Abstract:
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.