Jackson Mark. Machining with Abrasives

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shows a fractured small cup recessed grinding wheel. The speed at which failure

occurred was recorded and compared to the analytical calculation. Factors of safety

for each wheel were also recorded and the complete set of data is shown in

Table 3.3. Micrographs of fractured surfaces are shown in Figs. 3.42–3.44 at

various magniﬁcations. The images show that failure occurred by grain fracture

and by fracture at the bond bridge interface.

3.8.2 Large Cup Recessed Grinding Wheels

3.8.2.1 Computational Stress Analysis

The computational stress analysis was performed using a large cup grinding

wheel structure and the resu lts are shown in this section of the paper. MSC Patran

was used to calculate the maximum principal stress for a variety of large cup

recessed grinding wheels. Figure 3.45 shows a fringe plot for a wheel spinning at

45 m/s (P ¼200 and F ¼25). The maximum stress occurs at the bore and is

Fig. 3.34 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼100, F ¼10).

Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 165

Fig. 3.35 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼100, F ¼20).

Used with permission from Inderscience Publishers (2010)

Fig. 3.36 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼150, F ¼10).

Used with permission from Inderscience Publishers (2010)

166 M.J. Jackson

Fig. 3.37 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼150, F ¼20).

Used with permission from Inderscience Publishers (2010)

Fig. 3.38 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼150, F ¼30).

Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 167

Fig. 3.39 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼180, F ¼20).

Used with permission from Inderscience Publishers (2010)

Fig. 3.40 Fringe plot of small cup shaped grinding wheel spinning at 45 m/s (P ¼200, F ¼30).

Used with permission from Inderscience Publishers (2010)

168 M.J. Jackson

circumferential in nature. The magnitude of the stress is approximately 11.4p;MN/m

Figure 3.46 shows a fringe plot for a recessed wheel spinning at 45 m/s. The maximum

stress occurs at the bore and is approximately 19.2 MN/m

. The geometry of the cup

wheel is P ¼200 and F ¼50.

Figure 3.47 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 11 MN/m

. The geometry

of the cup wheel is P ¼250 and F ¼25. Used with permission from Inderscience

Publishers (2010).

Figure 3.48 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 15.6 MN/m

. The geome-

try of the cup wheel is P ¼250 and F ¼50.

Figure 3.49 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 15.1 MN/m

. The geome-

try of the cup wheel is P ¼250 and F ¼75.

Figure 3.50 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 13.8 MN/m

. The geome-

try of the cup wheel is P ¼275 and F ¼50. Used with permission from Inderscience

Publishers (2010).

Figure 3.51 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 13.9 MN/m

. The geome-

try of the cup wheel is P ¼325 and F ¼75. Once the rotational stresses were

calculated, the small cup grinding wheels were rotated to failure. The speed at

which failure occurred was recorded and compared to the analytical calculation.

Fig. 3.41 Exploded view of cup shaped vitriﬁed grinding wheel After Recovery From the Burst

Chamber. Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 169

Table 3.3 Mechanical property data concerning the operating conditions for various grinding

wheels with varying geometry and rotational speeds

Cup

geometry

(mm)

Maximum

circumferential

stress (MN/m

)

Factor of

safety x y f

Calculated

bursting speed

(m/s)

Experimental

bursting speed

(m/s)

Rotational speed ¼45 m/s

P ¼100,

F ¼10

3.51 16 0.4 0.25 1.16 480 456

P ¼100,

F ¼20

4.4 12.7 0.4 0.5 1.18 476 461

P ¼150,

F ¼10

3.08 18.2 0.6 0.25 1.3 452 460

P ¼150,

F ¼20

3.69 15.2 0.6 0.5 1.4 437 450

P ¼150,

F ¼30

3.19 17.5 0.6 0.75 1.39 439 452

P ¼180,

F ¼20

3.54 15.8 0.72 0.5 1.47 427 432

P ¼200,

F ¼30

3.39 16.5 0.8 0.75 1.46 428 430

Rotational speed ¼60 m/s

P ¼100,

F ¼10

6.25 9 0.4 0.25 1.16 480 456

P ¼100,

F ¼20

7.83 7.1 0.4 0.5 1.18 476 461

P ¼150,

F ¼10

5.47 10.24 0.6 0.25 1.3 452 460

P ¼150,

F ¼20

6.56 8.54 0.6 0.5 1.4 437 450

P ¼150,

F ¼30

5.66 9.9 0.6 0.75 1.39 439 452

P ¼180,

F ¼20

6.29 8.9 0.72 0.5 1.47 427 432

¼200,

F ¼30

6.03 9.3 0.8 0.75 1.46 428 430

Rotational speed ¼90 m/s

P ¼100,

F ¼10

14.1 4 0.4 0.25 1.16 480 456

P ¼100,

F ¼20

17.6 3.2 0.4 0.5 1.18 476 461

P ¼150,

F ¼10

12.3 4.5 0.6 0.25 1.3 452 460

P ¼150,

F ¼20

14.8 3.8 0.6 0.5 1.4 437 450

P ¼150,

F ¼30

12.7 4.4 0.6 0.75 1.39 439 452

P ¼180,

F ¼20

14.2 4 0.72 0.5 1.47 427 432

P ¼200,

F ¼30

13.6 4.1 0.8 0.75 1.46 428 430

Rotational speed ¼125 m/s

P ¼100,

F ¼10

27.1 2 0.4 0.25 1.16 480 456

(continued)

170 M.J. Jackson

Factors of safety for each wheel were also recorded and the complete set of data is

shown in Table 3.4.

For recessed grinding wheels, principal stresses were computed using a ﬁnite

element program were rotation depend ent terms were modiﬁed by adding structural

damping and stiffness matrices. It was discovered that the maximum principal

stress occurs at the bore of the small cup-shaped recessed grinding wheels and

Table 3.3 (continued)

Cup

geometry

(mm)

Maximum

circumferential

stress (MN/m

)

Factor of

safety x y f

Calculated

bursting speed

(m/s)

Experimental

bursting speed

(m/s)

P ¼100,

F ¼20

34 1.6 0.4 0.5 1.18 476 461

P ¼150,

F ¼10

23.7 2.4 0.6 0.25 1.3 452 460

P ¼150,

F ¼20

28.5 2 0.6 0.5 1.4 437 450

P ¼150,

F ¼30

24.6 2.3 0.6 0.75 1.39 439 452

P ¼180,

F ¼20

27.3 2 0.72 0.5 1.47 427 432

P ¼200,

F ¼30

26.2 2.15 0.8 0.75 1.46 428 430

Fig. 3.42 Micrograph of vitriﬁed grinding wheel magniﬁed at 5 magniﬁcation. Used with

permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 171

increases with rotational speed. Tables 3.3 and 3.4 show the magnitude of those

stresses for various geometry and peripheral operating speed. Convergence of the

ﬁnite element results occurred with a mesh that posse sses a global edge length of

0.012 mm with approximately 68,000 nodes for the recessed cup shape. The range

Fig. 3.44 Micrograph of cup shaped vitriﬁed grinding wheel magniﬁed at 10 magniﬁcation. The

micrograph shows a partially fractured abrasive grain. Used with permission from Inderscience

Publishers (2010)

Fig. 3.43 Micrograph of cup shaped vitriﬁed grinding wheel magniﬁed at 10 magniﬁcation.

Failure has occurred by failure at the bond bridge interface. The area that is focus shows the end of

a bond bridge that was previously connected to an abrasive grain. Used with permission from

Inderscience Publishers (2010)

172 M.J. Jackson

Fig. 3.46 Fringe plot of large cup shaped grinding wheel spinning at 45 m/s (P ¼200, F ¼50).

Used with permission from Inderscience Publishers (2010)

Fig. 3.45 Fringe plot of large cup shaped grinding wheel spinning at 45 m/s (P ¼200, F ¼25).

Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 173

of stresses increased from 3.51 MN/m

to a maximum of 34 MN/m

as a function of

rotational speed. The mea sured tensile strength of the abrasive grain material used

was approximately 56 MN/m

. This y ielded safety factors in the range between 1.6

and 18. The calculated bursting speeds are also shown in Table 3.3 and 3.4 and are

compared with experimentally determined bursting speed. The calcul ated bursting

speeds are within 5% of the measured bursting speeds and show remarkable

accuracy when using the method developed by Behrens and Kammle r [15]. Their

method takes account of stress concentrations that occur at the recesses of cup-

shaped grinding wheels. Polished micrographs of the structure o f cup wheels also

allowed the author to measure the critical pore size in order to measure the fracture

toughness of the cup wheels. The measured fracture toughness for the 80-grain size

cup wheel was 2.14 MPa m, and 1.8 MPa m for a 36-grain size cup wheel. The

standard deviation was 10 and 12%, respectively.

When considering conventional grinding wheels, a continuous layer of abrasive

adhered to the periphery of the hub material generates a maximum stress due to

centripetal loading that acts in the circumferential direction and is tensile in nature.

The circumferential stress is highest in all components of the grinding wheel, i.e.

abrasive layer, adhesive bonding layer, and the hub material. This magnitude of

stress is used to ascribe the maximum operating speed to a particular grinding

wheel. A clamping arrangement is used that prevents radial displacement at the

bore so that grinding wheels operate within their recommended safe operating

limits. The results obtained from ﬁnite element analyses of segmented grinding

wheels suggest that the maximum rotational stress in the grinding wheel is in the

radial direction and is tensile in nature. The change in direction is caused by the

Fig. 3.47 Fringe plot of large cup shaped grinding wheel spinning at 45 m/s (P ¼250, F ¼25).

Used with permission from Inderscience Publishers (2010)

174 M.J. Jackson