TRIPLE-WAVEENSEMBLES IN A THIN CYLINDRICAL SHELL
Kovriguine DA, Potapov AI
Introduction
Primitive nonlinear quasi-harmonic triple-wavepatterns in a thin-walled cylindrical shell are investigated. This task isfocused on the resonant properties of the system. The main idea is to trace thepropagation of a quasi-harmonic signal — is the wave stable or not? Thestability prediction is based on the iterative mathematical procedures. First,the lowest-order nonlinear approximation model is derived and tested. If thewave is unstable against small perturbations within this approximation, thenthe corresponding instability mechanism is fixed and classified. Otherwise, thehigher-order iterations are continued up to obtaining some definite result.
The theory of thin-walled shells based on theKirhhoff-Love hypotheses is used to obtain equations governing nonlinearoscillations in a shell. Then these equations are reduced to simplifiedmathematical models in the form of modulation equations describing nonlinearcoupling between quasi-harmonic modes. Physically, the propagation velocity ofany mechanical signal should not exceed the characteristic wave velocityinherent in the material of the plate. This restriction allows one to definethree main types of elemental resonant ensembles — the triads of quasi-harmonicmodes of the following kinds:
(i)high-frequency longitudinal and twolow-frequency bending waves (/>-typetriads);
(ii)high-frequencyshear and two low-frequency bending waves (/>);
(iii)high-frequencybending, low-frequency bending and shear waves (/>);
(iv)high-frequencybending and two low-frequency bending waves (/>).
Here subscripts identify the type of modes, namely (/>) — longitudinal, (/>) — bending, and (/>) — shear mode. The firstone stands for the primary unstable high-frequency mode, the other twosubscripts denote secondary low-frequency modes.
Triads of the first three kinds (i — iii) can beobserved in a flat plate (as the curvature of the shell goes to zero), whilethe />-type triads are inherentin cylindrical shells only.
Notice that the known Karman-type dynamicalgoverning equations can describe the />-typetriple-wave coupling only. The other triple-wave resonant ensembles, />, /> and />, which refer to thenonlinear coupling between high-frequency shear (longitudinal) mode andlow-frequency bending modes, cannot be described by this model.
Quasi-harmonic bending waves, whose group velocitiesdo not exceed the typical propagation velocity of shear waves, are stableagainst small perturbations within the lowest-order nonlinear approximationanalysis. However amplitude envelopes of these waves can be unstable withrespect to small long-wave perturbations in the next approximation. Generally,such instability is associated with the degenerated four-wave resonantinteractions. In the present paper the second-order approximation effects isreduced to consideration of the self-action phenomenon only. The correspondingmathematical model in the form of Zakharov-type equations is proposed todescribe such high-order nonlinear wave patterns.
Governing equations
We consider a deformed state of a thin-walledcylindrical shell of the length />,thickness />, radius />, in the frame ofreferences />. The />-coordinate belongs to aline beginning at the center of curvature, and passing perpendicularly to themedian surface of the shell, while /> and /> are in-plane coordinateson this surface (/>). Since thecylindrical shell is an axisymmetric elastic structure, it is convenient topass from the actual frame of references to the cylindrical coordinates, i.e. />, where /> and />. Let the vector ofdisplacements of a material point lying on the median surface be />. Here />, /> and /> stand for thelongitudinal, circumferential and transversal components of displacements alongthe coordinates /> and />, respectively, at the time/>. Then the spatialdistribution of displacements reads
/>
accordingly to the geometrical paradigm of theKirhhoff-Love hypotheses. From the viewpoint of further mathematicalrearrangements it is convenient to pass from the physical sought variables /> to the correspondingdimensionless displacements />. Letthe radius and the length of the shell be comparable values, i.e. />, while the displacementsbe small enough, i.e. />. Then thecomponents of the deformation tensor can be written in the form
/>
where /> is thesmall parameter; />; /> and />. The expression for thespatial density of the potential energy of the shell can be obtained usingstandard stress-straight relationships accordingly to the dynamical part of theKirhhoff-Love hypotheses:
/>
where /> is theYoung modulus; /> denotes thePoisson ratio; /> (the primesindicating the dimensionless variables have been omitted). Neglecting thecross-section inertia of the shell, the density of kinetic energy reads
/>
where /> is thedimensionless time; /> is typicalpropagation 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 ofthe cylindrical shell (the Donnell model):
(1)/>
(2)/>
Equations (1) and (2) are supplemented by theperiodicity conditions
/>Dispersion of linear waves
At /> thelinear subset of eqs.(1)-(2) describes a superposition of harmonic waves
(3)/>
Here /> is thevector of complex-valued wave amplitudes of the longitudinal, circumferentialand bending component, respectively; /> is thephase, where /> are the naturalfrequencies depending upon two integer numbers, namely /> (number of half-waves inthe longitudinal direction) and /> (numberof waves in the circumferential direction). The dispersion relation definingthis dependence /> has the form
(4)/>
where
/>
In the general case this equation possesses threedifferent roots (/>) at fixed valuesof /> and />. Graphically, thesesolutions are represented by a set of points occupied the three surfaces />. Their intersections witha plane passing the axis of frequencies are given by fig.(1). Any naturalfrequency /> corresponds to thethree-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 athin ring. The low-frequency branch corresponding generally to bending waves isapproximated by /> and />, while the high-frequencyazimuthal branch — /> and />. The bending and azimuthalmodes are uncoupled with the shear modes. The shear modes are polarized in thelongitudinal direction and characterized by the exact dispersion relation />.
Consider now axisymmetric waves (as />). The axisymmetric shearwaves are polarized by azimuth: />, whilethe other two modes are uncoupled with the shear mode. These high- andlow-frequency branches are defined by the following biquadratic equation
/>.
At the vicinity of /> thehigh-frequency branch is approximated by
/>,
while the low-frequency branch is given by
/>.
Let />, thenthe high-frequency asymptotic be
/>,
while the low-frequency asymptotic:
/>.
When neglecting the in-plane inertia of elasticwaves, the governing equations (1)-(2) can be reduced to the following set (theKarman model):
(5)/>
Here /> and /> are the differentialoperators; /> denotes the Airy stressfunction defined by the relations />, /> and />, where />, while />, /> and /> stand for the componentsof the stress tensor. The linearized subset of eqs.(5), at />, is represented by asingle equation
/>
defining a single variable />, whose solutions satisfythe following dispersion relation
(6)/>
Notice that the expression (6) is a goodapproximation of the low-frequency branch defined by (4).Evolution equations
If />, thenthe ansatz (3) to the eqs.(1)-(2) can lead at large times and spatialdistances, />, to a lack of the sameorder that the linearized solutions are themselves. To compensate this defect,let us suppose that the amplitudes /> be nowthe slowly varying functions of independent coordinates />, /> and />, although the ansatz tothe nonlinear governing equations conserves formally the same form (3):
/>
Obviously, both the slow /> andthe fast /> spatio-temporal scalesappear in the problem. The structure of the fast scales is fixed by the fastrotating phases (/>), while thedependence of amplitudes /> uponthe slow variables is unknown.
This dependence is defined by the evolutionequations describing the slow spatio-temporal modulation of complex amplitudes.
There are many routs to obtain the evolutionequations. Let us consider a technique based on the Lagrangian variationalprocedure. We pass from the density of Lagrangian function /> to its average value
(7)/>,
An advantage of the transform (7) is that theaverage Lagrangian depends only upon the slowly varying complex amplitudes andtheir derivatives on the slow spatio-temporal scales />, /> and />. In turn, the averageLagrangian does not depend upon the fast variables.
The average Lagrangian /> canbe formally represented as power series in />:
(8)/>
At /> theaverage Lagrangian (8) reads
/>
where the coefficient /> coincidesexactly with the dispersion relation (3). This means that />.
The first-order approximation average Lagrangian /> depends upon the slowlyvarying complex amplitudes and their first derivatives on the slowspatio-temporal scales />, /> and />. The correspondingevolution equations have the following form
(9)/>
Notice that the second-order approximation evolutionequations cannot be directly obtained using the formal expansion of the averageLagrangian />, since some corrections ofthe term /> are necessary. Thesecorrections are resulted from unknown additional terms /> of order />, which should generalizethe ansatz (3):
/>
provided that the second-order approximationnonlinear effects are of interest.Triple-wave resonant ensembles
The lowest-order nonlinear analysis predicts thateqs.(9) should describe the evolution of resonant triads in the cylindricalshell, provided the following phase matching conditions
(10)/>,
hold true, plus the nonlinearity in eqs.(1)-(2)possesses some appropriate structure. Here /> isa small phase detuning of order />, i.e. />. The phase matchingconditions (10) can be rewritten in the alternative form
/>
where /> is asmall frequency detuning; /> and /> are the wave numbers ofthree resonantly coupled quasi-harmonic nonlinear waves in the circumferentialand longitudinal directions, respectively. Then the evolution equations (9) canbe reduced to the form analogous to the classical Euler equations, describingthe motion of a gyro:
(11)/>.
Here /> is thepotential of the triple-wave coupling; /> arethe slowly varying amplitudes of three waves at the frequencies /> and the wave numbers /> and />; />are the group velocities; /> is the differentialoperator; /> stand for the lengths ofthe polarization vectors (/> and />); /> is the nonlinearitycoefficient:
/>
where />.
Solutions to eqs.(11) describe four main types ofresonant triads in the cylindrical shell, namely />-,/>-, /> — and />-type triads. Heresubscripts identify the type of modes, namely (/>)— longitudinal, (/>) — bending, and(/>) — shear mode. The firstsubscript stands for the primary unstable high-frequency mode, the other twosubscripts denote the secondary low-frequency modes.
A new type of the nonlinear resonant wave couplingappears in the cylindrical shell, namely />-typetriads, unlike similar processes in bars, rings and plates. From the viewpointof mathematical modeling, it is obvious that the Karman-type equations cannotdescribe the triple-wave coupling of />-, /> — and />-types, but the />-type triple-wave couplingonly. Since />-type triads are inherentin both the Karman and Donnell models, these are of interest in the presentstudy./>-triads
High-frequency azimuthal waves in the shell can beunstable with respect to small perturbations of low-frequency bending waves.Figure (2) depicts a projection of the corresponding resonant manifold of theshell possessing the spatial dimensions: /> and/>. The primaryhigh-frequency azimuthal mode is characterized by the spectral parameters /> and /> (the numerical values of /> and /> are given in the captionsto the figures). In the example presented the phase detuning />does not exceed onepercent. Notice that the phase detuning almost always approaches zero at somespecially chosen ratios between /> and />, i.e. at some specialvalues of the parameter/>. Almost all theexceptions correspond, as a rule, to the long-wave processes, since in suchcases the parameter /> cannot be small,e.g. />.
NB Notice that />-typetriads can be observed in a thin rectilinear bar, circular ring and in a flatplate.
NBThe wave modes entering />-type triads can propagatein the same spatial direction./>-triads
Analogously, high-frequency shear waves in the shellcan be unstable with respect to small perturbations of low-frequency bendingwaves. Figure (3) displays the projection of the />-typeresonant manifold of the shell with the same spatial sizes as in the previoussubsection. The wave parameters of primary high-frequency shear mode are /> and />. The phase detuning doesnot exceed one percent. The triple-wave resonant coupling cannot be observed inthe case of long-wave processes only, since in such cases the parameter /> cannot be small.
NBThe wave modes entering />-type triads cannotpropagate 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 triadsare essentially two-dimensional dynamical objects./>-triads
High-frequency bending waves in the shell can beunstable with respect to small perturbations of low-frequency bending and shearwaves. Figure (4) displays an example of projection of the />-type resonant manifold ofthe shell with the same sizes as in the previous sections. The spectralparameters of the primary high-frequency bending mode are /> and />. The phase detuning alsodoes not exceed one percent. The triple-wave resonant coupling can be observedonly in the case when the group velocity of the primary high-frequency bendingmode exceeds the typical velocity of shear waves.
NBEssentially, the spectral parameters of />-type triads fall near theboundary of the validity domain predicted by the Kirhhoff-Love theory. Thismeans that the real physical properties of />-typetriads can be different than theoretical ones.
NB/>-typetriads are essentially two-dimensional dynamical objects, since thenonlinearity parameter goes to zero, as all the waves propagate in the samedirection./>-triads
High-frequency bending waves in the shell can beunstable with respect to small perturbations of low-frequency bending waves.Figure (5) displays an example of the projection of the />-type resonant manifold ofthe shell with the same sizes as in the previous sections. The wave parametersof the primary high-frequency bending mode are /> and/>. The phase detuning doesnot exceed one percent. The triple-wave resonant coupling cannot also beobserved only in the case of long-wave processes, since in such cases theparameter /> cannot be small.
NBThe resonant interactions of />-type are inherent incylindrical shells only.Manly-Rawe relations
By multiplying each equation of the set (11) withthe corresponding complex conjugate amplitude and then summing the result, onecan reduce eqs.(11) to the following divergent laws
(12)/>
Notice that the equations of the set (12) are alwayslinearly dependent. Moreover, these do not depend upon the nonlinearitypotential />. In the case of spatiallyuniform wave processes (/>)eqs.(12) are reduced to the well-known Manly-Rawe algebraic relations
(13)/>
where /> arethe portion of energy stored by the quasi-harmonic mode number />; /> are the integrationconstants dependent only upon the initial conditions. The Manly-Rawe 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 ofenergy conservation
(14)/>.
Equation (14) predicts that the total energy of theresonant 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 tothe evolution equations (11) with spatially uniform initial conditions. Thesefirst integrals describe an unstable hyperbolic orbit behavior of triads nearthe stationary point />, or a stablemotion near the two stationary elliptic points />,and />.
In the case of spatially uniform dynamical processeseqs.(11), with the help of the first integrals, are integrated in terms ofJacobian elliptic functions [1,2]. In the particular case, as /> or />, the general analyticsolutions to eqs.(11), within an appropriate Cauchy problem, can be obtainedusing a technique of the inverse scattering transform [3]. In the general caseeqs.(11) cannot be integrated analytically.Break-up instability of axisymmetricwaves
Stability prediction of axisymmetric waves incylindrical shells subject to small perturbations is of primary interest, sincesuch waves are inherent in axisymmetric elastic structures. In the linearapproximation the axisymmetric waves are of three types, namely bending, shearand longitudinal ones. These are the axisymmetric shear waves propagatingwithout dispersion along the symmetry axis of the shell, i.e. modes polarizedin the circumferential direction, and linearly coupled longitudinal and bendingwaves.
It was established experimentally and theoreticallythat axisymmetric waves lose the symmetry when propagating along the axis ofthe shell. From the theoretical viewpoint this phenomenon can be treated withinseveral independent scenarios.
The simplest scenario of the dynamical instabilityis associated with the triple-wave resonant coupling, when the high-frequencymode breaks up into some pairs of secondary waves. For instance, let us supposethat an axisymmetric quasi-harmonic longitudinal wave (/> and />) travels along the shell.Figure (6) represents a projection of the triple-wave resonant manifold of theshell, with the geometrical sizes /> m; /> m; /> m, on the plane of wavenumbers. One can see the appearance of six secondary wave pairs nonlinearlycoupled with the primary wave. Moreover, in the particular case the triple-wavephase matching is reduced to the so-called resonance 2:1. This one can beproposed as the main instability mechanism explaining some experimentallyobserved patterns in shells subject to periodic cinematic excitations [4].
It was pointed out in the paper [5] that theresonance 2:1 is a rarely observed in shells. The so-called resonance 1:1 wasproposed instead as the instability mechanism. This means that the primaryaxisymmetric mode (with />) can beunstable one with respect to small perturbations of the asymmetric mode (with />) possessing a naturalfrequency closed to that of the primary one. From the viewpoint of theory ofwaves this situation is treated as the degenerated four-wave resonantinteraction.
In turn, one more mechanism explaining the loss ofstability of axisymmetric waves in shells based on a paradigm of the so-callednonresonant interactions can be proposed [6,7,8]. By the way, it was underlinedin the paper [6] that theoretical prognoses relevant to the modulationinstability are extremely sensible upon the model explored. This means that theKarman-type equations and Donnell-type equations lead to different predictionsrelated the stability properties of axisymmetric waves.Self-action
The propagation of any intense bending waves in along cylindrical shell is accompanied by the excitation of long-wavedisplacements related to the in-plane tensions and rotations. In turn, theselong-wave fields can influence on the theoretically predicted dependencebetween the amplitude and frequency of the intense bending wave.
Moreover, quasi-harmonic bending waves, whose groupvelocities do not exceed the typical propagation velocity of shear waves, arestable against small perturbations within the lowest-order nonlinearapproximation analysis. However amplitude envelopes of these waves can beunstable with respect to small long-wave perturbations in the nextapproximation.Amplitude-frequencycurve
Let us consider a stationary wave
/>
traveling along the single direction characterizedby the ”companion” coordinate />. Bysubstituting 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 nonlinearordinary differential equation of the fourth order:
(16)/>,
which describes simple stationary waves in thecylindrical shell (primes denote differentiation). Here
/>
where/> and /> are the integrationconstants.
If the small parameter />,and />, />, /> satisfies the dispersionrelation (4), then a periodic solution to the linearized equation (16) reads
/>
where /> arearbitrary constants, since />.
Let the parameter /> besmall enough, then a solution to eq.(16) can be represented in the followingform
(17)/>
where the amplitude /> dependsupon the slow variables />, while /> are small nonresonantcorrections. After the substitution (17) into eq.( 16) one obtains theexpression of the first-order nonresonant correction
/>
and the following modulation equation
(18)/>,
where the nonlinearity coefficient is given by
/>.
Suppose that the wave vector /> is conserved in thenonlinear solution. Taking into account that the following relation
/>
holds true for the stationary waves, one gets thefollowing modulation equation instead of eq.(18):
/>
or
/>,
where the point denotes differentiation on the slowtemporal scale />. This equationhas a simple solution for spatially uniform and time-periodic waves of constantamplitude />:
/>,
which characterizes the amplitude-frequency responsecurve of the shell or the Stocks addition to the natural frequency of linearoscillations:
(19)/>.
Spatio-temporalmodulation of waves
Relation (19) cannot provide information related tothe modulation instability of quasi-harmonic waves. To obtain this, one shouldslightly modify the ansatz (17):
(20)/>
where /> and /> denote the long-waveslowly varying fields being the functions of arguments /> and /> (these turn in constantsin the linear theory); /> is the amplitudeof the bending wave; />, /> and /> are small nonresonantcorrections. By substituting the expression (20) into the governing equations(1)-(2), one obtains, after some rearranging, the following modulationequations
(21)/>
where /> is thegroup velocity, and />. Notice thateqs.(21) have a form of Zakharov-type equations.
Consider the stationary quasi-harmonic bending wavepackets. 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 non-oscillatory in-plane wave fields aredefined by the following relations
/>
and
/>.
The theory of modulated waves predicts that theamplitude envelope of a wavetrain governed by eq.(22) will be unstable oneprovided the following Lighthill criterion
(23)/>
is satisfied.Envelope solitons
The experiments described in the paper [7] arisefrom an effort to uncover wave systems in solids which exhibit solitonbehavior. The thin open-ended nickel cylindrical shell, having the dimensions />cm, /> cm and /> cm, was made by anelectroplating process. An acoustic beam generated by a horn driver was aimedat the shell. The elastic waves generated were flexural waves which propagatedin the axial, />, andcircumferential, />, direction. Let /> and />, respectively, be theeigen numbers of the mode. The modes in which /> isalways one and /> ranges from 6 to32 were investigated. The only modes which we failed to excite (for unknownreasons) were />= 9,10,19. Aflexural wave pulse was generated by blasting the shell with an acoustic wavetrain typically 15 waves long. At any given frequency the displacement would begiven by a standing wave component and a traveling wave component. If thepickup transducer is placed at a node in the standing wave its response will belimited to the traveling wave whose amplitude is constant as it propagates.
The wave pulse at frequency of 1120 Hz wasgenerated. The measured speed of the clockwise pulse was 23 m/s and that of thecounter-clockwise pulse was 26 m/s, which are consistent with the valuecalculated from the dispersion curve (6) within ten percents. Theexperimentally observed bending wavetrains were best fitting plots of thetheoretical hyperbolic functions, which characterizes the envelope solitons.The drop in amplitude, in 105/69 times, was believed due to attenuation of thewave. The shape was independent of the initial shape of the input pulse envelope.
The agreement between the experimental data and thetheoretical curve is excellent. Figure 7 displays the dependence of thenonlinearity coefficient /> andeigen frequencies /> versus the wavenumber /> of the cylindrical shellwith the same geometrical dimensions as in the work [7]. Evidently, theenvelope solitons in the shell should arise accordingly to the Lighthillcriterion (23) in the range of wave numbers />=6,7,..,32,as />.
REFERENCES
[1]BrethertonFP (1964), Resonant interactions between waves, J. Fluid Mech., 20, 457-472.
[2]BloembergenK. (1965), Nonlinear optics, New York-Amsterdam.
[3]AblowitzMJ, H Segur (1981), Solitons and the Inverse Scattering Transform, SIAM,Philadelphia.
[4]KubenkoVD, Kovalchuk PS, Krasnopolskaya TS (1984), Nonlinear interaction of flexiblemodes of oscillation in cylindrical shells, Kiev: Naukova dumka publisher (inRussian).
[5]GinsbergJM (1974), Dynamic stability of transverse waves in circular cylindrical shell,Trans. ASME J. Appl. Mech., 41(1), 77-82.
[6]BagdoevAG, Movsisyan LA (1980), Equations of modulation in nonlinear dispersive mediaand their application to waves in thin bodies, .Izv. AN Arm.SSR, 3, 29-40 (inRussian).
[7]KovriguineDA, Potapov AI (1998), Nonlinear oscillations in a thin ring — I(II), ActaMechanica, 126, 189-212.