CN103200491A - Control method for bubble motion chaotization in sound field based on parameter perturbation - Google Patents

Abstract

The invention belongs to the field of non-linear acoustics, and particularly relates to a control method for bubble motion chaotization in a sound field based on parameter perturbation. The method comprises the steps: obtaining a radius and a pulsation speed of a three-dimension autonomous bubble at any time; measuring resonant frequency of the bubble, and estimating the size of a balance radius of the bubble; conducting the parameter perturbation on a bubble vibration system; and building a perturbation control parameter database under different balance radiuses of the bubble. According to the control method, a dynamics model of bubble motion under the parameter perturbation is built, the control method for the bubble motion chaotization based on the parameter perturbation is provided, limitation of a high-intensity and additionally incentive sound filed sound pressure level achieving the bubble motion chaotization is overcome, and requirements on devices such as underwater acoustic emission transducer and a power amplifier are reduced. Real-time picking of bubble vibration responses is avoided. The method improves the chaotization effect of the system, and achieves the optimal control on bubbles of the different radiuses.

Description

A kind of based on bubble motion chaos anti-control method in the sound field of parameter perturbation

Technical field

The invention belongs to the nonlinear acoustics field, be specifically related to a kind of based on bubble motion chaos anti-control method in the sound field of parameter perturbation.

Background technology

For the research of bubble motion state in sound field, mainly concentrate on the nonlinear kinetics behavior aspect of bubble vibrational system under the periodically outside sound field effect at present.Bubble can produce vibration under the acoustic wave excitation of certain frequency and amplitude, if only consider radial vibration, then can think the bubble radius response of system for this reason over time namely to show as contraction and the expansion of bubble.

From existing theoretical foundation, the vibration regularity of bubble satisfies the differential equation of a definite form in the liquid medium, can study motion of air bubbles state in the sound field by the solution of the differential equation.When the amplitude of vibrated was big, vibration characteristics had exceeded the category of linear theory, need set up the Mathematical Modeling of finite amplitude bubble Non-Linear Vibration.This Mathematical Modeling is generally nonlinear differential equation, can obtain the motion of air bubbles state by the dynamic behavior of research solution of equation.The method of analyzing the dynamic behavior of this type of solution of equation mainly contains two kinds, a kind of is by nonlinear dynamics theory equation to be carried out analytical analysis, another kind is to use the modern numerical calculation method that equation is carried out numerical analysis, and wherein the latter is the effective way that obtains the dynamic behavior of this type of differential equation solution fast.

Chaos phenomenon is the peculiar a kind of forms of motion of non linear system, is the random motion state of a deterministic system, shows as the initial value sensitiveness of system and the aperiodicity of motion.For bubble nonlinear kinetics system, because the incentive action of the outside sound field of periodicity, there is the chaotic motion form in its response.The confusion phenomena that chaos shows once had been considered to the object that needs suppress, but was useful in the generation of some occasion chaos, for example document " chaos vibration-isolation method research.Boats and ships mechanics, 2006,10:136-141 " propose chaos and in Nonlinear Vibration Isolation System, can be used for suppressing the low frequency spectrum lines noise.So the chaotic motion of underwater bubble has potential using value aspect the reduction target low frequency radiation noise, the chaotization method of bubble motion seems very necessary in the research sound field.

The research of bubble motion chaos anti-control method at present is also in the exploratory stage, the document " chaotic characteristic of bubble motion in the sound field.Acta Physica Sinica, 2011,60:104302 " dynamic behavior that adds sound field parameters underwater bubble when changing has been discussed, obtain the parameter region that bubble produces chaotic motion.The method adds sound field parameters by control, can reach the chaotization purpose of bubble motion.But the required outside sound field intensity of bubble generation chaotic motion has certain threshold value, usually this threshold value is very big, for example equilibrium radius is the bubble of 10 μ m, be under the acoustic wave excitation of 300kHz in frequency, the acoustic pressure threshold value that chaotic motion takes place is 250kPa, general underwater acoustic transducer is difficult to reach requirement, and alternative acoustic pressure chaos zone is also little.Document is " based on the line spectrum control technology research of discrete chaotization method.Vibration and impact, 2010,29:50-52 " utilize feedback chaos chemical control method processed, make Nonlinear Vibration Isolation System that stable chaos take place under harmonic excitation, and reduce the line spectrum composition in the radiation underwater sound.This type of feedback chaos method need be obtained the response of system in real time by transducer, calculates the input stimulus end that adds system after the feedback signal.But this method has its limitation for the vibrated system, on the one hand because the unsteadiness of bubble, be difficult to real-time acquisition vibrated response, when the high sound intensity sound wave is as the input stimulus of vibrated system on the other hand, because the nonlinear interaction of medium can produce wave distortion, and the method is very high to the required precision of feedback signal, makes sound wave have a negative impact to the chaotization effect of system as feedback excitation.

Summary of the invention

The objective of the invention is to propose a kind of restriction of avoiding high strength extrinsic motivated sound field sound pressure level, reduced the parameter threshold of system's generation chaos, improve bubble motion chaos anti-control method in the sound field of the chaotization validity of bubble motion in the sound field.

The object of the present invention is achieved like this:

The present invention includes following steps:

(1) set up the non-linear dynamic model of vibrated, model carry out autonomyization, obtain three-dimensional autonomous bubble radius and the fluctuation velocity of any time:

$\stackrel{\·}{R}=U$

$\stackrel{\&CenterDot;}{U}=[-\frac{{\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="HDA00002982539100021.png,HDA00002982539100031.png"\; state="\{\{state\}\}">U</figure-callout>}}^2}{2}(3-\frac{U}{c})+(1+(1-3\mathrm{\&kappa;})\frac{U}{c})\&times;(\frac{P_{\mathrm{stat}}-P_V}{\mathrm{\&rho;}}+\frac{2\mathrm{\&sigma;}}{\mathrm{\&rho;}{R}_n}){\left(\frac{R_n}{R}\right)}^{3\mathrm{\&kappa;}}-\frac{2\mathrm{\&sigma;}}{\mathrm{\&rho;R}}-\frac{4\mathrm{\&mu;}}{\mathrm{\&rho;}}\frac{U}{R}$

$-(1+\frac{U}{c})\frac{P_{\mathrm{stat}}+P_V+P_a\sin \left(2\mathrm{\&pi;\&theta;}\right)}{\mathrm{\&rho;}}-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00002982539100021.png,HDA00002982539100031.png"\; state="\{\{state\}\}">R</figure-callout>}\frac{2\mathrm{\&pi;}{v}_a{P}_a}{\mathrm{\&rho;c}}\cos \left(2\mathrm{\&pi;\&theta;}\right)]/[(1-\frac{U}{c})R+\frac{4\mathrm{\&mu;}}{\mathrm{\&rho;c}}]$

$\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=v_a$

Wherein, P _StatBe ambient atmosphere pressure, P _VFor steeping interior vapour pressure, p ₁For acting on the pressure of liquid on the walls, σ is surface tension of liquid, and μ is coefficient of viscosity, and c is the propagation velocity of sound wave in liquid, and ρ is fluid density, and κ is the polytropic index of gas in the bubble, R _nBe the bubble equilibrium radius, R is the bubble radius of any time,

Be the fluctuation velocity of bubble, P _aBe sound pressure amplitude, v _aBe frequency of sound wave, θ contains time t and eliminates and to turn to the variable that autonomous form is introduced for showing;

(2) the resonance frequency f of measurement bubble ₀, estimate bubble equilibrium radius R _nSize, the resonance frequency of bubble is:

${\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="HDA00002982539100041.png,HDA00002982539100071.png"\; state="\{\{state\}\}">f</figure-callout>}}_0=\frac{1}{2\mathrm{\&pi;}{R}_n\sqrt{\mathrm{\&rho;}}}\sqrt{3\mathrm{\&kappa;}(p_{\mathrm{stat}}+\frac{2\mathrm{\&sigma;}}{R_n}-p_V)-\frac{2\mathrm{\&sigma;}}{R_n}-\frac{{4\mathrm{\&mu;}}^2}{{\mathrm{\&rho;<figure-callout\; id="2"\; label="R\; n"\; filenames="HDA00002982539100021.png,HDA00002982539100031.png"\; state="\{\{state\}\}">R</figure-callout>}}_n^{}}};$

(3) according to system parameters R _nThe vibrated system is carried out parameter perturbation, wherein

R _n=R _psin(2πf _pt)；

According to time domain waveform, the phase path of response, the Poincare cross section, power spectrum, Lyapunov index judge whether system under parameter perturbation control chaotic motion takes place;

(4) control parameter perturbation frequency f _pWith perturbation amplitude R _p, chaotization parameter combinations in the parameter region of selecting system generation chaos is set up perturbation control parameter database under the different bubble equilibrium radius.

Beneficial effect of the present invention is:

The inventive method has been set up the kinetic model of the bubble motion under the parameter perturbation, bubble motion chaos anti-control method based on parameter perturbation has been proposed, overcome high strength extrinsic motivated sound field sound pressure level and realized the chaotization restriction of bubble motion, reduced the requirement to equipment such as underwater sound transmitting transducer and power amplifiers.Avoided picking up in real time the vibrated response.The method has improved the chaotization effect of system, has realized the optimal control to the different radii bubble.

Description of drawings

Fig. 1 is for before the parameter perturbation control, and system is with sound pressure amplitude P _aBifurcation graphs;

Fig. 2 is for after the parameter perturbation control, and system is with sound pressure amplitude P _aBifurcation graphs;

Fig. 3 is for after the parameter perturbation control, and system is with sound pressure amplitude P _aMaximum Lyapunov exponent figure;

Fig. 4 is different perturbation control parameter f _pFollowing system bifurcation graphs;

Fig. 5 is different perturbation control parameter f _pThe following maximum Lyapunov exponent figure of system;

Fig. 6 is different perturbation control parameters R _pFollowing system bifurcation graphs;

Fig. 7 is different perturbation control parameters R _pThe following maximum Lyapunov exponent figure of system;

Fig. 8 is based on bubble motion chaos anti-control method flow chart in the sound field of parameter perturbation.

Embodiment

The first step is set up the non-linear dynamic model of vibrated, carry out autonomyization, it is not shown contain the time, obtains three-dimensional autonomous bubble radius and the fluctuation velocity of any time.

Second step, the resonance frequency f of measurement bubble ₀, estimate bubble equilibrium radius R _nSize.

In the 3rd step, the vibrated system is carried out parameter perturbation, control parameter perturbation frequency f _pWith perturbation amplitude R _p, by f optimum in the parameter region of emulation selecting system generation chaos _pAnd R _pParameter.By periodic change system parameters R _n, make system produce chaos attractor.Adding under the incentive action of sound field, realizing the chaotization of bubble motion.

Concrete operations are as follows:

At first, derivation obtains bubble equilibrium radius R _nWhen changing according to harmonious rule, bubble is adding the three-dimensional autonomous variable element ordinary differential equation group that motion is satisfied under the sound field excitation.

Then, utilize computer programming language (Matlab, C) and modern numerical calculation method that the autonomous variable element ordinary differential equation of three-dimensional group is carried out numerical computations, the solution of equation is carried out chaos analysis, judge whether this solution chaos phenomenon occurs.

At last, regulate different parameter perturbation frequency f _pWith perturbation amplitude R _p, obtain the parameter region of system's generation chaos.

The best parameter combinations of chaotization effect in system's chaos parameter region, the equilibrium radius to bubble under this parameter combinations is carried out perturbation, and can realize the chaotization control of vibrated system periodically adding under the sound field effect this moment.

Below in conjunction with accompanying drawing the present invention is described further:

At first set up bubble in the liquid medium in the Mathematical Modeling that periodically adds under the sound field excitation, suppose that bubble only does radial vibration under sound field excitation, consider the vapour pressure of compressibility, viscosity, surface tension and the liquid of liquid, the bubble motion equation adopts the Keller-Miksis equation

$(1-\frac{\stackrel{\&CenterDot;}{R}}{c})R\stackrel{\&CenterDot;\&CenterDot;}{R}+\frac{3}{2}(1-\frac{\stackrel{\&CenterDot;}{R}}{3c}){\stackrel{\&CenterDot;}{R}}^2=(1+\frac{\stackrel{\&CenterDot;}{R}}{c})\frac{{\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA00002982539100041.png,HDA00002982539100051.png"\; state="\{\{state\}\}">p</figure-callout>}}_1}{\mathrm{\&rho;}}+\frac{R}{\mathrm{\&rho;<figure-callout\; id="1"\; label="c\; dp"\; filenames="HDA00002982539100041.png,HDA00002982539100051.png"\; state="\{\{state\}\}">c</figure-callout>}}\frac{{\mathrm{dp}}_{}}{}\frac{{}_1}{\mathrm{dt}}---\left(1\right)$

${\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA00002982539100041.png,HDA00002982539100051.png"\; state="\{\{state\}\}">p</figure-callout>}}_1=(P_{\mathrm{stat}}+\frac{2\mathrm{\&sigma;}}{R_n}-P_V){\left(\frac{R_n}{R}\right)}^{3\mathrm{\&kappa;}}-P_{\mathrm{stat}}+P_V-\frac{2\mathrm{\&sigma;}}{R}-\frac{4\mathrm{\&mu;}}{R}\stackrel{\&CenterDot;}{R}-P_a\sin \left(2\mathrm{\&pi;}{v}_{a}t\right)---\left(2\right)$

Wherein, P _StatBe ambient atmosphere pressure, P _VFor steeping interior vapour pressure, p ₁For acting on the pressure of liquid on the walls, σ is surface tension of liquid, and μ is coefficient of viscosity, and c is the propagation velocity of sound wave in liquid, and ρ is fluid density, and κ is the polytropic index of gas in the bubble, R _nBe the bubble equilibrium radius, R is the bubble radius of any time,

Be the fluctuation velocity of bubble,

Be fluctuation velocity rate of change, P _aBe sound pressure amplitude, v _aBe frequency of sound wave.The Keller-Miksis model is that Consideration is more comprehensive in several bubble radial vibration models with the meaning of representing, the scope of application is bigger one.

The Keller-Miksis equation can turn to following autonomous form through conversion:

$\stackrel{\&CenterDot;}{R}=U$

$\stackrel{\&CenterDot;}{U}=[-\frac{{\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="HDA00002982539100021.png,HDA00002982539100031.png"\; state="\{\{state\}\}">U</figure-callout>}}^2}{2}(3-\frac{U}{c})+(1+(1-3\mathrm{\&kappa;})\frac{U}{c})\&times;(\frac{P_{\mathrm{stat}}-P_V}{\mathrm{\&rho;}}+\frac{2\mathrm{\&sigma;}}{\mathrm{\&rho;}{R}_n}){\left(\frac{R_n}{R}\right)}^{3\mathrm{\&kappa;}}-\frac{2\mathrm{\&sigma;}}{\mathrm{\&rho;R}}-\frac{4\mathrm{\&mu;}}{\mathrm{\&rho;}}\frac{U}{R}$

(3)

$-(1+\frac{U}{c})\frac{P_{\mathrm{stat}}+P_V+P_a\sin \left(2\mathrm{\&pi;\&theta;}\right)}{\mathrm{\&rho;}}-\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00002982539100021.png,HDA00002982539100031.png"\; state="\{\{state\}\}">R</figure-callout>}\frac{2\mathrm{\&pi;}{v}_a{P}_a}{\mathrm{\&rho;c}}\cos \left(2\mathrm{\&pi;\&theta;}\right)]/[(1-\frac{U}{c})R+\frac{4\mathrm{\&mu;}}{\mathrm{\&rho;c}}]$

$\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=v_a$

Wherein,

Be the fluctuation velocity of bubble, θ contains time t elimination so that equation turns to the variable that autonomous form is introduced for showing.

The bubble equilibrium radius that the Keller-Miksis equation is suitable for is from several microns to several millimeters, the driving frequency scope that is suitable for from several hertz to several megahertzes, suitable excitation amplitude range from 0 handkerchief to several MPas.The expression formula of the resonance frequency of bubble is

${\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="HDA00002982539100041.png,HDA00002982539100071.png"\; state="\{\{state\}\}">f</figure-callout>}}_0=\frac{1}{2\mathrm{\&pi;}{R}_n\sqrt{\mathrm{\&rho;}}}\sqrt{3\mathrm{\&kappa;}(p_{\mathrm{stat}}+\frac{2\mathrm{\&sigma;}}{R_n}-p_V)-\frac{2\mathrm{\&sigma;}}{R_n}-\frac{{4\mathrm{\&mu;}}^2}{{\mathrm{\&rho;R}}_n^2}}---\left(4\right)$

Because bubble can produce attenuation to sound wave, when the frequency of sound wave was identical with the bubble resonance frequency, the response of bubble had maximum amplitude, and bubble is to the attenuation maximum of sound wave at this moment.If with sound waves of different frequencies bubble is encouraged, the frequency of the maximum sound wave correspondence that decays is the resonance frequency of bubble, brings the equilibrium radius that can calculate bubble in (4) formula into.

By the equilibrium radius R to the system parameters bubble _nCarry out perturbation, can change the dynamics of system, namely allow the attractor of system change, finally become chaos attractor, thereby make system enter chaos state.Because the equilibrium radius of the parameter bubble of system is a parameter that ratio is easier to control, can realize the control to the bubble equilibrium radius by control temperature, pressure.Consideration is to system parameters R in the equation (3) _nMode according to harmonious variation is carried out perturbation

R _n=R _psin(2πf _pt) (5)

Perturbation frequency is f _p, the perturbation amplitude is R _pThe reason of selecting the mode of harmonious variation to carry out perturbation is when considering by temperature, pressure means control equilibrium radius, the variation of radius can produce certain hysteresis effect, though if change the generation certain phase difference according to harmonious mode, but waveform can not change, and this perturbation method does not require the phase place of perturbation, therefore selects harmonious perturbation mode.This mode that system parameters is carried out periodically-varied also is a kind of exiting form, be referred to as parametric excitation, system is when being subjected to periodically adding sound field, if also be subjected to parametric excitation, the original cycle attractor of system will change in time so, and the possibility that finally becomes chaos attractor is arranged.

Bring (5) formula the One first-order ordinary differential equation group of into (4) formula, adopt the Runge-Kutta method to find the solution this equation group.Comprehensive service time, sequence, phase path, Poincare cross section, power spectrum, the multiple analysis means of maximum Lyapunov exponent utilized multiple chaos criterion to improve the accuracy of judgement system dynamics.If system's generation chaotic motion, the response of system will have aperiodic time domain waveform so, be limited in the phase path that certain zone has certain confusion degree, the Poincare cross section that infinite a plurality of points of a fixed structure are formed, the power spectrum of continuous distribution, and positive maximum Lyapunov exponent.Can judge accurately by above five kinds of chaos criterions whether system under parameter perturbation control chaotic motion has taken place.

Can obtain the perturbation parameter region of system's generation chaos by numerical computations, for the effect of this chaos anti-control method is described, below with R _n=40 μ m are that example is carried out numerical computations.The Keller-Miksis equation is carried out the chaotization control of parameter perturbation, and it is as follows to get the initial value design conditions: R| _T=0=R _n,

ρ=998kg/m ³, σ=0.0725N/m, μ=0.001Pas, P _Stat=1 * 10 ⁵Pa, P _V=2.33kPa, c=1500m/s, κ=4/3.

Fig. 1 and Fig. 2 provide R respectively _nBefore=40 μ m control and control back system is with external excitation sound pressure amplitude P _aBifurcation graphs, external excitation frequency of sound wave v wherein _a=200kHz can be f in the hope of the resonance frequency of bubble ₀=79.812kHz.In the represented system's bifurcation graphs of Fig. 1 and Fig. 2, abscissa is different excitation sound pressure amplitude P _a, the distribution that ordinate is put on the Poincare cross section of system responses under the sound pressure amplitude for this reason, if ordinate is distributed near the value, then illustrative system is done cycle one motion, namely only at driving frequency v _aAnd there is response at the frequency multiplication place; If the ordinate value continuous distribution is in a bounded domain, then system is experiencing the process of state unstability, and chaotic motion may take place, and greater than 0, then chaotic motion has taken place illustrative system as if maximum Lyapunov exponent at this moment.Fig. 1 is that along with the increase of external excitation acoustic pressure, chaos phenomenon never appears in system without the bifurcation graphs of when control system, and Fig. 2 is f for the control parameter _p=79.812kHz, R _pBifurcation graphs under=5 μ m, Fig. 3 is the maximum Lyapunov exponent figure of Fig. 2 correspondence, can see that system is at P _aMinimum is just chaos can take place on tens kPas the magnitude, has significantly reduced the parameter threshold of system's generation chaos.This shows that by the parameter perturbation chaos anti-control method, the sound wave that need not very big excitation density just can be controlled the generation of chaos.

Under the different control parameters, the dynamic behavior difference of system, the size of controlling this parameter then can realize the purpose that system is chaotization.Fig. 4 and Fig. 5 are for allowing parameters R _p=5 μ m immobilize, and change the control parameter f _pThe time system's bifurcation graphs and maximum Lyapunov exponent figure, wherein P _a=200kPa.From Fig. 4 and Fig. 5, can get, choose parameter f _pBe in 78～85kHz or the 145～175kHz scope time, can make parameter is P _a=200kPa, v _aBubble under the external excitation sound wave effect of=200kHz produces chaotic motion.Fig. 6 and Fig. 7 are for allowing parameter f _p=79.812kHz immobilizes, and changes the control parameters R _pThe time system's bifurcation graphs and maximum Lyapunov exponent figure thereof, can provide equally and choose parameters R _pBe in 3.2～10 mu m ranges time, also can make parameter is P _a=200kPa, v _aBubble under the external excitation sound wave effect of=200kHz produces chaotic motion.So just can provide different radii bubble chaotization perturbation parameter f under the different parameters acoustic wave excitation _p-R _pDatabase carries out perturbation to the equilibrium radius of bubble, thereby realizes the chaotization optimal control of bubble motion under the different equilibrium radius under attainable Reasonable Parameters combination.Particular flow sheet as shown in Figure 8.

This method has realized the chaotization control of bubble motion under little excitation intensity of acoustic wave by the bubble motion model being carried out parameter perturbation control.Verified the validity of the method by numerical computations, and designed the chaotization optimal control flow process of bubble motion under the different equilibrium radius, the result shows that the inventive method can realize effective control that bubble motion is chaotization, thereby this method can guarantee system to be carried out chaotization control lacking under the high-power experimental facilities situation, and engineering practice is had certain directive significance.

Claims (1)

1. one kind based on bubble motion chaos anti-control method in the sound field of parameter perturbation, it is characterized in that, comprises the steps:

(1) set up the non-linear dynamic model of vibrated, model carry out autonomyization, obtain three-dimensional autonomous bubble radius and the fluctuation velocity of any time:

$\stackrel{\&CenterDot;}{R}=U$

$\stackrel{\&CenterDot;}{U}=[-\frac{U^2}{2}(3-\frac{U}{c})+(1+(1-3\mathrm{\&kappa;})\frac{U}{c})\&times;(\frac{P_{\mathrm{stat}}-P_V}{\mathrm{\&rho;}}+\frac{2\mathrm{\&sigma;}}{\mathrm{\&rho;}{R}_n}){\left(\frac{R_n}{R}\right)}^{3\mathrm{\&kappa;}}-\frac{2\mathrm{\&sigma;}}{\mathrm{\&rho;R}}-\frac{4\mathrm{\&mu;}}{\mathrm{\&rho;}}\frac{U}{R}$

$-(1+\frac{U}{c})\frac{P_{\mathrm{stat}}+P_V+P_a\sin \left(2\mathrm{\&pi;\&theta;}\right)}{\mathrm{\&rho;}}-R\frac{2\mathrm{\&pi;}{v}_a{P}_a}{\mathrm{\&rho;c}}\cos \left(2\mathrm{\&pi;\&theta;}\right)]/[(1-\frac{U}{c})R+\frac{4\mathrm{\&mu;}}{\mathrm{\&rho;c}}]$

$\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=v_a$

Wherein, P _StatBe ambient atmosphere pressure, P _VFor steeping interior vapour pressure, p ₁For acting on the pressure of liquid on the walls, σ is surface tension of liquid, and μ is coefficient of viscosity, and c is the propagation velocity of sound wave in liquid, and ρ is fluid density, and κ is the polytropic index of gas in the bubble, R _nBe the bubble equilibrium radius, R is the bubble radius of any time,

Be the fluctuation velocity of bubble, P _aBe sound pressure amplitude, v _aBe frequency of sound wave, θ contains time t and eliminates and to turn to the variable that autonomous form is introduced for showing;

(2) the resonance frequency f of measurement bubble ₀, estimate bubble equilibrium radius R _nSize, the resonance frequency of bubble is:

$f_0=\frac{1}{2\mathrm{\&pi;}{R}_n\sqrt{\mathrm{\&rho;}}}\sqrt{3\mathrm{\&kappa;}(p_{\mathrm{stat}}+\frac{2\mathrm{\&sigma;}}{R_n}-p_V)-\frac{2\mathrm{\&sigma;}}{R_n}-\frac{{4\mathrm{\&mu;}}^2}{{\mathrm{\&rho;R}}_n^2}};$

(3) according to system parameters R _nThe vibrated system is carried out parameter perturbation, wherein

R _n=R _psin(2πf _pt)；

According to time domain waveform, the phase path of response, the Poincare cross section, power spectrum, Lyapunov index judge whether system under parameter perturbation control chaotic motion takes place;

(4) control parameter perturbation frequency f _pWith perturbation amplitude R _p, chaotization parameter combinations in the parameter region of selecting system generation chaos is set up perturbation control parameter database under the different bubble equilibrium radius.

Images