Дипломная работа: Triple-wave ensembles in a thin cylindrical shell

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 first-order approximation average Lagrangian  depends upon the slowly varying complex amplitudes and their first derivatives on the slow spatio-temporal scales ,  and . The corresponding evolution equations have the following form

(9)

Notice that the second-order 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 second-order approximation nonlinear effects are of interest.

Triple-wave resonant ensembles

The lowest-order 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 quasi-harmonic 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 triple-wave 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 high-frequency mode, the other two subscripts denote the secondary low-frequency 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 Karman-type equations cannot describe the triple-wave coupling of -, - and -types, but the -type triple-wave coupling only. Since -type triads are inherent in both the Karman and Donnell models, these are of interest in the present study.

-triads

High-frequency azimuthal waves in the shell can be unstable with respect to small perturbations of low-frequency bending waves. Figure (2) depicts a projection of the corresponding resonant manifold of the shell possessing the spatial dimensions:  and . The primary high-frequency 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 long-wave 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.

-triads

Analogously, high-frequency shear waves in the shell can be unstable with respect to small perturbations of low-frequency 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 high-frequency shear mode are  and . The phase detuning does not exceed one percent. The triple-wave resonant coupling cannot be observed in the case of long-wave 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 two-dimensional dynamical objects.

-triads

High-frequency bending waves in the shell can be unstable with respect to small perturbations of low-frequency 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 high-frequency bending mode are  and . The phase detuning also does not exceed one percent. The triple-wave resonant coupling can be observed only in the case when the group velocity of the primary high-frequency 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 Kirhhoff-Love theory. This means that the real physical properties of -type triads can be different than theoretical ones.

NB-type triads are essentially two-dimensional dynamical objects, since the nonlinearity parameter goes to zero, as all the waves propagate in the same direction.

-triads

High-frequency bending waves in the shell can be unstable with respect to small perturbations of low-frequency 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 high-frequency bending mode are  and . The phase detuning does not exceed one percent. The triple-wave resonant coupling cannot also be observed only in the case of long-wave processes, since in such cases the parameter  cannot be small.

NBThe resonant interactions of -type are inherent in cylindrical shells only.

Manly-Rawe relations

By 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 well-known Manly-Rawe algebraic relations

(13)

where  are the portion of energy stored by the quasi-harmonic mode number ;  are the integration constants 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 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.

Break-up instability of axisymmetric waves

Stability 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 triple-wave resonant coupling, when the high-frequency mode breaks up into some pairs of secondary waves. For instance, let us suppose that an axisymmetric quasi-harmonic longitudinal wave ( and ) travels along the shell. Figure (6) represents a projection of the triple-wave 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 triple-wave phase matching is reduced to the so-called 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 so-called 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 four-wave resonant interaction.

In turn, one more mechanism explaining the loss of stability of axisymmetric waves in shells based on a paradigm of the so-called 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 Karman-type equations and Donnell-type equations lead to different predictions related the stability properties of axisymmetric waves.

Self-action

The propagation of any intense bending waves in a long cylindrical shell is accompanied by the excitation of long-wave displacements related to the in-plane tensions and rotations. In turn, these long-wave fields can influence on the theoretically predicted dependence between the amplitude and frequency of the intense bending wave.

Moreover, 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.

Amplitude-frequency curve

Let 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 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 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 time-periodic waves of constant amplitude :

,

which characterizes the amplitude-frequency response curve of the shell or the Stocks addition to the natural frequency of linear oscillations:

(19).

Spatio-temporal modulation of waves

Relation (19) cannot provide information related to the modulation instability of quasi-harmonic waves. To obtain this, one should slightly modify the ansatz (17):

(20)

where  and  denote the long-wave 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 Zakharov-type equations.

Consider the stationary quasi-harmonic 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 non-oscillatory in-plane 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 solitons

The experiments described in the paper [7] arise from an effort to uncover wave systems in solids which exhibit soliton behavior. The thin open-ended 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 counter-clockwise 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 .

REFERENCES

[1]Bretherton FP (1964), Resonant interactions between waves, J. Fluid Mech., 20, 457-472.

[2]Bloembergen K. (1965), Nonlinear optics, New York-Amsterdam.

[3]Ablowitz MJ, H Segur (1981), Solitons and the Inverse Scattering Transform, SIAM, Philadelphia.

[4]Kubenko VD, Kovalchuk PS, Krasnopolskaya TS (1984), Nonlinear interaction of flexible modes of oscillation in cylindrical shells, Kiev: Naukova dumka publisher (in Russian).

[5]Ginsberg JM (1974), Dynamic stability of transverse waves in circular cylindrical shell, Trans. ASME J. Appl. Mech., 41(1), 77-82.

[6]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, 29-40 (in Russian).

[7]Kovriguine DA, Potapov AI (1998), Nonlinear oscillations in a thin ring - I(II), Acta Mechanica, 126, 189-212.