## 2 Acoustic Liner Design
### Vibration Design of Flexible Walls  
Analogously to a panel sound absorber, it is assumed that flexible walls inside each cavity improve the sound absorption of the acoustic liner due to the acoustic-structural interaction. Therefore, one or more walls of the honeycomb structure comprise rectangular cut-outs which are covered by flexible polymer films that exhibit significant material intrinsic damping (see Table 1).  
To approximate the damping characteristic of the novel HR liner concept, the flexible walls inside each cavity are considered as free plates without considering the impedances of the enclosed air volume. The films were modelled as three-dimensionally deformable body clamped at all sides. The size of the flexible walls is restricted by the size of the liner cavities (Figure 2b). To ensure a reliable bonding of the polymer films, sufficient adhesive surface is needed. The frame width on each side of the cut-out was \(b_{y}=b_{z}=2\) mm, which leads to cut-out with the dimension \(a_{y}=15\) mm and \(a_{z}=26\) mm. These lengths represent the size of the freely vibrating flexible wall. For the calculation of the eigenfrequencies, constant elastic properties and material damping of the polymer films are assumed (Table 1). For the investigation of the influence of the pre-stress, numerical modal analysis of the film's vibration behaviour was calculated by using the finite element method. Linear brick elements with reduced integration were used for discretisation. Linear mesh convergence studies (not reported here) have been done and indicate that the used mesh size yields acceptably accurate results for most of the values of interest. The film's sides were fixed in all directions.  
The coordinate system--which is used to describe the lateral deflection of the flexible wall--is depicted in Figure 2. The axes are parallel to the sides \(a_{y}\) and \(a_{z}\) of the cut-outs. The frequencies' modes \(m\) and \(n\) are associated to the \(y\)- and \(z\)-direction. To model the pre-stress of the flexible films, an additional load resulting in a constant deformation in y-direction was included. With an increasing uniaxial stress state the film thickness decreases due transversal contraction. The resulting modal  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties** & **Unit** & **TPU 1170 A** & **TPU 1195 A** \\ \hline Density \(\rho\) & g/cm\({}^{3}\) & 1.08 [26] & 1.15 [26] \\ Young’s modulus & MPa & 16 [1] & 91 [1] \\ Mechanical loss & - & 0.05 [27] & 0.13 [28] \\ coefficient \(tan\ \delta\) & - & 0.48 [29] & 0.48 [29] \\ Poisson’s ratio & - & 0.5 [30] & 0.4 \(\ldots\) 0.7 [29] \\ Moisture absorption & \% & 0.5 [30] & 0.4 \(\ldots\) 0.7 [29] \\ Nominal thickness \(t_{F}\) & mm & 0.3 & 0.1, 0.5 \\ \hline \hline \end{tabular} \({}^{1}\) Calculated with dynamic shear storage modulus, \({}^{*}\) Material property of TPU 1175 A.  
\end{table}
Table 1: Properties of polymer films under standard atmospheric conditions (i.e., 23 \({}^{\circ}\)C, 50% relative humidity).  
analysis shows increasing natural frequencies with an increasing pre-stress (Figure 4). The design point for the acoustic flow duct is the first eigenfrequency \(f_{11}\), which must be in the range up to 2000 Hz. The number of natural frequencies in the relevant range is very low for the sample 1195 A, \(t_{\rm F}=0.5\) mm, which indicates that film press stress is relevant (Figure 4).