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186 ALEKSANDR KORJAKIN ET AL. Damping Models ENERGY METHOD In this model a specific damping capacity (SDC) ψ is introduced as a parameter which characterises the damping properties of the material or structure [19]: ∆U ψ = 2πη= (39) U where η is the loss factor, ∆U is the strain energy dissipated during one cycle of vibration, and U is the total strain energy of the entire of sandwich shell at maximum of the displacement during the same cycle. Following the definition of the SDC, the total energy dissipation can be expressed by [19] 3 ∑ ∆ U = ∆U L= 1 ( L) (40) where ∆U (L) is the strain energy dissipated in the Lth layer ( L) 1 ZL T L ∆ U = { 0} [ ] { 0} 2 ∫∫ ε C dzdA A Z ψ ε L−1 (41) Here ⎡ L L L L ψ11C11 ψ12C12 0 0 0 ⎤ ⎢ ⎥ L L L L ⎢ψ21C21 ψ22C22 0 0 0 ⎥ L ⎢ [ ] L L ⎥ Cψ = ⎢ 0 0 ψ44C44 0 0 ⎥ ⎢ L L ⎥ ⎢ 0 0 0 ψ55C55 0 ⎥ ⎢ L L 0 0 0 0 66C ⎥ ⎣ ψ 66 ⎦ (42) and ψ L ij are the SDC of the corresponding layer [19]. WhereC L ij are the plane-stress reduced components of the Lth layer in the material axis (1,2,3) and {ε 0 } T = {ε 1 ,ε 2 ,ε 23 ,ε 13 ,ε 12 } are strain components in the same axis. Using the energy method it is assumed that the natural frequencies and modes of the damped system are the same as those of the associated undamped system. This expression is obtained for general case when damping matrix is not symmetrical. It is assumed that L L ψ12 = ψ21 in our algorithm. Solving the eigenproblem (38), the SDC to the nth mode can be calculated by ψ T ∆U ( n) H( n) ( n) = 2πη ( n) = = U T ( n) K( n) (43)
Free Damped Vibrations of Sandwich Shells of Revolution 187 where η (n) is the modal loss factor and H is the global damping matrix for shell. A detailed description of global damping matrix H is given in Reference [12]. NUMERICAL EXAMPLES In the examples the symmetry of the cross section is assumed. So we have the same isotropic material properties for the inner and outer layer, the core is made from orthotropic material. For the inner and outer layer characteristics the index t is used; for the core, c. Free Undamped Vibration of a Sandwich Conical Shell The first problem considered is a conical sandwich shell with clamped-clamped boundary conditions. The geometry of the shell is characterised by the following quantities: the thickness of outer layers h t = 0.000535 m and core h c = 0.00762 m, the radius of the cone at its small end R = 0.5702 m, the semi-vertex angle β = 5.07°, the meridional length of the cone L = 1.8415 m, Figure 6. The following properties of the material are assumed: outer layers from glass epoxy E t = 25.08 GPa, ν t = 0.20, p t = 2800 kg/m 3 and inner layer form aluminium honeycomb E1 c = 0.529 GPa,G12= c 0.2204 GPa,G13= c c c 0.126 GPa, ν = ν = 0.2, ρ c = 36.80 kg/m 3 . 12 13 Figure 6. The sandwich conical shell.
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