Dense modal response reconstruction method based on state transfer matrix

Abstract

The present invention provides a dense modal response reconstruction method based on a state transfer matrix, belonging to the field of structural response reconstruction, and comprising the following steps: S1, decomposing each order modal response, separating the initial phase from the initial amplitude in the response, and expressing each order modal response in matrix form; S2, constructing the state matrices of the measured point and the point to be reconstructed, respectively, and obtaining the modal mode ratio matrix of the measured point and the point to be reconstructed; S3, constructing the state transfer matrix between the point to be reconstructed and the measured point; S4, reconstructing the response of the point to be reconstructed according to the state transfer matrix. The dense modal response reconstruction method based on a state transfer matrix proposed by the present invention, based on the modal superposition method of linear systems, separates the initial phase from the initial matrix in the modal response, and reconstructs the response according to the state transfer matrix between the point to be reconstructed and the measured point. This method can handle the response reconstruction of structures with dense modal characteristics and ensure the accuracy of the response reconstruction.

Description

Technical Field

[0001] The present invention belongs to the field of structural response reconstruction, and in particular relates to a dense modal response reconstruction method based on a state transfer matrix. Background Art

[0002] Structural health monitoring (SHM) utilizes information from a series of sensors deployed on a target system to infer its structural integrity and quantify its performance degradation. However, in practical applications, economic constraints make it impossible to deploy sensors everywhere. Furthermore, the complexity of the system's structural geometry makes it difficult to place sensors in some critical structural areas. Therefore, when the system response in critical structural areas cannot be determined, dynamic response reconstruction techniques are used to reconstruct the system response in these areas using the measured system response data.

[0003] Current response reconstruction methods are mainly divided into two categories: frequency domain reconstruction methods and time domain reconstruction methods. Frequency domain methods generally require obtaining the excitation action position in advance and using more sensors than the number of excitations. Frequency domain methods also use data in the frequency response domain and use an inverse Fourier transform to convert the reconstructed frequency domain response back to the time domain, which introduces additional computational costs and numerical truncation errors. In contrast, time domain reconstruction methods have higher computational efficiency. Currently, response reconstruction methods based on empirical mode decomposition are more widely used. This method only requires one sensor to record the response data, extracts the modal responses of each order through a bandpass filter and empirical mode decomposition algorithm, and reconstructs the response through the mode ratio between the point to be reconstructed and the measured point. This method is suitable for dynamic response reconstruction based on measurements of different types of sensors, but it is difficult to handle structures with dense modal characteristics.

[0004] Dense modes are a challenging problem in structural parameter identification and structural response reconstruction. Dense modes have a significant impact on structural response reconstruction based on empirical mode decomposition (EMD). This is because if the two natural frequencies of a vibrating structure are close and their damping ratios are large, they appear as two overlapping peaks in the frequency domain, making it difficult to accurately extract the modal responses of each order using a bandpass filter. This reduces the accuracy of structural response reconstruction. Therefore, research on methods for reconstructing structural responses with dense modes is of great significance and promise. Summary of the Invention

[0005] In order to address the above-mentioned shortcomings of the prior art, the present invention proposes a dense modal response reconstruction method based on the state transfer matrix. The method first decomposes the modal response q(t) of each order, separates the initial phase and initial amplitude in the response, and expresses the response in the matrix form q(t) = APV(t); secondly, the state matrix S of the measured point and the point to be reconstructed are constructed respectively. k =φk AP, S u =φ u AP; Third, construct the state transfer matrix T between the point to be reconstructed and the measured point uk =V + φ u，k V; finally, the response reconstruction x of the point to be reconstructed is completed according to the state transfer matrix u =x k T uk Compared with traditional methods, the reconstruction method of the present invention can handle the response reconstruction of structures with dense modal characteristics and can ensure the accuracy of the response reconstruction.

[0006] Specifically, the present invention proposes a dense modal response reconstruction method based on a state transfer matrix, which includes the following steps:

[0007] S1. Decompose the modal responses of each order, separate the initial phase and initial amplitude in the response, and express the modal responses of each order in matrix form:

[0008] q(t)=APV(t)

[0009] Where q(t) is the modal response of each order, A is the initial amplitude matrix, P is the initial phase matrix, V is the vibration basis vector matrix, and t is time;

[0010] S2. Construct the state matrices of the measured points and the points to be reconstructed respectively and obtain the modal vibration ratio matrix of the measured points and the points to be reconstructed. The state matrix S of the measured points is k And the state matrix S of the point to be reconstructed u They are:

[0011] S k =φ k AP

[0012] S u =φ u AP

[0013] Where S is the state matrix, u and k represent the points to be reconstructed and the measured points respectively, and φ is the modal vibration matrix;

[0014] Based on the state matrices of the measured points and the points to be reconstructed, the modal shape ratio matrix of the measured points and the points to be reconstructed is obtained:

[0015] S u =S k φ u，k

[0016] Among them, φ u，k represents the modal shape ratio matrix;

[0017] S3: Constructing a state transfer matrix between the point to be reconstructed and the measured point, specifically including the following sub-steps:

[0018] S31. According to the modal superposition method, the response x of the free vibration of the structure to be reconstructed is converted to u Expressed as:

[0019] x u =φ u q(t)=φ u APV=S u V;

[0020] S32. Construct a relationship between the measured point and the point to be reconstructed based on the modal shape ratio matrix:

[0021] x u =S u V=S k φ u，k V;

[0022] S33, using the measured point information to solve the measured point state matrix S k :

[0023] S k =x k V +

[0024] Among them, + represents pseudo-inverse operation;

[0025] S34, based on the measured point state matrix S k The response x of the free vibration of the structure at the point to be reconstructed u Expressed as:

[0026] x u =S u V=S k φ u，k V=x k V + φ u，k V;

[0027] S35. Construct the state transfer matrix between the point to be reconstructed and the measured point as follows:

[0028] T uk =V + φ u，k V

[0029] Among them, T uk is the state transfer matrix between the point to be reconstructed and the measured point;

[0030] S4. Reconstruct the response of the point to be reconstructed according to the state transfer matrix:

[0031] xu =x k T uk ;

[0032] Based on the above formula, the frequency domain signal of the point to be reconstructed is obtained.

[0033] Preferably, step S1 specifically includes the following sub-steps:

[0034] S11. The modal response of the free vibration of the structure is:

[0035]

[0036] Among them, ξ i 、ω i Represents the damping ratio and natural frequency of each mode, u i and Represent the initial amplitude and initial phase of each order response, n is the number of truncated modes, and t is time;

[0037] S12. Separate the initial phase and initial amplitude in the response:

[0038]

[0039] S13. Express the separated modal responses in matrix form:

[0040] q(t)=APV(t)

[0041] in:

[0042]

[0043] q=[q1(t) … q n (t)] T

[0044]

[0045]

[0046] Preferably, step S2 specifically includes the following sub-steps:

[0047] S21. According to the modal superposition method, the responses of the free vibration of the structure at the measured point and the point to be reconstructed are expressed as:

[0048] x k =φ k q(t)=φ k APV

[0049] x u =φ u q(t)=φ u APV

[0050] Where u and k represent the points to be reconstructed and the measured points respectively, and φ represents the modal vibration matrix;

[0051] S22. Define the state matrix S of the structural vibration as:

[0052] S = φAP;

[0053] S23, the state matrix of the measured points and the points to be reconstructed is expressed as:

[0054] S k =φ k AP

[0055] S u =φ u AP

[0056] Among them, S k is the state matrix of the measured point, S u is the state matrix of the point to be reconstructed.

[0057] Preferably, step S21 specifically includes the following sub-steps:

[0058] S211. The motion equations of a multi-degree-of-freedom system are as follows:

[0059]

[0060] Where M is the mass matrix, C is the damping matrix, and K is the stiffness matrix. Both the column vector x and the force F(t) have degrees of freedom (DOF). If the number of degrees of freedom of the system is n, then the matrix size is , and the column vector x and the force F(t) have dimensions of n×1.

[0061] S212. Calculate system modal parameters:

[0062] The premise of modal superposition is to calculate the eigenfrequencies and the corresponding vibration modes, which are calculated using the following eigenvalue equation:

[0063] (K-ω 2 M)φ=0;

[0064] The calculation results are a set of natural frequencies ω i and the corresponding vibration mode φ i , where i ranges from 1 to n;

[0065] S213, System modal orthogonality:

[0066] The system stiffness matrix is ​​orthogonal:

[0067]

[0068]

[0069] Among them, m qi It is called the i-th order modal mass. Similarly, for the stiffness matrix:

[0070]

[0071]

[0072] Among them, K qi It is called the i-th order modal stiffness. When the system damping meets the requirements of proportional damping, C = aM + βK, where α and β are proportional coefficients. The vibration mode also meets the orthogonality with respect to the damping matrix:

[0073]

[0074]

[0075] Among them, C qi It is called the i-th order modal damping;

[0076] S214, System modal superposition method:

[0077] According to the orthogonality of vibration modes, we know that the vibration modes of each order are linearly independent. That is, the n vibration modes of an n-dimensional system constitute a set of bases of an n-dimensional vector space. The coordinate system based on this orthogonal basis is called the modal coordinate system. The displacement response x of the system is expressed as:

[0078] x=φq。

[0079] Compared with the prior art, the beneficial technical effects of the present invention are:

[0080] (1) The dense modal response reconstruction method based on the state transfer matrix proposed in this invention takes the modal superposition method of linear systems as its theoretical basis, separates the initial phase from the initial matrix in the modal response, constructs the system state matrix and the state transfer matrix, and reconstructs the response according to the state transfer matrix between the point to be reconstructed and the measured point.

[0081] (2) Compared with the traditional time-domain reconstruction method, the dense modal response reconstruction method based on the state transfer matrix proposed in the present invention can process the response reconstruction of a dense modal characteristic structure based on only one sensor response data, and can ensure the reconstruction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] Figure 1 Schematic diagram of the method flow of the dense modal response reconstruction method based on the state transfer matrix of the present invention;

[0083] Figure 2 A schematic diagram of a multi-degree-of-freedom system in a specific embodiment of the present invention;

[0084] Figure 3 Schematic diagram of time-frequency domain information of the measured point DOF-1 in a specific embodiment of the present invention;

[0085] Figure 4a Schematic diagram comparing the theoretical result and the reconstructed result of the DOF-2 displacement signal in a specific embodiment of the present invention;

[0086] Figure 4b Schematic diagram of the error between the theoretical result and the reconstruction result of the DOF-2 displacement signal in a specific embodiment of the present invention;

[0087] Figure 4c It is an enlarged schematic diagram comparing the theoretical result and the reconstructed result of the DOF-2 displacement signal in a specific embodiment of the present invention;

[0088] Figure 5a Schematic diagram comparing the theoretical result and the reconstructed result of the DOF-3 displacement signal in a specific embodiment of the present invention;

[0089] Figure 5b Schematic diagram of the error between the theoretical result and the reconstructed result of the DOF-3 displacement signal in a specific embodiment of the present invention;

[0090] Figure 5c It is an enlarged schematic diagram comparing the theoretical result and the reconstructed result of the DOF-3 displacement signal in a specific embodiment of the present invention. DETAILED DESCRIPTION

[0091] Specifically, the present invention proposes a dense modal response reconstruction method based on the state transfer matrix, such as Figure 1 As shown, it includes the following steps:

[0092] S1. Decompose the modal responses q(t) of each order, separate the initial phase and initial amplitude in the response and express them in matrix form:

[0093] q(t)=APV(t)

[0094] Where q(t) is the modal response of each order, A is the initial amplitude matrix, P is the initial phase matrix, V is the vibration basis vector matrix, and t is time.

[0095] This step specifically includes the following sub-steps:

[0096] S11. The modal response of the free vibration of the structure is:

[0097]

[0098] Among them, ξ i 、ω iRepresent the damping ratio and natural frequency of each mode respectively, and n is the number of truncated modes.

[0099] S12. Separate the initial phase and initial amplitude in the response:

[0100]

[0101] S13, expressed in matrix form:

[0102] q(t)=APV(t)

[0103] in:

[0104]

[0105] q=[q1(t) … q n (t)] T

[0106]

[0107]

[0108] S2. Construct the state matrix of the measured points and the points to be reconstructed:

[0109] The state matrices of the measured points and the points to be reconstructed are:

[0110] S k =φ k AP, S u =φ u AP.

[0111] This step specifically includes the following sub-steps:

[0112] S21. According to the modal superposition method, the response of the free vibration of the structure can be expressed as:

[0113] x k =φ k q(t)=φ k APV

[0114] x u =φ u q(t)=φ u APV

[0115] Where u and k represent the points to be reconstructed and the measured points, respectively, and φ represents the modal vibration matrix.

[0116] The step S21 specifically includes the following sub-steps:

[0117] S211. The motion equations of a multi-degree-of-freedom system are as follows:

[0118]

[0119] Where M is the mass matrix, C is the damping matrix, and K is the stiffness matrix. Both the column vector x and the force F(t) have degrees of freedom (DOF). If the number of DOFs of the system is n, then the matrix size is , and the column vector x and the force F(t) have dimensions of n × 1.

[0120] S212. Calculate system modal parameters:

[0121] The premise of modal superposition is to calculate the eigenfrequency and the corresponding vibration mode, which is usually calculated using the following eigenvalue equation:

[0122] (K-ω 2 M)φ=0

[0123] The calculation results are a set of natural frequencies ω i and the corresponding vibration mode φ i , where i ranges from 1 to n.

[0124] S213, System modal orthogonality:

[0125] The system stiffness matrix is ​​orthogonal:

[0126]

[0127]

[0128] Among them, m qi It is called the i-th order modal mass. Similarly, for the stiffness matrix:

[0129]

[0130]

[0131] Among them, K qi It is called the i-th order modal stiffness. When the system damping meets the requirements of proportional damping (C = αM + βK, where α and β are proportional coefficients), the vibration mode also meets the orthogonality with respect to the damping matrix:

[0132]

[0133]

[0134] Among them, C qi is called the i-th order modal damping.

[0135] S214, System modal superposition method:

[0136] According to the orthogonality of the vibration modes, the vibration modes of each order are linearly independent. That is, the n vibration modes of an n-dimensional system form a set of bases of an n-dimensional vector space. Any vector in this n-dimensional vector space can be represented by this set of orthogonal bases. The coordinate system based on this orthogonal base is called the modal coordinate system. Therefore, the displacement response x of the system can be expressed as:

[0137] x=φq。

[0138] S22. Define the state matrix S of the structural vibration as:

[0139] S=φAP.

[0140] S23, the state matrix of the measured points and the points to be reconstructed is expressed as:

[0141] S k =φ k AP

[0142] S u =φ u AP

[0143] Among them, S k is the state matrix of the measured point, S u is the state matrix of the point to be reconstructed.

[0144] S3. Construct the state transfer matrix between the point to be reconstructed and the measured point:

[0145] T uk =V + φ u，k V.

[0146] This step specifically includes the following sub-steps:

[0147] S31. According to the modal superposition method, the response x of the free vibration of the structure to be reconstructed is converted to u Expressed as:

[0148] x u =φ u q(t)=φ u APV=S u V.

[0149] S32. Construct a relationship between the measured point and the point to be reconstructed based on the modal shape ratio matrix:

[0150] x u =S u V=S k φ u，k V.

[0151] S33, using the measured point information to solve the measured point state matrix S k :

[0152] S k =x k V +

[0153] Here, + represents a pseudo-inverse operation.

[0154] S34, based on the measured point state matrix S k The response x of the free vibration of the structure at the point to be reconstructed u Expressed as:

[0155] x u =S u V=S k φ u，k V=x k V + φ u，k V.

[0156] S35. Construct the state transfer matrix between the point to be reconstructed and the measured point as follows:

[0157] T uk =V + φ u，k V

[0158] Among them, T uk is the state transfer matrix between the point to be reconstructed and the measured point.

[0159] S4. Reconstruct the response of the point to be reconstructed according to the state transfer matrix:

[0160] x u =x k T uk .

[0161] Based on the above formula, the frequency domain signal of the point to be reconstructed is obtained.

[0162] Specific application examples

[0163] In this example, the displacement signal of DOF-1 is used to reconstruct the displacement signal of DOF-2 and the displacement signal of DOF-3. Figure 2 A multi-degree-of-freedom system is shown, and the structural parameters of the system are as follows:

[0164]

[0165] Among them, the initial conditions of the multi-degree-of-freedom system are shown in Table 1:

[0166] Table 1

[0167]

[0168] Specifically, the signal reconstruction process of this embodiment includes the following steps:

[0169] S1. Decompose the modal responses of each order, separate the initial phase and initial amplitude in the response, and express the modal responses of each order in matrix form:

[0170] q(t)=APV(t)

[0171] Where q(t) is the modal response of each order, A is the initial amplitude matrix, P is the initial phase matrix, V is the vibration basis vector matrix, and t is time.

[0172] S2, construct the state matrix of the displacement signal of DOF-1 and the displacement signal of DOF-2 respectively and obtain the modal vibration ratio matrix of the measured point and the point to be reconstructed. The state matrix S of the measured point k And the state matrix S of the point to be reconstructed u They are:

[0173] S k =φ k AP

[0174] S u =φ u AP

[0175] Where S is the state matrix, u and k represent the points to be reconstructed and the measured points respectively, and φ is the modal vibration matrix.

[0176] Afterwards, the modal shape ratio matrix of the measured point and the point to be reconstructed is obtained based on the state matrix of the displacement signal of DOF-1 and the displacement signal of DOF-2:

[0177] S u =S k φ u，k

[0178] Among them, φ u，k represents the mode shape ratio matrix.

[0179] S3. Construct the state transfer matrix between the displacement signal of DOF-1 and the displacement signal of DOF-2:

[0180] T uk =V + φ u，k V.

[0181] This step specifically includes the following sub-steps:

[0182] S31, according to the modal superposition method, the free vibration response x of the displacement signal of DOF-2 is converted to u Expressed as:

[0183] x u=φ u q(t)=φ u APV=S u V.

[0184] S32. Construct a relationship between the displacement signal of DOF-1 and the displacement signal of DOF-2 according to the modal vibration shape ratio matrix:

[0185] x u =S u V=S k φ u，k V.

[0186] S33, using the displacement signal information of DOF-1 to solve the measured point state matrix S k :

[0187] S k =x k V + .

[0188] Wherein, + represents pseudo-inverse operation. In this embodiment, the time-frequency domain information of the point has been measured, that is, the time-frequency information of the DOF-1 displacement signal is as follows: Figure 3 shown.

[0189] S34, state matrix S based on the displacement signal of DOF-1 k The response x of the free vibration of the structure at the point to be reconstructed u Expressed as:

[0190] x u =S u V=S k φ u，k V=x k V + φ u，k V.

[0191] S35. Construct the state transfer matrix as follows:

[0192] T uk =V + φ u，k V

[0193] Among them, T uk is the state transfer matrix between the point to be reconstructed and the measured point.

[0194] S4. Reconstruct the response of the point to be reconstructed according to the state transfer matrix:

[0195] x u =x k T uk .

[0196] Then, the frequency domain signal is reconstructed by the above response reconstruction formula. In this embodiment, the displacement signal of DOF-2 reconstructed by the displacement signal of DOF-1 is as follows: Figure 4a-4c As shown. Figure 4a-4c It can be seen that the reconstruction method of this embodiment can process the response reconstruction of structures with dense modal characteristics based on the response data of only one sensor while ensuring reconstruction accuracy. Partial data of the displacement signal of DOF-1, the actual displacement signal of DOF-2, and the reconstructed displacement signal of DOF-2 are shown in Table 1:

[0197] DOF-1 response signal DOF-2 response signal (theoretical results) DOF-2 response signal (reconstruction result) 0 0 0.006587779 0.000255577 0.099313301 0.105816507 0.001654523 0.195969503 0.202356955 0.005061765 0.288156203 0.294397862 0.011184859 0.374272939 0.380340252 0.020525647 0.452997324 0.458863519 0.033345548 0.523336271 0.528976603 0.049646504 0.584659936 0.590051873 0.069168624 0.636716921 0.641840269 0.091404547 0.679630293 0.684467251

[0198] The displacement signal of DOF-3 is reconstructed using the above method. The displacement signal of DOF-3 reconstructed using the displacement signal of DOF-1 is as follows: Figure 5a-5c As shown. Figure 5a-5c It can be seen that the reconstruction method of this embodiment can process the response reconstruction of structures with dense modal characteristics based on the response data of only one sensor while ensuring reconstruction accuracy. Partial data for the displacement signal of DOF-1, the actual displacement signal of DOF-3, and the reconstructed displacement signal of DOF-3 are shown in Table 2:

[0199]

[0200]

[0201] In summary, it can be seen that the dense modal response reconstruction method based on the state transfer matrix proposed in the present invention, compared with the traditional time domain reconstruction method, can process the response reconstruction of a dense modal characteristic structure based on only one sensor response data, and can ensure the reconstruction accuracy.

[0202] The embodiments described above are merely descriptions of preferred implementations of the present invention and are not intended to limit the scope of the present invention. Without departing from the spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should fall within the scope of protection determined by the claims of the present invention.

Claims

1. A dense modal response reconstruction method based on a state transfer matrix, characterized by: It includes the following steps: S1. Decompose the modal responses of each order, separate the initial phase and initial amplitude in the response, and express the modal responses of each order in matrix form: q(t)=APV(t) Where q(t) is the modal response of each order, A is the initial amplitude matrix, P is the initial phase matrix, V is the vibration basis vector matrix, and t is time; S2. Construct the state matrices of the measured points and the points to be reconstructed respectively and obtain the modal vibration ratio matrix of the measured points and the points to be reconstructed. The state matrix S of the measured points is k And the state matrix S of the point to be reconstructed u They are: S k =φ k AP S u =φ u AP Where S is the state matrix, u and k represent the points to be reconstructed and the measured points respectively, and φ is the modal vibration matrix; Based on the state matrix of the measured point and the point to be reconstructed, the modal vibration ratio matrix of the measured point and the point to be reconstructed is obtained as follows: S u =S k φ u，k Among them, φ u，k represents the modal shape ratio matrix; S3: Constructing a state transfer matrix between the point to be reconstructed and the measured point, specifically including the following sub-steps: S31. According to the modal superposition method, the response x of the free vibration of the structure to be reconstructed is converted to u Expressed as: x u =φ u q(t)=φ u APV=S u V; S32. Construct a relationship between the measured point and the point to be reconstructed based on the modal shape ratio matrix: x u =S u V=S k φ u，k V; S33, using the measured point information to solve the measured point state matrix S k : S k =x k V + Among them, + represents pseudo-inverse operation; S34, based on the measured point state matrix S k The response x of the free vibration of the structure at the point to be reconstructed u Expressed as: x u =s u V=S k φ u，k V=x k V + φ u，k V; S35. Construct the state transfer matrix between the point to be reconstructed and the measured point as follows: T uk =V + φ u，k V Among them, T uk is the state transfer matrix between the point to be reconstructed and the measured point; S4. Reconstruct the response of the point to be reconstructed according to the state transfer matrix: x u =x k T uk Based on the above formula, the frequency domain signal of the point to be reconstructed is obtained.

2. The dense modal response reconstruction method based on the state transfer matrix according to claim 1, characterized in that: Step S1 specifically includes the following sub-steps: S11. The modal response of the free vibration of the structure is: Among them, ξ i 、ω i Represents the damping ratio and natural frequency of each mode, u i and Represent the initial amplitude and initial phase of each order response, n is the number of truncated modes, and t is time; S12. Separate the initial phase and initial amplitude in the response: S13. Express the separated modal responses in matrix form: q(t)=APV(t) in: q=[q1(t) … q n (t)] T 3. The dense modal response reconstruction method based on the state transfer matrix according to claim 1, characterized in that: Step S2 specifically includes the following sub-steps: S21. According to the modal superposition method, the responses of the free vibration of the structure at the measured point and the point to be reconstructed are expressed as: x k =φ k q(t)=φ k APV x u =φ u q(t)=φ u APV Where u and k represent the points to be reconstructed and the measured points respectively, and φ represents the modal vibration matrix; S22. Define the state matrix S of the structural vibration as: S = φAP; S23, the state matrix of the measured points and the points to be reconstructed is expressed as: S k =φ k AP S u =φ u AP Among them, S k is the state matrix of the measured point, S u is the state matrix of the point to be reconstructed.

4. The dense modal response reconstruction method based on the state transfer matrix according to claim 3 is characterized in that: Step S21 specifically includes the following sub-steps: S211. The motion equations of a multi-degree-of-freedom system are as follows: Where M is the mass matrix, C is the damping matrix, and K is the stiffness matrix. Both the column vector x and the force F(t) have degrees of freedom (DOF). If the number of degrees of freedom of the system is n, then the matrix size is , and the column vector x and the force F(t) have dimensions of n×1. S212. Calculate system modal parameters: The premise of modal superposition is to calculate the eigenfrequencies and the corresponding vibration modes, which are calculated using the following eigenvalue equation: (K-oh 2 M)φ=0; The calculation results are a set of natural frequencies ω i and the corresponding vibration mode φ i , where i ranges from 1 to n; S213, System modal orthogonality: The system stiffness matrix is ​​orthogonal: Among them, m qi It is called the i-th order modal mass. Similarly, for the stiffness matrix: Among them, K qi It is called the i-th order modal stiffness. When the system damping meets the requirements of proportional damping, C = αM + βK, where α and β are proportional coefficients. The vibration mode also meets the orthogonality with respect to the damping matrix: Among them, C qi It is called the i-th order modal damping; S214, System modal superposition method: According to the orthogonality of vibration modes, we know that the vibration modes of each order are linearly independent. That is, the n vibration modes of an n-dimensional system constitute a set of bases of an n-dimensional vector space. The coordinate system based on this orthogonal basis is called the modal coordinate system. The displacement response x of the system is expressed as: x=φq。

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