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Chaos in structural mechanics by Jan Awrejcewicz, Vadim Anatolevich Krys'ko

By Jan Awrejcewicz, Vadim Anatolevich Krys'ko

This quantity introduces and studies novel theoretical techniques to modeling strongly nonlinear behaviour of both person or interacting structural mechanical devices akin to beams, plates and shells or composite structures thereof.

The method attracts upon the well-established fields of bifurcation conception and chaos and emphasizes the thought of keep watch over and balance of gadgets and platforms the evolution of that's ruled via nonlinear usual and partial differential equations. Computational tools, specifically the Bubnov-Galerkin approach, are therefore defined intimately.

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1, has P–v dependence. It is possible then to determine Pcr value, but the curve defining the new equilibrium positions remains uninspected for P > Pcr . The essential feature of the Euler problem formulation is the introduction of an ideal system realization. For instance, after analysis of deflection a starting straight line of a rod is assumed and a compressing force should then be perfectly imposed along that rod’s axis. , by applying a static method, some real features (imperfections, inaccuracies, perturbations) such as initial deflections, initial non-ideal application of loads or any other additional external forces are introduced at the very beginning.

The third variant of the static method is based on the Lagrange theorem focusing on the conservation principle of minimums of the total potential energy of a system: Being in an equilibrium position, the total potential energy of a conservative system has some stationary value, but the position will be stable if and only if a minimum energy can be assigned for it [335]. A well known pictorial illustration of such an approach is the behavioral analysis of a ball lying on a smooth surface (Fig. 6).

We will assume a case of variation of function u. In the time interval (t0 − t1 ) the following relation holds: t1 t1 δu Kdt = Ω t0 t0 2hρ ∂ u δ g ∂t ∂u ∂t ds dt. 37) As a result of integration by parts one obtains t1 δu Kdt = Ω t1 t0 − t0 2hρ g Ω ∂u δ (u) ∂t t1 ds t0 2hρ ∂ 2 u δ (u) ds dt. 39) t1 ds t0 2hρ ∂ 2 w δ (w) ds dt. 41) where ε denotes the damping coefficient of a surrounding medium. Substituting all formulae previous to Eq. 43), which states the counterpart to the Hamilton principle, the following variational equations are obtained: 26 1 Theory of Non-homogeneous Shells t1 ∂ T11 ∂ T12 2hρ ∂ 2 u + + px − δ (u) ∂x ∂y g ∂ t2 Ω t0 ∂ T22 ∂ T12 2hρ ∂ 2 v + + + py − δ (v) ∂y ∂x g ∂ t2 2h3 E 3(1 − µ 2 ) + ∂ 2 w ∂ 2 (·) ∂ 2 w ∂ 2 (·) ∂ 2 w ∂ 2 (·) + 2 (1 − µ ) + 2 2 2 2 ∂x ∂x ∂y ∂y ∂ x∂ y ∂ x∂ y +µ ∂ 2 w ∂ 2 (·) ∂ 2 w ∂ 2 (·) + 2 − ∇2k F − L (w, F) ∂ x2 ∂ y2 ∂ y ∂ x2 + q− 2hρ g + ∂ 2w ∂w +ε ∂ t2 ∂t ∂ 2F ∂ 2F µ − ∂ y2 ∂ x2 a1 2h + 2 (1 + µ ) ∂ 2 (·) ∂ 2F ∂ 2F µ + − ∂ y2 ∂ x2 ∂ y2 ∂ 2 (·) ∂ x2 ∂ 2 F ∂ 2 (·) 1 + ∇2k w + L (w, w) δ (F) ∂ x∂ y ∂ x∂ y 2 ds dt t1 a − ε11 t0 0 + T22 + ε22 t0 0 T11 + Ω ∂ ∂ ε11 ∂ ε12 (δ F) − (δ F) + (δ F) + T22 (δ v) + T12 (δ u) ∂y ∂y ∂y ∂w ∂w + T12 δ (w) ∂y ∂x t1 b − δ (w) dx dt 0 ∂ ∂ ε22 ∂ (δ F) − (δ F) + ε12 (δ F) + T11 (δ u) + T12 (δ v) ∂x ∂x ∂y ∂w ∂w + T12 δ (w) ∂x ∂y 2hρ g b a dy dt 0 ∂u ∂v ∂w δ (u) + δ (v) + δ (w) ∂t ∂t ∂t t1 ds = 0.

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