BMe Research Grant
Harmful vibrations in the manufacturing industry can be a barrier to maximizing production. Therefore, the optimal tuning of machining parameters is an extremely important task for professional manufacturers, not only to increase productivity, but also to reduce financial costs [R1]. The biggest challenge of the research is thus the possible prediction of undesirable vibrations before their appearance. This requires the completion of the literature on the identification of machine tool vibrations by measurements.
Figure 1 Stability chart: stable and unstable conditions in milling operations
The main goal of this research is to characterize the dynamic behaviour of delayed periodic systems (such as the equations modelling the milling operation [R5]) without essential knowledge on the parameters of the underlying mechanical model. In this way, there is no need for modal parameter fitting, which is required for the traditional stability chart calculation methods, and is a major source of inaccurate predictions and model uncertainties. The main idea is to capture the so-called dominant spectral properties – which characterize the behavior – of the system from its impulse response directly [J3, C1].
The results of this main research goal can be used for example, in the operational stability prediction in milling processes [C2]. The new method introduced during the research is able to quantitatively measure the stability of milling operations based on measured resulting vibrations. In the prediction, the stability of the machining process can be characterized through the dominant spectral properties. Because the new method is capable of monitoring how the dominant spectral properties change as a result of technology parameter changes, a precise stability limit can be extrapolated while the manufacturing parameters remain in the safe region, thereby supporting the development of effective milling strategies.
The main research goals described above are therefore:
(i) to characterize dynamical systems without prior knowledge of the model and
(ii) to apply this method for predicting the stability conditions of machining operation.
For the above, two numerical methods have been developed and tested. First, the so-called Impulse Dynamic Subspace (IDS) description [R6] and, second, the Dynamic Mode Decomposition (DMD) [R7] methods.
IDS is an efficient mathematical description for the automatic parameter fitting of mechanical systems. Originally, the method was developed for the investigation of linear time-invariant systems, which is based on the evaluation of frequency response functions by using Green function representation of the homogeneous dynamics. However, in this work, the original ideas of the method had to be slightly modified for use for periodic systems, like milling.
The DMD method is a model reduction and fitting technique based on measurement data. It was originally developed to decompose fluid flows, and then it proved useful in various fields, like studying large-scale neural networks, investigating cyclic behavior of the stock market and pattern recognition in infectious diseases [R7]. The method is capable of approximating the so-called solution operator of periodic systems, which makes it possible to characterize the stability of milling processes.
The methods and their applicability were first tested on an artificial example before applying them in a real measurement environment. In the artificial test case, a time-periodic delay-differential equation (DDE), the so-called delayed Mathieu equation was examined by using time signals generated by numerical integration, as shown in Fig. 2a. Then the efficiency of the methods was demonstrated through a real case study.
Figure 2 The effect of discretization on the fitted spectral properties for both methods in absolute value and in the complex plane [C3].
Then, the above described methods were applied for milling operation in an experimental procedure [J3, C1, C2]. The schematic and the photo of the experimental setup are presented in Fig. 3a. The spectral properties were identified from the transient vibrations of the milling (see Fig. 3b) [J4], which gives a good approximation for the dynamical behaviour of the system. The obtained spectral properties were then compared to the theoretical ones for a set of different cutting speeds, and showed a good correlation, as illustrated in Fig. 3c. Finally, the stability boundary could be precisely determined by means of extrapolation from stable measurement points (see in Fig. 3d).
Figure 3 a: The experimental setup; b: time domain of the measured signal; c: identified dominant spectral properties in the complex plane; d: schematic figure representing the extrapolation method of the stability boundary [C2].
[J1] A.K. Kiss, D. Bachrathy, G. Stepan, Effects of varying dynamics of flexible workpieces in milling operations, J. Manuf. Sci. Eng., 2020, 142(1): 011005