On Heating of a Part During Drilling
Abstract Generated abstract
This paper analyzes the temperature field produced in a long cylindrical workpiece during drilling, modeling the drill contact as a point heat source moving along the cylinder axis at constant speed. Using a moving coordinate system, the nonstationary heat conduction problem is reduced to a quasi-stationary form with convective heat loss at the cylinder surface, then solved by separation of variables with Bessel functions and eigenvalues determined by a Biot-number boundary condition. The resulting series expression gives the temperature distribution in the part and indicates how the maximum temperature at the drill contact can be inferred when surface temperature data are available experimentally.
Full Text
ON HEATING OF A PART DURING DRILLING
V. V. IVANOV
(Presented by Prof. Dr. G. I. Fuks)
Knowledge of the temperature field of a part during its machining makes it possible to estimate the heating and thermal stresses for a given cutting regime. Finding the temperature distribution in this case can be reduced to the problem of a nonstationary heat flux caused by a moving heat source—the drill.
We shall investigate the temperature field during the drilling of a long cylinder (Fig. 1). The dimension along the \(z\)-axis is large in comparison with the radius \(R\) and is therefore considered infinite. The point of contact of the drill with the part will be regarded as a point heat source moving
Fig. 1.
along the \(z\)-axis with constant velocity \(u\). The heat-transfer coefficient \(\alpha\) from the surface of the cylinder to the surrounding medium, and the coefficients of thermal conductivity \(\lambda\) and thermal diffusivity \(a\) of the material of the part, are constant quantities. The machined part is in a medium with temperature equal to zero.
The temperature inside the cylinder must satisfy the differential equation
\[ \frac{1}{a}\frac{\partial \theta}{\partial \tau} = \frac{\partial^{2}\theta}{\partial r^{2}} + \frac{1}{r}\frac{\partial \theta}{\partial r} + \frac{\partial^{2}\theta}{\partial z^{2}}, \tag{1} \]
\[ 0<\tau<\infty,\quad 0<r<R,\quad -\infty<z<+\infty . \]
If we apply the moving coordinate system [1, 2]
\[ \tau'=\tau,\quad \varepsilon=z-u\tau, \]
then
\[ \frac{\partial \vartheta}{\partial \tau} = \frac{\partial \vartheta}{\partial \tau'} - u\frac{\partial \vartheta}{\partial \varepsilon}, \qquad \frac{\partial^2\vartheta}{\partial z^2} = \frac{\partial^2\vartheta}{\partial \varepsilon^2}. \]
The quantity \(\partial \vartheta/\partial \tau'\) is equal to zero from the point of view of an observer in the source [1, 2]; therefore equation (1) takes the quasi-stationary form
\[ -\frac{u}{a}\frac{\partial \vartheta}{\partial \varepsilon} = \frac{\partial^2\vartheta}{\partial r^2} + \frac{1}{r}\frac{\partial \vartheta}{\partial r} + \frac{\partial^2\vartheta}{\partial \varepsilon^2}. \tag{2} \]
We integrate (2) by the method of separation of variables, assuming that
\[ \vartheta(\varepsilon,r)=Q(\varepsilon)\cdot P(r), \]
in which \(Q\) depends only on \(\varepsilon\), and \(P\) only on \(r\). Then we obtain
\[ \frac{P''}{P} + \frac{1}{r}\frac{P'}{P} = -\frac{u}{a}\frac{Q'}{Q} - \frac{Q''}{Q} = -k^2, \]
\[ P''+\frac{1}{r}P'+k^2P=0, \tag{3} \]
\[ Q''+\frac{u}{a}Q'-k^2Q=0, \tag{4} \]
where \(k\) is the separation constant.
The solutions of equations (3) and (4) have the form
\[ P(r)=C_1J_0(kr)+C_2Y_0(kr), \]
\[ Q(\varepsilon) = C_3\exp\left[ -\frac{u}{2a} + \sqrt{\left(\frac{u}{2a}\right)^2+k^4} \right]\varepsilon + \]
\[ + C_4\exp\left[ -\frac{u}{2a} - \sqrt{\left(\frac{u}{2a}\right)^2+k^4} \right]\varepsilon . \]
Here \(C_1, C_2, C_3, C_4\) are arbitrary constants of integration. The general solution \(\vartheta(\varepsilon,r)\) must satisfy the boundary conditions
\[ -\lambda\frac{\partial\vartheta(\varepsilon,R)}{dr} = \alpha\vartheta(\varepsilon,R), \tag{5} \]
\[ \frac{\partial\vartheta(\infty,r)}{\partial \varepsilon}=0, \tag{6} \]
the symmetry condition and the condition of boundedness of the temperature on the cylinder axis
\[ \frac{\partial\vartheta(\varepsilon,0)}{dr}=0, \qquad \vartheta(\varepsilon,0)\ne\infty, \tag{7} \]
and the condition of maximum temperature at the point where the heat source is located
\[ \vartheta(0,0)=\vartheta_{\max}. \tag{8} \]
On the basis of (7), \(C_2=0\). Using (6), we obtain \(C_3=0\), \(\varepsilon>0\), and \(C_4=0\), \(\varepsilon<0\).
Therefore,
\[ \vartheta(\varepsilon,r)=C I_0(kr)\exp\left[-\frac{u}{2a}\pm\sqrt{\left(\frac{u}{2a}\right)^2+k^4}\right]\varepsilon . \tag{9} \]
Here the \(+\) sign applies to the case \(\varepsilon<0\), and the \(-\) sign to the case \(\varepsilon>0\). Condition (8) gives \(C=\vartheta_{\max}\). We now satisfy the particular solution (9) to the boundary condition (5)
\[ \frac{I_0(kR)}{I_1(kR)}=\frac{kR}{\mathrm{Bi}}, \qquad \mathrm{Bi}=\frac{\alpha R}{\lambda}. \]
The roots \(\upsilon=kR\) of this transcendental equation are given in [3].
Finally, the required temperature distribution is described by the relation
\[ \vartheta(\varepsilon,r)=\vartheta_{\max}\sum_{n=1}^{\infty} I_0\left(\upsilon_n\frac{r}{R}\right) \exp\left[-\frac{u}{2a}\pm\sqrt{\left(\frac{u}{2a}\right)^2+\left(\frac{\upsilon_n}{R}\right)^4}\right]\varepsilon . \]
On the basis of the last equation, one may, for example, determine the maximum temperature at the point of contact between the drill and the part, if the temperature on the surface of the part is known from experiment.
REFERENCES
- E. R. Eckert and R. M. Drake. Theory of Heat and Mass Transfer. Gosenergoizdat, 1961.
- P. Schneider. Engineering Problems of Heat Conduction. GIIL, 1960.
- A. V. Lykov. Theory of Heat Conduction. Gostekhizdat, 1952.