By Prof. Dr. Jan Awrejcewicz, Prof. Vadim A. Krys’ko, Prof. Alexander F. Vakakis (auth.)
This monograph is dedicated to contemporary advances in nonlinear dynamics of constant elastic platforms. an immense a part of the e-book is devoted to the research of non-homogeneous continua, e.g. plates and shells characterised by way of unexpected adjustments of their thickness, owning holes of their our bodies or/and edges, made up of diverse fabrics with diversified dynamical features and complex boundary stipulations. New theoretical and numerical ways for studying the dynamics of such continua are awarded, similar to the tactic of extra plenty and the tactic of right orthogonal decomposition. The provided hybrid method results in effects that can not be bought through different common theories within the box. The confirmed tools are illustrated through various examples of program.
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Extra info for Nonlinear Dynamics of Continuous Elastic Systems
90) The above equations deﬁne four independent functions w, γx , γy and F . An additional equation is obtained using the continuous deformation equation ∂ 2w ∂2w ∂ 2 ε22 ∂ 2 ε12 ∂ 2 ε11 − k1 2 − k2 2 = 0. 92) ∂ 2 (•) ∂ 2 (•) + k , 2 ∂y 2 ∂x2 ∂ 4 (•) ∂ 4 (•) ∂ 4 (•) + (a1122 + a2211 − a1212 ) 2 2 + a1111 . 85). Finally, we consider a second problem related to the vibrations of a shallow shell with added masses in the frame of the Kirchhoﬀ-Love theory. 94) ∂x ∂y which means that ε13 = ε23 = 0. In addition, we assume that, u i = u − zi ∂w , ∂x vi = v − zi ∂w , ∂y wi = w, ω3i = 0, i = 1, N .
Free support The other boundary conditions for the Kirchhoﬀ-Love model are given in the monograph . 3 Added Masses Stiﬀness The considerations described so far, are idealized, and can be used if the added masses stiﬀness is higher than the shell’s stiﬀness. Otherwise, the added masses stiﬀness should also be taken into account. Below we consider one possible variant of the discussed problem, where the shell carrying masses have variable thickness and stiﬀness characteristics. The application of the theory of generalized functions allows for eﬀective solutions to the corresponding vibration problem.
137), and Δ1 is the co-factor of the determinant’s element situated on the crossing point of the ﬁrst row and the ﬁrst column, and cik ˜j1 sin βn yi sin βn c˜j2 × mn = sin αm xj sin αm c ci1 c˜i2 , × sin αm xi sin βn yi ] π 2 mn˜ i, j = 1, N . 146) by an equivalent matrix with the polynomial elements, and then its reduction to an algebraic equation is not an eﬀective procedure due to of accuracy requirements. 146), there are imaginary roots of the characteristic equation, which is in contradiction to physical interpretation.