Random subsystem energy calculation method based on deterministic subsystem response

Abstract

A stochastic subsystem energy calculation method based on deterministic subsystem response comprises the following steps: firstly, dividing different components into stochastic subsystems and deterministic subsystems according to a vibration mode number N within a one-third octave bandwidth delta f of a structural component; secondly, according to a division result, establishing a statistical energy analysis model of a random subsystem and a finite element model of a deterministic subsystem, and establishing hybrid connection between the two models to obtain a hybrid finite element-statistical energy analysis model of the structure; and finally, according to the obtained hybrid finite element-statistical energy analysis model, calculating a random subsystem energy response by using a deterministic subsystem response. Measurement is responded by means of deterministic subsystem nodes, and the method has the advantages of being low in sensor number requirement, flexible in arrangement position and the like. Indirect measurement is carried out on the energy response of the random subsystem by means of the node response of the deterministic subsystem, and the problem that direct measurement of the energy response of the random subsystem is difficult to realize is solved.

Description

Technical Field

[0001] The invention belongs to the technical field of structural dynamic response measurement, and in particular relates to a random subsystem energy calculation method based on deterministic subsystem response. Background Art

[0002] The measurement of structural dynamic response is a key technology in the use of engineering structures. Its dynamic response is crucial to the optimal design, reliability analysis and high-precision prediction of the dynamic environment of the structure. For medium-frequency dynamic problems, the hybrid finite element-statistical energy analysis method is usually used to carry out dynamic analysis. This method divides the structure into a random subsystem and a deterministic subsystem. The dynamic response of the random subsystem is usually described by energy, and the dynamic response of the deterministic subsystem is usually described by parameters such as acceleration. Since the measurement of the energy response of the random subsystem requires a large number of sensors and has high requirements for the installation position of the sensors, it is very difficult to carry out measurements in actual engineering, and direct measurements are difficult to achieve in engineering; while the measurement of the node response of the deterministic subsystem has the advantages of low sensor quantity requirements and flexible layout positions. Indirect measurement of the energy response of the random subsystem with the help of the node response of the deterministic subsystem greatly reduces the measurement difficulty and has important engineering application value.

[0003] The existing technology for measuring the dynamic response of a structure is mainly to directly collect vibration signals after applying a load to the structure, or to establish a dynamic model of the structure and obtain simulation results of the response after applying the load. However, the load on the structure is usually unknown, which makes the above methods difficult to implement. For structures in working state, it is difficult to directly measure the energy response of the random subsystem due to the characteristics of the structure itself or the limitations of the sensor working conditions, as well as the number of sensors required to measure the energy response of the random subsystem and the requirements for the sensor installation position. Summary of the invention

[0004] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a method for calculating the energy of a random subsystem based on the response of a deterministic subsystem. First, a hybrid finite element-statistical energy analysis model of the structure is established; then, based on the model, the energy response of the random subsystem is calculated using the node response of the deterministic subsystem; thereby solving the problem that the energy response of the random subsystem is difficult to measure.

[0005] In order to achieve the above object, the technical solution adopted by the present invention can be described as follows:

[0006] A method for calculating the energy of a random subsystem based on the response of a deterministic subsystem comprises the following steps:

[0007] Step 1: Divide different components into random subsystems and deterministic subsystems according to the number of vibration modes N within the one-third octave bandwidth Δf of the structural components;

[0008] Step 2: According to the partitioning results, a statistical energy analysis model of the random subsystem and a finite element model of the deterministic subsystem are established, and a hybrid connection between the two models is established to obtain a hybrid finite element-statistical energy analysis model of the structure;

[0009] Step 3: Based on the model obtained in step 2, the energy response of the random subsystem is calculated using the deterministic subsystem response.

[0010] In one embodiment, the step 1, dividing different components into random subsystems and deterministic subsystems, is implemented by: according to the component composition of the structure, classifying the structure by components, dividing it into multiple subsystems, analyzing the number of vibration modes N within a one-third octave bandwidth Δf of these subsystems, dividing the subsystems with N<3 into deterministic subsystems, and dividing the subsystems with N≥3 into random subsystems;

[0011] Wherein, N=n(f)Δf, n(f) is the modal density of the subsystem, f is the center frequency of one-third octave band, and illustratively, the one-third octave band bandwidth Δf=0.232×f.

[0012] In one embodiment, the classification of the structures by components according to the components of the structures is based on one or more of the material properties, geometric features, connection methods and functional areas of the structures, and the implementation method is as follows:

[0013] Material properties: Parts made of different materials may exhibit different dynamic behaviors, for example, steel structures and parts made of aluminum alloy or composite materials can be classified as different components. Therefore, you can choose to define parts made of different materials as different components.

[0014] Geometric features: Parts with similar geometric characteristics can be grouped together as one component. Therefore, you can choose to define parts with different geometric characteristics as different components.

[0015] Connection method: The connection method between components (such as welding, hinged or free connection) affects the energy transfer, so it can be classified according to the connection characteristics. Therefore, it is possible to define parts with different connection characteristics as different components;

[0016] Functional area: The structure can be divided into different parts according to its functional use. Therefore, you can choose to define parts with different functional uses as different parts.

[0017] In one embodiment, the classification of components directly affects the number of subsystems and the way they are divided. The number of subsystems is usually related to the following factors: The number of components: Each component can be regarded as a subsystem, so the number of components determines the upper limit of the number of subsystems; Dynamic characteristics: The subsystem division follows the modal similarity criterion, that is, the vibration mode of the same subsystem must have similar dynamic characteristics. For example, a plate-shell structure can be divided into a bending subsystem, a stretching subsystem, and a shear subsystem. The components have different dynamic performances in different frequency ranges, and the subsystems may need to be further subdivided in a certain frequency band. Therefore, the division into multiple subsystems is implemented as follows:

[0018] Treat each component as a subsystem; or,

[0019] Components with the same or similar vibration modes are considered as a subsystem.

[0020] In one embodiment, in step 2, the statistical energy analysis model of the random subsystem is described by a power flow balance equation, which is expressed as follows:

[0021]

[0022] Where ω is the circular frequency; M is the number of random subsystems; n i is the modal density of the i-th random subsystem; η i is the internal loss factor of the i-th random subsystem; η ij is the coupling loss factor when energy is transferred from the i-th random subsystem to the j-th random subsystem; η d,i is the equivalent loss factor generated by the deterministic subsystem at the mixing boundary with the i-th random subsystem; E i is the energy of the i-th random subsystem; is the input power of the direct field to the i-th random subsystem; is the input power of the external environment to the i-th random subsystem, i, j = 1, ..., M, and i ≠ j;

[0023] The deterministic subsystem is described by a finite element model. The nodes of the deterministic subsystem are the nodes of the finite element model of the deterministic subsystem. The self-spectrum S of the node response of the deterministic subsystem is qq It is expressed as follows:

[0024]

[0025] The symbol <.> indicates the ensemble mean; the symbol -H Represents the conjugate transpose and inverse of a matrix; is the direct field dynamic stiffness matrix of the mth random subsystem, D totis the total dynamic stiffness matrix of the system, satisfying D d is the dynamic stiffness matrix of the deterministic subsystem, n m is the modal density of the mth random subsystem, E m is the energy of the mth random subsystem; is the power spectral density of the excitation force acting on the deterministic subsystem.

[0026] In one embodiment, a hybrid connection between the two models is established to obtain a hybrid finite element-statistical energy analysis model of the structure; the hybrid connection describes the energy transfer relationship between the deterministic subsystem and the random subsystem, and the direct field dynamic stiffness matrix Describe the direct field dynamic stiffness matrix Obtained by the boundary element method.

[0027] In one embodiment, in step 3, the energy response of the random subsystem is calculated using the deterministic subsystem response, and the method is as follows:

[0028] 1) Assume that the responses of the deterministic subsystem nodes 1, 2, ..., u are known, and the vector S composed of the self-spectrum of the deterministic subsystem node responses is [ 1,qq > 2,qq > … u,qq >] T ,in i,qq >, i=1,2,...u represents the response autospectrum of node i of the deterministic subsystem, and the vector composed of the energy responses of all random subsystems is represented as E;

[0029] 2) When there is a load on the random subsystem, the following relationship exists between S and E according to the system equation of the deterministic subsystem:

[0030] S=hE (3)

[0031] Where h=[h1 h2…h u ] T ,h k is an M-order row vector, k=1,2,...u,h k The mth element of

[0032] 3) The number of deterministic subsystem nodes with known responses is u, and the number of random subsystems is M. When u ≥ M, the energy response of the random subsystem is calculated by equation (3) and the known node responses of the deterministic subsystem.

[0033] ​​​​Based on the hybrid finite element-statistical energy analysis theory, the present invention derives the relationship between the node response of the deterministic subsystem and the energy response of the random subsystem. By taking advantage of the characteristics of the small number of sensors required and the low requirements on the location of the sensors in the process of measuring the node response of the deterministic subsystem, the energy response of the random subsystem is indirectly calculated. Compared with the previous direct measurement of the energy response of the random subsystem, the measurement of the energy response of the random subsystem requires sensors that are several times the number of random subsystems. The method of the present invention only requires the same number of sensors as the number of random subsystems or slightly more than the number of random subsystems to measure the energy response of all random subsystems; at the same time, the method of the present invention can indirectly measure the energy response of the difficult-to-measure random subsystem, which greatly reduces the difficulty of measurement. This provides a data basis for the dynamic optimization design, vibration reduction and noise reduction, and fatigue life assessment of the components of the concerned parts, and has important engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 Schematic diagram of the three-plate structure used in the embodiment of the present invention.

[0035] Figure 2 is the modal number of each subsystem in each frequency band of the three-board structure used in the embodiment of the present invention.

[0036] Figure 3 It is a hybrid finite element-statistical energy analysis model of a three-plate structure in an embodiment of the present invention, wherein (a) is a finite element model in the hybrid finite element-statistical energy analysis model, and (b) is a hybrid finite element-statistical energy analysis model of a three-plate structure.

[0037] Figure 4 The figure is a schematic diagram of the positions of acceleration sensors for obtaining the energy response of the random subsystem and the response of the deterministic subsystem through experiments in an embodiment of the present invention.

[0038] Figure 5 A comparison diagram of the energy responses of the random subsystem 1 directly measured by an embodiment of the present invention and indirectly settled through the deterministic subsystem response.

[0039] Figure 6 This is a comparison diagram of the energy responses of the random subsystem 2 that is indirectly settled through the deterministic subsystem response according to an embodiment of the present invention. DETAILED DESCRIPTION

[0040] The technical solution of the present invention is described in detail below with reference to the accompanying drawings, but the protection scope of the present invention is not limited to the embodiments.

[0041] In order to solve the problem that it is difficult to directly measure the energy response of the random subsystem, the present invention uses the advantages of low sensor quantity requirement and flexible layout of the deterministic subsystem node response measurement, as well as the node response of the deterministic subsystem to carry out indirect measurement of the energy response of the random subsystem.

[0042] In a specific embodiment of the present invention, a method for calculating the energy of a random subsystem based on the response of a deterministic subsystem is specifically performed as follows:

[0043] (1) Step 1: Figure 1 The three-plate structure shown is made of aluminum alloy with a density of 2700kg / m 3 , elastic modulus is 70Gpa, Poisson's ratio is 0.3. The structure consists of three plates, plate 1, plate 2, and plate 3 are the names of the three plates in the three-plate structure, and their specific positions and sizes are as follows Figure 1 As shown, plate one and plate two have the same structural dimensions, are located on both sides of plate three, and are connected to plate three by welding.

[0044] Each plate consists of three subsystems, namely bending subsystem, stretching subsystem and shear subsystem, a total of 9 subsystems. The modal number corresponding to the one-third octave center frequency in the range of 200 to 5000 Hz of each subsystem of the three-plate structure is analyzed. The results are as follows Figure 2 As shown. Figure 2 , the modal density of the bending subsystems of plate 1 and plate 2 is greater than 3 after 500Hz, and the modal density of the remaining subsystems is less than 3. Therefore, in the range of 500Hz to 5000Hz, the bending subsystems of plate 1 and plate 2 are divided into random subsystems, and the remaining 7 subsystems are divided into deterministic subsystems. The bending subsystem of plate 1 is recorded as random subsystem 1, and the bending subsystem of plate 2 is recorded as random subsystem 2.

[0045] (2) Step 2: Based on the partitioning results, a statistical energy analysis model of the random subsystem and a finite element model of the deterministic subsystem are established, as follows: Figure 3 As shown in (a) and (b), a hybrid connection between the two models is established to obtain a hybrid finite element-statistical energy analysis model of the structure. The power flow balance equation of the random subsystem can be expressed as:

[0046]

[0047] The self-spectrum S of the node response of the deterministic subsystem qq It is expressed as follows:

[0048]

[0049] (3) Step 3: Carry out the experiment and apply a random excitation perpendicular to the plate to plate 1, such as Figure 4 The schematic diagram of the acceleration sensor position is shown in Figure 1. A total of 7 acceleration sensors were used in the experiment. A1-A7 are the acceleration sensor numbers. The positions of A1-A7 are as follows: Figure 4As shown, the energy response of random subsystem 1 is directly measured by the measured responses of A1-A4 on plate 1 and equation (6):

[0050]

[0051] Where m i is the mass of the ith random subsystem, the spatially averaged mean square acceleration is the arithmetic mean of the acceleration responses at A1-A4. The response of the deterministic subsystem is obtained by the acceleration sensors at A5-A7 on board 3, and the energy response of the random subsystem is calculated by formula (3) and compared with the direct measurement results, as shown in Figure 5 As shown. Figure 5 It can be seen that the calculation results of the method of the present invention are almost consistent with the direct measurement results, and the number of sensors used is less than the number of sensors required for direct measurement. Moreover, the method of the present invention can simultaneously obtain the energy response of the random subsystem 1 and the random subsystem 2 using a small number of sensors, such as Figure 5 and Figure 6 As shown, the direct measurement method can only obtain the energy response of random subsystem one.

Claims

1. A method for calculating the energy of a random subsystem based on the response of a deterministic subsystem, characterized in that: The following steps are involved: Step 1: Divide different components into random subsystems and deterministic subsystems according to the number of vibration modes N within the one-third octave bandwidth Δf of the structural components; Step 2: According to the partitioning results, a statistical energy analysis model of the random subsystem and a finite element model of the deterministic subsystem are established, and a hybrid connection between the two models is established to obtain a hybrid finite element-statistical energy analysis model of the structure; Step 3: Based on the model obtained in step 2, the energy response of the random subsystem is calculated using the deterministic subsystem response.

2. The random subsystem energy calculation method based on deterministic subsystem response according to claim 1 is characterized in that: The step 1 is to divide different components into random subsystems and deterministic subsystems, and the implementation method is as follows: According to the components of the structure, the structure is classified into multiple subsystems according to the components, and the number of vibration modes N within the one-third octave bandwidth Δf of these subsystems is analyzed. The subsystems with N < 3 are classified as deterministic subsystems, and the subsystems with N ≥ 3 are classified as random subsystems. Where N = n(f)Δf, n(f) is the modal density of the subsystem, and f is the center frequency of one-third octave band.

3. The random subsystem energy calculation method based on deterministic subsystem response according to claim 2 is characterized in that: The classification of structures by components according to the components of the structure is based on one or more of the material properties, geometric characteristics, connection methods and functional areas of the structure. The implementation method is as follows: Material properties: define parts of different materials as different components; Geometric features: define parts with different geometric characteristics as different components; Connection method: define parts with different connection characteristics as different components; Functional area: Define parts with different functional uses as different components.

4. The random subsystem energy calculation method based on deterministic subsystem response according to claim 2 or 3, characterized in that: The division into multiple subsystems is implemented as follows: Treat each component as a subsystem; or, Components with the same or similar vibration modes are considered as a subsystem.

5. The random subsystem energy calculation method based on deterministic subsystem response according to claim 2 is characterized in that: The one-third octave bandwidth Δf=0.232×f.

6. The random subsystem energy calculation method based on deterministic subsystem response according to claim 1 is characterized in that: In step 2, the statistical energy analysis model of the random subsystem is described by the power flow balance equation, which is expressed as follows: Where ω is the circular frequency; M is the number of random subsystems; n i is the modal density of the i-th random subsystem; η i is the internal loss factor of the i-th random subsystem; η ij is the coupling loss factor when energy is transferred from the i-th random subsystem to the j-th random subsystem; η d,i is the equivalent loss factor generated by the deterministic subsystem at the mixing boundary with the i-th random subsystem; E i is the energy of the i-th random subsystem; is the input power of the direct field to the i-th random subsystem; is the input power of the external environment to the i-th random subsystem, i, j = 1, ..., M, and i ≠ j; The deterministic subsystem is described by a finite element model. The nodes of the deterministic subsystem are the nodes of the finite element model of the deterministic subsystem. The self-spectrum S of the node response of the deterministic subsystem is qq It is expressed as follows: The symbol <.> indicates the ensemble mean; the symbol -H Represents the conjugate transpose and inverse of a matrix; is the direct field dynamic stiffness matrix of the mth random subsystem, D tot is the total dynamic stiffness matrix of the system, satisfying D d is the dynamic stiffness matrix of the deterministic subsystem, n m is the modal density of the mth random subsystem, E m is the energy of the mth random subsystem; is the power spectral density of the excitation force acting on the deterministic subsystem.

7. The method for calculating random subsystem energy based on deterministic subsystem response according to claim 6, characterized in that: A hybrid connection between the two models is established to obtain a hybrid finite element-statistical energy analysis model of the structure; the hybrid connection describes the energy transfer relationship between the deterministic subsystem and the random subsystem through the direct field dynamic stiffness matrix Describe the direct field dynamic stiffness matrix Obtained by the boundary element method.

8. The random subsystem energy calculation method based on deterministic subsystem response according to claim 1 is characterized in that: In step 3, the energy response of the random subsystem is calculated using the deterministic subsystem response, as follows: 1) Assume that the responses of the deterministic subsystem nodes 1, 2, ..., u are known, and the vector S composed of the self-spectrum of the deterministic subsystem node responses is [ 1,qq > 2,qq >… u,qq >] T ,in i,qq >, i=1,2,...u represents the response autospectrum of node i of the deterministic subsystem, and the vector composed of the energy responses of all random subsystems is represented as E;​​​​ 2) When there is a load on the random subsystem, the following relationship exists between S and E according to the system equation of the deterministic subsystem: S=hE (3) Where h=[h1 h2 … h u ] T ,h k is an M-order row vector, k=1,2,...u,h k The mth element of 3) The number of deterministic subsystem nodes with known responses is u, and the number of random subsystems is M. When u ≥ M, the energy response of the random subsystem is calculated by equation (3) and the known node responses of the deterministic subsystem.

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