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. a tremendous a part of the e-book is devoted to the research of non-homogeneous continua, e.g. plates and shells characterised through surprising alterations of their thickness, owning holes of their our bodies or/and edges, made of assorted fabrics with different dynamical features and intricate boundary stipulations. New theoretical and numerical methods for studying the dynamics of such continua are offered, comparable to the strategy of extra plenty and the tactic of right orthogonal decomposition. The offered hybrid method results in effects that can not be acquired through different typical theories within the box. The validated equipment are illustrated via various examples of program.
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Additional info for Nonlinear Dynamics of Continuous Elastic Systems
82) where: u0 , v0 , w0 , γx0 , γy0 deﬁne displacement and rotation of the shell’s edge. In real constructions with shell elements diﬀerent supports are possible and therefore diﬀerent mathematical models can be formulated. Let us consider certain boundary conditions for the shell’s sides x = 0, x = a. 3 Analysis of Orthotropic Shallow Shells with Added Masses 29 1. Roller support w = γy = M1 = T1 = v = 0. 83) w = γy = M1 = T1 = S = 0. 84) u = v = w = M1 = γy = 0. 85) 2. Free support 3. Roller clamping The above mentioned boundary conditions do not exhaust all possibilities [117, 185].
1 m,D 8(m , n) '" L '" ... , 00 L m,D8(m. (m, n) 00 '" ... p" , . , "" L m,' -Xl L • 1... • ... ... (m, n) IX) L m. 1. 131). 3 Analysis of Orthotropic Shallow Shells with Added Masses 47 w = A sin αm x sin βn y sin ωmn t, γx = B cos αm x sin βn y sin ωmn t, γy = C sin αm x cos βn y sin ωmn t, F = D sin αm x sin βn y sin ωmn t. 83). 133) an algebraic system of linear equations with the unknowns A, B, C, D is obtained. Equalling its determinant to zero, we get the following characteristic equation 6 4 2 r1 ωmn + r2 ωmn + r3 ωmn + r4 = 0.
137) e44 = a1111 λ2 βn4 + 2 α4m + (a1122 + a2211 − a1212 ) α2m βn2 . 137) give three frequency spectra of free vibrations. For example, the relationship leading to the determination of free vibration frequencies of an orthotropic and rollerly supported shallow shell is as follows 2 ωmn = k2 βn2 1 A1111 4 α + 2 (A1122 + 2A1212 ) α2m βn2 + A2222 βn4 λ2 + k1 α2m + 12 λ2 m a2222 4 α + (a1122 + a2211 − a1212 ) α2m βn2 + a1111 βn4 λ2 . 138) it is concluded, that the correction brought by taking into consideration the tangential displacements depends on the shell dimensions, its stiﬀness characteristics and on have number of vibrationsin x and y directions.