Triple-wave ensembles in a thin cylindrical shell

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Kovriguine DA, Potapov AI




Primitive nonlinear quasi-harmonic triple-wave patterns in a thin-walled cylindrical shell are investigated. This task is focused on the resonant properties of the system. The main idea is to trace the propagation of a quasi-harmonic signal - is the wave stable or not? The stability prediction is based on the iterative mathematical procedures. First, the lowest-order nonlinear approximation model is derived and tested. If the wave is unstable against small perturbations within this approximation, then the corresponding instability mechanism is fixed and classified. Otherwise, the higher-order iterations are continued up to obtaining some definite result.

The theory of thin-walled shells based on the Kirhhoff-Love hypotheses is used to obtain equations governing nonlinear oscillations in a shell. Then these equations are reduced to simplified mathematical models in the form of modulation equations describing nonlinear coupling between quasi-harmonic modes. Physically, the propagation velocity of any mechanical signal should not exceed the characteristic wave velocity inherent in the material of the plate. This restriction allows one to define three main types of elemental resonant ensembles - the triads of quasi-harmonic modes of the following kinds:

high-frequency longitudinal and two low-frequency bending waves (-type triads);

high-frequency shear and two low-frequency bending waves ();

high-frequency bending, low-frequency bending and shear waves ();

high-frequency bending and two low-frequency bending waves ().

Here subscripts identify the type of modes, namely () - longitudinal, () - bending, and () - shear mode. The first one stands for the primary unstable high-frequency mode, the other two subscripts denote secondary low-frequency modes.

Triads of the first three kinds (i - iii) can be observed in a flat plate (as the curvature of the shell goes to zero), while the -type triads are inherent in cylindrical shells only.

Notice that the known Karman-type dynamical governing equations can describe the -type triple-wave coupling only. The other triple-wave resonant ensembles, , and , which refer to the nonlinear coupling between high-frequency shear (longitudinal) mode and low-frequency bending modes, cannot be described by this model.

Quasi-harmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowest-order nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small long-wave perturbations in the next approximation. Generally, such instability is associated with the degenerated four-wave resonant interactions. In the present paper the second-order approximation effects is reduced to consideration of the self-action phenomenon only. The corresponding mathematical model in the form of Zakharov-type equations is proposed to describe such high-order nonlinear wave patterns.


Governing equations


We consider a deformed state of a thin-walled cylindrical shell of the length , thickness , radius , in the frame of references . The -coordinate belongs to a line beginning at the center of curvature, and passing perpendicularly to the median surface of the shell, while and are in-plane coordinates on this surface (). Since the cylindrical shell is an axisymmetric elastic structure, it is convenient to pass from the actual frame of references to the cylindrical coordinates, i.e. , where and . Let the vector of displacements of a material point lying on the median surface be . Here , and stand for the longitudinal, circumferential and transversal components of displacements along the coordinates and , respectively, at the time . Then the spatial distribution of displacements reads



accordingly to the geometrical 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):





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



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







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





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):




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




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 o