# Passive Multi-Layer Composite Damper of Flat Belt Tensioner Idler
## 3 Results
Subsequently, homogenized \(E_{\text{pd}}\) was determined using the following relationship:  
\[E_{\text{pd}}=\frac{Fl^{3}}{3w_{\text{pd}}I} \tag{3}\]  
When comparing the bending stiffness (_E.I_ where \(I\) is the area moment of inertia) of passive dampers 2 and 3 with the same mass and damping effect, they differed by only 3%. Stiffness was not an advantage for either passive damper 2 or 3. To determine the material damping contribution of passive dampers 1, 2, and 3, the dependences of the RMS amplitude at resonance with respect to mass (Figure 7) were determined. Passive damper 2 had the minimum mass, i.e., \(m_{\text{pd}}=m_{0}+m_{2}=19.7\) g + 16.8 g = 35.7 g. If passive dampers 1 and 3 had the same mass as passive damper 2, their RMS amplitude would have been 25.5 and 28.6 g, respectively, meaning they would reduce the maximum amplitude at resonance by 23.0% and 13.4%, respectively. Moreover, if mass was equal, the damping contribution of passive damper 2 would be 5.7% and 15.3% higher than passive dampers 1 and 3, respectively. Values 15.3% and 5.7% express larger material damping of passive damper 2 compared to passive dampers 1 and 3, respectively. Thus, the flexible rubber foam and solid rubber layers (passive damper 3) provided the lowest damping per unit of mass and an unwanted sharp resonance peak. Compared with passive damper 3, the advantage of passive damper 2 is the material itself, as the stiffness of both was similar and even the mass of damper 2 was 56% lower. The vibration of the damped driving flat belt caused variability in the pressing force when driving the roller bearing. As a result, the changes in short-term rotational speed are fundamental to the bearing life, as they cause deceleration and acceleration of the rotation of the inner race and the rolling elements. Thus, the additional stress on the bearing cage appears, the kinematic slip of rolling elements increases, and additional torsional oscillations are generated. The average rotational speeds using a non-damped steel plate and a steel plate with passive damper 2 reached a maximum of 130,200 and 129,700 rpm, respectively, and the maximum change of rotational speeds was 410 and 380 rpm, respectively (a 7% decrease in the maximum change for the damped steel plate) (Figure 8). The obtained results are summarized in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Undamped** & **Type 1** & **Type 2** & **Type 3** \\ \hline Additional mass (grams) & - & 16.8 & 16.0 & 36.3 \\ \hline Mass of passive damper \(m_{\text{pd}}=m_{0}+m_{\text{i}}\) &
\begin{tabular}{c} \(m_{0}=19.7\) g \\ (mass of steel plate) \\ \end{tabular} & 36.5 & 35.7 & 56.0 \\ \hline RMS amplitude at resonance (g PTP) & 33.1 & 25.1 & 23.6 & 23.5 \\ \hline Reduction in RMS amplitude at resonance & - & 24.2\% & 28.7\% & 29.0\% \\ \hline Maximum amplitude (g Peak) & 50 & 30 & 27 & 26 \\ \hline Maximum amplitude reduction & - & 40\% & 46\% & 48\% \\ \hline Resonance peak shape & Undamped sharp & Damped flat & Damped flat & Damped sharp \\ \hline Frequency peaks (kHz) Sidebands & 7, 9, 13, 17, 22, 24 & suppressed & suppressed & suppressed \\ \hline Amplitude reduction efficiency \(E_{\text{a}}\) (-) & - & 9.4 & 11.1 & 5.22 \\ \hline Transmission coefficient \(T\) (-) & 2.00 & 1.14 & 1.01 & 1.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of results. The values of the amplitude reduction efficiency \(E_{\text{a}}\) in Table 3 were evaluated according to [19]  
\[E_{\text{a}}=\frac{A}{m_{\text{a}}} \tag{4}\]  
where \(A\) is the amplitude reduction relative to the undamped structure and \(m_{\text{a}}\) is the additional mass of the damper as a proportion of the native mass of the structure. The most efficient amplitude reduction was achieved with passive damper 2. The performance of one layer of a honeycomb-cored sandwich beam as a viscoelastic damper in [19] provided similar effects.