Modelling and evaluation of time-varying thermal errors in machine tool elements
This thesis addresses a comprehensive approach to understanding the time-varying thermal errors in machine tools. Errors in machine tools are generally classified as being time or spatial dependent. Thermal errors are strongly dependent on the continuously changing operating conditions of a machine and its surrounding environment. Uniform temperature rises or stable temperature gradients, which produce time-invariant thermal errors, are considered to be rare in ordinary shop floor environments. Difficulties in analysing time-varying thermal errors are that, first of all, the temperature distribution within the components of a machine should be evaluated, and secondly, the distribution is continuously changing with time. These difficulties can be overcome by introducing a point-wise description method with three thermal parameters. From the theoretical analysis of simple machine elements such as bars, beams and cylinders, and extensive finite-element simulation data for a straightedge subject to room temperature variations, three thermal parameters, i. e. time-delay, time-constant and gain, were identified to obtain a precise description of the thermal deformation of a point of a machine body. Time-delay is dependent largely on thermal diffusivity, and the heat transfer mechanism. The time-constant is governed by heat capacity, heat transfer mechanism and body size. Gain, on the other hand, is determined by the thermal expansion coefficient, heat transfer mechanism and mechanical constraint. The three thermal parameters, in turn, imply that thermal deformation of a point in a body can be described by a simple first- order differential equation. Regarding their dependence on the heat transfer mechanism, a more refined description requires a time-varying linear first-order differential equation. Such an equation can be applied to each point of interest of a machine body. The final form of modelling, using the parameters, is a state-space equation gathering the governing equations for the points of interest. By adopting the point-wise discrete modelling method, we can overcome the difficulty of the spatial distribution of the temperature. Indeed, the calibration of a machine tool is usually performed at discrete points. The completion of this approach was made by presenting the methods by which the three thermal parameters can be evaluated. The first method employs analytical tools based on simplifying assumptions about the shape and boundary conditions of machine components. The second method was to apply numerical techniques to complex machine components. Because there are many drawbacks in theoretical approaches, experimental techniques are essential to complement them. The three thermal parameters can be easily identified using popular parameter identification techniques which can be applied to time-varying cases by their recursive forms. The techniques described were applied to modelling the thermal errors in a single-point diamond turning research machine. It was found that the dominant error component was spindle axial growth. The predictive model for the time-constant was shown to be in agreement with both the machine and with the scaled physical model rig.