# Influence of Structural Stiffness and Loss Factor on Railroad Vehicle Comfort
## 5 Results  
The formulation in Section 2 allows flexible modes to be included. To assess the influence of structural stiffness, the beam cross-section area moment of inertia (\(I\)) will be allowed to vary within a range that will be specified in terms of the first bending frequency of the free-free beam. To be consistent with the numbering of modes in Equation (4), the first bending frequency is tagged \(f_{3}\). The relation between \(f_{3}\) and \(I\) is  
\[I=\frac{(f_{3}2\pi)^{2}m/l}{(E_{r}\beta_{3}/l)^{4}} \tag{19}\]  
where \(f_{3}\) is varied in the range from 6 Hz to 20 Hz, where the beam has been assumed to be made of aluminum, and thus \(E_{r}=70\,10^{9}\) N/m\({}^{2}\), and where \(\beta_{3}\approx 4.13/l\). The rest of the parameters take the same values as in the previous section (\(l=13\) m, \(v=300\) km/h, \(A_{v}=0.6\)\(\upmu\)m rad, \(m=13\),\(370\) kg, and \(f_{b}=1.3\) Hz), with \(\xi_{b}\) now fixed at \(\xi_{b}=0.05\). The material loss factor (\(\eta\)) has been set at a higher value (\(\eta=0.01\)) than that of aluminum to better account for typical structural damping (see Table 1). Figure 7 shows comfort indexes along the length of the beam versus bending frequency \(f_{3}\), whereas Figure 8 shows sections of this surface at four locations along the beam. It could be said that, as a general trend, comfort improves as the structure stiffens. Nevertheless, the statement cannot be taken for granted since the dependence of the comfort index  
Figure 5: Comfort index at four locations on the beam versus bounce frequency (\(f_{b}\)). Figure 6: Comfort index at four locations on the beam versus bounce damping factor (\(\xi_{b}\)). on stiffness is non-monotonic, thus proving the statement wrong every time the slope is positive in the \(c_{l}\) vs. \(f_{3}\) curve. This behavior makes the task of the designer particularly difficult. For example, stiffening the structure of the example under discussion beyond 9.3 Hz (\(f_{3}>9.3\) Hz, Figure 8) worsens comfort at a great rate (measured as the slope of \(c_{l}\) vs. \(f_{3}\) for a given \(x\)). In fact, comfort does not start to improve again until \(f_{3}\) reaches a value of approximately 15 Hz, and the improvement only happens for a short range of bending frequencies, starting to worsen again before \(f_{3}\) reaches 16 Hz. Despite the wavy evolution of comfort with stiffness, it is true that the values of the curves in Figure 8 lie above the corresponding values for the infinitely stiff beam analyzed in the previous section. This statement is supported by Figure 9 where the upper limit for the first bending frequency has been stretched beyond what is reasonable or even feasible (see Table 1). It can be seen that the non-monotonic comfort curves tend toward their corresponding rigid body asymptotes. These asymptotic values may be checked against those in Figure 5 for \(f_{b}=1.3\) Hz or against those in Figure 6 for \(\xi_{b}=0.05\). Figure 8: Comfort index at four locations on the beam versus the first bending frequency (\(f_{3}\)). Figure 7: Comfort index along the length of the beam versus the first bending frequency (\(f_{3}\)). The influence of structural damping is depicted in Figures 10 and 11. All parameters have the same values as in the previous analysis except for the bending frequency, which is now fixed at \(f_{3}=8\) Hz, and the previously fixed material loss factor \(\eta\), which is now free to vary from 0 to 10% (see Table 1). It can be seen that comfort improvements with an increasing loss factor are significant but not dramatic. These results indicate that railroad passenger car frames should have sufficient structural damping to improve comfort but that, at the same time, if the measures needed to increase the loss factor are too complicated, too costly, or too heavy (referring to the weight of the materials that would need to be added), the designer may pursue meeting specifications by other means rather than by increasing structural damping.