## 2 Experimental Methods
### Temperature Control  
The objective of the extended IET on beams and plates is to automatically measure the IRF at specified temperatures and programmed time intervals. It is essential that the temperature distribution within the sample remains homogeneous across all temperature steps. Heat conduction in composite material is a complex phenomenon. In a review article by [PERSON] et al. [45], different existing analytical solutions for heat conduction in multi-layer and composite materials were reviewed and classified in rectangular, cylindrical, spherical, and conical coordinates. This review highlights that despite decades of research, new studies continue to emerge, offering deeper insights into heat conduction in multi-layer composites. This emphasizes the challenges and shortcomings in current research, guiding future investigations. In this study, a challenge arises in transitioning the isothermal temperature distribution of a freely suspended sample from one value to another, as this process requires time. The transient heat conduction in a thin plate is influenced by the convection occurring at the surfaces. The temperature profile varies over time at different internal positions, with the surface temperature changing relatively quickly, while the temperature at the sample's mid-plane changes more slowly. Key control parameters include the convection heat transfer coefficient \(h\), the thermal conductivity \(k\), the specific heat capacity \(Cp\), and the density \(\rho\) of the composite material of the sample [46]. The practical question is estimating the duration required to transition from an isothermal state at an initial temperature (\(T\)i) to an isothermal state at another temperature (\(T\)). The time evolution of the temperature \(T\) at the center of a sample exposed to a surface temperature \(T_{S}\) is described by Formula (7):  
\[T=T_{S}+(T_{i}-T_{S})Ae^{-\frac{4h\omega^{2}}{D^{2}}t} \tag{8}\]  
Figure 9: Position for suspension wires and excitation position (red stars) for the flexural vibration of a beam and the torsion and breathing vibration modes of a Poisson plate. In (8), \(T\) represents the temperature at the center of the sample, \(t\) is time, \(D\) is the sample's thickness, \(T_{S}\) is the surface temperature, and \(\alpha\) is the thermal diffusivity. Coefficients \(A\) and \(\lambda\) are functions of the Biot number \(Bi\):  
\[Bi=\frac{hD}{2k} \tag{9}\]  
\[\alpha=\frac{k}{\rho Cp} \tag{10}\]  
The coefficients \(A\) and \(\lambda\) for specific Biot number values can be found in tables or computed using numerical finite element models [46]. The Resonyser software incorporates such thermal numerical models. When the value of center temperature T approaches TS, the sample can be considered isothermal. The resonance frequencies and damping ratios of the beams and plate can be measured by IET at the subsequent isothermal states. After measuring and identifying the engineering constants at this state, a new temperature step can be set. Before setting the temperature steps and timing values for the automated procedure, the delay time (see Figure 10) required to reach an isothermal state at different steps must be computed based on the assumed known thermal parameters of the sample. For given thermal properties of the test samples, the temperature in the climate chamber at different temperature steps and the time durations can be set by the automated control program. Figure 10 illustrates a simple temperature and time-stepping program starting at an isothermal state with temperature TA and progressing towards temperature TB. Theoretically, it takes an infinite amount of time to reach the new isothermal state. Therefore, a more elaborate approach with an overshoot temperature step is convenient (see Figure 11). Figure 10: A simple temperature and time-stepping program starting at an isothermal state with temperature TA towards temperature TB. An IET measurement is executed at the end of each temperature step (red circle).