STRESSES ARISING IN A COMPOSITE GRINDING WHEEL

V. A. Govorukhin

(Presented by the scientific seminar of the Department of Machine Tools and Metal Cutting and the Department of Mechanical-Engineering Technology)

To determine the distribution of stresses in a composite grinding wheel reinforced with a steel ring, we shall use equations (7–10) [1]. As an example, let us consider the calculation of stresses in a composite wheel whose abrasive ring is made from a 39A40ST2K5 grinding wheel. The material of the abrasive ring is characterized by a Poisson’s ratio equal to 0.2, an elastic modulus of \(5 \cdot 10^3\ \text{kg}/\text{mm}^2\), and a unit weight of \(2.4 \cdot 10^{-6}\ \text{kg}/\text{mm}^3\) [2]; the material of the steel ring is characterized by a Poisson’s ratio of 0.3, an elastic modulus of \(21 \cdot 10^3\ \text{kg}/\text{mm}^2\), and a unit weight of \(7.8 \cdot 10^{-6}\ \text{kg}/\text{mm}^3\). The calculation was performed for a circumferential speed of the grinding wheel of \(100\ \text{m/s}\).

The obtained curves of radial and tangential stresses are shown in Fig. 1 for an ordinary grinding wheel with an internal diameter of \(127\ \text{mm}\) and an external diameter of \(300\ \text{mm}\), and for three composite grinding wheels reinforced with steel rings having a constant internal diameter of \(127\ \text{mm}\) and a variable external diameter ranging from \(186\) to \(256\ \text{mm}\).

The tangential stresses decrease rather substantially from the inner surface of the metal ring to the outer surface, and at the adhesive joint they change discontinuously, which may be explained by the fourfold difference between the elastic moduli of the abrasive material and steel. Toward the outer surface of the abrasive ring the tangential stresses decrease smoothly. However, if in a solid grinding wheel the maximum tangential stresses amount to \(2.2\ \text{kg}/\text{mm}^2\), then in a composite wheel, in the abrasive ring, they decrease as the external diameter of the metal ring increases. For example, when the external diameter of the metal ring is \(168\ \text{mm}\), the maximum tangential stresses are \(1.28\ \text{kg}/\text{mm}^2\), whereas when the external diameter of the metal ring is \(256\ \text{mm}\), they are only \(0.7\ \text{kg}/\text{mm}^2\).

The radial stresses, on the contrary, increase in their maximum value as the external diameter of the metal ring increases. Thus, when the external diameter of the metal ring is \(168\ \text{mm}\), the maximum value of the radial stresses is \(0.6\ \text{kg}/\text{mm}^2\), whereas when the external diameter of the metal ring is \(256\ \text{mm}\), it is \(0.9\ \text{kg}/\text{mm}^2\).

As follows from the presented stress curves for the adopted ratios of the diameters of the abrasive and metal rings, tangen-

tangential stresses in magnitude in all cases exceed the radial stresses. Consequently, in this case the abrasive ring, as the least strong in comparison with the steel ring, will fail under the action of tangential stresses, and, with the tensile strength of the material of the abrasive ring not exceeding, as a rule, even for the strongest wheels, \(2\ \mathrm{kg/mm^2}\), it is expedient, in order to increase the peripheral working speed of the composite grinding wheel, to make the abrasive ring with the minimum wall thickness.

Fig. 1

At the same time, such a change in the dimensions of the composite grinding wheel is also desirable from the standpoint of the operating conditions of the adhesive joint, which fails under the action of radial stresses. For example, when the outside diameter of the metal ring is \(168\ \mathrm{mm}\), the radial stresses arising in the adhesive joint are \(0.6\ \mathrm{kg/mm^2}\), whereas already at a diameter of \(256\ \mathrm{mm}\) they are only \(0.3\ \mathrm{kg/mm^2}\).

A somewhat different picture of the variation of radial and tangential stresses is obtained in the case where the dimensions of the abrasive ring are not changed, and the change in the dimensions of the composite grinding wheel is effected by changing the thickness of the metal ring (Fig. 2). In this case, as the thickness of the metal ring increases, the tangential stresses in the abrasive ring decrease, but

the radial stresses increase both in the abrasive ring and in the adhesive joint, which is an undesirable phenomenon, since it is very difficult to select a composition for the adhesive such that its strength with the abrasive exceeds the tensile strength of the material of the abrasive ring.

From the standpoint of the operation of the adhesive joint, as the least strong of all the elements of a composite grinding wheel, the latter scheme for creating a high-speed grinding-wheel design is less acceptable than the first.

Fig. 2

At present, wheels of the PP shape, of size \(300 \times 40 \times 127\), on machines that do not have stepless variation of the rotational speed, are used at most by 64%, since the diameters of the flanges used for fastening the grinding wheel on the machine must be no less than 0.5 of the wheel diameter, and therefore the maximum diameter of the usable part of the wheel is \(206\ \mathrm{mm}\).

The indicated grinding-wheel size was chosen because the 312M model machine available to the investigators permits work precisely with wheels of these dimensions. However, the foregoing remains valid, since the standard sizes of wheels produced, owing to the presence of fastening flanges, are never used by more than 90%.

Thus, from the standpoint of the strength of the adhesive joint and the maximum utilization of the grinding wheels produced, the inner diameter of the abrasive ring of a composite grinding wheel

the wheel must be taken equal to the diameter of the flanges used to fasten the wheel.

At the same time, in the general case the composite grinding wheel must operate in such a way that the strength of the adhesive joint and the strength of the material of the abrasive ring are used to the maximum.

A composite grinding wheel may fail under the action of tangential stresses on the inner surface of the abrasive ring when they reach the strength of the material of the abrasive ring, or under the action of radial stresses when they reach the strength of the adhesive joint.

\[ \sigma_{t\,2B}=[\sigma_a], \qquad \sigma=[\sigma_k], \]

where

\[ [\sigma_a]\text{ is the tensile strength of the abrasive material;} \]
\[ [\sigma_k]\text{ is the tensile strength of the material of the adhesive composition.} \]

Consequently,

\[ \frac{\sigma_k}{\dfrac{M_2-M_1}{N_2+N_1}} = \frac{[\sigma_a]} { -\dfrac{M_2-M_1}{N_2+N_1} \left(\frac{1+\alpha_2^2}{1-\alpha_2^2}\right) + C_2(2+\alpha_2^2)-D_2\alpha_2^2 } \]

or

\[ C=\frac{[\sigma_a]}{[\sigma_k]} = -\frac{1+\alpha_2^2}{1-\alpha_2^2} + \frac{C_2(2+\alpha_2^2)-D_2\alpha_2^2} {\dfrac{M_2-M_1}{N_2+N_1}}. \tag{1} \]

According to equation (1), the optimum ratio of the dimensions of the abrasive and metal rings will depend on the ratio of the allowable tensile strengths of the materials of these rings.

By specifying the ratio of the strengths of the materials of the abrasive and metal rings and the ratio of the dimensions of one of the rings, it is possible to determine the desired ratio of the dimensions of the other ring at which, theoretically, failure of the composite wheel will occur both in the material of the abrasive ring and in the adhesive joint.

Figure 3 shows, for three cases of strength ratios of the ring materials, the change in the ratio of the dimensions of the abrasive ring as the ratio of the dimensions of the metal ring changes. It follows from the figure presented that, for a constant metal ring, as the strength of the adhesive joint increases, the outside diameter of the composite grinding wheel must decrease.

Fig. 3

For a constant ratio of the strengths of the adhesive joint and the material of the abrasive ring, a decrease in the dimensional ratios of the metal ring leads to an increase in the dimensional ratios of the abrasive ring, i.e., the abrasive ring becomes increasingly thin-walled and therefore of little suitability from the standpoint of its practical use.

Thus, proceeding from the outer and inner diameters of the composite grinding wheel, dictated by the design features of each grinding machine and by the strength of the material of the abrasive ring and the adhesive composition, it is very easy, according to the graphs in Fig. 3, to select the necessary ratio of the dimensions of the abrasive and reinforcing rings for the manufacture of a composite grinding wheel.

REFERENCES

1. V. A. Govorukhin. Strength calculation of composite grinding wheels. In the present collection.

2. G. M. Ippolitov. Abrasive-Diamond Machining. Moscow, Mashinostroenie Publishing House, 1969.