In this chapter the subject of statistical dynamics are discussed and non-parametric and parametric models for machining system identification are derived. The common characteristic for all discussed models is that they may be used for computing the operational dynamic parameters (ODP) of the closed loop machining system. Though the input to these models originates from the machining operations, not all models can be implemented for real-time identification. Generally, non-parametric models may be used solely for off-line identification, i.e., first recording the vibration signal from machining operations and then analysing the signal and identifying the nature of the excitation. Parametric models implemented in recursive algorithms are used for real-time identification of machining systems dynamic characteristics. 

The main objectives of the chapter are: (i) the development of parametric and non-parametric models based on identification techniques with the purpose of integrating into a single step within the estimation of dynamic parameters characterising the machining system, (ii) in non-parametric identification, implementing techniques for ODPs and random excitation estimation, (iii) in parametric identification, the development of the recursive computational model of the machining system based on the data obtained during the actual operational regime. Through these contributions, a step is taken beyond the classical approach to analyse the dynamics of a machining system by separately identifying the structural and process parameters. In the proposed process, the two substructures, tool/toolholder and workpiece/fixture, are coupled, in addition to the open loop (elastic structure of the machine tool), by a feedback loop closing the energy loop, through the thermoplastic chip formation mechanism.

The machining system can only be completely analysed only in closed loop i.e. in operational conditions since specially designed off-line experiments with controlled input, such as modal testing, give the response from only the open loop.