Jackson Mark. Machining with Abrasives

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Fig. 3.12 Maximum stress (circumferential) in the electroplated cBN grinding wheel rotating at

200 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 145

3.8 Recessed Grinding Wheels

3.8.1 Small Cup Recessed Grinding Wheels

Grinding wheels that are subjected to high rotational stresses can eventually “burst”

by breaking into smaller fragments that are dominated by bending moments at the

onset of fracture. This causes damage to the grinding machine and to the operator if

guards inadequately prevent the fragments from penetrating the case of the machine

tool. It is of prime importance to be able to calculate the bursting speed of the

grinding wheel accurately in order to prevent this from occurring. The energy

contained within a fragment of grinding wheel spinning at a certain speed was

calculated by Jackson et al. [10]. Methods used to calculate bursting speed that are

based on the bending strength of the vitriﬁed material are not accurate and deviate

from the true bursting speeds of rotating grinding wheels. Deviations of up to 20%

have been observed by Munnich [11]. Table 3.1 shows the deviations for a variety

of grinding wheels with constant bonding formulation, but different abrasive mesh

sizes. The grinding wheel was a parallel-sided vitriﬁed wheel with an outer dia-

meter, D, of 610 mm, inner diameter, H, of 304.8 mm, and a thickness, T, of 20 mm.

Thies [12] and Pompe et al. [13] considered the use of fracture mechanics

coupled with the stochastic nature of vitriﬁed compositions to improve the calcula-

tion of bursting speed, but unfortunately did not provide any improvement in

accuracy. Mewes et al. [14] developed correction functions using linear regression

Stress Factor Vs Node Numbers (V=200 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. 3.13 Stress factor at various nodal positions on the cross-section of the electroplated cBN

grinding wheel rotating at 200 m/s. Used with permission from Inderscience Publishers (2010)

146 M.J. Jackson

techniques, but they are not applicable because of non-uniform deviation of the

calculated bursting speed compared to observed bursting speeds. In a recent paper

by Behrens and Kammler [15], a procedure is described where bursting speed is

calculated using linear elastic fracture mechanics coupled with the determination of

rotational stresses using ﬁnite element calculations. In their approach, the critical

pore size that trigge rs failure is calculated for a 36-mesh and an 80-mesh vitriﬁed

grinding wheel. The stress intensity at the ﬂaw tip is calculated using the mode I

stress intensity factor, and is used to calculate the fracture toughness of the abrasive

body. The experimental fracture toughness is measured using a single edge v-notch

beam loaded in the four-point bending mode to failure. Th e results are then used in

concert with a set of modifying functions that are incorporated into Munnich’s

equation for calculating bursting speed. The modiﬁed equation is:

bursting

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

fracture

3 þ n þð1  nÞ:

:r:f

(3.17)

120 140 160 180 200 220 240

Grindin

wheel surface s

eed

(

m/s

)

Safety factor

Hyperbolic solid disc:

Steel

Hyperbolic solid disc :

Carbon Fibre

Trapezoidal solid disc :

Steel

Trapezoidal solid disc :

Carbon Fibre

Porous disc : Steel

Porous disc : Carbon

Fibre

Fig. 3.14 Effect of material

and geometric shape of the

reinforcing center-section on

the safety factor of vitriﬁed,

segmented grinding wheels.

Used with permission from

Inderscience Publishers

(2010)

3 Grinding Wheel Safety and Design 147

where v

bursting

is the bursting speed, s

fracture

is the fracture strength of the grinding

wheel, n is Poisson’s ratio, H is diameter of the bore of the grinding wheel, D is the

outer diameter of the grinding wheel, r is the density of the vitriﬁed material, and f

is the stress intensity function at the maximum tangential stress at the outer edge of

the bore as a function of the normalized diameter and the depth of the recess. The

function f

is calculated using the following equation:

¼3: 00102x

þ 4:49064x

 2:58489x þ0:640967y

 2:05676y

þ 4:98248x

 8:56909x

þ 6:31512xy þ 1:59627

(3.18)

where x ¼P/D, y ¼F/T, P is the diameter of the recess, T is the thickness of the

grinding wheel, F is the depth of the recess, x is the normalized recess diameter, and

y is the normalized recess depth. Equations (3.17) and (3.19 ) are valid for the range

of F/T between 0.13 and 0.88, and for the range of P/D between 0.36 and 0.96.

Fig. 3.15 Slotted grinding wheels showing mechanism of lubricant penetration (courtesy of

Dr. M. W. Bailey, De Beers Diamond Company)

148 M.J. Jackson

Equations (3.17) and (3.18) are used in the present work for calculating bursting

speeds for a variety of different recessed grinding wheels and are compared with

experimental bursting speed data.

3.8.1.1 Computational Analysis

Dynamic loads on a rotating grindin g wheel is deﬁned in the rotating co-ordinate

system and while the stiffness and damping terms are the same as those measured in

the stationary system, the terms due to inertial resistance are dependent upon the

rotation of the grinding wheel. Owing to the porous nature of the grinding wheel

body, the dynamic imbalance is magniﬁed in certain parts of the grinding wheel

when rotating at certain speeds. The inertial depend ent terms need to be determined

for a particular wheel structure and are then added to the total impedance of the

structure before static and dynamic analyses in the rotating wheel can be performed.

The approach in this study assumes that the co-ordinate system used will rotate

at a constant rate about a ﬁxed axis. Displacements associated with the structure and

forces applied to the structure are measured in the rotating system. The system is

accelerating relative to a stationary inertial system and as such, the mass dependent

impedance cannot be directly calculated in the rotating system. Therefore, the

impedance is calculated for the stationary system then transformed to the rotating

Fig. 3.16 Finite element model of a slotted grinding wheel showing boundary conditions and

elements in various orientations. The images show the inclusion of coolant holes in the body of the

wheel that are used to direct coolant into the slots that separate the abrasive segments. Used with

permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 149

system. The development of general transformations is required between stationary

and rotating systems. This is accomplished by describing the general vector trans-

formations between stationary and rotating systems, deﬁning the inertial terms in

the rotating co-ordinate system, applying these results to develop impedance in the

rotating system, and ﬁnally, the gyroscopic terms are added to the structural

matrices for analyzing the rotating system.

Fig. 3.17 Finite element hoop stress plots showing an aluminum body in various conﬁgurations at

maximum peripheral operating speed of (a) 45, (b) 100, (c) 200, and (d) 300 m/s. Used with

permission from Inderscience Publishers (2010)

150 M.J. Jackson

The general transformation of a time-dependent vector from a stationary to a

rotating co-ordinate is given by the following:

vðtÞ



¼ AðtÞ½vðtÞ



(3.19)

Fig. 3.18 Finite element hoop stress plots showing a carbon ﬁber body in various conﬁgurations

at maximum peripheral operating speed of (a) 45, (b) 100, (c) 200, and (d) 300 m/s. Used with

permission from Inderscience Publishers (2010)

3 Grinding Wheel Safety and Design 151

where {v(t)

} is the rotating co-ordi nate system of the time-dependent vector, [A(t)]

is the time-dependent transformation matrix from the rotating to the stationary

system, and {v(t)

} is the stationary co-ordinate system of the time-dependent

vector. The transformation is valid for any vector, both real and complex. Trans-

formations using complex identities for small angular deviations allows the

transformation vector to be described in the x-y-z co-ordinates:

Fig. 3.19 Finite element hoop stress plots showing a steel body in various conﬁgurations at

maximum peripheral operating speed of (a) 45, (b) 100, (c) 200, and (d) 300 m/s. Used with

permission from Inderscience Publishers (2010)

152 M.J. Jackson

Fig. 3.20 Calculated hoop stress versus peripheral velocity for the various types of slotted

grinding wheels. Used with permission from Inderscience Publishers (2010)

Fig. 3.21 Stress factor versus peripheral velocity for various types of slotted grinding wheels.

Used with permission from Inderscience Publishers (2010)

Table 3.1 Calculated and

measured bursting speeds

of a parallel-sided vitriﬁed

grinding wheel

Abrasive

mesh size

Calculated bursting

speed (m/s)

(Munnich [ 11])

Measured

bursting

speed (m/s)

36 140.7 119.6

54 156.5 133

70 166.9 142.8

100 183.8 153.9

150 196.9 164.6

3 Grinding Wheel Safety and Design 153

xðtÞ

yðtÞ

zðtÞ

;

1 i 0

i 10

000

iy

1 i 0

i 10

000

001



xðtÞ

yðtÞ

zðtÞ

;

(3.20)

Or,

xðtÞ

yðtÞ

zðtÞ

;

½þ

iy

½



þ T

½



xðtÞ

yðtÞ

zðtÞ

;

(3.21)

And the conjugate matrix is:

½T

¼

1 i 0

i 10

000

(3.22)

½T





1 i 0

i 10

000

(3.23)

½T

¼

000

001

(3.24)

Equation (3.21) can be re-formulated as a transformation from rotating to non-

rotating co-ordinates where the progressive and regress ive vectors can be written in

terms of a non-rotating system, thus,

xðtÞ

yðtÞ

zðtÞ

;

½



iy

½þT

½



xðtÞ

yðtÞ

zðtÞ

;

(3.25)

Inertial forces are cal culated using Newton’s laws of motion, whilst particles

contained within the body are deﬁned as position vectors. The inertial force on a

particle whose position is measured relative to a rotating co-ordinate system is,

154 M.J. Jackson