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finite element modeling of ball bearing 1
- Thread starter josh08
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I'd use the manufacturer's data, . Angular contact spindle bearings' radial stiffness can easily vary 2X or 3X by varying axial offset/preload just 0.001 inch, and even varies with subtleties like race curvature which is proprietery information.
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If you are modelling an angular contact ball bearing, the best representation is complete stiffness matrix attached between two reference nodes of the shaft and the housing (cross sections).
The manufacturers always provide the axial stiffnes, "Ka" for your bearing and its preload. Other members in the matrix can be approximately calculated if you know the contact angle "alph" and the mean bearing diameter "dm". I use these equations:
radial stiffness:
Kr=Ka/2*cotg^2(aplh)
rotational stiff along radial axis:
Kfi=Ka/8*dm^2
offdiagonal member radial-rotational:
Kfir=Ka/4*cotg(aplh)*dm
The sign of the offdiagonal member depends on the orientation of the bearing in your coordinate system and it is a bit tricky to introduce it correctly. However the same effect (as the off diagonal member in the matrix) has placing the two reference nodes at the point at which the contact axis of the bearing intersects the spindle axis (instead of placing it at the true axial bearing location).
My recomedation is the geometrical approach if you do not want to bother with structure of stiffness matrices too much. You can introduce the diagonal members Ka, Kr and Kfi parallelly as appropriate single "springs" between the two reference nodes.
Be aware that contact angles and stiffness change with preload, bearing radial press fit, speed, temperatures, misalignment,.... Particularly if speed of your application is high (n*dm > 0.5e6), everything will be much different compared to the static case.
The manufacturers always provide the axial stiffnes, "Ka" for your bearing and its preload. Other members in the matrix can be approximately calculated if you know the contact angle "alph" and the mean bearing diameter "dm". I use these equations:
radial stiffness:
Kr=Ka/2*cotg^2(aplh)
rotational stiff along radial axis:
Kfi=Ka/8*dm^2
offdiagonal member radial-rotational:
Kfir=Ka/4*cotg(aplh)*dm
The sign of the offdiagonal member depends on the orientation of the bearing in your coordinate system and it is a bit tricky to introduce it correctly. However the same effect (as the off diagonal member in the matrix) has placing the two reference nodes at the point at which the contact axis of the bearing intersects the spindle axis (instead of placing it at the true axial bearing location).
My recomedation is the geometrical approach if you do not want to bother with structure of stiffness matrices too much. You can introduce the diagonal members Ka, Kr and Kfi parallelly as appropriate single "springs" between the two reference nodes.
Be aware that contact angles and stiffness change with preload, bearing radial press fit, speed, temperatures, misalignment,.... Particularly if speed of your application is high (n*dm > 0.5e6), everything will be much different compared to the static case.
"radial stiffness: Kr=Ka/2*cotg^2(aplh)"
FAG says the radial rigidity (units N/um, just like stiffness) of a pair of 15 degree bearings is ~6 * axial rigidity.
I'm probably just out of touch with the proper rules for writing of algebraic expressions, but I'd write that expression like this to get a radial rigidity of 6.96 -
Kr=(Ka/2)cotg^2(alph)
When I first used to solve for lateral eigen-stuff with CosmosM I'd apply each bearing stiffness using a spring element to a single node at the bearing's line of action on the shaft core and housing. But my solutions were cluttered up with spurious very low frequency rigid body rotational modes. My solution was to use pairs of half-as-stiff spring elements for each bearing afixed to the sides of the spindle shaft, since that drove torsional modes artificially sky high without having to restrain the shaft model from rotating and risk inadvertently adding improper radial stiffness at some other location.
FAG says the radial rigidity (units N/um, just like stiffness) of a pair of 15 degree bearings is ~6 * axial rigidity.
I'm probably just out of touch with the proper rules for writing of algebraic expressions, but I'd write that expression like this to get a radial rigidity of 6.96 -
Kr=(Ka/2)cotg^2(alph)
When I first used to solve for lateral eigen-stuff with CosmosM I'd apply each bearing stiffness using a spring element to a single node at the bearing's line of action on the shaft core and housing. But my solutions were cluttered up with spurious very low frequency rigid body rotational modes. My solution was to use pairs of half-as-stiff spring elements for each bearing afixed to the sides of the spindle shaft, since that drove torsional modes artificially sky high without having to restrain the shaft model from rotating and risk inadvertently adding improper radial stiffness at some other location.
Structure of the model is constrained by possibilities of FEM software and users invent very individual approaches. Bearing application in particular is one of freeride zones in FEM modeling.
The ratio given by FAG coresponds to approx. alph=16 in the equation - which should be OK considering that nonzero axial preload slightly increases the contact angle.
The ratio given by FAG coresponds to approx. alph=16 in the equation - which should be OK considering that nonzero axial preload slightly increases the contact angle.
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