On the Selection of Grinding Conditions for a Specified Warpage Tolerance

D. E. Zaitsev, G. D. Del, M. G. Goldshmidt

Grinding is one of the finishing operations of the technological process. Therefore, in a large number of cases there arises the need to specify such machining conditions as would eliminate significant warpage of the product.

The magnitude of warpage depends mainly on the dimensions of the part, the material, and the magnitude of the residual stresses. By means of a simple experiment it proves possible to obtain sufficiently detailed information for selecting grinding conditions that ensure the manufacture of parts whose warpage lies within the specified tolerance.

From the material under study, 8–10 plates are made, with thickness from 2 to 8 mm, width \(b = 10\) mm, and length \(l = 200\) mm.

The plates are ground on all faces and then annealed in vacuum in the clamped state in order to eliminate warpage. The deviation of the annealed plates from flatness is checked by means of a dial-indicator fixture. Plates with a deflection \(\Delta_0 \le 0.03\) mm are admitted to the test. Subsequently, the specimens are placed with the convex side on the machine table and ground under the conditions being investigated (feed, speed, cooling, etc., are varied). The deflections \(\Delta'\) of the plates after grinding are measured by means of the same dial-indicator fixture. From the difference of the deflections \(\Delta = \Delta' - \Delta_0\), the radius of curvature is calculated:

\[ \rho = \frac{l^2}{8\Delta} \tag{1} \]

Bending is a consequence of the action of residual stresses, whose moment is equal to [1]

\[ M = \frac{EI}{\rho} \tag{2} \]

where \(E\) is the modulus of elasticity of the material,

\[ I = \frac{bh^3}{12} \]

is the moment of inertia of the section.

With known accuracy it may be assumed that, beginning with some thickness \(h_0\), the residual-stress epure does not depend on

thickness [2]. In this case the moment \(M\) is a linear function of the plate thickness (Fig. 1a)

\[ M=N\left(\frac{h}{2}-\delta\right), \tag{3} \]

where \(N\) is the resultant of the residual stresses,

\(\delta\) is the distance from the machined surface to the line of action of \(N\).

In Fig. 1a: neutral axis.

Fig. 1

In Fig. 1b this dependence of the moment on the thickness is shown schematically. For small thicknesses \(h<h_0\), the relationship between \(M\) and \(h\) is not linear (which is associated with penetration of the plastic zone beyond the neutral layer, better heat-removal conditions, etc.). Subsequently this relationship becomes linear. By extrapolating the straight line to the \(h/2\) axis, we obtain the displacement \(\delta\) of the resultant residual stresses from the surface. The tangent of the angle of inclination of the straight line, according to (3), is equal to the resultant \(N\).

Thus, knowing the warpage of a series of plates with different section heights, it is possible to determine the magnitude and position of the resultant residual stresses.

If a linear law of variation of the residual stresses [2, 3] is assumed, then the magnitude of the maximum stress at the surface can also be determined:

\[ \sigma_m=\frac{2}{3}\cdot\frac{N}{b\cdot\delta}. \tag{4} \]

The sign of the stress is determined as follows. If the plate is convex toward the ground surface, then the stresses are tensile. As is seen from Fig. 2, the value of \(\delta\) is determined with insufficient reliability; in this connection the error in determining the residual stresses may be significant.

Figure 2 shows the dependence of the bending moment of the residual stresses, referred to the width of the plate, on one half of its thickness,

Plates made of steel 45 were ground under various conditions. The cross feed was varied (Fig. 2 shows feeds in mm/double stroke), as were the cooling conditions (several batches were ground with cooling and three without cooling). The cutting speed was \(V = 27\) m/s; the longitudinal feed was \(S_{\text{pr.}} = 18\) m/min, with the exception of one batch processed at \(S_{\text{pr.}} = 2\) m/min.

Fig. 2

The bending moment referred to the width of the plate was calculated by the formula:

\[ m = \frac{M}{b} = \frac{2}{3} \cdot \frac{E h^3 \Delta}{l^2} \tag{5} \]

following from (1) and (2).

From the graphs obtained, the following was determined:

\[ n = \frac{N}{b} = \frac{(m_A - m_B)\cdot 2}{h_A - h_B} \tag{6} \]

(designation according to Fig. 1b).

As can be seen from Fig. 2, the experimental points are located sufficiently close to a straight line, which confirms the assumption that the epure of residual stresses is independent of the plate thickness.

Based on the results of the experiments performed, a nomogram was constructed (Fig. 3), showing the dependence of the magnitude of the resultant residual—

specific stresses, referred to the width of the plate, on the process parameters of grinding under study.

In calculating the warpage, we shall assume that the resultant of the residual stresses is directed along the table feed and acts along the ground surface (we neglect the quantity \(\delta\) in (3) in comparison with \(\frac{h}{2}\)). Consequently, the bending moment is

\[ M = N \cdot \frac{h}{2} \qquad \text{or} \qquad m = n \cdot \frac{h}{2}. \]

Fig. 3. Nomogram for selecting grinding conditions

Let us consider the deflection of the ground part after it is removed from the table. We shall consider the part thickness \(h\) to be constant. Its width \(b\) may be variable. The curvature of this part is

\[ \frac{1}{\rho} = \frac{M}{EI} = n \cdot \frac{h}{2} \cdot \frac{b}{EI} \tag{7} \]

For parts of constant thickness,

\[ I = \frac{b \cdot h^3}{12}. \]

(If the part has a hole, then \(b\) is the total width in the section under consideration.) Thus, the radius of curvature proves to be constant:

\[ \rho=\frac{E h^2}{6n} \tag{8} \]

This means that, for any configuration of a part having constant thickness, it bends along the arc of a circle.

The warpage of the part is evaluated by its deflection (1):

\[ \Delta=\frac{l^2}{8\rho}=\frac{3}{4}\cdot\frac{n}{E}\left(\frac{l}{h}\right)^2 \tag{9} \]

If the allowable warpage \([\Delta]\) is specified, then the resultant of the residual stresses, referred to the width of the part, must satisfy the inequality

\[ n\leq [n]=\frac{4}{3}E[\Delta]\left(\frac{h}{l}\right)^2 \tag{10} \]

Having calculated from (10) the allowable value of this quantity \([n]\), we select, with the aid of nomograms similar to that shown in Fig. 3, grinding conditions under which \(n\leq [n]\).

Consider an example. A ring of steel 45 is being ground, with an outside diameter of 200 and a thickness of 5 mm. The allowable warpage is 0.1 mm. We calculate

\[ [n]=\frac{4}{3}\cdot 2\cdot 10^6\cdot 0.01\left(\frac{0.5}{20}\right)^2=16.7\ \text{kg/cm} \]

From the nomogram in Fig. 3 we find that, when grinding with cooling, \(n\leq [n]\), i.e., the warpage will lie within the specified tolerance, at a feed of \(S_{\text{long}}\leq 0.055\ \text{mm/double stroke}\).

REFERENCES

1. V. I. Fedos’ev. Strength of Materials. Nauka Publishing House, 1964.

2. A. I. Isaev. Selection of the optimal specimen thickness in determining residual stresses in the surface layer. Vestnik Mashinostroeniya, No. 8, 1967.

3. A. V. Podzei. Regulation of residual stresses arising during the grinding of heat-resistant and titanium alloys. Machinability of Heat-Resistant and Titanium Alloys. Proceedings of the All-Union Interuniversity Conference. Kuibyshev, 1962.