A method for evaluating structural dynamic response based on transfer function

Abstract

The present application relates to a method for evaluating the dynamic response of a structure based on a transfer function, comprising the steps of: establishing a structural model and performing a modal analysis on the structural model; after completing the modal analysis, extracting the structural model features, obtaining the load mode matrix, calculating the modal force using the dynamic load, solving the single degree of freedom dynamic differential equation in the modal space after obtaining the modal force, and then using the response mode matrix to transform back to the physical space to obtain the response of interest. The present application uses a modal transfer method based on the transfer function between the input and output of a linear system, with faster calculation speed and less data storage, and can be used for multiple working conditions and real-time calculations of complex large structures.

Description

Technical Field

[0001] The present application relates to a structural dynamic response evaluation method, and in particular to a structural dynamic response evaluation method based on transfer function. Background Art

[0002] The response evaluation of structures under various loads is crucial for evaluating and predicting the health status of structures or real-time control, and is one of the core contents of structural monitoring and operation and maintenance. At present, the general method for calculating the dynamic response of structures is numerical simulation calculation, but it is difficult to meet the real-time or multi-condition calculation requirements of complex large structures due to its calculation time or data storage space. To solve this problem, there are currently four main strategies: the first is to introduce parallel computing, such as supercomputers, which can improve the speed to a certain extent, but it is difficult to calculate in real time; the second is to simplify complex models into multiple degrees of freedom, such as urban-scale building complex analysis, which can greatly simplify the calculation, but the accuracy is low and can only make response estimates; the third is to use an impedance model representing the input-output relationship. This method takes into account the complex mechanism of vibration propagation, predicts a small number of responses of simple structures with good accuracy and is easy to correct according to actual measurements, but it is difficult to establish an impedance model for large structures, and it is impossible to predict multiple responses at the same time; the fourth is to introduce artificial intelligence technology, but the physical interpretability is low, and the "black box" of the input-output relationship is obtained, and the training cost is huge.

[0003] When the dynamic load does not cause significant deformation or damage to the structure, the structure can be considered linear. For example, for structural vibration problems under the action of wind vibration, traffic, equipment, pedestrians, and small earthquakes, elastic analysis methods can be used. Common elastic analysis methods include direct integration method and modal superposition method. Among them, the direct integration method has good versatility, but it requires the gradual solution of the overall model, and the calculation efficiency is low; although the modal superposition method has vibration mode truncation errors, the modal theory maps the vibration behavior of the physical space to the modal space, revealing the essence of the vibration phenomenon.

[0004] With the development of urbanization, the service environment of construction projects is becoming increasingly complex. The structural dynamic response analysis must take into account multiple vibration sources, multiple working conditions and other conditions, and some high-precision, efficient and fast calculation methods are urgently needed. Summary of the invention

[0005] The purpose of this application is to solve the problem that existing methods are difficult to meet the needs of multiple working conditions and real-time calculations of complex large structures, and to provide a structural dynamic response evaluation method based on transfer function, which has the technical advantages of high precision, high efficiency and easy operation. This application improves the traditional modal superposition method based on the transfer function theory of linear systems, and proposes a modal transfer method with faster calculation speed and less data storage, which can be used for multiple working conditions and real-time calculations of complex large structures.

[0006] The present application relates to a structural dynamic response evaluation method based on a transfer function, and the structural dynamic response evaluation method comprises the following steps:

[0007] Step 1a: Establish a structural model and perform modal analysis on the structural model. The number of modes obtained is assumed to be N. M ;

[0008] Step 2a: After completing the modal analysis, extract the structural model features, which include the natural frequency and damping ratio, as well as N excited by the dynamic load p(t). P The load mode matrix Φ of the degrees of freedom p and the response mode matrix Φ of interest u , the load mode matrix Φ p The dimension is N P ×N M , the response mode matrix Φ u The dimension is N R ×N M Where N R is the number of structural responses of interest;

[0009] Step 3a: Get the load mode matrix Φ of step 2a p After that, the modal force f(t) is calculated using the dynamic load p(t). The calculation formula for the modal force f(t) is: f(t) = Φ p T p(t);

[0010] Step 4a: After obtaining the modal force f(t), solve the single degree of freedom dynamic differential equation in the modal space:

[0011]

[0012] Among them, ω r and r denote the natural frequency and damping ratio of the rth order mode, η r (t) and They represent the displacement response, velocity response and acceleration response of the r-th order mode respectively; f r (t) is the rth component of f(t);

[0013] Step 5a: After obtaining the modal displacement response η(t) described in step 4a, use the response mode matrix Φ of step 2a u Transform back to physical space to obtain the response u(t) of interest. The transformation formula is as follows:

[0014] u(t)=Φ u η(t)

[0015] The present application also relates to a structural dynamic response evaluation method based on a transfer function, the structural dynamic response evaluation method comprising the following steps:

[0016] Step 1b: Establish a structural model and perform modal analysis on the structural model. The number of modes obtained is assumed to be N. M , in N P Apply a unit force of 1 on each load excitation degree of freedom and complete N P A static analysis;

[0017] Step 2b: After completing the modal analysis, extract the structural model features, which include the natural frequency and damping ratio, as well as N excited by the dynamic load p(t). P The load mode matrix Φ of the degrees of freedom p and the response mode matrix Φ of interest u , the load mode matrix Φ p The dimension is N P ×N M , the response mode matrix Φ u The dimension is N R ×N M , the unit force response matrix F up The dimension is N R ×N P ;

[0018] Step 3b: Get the load mode matrix Φ of step 2b p After that, the modal force f(t) is calculated using the dynamic load p(t). The calculation formula for the modal force f(t) is: f(t) = Φ p T p(t);

[0019] Step 4b: After obtaining the modal force f(t), solve the single degree of freedom dynamic differential equation in the modal space:

[0020]

[0021] Among them, ω r and r denote the natural frequency and damping ratio of the rth order mode, η r (t) and They represent the displacement response, velocity response and acceleration response of the r-th order mode respectively; f r (t) is the rth component of f(t);

[0022] Step 5b: Obtain the modal velocity response described in step 4b and modal acceleration response Then, using the response mode matrix Φ from step 2bu Transform back to physical space to obtain the response u(t) of interest. The transformation formula is as follows:

[0023]

[0024] In the formula, u ps (t) = F up p(t), diag(·) represents a diagonal matrix with the elements in brackets as diagonal entries.

[0025] In step 4b, the natural frequency and damping ratio of each mode are extracted from the structural model according to step 2b; or, the damping ratio of each mode is calculated using Rayleigh damping according to the following formula:

[0026]

[0027] in,

[0028] Where (ω1,ξ1) and (ω2,ξ2) are the frequency bands of interest and the corresponding damping ratios.

[0029] The structural dynamic response evaluation method proposed in this application has the following technical advantages: for a large complex structure with N degrees of freedom, P degrees of freedom are subject to dynamic load excitation p(t), and only N R When a structure responds, there are usually N P 、N R The present application is based on the concept of transfer function, and reduces the degree of freedom dimension of the input end and the degree of freedom dimension of the output end from N to N respectively. P and N R , thereby greatly reducing the dimension of the calculation matrix, significantly improving the calculation speed, and significantly reducing the amount of data storage. In addition, the structural model characteristics obtained in this application can be saved offline as system characteristics, only need to be calculated once, and called during the calculation process. Based on the above two points, the method proposed in this application realizes the lightweight calculation and storage, making it possible to calculate multiple working conditions and real-time calculation of complex large structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The accompanying drawings are used to provide a further understanding of the present application and constitute a part of the specification. Together with the embodiments of the present application, they are used to explain the present application and do not constitute a limitation of the present application. In the accompanying drawings:

[0031] Figure 1 It is a flow chart of a structural dynamic response evaluation method based on transfer function of the present application.

[0032] Figure 2It is a schematic diagram of the structure of a residential building above a subway station taking into account soil-structure interaction in an embodiment of the present application.

[0033] Figure 3a-3b It is a typical time course and spectrum diagram of vehicle-induced excitation in the embodiment of the present application.

[0034] Figure 4 It is a comparison of the calculation results of the direct integration method and the modal superposition method for the displacement time history of the top measuring point of the structure in the embodiment.

[0035] Figure 5 It is a comparison of the calculation results of the direct integration method and the modal superposition method for the velocity time history of the top measuring point of the structure in the embodiment.

[0036] Figure 6 It is a comparison of the calculation results of the direct integration method and the modal superposition method for the acceleration time history of the top measuring point of the structure in the embodiment.

[0037] Figure 7a-7b It is a comparison of the calculation results of the vertical vibration acceleration level VAL value of the top floor of the structure in the embodiment using the direct integration method and the modal superposition method. DETAILED DESCRIPTION

[0038] In order to make the purpose, technical solutions and advantages of this application clearer, the technical solutions in this application will be clearly and completely described below in conjunction with the drawings in this application. Obviously, the described embodiments are part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0039] The present application relates to a structural dynamic response evaluation method, which provides a modal transfer method suitable for multiple working conditions and real-time calculation. Specifically, for a complex structure with a total number of degrees of freedom N, where N P degrees of freedom are subject to dynamic load excitation p(t), and only N R The structural dynamic response evaluation method of the present application can be divided into a modal displacement method and a modal acceleration method.

[0040] For the modal displacement method form, such as Figure 1 As shown, the structural dynamic response evaluation method of the present application includes the following steps:

[0041] Step 1a: Establish a structural model and perform modal analysis on the structural model. The number of modes obtained is assumed to be N. M .

[0042] Modal analysis is a method for studying the dynamic characteristics of structures in the field of engineering vibration. Mode refers to the natural vibration characteristics of engineering structures. Each mode has a specific natural frequency, damping ratio and modal vibration shape. The process of analyzing these modal parameters is called modal analysis, which can be performed numerically using finite element software, for example.

[0043] In order to reduce the modal truncation error, the Ritz vector method can be used according to the load arrangement and sufficient vibration modes can be considered during modal analysis. The Ritz vector method is a method based on the principle of energy minimization, which is used to solve the natural frequency and vibration mode of the structure. It assumes that the vibration mode of the structure can be represented by a linear combination of a set of suitable test functions, which are usually in the form of degrees of freedom of the structure. By selecting appropriate test functions, more accurate structural vibration characteristics can be obtained. The Ritz vector method converts the vibration problem of the structure into an eigenvalue problem. By solving the eigenvalues ​​and eigenvectors, the natural frequency and vibration mode of the structure can be obtained.

[0044] Step 2a, after completing the modal analysis, extract the structural model features.

[0045] Structural model characteristics include: natural vibration period or frequency, damping ratio, and N with dynamic load excitation. P The load mode matrix Φ of the degrees of freedom p , i.e., the vibration mode displacement. It also includes the response vibration mode matrix Φ of interest u , such as the mode displacement of certain points or the mode force of certain components. Among them, the load mode matrix Φ p The dimension is N P ×N M , the response mode matrix Φ u The dimension is N R ×N M For large and complex structures with a large number of degrees of freedom, manual extraction of the structural model features is inefficient, so step 2a can be implemented using the application programming interface API provided by the computing software. The structural model features can be saved offline as system characteristics, only need to be calculated once, and called in steps 3a to 5a.

[0046] Step 3a, obtain the load mode matrix Φ of step 2a p After that, the modal force f(t) can be calculated using the dynamic load excitation p(t). The calculation formula of the modal force f(t) is: f(t) = Φ p T p(t).

[0047] Step 4a, after obtaining the modal force f(t), solve the single degree of freedom dynamic differential equation in the modal space:

[0048]

[0049] Among them, ω r and r denote the natural frequency and damping ratio of the rth order mode, η r (t) and They represent the displacement response, velocity response and acceleration response of the rth order mode, respectively. r (t) is the rth component of f(t). The above solution process can be solved by various methods, such as frequency domain transformation method, Duhamel integral method or Newmark-β method. The natural frequency and damping ratio of each mode can be extracted from the structural model according to step 2a. Alternatively, the damping ratio of each mode can be calculated using Rayleigh damping according to the following formula:

[0050]

[0051] Where (ω1,ξ1) and (ω2,ξ2) are the frequency bands of interest and the corresponding damping ratios.

[0052] Step 5a: After obtaining the modal displacement response η(t) described in step 4a, use the response mode matrix Φ of step 2a u Transform back to physical space to get the response u(t) of interest. The transformation formula is as follows:

[0053] u(t)=Φ u η(t)

[0054] When the response component is velocity-dependent or acceleration-dependent, simply replace the modal displacement η(t) in the above formula with the modal velocity or modal acceleration That's it.

[0055] Step 6a, after obtaining the response of interest according to step 5a, the calculation results can be used for dynamic response evaluation and real-time analysis of the specific structure.

[0056] When the response of interest is related to displacement, such as deformation, component internal force, unit stress, etc., this method can be flexibly transformed into a modal acceleration method with smaller modal truncation error and faster convergence. For the modal acceleration method, the structural dynamic response evaluation method of this application includes the following steps:

[0057] Step 1b: Establish a structural model and perform modal analysis on the structural model. The number of modes obtained is assumed to be N. M .

[0058] In this step, in addition to completing the modal analysis in the modal displacement method, it is also necessary to P Apply a unit force of 1 on each load excitation degree of freedom and complete N P A static analysis.

[0059] Step 2b, after completing the modal analysis, extract the structural model features.

[0060] Structural model characteristics include: natural vibration period or frequency, damping ratio, and N with dynamic load excitation. P The load mode matrix Φ of the degrees of freedom p , i.e., the vibration mode displacement. It also includes the response vibration mode matrix Φ of interest u , such as the modal displacement of certain points or the modal force of certain components. The structural model features also include the unit force response matrix F obtained from the static analysis in step 1b. up . Among them, the load mode matrix Φ p The dimension is N P ×N M , the response mode matrix Φ u The dimension is N R ×N M , the unit force response matrix F up The dimension is N R ×N P For large and complex structures with a large number of degrees of freedom, manual extraction of the structural model features is inefficient, so step 2 can be implemented using the application programming interface API provided by the computing software. The structural model features can be saved offline as system characteristics, only need to be calculated once, and called in steps 3b to 5b.

[0061] Step 3b, obtain the load mode matrix Φ described in step 2b of this application p After that, the modal force f(t) can be calculated using the dynamic load excitation p(t). The calculation formula of the modal force f(t) is: f(t) = Φ p T p(t).

[0062] Step 4b, after obtaining the modal force f(t), solve the single degree of freedom dynamic differential equation in the modal space:

[0063]

[0064] Among them, ω r and r denote the natural frequency and damping ratio of the rth order mode, η r (t) and They represent the r-th order modal displacement response, modal velocity response and modal acceleration response, respectively, and f r(t) is the rth component of f(t). The above solution process can be performed by frequency domain transformation method, Duhamel integral method or Newmark-β method. The natural frequency and damping ratio of each mode can be extracted from the structural model according to step 2b. Alternatively, the damping ratio of each mode can be calculated using Rayleigh damping according to the following formula:

[0065]

[0066] Where (ω1,ξ1) and (ω2,ξ2) are the frequency bands of interest and the corresponding damping ratios.

[0067] Step 5b: Based on the modal velocity response obtained in step 4 and modal acceleration After that, we can use the response mode matrix Φ from step 2b u The unit force response matrix Fup is transformed back to the physical space to obtain the response u(t) of interest. The transformation formula is as follows:

[0068]

[0069] u ps (t) = F up p(t)

[0070] Where diag(·) represents a diagonal matrix with the elements in brackets as diagonal entries.

[0071] Step 6b, after obtaining the response of interest according to step 5b, the calculation results can be used for dynamic response evaluation and real-time analysis of the specific structure.

[0072] Example

[0073] like Figure 2 As shown in the figure, a residential building built above a subway is a reinforced concrete shear wall structure with 11 floors above ground, 1 floor underground, and 3 floors underground in some parts. As it is close to the subway, it is necessary to evaluate the impact of subway vibration on the building. A finite element model is established and the interaction between soil and structure is considered. In the model, shell elements are used for shear walls and floor slabs, with 17458 units; line elements are used for floor beams and piles, with 8133 units; solid elements are used for soil, with 22614 units; the total number of nodes is 38702, and the total number of unconstrained degrees of freedom exceeds 220,000. The load input is 10 nodes in the soil near the subway tunnel, and the response of interest is the vibration of each floor above ground. The vibration level caused by rail transit is usually low and will not cause obvious damage to the building structure, so elastic calculation can be performed for this project.

[0074] The vibration excitation time course of this embodiment is calculated based on the train-track model, and the sampling frequency of the excitation signal is 200 Hz and the duration is 25 s. Figure 3a-3b A set of typical time history diagrams and spectrum diagrams of vehicle-induced excitation are given respectively. Figure 3a The horizontal axis represents time in seconds, and the vertical axis represents the exciting force in N. Figure 3b The horizontal axis represents the frequency in Hz, and the vertical axis represents the frequency spectrum of the excitation force in N / Hz.

[0075] The modal transfer method (MTM) and the direct integration method (DIM) are used for comparative calculations. The modal transfer method is based on the two steps 1 and 2 of the present application. The modal analysis is first performed using finite element software. The Ritz vector method is used during the analysis and the 2000th-order vibration mode is considered. The OAPI interface provided by the finite element software is then used to extract the period, Φ p , Φ u The direct integration method is completed using finite element software, and the numerical algorithm is the Newmark-β method, where γ = 0.5 and β = 0.25. After the calculation is completed, the OAPI interface is used to extract the floor response of each floor.

[0076] The modal force f(t) is calculated according to the process described in step 3 of this application. After obtaining the modal force f(t), the single degree of freedom dynamic differential equation in the modal space is solved according to the process of step 4 to obtain the modal displacement η(t) and the modal velocity and modal acceleration The damping used in the analysis is Rayleigh damping, the upper and lower limits of the period are 1.4s and 0.1s respectively, and the corresponding damping ratio is specified as 2%. The modal displacement η(t) and modal velocity are obtained. and modal acceleration Finally, the vertical displacement, velocity and acceleration time history response of the floor are calculated according to step 5.

[0077] Figure 4 , Figure 5 and Figure 6 The displacement, velocity and acceleration response of a certain measuring point on the top layer using the modal transfer method (MTM) described in this application and the direct integration method (DIM) in the finite element software are compared. Figure 4-6 As shown, the horizontal axis is time, the unit is s, and the total coordinates are displacement, velocity and acceleration, the units are m, m / s and m / s respectively. 2 . Figure 7a-7b The vertical acceleration magnitude VAL values ​​of all nodes on the top floor are compared by modal analysis method and direct integration method. The results show that the calculation results of the two methods are in good agreement, indicating that the modal transfer method has sufficient accuracy.

[0078] The following is a comparison of the calculation time of the two methods in this embodiment. The modal transfer method proposed in this application needs to calculate the 2000th-order vibration mode to reduce the truncation error, so the modal analysis requires 2.9 hours. The extracted data includes frequency, 10 loading degree of freedom vibration mode displacements and 13401 floor vertical degree of freedom vibration mode displacements, which takes 3.0 hours. It should be pointed out that the above work is one-time, and the data obtained represents the inherent properties of the structure, which can be saved offline and called during calculation. The calculation time of the modal transfer method is only 3.5 seconds, while the direct integration method in the finite element software takes 2.5 hours to complete a single time-history calculation, and the time to extract the floor response from the complete result reaches 8.2 hours.

[0079]

[0080] For the embodiment of the present application, it only takes 3.5 seconds to apply the modal transfer method proposed in the present application, which is significantly more efficient than the 2.5 hours of the direct integration method. In addition, the output data of the modal transfer method is only the response of interest and no irrelevant data, so the spatial storage is much smaller than the direct integration method. Due to the above advantages of the present application, it is possible to calculate multiple working conditions and real-time calculations for large and complex structures.

[0081] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit it. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present application.

Claims

1. A structural dynamic response evaluation method based on transfer function, characterized in that: The structural dynamic response evaluation method comprises the following steps: Step 1a: Build a model of the structure to be evaluated and perform modal analysis on the model of the structure. The number of modes obtained is assumed to be N. M ; Step 2a: After completing the modal analysis, extract the structural model features, which include the natural frequency and damping ratio, as well as N excited by the dynamic load p(t). P The load mode matrix Φ of the degrees of freedom p and the response mode matrix Φ of interest u , the load mode matrix Φ p The dimension is N P ×N M , the response mode matrix Φ u The dimension is N R ×N M Where N R is the number of structural responses of interest; Step 3a: Get the load mode matrix Φ of step 2a p After that, the modal force f(t) is calculated using the dynamic load p(t). The calculation formula for the modal force f(t) is: f(t) = Φ p T p(t); Step 4a: After obtaining the modal force f(t), solve the single degree of freedom dynamic differential equation in the modal space: Among them, ω r and r denote the natural frequency and damping ratio of the rth order mode, η r (t) and They represent the displacement response, velocity response and acceleration response of the r-th order mode respectively; f r (t) is the rth component of f(t); Step 5a: After obtaining the modal displacement response η(t) from step 4a, use the response mode matrix Φ from step 2a u Transform back to physical space to obtain the response u(t) of interest. The transformation formula is as follows: u(t)=Φ u n(t).

2. The structural dynamic response evaluation method according to claim 1, characterized in that: In step 4a, the natural frequency and damping ratio of each mode are extracted from the structural model according to step 2a; or, the damping ratio of each mode is calculated using Rayleigh damping according to the following formula: in, Where (ω1,ξ1) and (ω2,ξ2) are the frequency bands of interest and the corresponding damping ratios.

3. A structural dynamic response evaluation method based on transfer function, characterized in that: The structural dynamic response evaluation method comprises the following steps: Step 1b: Build a model of the structure to be evaluated and perform modal analysis on the model of the structure. The number of modes obtained is assumed to be N. M , in N P Apply a unit force of 1 on each load excitation degree of freedom and complete N P A static analysis; Step 2b: After completing the modal analysis, extract the structural model features, which include the natural frequency and damping ratio, as well as N excited by the dynamic load p(t). P The load mode matrix Φ of the degrees of freedom p and the response mode matrix Φ of interest u , the load mode matrix Φ p The dimension is N P ×N M , the response mode matrix Φ u The dimension is N R ×N M , unit force response matrix F up The dimension is N R ×N P ; Step 3b: Get the load mode matrix Φ of step 2b p After that, the modal force f(t) is calculated using the dynamic load p(t). The calculation formula for the modal force f(t) is: f(t) = Φ p T p(t); Step 4b: After obtaining the modal force f(t), solve the single degree of freedom dynamic differential equation in the modal space: Among them, ω r and r denote the natural frequency and damping ratio of the rth order mode, η r (t) and They represent the displacement response, velocity response and acceleration response of the r-th order mode respectively; f r (t) is the rth component of f(t); Step 5b: Get the modal velocity response of step 4b and modal acceleration response Then, using the response mode matrix Φ from step 2b u Transform back to physical space to obtain the response u(t) of interest. The transformation formula is as follows: Where: u ps (t) = F up p(t), diag(·) represents a diagonal matrix with the elements in brackets as diagonal entries.

4. The structural dynamic response evaluation method according to claim 3, characterized in that: In step 4b, the natural frequency and damping ratio of each mode are extracted from the structural model according to step 2b; or, the damping ratio of each mode is calculated using Rayleigh damping according to the following formula: in, Where (ω1,ξ1) and (ω2,ξ2) are the frequency bands of interest and the corresponding damping ratios.

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