, Fy, Fz)/ ∂Fy). So, the first curve is the main diagonal of a 5x5-stiffness matrix, the second curve is the reciprocal value of the main diagonal of the 5x5 compliance matrix and the third value is the reciprocal of the main diagonal of the 3x3 compliance matrix for a bearing with fixed tilting angle. All curves show a ratio of force difference over deflection difference. Just the conditions for other parameters are different, having either fixed displacements or fixed forces.
Why is the stiffness reduced under increased radial force? Tis is because
FIG 3. Four calculation models for housing stiffness Design for a wind turbine gearbox
of constant axial force, like with spring pretension and the load distribution changes. For pure axial load all rolling elements have the same load. With rising radial load some rolling elements are unloaded, and this leads to the initial decrease of radial stiffness. For the spindle in Fig. 2 a radial and axial stiffness for the left end should be calculated. A measurement would probably be done by applying a force and measuring the deflection. Tis would relate to the main diagonal element of a reduced compliance matrix as other deflections and tilting angles are not restricted. Te main diagonal element of the stiffness matrix would show much larger values. Te catalogue stiffness for bearings is only taking into account geometry and manufactured pretension. Te stiffness is then affected by mounting conditions such as interference fits, or operating conditions such as rotation speed and temperatures. If there is clearance between the outer ring and the housing, the outer ring can expand radially, which will decrease the stiffness. Fig. 2 shows the influence on bearing and spindle stiffness as an example. In case of the temperature difference the spindle stiffness
decreases because the axial expansion of the shaft leads to a reduced bearing pretension. Elastic deformations of the housing or other elastic surroundings will also have an influence on the stiffness of shaft-bearing systems. A calculation model for housing stiffness is just more difficult. Often FEA models of housings are statically reduced to consider their stiffness. Fig. 3 shows four modelling approaches for the reduction of housing stiffness. Te first is a reduction to a central node using rigid constraints. Tis will stiffen the housing. Te second approach is a reduction to a central node using averaging. Ten the housing is not stiffened, but the displacements will not match. Te third approach couples the housing deformations with deformations of the bearing ring. Note that there is a vertical movement in the example because of the deformed shape. Te fourth adds a contact model between housing and bearing ring. Increased model complexity increases accuracy but also calculation time.
Markus Raabe is the managing director of Mesys.
www.mesys.ag
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