Optimum Spacing between Spindle Supports

The ratio 'τ' is an significant parameter in spindle design. In which, τ = a/m. The optimum value of this ratio is the one that ensures that minimum total deflection 'd' can be ascertained from differentiating it partially with respect to τ.

For minimum deflection d, dd/ d τ = 0.

The point of the minimum of the d₁ + d₂ curve, provides the optimum value of ratio a/m that generally lie between 3 and 5. The value of τ_opt depends on ratio of stiffness of the front and rear bearings,

η= S_E / S_G and factor J = S_G I_m / S_C I_a

Where S_C = 3EI_m / m³ = bending stiffness of the cantilever.

I_m = average moment of inertia of the spindle over cantilever.

I_a = average moment of inertia of the spindle over the supported length.

An opposite constraint on maximum span stems from the need that for normal functioning of the spindle driving gear, stiffness of the span should not be less than

245-260 N/µm. This constraint is expressed by the following relationship:

In which D_a = average diameter of the supported length of the spindle,

i = 0.05 for normal accuracy machine tools, and

= 0.1 for precision machine tools.

While the spindle is supported on hydrostatic journal bearings, the maximum deflection at the middle of the span should satisfy the following condition:

d_{a max} ≤ 10 ^-4 a

And the maximum span length a max has to be limited by the above constraint. This constraint is based upon the need that the maximum misalignment because of deflection of the journals should not exceed one third of the bearing gap.