Jackson Mark. Machining with Abrasives

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loaded at very high spindle speeds. The grinding wheel industry uses a factor of

safety of three unless the wheel is completely guarded, then a factor of safety of two

is used.

3.4 Segmented Grinding Wheels

The magnitude of the circumferential stress due to centripetal loading is a direct

result of radial expansion that leads to “circumferential stretching” of the abra sive

layer. The magnitude of circumferential stretching can be reduced by replacing the

continuous layer of abrasive, which is attached to the outer surface of the center

section, with a set of discontinuous segments.

Although the linear theory of elasticity can be used to predict the level of stress

due to centripetal loading in a continuo us grinding wheel, a numerical technique

such as the ﬁnite element method is required to predict stress levels in a segmented

grinding wheel. In order to allow the penetration of lubricants to the grinding zone in

creep feed grinding, conven tional wheels are slotted and give rise to localized stress

concentrations at the root of the slot that is far away from the bore. An example of

this is shown in Fig. 3.1. Here, the slots are molded into the face of the grinding

wheel and are usually present in conventional aluminum oxide grinding wheels.

The design of these particular wheels are subjected to special treatment and are

not discussed here as they are limited to maximum operating speeds of 35 m/s at

their surface. However, abrasive segments bonded to metal discs are considered

here as they operate at much higher speeds. These are arranged as a series of

segments adhesively bonded to the solid disc that is usually metallic. In order to

examine the inﬂuence of the number and the size of the abrasive segment on the

Fig. 3.1 Peripheral slotting on the surface of a conventional grinding wheel (courtesy of S. A.

Jones, Unicorn Abrasives)

3 Grinding Wheel Safety and Design 135

levels of stress, a ﬁnite element model of a segmented grinding wheel was created.

The abrasive material is a vitreous-bonded cubic boron nitride that has a density of

2,270 kg/m

, a modulus of elasticity of 8.6 GN/m

, and a Po isson’s ratio of 0.2. The

material used for the reinforcing center section was En 24T steel.

The segments were bonded to the steel center section using an adhesive with a

density of 1,700 kg/m

, modulus of elasticity of 1.5 GN/m

, and a Poisson’s ratio of

0.4. The edges of the model shown were constrained from movement in the “Y” and

“Z” directions in order to simulate connectivity with the remaining parts of the

wheel. All other nodes were allowed to move freely, and the size and number of

segments were varied according to the nature of the grinding operation. Figure 3.2

shows the inﬂuence of the depth of the abrasive segments on the circumferential

and radial stresses in the abrasive part of the grinding wheel. The number of

segments was constant at this stage of the analysis.

The number of segments used in industrial practice for this particular application

was 60. Figure 3.3 shows the inﬂuence of the number of segments on the circum-

ferential and radial stresses in the abrasive part of the wheel. The depth of the

abrasive segment has a signiﬁcant effect on the levels of stress in a segmented

grinding wheel [5, 7]. For this reason, the depth of the abrasive segment was 4 mm.

The abrasive segment consists of a 3-mm thickness usable layer and a 1-mm

backing layer that has a coefﬁcient of thermal expansion that is matched to that

of the abrasive layer.

For the grinding wheel considered here, it is shown that the smaller the depth

of the abrasive segment and the higher the number of segments leads to a reduction

in the level of stress in the abrasive part of the grinding wheel. The greatest effect on

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

345678910

Depth of segment (mm)

Stress factor

Radial stress

Circumferential stress

Fig. 3.2 Effect of depth of

the segments on the stress

levels in the abrasive part of

the grinding wheel. Used with

permission from Inderscience

Publishers (2010)

136 M.J. Jackson

the magnitude of stresses in segmented grinding wheels is clearly the number of

segments. It can be shown that the maximum stress in the abrasive segments is in

the radial direction, whereas the maximum stress in the continuous layer of abrasive

is in the circumferential direction.

3.5 High Speed Segmented Grinding Wheels

3.5.1 Grinding Wheels Capable of Being Dressed

Very high grinding wheel speeds can be achieved (up to 500 m/s) if more exotic

designs are used that effectively concentrate mass at, or near to, the bore of the

wheel [5]. Figure 3.4 shows a wheel design that was developed to reduce the mass

of the reinforced center section. This design allows a wider segment to be used that

allows direct plunge-grinding operations to be undertaken at higher metal removal

rates. Hyperbolic- and trapezoidal-shaped grinding wheels use narrow abrasive

segments, which means that plunge grinding operations are accompanied by tra-

verse movements in order to rem ove the same stock of material compared to the

ultra high speed grinding wheel shown in Fig. 3.4. The ﬁnite element model

represents one half of the wheel that has symmetry about the “Y” axis. The corners

of the wheel are restrained from movement in the “Y” direction in order to simulate

connectivity. The properties of the grinding wheel are the same as those stated

earlier, in order to compare with the segmented grinding wheel.

Figure 3.5 shows the results of the ﬁnite element analysis carried out on this

particular geometry. The results are compared with the results of the segmented

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

8 1624324048566472

Number of segments

Stress factor

Radial stress

Circumferential stress

Fig. 3.3 Effect of the number of segments on the stress levels in the abrasive layer of the grinding

wheel. Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 137

grinding wheel. Figure 3.5 show s the effect of the number of segments on the

circumferential and radial stress in the abrasive part of the grinding wheel shown in

Fig. 3.4.

The depth of the abrasive segments was maintained at 4 mm. It is shown that the

ultra high-speed grindin g wheel with the “porous” reinforced steel center exhibits a

lower stress factor due to centripetal loading. For the same level of safety experi-

enced by segmented grinding wheels, higher wheel speeds can be achieved using

the novel design shown in Fig. 3.4. Once again, the maximum stress observed is

in the radial direction and is tensile in nature.

3.5.2 Electroplated Grinding Wheels

Electroplated wheels can be rotated at much higher speeds than vitriﬁed wheels

owing to the nature of the grinding layer that is on the wheel body itself. The

stochastic nature of strength in vitriﬁed segments means that speeds are limited due

to the brittleness of the abrasive layer. However, in practice speeds are lower so that

tolerances are achieved on engineering components. Higher speeds lead to distor-

tion of the body that impairs tolerances on components especially run-out toler-

ances on cam shafts and crankshafts. Figure 3.6 shows a typical cross-section of a

Fig. 3.4 Finite element model of the ultra high-speed grinding wheel showing abrasive segments

bonded to the “porous” steel reinforced center. Used with permission from Inderscience Publishers

(2010)

138 M.J. Jackson

single-layer electroplated grinding wheel that supplies coolant directly through the

wheel and into the contact zone. The coolant channel is drilled with a series of

through holes and delivers coolant between the spaces of the abrasive grain.

Figure 3.7 shows the node numbers of the ﬁnite element model of the grinding

wheel. These numbers are used to identify points of stress when the wheel is rotated

at various speeds using different materials.

Figure 3.8 shows the magnitude of maximum stress (circumferential, or hoop

stress) assuming that the wheel is rotating freely at the bore, i.e. the maximum stress

condition. The speed of rotation is 45 m/s. At this speed, there is no appreciable

difference in the stress factor as the material of the wheel changes (Fig. 3.9).

Figure 3.9 also shows that the maximum stress occurs at nodes 2 and 48, which

are the positions of the bore and the point at which the coolant is channeled to the

contact zone. At 100 m/s, the stresses have increased (Fig. 3.10) and the stress

factors at node 2 and 48 are still highest (Fig. 3.11).

However, it is shown that the use o f carbon ﬁber wheel material has slightly

decreased the stress factor experienced by the rotating wheel. A further increase in

speed to 200 m/s has raised the stresses signiﬁcantly (Fig. 3.12) and the maximum

stress still occurs at node positions 2 and 48. The use of carbon ﬁber has again

reduced the stress factor signiﬁcantly (Fig. 3.13) at the highest speed of 200 m/s.

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

8 1624324048566472

Number of segments

Stress factor

Radial stress (segmented wheel)

Circumferential stress (segmented wheel)

Radial stress (ultra high speed wheel)

Circumferential stress (ultra high speed wheel)

Fig. 3.5 Inﬂuence of number of segments on stress levels in the abrasive layer for segmente d

and ultra high speed grinding wheels. Used with permission from Inderscience Publishers

(2010)

3 Grinding Wheel Safety and Design 139

Fig. 3.6 Finite element mesh of the cross-section of the electroplated cBN grinding wheel. Used

with permission from Inderscience Publishers (2010)

Fig. 3.7 Node numbers associated with the ﬁnite element model shown in Fig. 3.5. Used with

permission from Inderscience Publishers (2010)

140 M.J. Jackson

Fig. 3.8 Maximum stress (circumferential) in the electroplated cBN grinding wheel rotating at

45 m/s: (a) aluminum alloy center; (b) carbon ﬁber center; and (c) steel center. Stress scale is in N/

. Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 141

3.6 Safety of Grinding Wheels

The results of the F. E. analysis provide the design engineer with the ability to

calculate the factor of safety at a particular operating speed . When using certain

yield criteria, safety factors can be calculated for each part of the grinding wheel.

It is usual to apply the maximum principal stress theory, sometimes referred to as

the Rankine theory, to the abrasive segments as they are classed as brittle materials.

The maximum shear strain energy per unit volume, or von Mises, theory is applied

to ductile materials that allow some form of plastic deformation before fracture

occurs. This theory can be applied to the reinforcing center section of the seg-

mented grinding wheel, and to the adhesive layer that bonds the abrasive segments

to the center section.

These two theories allow the designer to calculate the yielding strength of the

composite materials, and can be compared to the operating stress at a particular

operating speed to give a factor of safety. In order to provide a comparison between

materials used for the reinforcing center section, Fig. 3.14 shows the effect of using

different reinforcing materials on the safety factor exhibited by the standard,

segmented grinding wheels and porous, segmented grinding wheels at different

surface speeds.

Stress Factor Vs Node Numbers (V=45 m/s)

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

104

116

119

Node Numbers

Stress Factor

Steel Aluminium Carbon Fibre

Fig. 3.9 Stress factor at various nodal positions on the cross-section of the electroplated cBN

grinding wheel rotating at 45 m/s. Used with permission from Inderscience Publishers (2010)

142 M.J. Jackson

Fig. 3.10 Maximum stress (circumferential) in the electroplated cBN grinding wheel rotating at

100 m/s: ( a) aluminum alloy center; (b) carbon ﬁber center; and (c) steel center. Stress scale is in

N/mm

. Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 143

3.7 Slotted Grinding Wheels

Slotted grinding wheels are very useful in that they allow coolant to penetrate the

gap between abrasive segment and workpiece surface. There are various designs of

slotted grinding wheels, but the most popular type is shown in Fig. 3.15.

Figure 3.16 shows a three-dimensional computer aided design model with

boundary conditions applied at the bore of the wheel. It is assumed that the wheel

is freely rotating on its spindle with no clamping, i.e., wheel slippage is occurring

that generates the highest circumferential, or hoop, stresses at the bore.

Figures 3.17–3.19 show the magnitude of maximum hoop stress at the bore for

wheel peripheral speeds of 45, 100, 200 and 300 m/s. There appears to be a

concentration of stress at positions of the wheel where there are discontinuities in

geometry especially around the holes that supply coolant. The ﬁgures represent the

use of different materials for the reinforcing body, i.e., aluminum, carbon ﬁber and

steel. Figure 3.20 shows the magnitude of hoop stress as a function of speed for the

different body materials, whilst Fig. 3.21 shows the magnitude of stress factor also

as a function of speed for different wheel body materials. It is shown that carbon

ﬁber bodies generate the least amount of hoop stresses as the speed increases, but

has the highest stress factor at low grinding wheel speed s. This implies that carbon

ﬁber bodies must used at peripheral speeds in excess of 100 m/s (or signiﬁcantly

re-design the body to concentrate its mass at the bore), whilst steel and aluminum be

used at much lower peripheral speeds.

Stress Factor Vs Node Numbers (V=100 m/s)

0.1

0.2

0.3

0.4

0.5

0.6

104

116

119

Node Numbers

Stress Factor

Steel Aluminium Carbon Fibre

Fig. 3.11 Stress factor at various nodal positions on the cross-section of the electroplated cBN

grinding wheel rotating at 100 m/s. Used with permission from Inderscience Publishers (2010)

144 M.J. Jackson