Method, device, medium and product for analyzing the harmonic resonance characteristics of cylindrical bubbles

Abstract

The present application discloses a method, device, medium and product for analyzing the harmonic resonance characteristics of cylindrical bubbles, which relates to the field of fluid mechanics technology. The method includes: constructing a dimensionless bubble wall motion equation and a second-order accuracy analytical solution model when a cylindrical bubble undergoes harmonic resonance with an external acoustic excitation; harmonic resonance includes: 2nd-order superharmonic resonance and 1 / 2-order subharmonic resonance; based on the Lyapunov stability theory and the Routh-Hurwitz stability criterion, the dimensionless bubble wall motion equation and the second-order accuracy analytical solution model when a cylindrical bubble undergoes harmonic resonance with an external acoustic excitation are quantitatively analyzed for the local stability of the steady-state oscillation solution and the global stability of the bubble oscillation, and analysis results are obtained; based on the analysis results, the influence mechanism of the core parameters on the harmonic resonance characteristics of the cylindrical bubble is determined. The present application realizes the precise analysis of bubble oscillations and analyzes the local stability and global stability of bubble oscillations.

Description

Technical Field

[0001] The present application relates to the field of fluid mechanics technology, and in particular to a method, device, medium and product for analyzing the harmonic resonance characteristics of cylindrical bubbles. Background Art

[0002] The collapse of cavitation is accompanied by high-speed bubble wall motion and the release of high energy, making it widely applicable in fields such as chemistry, biomedicine, and industry. With the advancement of production technology and the deepening of bubble dynamics research, researchers have discovered that cylindrical cavitation bubbles appear in complex scenarios such as underwater directional blasting and the jamming of underwater weapons. The cavitation pulsation generated during the explosion plays a significant role. Therefore, a method is needed to effectively analyze the vibration characteristics of cylindrical cavitation bubbles.

[0003] The cavitation oscillation system can be considered a typical complex nonlinear oscillation system. When an external excitation is applied, and the excitation frequency is a multiple (integer or fractional) of the cavitation bubble's natural frequency (also known as the resonant frequency), the cavitation bubble will resonate with the external excitation. When this resonance occurs, the cavitation bubble's dynamic characteristics may exhibit nonlinear phenomena such as period-doubling bifurcations and chaos. Therefore, in-depth research on its resonance characteristics is necessary.

[0004] Based on existing research, the resonance characteristics and stability analysis of bubbles have received extensive attention and research. However, current research focuses primarily on spherical bubbles. Research on the dynamic characteristics of cylindrical bubbles started relatively late, and the depth and progress of research have lagged behind. Currently, there is no accurate analytical method for nonlinear equations such as bubble oscillations. Summary of the Invention

[0005] The purpose of this application is to provide a method, device, medium and product for analyzing the harmonic resonance characteristics of cylindrical bubbles, which can achieve accurate analysis of bubble oscillations and analyze the local stability and global stability of bubble oscillations.

[0006] To achieve the above objectives, this application provides the following solutions:

[0007] In a first aspect, the present application provides a method for analyzing the harmonic resonance characteristics of a cylindrical bubble, comprising:

[0008] Constructing a dimensionless bubble wall motion equation and a second-order accuracy analytical solution model when a cylindrical bubble undergoes harmonic resonance with external acoustic excitation; the harmonic resonance includes: second-order superharmonic resonance and 1 / 2-order subharmonic resonance;

[0009] Based on the Lyapunov stability theory and the Routh-Hurwitz stability criterion, the dimensionless bubble wall motion equation and the second-order accurate analytical solution model for the harmonic resonance of a cylindrical bubble with external acoustic excitation are quantitatively analyzed for the local stability of the steady-state oscillation solution and the global stability of the bubble oscillation.

[0010] According to the analysis results, the influence mechanism of core parameters on the harmonic resonance characteristics of cylindrical bubbles is determined; the core parameters include: excitation amplitude, viscosity parameter, nonlinear parameter and bubble equilibrium radius.

[0011] Optionally, a dimensionless bubble wall motion equation and a second-order accurate analytical solution model for a cylindrical bubble in harmonic resonance with external acoustic excitation are constructed, including:

[0012] Construct the dimensionless bubble wall motion equation when a cylindrical bubble undergoes harmonic resonance with external acoustic excitation;

[0013] Using a multi-scale method, according to the dimensionless bubble wall motion equation, a second-order accuracy solution of the resonance system is solved to obtain an approximate equation set; the approximate equation set includes: a zero-order approximate equation, a first-order approximate equation, and a second-order approximate equation;

[0014] According to the approximate equation set, the conditions for harmonic resonance between the cylindrical bubble and the external acoustic excitation are solved;

[0015] On the basis of satisfying the conditions for harmonic resonance, using the tuning parameters, the solution of the zero-order approximate equation and the solution of the first-order approximate equation, a second-order superharmonic resonance approximate equation and a 1 / 2-order subharmonic resonance approximate equation are obtained;

[0016] According to the second-order superharmonic resonance approximate equation, the zero-order approximate equation and the first-order approximate equation, a second-order frequency response equation is obtained; based on the second-order frequency response equation, the solution of the zero-order approximate equation and the solution of the first-order approximate equation, a superharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation is obtained;

[0017] Determining a steady-state response solution of the second-order superharmonic resonance based on the second-order frequency response equation and the second-order solution of the superharmonic resonance;

[0018] According to the 1 / 2 order subharmonic resonance approximate equation, the zero-order approximate equation and the first-order approximate equation, a 1 / 2 order frequency response equation is obtained; based on the solutions of the 1 / 2 order frequency response equation, the zero-order approximate equation and the first-order approximate equation, a subharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation is obtained;

[0019] A steady-state response solution of the 1 / 2 order subharmonic resonance is determined based on the 1 / 2 order frequency response equation and the second-order solution of the subharmonic resonance.

[0020] Optionally, the dimensionless bubble wall motion equation is:

[0021]

[0022] in, is the second-order derivative of u with respect to time; ω0 is the natural frequency of the vibration system; u is the small disturbance generated by external acoustic excitation; P is the excitation adjustment parameter; Ω is the external acoustic excitation frequency; τ is the dimensionless parameter of the influence of time on the system vibration; is the first-order derivative of u with respect to time; B is the adjustment parameter of the dissipation term; a1 and a2 are the coefficients of the nonlinear term; ε is a dimensionless small parameter.

[0023] Optionally, the zero-order approximate equation is:

[0024]

[0025] Where D0 is the constant term in the first-order derivative expression with respect to time t; u0 is the solution of the zero-order approximate equation; T0 is the time variable at the zero-order scale;

[0026] The first-order approximate equation is:

[0027]

[0028] The second-order approximate equation is:

[0029]

[0030] Wherein, u1 is the solution of the first-order approximate equation; D1 is the coefficient of the linear term in the first-order derivative expression with respect to time t; D2 is the coefficient of the quadratic term in the first-order derivative expression with respect to time t.

[0031] Optionally, the conditions for harmonic resonance to occur include:

[0032] c1Ω+c2ω0=ω0+O(ε);

[0033] |c1|+|c2|=2c1,c2∈Z;

[0034] Where c1 is the external acoustic excitation frequency coefficient; c2 is the cylindrical bubble natural frequency coefficient; O(ε) is a high-order infinitesimal with respect to the dimensionless small parameter ε; and Z is an integer.

[0035] Optionally, the second-order superharmonic resonance approximation equation is:

[0036]

[0037] Where i is the imaginary unit; A is a complex function; T1 is the time variable at the first-order scale; m1 is the coefficient of the iexp(iΩT0) term in the first-order approximate equation solution; Λ is the coefficient of the exp(iΩT0) term in the zero-order approximate equation solution; δ is the tuning parameter; n2 is the coefficient of the square term of the complex function modulus in the first-order approximate equation solution; m2 is A in the first-order approximate equation solution 2 exp(2iω0T0) term coefficient; is the complex conjugate of A; m3 is the solution of the first-order approximate equation term coefficient; m4 is the coefficient of the Aexp(i(Ω+ω0)T0) term in the solution of the first-order approximate equation; n1 is the constant term of the solution of the first-order approximate equation.

[0038] Alternatively, the second-order frequency response equation is:

[0039]

[0040] Among them, g3 is The cubic coefficient of α in the expression; α is a real number related to T1 and T2; T2 is the time variable under the second-order scale; g4 is The coefficient of the α term in the expression; g0 is The coefficient of the sine function term in the expression; g2 is The coefficient of the sine term in the expression; g1 is The coefficient of the cosine term in the expression.

[0041] In a second aspect, the present application provides a computer device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement any of the above-described methods for analyzing the harmonic resonance characteristics of cylindrical bubbles.

[0042] In a third aspect, the present application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements any of the above-mentioned methods for analyzing the harmonic resonance characteristics of cylindrical bubbles.

[0043] In a fourth aspect, the present application provides a computer program product, comprising a computer program, which, when executed by a processor, implements any of the above-mentioned methods for analyzing the harmonic resonance characteristics of cylindrical bubbles.

[0044] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0045] The present application discloses a method, device, medium and product for analyzing the harmonic resonance characteristics of cylindrical bubbles. Through multi-scale analytical methods and Lyapunov stability theory, the nonlinear dynamic characteristics and stability of the harmonic resonance of cylindrical bubbles under external sound field excitation are studied. By constructing a dimensionless bubble wall motion equation and its second-order accuracy analytical solution model when a cylindrical bubble undergoes second-order superharmonic resonance and 1 / 2-order subharmonic resonance with external sound excitation, a large number of tedious experimental explorations with high equipment and technical barriers are avoided, saving experimental costs. Through the frequency response curve, the influence mechanism of core parameters such as excitation amplitude, viscosity parameter, nonlinear parameter and bubble equilibrium radius on the harmonic resonance characteristics of cylindrical bubbles is explored, and the nonlinear dynamic characteristics of single-frequency resonance of cylindrical bubbles can be analyzed. Based on the Lyapunov stability theory and the Routh-Hurwitz stability criterion, the local stability and global stability of bubble harmonic resonance can be predicted. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0047] Figure 1 A schematic flow chart of a method for analyzing the harmonic resonance characteristics of cylindrical bubbles provided in an embodiment of the present application;

[0048] Figure 2 A schematic diagram of the structure of a computer device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0049] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0050] The purpose of this application is to provide a method, device, medium and product for analyzing the harmonic resonance characteristics of cylindrical bubbles, aiming to achieve accurate analysis of bubble oscillations and analyze the local stability and global stability of bubble oscillations.

[0051] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0052] In an exemplary embodiment, Figure 1 As shown, the cylindrical bubble harmonic resonance characteristic analysis method in this embodiment includes:

[0053] Step 1: Construct the dimensionless bubble wall motion equation and second-order accuracy analytical solution model when a cylindrical bubble undergoes harmonic resonance with external acoustic excitation; harmonic resonance includes: second-order superharmonic resonance and 1 / 2-order subharmonic resonance.

[0054] As an optional implementation, step 1 includes:

[0055] Step 11: Construct the dimensionless bubble wall motion equation when the cylindrical bubble undergoes harmonic resonance with external acoustic excitation.

[0056] Specifically, when the bubbles resonate harmonically with the external acoustic excitation, in order to ensure that the acoustic excitation is strong, ξ = O(ε) is adjusted, and the excitation term and dissipation term are adjusted to:

[0057] β=εB (1)

[0058] ξ=εP (2)

[0059] Among them, β and ξ are dimensionless small parameters of the same order or higher than ε; O(ε) is a high-order infinitesimal with respect to the dimensionless small parameter ε; B is the adjustment parameter of the dissipation term; P is the adjustment parameter of the excitation term; and ε is a dimensionless small parameter.

[0060] Substituting the excitation term and dissipation term into the dimensionless second-order bubble wall motion equation, the dimensionless bubble wall motion equation of harmonic resonance is obtained:

[0061]

[0062] in, is the second-order derivative of u with respect to time; ω0 is the natural frequency of the vibration system; u is the small disturbance generated by external acoustic excitation; P is the excitation adjustment parameter; Ω is the external acoustic excitation frequency; τ is the dimensionless parameter of the influence of time on the system vibration; is the first-order derivative of u with respect to time; B is the adjustment parameter of the dissipation term; a1 and a2 are the coefficients of the nonlinear term; ε is a dimensionless small parameter.

[0063] Step 12: Using the multi-scale method, according to the dimensionless bubble wall motion equation, solve the second-order accuracy solution of the resonance system to obtain an approximate equation set; the approximate equation set includes: a zero-order approximate equation, a first-order approximate equation, and a second-order approximate equation.

[0064] Specifically, according to the dimensionless bubble wall motion equation, the second-order accuracy solution of the resonance system is solved, and based on the multi-scale method, the dimensionless small parameter ε is obtained. 0 , ε 1 and ε 2The corresponding zero-order, first-order and second-order approximate equations.

[0065] As an optional implementation, the zero-order approximate equation is:

[0066]

[0067] Where D0 is the constant term in the first-order derivative expression with respect to time t; u0 is the solution of the zero-order approximate equation; and T0 is the time variable at the zero-order scale.

[0068] The first-order approximate equation is:

[0069]

[0070] The second-order approximate equation is:

[0071]

[0072] Wherein, u1 is the solution of the first-order approximate equation; D1 is the coefficient of the linear term in the first-order derivative expression with respect to time t; D2 is the coefficient of the quadratic term in the first-order derivative expression with respect to time t.

[0073]

[0074] Among them, T υ is the time variable at the υ-order scale.

[0075] Step 13: Based on the approximate equation set, solve the conditions for harmonic resonance between the cylindrical bubble and the external acoustic excitation.

[0076] Specifically, the conditions for harmonic resonance between the cylindrical bubble and the acoustic excitation are solved by substituting the solution of the zero-order approximate equation into the first-order approximate equation and allowing the cylindrical bubble to resonate harmonically with the external acoustic excitation:

[0077] The solution of the zero-order approximate equation can be expressed as the following formula:

[0078] u0=A(T1,T2)exp(iω0T0)+Λexp(iΩT0)+cc(8)

[0079] Where u0 is the solution of the zero-order approximate equation; A(T1,T2) is a complex function with respect to T1 and T2; T1 is the time variable at the first-order scale; T2 is the time variable at the second-order scale; Λ is the coefficient of the exp(iΩT0) term in the solution of the zero-order approximate equation; i is the imaginary unit; cc is the complex conjugate of all terms in formula (8).

[0080]

[0081] Substituting the solution of the zero-order approximate equation into the first-order approximate equation, we can obtain the following formula:

[0082]

[0083] in, is the complex conjugate of Λ.

[0084] When equation (10) satisfies the following conditions (i.e., the conditions for harmonic resonance), the cylindrical bubble and the acoustic excitation undergo harmonic resonance:

[0085] c1Ω+c2ω0=ω0+O(ε) (11)

[0086] |c1|+|c2|=2 c1,c2∈Z (12)

[0087] Where c1 is the external acoustic excitation frequency coefficient; c2 is the cylindrical bubble natural frequency coefficient; O(ε) is a high-order infinitesimal with respect to the dimensionless small parameter ε; and Z is an integer.

[0088] Step 14: On the basis of satisfying the conditions for harmonic resonance, the second-order superharmonic resonance approximate equation and the 1 / 2-order subharmonic resonance approximate equation are obtained using the tuning parameters, the solution of the zero-order approximate equation and the solution of the first-order approximate equation.

[0089] Specifically, on the basis of satisfying the conditions for harmonic resonance, the external acoustic excitation frequency Ω is expressed by using the tuning parameter δ, eliminating the secular terms of the first-order approximate equation and the second-order approximate equation; by combining the solution of the zero-order approximate equation with the solution of the first-order approximate equation, the second-order superharmonic resonance approximate equation is obtained. Specifically, it includes:

[0090] Using the tuning parameter δ, the second-order superharmonic resonance acoustic excitation frequency Ω is expressed as follows:

[0091] 2Ω=ω0+εδ(13)

[0092] Use 2ΩT0=(ω0+εδ)T0=ω0T0+δT1 to process the terms containing exp(2iΩT0) in the first-order approximate equation, and set all terms containing exp(iω0T0) in the first-order approximate equation to 0, eliminating the duration term, and then we have:

[0093]

[0094] Let the expression of complex function A be:

[0095]

[0096] Where λ is a real number related to T1 and T2.

[0097] Let γ = δT1 - λ and separate the real and imaginary parts to obtain:

[0098]

[0099] Among them, g0 is The coefficients of the sine function terms in the expression.

[0100]

[0101] After eliminating the duration term, the solution to the first-order approximate equation can be expressed as follows:

[0102]

[0103] Among them, m1 is the coefficient of iexp(iΩT0) in the solution of the first-order approximate equation; m2 is the coefficient of A in the solution of the first-order approximate equation 2 exp(2iω0T0) term coefficient; is the complex conjugate of A; m3 is the solution of the first-order approximate equation term coefficient; m4 is the coefficient of the Aexp(i(Ω+ω0)T0) term in the first-order approximate equation solution; n1 is the constant term in the first-order approximate equation solution; n2 is the coefficient of the square term of the complex function modulus in the first-order approximate equation solution.

[0104]

[0105] Substituting the solutions of the zero-order approximate equation and the first-order approximate equation into the second-order approximate equation, and setting the duration terms containing exp(iω0T0) and exp(2iΩT0) to 0, the second-order superharmonic resonance approximate equation is obtained. The second-order superharmonic resonance approximate equation is:

[0106]

[0107] Where i is the imaginary unit; A is a complex function; T1 is the time variable at the first-order scale; m1 is the coefficient of the i exp(iΩT0) term in the first-order approximate equation solution; Λ is the coefficient of the exp(iΩT0) term in the zero-order approximate equation solution; δ is the tuning parameter; n2 is the coefficient of the square term of the complex function modulus in the first-order approximate equation solution; m2 is A in the first-order approximate equation solution 2 exp(2iω0T0) term coefficient; is the complex conjugate of A; m3 is the solution of the first-order approximate equation term coefficient; m4 is the coefficient of the Aexp(i(Ω+ω0)T0) term in the solution of the first-order approximate equation; n1 is the constant term of the solution of the first-order approximate equation.

[0108] Specifically, on the basis of satisfying the conditions for harmonic resonance, the tuning parameter δ is used to express the 1 / 2 order subharmonic resonance acoustic excitation frequency Ω, eliminating the secular terms of the first-order approximate equation and the second-order approximate equation; by combining the solution of the zero-order approximate equation and the solution of the first-order approximate equation, the 1 / 2 order subharmonic resonance approximate equation is obtained. Specifically, it includes:

[0109] According to the tuning parameter δ, the expression of the external acoustic excitation frequency Ω is obtained, and its duration term is set to 0. The solution process of the approximate equation of the second-order superharmonic resonance is consistent with that of the first-order approximate equation, which can be expressed as the following formula:

[0110]

[0111] Substituting the solutions of the zero-order approximate equation and the first-order approximate equation into the second-order approximate equation and setting the secular term to 0, we obtain the 1 / 2-order subharmonic resonance approximate equation. The 1 / 2-order subharmonic resonance approximate equation is:

[0112]

[0113]

[0114] Step 15: According to the second-order superharmonic resonance approximate equation, the zero-order approximate equation and the first-order approximate equation, the second-order frequency response equation is obtained. Based on the solutions of the second-order frequency response equation, the zero-order approximate equation and the first-order approximate equation, the superharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation is obtained.

[0115] Specifically, based on the second-order superharmonic resonance approximate equation, the expressions after separating the real and imaginary parts of the first-order approximate equation and the second-order approximate equation are combined and sorted out to obtain the second-order frequency response equation; combined with the solution of the zero-order approximate equation and the solution of the first-order approximate equation, the superharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation is obtained. Specifically including:

[0116] Eliminate the and Substitute the expression for A into the above equation and separate the real and imaginary parts:

[0117]

[0118]

[0119] Among them, g0 is The coefficient of the sine function term in the expression; g1 is The cosine term coefficient in the expression; g2 is The coefficient of the sine term in the expression; g3 is The cubic coefficient of α in the expression; g4 is The coefficient of the α term in the expression.

[0120] Combining the first-order approximate equation and the second-order approximate equation after separating the real and imaginary parts, the second-order transient response equation group can be obtained:

[0121]

[0122] Among them, τ is a dimensionless parameter that affects the vibration of the system.

[0123] In order to obtain the steady-state oscillation solution of the second-order superharmonic resonance of the cylindrical bubble, let dα / dτ=dγ / dτ=0, and rearrange the second-order transient response equations to obtain the rearranged second-order transient response equations:

[0124]

[0125] -αδ+εg3α 3 +εg4α=(g0+εg2δ)cos(γ)-εg1sin(γ)(44)

[0126] The squares of the second-order transient response equations are added together to obtain the second-order frequency response equation. The second-order frequency response equation is:

[0127]

[0128] Among them, g3 is The cubic coefficient of α in the expression; α is a real number related to T1 and T2; T2 is the time variable under the second-order scale; g4 is The coefficient of the α term in the expression; g0 is The coefficient of the sine function term in the expression; g2 is The coefficient of the sine term in the expression; g1 is The coefficient of the cosine term in the expression.

[0129] Combining the solutions of the zero-order and first-order approximate equations, the superharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation can be obtained as:

[0130]

[0131] Step 16: Determine the steady-state response solution of the second-order superharmonic resonance based on the second-order frequency response equation and the second-order solution of the superharmonic resonance.

[0132] Specifically, by solving the sixth-order equation of α for the second-order frequency response equation, the obtained result is substituted into the second-order solution of superharmonic resonance to obtain the steady-state response solution of the second-order superharmonic resonance.

[0133] Step 17: According to the 1 / 2 order subharmonic resonance approximate equation, the zero order approximate equation and the first order approximate equation, the 1 / 2 order frequency response equation is obtained. Based on the solutions of the 1 / 2 order frequency response equation, the zero order approximate equation and the first order approximate equation, the subharmonic resonance second order solution of the cylindrical bubble approximate oscillation equation is obtained.

[0134] Specifically, by eliminating the first-order and second-order differential terms in the formula and separating the real and imaginary parts, and then combining the first-order and second-order approximate equations to separate the real and imaginary parts, the transient response equations can be obtained:

[0135]

[0136] In order to obtain the analytical solution of the bubble 1 / 2 order subharmonic resonance steady-state oscillation, the transient response equations are sorted and squared and added to obtain the 1 / 2 order frequency response equation:

[0137]

[0138] Only when the following two equations hold true can there be a non-trivial resonance solution:

[0139] (2g0+2εg2δ) 2 +(2εg1) 2 ≥(3B) 2 (51)

[0140]

[0141] Combining the solutions of the zero-order approximate equation and the first-order approximate equation, the subharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation can be obtained as:

[0142]

[0143] Step 18: Determine the steady-state response solution of the 1 / 2 order subharmonic resonance based on the 1 / 2 order frequency response equation and the second-order solution of the subharmonic resonance.

[0144] Specifically, according to the 1 / 2 order frequency response equation and the second-order solution of subharmonic resonance, the 1 / 2 order frequency response equation is solved and the obtained result is substituted into the second-order solution of subharmonic resonance to obtain the steady-state response solution of 1 / 2 order subharmonic resonance.

[0145] Step 2: Based on the Lyapunov stability theory and the Routh-Hurwitz stability criterion, the dimensionless bubble wall motion equation and the second-order analytical solution model when the cylindrical bubble undergoes harmonic resonance with external acoustic excitation are quantitatively analyzed for the local stability of the steady-state oscillation solution and the global stability of the bubble oscillation. The analytical results are obtained.

[0146] Specifically, based on the Lyapunov stability theory, the local stability conditions of the superharmonic resonance and subharmonic resonance of cylindrical bubbles are analyzed, and the transient response equations are linearized at the steady-state solution:

[0147] α=α0+α1 (55)

[0148] γ=γ0+γ1 (56)

[0149] Substituting the above formula into the transient response equations, performing Taylor expansion on the small quantities α1 and γ1, ignoring the high-order small quantities of α1 and γ1, the characteristic equation corresponding to the linear approximation equation is obtained as follows:

[0150] χ 2 +3εBχ+Γ=0 (57)

[0151] Where χ is the solution of the characteristic equation; Γ is the constant term of the characteristic equation.

[0152] In the 2nd order superharmonic resonance:

[0153]

[0154] In 1 / 2 order subharmonic resonance:

[0155]

[0156] According to the Routh-Hurwitz stability criterion, the solution is stable when all the characteristic roots have negative real parts. Therefore, the necessary and sufficient condition for the stability of the steady-state oscillation solution of the cylindrical bubble at the second-order superharmonic resonance and the 1 / 2-order subharmonic resonance is Γ>0.

[0157] Step 3: Based on the analysis results, determine the mechanism of the influence of core parameters on the harmonic resonance characteristics of cylindrical bubbles; core parameters include: excitation amplitude, viscosity parameter, nonlinear parameter and bubble equilibrium radius.

[0158] Furthermore, the influence mechanism of core parameters on the harmonic resonance characteristics of cylindrical bubbles is used to optimize the structure of the fluid mechanical system in a confined space to ensure system stability.

[0159] In an exemplary embodiment, a computer device is provided, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor. The processor executes the computer program to implement the cylindrical bubble harmonic resonance characteristic analysis method in Example 1.

[0160] In an exemplary embodiment, a computer-readable storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the method for analyzing the harmonic resonance characteristics of cylindrical bubbles in Example 1 is implemented.

[0161] In an exemplary embodiment, a computer program product is provided, including a computer program. When the computer program is executed by a processor, the method for analyzing the harmonic resonance characteristics of cylindrical bubbles in Example 1 is implemented.

[0162] In an exemplary embodiment, a computer device is provided. The computer device may be a server or a terminal. The internal structure diagram thereof may be as follows: Figure 2 As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O) and a communication interface. The processor, the memory and the input / output interface are connected via a system bus, and the communication interface is connected to the system bus via the input / output interface. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal via a network connection. When the computer program is executed by the processor, a method for analyzing the harmonic resonance characteristics of a cylindrical bubble is implemented.

[0163] Those skilled in the art will understand that Figure 2 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0164] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant regulations.

[0165] Those skilled in the art will understand that all or part of the processes in the above-mentioned embodiment methods can be implemented by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to memory, database or other media used in the embodiments provided in this application may include at least one of non-volatile and volatile memory. Non-volatile memory may include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory may include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM may be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0166] The databases involved in the various embodiments provided herein may include at least one of a relational database and a non-relational database. Non-relational databases may include, but are not limited to, distributed databases based on blockchains. The processors involved in the various embodiments provided herein may include, but are not limited to, general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic units, data processing logic units based on quantum computing, and the like.

[0167] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0168] This document uses specific examples to illustrate the principles and implementation methods of this application. The description of the above examples is only intended to help understand the method and core concept of this application. At the same time, for those skilled in the art, based on the concept of this application, there may be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A method for analyzing the harmonic resonance characteristics of cylindrical bubbles, characterized in that: The method comprises: Constructing a dimensionless bubble wall motion equation and a second-order accuracy analytical solution model when a cylindrical bubble undergoes harmonic resonance with external acoustic excitation; the harmonic resonance includes: second-order superharmonic resonance and 1 / 2-order subharmonic resonance; Based on the Lyapunov stability theory and the Routh-Hurwitz stability criterion, the dimensionless bubble wall motion equation and the second-order accurate analytical solution model for the harmonic resonance of a cylindrical bubble with external acoustic excitation are quantitatively analyzed for the local stability of the steady-state oscillation solution and the global stability of the bubble oscillation. Based on the analysis results, the influence mechanism of core parameters on the harmonic resonance characteristics of cylindrical bubbles is determined; the core parameters include: excitation amplitude, viscosity parameter, nonlinear parameter and bubble equilibrium radius; Construct a dimensionless bubble wall motion equation and a second-order accurate analytical solution model for the harmonic resonance of a cylindrical bubble with external acoustic excitation, including: Construct the dimensionless bubble wall motion equation when a cylindrical bubble undergoes harmonic resonance with external acoustic excitation; Using a multi-scale method, according to the dimensionless bubble wall motion equation, a second-order accuracy solution of the resonance system is solved to obtain an approximate equation set; the approximate equation set includes: a zero-order approximate equation, a first-order approximate equation, and a second-order approximate equation; According to the approximate equation set, the conditions for harmonic resonance between the cylindrical bubble and the external acoustic excitation are solved; On the basis of satisfying the conditions for harmonic resonance, using the tuning parameters, the solution of the zero-order approximate equation and the solution of the first-order approximate equation, a second-order superharmonic resonance approximate equation and a 1 / 2-order subharmonic resonance approximate equation are obtained; According to the second-order superharmonic resonance approximate equation, the zero-order approximate equation and the first-order approximate equation, a second-order frequency response equation is obtained; based on the second-order frequency response equation, the solution of the zero-order approximate equation and the solution of the first-order approximate equation, a superharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation is obtained; Determining a steady-state response solution of the second-order superharmonic resonance based on the second-order frequency response equation and the second-order solution of the superharmonic resonance; According to the 1 / 2 order subharmonic resonance approximate equation, the zero-order approximate equation and the first-order approximate equation, a 1 / 2 order frequency response equation is obtained; based on the solutions of the 1 / 2 order frequency response equation, the zero-order approximate equation and the first-order approximate equation, a subharmonic resonance second-order solution of the cylindrical bubble approximate oscillation equation is obtained; A steady-state response solution of the 1 / 2 order subharmonic resonance is determined based on the 1 / 2 order frequency response equation and the second-order solution of the subharmonic resonance.

2. The cylindrical bubble harmonic resonance characteristic analysis method according to claim 1, characterized in that: The dimensionless bubble wall motion equation is: in, is the second-order derivative of u with respect to time; ω0 is the natural frequency of the vibration system; u is the small disturbance generated by external acoustic excitation; P is the excitation adjustment parameter; Ω is the external acoustic excitation frequency; τ is the dimensionless parameter of the influence of time on the system vibration; is the first-order derivative of u with respect to time; B is the adjustment parameter of the dissipation term; a1 and a2 are the coefficients of the nonlinear term; ε is a dimensionless small parameter.

3. The cylindrical bubble harmonic resonance characteristic analysis method according to claim 2, characterized in that: The zero-order approximate equation is: Where D0 is the constant term in the first-order derivative expression with respect to time t; u0 is the solution of the zero-order approximate equation; T0 is the time variable at the zero-order scale; The first-order approximate equation is: The second-order approximate equation is: Wherein, u1 is the solution of the first-order approximate equation; D1 is the coefficient of the linear term in the first-order derivative expression with respect to time t; D2 is the coefficient of the quadratic term in the first-order derivative expression with respect to time t.

4. The cylindrical bubble harmonic resonance characteristic analysis method according to claim 3, characterized in that: The conditions for harmonic resonance to occur include: c1Ω+c2ω0=ω0+O(ε); |c1|+|c2|=2 c1,c2∈Z; Where c1 is the external acoustic excitation frequency coefficient; c2 is the cylindrical bubble natural frequency coefficient; O(ε) is a high-order infinitesimal with respect to the dimensionless small parameter ε; and Z is an integer.

5. The cylindrical bubble harmonic resonance characteristic analysis method according to claim 4, characterized in that: The second-order superharmonic resonance approximate equation is: Where i is the imaginary unit; A is a complex function; T2 is the time variable at the second-order scale; T1 is the time variable at the first-order scale; m1 is the coefficient of the iexp(iΩT0) term in the first-order approximate equation solution; Λ is the coefficient of the exp(iΩT0) term in the zero-order approximate equation solution; δ is the tuning parameter; n2 is the coefficient of the square term of the complex function modulus in the first-order approximate equation solution; m2 is A in the first-order approximate equation solution 2 exp(2iω0T0) term coefficient; is the complex conjugate of A; m3 is the solution of the first-order approximate equation term coefficient; m4 is the coefficient of the Aexp(i(Ω+ω0)T0) term in the solution of the first-order approximate equation; n1 is the constant term of the solution of the first-order approximate equation.

6. The method for analyzing the harmonic resonance characteristics of cylindrical bubbles according to claim 5, wherein: The second-order frequency response equation is: Among them, g3 is The cubic coefficient of α in the expression; α is a real number related to T1 and T2; g4 is The coefficient of the α term in the expression; g0 is The coefficient of the sine function term in the expression; g2 is The coefficient of the sine term in the expression; g1 is The coefficient of the cosine term in the expression.

7. A computer device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method for analyzing the harmonic resonance characteristics of cylindrical bubbles according to any one of claims 1 to 6.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method for analyzing the harmonic resonance characteristics of cylindrical bubbles according to any one of claims 1 to 6 is implemented.

9. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the method for analyzing the harmonic resonance characteristics of cylindrical bubbles according to any one of claims 1 to 6 is implemented.

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