Total Rigidity of a Linear Drive
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Elasticity of a linear drive includes elastic deformations of thrust bearings, ball screw shaft and ball nut. This total, based on the "inverse" equation below, is what matters to the performance of the drive. But only the nut stiffness is normally given in the technical data of a ball screw.
Rt Total rigidity
Rnu,ar Actual nut rigidiy
Rs Stiffness of the screw shaft
Rb Axial stiffness of the thrust bearing
The rigidity of the thrust bearing can be obtained from the literature of the manufacturer. Rigidity of the screw shaft can be calculated from the elasticity modulus of steel (210,000 N/mm3), the shaft's cross section and the length of the loaded portion of the shaft. For screws with large lead / diameter ratio, torsion also plays an important role and must be considered as well.
If you want to do the calculation of axial and torsional stiffness yourself, just use the nominal diameter of the ball screw minus the ball diameter to calculate the cross section and moment of interia. Or, contact us and we will do this for you.
Installing thrust bearings at both ends of the screw yields four times the axial shaft stiffness compared to a single thrust bearing at one end and no support on the opposite end. A factor of 2 results because forces are transmitted through the shaft on both sides of the nut. A second factor of 2 also applies because the weakest point is now in the middle of the shaft rather than at the extreme end.
Thrust bearings at both ends normally require pre-tensioning the screw, against the support bearings, in order to avoid compressive loads on the shaft from thermal expansion (which may cause buckling). Make sure to check the impact this additional force has on the bearing life.
Linear drives with rotating nut and stationary screw allow a simple way of increasing torsional stiffness of the shaft by transmitting moments into the surrounding structure at both ends of the shaft. Then the same effect as described above applies to the torsional stiffness: A factor of 2 for twice the cross section, and again a factor of 2 for half the distance to the weakest point.