Submitted to HEJ Manuscript no.: MET-990617-A

The character of bilge in the model applied

If $k\neq0$, for calculation of ideal and friction power, bilge character also should be taken into account. For the boundary of integration (relation (16)) and for definition of the radius $R$ of friction surface (relation (26)), the function $r=f(z)$, describing the instantaneous bilge form, is necessary. It makes the exact definition of the function more difficult because that it depends on time also, that is $r = f(z, t)$. Considering the displacement of a point of the function under an elementary $dt$ period, the following equations hold:

$$\,f(z + w_z(z) dt, t+dt)= f(z,t) + w_r(r=f(z,t), z) dt$$ (30)

Expanding the left-hand in terms of Taylr series, we obtain a differential equation which describes the the change of the curve in time:

 

$$\frac{\partial f(z,t)}{\partial z}\cdot w_z (z) + \frac{\partial f(z,t)}{\partial t}\, = w_r( r=f(z,t), z)$$ (31)

If the curvature of the surface is small enough, the first term of the left-hand side is negligible. Assuming that the initial condition is cylindrical (i.e. $f(z, t=0) = \mbox{constant}$), and taking into account that $w_r$ is a quadratic function of $z$, the solution of differential equation (31) is also quadratic in $z$. Therefore we approximated the profile curve by a second degree regression curve. Approximation with a second degree curve is quite usual (see e.g. [6])). Table 1. shows the errors of the approximation relative to the average radius at different values of the coefficient of friction ($\mu$) and the upset height ($h$) see [13]. The initial sizes of the work piece are: $H_0 = 5.3 \mbox{mm}$, $R_0 = 8 \mbox{mm}$.

 

Table 1: Relative error of approximation

The height of the upset work piece	Relative error
$h$	$\mu = 0.05$	$\mu = 0.1$	$\mu = 0.15$	$\mu = 0.25$
2.5	--	6.70768E-05	1.73792E-04	5.35802E-04
3	1.29048E-05	5.45206E-05	1.24832E-04	3.51149E-04
3.5	1.17484E-05	3.84116E-05	8.03957E-05	2.10181E-04
4	8.11444E-06	2.23918E-05	4.37545E-05	1.07693E-04
4.5	3.80848E-06	9.30945E-06	1.72268E-05	4.03032E-05
5	6.40259E-07	1.43069E-06	2.53462E-06	5.68641E-06

Table 1. clearly shows that:

• the relative error increases with the coefficient of friction and with the degree of upsetting;
• for practical purposes, the profile of the work piece can be approximated with a function of second degree with good accuracy.

Lnowing the profile, one can determine the current volume of the work piece and the force requirement of the upsetting. The control calculations justified the volume-consistence with the accuracy of 1-2%.

Submitted to HEJ Manuscript no.: MET-990617-A