Solution of the Analysis Problem of a Machine Assembly Distributed System with Time-Dependent Heat Transfer Coefficient

It is well known that temperature is a factor that significantly influences the accuracy of machine tools. Compensation enables machine errors to be reduced, even for a moderately accurate machine tool. The first step in compensation is to estimate the thermal characteristics of the machine part. Thermal models with distributed parameters provide high accuracy of estimation. In these thermal models, the environmental thermal fluctuations influencing the temperature may be taken into account as the time-dependent heat-transfer coefficient. The finite elements method facilitates simulation of the machine system geometry, but is computationally expensive. One approach is to use the simplified thermal model at an early stage of development, which allows the investigation of the temperature field and the possible influence of the environment at any point of the model. In this article, it is proposed to use the spectral method based on the expansion of the temperature function in a Fourier series to analyze the thermal model distributed along the axial coordinate presented in PDE form. To maintain the similarity of thermal processes and the model, the dimension parameters of the model should be chosen such that the Biot and Fourier coefficients would be the same for the model and the machine part. The proposed method allows the PDE to be represented as an indefinite system of linear algebraic equations for the coefficients of the Fourier series, which are the amplitudes of the space–time modes of the temperature function. The solution has the advantage of an analytical solution because it provides information about the model’s temperature at any point.