A sound field calculation method based on coupling of structural dynamic stiffness and acoustic boundary element

Abstract

The application discloses a sound field calculation method based on coupling of structural dynamic stiffness and acoustic boundary element, relates to the technical field of sound field prediction, and considers coupling of structural dynamic stiffness and acoustic boundary element from a numerical technical implementation means, provides two kinds of ideas, one method is that a plate shell transverse displacement response considering fluid load is obtained through a plate structure dynamic stiffness equation, is taken as input and boundary conditions of acoustic boundary element sound field calculation, and finally sound field analysis results are obtained, the method is direct, simple and clear, and strong in realizability; another method establishes a physical relationship of direct coupling of transverse displacement and acoustic boundary element, namely, simultaneous equations of a forced vibration control equation of transverse displacement of plate shell vibration and a deformation equation in the form of a constant acoustic boundary element matrix are formed, a displacement-sound field coupling equation group is formed, and sound field analysis results are solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of sound field prediction, and particularly relates to a sound field calculation method based on coupling of structural dynamic stiffness and acoustic boundary element. BACKGROUND

[0002] There are many excitation sources with different spectral characteristics on large ship equipment, and the induced structural vibration and noise contain rich low-frequency line spectrum components and wideband continuous spectrum components. Therefore, full-band vibration and noise prediction technology is a real demand in various stages of ship structure acoustic design. Unfortunately, existing mature numerical technologies, such as finite element method, boundary element method and statistical energy analysis, are only applicable to specific frequency bands due to different discretization methods and analysis variables. In recent years, the dynamic stiffness method is expected to become a powerful potential means for full-band vibration and noise prediction because its displacement function adopts an analytical solution and the discretization error is independent of frequency.

[0003] The boundary element method is another efficient and practical numerical method after the finite element method, and is particularly suitable for the dynamics prediction problem of fluid-structure coupling system in infinite domain and semi-infinite domain. It converts the three-dimensional differential equation in the infinite domain into a two-dimensional integral equation on the finite boundary, and realizes dimension reduction through discretization on the boundary surface, which can effectively simplify the calculation. Compared with the finite element method, the boundary element method has three advantages of dimension reduction, high solution accuracy and suitability for infinite domain problems.

[0004] The coupling method of structural finite element and acoustic boundary element is relatively mature, but few people propose the coupling method of structural dynamic stiffness and acoustic boundary element. The coupling of the two methods is difficult because the two methods are analytical method and numerical method respectively, have different grid discretization requirements, and the dynamic stiffness takes the free edge displacement as the degree of freedom, and the boundary element takes the grid node displacement as the degree of freedom, which cannot be synchronized and bidirectionally coupled like the structural finite element-acoustic boundary element which can independently form the stiffness or coefficient matrix. At present, there is no public literature reporting the numerical model of direct bidirectional coupling of structural dynamic stiffness and acoustic boundary element. SUMMARY

[0005] The present application relates to the field of sound field prediction, and particularly relates to a sound field calculation method based on coupling of structural dynamic stiffness and acoustic boundary element.

[0006] A sound field calculation method based on coupling of structural dynamic stiffness and acoustic boundary element, comprising the following steps:

[0007] Constructing a plate transverse vibration control equation and solving the undetermined variable by using series expansion method;

[0008] Deriving a plate transverse vibration dynamic stiffness equation according to the undetermined variable;

[0009] According to the plate transverse vibration dynamic stiffness equation and the Helmholtz integral equation of the smooth structure fluid interface, the sound pressure value of any field point is calculated.

[0010] The beneficial technical effect of the present application is:

[0011] The present application considers coupling the dynamic stiffness and the acoustic boundary element from the numerical technical implementation means, and provides two ideas for implementation. One method is to obtain the plate shell transverse displacement response considering the fluid load through the plate structure dynamic stiffness equation as the input and boundary condition of the acoustic field calculation of the acoustic boundary element, and finally obtain the acoustic field analysis result. This method is direct, simple and easy to implement. Another method establishes the physical relationship of direct coupling of transverse displacement and acoustic boundary element, that is, the forced vibration control equation of the transverse displacement of the plate shell vibration and the deformation equation in the form of the constant acoustic boundary element matrix are solved to form the displacement-acoustic field coupling equation set, so as to solve the acoustic field analysis result. BRIEF DESCRIPTION OF DRAWINGS

[0012] Figure 1 It is the flow chart of the acoustic field calculation method based on the coupling of the structural dynamic stiffness and the acoustic boundary element provided by the embodiment one of the present application.

[0013] Figure 2 It is a schematic diagram of dividing the surface of the plate into a plurality of grid units provided by the present application.

[0014] Figure 3 It is the flow chart of the acoustic field calculation method based on the coupling of the structural dynamic stiffness and the acoustic boundary element provided by the embodiment two of the present application. DETAILED DESCRIPTION

[0015] The specific embodiments of the present application will be further described below in combination with the drawings.

[0016] Embodiment one:

[0017] As shown in the figure, the present embodiment provides an acoustic field calculation method based on the coupling of the structural dynamic stiffness and the acoustic boundary element, including the following steps: Figure 1 Step 1: Construct the plate transverse vibration control equation.

[0018] When the structure surface vibrates, the fluid load needs to be introduced, and the expression of the plate transverse vibration control equation considering the fluid load is:

[0019]

[0020]

[0021]

[0022] Wherein, D=Eh 3 / 12(1-ν 2 ​), ω is the circular frequency, E is the Young's modulus, v is the Poisson's ratio, h is the structure thickness; the structure surface density m s = ρh, ρ is the structure density, ρ0 is the fluid density; w as an undetermined variable represents the transverse displacement, which is the displacement perpendicular to the direction of the plate; k b = (ρhω 2 / D) 1 / 4 , k0= ω / c, c is the propagation speed of acoustic wave in the fluid.

[0023] Optionally, when the acoustic medium is a light fluid (such as air, etc.), the acoustic pressure on the structure reaction force can be ignored, and ε f is simplified, that is, it is equal to 0; when the acoustic medium is a heavy fluid (such as water, etc.), the acoustic pressure on the structure reaction force cannot be ignored, and the fluid load needs to be introduced into the structure vibration control equation to obtain the structure vibration response considering the additional mass of the fluid. The subsequent sound field prediction method is the same as that of the light fluid.

[0024] Step 2: Solve the undetermined variable by using series expansion method.

[0025] Using the improved Gorman superposition method, the transverse displacement solution of formula (1) under any boundary condition is obtained as:

[0026] w = w1 + w2 (3)

[0027]

[0028]

[0029] Where, a and b are the length and width of the plate; M is the number of cos function expansion terms; (x, y) are the coordinates of any point on the plate; X m (x) and Y m (y) are undetermined convolution terms, which are expressed as:

[0030] X m (x) = A 1,m coshβ 1,m x + B 1,m sinhβ 1,m x + C 1,m cosγ 1,m x + D 1,m sinγ 1,m x (6)

[0031]

[0032]

[0033] Y m (y) = A 2,mcosh β 2,m y + B 2,m sinh β 2,m y + C 2,m cos γ 2,m y + D 2,m sin γ 2,m y (9)

[0034]

[0035] where λ 4 = p h (1 + ε f ) ω 2 / D, A 1,m , B 1,m , C 1,m , D 1,m , A 2,m , B 2,m , C 2,m , D 2,m are undetermined coefficients before each factor.

[0036] Step 3: Derive the lateral vibration stiffness equation of the plate according to the undetermined variables.

[0037] Step 3.1: Determine the displacement array q out and the external load array Q out of the four free edges of the plate according to the lateral displacement, including:

[0038] Use the lateral displacement w to obtain the rotation angle φ x , φ y , the shear force V x and V y , and the bending moment M x and M y :

[0039]

[0040]

[0041]

[0042] Then the displacement array q out and the external load array Q out of the four free edges of the plate are respectively expressed as:

[0043]

[0044] where the superscript out represents the out-of-plane vibration.

[0045] Step 3.2: Perform the following operations on the displacement array q out and the external load array Q outThe displacement array q out and the external load array Q out respectively are projected on the projection basis, and the projected displacement array q out is obtained

[0046]

[0047] wherein, is the displacement array q out is the convolution coefficient when the series expansion is performed, i.e. the convolution coefficient on the cosine orthogonal basis in the formula (4), (5); is the external load array Q out is the convolution coefficient when the series expansion is performed, i.e. the convolution coefficient on the cosine orthogonal basis in the formula (4), (5); C out is the array composed of the undetermined coefficients A 1,m , B 1,m , C 1,m , D 1,m , A 2,m , B 2,m , C 2,m , D 2,m , can be written as the expression about the undetermined coefficient array C out ; is the coefficient matrix of , is the coefficient matrix of , which is a known quantity, both of which are related to the undetermined coefficients in the transverse displacement solution and can be solved by mathematical calculation, and the calculation steps are not described here.

[0048] Optionally, in order to cancel the spatial dependence of the displacement, the displacement array q out and the external load array Q out are projected on the projection basis respectively according to the projection rule, and the projected displacement on the projection basis is represented as:

[0049]

[0050] wherein,

[0051]

[0052]

[0053] The external load vector corresponding to each displacement is also projected on the same projection basis, and the description is not repeated here due to the similar representation as described above.

[0054] Step 3.3: Eliminate C out, the plate transverse vibration dynamic stiffness equation with fluid load is obtained as

[0055] wherein K out is the plate transverse vibration dynamic stiffness matrix, and

[0056] Step 4: According to the plate transverse vibration dynamic stiffness equation and the Helmholtz integral equation of the smooth structure-fluid interface, the sound pressure value of any field point is calculated.

[0057] Step 4.1: Discretize the plate surface to obtain a plurality of grid elements; when the external load is known, the average vibration velocity of each grid element is determined according to the plate transverse vibration dynamic stiffness equation with fluid load and the node coordinates of the grid element including:

[0058] When the external load is known, the is calculated by convolution according to the projection rule Substitute into formula (20) to obtain The undetermined coefficient matrix C is obtained by formula (18) out In this embodiment, the reason for reversing C out in formula (18) is that the known external load may not be one-to-one corresponding to Q out in the formula, in order to improve the calculation accuracy, formula (18) is used for calculation.

[0059] Discretize the plate surface to obtain a plurality of grid elements, such as Figure 2 p elements and q elements as shown. Substitute the node coordinates of the grid element and the undetermined coefficient matrix C out into the equation for solving the transverse displacement solution of the plate transverse vibration control equation with fluid load in a series expansion manner, that is, substitute into formulas (4)-(10) to obtain the transverse displacement values of each node of the grid element, forming the transverse displacement matrix d of the grid element; wherein each node of the grid element is the intersection point of the edges of the grid element. Take the p element in Figure 2 for example, the coordinates of nodes (i), (j), (k), (l) are (x i , y i ), (x j , y j ), (x k , y k ), (x k , y k ), respectively. Then the transverse displacement matrix of the p element is:

[0060] d p = [w i , w j , wk ,w l ]。

[0061] According to the relationship between displacement and velocity in frequency domain, the vibration velocity vector V = iωd of each node of the grid element is obtained, and the average vibration velocity of each grid element is obtained by averaging the vibration velocity of each node of the grid element Wherein, i is the imaginary part.

[0062] Step 4.2: for the smooth structure fluid interface, the corresponding Helmholtz integral equation is:

[0063]

[0064] In the formula, S, E and V respectively represent the outer surface, outer domain and inner domain of the structure; Q is the source point, and P is the field point; v n is the outer normal velocity at the node on the surface of the structure; p(P) is the sound pressure of the field point; D(Q, P) is the fundamental solution at the Q point, and has:

[0065]

[0066] In the formula, r is the distance between the source point and the field point, r = |P-Q|.

[0067] After the numerical dispersion of the sound pressure on the surface of the plate, the equation in the form of constant acoustic boundary element matrix is:

[0068]

[0069] Wherein, H and C are the coefficient matrices of the boundary element, and the expression of the elements in the matrix can be referred to relevant boundary element books, which will not be expanded here; is the average vibration velocity of each grid element The average vibration velocity of the plate surface composed of

[0070] Array, NEL is the number of elements.

[0071] The sound pressure p on the surface of the plate is obtained by solving formula (23), and the sound pressure value of any field point is obtained according to the boundary integral equation (21). The boundary element method is relatively mature, and will not be repeated here.

[0072] In the embodiment, the dynamic stiffness method of the rectangular plate with arbitrary boundary conditions on four sides can form the dynamic stiffness of the combined closed structure, so as to realize the vibration harmonic response analysis of the structure. The analytical property can guarantee the accuracy of the calculation result in a wide frequency range, and the grid discretization requirement is low, and the discretization is only required according to the physical boundary and the material property; the acoustic boundary element is based on the boundary integral equation, and the Sommerfeld boundary condition is automatically satisfied, and the sound field can be calculated and analyzed only by discretizing the structure surface without discretizing the sound field calculation domain. The coupling method of the structure dynamic stiffness and the sound field boundary element provided in the embodiment is a one-way coupling, which can obtain relatively accurate results for the sound field prediction of light fluid medium in a wide frequency range, and there is a certain numerical error for the sound field prediction of heavy fluid medium, but it also meets the accuracy requirement of engineering calculation.

[0073] Embodiment two:

[0074] As shown in Figure 3 , the embodiment provides a sound field calculation method based on the coupling of the structure dynamic stiffness and the acoustic boundary element, which establishes a physical relationship of directly coupling the transverse displacement and the acoustic boundary element, and the mathematical implementation difficulty is greater than that of the method provided in the embodiment one. Taking a single rectangular plate as an example, the method for the combined plate is similar. The method comprises the following steps.

[0075] Step 1: constructing a transverse vibration control equation of the plate.

[0076] Based on the transverse free vibration control equation of the plate, a transverse forced vibration control equation of the plate is constructed, and the expression is as follows:

[0077]

[0078] Wherein, w represents the transverse displacement, the transverse displacement is the displacement perpendicular to the plate; λ 4 = ρhω 2 / D, D = Eh 3 / 12(1-ν 2 ), ω is the circular frequency, E is the Young's modulus, ν is the Poisson's ratio, h is the structure thickness, ρ is the structure density; p(x, y) is a to-be-determined variable representing the distributed sound pressure load acting on the plate.

[0079] Step 2: solving the to-be-determined variable by using the series expansion method.

[0080] Since the transverse displacement is solved by using the series expansion method in the free vibration, as shown in formulas (3)-(5). Similarly, the sound pressure load p(x, y) is expanded on the cos function base to obtain:

[0081]

[0082] Wherein, a and b are the length and width of the plate; M is the number of cos function expansion terms. (x, y) is the coordinate of any point on the plate.

[0083] Step 3: Derive the plate transverse vibration dynamic stiffness equation according to the undetermined variables.

[0084] According to the steps of forming the plate transverse vibration dynamic stiffness provided in Example 1, the plate transverse forced vibration dynamic stiffness equation is:

[0085]

[0086] where K out is the plate transverse forced vibration dynamic stiffness matrix, which is a known quantity; is the convolution coefficient when the sound pressure is expanded in series, that is, the convolution coefficient of the sound pressure load components p1, p2 on the cosine orthogonal basis in equation (25) Form the matrix is the displacement column array q out is the convolution coefficient when the series expansion is performed, that is, the convolution coefficient on the cosine orthogonal basis in equations (4), (5).

[0087] Step 4: Calculate the sound pressure value at any field point according to the plate transverse vibration dynamic stiffness equation and the Helmholtz integral equation of the smooth structure-fluid interface, including:

[0088] Step 4.1: Rewrite the corresponding Helmholtz integral equation of the smooth structure-fluid interface as:

[0089]

[0090] This equation can be solved by Gaussian integration.

[0091] After discretizing the plate surface sound pressure values, the equation in the form of constant acoustic boundary element matrix becomes:

[0092]

[0093] where, is the boundary element coefficient matrix, which can be obtained by Gaussian integration.

[0094] Step 4.2: Solve the unknown displacement and the sound pressure by simultaneously solving the plate transverse forced vibration dynamic stiffness equation (26) and the transformed equation (28) in the form of constant acoustic boundary element matrix, and then obtain the sound pressure value at any field point according to the rewritten boundary integral equation (27).

[0095] In the present embodiment, by simultaneously solving the transverse displacement forced vibration control equation of the panel shell and the deformation equation in the form of the constant acoustic boundary element matrix, a displacement-sound field coupling equation set is formed to solve the sound field analysis result.

[0096] The above only is the preferred embodiment of the present application, the present application is not limited to the above examples. It can be understood that other improvements and changes directly derived or conceived by those skilled in the art without departing from the spirit and concept of the present application should be considered to be included in the protection scope of the present application.

Claims

1. A sound field calculation method based on coupling of structural dynamic stiffness and acoustic boundary elements, characterized in that, The method comprises: The method for constructing the control equation of the transverse vibration of the plate comprises: The method for deriving the dynamic stiffness equation of the transverse vibration of the plate according to the undetermined variable comprises: The method for deriving the dynamic stiffness equation of the transverse vibration of the plate according to the undetermined variable comprises: The flat plate surface is discretely processed to divide a plurality of grid units; when the external load is known, the average vibration speed of each grid unit is determined according to the flat plate transverse vibration dynamic stiffness equation and the node coordinates of the grid units ​ The method for deriving the dynamic stiffness equation of the transverse vibration of the plate according to the undetermined variable comprises: ； wherein is the circular frequency, is the fluid density, i is the imaginary part; For the smooth structure-fluid interface, the corresponding Helmholtz integral equation is: denote the outer surface, the outer region and the inner region of the structure, respectively; is the source point, is the field point; is the outward normal velocity at the node on the surface of the structure; p P is the sound pressure at the field point; is the fundamental solution at the point, which is given by​ ； wherein is the distance between the source point and the field point, ; , c is the speed of sound propagation in the fluid; S, E, V ； wherein H , C is a boundary element coefficient matrix, is the average velocity of each mesh element is the average velocity of each mesh element , After the numerical values of the sound pressure on the plate surface are discretized, the equation in the form of the constant acoustic boundary element matrix is: is the number of elements; Solving the equation in the form of the constant acoustic boundary element matrix gives the sound pressure on the surface of the plate p The sound pressure at any field point is then obtained from the boundary integral equation.

2. The method of claim 1, wherein, NEL The method for constructing the control equation of the transverse vibration of the plate comprises: ； ； wherein, , is the Young's modulus, is the Poisson's ratio, h is the structure thickness; structure surface density , is the structure density; w denotes the lateral displacement, which is the displacement perpendicular to the direction of the plate, as an undetermined variable; .

3. The method of claim 2, wherein, The method for constructing the control equation of the transverse vibration of the plate comprises: The method for solving the undetermined variable by using the series expansion method comprises: ； ； ； wherein, a , b is the length and width of the panel; M is the number of terms in the cosine expansion; X m x Y m y is the pending convolution term, x y is the coordinate of any point on the panel.​​​​ 4. The method of claim 2, wherein, The method for solving the undetermined variable by using the series expansion method comprises: determining a displacement array of four free edges of the flat plate according to the lateral displacement and an external load array ; projecting the displacement matrix and the external load matrix on the projection basis respectively, to obtain the displacement matrix and the external load matrix respectively, the relationship equation of the undetermined coefficient matrix is: ， ； wherein is a displacement array the convolution coefficient when performing a cos series expansion; is an external load array the convolution coefficient when performing a cos series expansion; is the coefficient matrix of is the coefficient matrix of, is a known quantity; Solving the two equations simultaneously eliminates , and the equation for the lateral vibration stiffness of the plate with fluid loading is obtained as ; wherein is the flat panel lateral vibration dynamic stiffness matrix, and .

5. The method of claim 4, wherein, Determining a displacement array of four free edges of a flat plate from the lateral displacement and an external load array comprises: With lateral displacement w Get the corner , , shear V x and V y , and bending moment M x and M y : ； ； ； The displacement column of the four free edges of the plate and the external load column are respectively expressed as ， ； where the superscript The method for deriving the dynamic stiffness equation of the transverse vibration of the plate according to the undetermined variable comprises: represents out-of-plane vibration.

6. The method of claim 1, wherein, When the external load is known, the average vibration velocity of each grid element is determined according to the plate transverse vibration dynamic stiffness equation of the introduced fluid load and the node coordinates of the grid element The method comprises: When the external load is known, it is obtained by convolution calculation according to the projection rule. Then, substituting this into the equation for the lateral vibration dynamic stiffness of the plate under fluid load, we obtain... Using relational equations Obtain the matrix of undetermined coefficients ; the node coordinates of the grid element and the pending coefficient matrix into the equation for solving the transverse displacement solution of the transverse vibration control equation of the plate with the introduced fluid load by using the series expansion method, to obtain the transverse displacement values of each node of the grid element, and form a transverse displacement matrix of the grid element ; wherein each node of the grid element is an intersection point of edges of the grid element; According to the relationship between displacement and velocity in frequency domain, the vibration velocity vector of each node of the grid element is obtained The vibration velocity of each node of the grid element is averaged to obtain the average vibration velocity of each grid element .

7. The method of claim 1, wherein, out The method for constructing the control equation of the transverse vibration of the plate comprises: ； wherein, w represents a lateral displacement, which is a displacement perpendicular to the direction of the plate; , , is the Young's modulus, is the Poisson's ratio, h is the structure thickness, is the structure density; p ( The method for constructing the control equation of the transverse vibration of the plate comprises: ) represents a distributed sound pressure load acting on the plate as an undetermined variable.

8. The method of claim 7, wherein, x, y The acoustic pressure load p ( The method for solving the undetermined variable by using the series expansion method comprises: ) is obtained by expanding the cos function basis ； wherein, a , b is the length and width of the flat plate; M is the number of terms of the cos function expansion; , is the pending convolution term, x , y is the coordinate of any point on the flat plate.

9. The method of claim 7, wherein, x, y The method for deriving the dynamic stiffness equation of the transverse vibration of the plate according to the undetermined variable comprises: ； wherein, is the plate transverse forced vibration stiffness matrix, which is a known quantity; is the convolution coefficient when the sound pressure is expanded in a cos series; is the displacement column vector is the convolution coefficient when the sound pressure is expanded in a cos series; The dynamic stiffness equation of the forced transverse vibration of the plate is: ； The corresponding Helmholtz integral equation of the smooth structure-fluid interface is also rewritten according to the series expansion as: After the numerical values of the sound pressure on the plate surface are discretized, the equation in the form of the constant acoustic boundary element matrix is: ； wherein , is a boundary element coefficient matrix; Solving the transversely forced vibration of a plate by using the matrix form of the modified equation of deformation and the dynamic stiffness equation of the plate and The sound pressure value of any field point is obtained according to the modified boundary integral equation.

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