PSI - Issue 81

Igor Stoiko et al. / Procedia Structural Integrity 81 (2026) 447–454

451

dimensional Gaussian distribution. In the absence of a dominant error, the total error distribution is subordinated to Gauss's law regardless of the distribution of component errors. This means that: ()()()()() 2 2 2 2 2 . b b k k K      = + + + + (4) Coefficient K indicates how many times the actual distribution of error values differs from the distribution of this error under normal distribution with the same mean value. Conducting a precise analytical calculation of part locating error is practically impossible. Therefore, for (2), it is necessary to introduce a new concept – the locating error coefficient for offset centers K ZM , which replaces the mathematical expression K ZM = sinβ cosψ . In this case, (2) takes the following form: ( ) . i ZM a L b К L  = − (5) To determine the accuracy of obtaining the axis angle for this locating scheme, we conditionally equate the total difference vector magnitude of locating error εᵢ to the offset of one of the centers, while other offsets equal zero. The calculation scheme has the form presented in Fig. 4.

Fig. 4. Calculation scheme for determining the curved axis angle for locating using parallel-offset, angular and axial centers.

From (5), it is evident that the maximum locating error will occur at b=0 , that is, in the plane of offset center 2: max . i i ЗМ a К  = =  The magnitude of the axis inclination angle for part locating relative to the spindle rotation axis α equals: tg , a L = arctg ; a L   =    

(6)

(7)

sin a

a x

.

x =

(8)

sin , = 



The actual magnitude of locating angle α′ considering (2), (5): ( ) 1 a К a + +

(

)

ZM

sin

sin 1

sin . 

К = + 

 =

=



i

(9)

ZM

x

a

From formula (9), the following equation is obtained:

(

)

arcsin 1 

sin . ZM К  

 =

+

  The difference between angles α and α′ constitutes the angular locating error, meaning: ( ) .    =  − The maximum angular error Δγ for single-pass machining (roughing and finishing on one machine) considering part repositioning for machining both ends and (7) and (9): ( ) 2 2 arcsin 1 sin arctg . ZM a К L      =  = +  −         (10) For preliminary and finish machining (turning and grinding with two-part setups) under conditions of vector summation of errors for two repositionings on different technological equipment and tooling equals: ( ) 2 2 2 2 arcsin 1 sin arctg . ZM a К L      =   =   +  −         (11) 3.2 Achievable part accuracy for locating using two perpendicularly offset and axial centers The second locating scheme for CA involves locating using two perpendicularly offset and axial centers (Fig. 5). Locating is performed in three centers of the technological tooling for each machining end, two of which are coaxial and positioned in a plane perpendicular to the axis of rotation. In this case, six center holes must be prepared for part locating and machining is performed sequentially using center holes 1-2-6 and 4-3-5. Furthermore, only one center (5 or 6, respectively) is rigid, while two centers (3 and 4 or 1 and 2, respectively) are movable. Since the axis of part rotation during machining does not coincide with the installation axis and they are positioned at an angle α to each other, the locating error εᵢ for different surfaces will vary and changes from 0 at the tailstock center 4 to max i  at center 5.