Acoustic black hole beam dynamics analysis method

Abstract

The invention discloses an acoustic black hole beam dynamic analysis method, which relates to the technical field of structural dynamics and acoustics, and comprises the following steps: step 1, establishing an acoustic black hole beam theoretical model; 2, according to the Euler-Bernoulli beam theory, the displacement field components of the moment t and the acoustic black hole beam point x can be expressed as # imgabs0 #; and 3, the NURBS function is used for describing the geometry and displacement of the acoustic black hole beam. According to the acoustic black hole beam vibration characteristic and energy transfer analysis method based on isogeometric analysis and the finite element thought, the calculation efficiency is improved while the geometric accuracy is guaranteed, a powerful tool is provided for optimization design and performance evaluation of an acoustic black hole structure, and through deep analysis of energy transfer and distribution characteristics, the method has the advantages that the calculation efficiency is improved; the mechanism of the acoustic black hole effect can be further revealed, and a new thought and method are provided for development of the vibration and noise reduction technology.

Description

Technical Field

[0001] The invention relates to the field of structural dynamics and acoustic technology, and in particular to an acoustic black hole beam dynamics analysis method. Background Art

[0002] Acoustic black hole structures have attracted much attention due to their efficient vibration and noise reduction capabilities. However, in order to further optimize their performance, scholars are still exploring new analysis methods. The traditional finite element method has the problem of time-consuming meshing when dealing with complex geometric structures, which limits its efficiency in engineering applications. Therefore, it is particularly important to develop an analysis method that can both ensure geometric accuracy and improve computational efficiency.

[0003] In recent years, many scholars have proposed many solutions to the vibration problems and solutions of acoustic black hole beam structures, such as the improved Fourier method, the finite element method and the Chebyshev method. However, most of the solutions in the acoustic black hole region will make the vibration control differential equations complicated, and the finite element method has the problems of geometric precision loss and unsatisfactory mesh division when solving acoustic black hole beams. In order to meet the requirements of accurate modeling of acoustic black hole beam dynamics, this method provides an acoustic black hole beam vibration analysis method based on isogeometric analysis and finite element thinking. This method has the characteristics of fast convergence speed and high computational efficiency in solving the vibration frequency of acoustic black hole beam structures. Summary of the invention

[0004] The purpose of the present invention is to provide an acoustic black hole beam vibration characteristic analysis method based on isogeometric analysis and finite element thinking. The method establishes a theoretical model of the acoustic black hole beam, describes the geometry and displacement of the uniform region and the black hole region of the beam using NURBS spline basis functions, and then establishes the vibration control equation of the beam. By solving the vibration control equation, the vibration characteristics of the acoustic black hole beam can be obtained.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0006] An acoustic black hole beam dynamics analysis method, the acoustic black hole beam dynamics analysis method comprising the following steps:

[0007] Step 1: Establish the theoretical model of acoustic black hole beam;

[0008] Step 2: According to the Euler-Bernoulli beam theory, the displacement field components at time t and acoustic black hole beam point x can be expressed as:

[0009]

[0010] Step 3: Use NURBS functions to describe the geometry and displacement of the acoustic black hole beam;

[0011] Step 4: Establish the potential energy V and kinetic energy T of the acoustic black hole system;

[0012] Step 5: Based on Hamilton's principle, derive the vibration control equation of the acoustic black hole beam;

[0013] Step 6: Solve the vibration control equation to obtain the dynamic characteristics of the acoustic black hole beam.

[0014] The further improvement of the technical solution of the present invention is that: the step 1, establishing the acoustic black hole beam theoretical model specifically includes:

[0015] The thickness distribution function h(x) of the beam is set according to the piecewise function of the acoustic black hole characteristics. The entire acoustic black hole beam contains two parts in the length direction: uniform thickness and acoustic black hole. The thickness variation function can be expressed as:

[0016]

[0017] In the formula, ε and γ are the coefficient and exponent of the power function, respectively, and the length of the uniform part is L uni The total length L of the acoustic black hole contour in the ideal state without truncation total and the cut-off thickness h0, satisfying the following relationship:

[0018]

[0019] The further improvement of the technical solution of the present invention is that in step 2, according to the Euler-Bernoulli beam theory, the displacement field component at time t and acoustic black hole beam point x can be expressed as:

[0020]

[0021] The normal strain at any point on the acoustic black hole beam is expressed as:

[0022]

[0023] According to Hooke's law, the relationship between stress and strain of the acoustic black hole beam is as follows:

[0024] σ x =E b ε x (5)

[0025] The further improvement of the technical solution of the present invention is that: in step 3, the NURBS function is used to describe the geometry and displacement of the acoustic black hole beam:

[0026]

[0027] Where x i and wi Represent the geometric coordinates and displacement of the control point respectively.

[0028] A further improvement of the technical solution of the present invention is that the potential energy V and kinetic energy T of the acoustic black hole system in step 4 can be expressed as follows:

[0029]

[0030] In the formula, and Ib =2bh 3 (x) / 3 represents the moment of inertia of the damping layer and the beam about the x-axis; ρ b and ρ d are the densities of the beam and the damping material, respectively.

[0031] The further improvement of the technical solution of the present invention is that: in step 5, based on Hamilton's principle, the vibration control equation of the acoustic black hole beam is derived as follows:

[0032]

[0033] Substituting equation (12) into the dynamic equation of the acoustic black hole beam and combining it with the finite element method, the control equation of the acoustic black hole beam is obtained, which is expressed as:

[0034] (K-ω 2 M)u=0 (14)

[0035] Where ω is the natural frequency of the acoustic black hole beam, K and M are the global stiffness matrix and mass matrix, respectively, and u is the displacement vector of the control point.

[0036] Due to the adoption of the above technical solution, the present invention has the following technical advances compared with the prior art:

[0037] 1. The present invention provides an acoustic black hole beam dynamics analysis method. By providing an acoustic black hole beam vibration characteristic and energy transfer analysis method based on isogeometric analysis and finite element thinking, this method improves the calculation efficiency while ensuring geometric accuracy, and provides a powerful tool for the optimal design and performance evaluation of acoustic black hole structures.

[0038] 2. The present invention provides an acoustic black hole beam dynamics analysis method, which helps to further reveal the mechanism of the acoustic black hole effect by deeply analyzing the energy transfer and distribution characteristics, and provides new ideas and methods for the development of vibration and noise reduction technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 It is a schematic diagram of the process of the present invention;

[0040] Figure 2 A model diagram of the acoustic black hole beam in the present invention;

[0041] Figure 3 It is a comparison diagram of the normalized modal vibration shapes of the ABH beam using the finite element method of the present invention and the method in this paper. DETAILED DESCRIPTION

[0042] The present invention is further described in detail below in conjunction with embodiments:

[0043] Example

[0044] The acoustic black hole beam studied in the present invention is as follows Figure 1 As shown in the figure, the origin of the coordinate is located at the left end of the beam, and the entire beam structure is symmetrical about the x-axis. The length of the acoustic black hole beam is L, and the width is b. The external force with an amplitude of F(t) is applied to the x-axis of the beam along the negative direction of the z-axis. f Position, on the beam [x d The symmetrical bonding thickness in the interval of [L] is h d , with a width of b d The damping layer is assumed to be fully coupled to the beam. Two sets of stiffness are set at the left end of the beam, namely k b , k w The spring stiffness is set to simulate different boundary conditions. The beam and damping material use the complex Young's modulus, respectively. and Indicates that E b 、E d are the Young's modulus of the beam and the damping material, η b and η d are the loss factors of the beam and the damping material, respectively.

[0045] like Figure 1-2 As shown, the present invention provides an acoustic black hole beam dynamics analysis method, the acoustic black hole beam dynamics analysis method comprises the following steps:

[0046] Step 1: Establish the theoretical model of acoustic black hole beam;

[0047] The theoretical model is as follows Figure 2 Shown

[0048] The thickness distribution function h(x) of the beam is set according to the piecewise function of the acoustic black hole characteristics. The entire acoustic black hole beam contains two parts in the length direction: uniform thickness and acoustic black hole. The thickness variation function can be expressed as:

[0049]

[0050] In the formula, ε and γ are the coefficient and exponent of the power function, respectively, and the length of the uniform part is L uniThe total length L of the acoustic black hole contour in the ideal state without truncation total and the cut-off thickness h0, satisfying the following relationship:

[0051]

[0052] Table 1 shows the material parameters and geometric parameters of the acoustic black hole beam used in this paper.

[0053] Table 1 Material parameters and geometric parameters of acoustic black hole beam structure

[0054]

[0055] Step 2: According to the Euler-Bernoulli beam theory, the displacement field components at time t and acoustic black hole beam point x can be expressed as:

[0056]

[0057] The normal strain at any point on the acoustic black hole beam is expressed as:

[0058]

[0059] According to Hooke's law, the relationship between stress and strain of the acoustic black hole beam is as follows:

[0060] σ x =E b ε x (5)

[0061] Step 3: Given a monotonically non-decreasing node vector E = {ξ1,ξ1,…,ξ i ,…,ξ n+p+1}, where ξ i is the i-th node, n and p are the number and order of basis functions respectively, and the B-spline basis function N i,p (ξ) is defined as:

[0062]

[0063] The one-dimensional NURBS basis function is defined as:

[0064]

[0065] In the formula, is a one-dimensional weight factor; ξ is the parameter space; N i,p (ξ) is the B-spline basis function in the ξ direction.

[0066] The NURBS function is used to describe the geometry and displacement of the acoustic black hole beam:

[0067]

[0068] Where x i and w i Represent the geometric coordinates and displacement of the control point respectively

[0069] Step 4: Establish the potential energy V and kinetic energy T of the acoustic black hole system:

[0070]

[0071] In the formula, and Ib =2bh 3 (x) / 3 represents the moment of inertia of the damping layer and the beam about the x-axis; ρ b and ρ d are the densities of the beam and the damping material, respectively.

[0072] Step 5: Based on Hamilton's principle, the vibration control equation of the acoustic black hole beam can be derived as follows:

[0073]

[0074] Substituting equation (12) into the dynamic equation of the acoustic black hole beam and combining it with the finite element method, the control equation of the acoustic black hole beam is obtained, which is expressed as:

[0075] (K-ω 2 M)u=0 (14)

[0076] Where ω is the natural frequency of the acoustic black hole beam, K and M are the global stiffness matrix and mass matrix, respectively, and u is the displacement vector of the control point.

[0077] The element stiffness matrix and mass matrix of the undamped zone are defined as:

[0078]

[0079] Where L k , L m1 , L m2 , They are defined as:

[0080] L k =-R ,xx ,L m1 =-R ,x ,L m2 =R (16)

[0081]

[0082] Where R is the vector composed of NURBS basis functions in the unit, R,x and R ,xx They represent the vectors composed of the first-order derivatives and second-order derivatives of the NURBS basis functions in the unit respectively. The superscript T is the transpose of the matrix.

[0083] The element stiffness matrix and mass matrix of the damping zone are defined as:

[0084]

[0085] In the formula They are defined as:

[0086]

[0087] The contribution of the boundary springs to the element stiffness matrix is:

[0088]

[0089] In the formula Re＝1(0) and Re＝1,x(0) They are respectively represented as the vector composed of NURBS functions and the vector composed of NURBS first-order derivatives when ξ=0 in the first unit;

[0090] Step 6: Solve the vibration control equation to obtain the dynamic characteristics of the acoustic black hole beam.

[0091] Table 2 gives the comparison results of the first 10 natural frequencies of the acoustic black hole beam calculated by the proposed method and the finite element method, as well as the relative error (|f 本文方法 -f 有限元 | / f 有限元 )×100%. It can be seen from the comparison that the results of this method and the finite element calculation are in good agreement, which shows that the acoustic black hole beam model constructed by this method can well predict the dynamic characteristics of the acoustic black hole beam.

[0092] Table 2 Comparison of natural frequencies of acoustic black hole beams obtained by this method and FEM

[0093]

[0094] In addition to verifying the natural frequency of the acoustic black hole beam by finite element method, Figure 3 The 1st, 5th and 15th normalized modal vibration shapes of the acoustic black hole beam are given respectively. By comparing the normalized modal vibration shapes, it can be found that even at higher orders, the method in this paper can well predict the modal vibration shapes of the acoustic black hole beam structure. At the same time, it can be seen that in high-order modes, the wavelength becomes shorter and the wave energy is concentrated in the acoustic black hole region, resulting in wave compression in this region and an increase in the amplitude of the acoustic black hole part.

[0095] In summary, the effect of the present invention is: the present invention provides an acoustic black hole beam vibration characteristics and energy transfer analysis method based on isogeometric analysis and finite element thinking. This method improves calculation efficiency while ensuring geometric accuracy, and provides a powerful tool for the optimal design and performance evaluation of acoustic black hole structures. Through in-depth analysis of energy transfer and distribution characteristics, it helps to further reveal the mechanism of the acoustic black hole effect, and provide new ideas and methods for the development of vibration reduction and noise reduction technology.

[0096] The above generally describes the present invention in detail, but it is obvious to a person skilled in the art that some modifications or improvements can be made to the present invention. Therefore, modifications or improvements that do not depart from the spirit of the present invention are within the scope of protection of the present invention.

Claims

1. An acoustic black hole beam dynamics analysis method, characterized in that: The acoustic black hole beam dynamics analysis method comprises the following steps: Step 1: Establish the theoretical model of acoustic black hole beam; Step 2: According to the Euler-Bernoulli beam theory, the displacement field components at time t and acoustic black hole beam point x can be expressed as: Step 3: Use NURBS functions to describe the geometry and displacement of the acoustic black hole beam; Step 4: Establish the potential energy V and kinetic energy T of the acoustic black hole system; Step 5: Based on Hamilton's principle, derive the vibration control equation of the acoustic black hole beam; Step 6: Solve the vibration control equation to obtain the dynamic characteristics of the acoustic black hole beam.

2. The acoustic black hole beam dynamics analysis method according to claim 1, characterized in that: The step 1, establishing the acoustic black hole beam theoretical model specifically includes: The thickness distribution function h(x) of the beam is set according to the piecewise function of the acoustic black hole characteristics. The entire acoustic black hole beam contains two parts in the length direction: uniform thickness and acoustic black hole. The thickness variation function can be expressed as: In the formula, ε and γ are the coefficient and exponent of the power function, respectively, and the length of the uniform part is L uni The total length L of the acoustic black hole contour in the ideal state without truncation total and the cut-off thickness h0, satisfying the following relationship:

3. The acoustic black hole beam dynamics analysis method according to claim 2, characterized in that: In step 2, according to the Euler-Bernoulli beam theory, the displacement field component at time t and acoustic black hole beam point x can be expressed as: The normal strain at any point on the acoustic black hole beam is expressed as: According to Hooke's law, the relationship between stress and strain of the acoustic black hole beam is as follows: s x =E b e x (5)。 4. The acoustic black hole beam dynamics analysis method according to claim 3, characterized in that: The third step is to use NURBS functions to describe the geometry and displacement of the acoustic black hole beam: Where x i and w i Represent the geometric coordinates and displacement of the control point respectively.

5. The acoustic black hole beam dynamics analysis method according to claim 4, characterized in that: The fourth step, establishing the potential energy V and kinetic energy T of the acoustic black hole system, can be expressed as follows: In the formula, and I b =2bh 3 (x) / 3 represents the moment of inertia of the damping layer and beam about the x-axis; ρ b and ρ d are the densities of the beam and the damping material, respectively.

6. The acoustic black hole beam dynamics analysis method according to claim 5, characterized in that: In step 5, based on Hamilton's principle, the vibration control equation of the acoustic black hole beam is derived as follows: Substituting equation (12) into the dynamic equation of the acoustic black hole beam and combining it with the finite element method, the control equation of the acoustic black hole beam is obtained, which is expressed as: (K-ω 2 M)u=0 (14) Where ω is the natural frequency of the acoustic black hole beam, K and M are the global stiffness matrix and mass matrix, respectively, and u is the displacement vector of the control point.

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