Transfer path analysis method based on modal parameter reconstruction of frequency response function

Abstract

Disclosed is a transfer path analysis method based on modal parameter reconstruction of frequency response function, in which, according to the mechanical structure characteristics of a to-be-tested object and the arrangement of measuring points of the transfer path, one row or one column of a system-level coupled frequency response function matrix is obtained by hammer percussion test and modal parameters are solved; the modal parameters and modal shapes are reconstructed into system unmeasurable frequency response functions by minimum norm least square method, and a system-level coupled frequency response function matrix is obtained; the system-level coupled frequency response function matrix is virtually decoupled by inverse substructure method, and a passive substructure frequency response function matrix is obtained; the interface force between the main and passive substructures and the target point response are inversely solved by using the passive substructure frequency response function matrix, and contribution evaluation is performed.

Description

Technical Field

[0001] This invention relates to the field of vibration analysis technology for mechanical equipment, and in particular to a method for transmission path analysis based on modal parameter reconstruction of frequency response function. Background Technology

[0002] The transmission systems of high-end equipment are characterized by numerous internal components and complex coupling connections. Vibrations generated by the active components during operation are transmitted to the entire equipment via connecting parts. If an unexplained fault occurs in an internal component, the vibration signal carrying the fault will be transmitted and attenuated through the internal multi-layered coupling structure. This makes it difficult for external measuring points to detect the fault signal, allowing continued operation to exacerbate the fault threat and lead to unpredictable consequences. The general method for assessing the vibration transmission contribution from the vibration source to the target point in a mechanical structure and clarifying the fault transmission contribution is transmission path analysis. Transmission path analysis can be divided into frequency response function-based and transmission path analysis methods based on different transmission characteristics. Generally, for transmission systems with high correlation, frequency response function-based transmission path analysis is more reliable.

[0003] Therefore, obtaining the frequency response function is a crucial step in transfer path analysis. A common method involves exciting one point of the mechanical structure with a hammer or vibrator while simultaneously measuring the response at another point, estimating the frequency response function through the excitation and response. However, for complex mechanical systems, the spatial location for obtaining the frequency response function is often narrow, making hammer or vibrator excitation difficult, thus rendering the frequency response function unmeasurable. Furthermore, the active and passive structures of the mechanical structure are often coupled; traditional transfer path analysis methods require dismantling the active structure to decouple the frequency response function, which is time-consuming and labor-intensive.

[0004] The information disclosed in the background section is only intended to enhance the understanding of the background of the present invention, and therefore may contain information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide a transmission path analysis method based on modal parameter reconstruction of frequency response function, which realizes transmission path analysis of complex mechanical structures, obtains the main transmission path of fault vibration signal, and thus provides a theoretical basis for obtaining the main path location of external sensors for detecting faults in mechanical systems, and solves the problems of unmeasurable frequency response function and decoupling of active and passive structures.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] The present invention provides a method for propagation path analysis based on modal parameter reconstruction of frequency response function, comprising:

[0008] Step 1: Determine the coupling relationship and coupling points of the active substructure containing the excitation source and the passive substructure without the excitation source in the test object. The target point is the location where the sensor is installed on the passive substructure to obtain response data. The measurement points are arranged near the coupling point and the target point. Use a hammer to strike the single-point sensor to pick up the multi-point response. Use power spectral density estimation to obtain a row or column of the system-level coupled frequency response function matrix. Use the least squares complex frequency domain method to solve for the modal parameters and mode shapes.

[0009] Step 2: Reconstruct the unmeasurable frequency response function of the system using the least norm least squares method on the modal parameters and mode shapes. The unmeasurable frequency response function of the system and a row or column of the system-level coupled frequency response function matrix together constitute the system-level coupled frequency response function matrix.

[0010] Step 3: Perform virtual decoupling using the inverse substructure method on the system-level coupled frequency response function matrix to obtain the passive substructure frequency response function matrix;

[0011] Step 4: Using the frequency response function matrix of the passive substructure, inversely solve the interface force and target point response between the main and passive substructures, and evaluate their contribution.

[0012] In the method described above, in step 1, a hammer is used to strike the measuring point to collect the multi-point response, and the average value of three or more strikes is used for calculation to obtain a row or column of the system-level coupled frequency response function matrix to solve for the modal parameters.

[0013] In the method described above, step 2) involves using the least squares method to fit the first N modal representations of the unmeasurable frequency response function of the system.

[0014] The vibration equation of an Nth-order mechanical system is expressed as:

[0015]

[0016] Based on complex modal theory, modal analysis yields 2N complex conjugate eigenvalues ​​and complex conjugate eigenvectors. Introducing the state variable y, the above equation is rewritten as follows:

[0017]

[0018] in,

[0019] The equations after modal decoupling are:

[0020]

[0021] in, R(t) = Ψ T Q(t), y=Ψη, where Ψ is the eigenvector of the rewritten equation, expressed as: Φ = [Φ1 … Φ r …Φ N ] is the modal vector matrix, Let Φ be a complex conjugate vector matrix, and Λ = diag(p1, ..., p2). r , ..., p N ) is the modal eigenvalue matrix, Let Λ be the complex conjugate matrix.

[0022] The solution to the modal decoupling equation is:

[0023] η=(jωA p +B p ) -1 R(t)

[0024] Then y = Ψη = Ψ(jωA) p +B p ) -1 R(t)=Ψ(jωA p +B p ) -1 Ψ T Q(t) represents the relationship between the response and the stimulus:

[0025]

[0026] Because A p and B p They are all diagonal arrays, so According to frequency response function theory, x = HF. Therefore, the frequency response function can be expressed in terms of modal parameters as follows:

[0027]

[0028] It can be expressed in the form containing pole parameters as follows:

[0029]

[0030] in r = 1, 2, ..., N, where r is the r-th order mode shape vector, m is the number of channels, and T represents matrix transpose. Φ is the vector of the r-th mode shape. r The conjugate vector of p r For the r-th order pole, For the r-th conjugate pole, the modal parameters include pole parameters and mode shape parameters, α r With β r The parameters to be solved;

[0031] The frequency response function obtained from step 1 is H. o (ω), which represents the o-th row of the frequency response function matrix H(ω), mathematically expressed as:

[0032]

[0033] Among them, let

[0034]

[0035]

[0036] Then the frequency response function of the o-th row above is expressed as:

[0037] H o =D o X

[0038] The above equation is a system of linear equations, and its least norm least squares solution is:

[0039]

[0040] The reconstructed system-level coupled frequency response function matrix is ​​then...

[0041]

[0042] in

[0043]

[0044] In the method described, in step 3, the frequency response function matrix of the passive substructure is obtained by solving the inverse substructure method.

[0045] The system-level coupling frequency response function matrix and the passive substructure frequency response function matrix are expressed by the following formulas:

[0046]

[0047]

[0048] Among them, [H S ] = H(jω) represents the system-level coupling frequency response function matrix, [H A Let be the frequency response function matrix of the passive substructure, where o(a), i(a), and c(a) represent the output coordinates, input coordinates, and coupling coordinates of substructure A, respectively. Similarly, o(b), i(b), and c(b) represent the output coordinates, input coordinates, and coupling coordinates of substructure B, respectively. The frequency response function of the substructure can be expressed as:

[0049]

[0050]

[0051] [H A ] o(a)c(a) =[HS ] o(a)c(a) ([I]-[P]1([P]1+[P] b [P]1([P] a +[I])) -1 ) -1

[0052] [H A ] o(a)i(a) =[H S ] o(a)i(a) +[H A ] o(a)c(a) [C][H A ] c(a))i(a)

[0053] in

[0054]

[0055] [P] a =([H S ] c(a)c(a) -[H S ] c(a)c(b) )[H s ] c(a)c(b)

[0056]

[0057] In the method described above, step 4, the inversion solution of the interface forces between the active and passive substructures adopts a condition number threshold regularization method, the formula of which is as follows:

[0058]

[0059] Where F represents the interfacial force, H represents the frequency response function of the passive substructure, and H H h represents the conjugate transpose of the frequency response function. c =cond(H)=||H||||H -1 || represents the condition number of the frequency response function matrix, h cn The condition number threshold is represented as: m is the number of input channels, n is the number of output channels, and singular value decomposition of H yields U, S, and V, expressed as: H = U / (USV) H X represents the indicator point response.

[0060] In the method described, step 4, the contribution assessment includes overall contribution assessment, frequency band contribution assessment, and characteristic frequency contribution assessment, wherein...

[0061]

[0062] Wherein, the contribution y of target point kk (ω) represents the contribution of paths 1 to n, where the contribution of the i-th path is y. ki (ω), then the full frequency band ω0~ω n The overall contribution is:

[0063]

[0064] The frequency band p contribution is:

[0065]

[0066] The starting frequency of frequency band p is ω p The termination frequency is ω p+1 For characteristic frequencies The contributions are as follows:

[0067]

[0068] In the method described, the modal parameters and mode shapes are reconstructed based on one column or one row of the frequency response function matrix.

[0069] Beneficial effects

[0070] The transmission path analysis method based on modal parameter reconstruction of frequency response function can achieve efficient and accurate reconstruction of unmeasurable frequency response function inside mechanical system, and decouple the active and passive substructures without removing the active end. It provides a new idea for transmission path analysis based on frequency response function, and conducts transmission path analysis of unmeasurable frequency response function reconstruction of mechanical system based on modal parameter reconstruction, which can guide the vibration control of mechanical system and the identification of sensitive paths of excitation source characteristics.

[0071] The above description is merely an overview of the technical solution of the present invention. In order to make the technical means of the present invention clearer and more understandable, so that those skilled in the art can implement it according to the contents of the specification, and in order to make the above and other objects, features and advantages of the present invention more obvious and understandable, specific embodiments of the present invention are described below. Attached Figure Description

[0072] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0073] Various other advantages and benefits of the present invention will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiments below. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort. Furthermore, the same reference numerals denote the same parts throughout the drawings.

[0074] In the attached diagram:

[0075] Figure 1(a) is a schematic diagram of the simulation settings in an embodiment of the present invention, and Figures 1(b) and 1(c) are schematic diagrams of the applied excitation spectrum and its characteristic frequencies in an embodiment of the present invention.

[0076] Figure 2 This is a modal steady-state diagram in an embodiment of the present invention;

[0077] Figure 3 This is a diagram showing the MAC mode confidence matrix results in an embodiment of the present invention;

[0078] Figure 4 This is a graph showing the average relative error of each channel in an embodiment of the present invention;

[0079] Figure 5 This refers to the average relative error of each frequency in the embodiments of the present invention;

[0080] Figures 6(a), 6(b), and 6(c) are comparison diagrams of the reconstructed frequency response function and the theoretical frequency response function in some embodiments of the present invention, as well as the relative error analysis diagrams of each frequency.

[0081] Figures 7(a), 7(b), and 7(c) are comparison diagrams of the results of solving the passive substructure frequency response function and the theoretical substructure frequency response function in some embodiments of the present invention;

[0082] Figures 8(a), 8(b), and 8(c) are comparison diagrams of the results of solving the interface force and the theoretical interface force in the embodiments of the present invention;

[0083] Figure 9 This is a comparison chart of the target point reconstruction response and the target point theoretical response results in an embodiment of the present invention;

[0084] Figures 10(a), 10(b), and 10(c) respectively show the overall contribution evaluation, frequency band contribution evaluation, and characteristic frequency contribution evaluation of each path in the embodiments of the present invention.

[0085] Figure 11 This is a schematic diagram of the coupling relationship and transmission path analysis model of the active and passive substructures in an embodiment of the present invention.

[0086] The present invention will be further explained below with reference to the accompanying drawings and embodiments. Detailed Implementation

[0087] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0088] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0089] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0090] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0091] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0092] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0093] In this invention, unless otherwise explicitly specified and limited, "above" or "below" the second feature can include direct contact between the first and second features, or contact between the first and second features through another feature between them. Furthermore, "above," "over," and "on top" of the second feature includes the first feature directly above or diagonally above the second feature, or simply indicates that the first feature is at a higher horizontal level than the second feature. "Below," "below," and "under" the second feature includes the first feature directly below or diagonally below the second feature, or simply indicates that the first feature is at a lower horizontal level than the second feature.

[0094] To enable those skilled in the art to better understand the technical solutions of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings, and the drawings do not constitute a limitation on the embodiments of the present invention.

[0095] In one embodiment, as shown in Figures 1 to 12... Figure 11 As shown, this disclosure provides a transfer path analysis method based on modal parameter reconstruction of the frequency response function, including the following steps:

[0096] Step 1: Determine the coupling relationship and coupling points between the active substructure containing the excitation source and the passive substructure not containing the excitation source in the test object. Specifically, this involves the cooperation between the active and passive substructures. For example, in a gearbox, the gears and shafts containing the excitation source are the active substructures, while the external housing and bearing housings are the passive substructures. The coupling relationship between the active and passive substructures is bearing stiffness coupling, and the coupling points are the positions of the bearings that mate with the housing. Next, determine the target points. The target points are generally located where sensors can be easily installed on the passive substructures to obtain response data. Measurement points are arranged near the coupling points and the target points. For example... Figure 11 As shown. A single-point sensor is struck by a hammer to pick up the multi-point response. A row or column of the system-level coupled frequency response function matrix is ​​obtained by power spectral density estimation. The modal parameters and mode shapes are solved by the least squares complex frequency domain method.

[0097] Step 2: Reconstruct the unmeasurable frequency response function of the system using the least norm least squares method based on the modal parameters and mode shapes. The unmeasurable frequency response function of the system, together with a row or column of the system-level coupled frequency response function matrix, constitutes the system-level coupled frequency response function matrix. Specifically, while a row or column of the system-level coupled frequency response function is obtained through testing, the values ​​of other rows and columns cannot be obtained. These values ​​are unmeasurable because they cannot be obtained in actual testing due to positional and directional limitations caused by hammer impact. These unmeasurable values ​​are obtained by solving for the modal parameters and mode shapes, and together they constitute the system-level frequency response function. In this embodiment, the system-level coupled frequency response function matrix is ​​8 rows and 8 columns. The first row is actually obtained; it is assumed that the other 7 rows are unmeasurable. The modal parameters and mode shapes are obtained by solving for them, and the values ​​of those 7 unmeasurable rows are reconstructed. These values, together with the measured values ​​of the first row, constitute an 8-row, 8-column matrix.

[0098] Step 3: Perform virtual decoupling using the inverse substructure method on the system-level coupled frequency response function matrix to obtain the passive substructure frequency response function matrix;

[0099] Step 4: Using the frequency response function matrix of the passive substructure, inversely solve the interface force and target point response between the main and passive substructures, and evaluate their contribution.

[0100] In a preferred embodiment of the method, in step 1, a hammer is used to strike the measuring point to collect the multi-point response, and the average value of three or more strikes is used for calculation. A row or column of the system-level coupled frequency response function matrix is ​​obtained by power spectral density estimation, and the modal parameters and mode shapes are solved by the least squares complex frequency domain method.

[0101] In a preferred embodiment of the method, in step 2), the first N modal representations of the unmeasurable frequency response function of the system are fitted using the least squares method.

[0102] The vibration equation of an Nth-order mechanical system is expressed as:

[0103]

[0104] Where M, C, and K represent the system's mass matrix, damping matrix, and stiffness matrix, respectively. F(t) represents the external excitation vector acting on the system. x represents the system's response vector.

[0105] Based on complex modal theory, modal analysis yields 2N complex conjugate eigenvalues ​​and complex conjugate eigenvectors. Introducing the state variable y, the above equation is rewritten as follows:

[0106]

[0107] in,

[0108] The equations after modal decoupling are:

[0109]

[0110] in, R(t) = Ψ T Q(t), y=Ψη, where Ψ is the eigenvector of the rewritten equation, expressed as: Φ = [Φ1 … Φ r …Φ N ] is the modal vector matrix, Let Φ be a complex conjugate vector matrix, and Λ = diag(p1, ..., p2). r , ..., p N ) is the modal eigenvalue matrix, Let Λ be the complex conjugate matrix.

[0111] The solution to the modal decoupling equation is:

[0112] η=(jωA p +B p ) -1 R(t)

[0113] Then y = Ψη = Ψ(jωA) p +B p ) -1 R(t)=Ψ(jωA p +B p ) -1 Ψ T Q(t) represents the relationship between the response and the stimulus:

[0114]

[0115] Because A p and B p They are all diagonal arrays, so According to frequency response function theory, x = HF. Therefore, the frequency response function can be expressed in terms of modal parameters as follows:

[0116]

[0117] In a multi-substructure coupled system, this is the system-level coupled frequency response function, which can be expressed in the form containing pole parameters as follows:

[0118]

[0119] in r = 1, 2, ..., N, where r is the r-th order mode shape vector, m is the number of channels, and T represents matrix transpose. Φ is the vector of the r-th mode shape. r The conjugate vector of p r For the r-th order pole, For the r-th conjugate pole, the modal parameters include pole parameters and mode shape parameters, α r With β r The parameters to be solved;

[0120] The frequency response function obtained from step 1 is H. o (ω), which represents the o-th row of the frequency response function matrix H(ω), mathematically expressed as:

[0121]

[0122] Among them, let

[0123]

[0124]

[0125] Then the frequency response function of the o-th row above is expressed as:

[0126] H o =D o X

[0127] The above equation is a system of linear equations, and its least norm least squares solution is:

[0128]

[0129] The reconstructed system-level coupled frequency response function matrix is ​​then...

[0130]

[0131] in

[0132]

[0133] In a preferred embodiment of the method, in step 3, the frequency response function matrix of the passive substructure is obtained by solving the inverse substructure method.

[0134] The system-level coupling frequency response function matrix and the passive substructure frequency response function matrix are expressed by the following formulas:

[0135]

[0136]

[0137] Among them, [H S ] = H(jω) represents the system-level coupling frequency response function matrix, [H ALet be the frequency response function matrix of the passive substructure, where o(a), i(a), and c(a) represent the output coordinates, input coordinates, and coupling coordinates of substructure A, respectively. Similarly, o(b), i(b), and c(b) represent the output coordinates, input coordinates, and coupling coordinates of substructure B, respectively. The frequency response function of the substructure can be expressed as:

[0138]

[0139]

[0140] [H A ] o(a)c(a) =[H S ] o(a)c(a) ([I]-[P]1([P]1+[P] b [P]1([P] a +[I])) -1 ) -1

[0141] [H A ] o(a)i(a) =[H S ] o(a)i(a) +[H A ] o(a)c(a) [C][H A ] c(a)i(a)

[0142] Intermediate variables [P]1, [P] a and [P] b They are represented as follows:

[0143]

[0144] [P] a =([H S ] c(a)c(a) -[H S ] c(a)c(b) ) -1 [H S ] c(a)c(b)

[0145]

[0146] In a preferred embodiment of the method, in step 4, the inversion solution of the interface force between the active and passive substructures adopts a condition number threshold regularization method, the formula of which is as follows:

[0147]

[0148] Where F represents the interfacial force, H represents the frequency response function of the passive substructure, and H H h represents the conjugate transpose of the frequency response function.c =cond(H)=||H||||H -1 || represents the condition number of the frequency response function matrix, h cn The condition number threshold is represented as: m is the number of input channels, n is the number of output channels, and singular value decomposition of H yields U, S, and V, expressed as: H = U / (USV) H X represents the indicator point response.

[0149] In a preferred embodiment of the method, step 4, the contribution assessment includes overall contribution assessment, band contribution assessment, and characteristic frequency contribution assessment, wherein...

[0150]

[0151] Wherein, the contribution y of target point k k (ω) represents the contribution of paths 1 to n, where the contribution of the i-th path is y. ki (ω), then the full frequency band ω0~ω n The overall contribution is:

[0152]

[0153] The frequency band p contribution is:

[0154]

[0155] The starting frequency of frequency band p is ω p The termination frequency is ω p+1 For characteristic frequencies The contributions are as follows:

[0156]

[0157] In a preferred embodiment of the method, the modal parameters and mode shapes are reconstructed based on one column or one row of the frequency response function matrix.

[0158] In one embodiment, the object to be tested includes a spring-mass structure.

[0159] In one embodiment, a transfer path analysis method based on modal parameters to reconstruct an unmeasurable frequency response function includes the following steps:

[0160] Step 1) Arrange measurement points according to the coupling relationship between the active substructure containing the excitation source and the passive substructure not containing the excitation source of the test object, as well as the transmission path between the coupling point and the target point, and use the least squares complex frequency domain method to solve the modal parameters and mode shapes.

[0161] Specifically, when obtaining a row or column of the system-level frequency response function matrix, a hammer is used to strike the reference point to pick up the multi-point response, and the average value of three or more strikes is used for calculation. The modal parameters and mode shapes are solved using the least squares complex frequency domain method.

[0162] Step 2) The modal parameters and mode shapes obtained in Step 1) are used to reconstruct the unmeasurable frequency response function of the system using the least norm least squares method. The unmeasurable frequency response function of the system, together with a row or a column of the system-level coupled frequency response function matrix obtained in Step 1, constitute the complete system-level coupled frequency response function matrix.

[0163] Specifically, the frequency response function reconstruction method uses the least squares method to fit the first N modes of the frequency response function.

[0164] The vibration equation of a general Nth-order mechanical system can be expressed as:

[0165]

[0166] Where M, C, and K represent the system's mass matrix, damping matrix, and stiffness matrix, respectively. F(t) represents the external excitation vector acting on the system. x represents the system's response vector.

[0167] According to complex modal theory, modal analysis yields 2N complex conjugate eigenvalues ​​and complex conjugate eigenvectors. Introducing the state variable y rewrites the above equation as follows:

[0168]

[0169] in,

[0170] The equations after modal decoupling are:

[0171]

[0172] in, R(t) = Ψ T Q(t), y=Ψη, where Ψ is the eigenvector of the rewritten equation, expressed as: Φ = [Φ1 … Φ r …Φ N ] is the modal vector matrix, Let Φ be a complex conjugate vector matrix, and Λ = diag(p1, ..., p2). r , ..., p N ) is the modal eigenvalue matrix, Let be the complex conjugate matrix of Λ.

[0173] The solution to the modal decoupling equation is:

[0174] η=(jωA p +Bp ) -1 R(t)

[0175] Then y = Ψη = Ψ(jωA) p +B p ) -1 R(t)=Ψ(jωA p +B p ) -1 Ψ T Q(t) gives the following relationship between response and stimulus:

[0176]

[0177] Because A p and B p They are all diagonal arrays, so According to frequency response function theory, x = HF. Therefore, the frequency response function can be expressed in terms of modal parameters as follows:

[0178]

[0179] In a multi-substructure coupled system, this is the system-level coupled frequency response function, which can be expressed in the form containing pole parameters as follows:

[0180]

[0181] in r = 1, 2, ..., N, where r is the r-th order mode shape vector, m is the number of channels, and T represents matrix transpose. Φ is the vector of the r-th mode shape. r The conjugate vector of p r For the r-th order pole, α represents the r-th order conjugate pole, and its modal parameters include pole parameters and mode shape parameters. r With β r These are the parameters to be solved.

[0182] In step 1), the obtained frequency response function is H. o (ω), which represents the o-th row of the frequency response function matrix H(ω). Mathematically, it is represented as:

[0183]

[0184] Among them, let

[0185]

[0186]

[0187] Then the frequency response function of the o-th row above can be expressed as:

[0188] H o =D o X

[0189] The above equation is a system of linear equations, and its least norm least squares solution is:

[0190]

[0191] The reconstructed unmeasurable system-level coupled frequency response function matrix is ​​then...

[0192]

[0193] in

[0194]

[0195]

[0196] Step 3) Perform virtual decoupling using the inverse substructure method on the system-level coupled frequency response function obtained in Step 2) to obtain the complete passive substructure frequency response function matrix;

[0197] Specifically, the inverse substructure method is used to obtain the frequency response function matrix of the entire substructure.

[0198] The system-level coupling frequency response function matrix and the passive substructure frequency response function matrix are expressed as follows:

[0199]

[0200]

[0201] Among them, [H S ] = H(jω) represents the system-level coupling frequency response function matrix, [H A Let be the frequency response function matrix of the substructure, where o(a), i(a), and c(a) represent the output coordinates, input coordinates, and coupling coordinates of substructure A, respectively. Similarly, o(b), i(b), and c(b) represent the output coordinates, input coordinates, and coupling coordinates of substructure B, respectively. The frequency response function of the substructure can be expressed as:

[0202]

[0203]

[0204] [H A ] o(a)c(a) =[H S ] o(a)c(a) ([I]-[P]1([P]1+[P] b [P]1([P] a +[I])) -1 )-1

[0205] [H A ] o(a)i(a) =[H S ] o(a)i(a) +[H A ] o(a)c(a) [C][H A ] c(a)i(a)

[0206] Intermediate variables [P]1, [P] a and [P] b They are represented as follows:

[0207] [P] a =([H S ] c(a)c(a) -[H S ] c(a)c(b) ) -1 [H S ] c(a)c(b)

[0208]

[0209] Step 4) Using the passive substructure frequency response function matrix obtained in Step 3), invert and solve the interface force and target point response between the main and passive substructures, and evaluate their contribution.

[0210] Furthermore, the inversion solution for the interface forces between the active and passive substructures employs a condition number threshold regularization method. The specific formula is as follows:

[0211]

[0212] Where F represents the interfacial force, H represents the frequency response function of the passive substructure, and H H h represents the conjugate transpose of the frequency response function. c =cond(H)=||H||||H -1 || represents the condition number of the frequency response function matrix, h cn The condition number threshold is represented as: m is the number of input channels, n is the number of output channels, and singular value decomposition of H yields U, S, and V, expressed as: H = U / (USV) H X represents the indicator point response.

[0213] Furthermore, the contribution assessment includes overall contribution assessment, band contribution assessment, and characteristic frequency contribution assessment. The specific formulas are as follows:

[0214]

[0215] Wherein, the contribution y of target point k k (ω) represents the contribution of paths 1 to n, where the contribution of the i-th path is y. ki (ω). Then the full frequency band ω0~ω n The overall contribution is:

[0216]

[0217] The frequency band p contribution is:

[0218]

[0219] The starting frequency of frequency band p is ω p The termination frequency is ω p+1 For characteristic frequencies The contributions are as follows:

[0220]

[0221] Example 1:

[0222] 1) An 8-DOF lumped mass simulation model is used. Its schematic diagram and excitation source locations are shown below. Figures 1(a) to 1(c) As shown in Figure 1(a), the model consists of eight mass blocks connected by ten sets of springs and dampers. The connection and coupling relationship is shown in Figure 1(a), with the reference point at mass block m1. Specific parameters are shown in the table below.

[0223] mass m <![CDATA[m1]]> <![CDATA[m2]]> <![CDATA[m3]]> <![CDATA[m4]]> <![CDATA[m5]]> Value (kg) 20 12 15 25 15 mass m <![CDATA[m6]]> <![CDATA[m7]]> <![CDATA[m8]]> / / Value (kg) 30 40 35 / / Stiffness k <![CDATA[k 12 ]]> <![CDATA[k 13 ]]> <![CDATA[k 14 ]]> <![CDATA[k 25 ]]> <![CDATA[k 36 ]]> <![CDATA[Value (10 8 N / m)]]> 10 8 4 2 8 Stiffness k <![CDATA[k 47 ]]> <![CDATA[k 58 ]]> <![CDATA[k 68 ]]> <![CDATA[k 78 ]]> <![CDATA[k 80 ]]> <![CDATA[Value (10 8 N / m)]]> 6 8.3 5.8 7.8 10 Damping c <![CDATA[c 12 ]]> <![CDATA[c 13 ]]> <![CDATA[c 14 ]]> <![CDATA[c 25 ]]> <![CDATA[c 36 ]]> Value (N·s / m) 700 410 200 300 300 Damping c <![CDATA[c 47 ]]> <![CDATA[c 58 ]]> <![CDATA[c 68 ]]> <![CDATA[c 78 ]]> <![CDATA[c 80 ]]> Value (N·s / m) 400 350 360 580 740

[0224] The system composed of mass blocks m1 to m4 is the active substructure B, and the system composed of mass blocks m5 to m8 is the passive dynamic substructure A. The coupling between them is stiffness k. 25 k 36 k 47 and damping c 25 c 36 c 47 The excitation F0 applied to the active substructure at m1 is a periodic pulse excitation, and its mathematical expression is:

[0225] F0(t)=5000sin 1024 (70πt)

[0226] The amplitude-frequency diagram of F0 is shown in Figure 1(b), and its characteristic frequency and magnified diagram are shown in Figure 1(c). Its characteristic frequency is 70Hz.

[0227] The subscripts for stiffness and damping are the stiffness and damping values ​​between the two measurement point numbers. Taking mass block m1 as the reference point, obtain one row of its system-level frequency response function matrix: H1 = [H... 11 … H 18The values ​​in the other 7 rows are obtained by reconstructing modal parameters and mode shapes.

[0228] Modal parameters were extracted using the least squares complex frequency domain method based on H1, and the modal fitting steady-state diagram is shown below. Figure 2 As shown in the figure, 's' represents all parameters being stable, 'f' represents frequency stability, 'd' represents damping stability, and 'o' represents pole instability. The right-hand axis (3-40) represents the order of the mathematical model fitting. The modal parameters are shown in the table below.

[0229]

[0230]

[0231] The modal parameters are subjected to a MAC confidence test. Specifically, the correlation of the modal shape vectors is evaluated using the modal confidence matrix, and the formula is expressed as follows:

[0232]

[0233] Where, Φ i Φ j Let be the i-th and j-th order vectors of the mode shape matrix, respectively. The value of MAC is between 0 and 1. The closer the value is to 0, the less correlated the two mode shape vectors are; the closer the value is to 1, the more correlated the two vectors are. The modal confidence matrix results are as follows: Figure 3 As shown.

[0234] 2) Reconstruct the frequency response function using the least norm least squares method based on the obtained modal parameters and mode shapes to obtain the complete frequency response function matrix;

[0235] Specifically, the frequency response function reconstruction method uses the least squares method to fit the first N modes of the frequency response function, where N = 8 in this embodiment.

[0236] In step 1), the obtained frequency response function is H1(ω), which represents the first row of the frequency response function matrix H(ω). Mathematically, it is represented as:

[0237]

[0238] Let X = [α1, ... α r ,…α N ,β1,…β r ,…β N , T 2N×1 ,

[0239]

[0240]

[0241] Then the frequency response function in the first row above can be expressed as:

[0242] H1 = D1X

[0243] The above equation is a system of linear equations, and its least norm least squares solution is:

[0244]

[0245] The reconstructed unmeasurable system-level coupled frequency response function matrix is:

[0246]

[0247] in,

[0248]

[0249]

[0250] By solving the coefficient matrix X, the frequency response function matrix H(jω) is reconstructed, and error analysis is performed on the reconstructed frequency response function. The formula for the average error of each channel is as follows:

[0251]

[0252] Among them, h ij (jω) are the i and j elements of the reconstructed frequency response function matrix, h ij (jω) theory The i and j elements of the theoretical frequency response function matrix, and the average relative error result are as follows: Figure 4 As shown, the results indicate that the average relative error of each channel is mostly less than 4%, and in some cases less than 10%. The relative error analysis of each frequency of the reconstructed frequency response function is presented in the following formula:

[0253]

[0254] The results of relative error analysis of each frequency of the reconstructed frequency response function are as follows: Figure 5 As shown, the results indicate that the error is less than 10% at most frequency points, with a few frequency points having an error greater than 10%, but the overall average error is less than 30%. The error is relatively large at 808.125Hz, at 79.7%, which is the anti-resonance peak. A relative error analysis of the reconstructed frequency response function is performed, and the formula is as follows:

[0255]

[0256] The reconstructed frequency response functions and relative error analysis results are shown in Figures 6(a), 6(b), and 6(c). The results show that the reconstructed frequency response functions are in good agreement with the theoretical frequency response functions, with errors less than 5% at most frequency points and the resonant frequency. However, at the anti-resonant frequency, the relative errors of the frequency response functions of each channel are relatively large.

[0257] 3) The reconstructed system-level coupled frequency response function is virtually decoupled using the inverse substructure method to obtain the complete passive substructure frequency response function matrix; the formulas for the system-level coupled frequency response function matrix and the passive substructure frequency response function matrix are as follows. The comparison between the calculated and theoretical results is shown in Figures 7(a), 7(b), and 7(c). The results show that the calculated values ​​and theoretical values ​​are basically in agreement, with errors at a few frequencies.

[0258] The system-level frequency response function and the frequency response function of substructure A can be expressed as follows:

[0259]

[0260]

[0261] In this embodiment, the input coordinate is 1, the output coordinate is 8, and the coupling coordinates of substructures A and B are 5, 6, 7 and 2, 3, 4 respectively. Therefore, their correspondence is expressed as follows:

[0262]

[0263]

[0264] Among them, [H S ] = H(jω) represents the system-level coupling frequency response function matrix, [H A ] is the frequency response function matrix of the substructure, where,

[0265]

[0266]

[0267] [H A ] o(a)c(a) =[H S ] o(a)c(a) ([I]-[P]1([P]1+[P] b [P]1([P] a +[I])) -1 ) -1

[0268] [H A ] o(a)i(a) =[H S ] o(a)i(a) +[H A] o(a)c(a) [C][H A ] c(a)i(a)

[0269] Intermediate variables [P]1, [P] a and [P] b They are represented as follows:

[0270] [P] a =([H S ] c(a)c(a) -[H S ] c(a)c(b) ) -1 [H S ] c(a)c(b)

[0271]

[0272] 4) Obtain the frequency response function matrix of the passive substructure and use it to solve for the interfacial forces between the main and passive substructures;

[0273] Furthermore, the inversion solution for the interfacial forces between the active and passive substructures employs a condition number threshold regularization method. The specific formula is as follows:

[0274]

[0275] Where F represents the interfacial force, H represents the frequency response function of the passive substructure, and H H h represents the conjugate transpose of the frequency response function. c =cond(H)=||H||||H -1 || represents the condition number of the frequency response function matrix, h cn The condition number threshold is represented as: m is the number of input channels, n is the number of output channels, and singular value decomposition of H yields U, S, and V, expressed as: H = U / (USV) H X represents the response of the indicator point. In this embodiment, the indicator points are m5 to m7, and the target point is m8.

[0276] Inversion solution for the passive substructure coupling position m5~ m The interfacial forces F5, F6, and F7 are shown in Figures 8(a), 8(b), and 8(c).

[0277] The target point response is obtained from the acquired substructure frequency response function matrix and the interface force, and its contribution is evaluated. The calculated target point response is compared with the theoretical response, for example... Figure 9 As shown in the figure, the results show that the theoretical values ​​and the calculated values ​​are in good agreement.

[0278] The contribution assessment includes overall contribution assessment, frequency band contribution assessment, and characteristic frequency contribution assessment. The specific formulas are as follows:

[0279]

[0280] Wherein, the contribution y of target point k k (ω) represents the contribution of paths 1 to n, where the contribution of the i-th path is y. ki (ω). Then the full frequency band ω0~ω n The overall contribution is:

[0281]

[0282] The frequency band p contribution is:

[0283]

[0284] The starting frequency of frequency band p is ω p The termination frequency is ω p+1 For characteristic frequencies The contributions are as follows:

[0285]

[0286] The results are shown in Figures 10(a), 10(b), and 10(c). The overall contribution ranking of the paths is: Path 1 > Path 3 > Path 2. The frequency band with the largest contribution for each path is 750Hz to 1500Hz. At the characteristic frequency of 70Hz, the contribution ranking is: Path 2 > Path 3 > Path 1. Therefore, if vibration isolation measures are to be implemented for overall vibration, vibration isolation measures for Path 1 in the 750Hz to 1500Hz frequency band should be given priority. If feature extraction is to be performed, feature extraction for Path 2 should be given priority.

[0287] Although embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the specific embodiments and application fields described above. The specific embodiments described above are merely illustrative and instructive, and not restrictive. Those skilled in the art can make many other forms based on the guidance of this specification and without departing from the scope of protection of the claims of the present invention, and all of these are within the scope of protection of the present invention.

Claims

1. A method for analyzing the transfer path of a frequency response function based on modal parameter reconstruction, characterized in that, It includes the following steps: Step 1: Determine the coupling relationship and coupling points of the active substructure containing the excitation source and the passive substructure without the excitation source in the test object. The target point is the location where the sensor is installed on the passive substructure to obtain response data. The measurement points are arranged near the coupling point and the target point. Use a hammer to strike the single-point sensor to pick up the multi-point response. Use power spectral density estimation to obtain a row or column of the system-level coupled frequency response function matrix. Use the least squares complex frequency domain method to solve for the modal parameters and mode shapes. Step 2: Reconstruct the unmeasurable frequency response function of the system using the least norm least squares method on the modal parameters and mode shapes. The unmeasurable frequency response function of the system and a row or column of the system-level coupled frequency response function matrix together constitute the system-level coupled frequency response function matrix. Step 3: Perform virtual decoupling using the inverse substructure method on the system-level coupled frequency response function matrix to obtain the passive substructure frequency response function matrix; Step 4: Using the frequency response function matrix of the passive substructure, invert and solve the interface force and target point response between the main and passive substructures, and evaluate their contribution. In step 4, the interface forces between the active and passive substructures are solved by a condition number threshold regularization method, the formula of which is as follows: ； in, Indicates interfacial force. This represents the frequency response function of the passive substructure. This represents the conjugate transpose of the frequency response function. The condition number represents the frequency response function matrix. The condition number threshold is represented as: , Input the number of channels. For the number of output channels, Singular value decomposition yields , , , is represented as: , For the indicator point response; In step 4, the contribution assessment includes overall contribution assessment, band contribution assessment, and characteristic frequency contribution assessment, wherein... ； Among them, the contribution of target point k The contribution is divided into paths 1 to n, where the th path... The contribution of each path is Then the entire frequency band ~ The overall contribution is: ； The frequency band p contribution is: ； The starting frequency of the frequency band p is The termination frequency is For characteristic frequencies The contributions are as follows: ； The modal parameters and mode shapes are used to reconstruct the entire frequency response function matrix based on one column or one row of the frequency response function matrix.

2. The method according to claim 1, characterized in that, In step 1, a hammer is used to strike the measuring point to collect the multi-point response, and the average value of three or more strikes is used for calculation to obtain a row or column of the system-level coupled frequency response function matrix to solve for the modal parameters.

3. The method according to claim 1, characterized in that, In step 3, the frequency response function matrix of the passive substructure is obtained by solving the inverse substructure method. The system-level coupling frequency response function matrix and the passive substructure frequency response function matrix are expressed by the following formulas: ； ； in, Represents the system-level coupled frequency response function matrix. Let be the frequency response function matrix of the passive substructure, where ； ； ； ； in ； ； 。

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