## 3 Acoustic Analysis  
A BEM analysis was carried out to provide the acoustic emission of the chain cover. It is important to underline that the acoustic calculation evaluates the noise emission solely due to structurally transmitted excitation vibrations (structural born noise). The airborne noise and that emitted by other sources is not taken into account in any way.  
The BEM analysis was performed, resorting to the indirect BEM [18] (quantities at the surface are jump of pressure and jump of normal velocity) with a variational approach for the numerical resolution [19; 20].  
The acoustic calculation is performed on a modal basis and involves different stages [17]:  
* creation of a BEM mesh;
* calculation of the ATVs;
* introduction of experimentally-measured vibration velocities on the BEM mesh;
* calculation of the sound pressure and sound power.  
As mentioned, the BEM mesh represents the mesh of the component surface.  
The ATVs [21] are transfer functions allowing the calculation of the sound pressure in the chain cover surrounding environment, starting from the structure vibration.  
The ATVs depend on the geometry and impedance of the vibrating surface, the microphones position, the frequency, and the physical properties of the domain in which the sound propagates (temperature and density).  
The calculation of the ATVs is independent from the vibration result, and consequently, it does not depend on the loading conditions and damping values used in the structural calculation.  
If the vibrating surface is divided into a discrete number of elements, the relationship between their normal velocities and the sound pressure in multiple domain points can be expressed as follows:  
\[\{p(\omega)\}=[ATV(\omega)]\{v_{n}(\omega)\} \tag{1}\]  
where \(\{p(\omega)\}\) is the column vector containing the acoustic pressure in the different points of interest, \([ATV(\omega)]\) is the transfer vectors matrix, and \(\{v_{n}(\omega)\}\) is the column vector containing the normal velocities of vibrating elements (as obtained by experimental measurements).  
In particular, ATV is the output of a BEM problem solution, and represents the transfer function relating the vibration velocity in a node of the structure to the sound measured in a point of the acoustic domain.  
The aforementioned computational processes represent a quite easy tool for the acoustic calculation. In fact, once the mesh of the component has been defined, it turns to the ATVs computation.  
The ATVs' independence from the structural modal response entails a remarkable advantage in terms of computational times reduction, since any further modification to the component--which does not require a variation of its outer surface (acoustic emission side)--can be processed without the need for an ex-novo ATVs computation.  
ATVs would only change if the external (emission side) chain cover design changed, whereas they would not be affected by structural changes on the remaining part of engine or changes in the excitations.  
The main interest of this work stands in the assessment of chain cover emission in order to optimize its shape. The idea is to apply ribs on the chain cover internal side in order to stiffen it and correspondingly reduce the acoustic emission versus variable engine regimes.  
The boundaries of the BE model enclose a domain; to avoid the presence of anomalous peaks in correspondence of the eigenfrequencies of such enclosed acoustic cavity, a specific impedance was assigned to the internal surface of the elements (and therefore to the material), equal to that of air--that is, 416.5 Rayl (kg\(\cdot\)s\({}^{-1}\cdot\)m\({}^{-2}\)).  
Having placed the microphone in the chain cover near field, it is reasonable to assume that the contribution from parts other than the chain cover to the recorded sound pressure level turns out to be negligible.