A method for constructing a strain modal expansion matrix based on a limited identification domain

Abstract

The application discloses a kind of based on the construction method of strain modal extension domain matrix of limited identification domain, which includes the mathematical representation model array between the system global strain response and the identified subdomain strain response, unidentified subdomain strain response;The mathematical representation model array between the system global strain response and the system global modal shape is established;The mathematical representation model array between the system global strain response and the identified subdomain modal shape, unidentified subdomain modal shape is established;System global strain modal extension domain matrix based on limited identification domain is constructed;Further construct strain frequency response function matrix, extract the inherent frequency, damping, modal shape of system from frequency response function matrix;The application realizes the modal parameter identification of system global, provides a new idea, new means for the system global modal parameter that cannot be directly identified, has important engineering application value and scientific significance, and lays a foundation for scientific analysis of structure dynamics characteristics.

Description

Technical Field

[0001] This invention relates to a method for constructing a strain mode extended domain matrix based on a finite identification domain. Background Technology

[0002] With the rapid development of my country's modern construction, industry, and defense sectors, especially the design of large buildings, bridges, engineering structures, and various aircraft structures—such as the design optimization of aircraft wings, missile control surfaces, and other navigation structures—relies on reliable structural modal parameters. These parameters include natural frequencies, damping, and mode shapes, serving as input parameters for modifying structural designs. Currently, the identification of structural modal parameters is limited to obtaining them from a finite number of measurement points. However, this limited number of measurement points has a significant drawback: when the system's modal order coincides with the system's nodes, the system's second-order modal parameters disappear, making it impossible to identify the true second-order modal parameters. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides a method for constructing a strain mode extended domain matrix based on a finite identification domain. This method extends the modal parameters of the entire system and uses the finite identification domain to construct the strain response function matrix of the entire system.

[0004] The present invention is achieved through the following technical solutions.

[0005] This invention provides a method for constructing a strain mode extended field matrix based on a finite identification domain, comprising the following steps:

[0006] ①Establish a mathematical representation model matrix between the global strain response of the system and the strain responses of the identified and unidentified subdomains;

[0007] ② Assuming the system exhibits linear variation characteristics, the mathematical representation model matrix between the global strain response and the system modal coordinates and the contribution of modal parameters of each order of the system can be obtained by the modal superposition principle;

[0008] ③ Construct a mathematical representation model matrix between the strain response of the identified subdomains of the system and the modal vibration and modal coordinates of the identified subdomains;

[0009] ④ Since the modal mode matrix of the identified subdomain is not necessarily a square matrix, a square matrix of the identified subdomain modal modes is constructed by multiplying both sides of the mathematical representation model equation built in step ③ through the transpose of the identified subdomain modal mode matrix; the obtained system modal coordinate mathematical model is coupled with the mathematical representation model matrix of step ② to construct the mathematical representation model matrix between the system global strain response and the system global modal modes and the identified subdomain strain response and identified subdomain modal modes;

[0010] ⑤ Perform similarity mapping between the model matrix constructed in step ④ and the model matrix constructed in step ① to obtain the extended domain model matrix of modal parameters based on the finite recognition domain;

[0011] ⑥ The strain frequency response function matrix of the identified subdomain is extended using the constructed extended domain matrix to obtain the strain frequency response function matrix of the entire system, and the modal parameters of the entire system are extracted to achieve modal parameter identification of the entire system.

[0012] In step ⑥, the modal parameters include natural frequency, damping, and mode shape.

[0013] The algorithm in step ① is as follows:

[0014]

[0015] in,

[0016]

[0017]

[0018]

[0019] [ε N ],[ε A ],[ε D ],[ε EF ] respectively represent the global strain response, the strain response of the identified subdomain, the strain response of the unidentified subdomain, and the extended domain matrix.

[0020] The algorithm in step ② is as follows:

[0021] [ε N ] = [Q N ]·[q]

[0022] in,

[0023]

[0024]

[0025] [ε N ], [Q N ], [q] represent the global strain response, global mode shape, and contribution of each subdomain strain response, respectively.

[0026] The algorithm in step ③ is as follows:

[0027]

[0028] in,

[0029]

[0030]

[0031] [ε N ], [Q A ], [Q Ｄ ] respectively represent the global strain response of the system, the mode shape of the identified subdomain, and the mode shape of the identified subdomain.

[0032] The algorithm in step ④ is as follows:

[0033] [ε A ] = [Q A ]·[q]

[0034] [Q A ] T ·[ε A ] = [Q A ] T ·[Q A ]·[q]

[0035] [q] = ([Q] A ] T ·[Q A ]) -1 ·([Q A ] T ·[ε A ])

[0036] in,

[0037]

[0038] [ε A [QA] and [q] represent the strain response of the identified domain, the mode shape of the identified subdomain, and the contribution of each subdomain to the strain response, respectively.

[0039] The algorithm in step ⑤ is as follows:

[0040] [ε N] = [Q N ]·([Q A ] T ·[Q A ]) -1 ·([Q A ] T ·[ε A ])

[0041] [ε EF ] = [Q N ]·([Q A ] T ·[Q A ]) -1 ·[Q A ]

[0042] in,

[0043]

[0044]

[0045] [ε N ], [Q N ], [Q A ] respectively represent the global strain response, the global mode shape, and the mode shape of the identified subdomain.

[0046] The algorithm in step ⑥ is as follows:

[0047] [E N ] = [ε EF ]·[E A ]

[0048] in,

[0049]

[0050]

[0051] [E] Ｎ ], [ε EＦ ], [E Ａ ] respectively represent the system's global strain frequency response function matrix, extended domain matrix, and identified subdomain strain response frequency response function matrix.

[0052] The beneficial effects of this invention are as follows: by constructing an extended domain matrix, it becomes possible to construct modal parameter identification of the entire system domain through a finite identification domain. By cleverly utilizing the constructed extended domain matrix, the modal parameters of the entire system domain can be extended. This provides strong data support for the comprehensive analysis of the system's structural characteristics and dynamic characteristics, and the optimization of structural design. It lays the foundation for the identification of modal parameters of the entire system domain and also provides a new scientific means for structural health monitoring. Attached Figure Description

[0053] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0054] The technical solution of the present invention is further described below, but the scope of protection is not limited to what is described.

[0055] like Figure 1 The method for constructing a strain mode extension field matrix based on a finite identification domain, as shown, includes the following steps:

[0056] ①Establish a mathematical representation model matrix between the global strain response of the system and the strain responses of the identified and unidentified subdomains;

[0057] ② Assuming the system exhibits linear variation characteristics, the mathematical representation model matrix between the global strain response and the system modal coordinates and the contribution of modal parameters of each order of the system can be obtained by the modal superposition principle;

[0058] ③ Construct a mathematical representation model matrix between the strain response of the identified subdomains of the system and the modal vibration and modal coordinates of the identified subdomains;

[0059] ④ Since the modal mode matrix of the identified subdomain is not necessarily a square matrix, by transposing the modal mode matrix of the identified subdomain, we can expand and multiply both sides of the mathematical representation model equation constructed in step ③ to construct a square matrix of the modal modes of the identified subdomain. This is beneficial for inverting the square matrix. The obtained system modal coordinate mathematical model is coupled with the mathematical representation model matrix of step ② to construct the mathematical representation model matrix between the system global strain response and the system global modal mode and the strain response and modal mode of the identified subdomain.

[0060] ⑤ Perform similarity mapping between the model matrix constructed in step ④ and the model matrix constructed in step ① to obtain the extended domain model matrix of modal parameters based on the finite recognition domain;

[0061] ⑥ The strain frequency response function matrix of the identified subdomain is extended using the constructed extended domain matrix to obtain the strain frequency response function matrix of the entire system, and the modal parameters of the entire system are extracted to achieve modal parameter identification of the entire system.

[0062] In step ⑥, the modal parameters include natural frequency, damping, and mode shape.

[0063] The algorithm in step ① is as follows:

[0064]

[0065] in,

[0066]

[0067]

[0068]

[0069] [ε D ],[ε A ],[ε D ],[ε EF ] respectively represent the global strain response, the strain response of the identified subdomain, the strain response of the unidentified subdomain, and the extended domain matrix.

[0070] The algorithm in step ② is as follows:

[0071] [ε N ] = [Q N ]·[q] (2)

[0072] in,

[0073]

[0074]

[0075] [ε N ], [Q N ], [q] represent the global strain response, global mode shape, and contribution of each subdomain strain response, respectively.

[0076] The algorithm in step ③ is as follows:

[0077] Combine (1) and (2)

[0078]

[0079] in,

[0080]

[0081]

[0082] [ε N ], [QA ], [Q Ｄ ] respectively represent the global strain response of the system, the mode shape of the identified subdomain, and the mode shape of the identified subdomain.

[0083] The algorithm in step ④ is as follows:

[0084] [ε A ] = [Q A ]·[q]

[0085] [Q A ] T ·[ε A ] = [Q A ] T ·[Q A ]·[q]

[0086] [q] = ([Q] A ] T ·[Q A ]) -1 ·([Q A ] T ·[ε A (4)

[0087] in,

[0088]

[0089] [ε A ], [Q A ], [q] represent the strain response of the identified domain, the mode shape of the identified subdomain, and the contribution of the strain response of each subdomain (i.e., the modal coordinates of each order of the system), respectively.

[0090] The algorithm in step ⑤ is as follows:

[0091] [ε N ] = [Q N ]·([Q A ] T ·[Q A ]) -1 ·([Q A ] T ·[ε A (5)

[0092] By comparing this equation with equation (1), the extended domain model matrix can be obtained.

[0093] [ε EF ] = [Q N ]·([Q A ] T ·[Q A ])-１ ·[Q A ] T

[0094] in,

[0095]

[0096]

[0097] [ε N ], [Q N ], [Q A ] respectively represent the global strain response, the global mode shape, and the mode shape of the identified subdomain.

[0098] The algorithm in step ⑥ is as follows:

[0099] [E N ] = [ε EF ]·[E A (6)

[0100] in,

[0101]

[0102]

[0103] [E] Ｎ ], [ε ＦＦ ], [E Ａ ] respectively represent the system's global strain frequency response function matrix, extended domain matrix, and identified subdomain strain response frequency response function matrix.

[0104] This invention can be applied to: expanding the modal parameters of the entire system. By identifying the strain response of the subdomains, the strain frequency response function matrix of the entire system can be reconstructed based on the expanded domain matrix. Furthermore, the natural frequency, damping, mode shape and other parameters of the entire system can be obtained, providing strong data support for the comprehensive analysis of the structural characteristics and dynamic characteristics of the system and optimizing the structural design, and laying the foundation for the identification of the modal parameters of the entire system.

[0105] In summary, this invention can construct a mathematical representation model matrix between the global strain response of the system and the strain responses of the identified and unidentified subdomains; it can construct a mathematical representation model matrix between the global modal shapes of the system and the modal shapes of the identified and unidentified subdomains and the system modal coordinates; and it can effectively construct the strain frequency response function matrix of the global system through the strain response of the finite identification domain.

Claims

1. A method for constructing a strain mode extended field matrix based on a finite identification domain, characterized in that: Includes the following steps: ①Establish a mathematical representation model matrix between the global strain response of the system and the strain responses of the identified and unidentified subdomains; ② Assuming the system exhibits linear variation characteristics, the mathematical representation model matrix between the global strain response and the system modal coordinates and the contribution of modal parameters of each order of the system can be obtained by the modal superposition principle; ③ Construct a mathematical representation model matrix between the strain response of the identified subdomains of the system and the modal vibration and modal coordinates of the identified subdomains; ④ Since the modal mode matrix of the identified subdomain is not necessarily a square matrix, a square matrix of the identified subdomain modal modes is constructed by multiplying both sides of the mathematical representation model equation built in step ③ through the transpose of the identified subdomain modal mode matrix; the obtained system modal coordinate mathematical model is coupled with the mathematical representation model matrix of step ② to construct the mathematical representation model matrix between the system global strain response and the system global modal modes and the identified subdomain strain response and identified subdomain modal modes; ⑤ Perform similarity mapping between the model matrix constructed in step ④ and the model matrix constructed in step ① to obtain the extended domain model matrix of modal parameters based on the finite recognition domain; ⑥ The strain frequency response function matrix of the identified subdomain is extended by the constructed extended domain matrix to obtain the strain frequency response function matrix of the whole system, and the modal parameters of the whole system are extracted to realize the identification of modal parameters of the whole system. In step ⑥, the modal parameters include natural frequency, damping, and mode shape.

2. The method for constructing the strain mode extended field matrix based on a finite identification domain as described in claim 1, characterized in that: The algorithm in step ① is as follows: in, , , , These represent the global strain response, the strain response of the identified subdomains, the strain response of the unidentified subdomains, and the extended domain matrix, respectively.

3. The method for constructing the strain mode extended field matrix based on a finite identification domain as described in claim 1, characterized in that: The algorithm in step ② is as follows: in, , , These represent the global strain response, the global modal shape, and the contribution of each subdomain strain response, respectively.

4. The method for constructing the strain mode extended field matrix based on a finite identification domain as described in claim 1, characterized in that: The algorithm in step ③ is as follows: in, , , These represent the global strain response of the system, the mode shape of the identified subdomain, and the mode shape of the identified subdomain, respectively.

5. The method for constructing the strain mode extended field matrix based on a finite identification domain as described in claim 1, characterized in that: The algorithm in step ④ is as follows: in, , , These represent the strain response of the identified domain, the mode shape of the identified subdomain, and the contribution of the strain response of each subdomain, respectively.

6. The method for constructing the strain mode extended field matrix based on a finite identification domain as described in claim 1, characterized in that: The algorithm in step ⑤ is as follows: in, , , These represent the global strain response, the global mode shape, and the mode shape of the identified subdomains, respectively.

7. The method for constructing the strain mode extended field matrix based on a finite identification domain as described in claim 1, characterized in that: The algorithm in step ⑥ is as follows: in, , , , These represent the system's global strain frequency response function matrix, extended domain matrix, and identified subdomain strain response frequency response function matrix, respectively.

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