Triple-wave ensembles in a thin cylindrical shell — страница 2

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paradigm of the Kirhhoff-Love hypotheses. From the viewpoint of further mathematical rearrangements it is convenient to pass from the physical sought variables to the corresponding dimensionless displacements . Let the radius and the length of the shell be comparable values, i.e. , while the displacements be small enough, i.e. . Then the components of the deformation tensor can be written in the form where is the small parameter; ; and . The expression for the spatial density of the potential energy of the shell can be obtained using standard stress-straight relationships accordingly to the dynamical part of the Kirhhoff-Love hypotheses: where is the Young modulus; denotes the Poisson ratio; (the primes indicating the dimensionless variables have been omitted). Neglecting the

cross-section inertia of the shell, the density of kinetic energy reads where is the dimensionless time; is typical propagation velocity. Let the Lagrangian of the system be . By using the variational procedures of mechanics, one can obtain the following equations governing the nonlinear vibrations of the cylindrical shell (the Donnell model): (1) (2) Equations (1) and (2) are supplemented by the periodicity conditions Dispersion of linear waves At the linear subset of eqs.(1)-(2) describes a superposition of harmonic waves (3) Here is the vector of complex-valued wave amplitudes of the longitudinal, circumferential and bending component, respectively; is the phase, where are the natural frequencies depending upon two integer numbers, namely (number of half-waves in the

longitudinal direction) and (number of waves in the circumferential direction). The dispersion relation defining this dependence has the form (4) where In the general case this equation possesses three different roots () at fixed values of and . Graphically, these solutions are represented by a set of points occupied the three surfaces . Their intersections with a plane passing the axis of frequencies are given by fig.(1). Any natural frequency corresponds to the three-dimensional vector of amplitudes . The components of this vector should be proportional values, e.g. , where the ratios and are obeyed to the orthogonality conditions as . Assume that , then the linearized subset of eqs.(1)-(2) describes planar oscillations in a thin ring. The low-frequency branch corresponding

generally to bending waves is approximated by and , while the high-frequency azimuthal branch — and . The bending and azimuthal modes are uncoupled with the shear modes. The shear modes are polarized in the longitudinal direction and characterized by the exact dispersion relation . Consider now axisymmetric waves (as ). The axisymmetric shear waves are polarized by azimuth: , while the other two modes are uncoupled with the shear mode. These high- and low-frequency branches are defined by the following biquadratic equation . At the vicinity of the high-frequency branch is approximated by , while the low-frequency branch is given by . Let , then the high-frequency asymptotic be , while the low-frequency asymptotic: . When neglecting the in-plane inertia of elastic waves, the

governing equations (1)-(2) can be reduced to the following set (the Karman model): (5) Here and are the differential operators; denotes the Airy stress function defined by the relations , and , where , while , and stand for the components of the stress tensor. The linearized subset of eqs.(5), at , is represented by a single equation defining a single variable , whose solutions satisfy the following dispersion relation (6) Notice that the expression (6) is a good approximation of the low-frequency branch defined by (4). Evolution equations If , then the ansatz (3) to the eqs.(1)-(2) can lead at large times and spatial distances, , to a lack of the same order that the linearized solutions are themselves. To compensate this defect, let us suppose that the amplitudes be now the

slowly varying functions of independent coordinates , and , although the ansatz to the nonlinear governing equations conserves formally the same form (3): Obviously, both the slow and the fast spatio-temporal scales appear in the problem. The structure of the fast scales is fixed by the fast rotating phases (), while the dependence of amplitudes upon the slow variables is unknown. This dependence is defined by the evolution equations describing the slow spatio-temporal modulation of complex amplitudes. There are many routs to obtain the evolution equations. Let us consider a technique based on the Lagrangian variational procedure. We pass from the density of Lagrangian function to its average value (7), An advantage of the transform (7) is that the average Lagrangian depends only