## 1. Introduction

Several factors affect the productivity of manufacturing by milling processes. One key component is the material removal rate, which cannot be arbitrarily increased due to undesirable vibrations. The most critical form of vibrations is called chatter, which can result in unacceptable surface quality and possible damage in the milling tool and/or in machining centre components [

1,

2,

3,

4,

5].

The preferable machining parameters, where chatter does not occur, are usually illustrated on the so-called stability chart, from which one can select optimal technological parameters in order to achieve acceptable surface quality and the highest available material removal rate [

6,

7,

8]. Usually, the most productive parameter regions are located near the pockets of the so-called stability lobes. In these regions, however, large-amplitude stationary vibrations can occur due to resonant excitation, which results in significant surface location errors [

9], but it can be compensated with a well-defined tool path correction.

According to [

10], over the past six decades, an extensive number of studies have addressed the investigation of productivity and chatter, still, the relationship between tool wear and chatter is complex and is still an open field of research.

In [

11] it is shown that the vibration amplitude has a direct influence the surface quality and on tool life; consequently, even in the case of stable milling processes, tool wear can still be progressed due to the large-amplitude stationary vibration, in contrast to turning, where stable operation results in continuous and smooth chips. Therefore, not only stability but also large-amplitude stationary vibrations must be considered during the analysis of tool wear. In [

12] the authors introduce the so-called superchart, which is the stability lobe diagram presented together with the surface location error. This surface error relates to the large-amplitude stationary vibrations, thereby one can select stable machining conditions while obtaining indirect information about tool wear. A comprehensive literature review about tool wear associated with chatter can be found in [

10,

13] and all the references therein.

In addition, according to [

14], at lower spindle speeds, it is more likely that chatter appears for a new tool, while chatter may vanish later as tool wear starts to appear. The most accepted explanation for this is the effect of process damping when the flank face of the tool is in contact with the workpiece, thereby increasing the achievable stability domain. The explanation of process damping and tool wear is out of the scope of this paper, although there exist several models in the literature [

15,

16,

17,

18].

It is well-known that the dynamic properties of the workpiece and the machining centre greatly affect the performance of the milling operation. The position of the lobes in the stability chart is primarily influenced by the natural frequencies of the system. Consequently, there is an essential need to accurately determine these dynamic parameters in the interest of precise stability chart production. An additional challenge in this task is that dynamic behaviour can vary significantly during the machining process itself. The dynamic characteristics of a tool may vary, depending on the actual configurations of the machine tool structure within the workspace [

19,

20,

21,

22] and it may also change for different spindle speeds (e.g., Campbell diagram) [

23]. In addition, the changing dynamic behaviour can also correspond to the workpiece, as its geometry varies according to the removed material quantity. These phenomena can be substantial and can result in a significant change in the stability lobe diagrams [

24,

25,

26,

27,

28] for thin-walled parts where sometimes 90% of the original material is removed, such as in case of turbine blade manufacturing, cutting tubular parts, pocket milling and rib machining. It is the varying workpiece dynamics which is in the focus of this study.

To model thin-walled parts, a reasonable choice is the Finite Element Analysis (FEA). Although FEA can be applied to any complex structure, the geometry of the model and the corresponding mesh have to be frequently updated due to the removed material. This procedure requires considerable computational effort and time. In addition, despite the fact that the stiffness of a structure can accurately be determined with a properly created FE model, the modelling of the contact surfaces makes the results for the natural frequencies very sensitive for parameter tuning. For accurate modelling of contact surfaces (e.g., clampings), spring-damper elements can be inserted; however, the parameters must be tuned based on measurement results.

The receptance coupling method is widely used to properly update the model and to reduce the computational time [

29,

30,

31]. It can be based on both FEA and real measurement. However, it should be noted that complete modal testing requires a trained expert and it is very time-consuming as well. Even with the receptance coupling method, it is still difficult to properly model the entire operation, since the model errors are accumulating during the material removal phase.

A possible solution is to “pause” the milling process at certain steps and perform modal testing in the meantime. This allows the model to be corrected and adapted to the measurement results step by step. This lengthy process could be accepted during the fine-tuning of the milling operation, but it is not feasible in mass production; the modal parameters can change slowly in time by the ageing of the components, or simply by using a new tool. Modal testing during the cutting process presents additional challenges [

32], since, in case of traditional modal testing, the Frequency Response Function (FRF) can be determined from the response of impulse excitation induced by a modal hammer [

33]. However, during the hammer excitation of thin-walled workpieces, the so-called multiple-hitting (prall) phenomenon can easily occur causing non-ideal input force, which should be avoided.

A ball shooter device can replace the modal hammer and eliminate many of its disadvantages. Just to name a few: the shorter impact time of pellet shots reduces the multiple-hitting effect, leading to a wide achievable frequency bandwidth, therefore vibration modes with very high frequencies can be excited; it can access hidden locations of the workpiece; the use of traditional modal hammer poses safety risk during the rotating milling tool; the ball shooter has high repeatability capabilities and it can be triggered precisely by external signals; combining the ball shooter with an automatic unmanned expert monitoring system, the milling process does not need to be paused [

34]. Please note that a detailed investigation of pellet impacts for impulse excitation can be found in [

35].

Our goal in this paper is to measure the varying natural frequencies automatically during the milling process by means of using a ball shooter device. As a first step, this may be enough to avoid chatter at a higher material removal rate to increase the productivity, since it makes it possible to keep track of optimal positions of the varying stability lobes [

36,

37]. With this on-line information, the spindle speed could be fine-tuned properly, since the preferable technological parameters are usually close to the resonant speeds.

One current drawback of the ball shooter device is that we have only partial information about the force signal, which is essential for the modal analysis. In this study, it is shown that the spectrum of the excitation is so large that even a single acceleration signal is satisfactory to determine the natural frequencies accurately.

The paper is organized as follows: in

Section 2.1, the investigated blade-type workpiece is introduced. Then, in

Section 2.2, an analytical approach is presented to give an insight into the underlying effect of the pulsating natural frequencies.

Section 2.3 presents the results of the FEA. In

Section 3. the measurement results of the impulse tests, performed with a micro hammer and the ball shooter, are analysed. Finally, an improved FEA model is provided in order to fit the model to the measurement results. Conclusions are presented, and we discuss future research directions.