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Triplewave ensembles in a thin cylindrical shell
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Kovriguine DA, Potapov AI
IntroductionPrimitive nonlinear quasiharmonic triplewave patterns in a thinwalled 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 quasiharmonic signal  is the wave stable or not? The stability prediction is based on the iterative mathematical procedures. First, the lowestorder 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 higherorder iterations are continued up to obtaining some definite result.
The theory of thinwalled shells based on the KirhhoffLove 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 quasiharmonic 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 quasiharmonic modes of the following kinds:
highfrequency longitudinal and two lowfrequency bending waves (type triads);
highfrequency shear and two lowfrequency bending waves ();
highfrequency bending, lowfrequency bending and shear waves ();
highfrequency bending and two lowfrequency bending waves ().
Here subscripts identify the type of modes, namely ()  longitudinal, ()  bending, and ()  shear mode. The first one stands for the primary unstable highfrequency mode, the other two subscripts denote secondary lowfrequency 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 Karmantype dynamical governing equations can describe the type triplewave coupling only. The other triplewave resonant ensembles, , and , which refer to the nonlinear coupling between highfrequency shear (longitudinal) mode and lowfrequency bending modes, cannot be described by this model.
Quasiharmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowestorder nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small longwave perturbations in the next approximation. Generally, such instability is associated with the degenerated fourwave resonant interactions. In the present paper the secondorder approximation effects is reduced to consideration of the selfaction phenomenon only. The corresponding mathematical model in the form of Zakharovtype equations is proposed to describe such highorder nonlinear wave patterns.
Governing equationsWe consider a deformed state of a thinwalled 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 inplane 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 KirhhoffLove 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 stressstraight relationships accordingly to the dynamical part of the KirhhoffLove hypotheses:
where is the Young modulus; denotes the Poisson ratio; (the primes indicating the dimensionless variables have been omitted). Neglecting the crosssection 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 wavesAt the linear subset of eqs.(1)(2) describes a superposition of harmonic waves
(3)
Here is the vector of complexvalued 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 halfwaves 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 threedimensional 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 lowfrequency branch corresponding generally to bending waves is approximated by and , while the highfrequency 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 lowfrequency branches are defined by the following biquadratic equation
.
At the vicinity of the highfrequency branch is approximated by
,
while the lowfrequency branch is given by
.
Let , then the highfrequency asymptotic be
,
while the lowfrequency asymptotic:
.
When neglecting the inplane 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 lowfrequency branch defined by (4).
Evolution equationsIf , 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 spatiotemporal 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 spatiotemporal 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 upon the slowly varying complex amplitudes and their derivatives on the slow spatiotemporal scales , and . In turn, the average Lagrangian does not depend upon the fast variables.
The average Lagrangian can be formally represented as power series in :
(8)
At the average Lagrangian (8) reads
where the coefficient coincides exactly with the dispersion relation (3). This means that .
The firstorder approximation average Lagrangian depends upon the slowly varying complex amplitudes and their first derivatives on the slow spatiotemporal scales , and . The corresponding evolution equations have the following form
(9)
Notice that the secondorder approximation evolution equations cannot be directly obtained using the formal expansion of the average Lagrangian , since some corrections of the term are necessary. These corrections are resulted from unknown additional terms of order , which should generalize the ansatz (3):
provided that the secondorder approximation nonlinear effects are of interest.
Triplewave resonant ensemblesThe lowestorder nonlinear analysis predicts that eqs.(9) should describe the evolution of resonant triads in the cylindrical shell, provided the following phase matching conditions
(10),
hold true, plus the nonlinearity in eqs.(1)(2) possesses some appropriate structure. Here is a small phase detuning of order , i.e. . The phase matching conditions (10) can be rewritten in the alternative form
where is a small frequency detuning; and are the wave numbers of three resonantly coupled quasiharmonic nonlinear waves in the circumferential and longitudinal directions, respectively. Then the evolution equations (9) can be reduced to the form analogous to the classical Euler equations, describing the motion of a gyro:
(11).
Here is the potential of the triplewave coupling; are the slowly varying amplitudes of three waves at the frequencies and the wave numbers and ; are the group velocities; is the differential operator; stand for the lengths of the polarization vectors ( and ); is the nonlinearity coefficient:
where .
Solutions to eqs.(11) describe four main types of resonant triads in the cylindrical shell, namely , ,  and type triads. Here subscripts identify the type of modes, namely ()  longitudinal, ()  bending, and ()  shear mode. The first subscript stands for the primary unstable highfrequency mode, the other two subscripts denote the secondary lowfrequency modes.
A new type of the nonlinear resonant wave coupling appears in the cylindrical shell, namely type triads, unlike similar processes in bars, rings and plates. From the viewpoint of mathematical modeling, it is obvious that the Karmantype equations cannot describe the triplewave coupling of ,  and types, but the type triplewave coupling only. Since type triads are inherent in both the Karman and Donnell models, these are of interest in the present study.
triadsHighfrequency azimuthal waves in the shell can be unstable with respect to small perturbations of lowfrequency bending waves. Figure (2) depicts a projection of the corresponding resonant manifold of the shell possessing the spatial dimensions: and . The primary highfrequency azimuthal mode is characterized by the spectral parameters and (the numerical values of and are given in the captions to the figures). In the example presented the phase detuning does not exceed one percent. Notice that the phase detuning almost always approaches zero at some specially chosen ratios between and , i.e. at some special values of the parameter. Almost all the exceptions correspond, as a rule, to the longwave processes, since in such cases the parameter cannot be small, e.g. .
NB Notice that type triads can be observed in a thin rectilinear bar, circular ring and in a flat plate.
NBThe wave modes entering type triads can propagate in the same spatial direction.
triadsAnalogously, highfrequency shear waves in the shell can be unstable with respect to small perturbations of lowfrequency bending waves. Figure (3) displays the projection of the type resonant manifold of the shell with the same spatial sizes as in the previous subsection. The wave parameters of primary highfrequency shear mode are and . The phase detuning does not exceed one percent. The triplewave resonant coupling cannot be observed in the case of longwave processes only, since in such cases the parameter cannot be small.
NBThe wave modes entering type triads cannot propagate in the same spatial direction. Otherwise, the nonlinearity parameter in eqs.(11) goes to zero, as all the waves propagate in the same direction. This means that such triads are essentially twodimensional dynamical objects.
triadsHighfrequency bending waves in the shell can be unstable with respect to small perturbations of lowfrequency bending and shear waves. Figure (4) displays an example of projection of the type resonant manifold of the shell with the same sizes as in the previous sections. The spectral parameters of the primary highfrequency bending mode are and . The phase detuning also does not exceed one percent. The triplewave resonant coupling can be observed only in the case when the group velocity of the primary highfrequency bending mode exceeds the typical velocity of shear waves.
NBEssentially, the spectral parameters of type triads fall near the boundary of the validity domain predicted by the KirhhoffLove theory. This means that the real physical properties of type triads can be different than theoretical ones.
NBtype triads are essentially twodimensional dynamical objects, since the nonlinearity parameter goes to zero, as all the waves propagate in the same direction.
triadsHighfrequency bending waves in the shell can be unstable with respect to small perturbations of lowfrequency bending waves. Figure (5) displays an example of the projection of the type resonant manifold of the shell with the same sizes as in the previous sections. The wave parameters of the primary highfrequency bending mode are and . The phase detuning does not exceed one percent. The triplewave resonant coupling cannot also be observed only in the case of longwave processes, since in such cases the parameter cannot be small.
NBThe resonant interactions of type are inherent in cylindrical shells only.
ManlyRawe relationsBy multiplying each equation of the set (11) with the corresponding complex conjugate amplitude and then summing the result, one can reduce eqs.(11) to the following divergent laws
(12)
Notice that the equations of the set (12) are always linearly dependent. Moreover, these do not depend upon the nonlinearity potential . In the case of spatially uniform wave processes () eqs.(12) are reduced to the wellknown ManlyRawe algebraic relations
(13)
where are the portion of energy stored by the quasiharmonic mode number ; are the integration constants dependent only upon the initial conditions. The ManlyRawe relations (13) describe the laws of energy partition between the modes of the triad. Equations (13), being linearly dependent, can be always reduced to the law of energy conservation
(14).
Equation (14) predicts that the total energy of the resonant triad is always a constant value , while the triad components can exchange by the portions of energy , accordingly to the laws (13). In turn, eqs.(13)(14) represent the two independent first integrals to the evolution equations (11) with spatially uniform initial conditions. These first integrals describe an unstable hyperbolic orbit behavior of triads near the stationary point , or a stable motion near the two stationary elliptic points , and .
In the case of spatially uniform dynamical processes eqs.(11), with the help of the first integrals, are integrated in terms of Jacobian elliptic functions [1,2]. In the particular case, as or , the general analytic solutions to eqs.(11), within an appropriate Cauchy problem, can be obtained using a technique of the inverse scattering transform [3]. In the general case eqs.(11) cannot be integrated analytically.
Breakup instability of axisymmetric wavesStability prediction of axisymmetric waves in cylindrical shells subject to small perturbations is of primary interest, since such waves are inherent in axisymmetric elastic structures. In the linear approximation the axisymmetric waves are of three types, namely bending, shear and longitudinal ones. These are the axisymmetric shear waves propagating without dispersion along the symmetry axis of the shell, i.e. modes polarized in the circumferential direction, and linearly coupled longitudinal and bending waves.
It was established experimentally and theoretically that axisymmetric waves lose the symmetry when propagating along the axis of the shell. From the theoretical viewpoint this phenomenon can be treated within several independent scenarios.
The simplest scenario of the dynamical instability is associated with the triplewave resonant coupling, when the highfrequency mode breaks up into some pairs of secondary waves. For instance, let us suppose that an axisymmetric quasiharmonic longitudinal wave ( and ) travels along the shell. Figure (6) represents a projection of the triplewave resonant manifold of the shell, with the geometrical sizes m; m; m, on the plane of wave numbers. One can see the appearance of six secondary wave pairs nonlinearly coupled with the primary wave. Moreover, in the particular case the triplewave phase matching is reduced to the socalled resonance 2:1. This one can be proposed as the main instability mechanism explaining some experimentally observed patterns in shells subject to periodic cinematic excitations [4].
It was pointed out in the paper [5] that the resonance 2:1 is a rarely observed in shells. The socalled resonance 1:1 was proposed instead as the instability mechanism. This means that the primary axisymmetric mode (with ) can be unstable one with respect to small perturbations of the asymmetric mode (with ) possessing a natural frequency closed to that of the primary one. From the viewpoint of theory of waves this situation is treated as the degenerated fourwave resonant interaction.
In turn, one more mechanism explaining the loss of stability of axisymmetric waves in shells based on a paradigm of the socalled nonresonant interactions can be proposed [6,7,8]. By the way, it was underlined in the paper [6] that theoretical prognoses relevant to the modulation instability are extremely sensible upon the model explored. This means that the Karmantype equations and Donnelltype equations lead to different predictions related the stability properties of axisymmetric waves.
SelfactionThe propagation of any intense bending waves in a long cylindrical shell is accompanied by the excitation of longwave displacements related to the inplane tensions and rotations. In turn, these longwave fields can influence on the theoretically predicted dependence between the amplitude and frequency of the intense bending wave.
Moreover, quasiharmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowestorder nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small longwave perturbations in the next approximation.
Amplitudefrequency curveLet us consider a stationary wave
traveling along the single direction characterized by the ''companion'' coordinate . By substituting this expression into the first and second equations of the set (1)(2), one obtains the following differential relations
(15)
Here
while
where and .
Using (15) one can get the following nonlinear ordinary differential equation of the fourth order:
(16),
which describes simple stationary waves in the cylindrical shell (primes denote differentiation). Here
where and are the integration constants.
If the small parameter , and , , satisfies the dispersion relation (4), then a periodic solution to the linearized equation (16) reads
where are arbitrary constants, since .
Let the parameter be small enough, then a solution to eq.(16) can be represented in the following form
(17)
where the amplitude depends upon the slow variables , while are small nonresonant corrections. After the substitution (17) into eq.( 16) one obtains the expression of the firstorder nonresonant correction
and the following modulation equation
(18),
where the nonlinearity coefficient is given by
.
Suppose that the wave vector is conserved in the nonlinear solution. Taking into account that the following relation
holds true for the stationary waves, one gets the following modulation equation instead of eq.(18):
or
,
where the point denotes differentiation on the slow temporal scale . This equation has a simple solution for spatially uniform and timeperiodic waves of constant amplitude :
,
which characterizes the amplitudefrequency response curve of the shell or the Stocks addition to the natural frequency of linear oscillations:
(19).
Spatiotemporal modulation of wavesRelation (19) cannot provide information related to the modulation instability of quasiharmonic waves. To obtain this, one should slightly modify the ansatz (17):
(20)
where and denote the longwave slowly varying fields being the functions of arguments and (these turn in constants in the linear theory); is the amplitude of the bending wave; , and are small nonresonant corrections. By substituting the expression (20) into the governing equations (1)(2), one obtains, after some rearranging, the following modulation equations
(21)
where is the group velocity, and . Notice that eqs.(21) have a form of Zakharovtype equations.
Consider the stationary quasiharmonic bending wave packets. Let the propagation velocity be , then eqs.(21) can be reduced to the nonlinear Schrцdinger equation
(22),
where the nonlinearity coefficient is equal to
,
while the nonoscillatory inplane wave fields are defined by the following relations
and
.
The theory of modulated waves predicts that the amplitude envelope of a wavetrain governed by eq.(22) will be unstable one provided the following Lighthill criterion
(23)
is satisfied.
Envelope solitonsThe experiments described in the paper [7] arise from an effort to uncover wave systems in solids which exhibit soliton behavior. The thin openended nickel cylindrical shell, having the dimensions cm, cm and cm, was made by an electroplating process. An acoustic beam generated by a horn driver was aimed at the shell. The elastic waves generated were flexural waves which propagated in the axial, , and circumferential, , direction. Let and , respectively, be the eigen numbers of the mode. The modes in which is always one and ranges from 6 to 32 were investigated. The only modes which we failed to excite (for unknown reasons) were = 9,10,19. A flexural wave pulse was generated by blasting the shell with an acoustic wave train typically 15 waves long. At any given frequency the displacement would be given by a standing wave component and a traveling wave component. If the pickup transducer is placed at a node in the standing wave its response will be limited to the traveling wave whose amplitude is constant as it propagates.
The wave pulse at frequency of 1120 Hz was generated. The measured speed of the clockwise pulse was 23 m/s and that of the counterclockwise pulse was 26 m/s, which are consistent with the value calculated from the dispersion curve (6) within ten percents. The experimentally observed bending wavetrains were best fitting plots of the theoretical hyperbolic functions, which characterizes the envelope solitons. The drop in amplitude, in 105/69 times, was believed due to attenuation of the wave. The shape was independent of the initial shape of the input pulse envelope.
The agreement between the experimental data and the theoretical curve is excellent. Figure 7 displays the dependence of the nonlinearity coefficient and eigen frequencies versus the wave number of the cylindrical shell with the same geometrical dimensions as in the work [7]. Evidently, the envelope solitons in the shell should arise accordingly to the Lighthill criterion (23) in the range of wave numbers =6,7,..,32, as .
REFERENCESBretherton FP (1964), Resonant interactions between waves, J. Fluid Mech., 20, 457472.
Bloembergen K. (1965), Nonlinear optics, New YorkAmsterdam.
Ablowitz MJ, H Segur (1981), Solitons and the Inverse Scattering Transform, SIAM, Philadelphia.
Kubenko VD, Kovalchuk PS, Krasnopolskaya TS (1984), Nonlinear interaction of flexible modes of oscillation in cylindrical shells, Kiev: Naukova dumka publisher (in Russian).
Ginsberg JM (1974), Dynamic stability of transverse waves in circular cylindrical shell, Trans. ASME J. Appl. Mech., 41(1), 7782.
Bagdoev AG, Movsisyan LA (1980), Equations of modulation in nonlinear dispersive media and their application to waves in thin bodies, .Izv. AN Arm.SSR, 3, 2940 (in Russian).
Kovriguine DA, Potapov AI (1998), Nonlinear oscillations in a thin ring  I(II), Acta Mechanica, 126, 189212.
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