Modelling, identification and control of thermal deformation of machine tool structures, part 3: Real-time estimation of heat sources

Compensation of thermal deformation of machine tools requires real-time est imation of the heat input to the structure in order to fully describe its t hermoelastic response. Available solutions of the inverse heat conduction p roblem IHCP are not suitable for real-time feedback control applications, s ince they are too slow and/or rely on future data to stabilize the solution . A new real-time IHCP solver is derived in the form of a convolution integ ral of the inverse thermal transfer function G(-1) (s) and the measured tem perature difference at two points near the heat source. An expression for G (-1) (s) is derived for multi-dimensional structural components. To transfo rm G(-1) (s) to the time domain, a special consideration is given to the tr eatment of its complex singularity functions. Analytical approach was follo wed to identify these functions and to determine their time-domain represen tation. Computer-simulation test cases were conducted using a finite elemen t model of a three-dimensional structure. The random temperature measuremen t errors, which can lead to non-uniqueness and instability problems, have a lso been simulated. The test results showed that the computation time can s ignificantly be improved to achieve a control cycle of less than one second , without compromising the accuracy and stability requirements.