Jackson Mark. Machining with Abrasives

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FðtÞ

inertial



¼ M½

rðtÞ





 O B

½

d rðtÞ





 O

½rðtÞ





(3.26)

where,

M½¼

m 00

0 m 0

00m

(3.27)

½¼

02m 0

2m 00

000

(3.28)

½¼

m 00

0 m 0

000

(3.29)

The structural impedance of the rotating structure requires inertial force terms that

are functions of the time varying position of mass of the particle. Therefore, the

position vector can be written in terms of an initial component and a time-varying

component. The equation of motion for a particle in a rotating elastic disk with no

externally applied forces is given by,

½r

¼ M½

uðtÞ





þð B½O B

½Þ

duðtÞ





þð K½O

½ÞuðtÞ





(3.30)

Equation (3.30) shows that the centripetal loading is always present in the rotating

structure. For accurate analysis of the rotating structure, a differential stiffness term

is added,

M½

DrðtÞ





þ B½O B

½ðÞ

d DrðtÞ





þ K½þO



O

½DrðtÞ



¼0

(3.31)

Equation (3.31) is used to determine the motion of a rotating structure about the

deformation due to centripetal loading. For the case of a rotating recessed grinding

wheel, ﬁnite element computations are modeled as rigid bodies with added gyro-

scopic effects at critical speeds of rotation. At selected speeds, dynamic out-of-

balance forces are predicted using rotation-dependent terms. The ﬁnite element

program used in the present work is MSC. Nastran™ and the procedure to add

rotation dependent terms is done by calculating the terms then adding them to the

3 Grinding Wheel Safety and Design 155

structural damping and stiffness matrices, and by starting the analysis with the

modiﬁed matrices.

3.8.1.2 Experimental Methods

The experimental part of the current work is focused on demonstrating the mag ni-

tude of stresses in recessed vitriﬁed grinding wheels, calculating bursting speeds of

the said wheels, and comparing those speeds with experimentally determined

bursting speeds. The experimental section also focuses on measuring selected

mechanical properties in order the calculate factor s of safety for a variety of

small cup grinding wheels.

Computational Stress Analysis

A solid model of a parallel-sided grinding wheel was created using the Solid

Works™ software package. The dimensions of the wheels and associated solid

models are shown in Fig. 3.22. The grinding wheel possessed an oute r diameter of

508 mm and an inner diameter of 304.8 mm. The grinding wheel was composed of a

mixture of chromium-doped alumina abrasive grain mixed with a glass-clay bond-

ing agent. The grade of the wheel was RA36J8V, indicating a 36-mesh grain size,

fairly open eight structure, J-grade vitriﬁ ed structure. The model was imported into

the MSC. Patran™ software program in the form of a .xmt ﬁle, or Parisolid Model.

The model was then meshed and the global edge length to optimize the mesh

density of the solid was determined by means of convergence. The convergence

model is shown in Fig. 3.23. Material properties (Table 3.2) were assigned to the

solid model of the wheel and constraints and bo undary conditions were applied to

the model. The center of the wheel was constrained in the translational x-, y-, and z-

directions as well as in the rotational x- and z-directions. The y-rotational direction

was left unconstrained because it is the axis in which the wheel rotates. An inertial

velocity was then applied to the entire solid in order to simulate the rotation of the

grinding wheel about the y-axis. Visual results, or fringe plots, showing where the

maximum principal stress occurred and the quantity of that stress, was the ﬁnal

output of the analysis. The result s are shown in Sect. 3.8.1.3 of the chapter.

Figure 3.24 shows the dimensions and the solid model of a recessed, small cup

vitriﬁed grinding wheel. The grinding wheel possessed an outer diameter of

250 mm and an inner diameter of 76 mm. The grinding wheel was composed of a

mixture of chromium-doped alumina abrasive grain mixed with a glass-clay bond-

ing agent. The grade of the wheel was RA36J8V, indicating a 36-mesh grain size,

fairly open eight structure, J-grade vitriﬁed structure. The recessed model was also

imported into the MSC. Patran ™ software program in the form of a .xmt ﬁle, or

Parisolid Model. Again, material properties (Table 3.1) were assigned to the solid

model of the wheel and constraints and boundary conditions were applied to the

156 M.J. Jackson

model. The center of the wheel was constrained in the translational x-, y-, and

z-directions as well as in the rotational x- and z-directions.

The y-rotational direction was left unconstrained because it is the axis in which

the wheel rotates. An inertial velocity was then applied to the entire solid in order to

simulate the rotation of the grinding wheel about the y-axis. Visual results, or fringe

Fig. 3.22 (a)SolidWorks™ model of the parallel-sided grinding wheel, (b) and dimensions associated

with the parallel-sided grinding wheel. Used with permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 157

plots, showing where the maximum principal stress occurred and the quantity of

that stress, was the ﬁnal output of the analysis.

Determination of Bursting Speed

In order to measure the bursting speed of a grinding wheel, it is necessary to rotate

the wheel at its maximum speed when failure occurs. In order to achieve this in a

safe way, the grinding wheel is located in rotating frame that it buried in the ground

surrounded by reinforced concrete bafﬂes that absorbs the energy of impacting parts

of the broken grinding wheel. The containment structure was lined with an energy

absorbing foam, of which the top could be opened for access. The enclosure

contained a roll grinding whe el head with the workpiece drive and in-feed mecha-

nism removed. The wheel head was modiﬁed and ﬁtted with a spindle cartridge

designed for speeds up to 3,000 rpm. Once the grinding wheels were mounted and

bolted on the spindle housing, the speed of the wheel was increased until the wheel

fractured about the spindle, usually into three to four parts. The grinding wheel was

inspected for failure and the failure speed was recorded. The grinding wheel

was photographed and points of failure recorded for posterity.

Convergence of Mesh for a Plain Sided Fixed

Rotating Grinding Wheel

5.5

5.7

5.9

6.1

6.3

6.5

6.7

6.9

7.1

7.3

0 20000 40000 60000 80000 100000 120000

Number of Nodes

Max Principal Stress (MN/m^2)

Optimum Mesh (Global

Edge Length of 0.012)

0.025

0.02

0.015 0.011

Fig. 3.23 Convergence of the solid model mesh showing an optimized global edge length of

0.012 mm. Used with permission from Inderscience Publishers (2010)

Table 3.2 Properties

of the vitriﬁed grinding

wheel structure

Property Value

Young’s modulus of elasticity (N/m) 57.2 10

Poisson’s ratio 0.2

Density (kg/m

) 2,257

158 M.J. Jackson

Determination of Mechani cal Properties of Grinding Wheels

Two types of grinding wheels were used to determine the mechanical properties of

the abrasive structures. The experimental determination of density and Poisson’s

Fig. 3.24 (a) Solid Works™ model of the recessed, small cup grinding wheel, (b) and dimensions

associated with the recessed, small cup grinding wheel. Used with permission from Inderscience

Publishers (2010)

3 Grinding Wheel Safety and Design 159

ratio was demonstrated in a previous paper by the author [10], and involves using a

pycnometer and a universal tensile testing machine. The grinding wheel structures

were manufactured in the form of tensile test bars and parallel-sided and recessed

grinding wheels. The grains of abrasive were mixed with a clay-glass bonding

formulation, molded, and pressed to the correct density to provide the correct

amount of porosity and structure. The samples were ﬁred in an electric furnace to

a temperature of approximately 1,080



C. The samples were cooled slowly until

their appropriate hardness is reached. Hardness was measured by measuring the

torque resistance of an indenting chisel that was rotated at a prescribed force. Once

cooled, samples of grinding wheel were polished and mounted in resin moulds.

Figure 3.25 shows the polished microstructure of a vitriﬁed grinding wheel of

known speciﬁcation, RA36J8V, whilst Fig. 3.26 shows a microstructure with a

known speciﬁcation of RA80J8V. From the micrographs the critical pore size is

found by measuring the distance between abrasive grains. The critical pore size

allows one to calculate the fracture toughness of the abrasive structures. The

measured fracture toughness is determined by stressing an abrasive structure to

failure using notched test bars in the four-point bending mode I using a universal

testing machine.

3.8.1.3 Experimental Results

Computational Stress Analysis

The computational stress analysis was performed on both parallel-sided and small

cup grinding wheel structures and the results are shown in this section of the paper.

Figure 3.27 shows the computed circumferential stress as a function of distance

from the bore to the edge of the wheel spinning at 35 m/s.

Parallel-Sided Grinding Wheels

The maximum stress at the bore is approximately 3.75 MN/m

. Figure 3.28 shows

the ﬁnite element mesh and boundar y conditions for the parallel-sided grinding

wheel spinning at 45 m/s. The associated fringe plot showing the magnitude of

circumferential stress is shown in Fig. 3.29. The indicated maximum stress is

approximately 7.14 MN/m

Figures 3.30–3.33 show the magnitudes of stress throughout a parallel-sided

grinding wheel for a wheel rotating at 45, 60, 80, 100, and 120 m/s. Th e maximum

circumferential stress occurs at the bore and the stresses are approximately, 6, 13,

19, 30, and 43 MN/m

, respectively.

160 M.J. Jackson

Fig. 3.26 Polished microstructure of the RA36J8V grinding wheel. Light areas indicate the

abrasive grain, grey areas show the vitriﬁed bonding system, and the darker areas show porosity.

Bubbles within the pores are features associated with the resin that has inﬁltrated the open pores.

Closed pores are shown with the abrasive grains and the bonding bridges. Average distance

between grains is approximately 0.7 mm. Used with permission from Inderscience Publishers

(2010)

Fig. 3.25 Polished microstructure of the RA80J8V grinding wheel. Light areas indicate the

abrasive grain, grey areas show the vitriﬁed bonding system, and the darker areas show porosity.

Bubbles within the pores are features associated with the resin that has inﬁltrated the open pores.

Closed pores are shown with the abrasive grains and the bonding bridges. Average distance

between grains is approximately 1.8 mm. Used with permission from Inderscience Publishers

(2010)

3 Grinding Wheel Safety and Design 161

Small Cup Recessed Grinding Wheels

In contrast to parallel-sided grinding wheels, MSC Patran was used to calculate

the maximum principal stress for a variety of small cup recessed grinding whe els.

Figure 3.34 shows a fringe plot for a wheel spinning at 45 m/s (P ¼100 and F ¼10).

The maximum stress occurs at the bore and is circum ferential in nature.

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

Distance from Center of Bore (m)

Stress (N/m^2)

Fig. 3.27 Calculated circumferential stress distribution as a function of distance from the center

of the bore of the grinding wheel spinning at 35 m/s. Used with permission from Inderscience

Publishers (2010)

Fig. 3.28 Fringe plot of a plain sided grinding wheel spinning at 45 m/s. Used with permission

from Inderscience Publishers (2010)

162 M.J. Jackson

1000000

2000000

3000000

4000000

5000000

6000000

7000000

0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27

Distance from Center of Bore (m)

Stress (N/m^2)

Fig. 3.29 Calculated

circumferential stress

distribution as a function of

distance from the center of the

bore of the grinding wheel

spinning at 45 m/s. Used with

permission from Inderscience

Publishers (2010)

Fig. 3.30 Calculated

circumferential stress

distribution as a function of

distance from the center of the

bore of the grinding wheel

spinning at 60 m/s. Used with

permission from Inderscience

Publishers (2010)

Fig. 3.31 Calculated

circumferential stress

distribution as a function of

distance from the center of the

bore of the grinding wheel

spinning at 80 m/s. Used with

permission from Inderscience

Publishers (2010)

3 Grinding Wheel Safety and Design 163

The magnitude of the stress is approximately 3.5 MN/m

. Figure 3.35 shows a

fringe plot for a recessed wheel spinning at 45 m/s. The maximum stress occurs at

the bore and is approximately 4.4 MN/m

. The geometry of the cup wheel is

P ¼100 and F ¼20.

Figure 3.36 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 3.8 MN/m

. The geometry

of the cup wheel is P ¼150 and F ¼10.

Figure 3.37 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 3.69 MN/m

. The geome-

try of the cup wheel is P ¼150 and F ¼20.

Figure 3.38 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 3.19 MN/m

. The geome-

try of the cup wheel is P ¼150 and F ¼30.

Figure 3.39 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 3.54 MN/m

. The geome-

try of the cup wheel is P ¼180 and F ¼20.

Figure 3.40 shows a fringe plot for a recessed wheel spinning at 45 m/s. The

maximum stress occurs at the bore and is approximately 3.39 MN/m

. The geome-

try of the cup wheel is P ¼200 and F ¼30. Once the rotational stresses were

calculated, the small cup grinding wheels were rotated to failure. Figure 3.41

Fig. 3.32 Calculated

circumferential stress

distribution as a function of

distance from the center of the

bore of the grinding wheel

spinning at 100 m/s. Used

with permission from

Inderscience Publishers

(2010)

Fig. 3.33 Calculated

circumferential stress

distribution as a function of

distance from the center of the

bore of the grinding wheel

spinning at 120 m/s. Used

with permission from

Inderscience Publishers

(2010)

164 M.J. Jackson