In order to verify the practical usability of the developed compensation method in an industrial environment, the behavior during the application of milling forces will be investigated. For this purpose, a standard 2-DOF milling force model [
1] is implemented on the real-time system with configurable process parameters, which is used to control the linear direct drive. Since significantly higher disturbance forces are caused in peripheral milling compared to face milling due to the same direction of all force vectors, the critical case of peripheral milling in the feed direction will be addressed below. The process force of a single cutting edge of the milling cutter
$$\begin{aligned} F_P(\varphi ,a_p)=f(\varphi )\left( -F_t(\varphi ,a_p)\mathrm {cos}(\varphi )-F_r(\varphi ,a_p)\mathrm {sin}(\varphi ) \right) \end{aligned}$$
(9)
is composed of a cutting force
$$\begin{aligned} F_t(\varphi ,a_p)=a_pk_ch_z(\varphi )^{1-m_c} \end{aligned}$$
(10)
and a radial force
$$\begin{aligned} F_r(\varphi ,a_p)=a_pk_nh_z(\varphi )^{1-m_n}, \end{aligned}$$
(11)
whereby the infeed depth is captured via the parameter
\(a_p\). The Heaviside function
f, which depends on the cutter geometry, indicates whether the cutting edge is in engagement
$$\begin{aligned} f (\varphi ) = \left\{ \begin{array}{ll} 1 ~~ &{} \text {if intervention,} \\ 0 ~~ &{} \text {otherwise.} \\ \end{array}\right. \end{aligned}$$
(12)
The chip thickness
$$\begin{aligned} h_z(\varphi )=f_z\mathrm {sin}(\varphi ) \end{aligned}$$
(13)
contained in the partial forces depends on the current rotational angle of the cutter
\(\varphi\) and the feed per tooth
\(f_z\). The remaining parameters for the specific cutting force
\(k_c\)/
\(k_n\) and the material constants
\(m_c\)/
\(m_n\) in eqs. (
10) and (
11) are material-dependent cutting parameters and can be taken from tables, thus they will not be discussed further [
10,
26]. Taking into account the number of cutting edges
n of the cutter and the corresponding angular offset between the individual cutting edges
\(\Delta \phi\), the total force acting in the feed direction can finally be calculated as
$$\begin{aligned} F_s(\varphi ,\Delta \phi ,a_p)=\sum _{k=0}^{n-1}F_P(\varphi +k\Delta \phi ,a_p). \end{aligned}$$
(14)
As an application scenario, a
\(2\,\mathrm {m}\) long,
\(3\,\mathrm {mm}\) deep and
\(30\,\mathrm {mm}\) wide material removal is to be considered, with a cutting speed of
\(150\,\mathrm {m/min}\) at a feed rate of
\(390\,\mathrm {mm/min}\). All parameters used for the milling force model are presented in Table
4. The choice of the mentioned parameters is based on reference values from table books [
10] (low strength steels, milling with carbide). The process force resulting from the milling force model for the application described is shown in Fig.
10. The generated milling force leads to a broadband excitation of the drive system, which can be recognized, by the additionally shown one-sided spectrum of the milling force.
Table 4
Process parameters of the generated milling force
Groove length | l | 2000 | mm |
Cutting speed | \(v_c\) | 150 | m/min |
Infeed | \(a_p\) | 3 | mm |
Width of engagement | \(a_e\) | 30 | mm |
Cutting edge number | n | 3 | – |
Feed rate | \(v_F\) | 390 | mm/min |
Specific cutting force | \(k_c\) | 1717 | \({{{\rm N/mm}}^2}\) |
| \(k_n\) | 928 | \({{{\rm N/mm}}^2}\) |
Material constant | \(m_c\) | 0.23 | – |
| \(m_n\) | 0.39 | – |
Furthermore, the error curve on the output side according to eq. (
8), as well as the resulting quadratic error summation
$$\begin{aligned} J_s = \int _{0}^{t_f}e(t)^2\mathrm {d}t \end{aligned}$$
(15)
over the duration of the milling process
\(t_f\) is shown. As shown in the figure, the use of an additional acceleration-based disturbance compensator contributes to a significant reduction of the output side positioning error.
In addition to the objective of increasing accuracy, the efficiency of drive systems is becoming equally important. For this reason, the electrical energy consumption
$$\begin{aligned} E=\int _{0}^{t_f}P(t)\mathrm {d}t \end{aligned}$$
(16)
via the integral of the absorbed power
P of the electric motor as well as the mechanical load [
8]
$$\begin{aligned} B=\int _{0}^{t_f}|M(t)|^3|\dot{\phi }(t)|\mathrm {d}t \end{aligned}$$
(17)
should also be taken into account. The latter is related to the wear and thus the durability of the feed drive system and is calculated via the motor torque
M acting on the drive system as well as the angular velocity oft the motor
\(\dot{\phi }\). As shown in Table
5 with the listed performance indicators, the use of the disturbance compensator contributes to a significant increase in efficiency. The electrical power consumption of the motor is reduced by the counterphase compensation, since the cascaded drive controller has to operate less against the disturbance variables that occur. Additionally, the mechanical load to the drive system is reduced due to the damping effect, which leads to a reduction in the acceleration peaks.
Table 5
Performance characteristics during milling forces
Sq. error \(J_s {({\rm mm}^{2{\rm s}})}\) | 0.510 | 0.195 (\(-61.8\,\mathrm {\%}\)) |
El. Energy \(E {\rm (Wh)}\) | 4.857 | 2.638 (\(-45.8\,\mathrm {\%}\)) |
Load \(B {({\rm Nm}^{3{\rm rad/s}})}\) | 3435.55 | 1040.32 (\(-69.8\,\mathrm {\%}\)) |