Tham khảo tài liệu 'computational mechanics of composite materials part 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Multiresolutional Analysis 345 Free Vibrations Analysis The main idea of homogenisation problem solution now is a separate calculation of the effective elastic modulus and spatial averaging of the mass density where the first part only needs multiresolutional approach 189 . The alternative wavelet-based methodology is presented in 328 329 for a plate wave propagation in 152 whereas some classical unidirectional examples are contained in 330 . Let us consider the following differential equilibrium equation -dI e x I x du x Ì M x dx I dx I x e 0 1 where e x defining material properties of the heterogeneous medium varies arbitrarily on many scales together with the inertia momentum I x . A multiresolutional homogenisation starts now from the following decomposition of the equilibrium equation dv x -M x dx d v x u x dx e x I x to determine the homogenised coefficient e eff constant over the interval x e 0 1 which takes the integral form u x u 0 x Ỷ v x 7 v 0 J 7 0 0 0 e í -11 t 0 1 i Ì Ì . v t - M t 7 7 7 dt On the other hand the reduction algorithm between multiple scales of the . . . f . . iejj i eff eff composite consists in determination of such effective tensors B A p and qef such that i B ef u x x q eff  i A eff u t A p eff .v x 7 2 J 0 I Iv t 7 7 dt It can be shown that where B eff 0 0 Ì A eff 0 C1 2C2 Ì 00 J 0 0 7 346 Computational Mechanics of Composite Materials C f dt . C 1 t - 2d 1 00 e t I t 2 0 e t I t Furthermore for f x 0 there holds p eff q ff 0 while in a general case B ff and A eef do not depend on p and q. Finally the homogenised ODEs are obtained as d x f -ị_u x C1-2C v x which is essentially different to the classical result of the asymptotic homogenisation shown previously. Effective mass density of a composite can be derived by a spatial averaging method which is completely independent from the space configuration and periodicity conditions of a composite structure. The relation is used for classical and .
