Formulation for Interaction Analysis Between Continuous and Discrete Structures Subject to Moving Loads Applied to Turning

A control oriented formulation of continuous structures interacting with moving and/or fixed structures is presented. Machining of long parts in a lathe is presented in order to illustrate the methodology. 

The system is time-varying in nature due to the movement of the cutting tool during the machining process. However the invariant part corresponding to the structure dynamics (tool and workpiece) and the time varying part corresponding to the movement of the cutting point can be separated. The link between the two is given by the eigenfunctions relating the output of the dynamic part with the actual displacement within the workpiece continuum. This splitted formulation allows modular design approaches, facilitating the inclusion and analysis of further elements such as flexible supports and/or allowing direct modifications of the characteristics of the different structural elements individually (considered vibration modes, etc.). For the presented turning example the implications of workpiece flexibility and the inclusion of flexible supports at different locations on regenerative chatter stability are also discussed.