# Implicit Subspace Iteration to Improve the Stability Analysis in Grinding Processes
## 1 Introduction  
Grinding is a finishing process characterized by its ability to achieve workpieces with smooth surfaces and fine tolerances. One of the most important limitations in grinding processes is the self-excited vibration or chatter. The appearance of this regenerative effect leads to lower form accuracy and worse surface finish of the ground parts, loss of productivity, grinding machine damage or even wheel breakage. Chatter in grinding occurs when the grinding forces change during the process due to the variations of the chip thickness in the presence of vibrations, then the corresponding delayed responses to the forces increase the undulations over the workpiece surface. The generative effect can appear on both the workpiece and the grinding wheel, adding complexity to the dynamic behavior. The undulations generated on the workpiece surface grow quite rapidly, whereas those generated in the grinding wheelsurface grow more gradually [1]. Besides, when the vibration amplitude builds up to a certain limit, those undulations on the grinding wheel surface are removed through the dressing process. For these reasons, this paper is only focused on the workpiece regenerative chatter. Different approaches have been developed to analyze this phenomenon and devise practical methods for suppressing these vibrations [2]. The first discussion of regenerative stability of a grinding process was done by [PERSON] using the Nyquist criteria [3]. Then, [PERSON] and [PERSON] [4] performed a similar analysis by means of a doubly regenerative closed-loop block diagram and obtained the characteristic equation of the process and a stability criterion. [PERSON] proposed an alternative approach obtaining solutions of double regenerative chatter for the stability boundary [5] and chatter growth [6]. Later, different time-domain simulation approaches were applied to grinding stability analysis and regenerative chatter prediction [7, 8], considering also non-linear effects [9, 10]. [PERSON] et al. applied the semi-discretization method proposed by [PERSON] et al. for milling [11] and turning [12]--later used for selection of variable pitch for chatter suppression in face milling [13] to infeed cylindrical grinding process [14] and traverse cylindrical grinding process [15]--demonstrating the potential of this approach to analyze the application of variable speed to chatter avoidance with an efficient computing effort comparing to previous time-domain simulations. This periodically variable speed is one of the most extended techniques for chatter suppression in machining processes in general [12, 16, 17], and in grinding processes in particular. [PERSON] [18] was the first to simulate the workpiece sinusoidal speed variation effect, concluding that chatter can be avoided with short periods and large amplitudes of variation. [PERSON] et al. [19] and [PERSON] et al. [20] demonstrated the application of this technique for infeed and through-feed centerless grinding processes, respectively, by devising a time-domain dynamic approach for generating stability maps whose axes were the amplitude and frequency of the sinusoidal variation. The analysis concluded that the sinusoidal shape is the optimal speed variation signal. Those maps were validated experimentally, but the most important limitation of that method was the very long time required for obtaining a complete stability map due to the large amount of simulations needed. Therefore, the goal of this paper is to obtain a method for computation of the instability lobes in the time domain with much lower computing effort compared to traditional ones. The method is based on the implicit subspace iteration presented by [PERSON] et al. [21, 22] for milling process analysis. The improvement comparing with the semi-discretization method is achieved by removing the need for calculating the complete transition matrix whose eigenvalues give the stability degree of parameter combinations.