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 structures. a huge a part of the publication is devoted to the research of non-homogeneous continua, e.g. plates and shells characterised by means of unexpected alterations of their thickness, owning holes of their our bodies or/and edges, made of diversified fabrics with various dynamical features and intricate boundary stipulations. New theoretical and numerical techniques for interpreting the dynamics of such continua are awarded, equivalent to the strategy of additional plenty and the strategy of right orthogonal decomposition. The awarded hybrid method results in effects that can't be got via different average theories within the box. The verified equipment are illustrated through various examples of program.
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Extra info for Nonlinear Dynamics of Continuous Elastic Systems
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 . 97) ∂ ∂2u ∂3w 2 ∂ δ(x−x ) δ(y−y )−z δ(x−xi ) × i i i ∂x ∂t2 ∂x ∂x∂t2 ∂ ∂2v ∂ ∂3w δ(y − yi ) δ(x − xi ) − zi2 δ(y − yi ) × 2 ∂y ∂t ∂y ∂y∂t2 ∂3w ∂ 2w i ∂ δ(x − xi )δ(y − yi ) − Jxx δ(y − yi ) δ(x − xi )− 2 ∂t ∂y ∂y∂t2 ∂ ∂3w ∂3w i ∂ δ(x − x ) δ(y − y ) − J δ(y − yi ) δ(x − xi )− i i xy ∂x ∂x∂t2 ∂y ∂x∂t2 i Jxy ∂ ∂3w δ(x − xi ) δ(y − yi ) .
M ,n L"" m. 2. 142). 3 Analysis of Orthotropic Shallow Shells with Added Masses 51 input parameters and the frequency ω. e. occurs as a division of two polynomials of diﬀerent degrees. 142) can not be found in the closed form, approximate methods will be used in what follows. 5 Algorithms and Numerical Results In the previous chapters a general approach, using generalized functions to analyse the vibrations of plates and shallow shells, has been presented.
5) sin αm xi cos βn yi , i = 1, N . 142) obtained from the 7N dimensional and 5N dimensional determinants, correspondingly. The elements of the determinant are inﬁnite series of function dependent on all II I p i Al I 1 A2 d(01, n) P 21 I P 21 nJ I nJ 1 A2 ~(m, 12 p i Al m. , ~(m,