The system for adaptive control of axial tool oscillations in vibratory drilling: description and experimental study

2.1. Closed-loop dynamic system structure

Fig. 4 shows the structure of the closed-loop dynamic system “cutting – VDH – control system”. The interactions between the main elements of the system are described below. For brevity, the presented block diagram does not show a measurement system, an amplifier, a signal filtration block, and some other elements.

The displacement $x$ of the moving part of the VDH is determined by the cutting force $F_c$, actuator force $F_a$, mass and stiffness of the elastic element and damping factor. The cutting force $F_c$ is generated in drilling process. Generally, tool vibrations cause an alternating value of $F_c$. It should be noted that $F_c$ is a function of not only the vibration displacements of the cutting edges, but also the surface obtained during previous passes.

The control system determines the actuator force $F_a$ as follows. The measured signals are supplied to two control loops: the main loop and the adaptation loop. In the main loop, the vibration velocity signal is multiplied by the feedback gain $b$. The controller sets the gain $b$ in the adaptation loop according to the algorithm described in Subsection 2.3. If $b>$0, the vibrations are amplified; if $b<$0, the vibrations are damped. The product of $b$ and $\dot{x}$, the control output $u$, is supplied to the actuator generating the force $F_a$.

The described system thus considers the influence of vibration on cutting forces and vice versa. The adaptation of the coefficient $b$ allows the adjustment of the amplitude of the vibration process.

Fig. 4Block diagram of the closed-loop system “cutting – VDH – control system”

2.2. Control objective

Evidently, vibration amplitudes over a certain threshold value cause chip segmentation. This threshold is determined by the feed per cutting edge and the ratio of vibration frequency to the tooth pass frequency [2], [32]; it thus can be either calculated or found experimentally. If the proper amplitude is known and the control system maintains it, then chip segmentation is ensured. As the mean value of vibration displacement is generally not zero, it would be easier to consider not the displacement amplitude but the PTP displacement value. This is defined as the difference between the maximum and minimum values of the x coordinate for the considered time interval.

It should be noted that the alternative variant of control based on the cutting continuity coefficient [9], [27] is promising. However, as mentioned in the Introduction, this is currently unfeasible.

2.3. Control strategy

This paper proposes a new adaptive control strategy of the vibration process. Although the system was developed for the vibration drilling process, it could be applied to any monoharmonic vibration process.

The assumptions for developing the control system are as follows:

1) The hardware part of the control system and the actuator do not make any serious changes in the phase of the signal $u$ (Fig. 4). This assumption is elaborated upon in the conclusion.

2) The steady-state PTP displacement value will grow with increased values of the feedback gain $b$, and it will recede if the feedback gain decreases. In other words, the coefficient $b$ can be considered as negative damping.

3) System vibrations are close to monoharmonic. This assumption is proved by the experimental results of this study.

4) The computation time of the controller (additional delay) is negligible compared to the VDH vibration period; it does not lower the control quality.

The control objective is to maintain the target value of the vibration displacement PTP value $A_0$. This requirement is met by the following heuristic adaptation algorithm:

where $b_c$ is dimensionless feedback gain; $c_1$ and $c_2$ are dimensionless adaptation law constants; $A$ is the PTP displacement value, mm; $A_0$ is the target value of $A$, mm; and $T_b$ is the time constant of the adaptation law, s. The first addend of Eq. (1) ensures the growth of $b_c$, if $A<A_0$, and the reduction of $b_c$, if $A>A_0$. The second addend is introduced to stabilize the closed-loop system. It should be noted that $b_c$ differs from $b$ shown in Fig. 4.

The inequality (2) is introduced because the practical voltage levels in the electronic circuits are limited. For normalization purposes, the maximum value of $b_c$ is set to 1. In this study, the control system is organized in such a way that $b_c=$1 corresponds to 5 V at the controller output, and $b_c=$ –1 corresponds to 0 V.

Unfortunately, testing revealed that the direct application of Eq. (1) was not useful. First, the process of settling $A$ to the target value of $A_0$ could last a long time – at worst, longer than the drilling process itself. This may cause the drill to jam, as PTP displacements would not be sufficiently high to fragment chips. Second, the vibration drilling process is characterized by the super-fast dynamics of the change of integral process characteristics. According to our experience, the PTP vibration displacements often grow from zero to a steady-state value greater than the target value $A_0$ and back in around 0.1…0.2 s. This phenomenon is related to encountering hard particles in the cutting zone plus the subcritical character of the bifurcation that manifests in leaping fluctuations of the vibration amplitude. The smooth change of $b_c$ prescribed by Eq. (1) cannot react fast enough to these fluctuations. Low values of $c_1$ and $c_2$ do not allow the PTP value $A$ to return to the target value $A_0$ in the required time in case of large deviations, while large values of $c_1$ and $c_2$ cause instability in the closed-loop system “cutting-VDH-control system”.

Thus, Eq. (1) was modified as follows:

where $g_1$ and $g_2$ are dimensionless parameters, $g_{\mathrm{1,2}}<$ 1. These parameters determine the width of the PTP displacement range within which the equality $b=b_c$ is used.

Therefore, when the tool starts cutting, $A$ is around zero, and the control system sets the maximum feedback gain $b$ to 1. Then, as the system approaches $A_0$, the feedback ratio is set to $b_c$. Using the value $b_c$ determined by Eq. (1) allows the automatic regulation of the vibration process in case of a slight deviation of the PTP displacement $A$ from its target value $A_0$. If the PTP displacement $A$ is too far from $A_0$, Eq. (3) sets $b=$ ±1 depending on the deviation sign.

The value of the coefficient $b$ can only be in a discrete set of values determined by the bitness of the digital-to-analog converter. For brevity, this issue is not considered in this study.

2.4. Calculating peak-to-peak displacements

The PTP vibration displacement $A$ is an important term of Eqs. (1-3). This subsection describes the calculation method. By definition, the PTP displacement can be calculated as follows:

where $T_A$ is the duration of the considered time interval, s; $t_f$ is the end of the considered time interval, s.

In practice, calculations according to Eq. (4) require time-consuming operations of finding the maximum and minimum values of the array, which does not allow the implementation of the control action on time.

Therefore, this study uses another expression whose brief derivation is given in Appendix:

where $A$ is the PTP displacement estimate, mm; ${\omega }_c$ is the chatter frequency, rad/s. For smoothing, signal $A$ goes through a digital low-pass filter.

Eq. (5) is valid because the vibration process is close to monoharmonic. However, only acceleration $\ddot{x}$ is measured directly in Eq. (5); velocity is determined as integrated acceleration.

Chatter frequency ${\omega }_c$ in Eq. (5) is determined as follows. All time points where the vibration velocity signal crosses zero from negative to positive are determined. The distance between the last two points is the estimate of the vibration period $T_c$. This value is used to calculate chatter frequency:

where ${\omega }_c$ is the chatter frequency estimate, rad/s; $T_c$ is the vibration period, s. The signal ${\omega }_c$ goes through a digital low-pass frequency for smoothing.

2.5. Description of the control system operation

Fig. 5 displays the hardware of the control system. Vibration velocities are calculated by integrating vibration accelerations. Based on both signals, the controller calculates $b$, which is then used as the velocity feedback gain.

Fig. 5Diagram of the hardware part of the control system. LPF – low-pass filter, HPF – high-pass filter, ux¨ – vibration acceleration signal, ux˙ – vibration velocity signal, ADC – analog-to-digital converter, DAC – digital-to-analog converter

The controller algorithm is as follows:

1) Initialization of variables: ${\omega }_c$ is equal to the eigenfrequency of the VDH, $A$ and $b_c$ are 0, and $b$ is 1.

2) Time step $j$.

3) Input the current values of vibration acceleration $\ddot{x}$ and vibration velocity $\dot{x}$.

4) If ${\dot{x}}_j>0,{\dot{x}}_{j-1}<0$, adjust ${\omega }_c$, using Eq. (6) and the low-pass filter.

5) Calculate PTP displacement using Eq. (5) and the low-pass filter.

6) Calculate $b_c$ using the equation derived from Eq. (1):

where $\mathrm{\Delta }t$ is the time step, s.

7) Limit $b_c$ according to Eq. (2).

8) Calculate $b$ using Eq. (3).

9) Set a new value of $b$ at the controller output.

10) $j=j+1$, go to step 2.

4.1. Results of measuring axial force during conventional drilling

To study the influence of the chisel edge on the cutting force, drilling without vibrations was performed. In these tests, the workpiece was rigidly fixed to a Kistler Type 9257B dynamometer instead of a vibration drilling setup (Fig. 7). Table 3 shows the results of axial force measurements for different feed values for drilling with and without a pilot hole. The spindle speed was 3600 rpm. Table 3 shows that around 80 % of the axial force was produced at the chisel edge.

4.2. Drilling with a pilot hole (${\mathit{d}}_0=$ 3 mm)

Fig. 9 shows the test results for drilling without control; spindle speed was $\mathrm{\Omega }=$ 2500 rpm. PTP displacements reached significant values of about 0.6 mm, and the vibration character was almost monoharmonic. The vibration frequency of 72.38 Hz was slightly higher than the eigenfrequency of the elastic system (70.86 Hz). Fig. 10 shows the results for drilling with control. The vibration frequency did not practically change, and the PTP displacement value was around the target value of 0.06 mm. Here, the control system limited the self-excited vibrations of the system by the target level. The presented spectra confirm the monoharmonic signal character assumption that was made to derive Eq. (5).

Fig. 9Vibrations for drilling with a pilot hole without control, spindle speed 2500 rpm

a) Vibration displacements

b) Vibration displacements (enlarged)

c) Spectrum

Fig. 10Vibrations for drilling with a pilot hole with control, spindle speed 2500 rpm

a) Vibration displacements

b) Vibration displacements (enlarged)

c) Spectrum

Table 4Time histories of vibration displacements for drilling with a pilot hole

	Without control	With control
$\mathrm{\Omega }=\mathrm{}$2000 rpm
$\mathrm{\Omega }=\mathrm{}$3000 rpm
$\mathrm{\Omega }=\mathrm{}$3500 rpm
$\mathrm{\Omega }=\mathrm{}$4000 rpm
$\mathrm{\Omega }=\mathrm{}$4500 rpm
$\mathrm{\Omega }=\mathrm{}$5000 rpm

Table 4 shows all experimental results when drilling with a pilot hole. For $\mathrm{\Omega }=$ 2000 rpm, there was no self-excitation of vibrations without control, but turning on the control provided the excitation of vibrations at the required level. In contrast, the control system limited vibrations for $\mathrm{\Omega }=$ 3000, 3500 and 4000 rpm. For $\mathrm{\Omega }=$ 4500 and 5000 rpm, the control system did not significantly change the characteristics of the vibration process because its characteristics were already close to the required values without any control.

Fig. 11 generalizes the conducted experiments in the form of a Poincare map. It was built as follows. The $X$-axis represents spindle speed $\mathrm{\Omega }$. For a given value of $\mathrm{\Omega }$, vibrations were measured, and vibration displacement peaks were determined on the steady-state time interval. These peaks were plotted on a vertical line corresponding to the given $\mathrm{\Omega }$ on the Poincare map.

The diagram comparison shows that introducing control ensured vibrations with PTP displacements close to the target value 0.06 mm for the whole investigated spindle speed range. Thus, in contrast to vibratory drilling without control [8], [18], with the developed novel control system, the vibration amplitude is almost not sensitive to the spindle speed.

Fig. 11Poincare maps of vibration displacements when drilling with a pilot hole

a) Control off

b) Control on

Fig. 12Vibrations for full hole drilling without control with a spindle speed of 5000 rpm

a) Vibration displacements

b) Spectrum

Fig. 13Vibrations for full hole drilling with control with a spindle speed of 5000 rpm

a) Vibration

b) Vibration (enlarged)

c) Spectrum

Table 5Time histories of vibration displacements for full hole drilling

	Without control	With control
$\mathrm{\Omega }=\mathrm{}$2000 rpm
$\mathrm{\Omega }=\mathrm{}$2500 rpm
$\mathrm{\Omega }=\mathrm{}$3000 rpm
$\mathrm{\Omega }=\mathrm{}$3500 rpm
$\mathrm{\Omega }=\mathrm{}$4000 rpm
$\mathrm{\Omega }=\mathrm{}$4500 rpm

4.3. Full hole drilling

Fig. 12 shows example measurements without control. The PTP displacement was much less than the target value of 0.06 mm and even the feed value of 0.02 mm/cutting edge. Vibrations were polyharmonic. Results when drilling with control are shown in Fig. 13. Here, the control ensures the vibration of the system with a PTP displacement value close to the target value 0.06 mm; the control system also enables chip segmentation. The spectrum (Fig. 13(c)) shows that the process is not strictly monoharmonic, but there is a single distinct peak around 101.5 Hz.

Table 5 shows all experimental results for full hole drilling. For $\mathrm{\Omega }=$ 4000 and 4500 rpm, regenerative chatter was self-excited and PTP displacements were greater than the target values. With control, the PTP vibration displacements were limited to the target values. For other process regimes, $\mathrm{\Omega }=$ 2000, 2500, 3000 and 3500 rpm, there was no self-excitation, and vibrations were low without control, so control enabled the excitation of vibrations to the required levels.

The experimental results are generalized as Poincare maps in Fig. 14. By comparing the results for this case with the results for drilling with a pilot hole, Fig. 11, it can be concluded that the chisel edge has a fundamental influence on energy dissipation in the system. When the chisel edge did not take part in drilling, the self-excitation of vibrations without control occurred for all regimes except one ($\mathrm{\Omega }=$ 2000 rpm). For full hole drilling, there was no pilot hole and the chisel edge was not excluded from the cutting process, which caused an inverse effect: The excitation of vibration with PTP displacement close to the target value of 0.06 mm took place for two regimes only. For other regimes, there were no notable vibrations.

The second effect of the drill chisel edge was a significant fluctuations of the vibration magnitudes for all regimes, even with control (Fig. 14(b)). Furthermore, for two regimes ($\mathrm{\Omega }=$ 2500 and 3000 rpm), the control system did not ensure a stable vibration process; one time interval demonstrated a temporary suppression of vibrations. This interval indicates the significant non-stationary nature of the characteristics of energy dissipation in the chisel edge and indicates the need to improve the control system, which is discussed in the conclusion.

Fig. 14Poincare maps of vibration displacements, for full hole drilling

a) Without control

b) With control

4.4. Actuator force measurements

Fig. 15 presents the results of measuring the actuator force $F_a$. Force was measured using a sensor installed in the rod between the actuator (Fig. 7(b)) and the movable part of the experimental setup for vibration drilling (Fig. 7(a)). The cases of drilling with a pilot hole (Fig. 15(а)) and full hole drilling (Fig. 15(b)) were considered. Fig. 15 indicates that the chisel edge caused a significant (approximately four times) increase of the maximum actuator force. This conclusion agrees with the increased axial force of full hole conventional drilling compared to the drilling with a pilot hole observed in Subsection 4.1.

Fig. 15Poincare maps of the actuator force with control on

a) Full hole drilling

b) Drilling with a pilot hole

4.5. Repeatability of the experimental results

An important observation made by the authors was that the vibration levels obtained without control in different tests under the same test conditions were different. When drilling with a pilot hole (when the chisel edge does not contribute to the process), the difference was in an insignificant fluctuation of vibration amplitude and frequency. When full hole drilling (when the chisel edge contributes to the cutting), there was a qualitative difference between vibrations obtained for the same process regimes. The self-excitation of vibrations was observed for only some experiments. A negative influence of tool wear on the self-excitation of vibrations was observed. The results for $\mathrm{\Omega }=$ 4000 and 4500 rpm (Table 5 (left pictures)) were obtained on a practically new drill with sharp edges. For the second series of tests, there was no self-excitation of vibrations. Replacing the drill with a new one allowed self-vibrations to be generated again, but only once. This observation is consistent with a known dependence between tool wear and vibration energy dissipation in the cutting zone [20]. With control, this phenomenon was not observed: the required level of vibrations was maintained for both the new and the worn drill.

Tables 6 and 7 show the full reports on the excitation of vibrations and the repeatability of the results. When the chisel edge did not transfer force, as shown in Table 6, there was a qualitative repeatability of results from test to test. When the chisel edge transferred load, as shown in Table 7, the qualitative repeatability of results was only observed with control. If the control was off and the chisel edge contributed to cutting, there was no repeatability of the self-excitation of vibrations.

4.6. Vibration frequency

Table 8 presents the frequency values corresponding to peaks of the vibration displacement spectra. The case of drilling with control on was considered. With control off, the results were qualitatively identical.

Let us consider the cases of drilling with a pilot hole and full hole drilling. It can be seen that the vibration frequency for drilling with a pilot hole is greater than the eigenfrequency of the elastic fixture, as shown in Table 1. This increased frequency is related to the influence of the stiffness of the process, and this is consistent with the established fact that the chatter frequency is higher than the eigenfrequency of the elastic system [11]. For full hole drilling, the vibration frequencies for all regimes were greater than those when drilling with a pilot hole. This phenomenon is related to the increased stiffness of the process when the chisel edge is introduced in the cutting process. This observation is confirmed by an increased axial cutting force when a chisel edge is included in the process (Subsection 4.1, as well as by [33]).

The second observation is a significant difference between the vibration frequencies and the eigenfrequency of the elastic system (see Table 1). Therefore, the use of a chatter frequency estimate (Eq. (6)) based on data acquired during drilling is justified.

Aside from the data shown in Table 8, the authors observed significant vibration frequency changes for different tests under the same conditions. For full hole drilling, the change in the frequency measured experimentally in a subsequent test reached 20 Hz. This could have been related to the significant influence of tool wear on the cutting force at the chisel edge and, consequently, on the stiffness of the cutting process.