## 2 Mathematical Model of the System  
The mathematical model of transient heat transfer is based on a boundary value problem with some restrictions:  
1. The spindle bearings are the source of the most active heat generation and heat supply to the spindle. They are considered in the model as boundary conditions of the second kind.
2. Heat radiation to the surrounding is ignored.
3. The temperature depends only on the axial spatial variable. It is possible to expand the model and take into account the dependence of temperature on the radial coordinate [22]. In this case, the mathematical description of the complex shape is represented as the system of PDEs with the corresponding boundary conditions. This restriction does not limit the proposed analysis method but greatly simplifies calculations and allows a graphic result to be obtained.  
To ensure the similarity of thermal processes and the model under the third assumption, the dimension parameters of the model should be chosen such that the Biot and Fourier coefficients are the same for the model and the structure [23].  
Thermal model length and spindle length are equal. The following expression allows calculation of the diameter of the equivalent cylinder used in the model [23]:  
\[d=\frac{\sum\limits_{i=1}^{k}D_{i}^{2}l_{Di}-\sum\limits_{j=1}^{m}d_{j}^{2}l_{ dj}}{\sum\limits_{i=1}^{k}D_{i}l_{Di}+\sum\limits_{j=1}^{m}d_{j}l_{dj}}\, \tag{1}\]  
where \(k\) is the number of spindle sections of the same outer diameter, \(m\) is the number of spindle sections of the same inner diameter, \(D_{i}\) is the outer diameter of the \(i\)-th section, \(d_{j}\) is the inner diameter of the \(j\)-th section, \(l_{Di}\), \(l_{dj}\) is the section length with the same inner or outer diameter.  
One of the possible spindle configurations is shown in Figure 1. In the case of \(D_{1}=0.108\) m, \(D_{2}=0.145\) m, \(D_{3}=0.235\) m, \(D_{4}=0.195\) m, \(l_{D1}=0.022\) m, \(l_{D2}=0.063\) m, \(l_{D3}=0.045\) m, \(l_{D4}=0.370\) m, \(d_{1}=0.038\) m, \(l_{1}=0.500\) m, the calculations by Expression (1) give \(d=0.150\) m.  
Figure 1: Configuration and geometry dimensions of the spindle.  
Under the mentioned restrictions, the mathematical model of heat distribution is a heat-conduction equation [23]:  
\[\begin{array}{c}\frac{\partial T(x,t)}{\partial t}=a\frac{\partial^{2}T(x,t)}{ \partial x^{2}}-h(t)T(x,t)\\ h(t)=\frac{4\alpha(t)}{\lambda d}a,\;\mathrm{x}\in(0,L),\;\mathrm{t}\in(0,t^{*} )\end{array} \tag{2}\]  
with boundary conditions of the second kind:  
\[\left.\frac{\partial T(x,t)}{\partial x}\right|_{x=0}=-\frac{q_{1}(t)}{F \lambda},\left.\frac{\partial T(x,t)}{\partial x}\right|_{x=L}=\frac{q_{2}(t) }{F\lambda},\;\mathrm{t}\in[0,t^{*}]\,, \tag{3}\]  
and the initial condition:  
\[T(x,t)\big{|}_{t=0}=0,\;\mathrm{x}\in[0,L]\,. \tag{4}\]  
where \(T\) is the temperature, \(t\) is the time, \(x\) is the spatial variable, \(L\) is the length of the model, \(a\) is the thermal diffusivity, \(\alpha(t)\) is the time-dependent heat-transfer coefficient, \(\lambda\) is the thermal conductivity, \(q_{1},q_{2}\) are heat flux in the rear and front supports, \(F\) is the cross-sectional area of a cylinder, \(d\) is the diameter of the model.  
The representation of the Equations (2)-(4) in the dimensionless form:  
\[\begin{array}{c}\frac{\partial\theta(\xi,\tau)}{\partial\tau}=\frac{a\cdot t ^{*}}{L^{2}}\frac{\partial^{2}\theta(\xi,\tau)}{\partial\xi^{2}}-k(\tau)\theta (\xi,\tau)\,,\\ \xi\in(