# The Bivariate Empirical Mode Decomposition and Its Contribution to Grinding Chatter Detection
## 1 Introduction  
Grinders, which are often high-precision, large-tonnage machines, are essential in the equipment manufacturing industry. Monitoring their condition and diagnosing their faults ensures their safe operation, which has great practical and economic value [1; 2]. Most notably, grinders can enter a \"chatter\" state during machining operations, which causes a series of negative effects [3] such as machined surface undulations, reduced tool life, poor surface finish, noise, and reduced productivity. Friction, regeneration and mode coupling are three well-known mechanisms that cause grinding chatter. Typically, regeneration chatter is the most common type of chatter encountered in machining processes. It occurs when oscillations in the grinder cause undulations in the machined surface that are regenerated with subsequent passes of the grinder. As numerous experiments have shown, the amplitude of vibration signals increases substantially when a stable grinding state turns into a chatter state [4]. Additionally, there is a transitional phase in this process, which contains considerable information about grinding status [5]. Thus, we can discover the mechanism of grinding chatter and predict it by analyzing transition-phase signals. It is worth noting that the transitional phase occurswithin a very short time and is easily affected by many random factors. In order to alleviate the problem of chatter, reliable chatter detection and identification methods are essential so that effective chatter suppression can be applied in a timely manner [6]. Vibration detection in grinding is usually performed by using accelerometers, acoustic emission (AE) sensors or dynamometers. A considerable amount of research work describes these different methods. For example, [PERSON] et al. used acceleration signals from a lathe tailstock to detect chatter during turning operations, and used an s-transformation to extract a damping index as a descriptive feature of chatter [7]. [PERSON] et al. proposed two-dimensional feature vectors for chatter detection based on the standard deviation of wavelet transform of drilling machining, which allowed rapid chatter identification [8]. Furthermore, [PERSON] et al. equipped a lathe with eddy current sensors to monitor vibration. These were processed by a fuzzy method to classify the tests [9]. Moreover, many researchers have predicted cutting chatter by using dynamic forces. [PERSON] et al. employed a coarse-grained entropy rate as a chatter index in a cutting process, which exhibited a drastic drop at the onset of chatter [10]. [PERSON] et al. presented the usage of permutation entropy, a computer-based, rapid measurement of chatter onset using sound signals recorded by a unidirectional microphone [11]. [PERSON] et al. proposed a method using feature vectors incorporating signal variance and a one-step self-correlation function as a chatter indicator for detecting cutting state [12]. Acoustic emission sensors were used by [PERSON] et al. to detect chatter; the specific peculiarities of acoustic emission sensors can effectively reject external disturbances [13]. In summary, wavelet transform [14; 15], s-transformation [16], singular value decomposition (SVD) [17], and artificial intelligence [18], are the most popular methods used to process vibration signals. These analysis methods are mostly based on the theory of Fourier transform. However, grinder vibration signals are mostly nonstationary and nonlinear, such that these conventional methods cannot effectively extract signal features [19]. Subsequently, empirical mode decomposition (EMD), proposed by [PERSON] et al., has been demonstrated as an appropriate approach for overcoming the shortcomings of conventional techniques [20]. The EMD approach can decompose signals into several intrinsic mode functions (IMFs) which distribute from high frequency to low frequency. All IMFs derived from grinding signals are characterized as zero-mean oscillations.