The grinding system consists of several elastic elements in series, such as motors, bearings, feed axis, spindle, grinding wheel, etc. [KING86, p. 171]. The overall stiffness, kt, adds up reciprocally from the stiffness of each element (Eq. 6.14)
Depth of cut ae
Fig. 6.7 Decrease in cutting edge distance at higher workpiece speed, vw2, after [PEKL57, p. 114]
[KING86, p. 171]. The element with the lowest rigidity will dominate system rigidity.
kt total stiffness
ki stiffness of each element in the grinding system
Geometry of the contact zone between grinding wheel and workpiece depends on the tool and workpiece diameter. To compare kinematics and engagement conditions of different grinding processes, the equivalent grinding wheel diameter is calculated from Eqs. 6.15, 6.16 and 6.17 [KLOC05a, p. 190, STEF83, p. 21].
Hahn calculates the curvature between tool and workpiece for these different cases depending on the speed ratio, q [HAHN55]. Depending on the load conditions, either the tool reproduces its own circumference (low workpiece speeds) or the curvature difference approaches the inverse of the equivalent radius [HAHN55].
deq equivalent grinding wheel diameter dw workpiece diameter ds grinding wheel diameter
The contact zone deformation of two touching cylinders increases with the normal forces [SCHR71, p. 40]. Higher contact stiffness leads to smaller elastic deformation (Fig. 6.8).
Gerhardt relates the grinding wheel contact stiffness to the individual grit spring constant and the preload force (Eq. 6.18) [GERH70]. The wheel stiffness can be measured by applying both a static and a dynamic force and measuring the local deformation [GERH70]. King and Hahn give Eq. 6.19 as example for a specific grinding wheel stiffness of a vitrified grinding wheel (2A60K6VLE, deq = 4.93 in. = 125.2 mm) [KING86].
|
k* = A • k1-b • (F;)b |
(6-18) |
|
:* = k0 • deq0’25 • (f;)0-75 |
(6-19) |
k* grinding wheel stiffness per unit width of contact zone in [lb/in.2]
A constant in [1/in.]
k spring constant of individual grit in [lb/in.] b constant depending on cutting edge distribution Fn normal force per unit width in [lb/in.]
k0 constant (0.58 x 105 lb/in. 2)
deq equivalent grinding wheel diameter (Eqs. 6.15, 6.16 and 6.17) in [in.]