Deflection due to Compliance of the Tapered Joint

The spindle ends of several machine tools have a tapered hole for accommodating a center (in lathe) or cutting tool shanks (in milling and drilling machine). The deflection of center or shank at a distance d from the spindle axis in which the force F is acting is expressed by equation:

Where  

d₃ = Δ+ φ d

Δ stands for displacement of the shank or center at the edge of taper because of contact compliance, and

φ stands for angle of slope of the shank or at the edge of taper.

 If manufacturing the errors of the taper are ignored, Δ and φ can be computer from subsequent equations:

Δ = 4ΨDC₁ / πD (Ψd C₂ + C₃), μm

Φ = 4FΨ²C₁ / πD (2Ψd C₄ + C₂)

Where 

C₁ stands for coefficients of contact compliance

C₂, C₃, C₄ stands for coefficients that account for the diameter variation along the length of taper.

Ψ = (1/2.3C₁D⁴)^1/4 , cm^-1

D and d are expressed in cm (centimeter).

Usually displacement Δ because of contact compliance can be ignored in comparison with the displacement because of bending of the shank or center.

Equation

Hence