# Investigation of the Mechanical Properties of Spur Involute Gearing by Infrared Thermography
## 2 Thermoelastic Analysis and Lock-In Method  
As already mentioned, the mechanical properties of non-metallic wheels are important in deciding whether these wheels will replace the original metal wheels. Methods are developed to model the wheel footprint and to learn the strain and stress fields. It is useful if the results of computational simulations are compared with the results of experimental measurements. These measurements use, for example, strain gauges installed in the tooth root [16], elasticity-tymetry [17,18] or the PhotoStress method [19]. Thermoelastic measurements are performed at the workplace of the authors of the paper, for example [20]. Therefore, the thermoelastic analysis method was chosen to determine the stress field on the loaded wheel tooth or in its surroundings. Let us describe the thermoelastic analysis as follows. When a material is loaded in compression or tension in the elastic region, there is a reversible conversion between mechanical and thermal energy. This method of loading causes a change in temperature. In adiabatic action, there is a linear relationship between the change in temperature and the change in the first invariant of the mechanical stress and it is independent of the loading frequency. After adjustment, we can use the relationship in the thermoelastic region [21]:  
\[\Delta T=-\frac{\alpha}{\rho c_{p}}T_{0}\Delta\sigma_{ii}\;for\;i=1,2 \tag{1}\]  
where:  
\(\alpha\)--linear coefficient of thermal expansion of the material,  
\(c_{p}\)--heat capacity under constant pressure,  
\(\rho\)--density,  
\(\sigma_{ii}\)--stress tensor and \(\Delta\sigma_{ii}\) is the change of the 1 st invariant of the tensor,  
\(\Delta T\)--change of temperature,  
\(T_{0}\)--absolute temperature. \begin{table}
\begin{tabular}{c c} \hline
**3D Printing Technology** & **Tolerance** \\ \hline SLS & 0.06 mm \\ FDM & 0.25 mm \\ Polylet & 0.016 mm \\ \hline \end{tabular}
\end{table} 
Table 1: 3D printing technology and its manufacturing tolerances [15]. The detected temperature change (more precisely the radiation incident on the camera detector) is very small and determining its value by direct measurement is almost impossible. In order to measure the induced thermal modulations generated by the input pulse, they must exceed the noise level of the camera. This requires high excitation pulse energy [22]. Lock-in thermography, also known as thermal wave imaging technique, is offered as an alternative technique because it can be described by oscillatory temperature wave theory [23; 24; 25; 26]. The excitation of heat in the body occurs periodically with a certain frequency [22]. Using the lock-in method it is possible to increase the signal-to-noise ratio many times [27; 28; 29]. This takes advantage of the fact that if the excitation signal (changing wheel load) is harmonic with period \(f_{lock-in}\) then also the response of the object to this load is harmonic with the same frequency \(f_{lock-in}\), and hence the energy change that is detected by the camera will be periodic. If we perform a Fourier transform for the registered temperature (or radiation) values and express the result in the form of a series whose lowest frequency is \(f_{lock-in}\), then the amplitude of the series member for the frequency \(f_{lock-in}\) will correspond to the detected temperature change \(\Delta T\). Let the excitation source operate at the excitation frequency \(f_{lock-in}\). It produces a response, which we express by a harmonic function \(s(t)\), with amplitude \(A\), with the same frequency \(f_{lock-in}\) and with phase shift \(\varphi\):  
\[s(t)=Asin(2\pi f_{lock-in}t+\varphi) \tag{2}\]  
Let there be in the signal \(S(t)\), that is detected by the camera is included in the response \(s(t)\) and noise \(N(t)\) in the form of:  
\[S(t)=s(t)+N(t), \tag{3}\]  
\(S(t)\) we express using Fourier series as:  
\[