Quasi-periodic structures have been widely studied for its atom dynamics, Photonic, magnetic and electronic wave propagation(Ashraff and Stinchcombe (1989), Vekilov et al. (1999), Szallas and Jagannathan (2008)). These structures exhibit frequency ranges in which no propagative wave exist(Bayindir et al. (2001) and Engel et al. (2007)). Band gap can lead to interesting applications in various domain (Florescu et al. (2009)). The recent progress in the additive manufacturing open new possibilities for using quasi-periodic structures by allowing the printing of complex meta-materials, in a consistent manner. Additive manufactured meta material can be designed to exhibit unusual macroscopic behavior due to their internal structure as in (Bückmann et al. (2012) and Claeys et al. (2014)). Therefore the possibility of creating meta materials having the same properties than the quasi periodic atomic structures can be highly interesting and allow to get rid of unwanted complex behaviors (Engel et al. (2007)). Such meta material could create band gap in their virbational mechanical response while being isotropic. 

The mechanical and vibrational properties of quasi-periodic and of amorphous structures are related to complex mathematical problems due to the impossibility of periodic simplifications. Therefore, in order to solve these problems, big size matrix problems have to be dealed with. In this paper a beam structure based on a octohedric quasi-periodic Pensore lattice approximant is studied. The octohedric Penrose tilling is chosen for his ability to create a periodic approximant to the quasi periodic tilling thus allowing the use of periodic boundary conditions as suggested in (Duneau (1989)).It has been shown in Sørensen et al. (1991) that, for ferromagnetic properties, the approximant with periodic boundary conditions closely mimic the in finite lattice properties. Numerical methods for the vibrational study of big systems have been applied from the atomic vibration domain up to finite elements modelization. Kernel Polynomial Method (KPM) is used herein to calculate the Vibrational Density Of States (VDOS) and the Dynamical Structure Factor (DSF) without exact diagonalization of the dynamical matrix. The KPM method is detailled by Lin et al. (2016) and was adapted to the study of to the vibrational properties of large-size systems by Beltukov et al. (2016b). These methods allow better understanding of vibrational response of quasi-periodic structures.