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ABSTRACT
The predictility of dynamics of the machine tool spindles is essential for machining precision. During machining, the machine tool components and the cutting process interact with each other. Accordingly, it is necessary to take the process-machine interaction effects into account in order to predict the spindle's dynamics accurately. This paper presents an integrated model for the prediction of a spindle's dynamics. The model synthesizes the interactive influence between machine dynamics and forces in grinding process. The thermo-mechanical model of the spindle with angular contact ball bearings was built by using the finite-element method. The analytical model was used to calculate the process forces. A coupled simulation was adopted to accomplish the interactive process between the two models. Basing on the integrated model, the bearing stiffness, the natrual frequency, the spindle tip stiffness and deformations of a grinder's spindle were investigated. The prediction of the deformation fluctuations at the spindle tip due to process-machine interaction was also achieved.
Nomenclature
| Hf | = | heat generation of bearing, W |
| n | = | spindle speed, r/min |
| M | = | total friction torque of bearing, Nmm |
| Ml | = | load torque, Nmm |
| Mv | = | viscous torque, Nmm |
| Dm | = | mean diameter of the bearing, mm |
| pl | = | factor depending on the the magnitude and direction of the bearing load |
| fν | = | factor related to bearing type and lubrication method |
| hc | = | coefficient of convection heat transfer of spindle surface, W/(m2K) |
| λ | = | thermal conductivity, W/(m2K) |
| Re | = | Reynolds number of the air |
| Pr | = | Prandtl number of the air |
| uc | = | velocity of flow, mm/s |
| vair | = | kinematic viscosity of the air, mm2/s |
| l | = | cross-section perimeter of the spindle, mm |
| ϵr, ϵa | = | thermal expansion of inner ring in radial direction and axial direction, respectively, mm |
| αs | = | linear expansion coefficient of the ball and the spindle shaft, respectively, 1/K |
| ΔTs, ΔTh | = | temperature rise of the shaft and the housing/spacer, respectively, ℃ |
| Dio, Doi | = | inner diameter of outer ring and outer diameter of inner ring, respectively, mm |
| Li, Lo | = | distance of the contact points between the paired bearings of inner rings and outer rings, respectively, mm |
| k | = | index of bearing balls |
| Ki, Ko | = | contact coefficients between bearing balls and inner rings, outer rings, respectively |
| δxi, δyi, δzi, γyi, γzi | = | displacements of inner ring, mm |
| δxo, δyo, δzo, γyo, γzo | = | displacements of outer ring, mm |
| ϕk | = | 2πk/N |
| ri, ro | = | radii of the inner and outer ring grooves, respectively, mm |
| B | = | fo + fi − 1 |
| D | = | diameter of the bearing ball, mm |
| N | = | number of bearing balls for each bearing |
| fi | = | |
| fo | = | |
| ϵa, ϵr | = | relative axial and radial thermal expansion of inner ring with respect to the outer ring respectively, mm |
| uic | = | expansion of inner ring under centrifugal forces, mm |
| θik, θok | = | inner and outer ring contact angles of the bearing, respectively, deg. |
| Ms, Md | = | mass matrix of spindle shaft and disk, respectively |
| Mc | = | mass matrix used for computing the centrifugal force |
| Ks, Kb | = | spindle shaft and radial bearing stiffness matrices, respectively |
| F(t) | = | external load on the spindle, N |
| Ft | = | tangential grinding force, N |
| Fn | = | normal grinding force, N |
| Cp | = | specific grinding energy, N/mm2 |
| v | = | feed speed, m/s |
| Δ | = | grinding depth, mm |
| V | = | linear velocity of the wheel, m/s |
| b | = | grinding width, mm |
| 2α | = | tip angle of grain cutting edge, deg. |
| me | = | eccentric mass, kg |
| de | = | eccentric distance, m |
| ω | = | angular velocity of spindle, rad/s |
| Fc | = | centrifugal force acting on spindle section, N |