180 M. Krifa et al. The present study employs this approach to calculate the loss factor associated with the localized linear interfaces of a globally linear structure. In practice, the response of the system is not calculated explicitly due to the high computational burden. When assuming a linear and dissipative problem of assembled structures, the discrete form of the damped vibration problem may be governed by the equation: MRy.t/ CC Py.t/ CKy.t/ Df.t/ (17.1) where K, M and C are respectively the stiffness, mass and damping matrices, f is the vector of the external loads. The objective of the energetic method is to determine the damping factor corresponding to each vibration mode of the structure. It is based on the concept of the dissipated energy in the interfaces for which the close form expression of the loss factor is the ratio between dissipated energy and maximal potential energy, over a cycle of periodic vibration. as shown in this relation: D 1 4 Ediss E pot I D1;2;::::;n (17.2) where Ediss and E pot are respectively the dissipated energy and maximal potential energy. In the following we will present the necessary formulas for calculating these energies. The estimation of the dissipated and maximal potential energies requires an accurate characterization of the response levels of the system and the latter remain an approximation since they depends a priori on a knowledge of the different dissipation mechanisms. So, dissipated energy is calculated by the following expression: E diss D T Z 0 Py.t/Tfc.t/dt (17.3) Where • T D2 ! the cycle of periodic vibration • fc.t/ DC Py.t/: dissipated force • Py.t/ DRe.j! y.! /e j! t / the velocity of harmonic response Potential energy is calculated as follow: E pot D 1 2f y.w /g TKfy.w /g (17.4) where y.w / is the frequency response of the system. We distinguish two cases: proportional damping and localized damping. In the first case the frequency response of the system is equal to y.w / D n X D1 q (17.5) If the response is projected on a single mode of vibration then Eq. (17.5) becomes y.! / D q (17.6) where and q are respectively the eigenmode and modal amplitude corresponding to eigenfrequency ! . The exact frequency response is equal to: y.!/ D.KCj!C !2M/ 1 f (17.7) In the case of proportional damping, assuming that the eigenmodes are orthonormalized with respect to the mass matrix M, i.e. TM k l Dıkl, Eqs. (17.4) and (17.5) allow the total strain energy can be derived in the form: E pot D 1 2 n X D1 !2 q 2 (17.8)
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