A calculation method for vibration wave propagation based on WPA method and its application

Abstract

This invention provides a vibration wave propagation calculation method based on the WPA method and its application. The calculation method includes: Step S1, supplementing the propagation and reflection parameters of vibration waves at the beam-slab-column connection based on the classical theory of the WPA method; Step S2, using MATLAB software to write code to calculate the response of the vibration wave at any time and any position within the structure based on the obtained propagation and reflection parameters; Step S3, using the written code to calculate the actual structure and conducting field tests to verify the accuracy of the code calculation, and calculating the response of the vibration wave at any time and any position within the structure based on the verified code; Step S4, calculating the peak acceleration at different positions in each layer under different point excitations, thereby regressing the vibration propagation law of the vibration wave between the same layer and different layers. The vibration wave propagation analysis and calculation of the technical solution of this invention is extended to the entire structure, which is closer to the actual situation.

Description

Technical Field

[0001] This invention relates to the field of building vibration analysis technology, and in particular to a vibration wave propagation calculation method based on the WPA method and its application. Background Technology

[0002] Existing research on vibration wave propagation calculations mainly focuses on single components, such as the propagation of vibration waves within beams, slabs, or columns in a structure. Commonly used methods include the transfer matrix method with boundary approximation, the admittance method using indirect parameters, and spectral analysis in the frequency domain. These methods can calculate the vibration response of vibration waves at any time and location within a single component. However, existing methods have some problems: (1) They are less relevant to structures composed of multiple components, or they neglect the reflection and refraction of vibration waves at the connection points of beams, slabs, and columns; (2) For methods that specifically use boundary approximation, there are often differences between the actual boundary conditions and the actual boundary conditions, resulting in inaccurate calculation results; (3) For methods that specifically use indirect parameters, the selection of the indirect parameters has a significant impact on the accuracy of the calculation, and the selection of these parameters often has errors; (4) For analysis methods in the frequency domain, the calculations are relatively complex, and the requirements for boundary conditions are vague, making them more suitable for calculations of single components. Summary of the Invention

[0003] To address the above technical problems, this invention discloses a vibration wave propagation calculation method based on the WPA method and its application. The obtained vibration wave propagation law between the same layer and different layers is more accurate, and the response at the target location can be quickly determined.

[0004] The technical solution adopted by this invention is as follows:

[0005] A method for calculating vibration wave propagation based on the WPA method includes the following steps:

[0006] Step S1: Based on the WPA method, supplement the propagation and reflection parameters of vibration waves at the beam-slab-column connection according to the dynamic characteristics of the connection.

[0007] Step S2: Based on the WPA method, code is written using MATLAB software to calculate the response of the vibration wave at any time and any position within the structure based on the obtained propagation and reflection parameters of the vibration wave.

[0008] Step S3: Use the written code to calculate the actual structure and conduct field tests to verify the accuracy of the code calculation. Calculate the response of the vibration wave at any time and any position within the structure based on the verified code.

[0009] Step S4: Calculate the peak acceleration at different positions in each layer under different point excitation, and regress to obtain the vibration propagation law of the vibration wave between the same layer and different layers.

[0010] This technical solution takes into account the reflection and refraction of vibration waves at the connection points of beams, slabs and columns, so that the analysis and calculation of vibration wave propagation is not limited to the components, but extended to the entire structure, making the calculated laws more consistent with the actual situation.

[0011] As a further improvement of the present invention, in step S1, the beam and slab at the connection are divided into cases such as simply supported in the middle, simply supported at the end and fixed at the end.

[0012] Among them, the amplitudes of the four structural waves in the simply supported intermediate condition of the slab-beam structure are as follows:

[0013]

[0014] Among them, A r It is the amplitude of the reflected wave, A nl It is the amplitude of the reflected near-field wave, A t A nr These are the amplitudes of the transmitted propagating wave and the near-field wave, respectively. i It is the amplitude of the normally incident curved wave, and j is the initial phase angle;

[0015] The amplitudes A of the two reflected waves in the simply supported end case of the slab-beam structure r A nl They are respectively:

[0016]

[0017] Among them, A i It is the amplitude of a normally incident curved wave;

[0018] The amplitudes A of the two reflected waves in the case of fixed end supports of the slab-beam structure r A nl They are respectively:

[0019]

[0020] Among them, A i It is the amplitude of the normally incident curved wave, and j is the initial phase angle.

[0021] As a further improvement of the present invention, in step S1, for the plate and column at the connection point, if the plate and column are rigidly connected at the connection point, the reflection of the wave is ignored;

[0022] If the connection between the plate and column is semi-rigid, the reflected wave parameters can be obtained by solving the following equation:

[0023]

[0024] w 板 =w 柱

[0025] M 板 -M 柱 =K θ (θ 柱 -θ 板 )

[0026] Among them, M 连接 Let θ be the bending moment at the connection. 柱 θ 板 This refers to the angle at the joint of the slab column.

[0027] If the plate and column are hinged at the connection point, allowing rotation but not transmitting bending moment, only shear force and axial force, the reflected wave parameters can be obtained by solving the following equations:

[0028] M 连接 =0

[0029] w 板 =w 柱

[0030] M 板 =0,M 柱 =0

[0031] Among them, M 连接 This represents the bending moment at the connection.

[0032] As a further improvement to the present invention, step S2 is calculated using the following steps:

[0033] Step S21: Define the physical and geometric parameters of beams, slabs and columns based on the damping loss factor and damping bending stiffness;

[0034] Step S22: Define excitation parameters and define the location of the extraction measurement point. The excitation parameters include excitation load type, frequency, amplitude, and location.

[0035] Step S23: Perform mesh generation, set the number of modes, initialize the displacement response matrix, calculate the mechanical impedance of beams, plates, and columns, and initialize the displacement and acceleration matrices;

[0036] Step S24: Determine the boundary conditions at the beam-slab connection and calculate the reflection and transmission coefficients between the beam-slab and the slab-column connection.

[0037] Step S25: Calculate the displacement response;

[0038] Step S26, calculate acceleration;

[0039] Step S27: Calculate the peak acceleration.

[0040] As a further improvement of the present invention, step S3 includes: for the standard layer plan of the structure to be tested, using the steps...

[0041] The S2 code applies a simple harmonic wave load to one layer of the structure, obtaining acceleration time history curves for each layer of the structure under test. The same simple harmonic wave load as in the code is applied using a vibrator, and field tests are conducted. Acceleration time history curves at the same points in each layer are collected and compared with the acceleration time history curves obtained by the code to verify the accuracy of the code's calculation.

[0042] As a further improvement of the present invention, in step S4, the peak acceleration of each layer at different positions under different point excitation of each layer of the structure under test is calculated according to the verified code, and then the vibration propagation law of the vibration wave between the same layer and different layers is regressed to obtain the prediction equation of the peak acceleration of the vibration wave at any time and at any position in the structure.

[0043] As a further improvement of the present invention, the peak acceleration prediction equation is as follows:

[0044] α = -7.868Y 2 -9.796Z 2 -17.799x 2 -0.758y 2 -9.969z 2 +5.657XY+56.854ZY+12xY-0.955yY+0.794zY+0.928XZ+0.857YZ+1.133xZ-1.4yZ+19.691zZ+0.128Xx-0.982xY+1.281xZ Where α is the peak acceleration, x, y, z are the coordinates of the measuring point, and X, Y, Z are the coordinates of the excitation source.

[0045] This invention also discloses the application of the vibration wave propagation calculation method based on the WPA method described above, which is used to determine the response at a target location.

[0046] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0047] The technical solution of this invention supplements the existing classical WPA theory with the analysis theory of reflection and refraction of vibration waves at the connection of beams, slabs and columns, so that the vibration wave propagation analysis and calculation is not limited to the component but extended to the whole structure, which is closer to the actual situation. The MATLAB code is used to realize the calculation of arbitrary working conditions more accurately and efficiently. Furthermore, based on the obtained peak acceleration prediction equation, the response at the target location can be quickly determined. Attached Figure Description

[0048] Figure 1 This is a flowchart of a vibration wave propagation calculation method based on the WPA method according to an embodiment of the present invention.

[0049] Figure 2This is a schematic diagram of wave propagation at discontinuities in a plate-beam structure according to an embodiment of the present invention.

[0050] Figure 3 This is a schematic diagram of wave propagation in a plate-beam structure with simple support in the middle, according to an embodiment of the present invention.

[0051] Figure 4 This is a schematic diagram of wave propagation in a plate beam structure with simply supported ends, according to an embodiment of the present invention.

[0052] Figure 5 This is a schematic diagram of wave propagation in a plate beam structure with fixed supports at the ends, according to an embodiment of the present invention.

[0053] Figure 6 This is a flowchart of the code calculation process according to an embodiment of the present invention.

[0054] Figure 7 This is a code content diagram of an embodiment of the present invention.

[0055] Figure 8 This is a standard layer plan view of the actual structure of an embodiment of the present invention.

[0056] Figure 9 These are the acceleration time history curves at 8 points on layers 1-13 calculated according to an embodiment of the present invention.

[0057] Figure 10 This is a comparison of the acceleration curves obtained by code calculation and test results in the embodiments of the present invention.

[0058] Figure 11 This is the regression prediction peak acceleration diagram obtained from an embodiment of the present invention.

[0059] Figure 12 It is the residual of the regression prediction obtained in the embodiment of the present invention. Detailed Implementation

[0060] The preferred embodiments of the present invention will be described in further detail below.

[0061] A method for calculating vibration wave propagation based on the WPA method, such as Figure 1 As shown, it includes the following steps:

[0062] S1. Based on the classical theory of WPA method, the dynamic characteristics of the connection between beams, slabs and columns are considered to describe the propagation and reflection of vibration waves as accurately as possible;

[0063] S2. Use MATLAB software to write code that can calculate the response of vibration waves at any time and position within the structure under simple harmonic load.

[0064] S3. Use the above code to calculate the actual structure and conduct field tests to verify the accuracy of the code calculations;

[0065] S4. Based on the verified code, the vibration propagation law of the vibration wave between the same layer and different layers can be calculated, thereby realizing the response calculation of the vibration wave at any time and any position in the structure.

[0066] Specifically, step S1 includes:

[0067] (I) Analysis of beam-slab connections

[0068] Assuming the beam-slab structure is discontinuous at x=0, such as Figure 2 As shown. An incident curved wave occurs in the region x < 0. for

[0069]

[0070] Among them, A i Let w be the amplitude of the incident curved wave, t be the angular frequency, k be the time, and x be the proportionality coefficient.

[0071] Because the beam-slab structure is discontinuous at x=0, structural waves are reflected and transmitted at this point. The reflected wave can be described as follows:

[0072]

[0073] In the formula A r It is the amplitude of the reflected wave.

[0074] The near-field wave generated by the reflecting interface is

[0075]

[0076] In the formula A nl It is the amplitude of the reflected near-field wave.

[0077] In the region x>0, the transmitted propagating wave and the near-field wave are respectively

[0078]

[0079] In the formula A t A nr These are the amplitudes of the transmitted propagating wave and the near-field wave, respectively.

[0080] Therefore, the motion of the beam-slab structure in the regions around x=0 can be described as follows:

[0081]

[0082] In beam-slab structures, the boundary conditions at discontinuities can be categorized into the following cases:

[0083] (1) Consider a beam-slab model with simple support in the middle, such as Figure 3 As shown, there is a positively incident curved wave It propagates within the plate. According to the equilibrium condition, we have...

[0084]

[0085] M + With M - The left and right bending moments are the points of discontinuity.

[0086] Substituting the wave expression into the continuity condition, we can obtain the amplitudes A of the four structure waves. r A nl A t A nr They are respectively

[0087]

[0088] (2) Considering the simply supported plate at the end, the model is as follows: Figure 4 As shown, there is a positively incident curved wave It propagates within the plate. Similarly, according to the equilibrium condition, we have...

[0089]

[0090] Substituting the wave expression into the continuity condition, we can obtain the amplitudes A of the two reflected waves. r A nl , respectively

[0091]

[0092] (3) Considering the end fixed plate, the model is as follows: Figure 5 As shown, there is a positively incident curved wave It propagates within the plate. Similarly, according to the equilibrium condition, we have...

[0093]

[0094] Substituting the wave expression into the continuity condition, we can obtain the amplitudes A of the two reflected waves. r A nl , respectively

[0095]

[0096] (II) Analysis of the connection between slab and column

[0097] (1) If the plate and column are rigidly connected at the joint, then both displacement and rotation are continuous. For the connection surface z = 0 between the plate and column, there is displacement continuity, and the deflection w(x,y,t) of the plate is equal to the lateral displacement v(0,t) at the top of the column, i.e.

[0098] w(x,y,t)=v(0,t) (13)

[0099] Furthermore, rotational continuity exists; the rotation angle of the plate is equal to the rotation angle of the column top, that is...

[0100]

[0101] θ(0,t) is the rotation angle at the top of the column.

[0102] In this case, wave reflection is minimal and can be ignored.

[0103] (2) If the plate and column are semi-rigid at the connection, it can be simulated by rotation spring (bending spring) and shear spring, indicating that the connection has limited stiffness and allows a certain degree of rotation and displacement.

[0104] The horizontal and vertical displacements at the plate-column connection must still be continuous. However, due to the presence of a rotational spring in the connection, the rotation angle is no longer strictly continuous, but rather satisfies...

[0105] M 连接 =K θ (θ 柱 -θ 板 (15)

[0106] In the formula M 连接 Let θ be the bending moment at the connection. 柱 θ 板 This refers to the angle at the joint of the slab column.

[0107]

[0108] Substitute boundary conditions

[0109] w 板 =w 柱 (18)

[0110] M 板 -M 柱 =K θ (θ 柱 -θ 板 (19)

[0111] The reflected wave can be obtained by solving the equation.

[0112] (3) If the plate and column are hinged at the connection, rotation is allowed, but bending moment is not transmitted; only shear force and axial force are transmitted.

[0113] Horizontal and vertical displacements at the slab-column connection must remain continuous. Discontinuous rotation angles are permissible, but abrupt changes in rotation are allowed.

[0114] M 连接 =0 (20)

[0115] According to boundary conditions

[0116] w 板 =w 柱 (twenty one)

[0117] M 板 =0,M 柱 =0 (22)

[0118] The reflected wave can be obtained by solving the equation.

[0119] In step S2, based on the improved WPA theory described above, code is written using MATLAB software to calculate the response of the vibration wave at any time and location within the structure. The flowchart is as follows. Figure 6 As shown, the code content is as follows Figure 7 As shown, Figure 7 Each line of code has comments, and the parameters, extraction point locations, and grid divisions in the code can be changed according to specific circumstances.

[0120] In step S3, Figure 8 Based on the standard floor plan of the actual structure, a simple harmonic wave load was applied to 8 points on the first floor of the structure under test according to the above code, resulting in acceleration time history curves for 8 points on floors 1-13, as shown below. Figure 9 As shown. Field tests were conducted, applying the same simple harmonic wave load as in the code using a vibrator. Acceleration time history curves at the same points on each floor were collected using an acquisition system and compared with the curves obtained from the code. The results are as follows. Figure 10 As shown, the accuracy of the code's calculations has been verified.

[0121] In step S4, based on the verified code, the peak acceleration values ​​at different locations within each layer under different point excitations can be calculated. This allows for the regression analysis of the vibration propagation patterns of the vibration wave within the same layer and between different layers. Table 1 shows the statistical results of the peak acceleration values ​​at different points within each layer under three-layer, eight-point excitation. Figure 11 and Figure 12 The data is calculated based on the code and then used for regression prediction.

[0122] Table 1

[0123]

[0124] Finally, the prediction equation for the peak acceleration of the vibration wave at any time and any position within the structure can be derived as follows:

[0125] α = -7.868Y 2 -9.796Z 2 -17.799x 2 -0.758y 2 -9.969z 2+5.657XY+56.854ZY+12xY-0.955yY+0.794zY+0.928XZ+0.857YZ+1.133xZ-1.4yZ+19.691zZ+0.128Xx-0.982xY+1.281xZ (23)

[0126] In the formula, α is the peak acceleration, x, y, z are the coordinates of the measuring point, and X, Y, Z are the coordinates of the excitation source.

[0127] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A method for calculating vibration wave propagation based on the WPA method, characterized in that: The steps include: Step S1: Based on the WPA method, supplement the propagation and reflection parameters of vibration waves at the beam-slab-column connection according to the dynamic characteristics of the connection. Step S2: Based on the WPA method, code is written using MATLAB software to calculate the response of the vibration wave at any time and any position within the structure based on the obtained propagation and reflection parameters of the vibration wave. Step S3: Use the written code to calculate the actual structure and conduct field tests to verify the accuracy of the code calculation. Calculate the response of the vibration wave at any time and any position within the structure based on the verified code. Step S4: Calculate the peak acceleration at different positions in each layer under different point excitation, and regress to obtain the vibration propagation law of the vibration wave in the same layer and between different layers; In step S1, for the beam-slab connection, the amplitudes of the four structural waves in the case of the beam-slab structure under the condition of intermediate simple support are as follows: Among them, A r It is the amplitude of the reflected wave, A nl It is the amplitude of the reflected near-field wave, A t 、A nr These are the amplitudes of the transmitted propagating wave and the near-field wave, respectively. i It is the amplitude of the normally incident curved wave, and j is the initial phase angle; The amplitudes A of the two reflected waves in the simply supported end case of the slab-beam structure r 、A nl They are: Among them, A i It is the amplitude of a normally incident curved wave; The amplitudes A of the two reflected waves in the case of fixed end supports of the slab-beam structure r 、A nl They are: Among them, A i It is the amplitude of the normally incident curved wave, and j is the initial phase angle; Step S2 is calculated using the following steps: Step S21: Define the physical and geometric parameters of beams, slabs and columns based on the damping loss factor and damping bending stiffness; Step S22: Define excitation parameters and define the location of the extraction measurement point. The excitation parameters include excitation load type, frequency, amplitude, and location. Step S23: Perform mesh generation, set the number of modes, initialize the displacement response matrix, calculate the mechanical impedance of beams, plates, and columns, and initialize the displacement and acceleration matrices; Step S24: Determine the boundary conditions at the beam-slab connection and calculate the reflection and transmission coefficients between the beam-slab and the slab-column connection. Step S25: Calculate the displacement response; Step S26, calculate acceleration; Step S27: Calculate the peak acceleration.

2. The vibration wave propagation calculation method based on the WPA method according to claim 1, characterized in that: In step S1, for the plate and column at the connection point, if the plate and column are rigidly connected at the connection point, wave reflection is ignored; If the connection between the plate and column is semi-rigid, the reflected wave parameters can be obtained by solving the following equation: In 板 =in 柱 M 板 -M 柱 =K θ (i 柱 -θ 板 ) Among them, M 连接 Let θ be the bending moment at the connection. 柱 ,θ 板 This refers to the corner of the slab column at the joint; If the plate and column are hinged at the connection point, allowing rotation but not transmitting bending moment, only shear force and axial force, the reflected wave parameters can be obtained by solving the following equations: M 连接 ＝0 In 板 =in 柱 M 板 =0,M 柱 =0 Among them, M 连接 This represents the bending moment at the connection.

3. The vibration wave propagation calculation method based on the WPA method according to claim 2, characterized in that: Step S3 includes: for the standard floor plan of the structure under test, using the code in step S2, applying a simple harmonic wave load to one floor of the structure to obtain the acceleration time history curves of each floor of the structure under test; applying the same simple harmonic wave load as in the code using an exciter, conducting field tests, collecting the acceleration time history curves of the same points in each floor, and comparing them with the acceleration time history curves obtained by the code to verify the accuracy of the code calculation.

4. The vibration wave propagation calculation method based on the WPA method according to claim 3, characterized in that: In step S4, based on the verified code, the peak acceleration values ​​at different positions of each layer under different point excitation of each layer of the structure under test are calculated, and then the vibration propagation law of the vibration wave between the same layer and different layers is regressed to obtain the prediction equation of the peak acceleration of the vibration wave at any time and any position in the structure.

5. The vibration wave propagation calculation method based on the WPA method according to claim 4, characterized in that: The peak acceleration prediction equation is: α=-7.868Y 2 -9.796Z 2 -17,799x 2 -0.758y 2 -9.969z 2 +5.657XY+56.854ZY+12xY-0.955yY+0.794zY+0.928XZ+0.857YZ+1.133xZ-1.4yZ+19.691zZ+0.128Xx-0.982xY+1.281xZ In the formula, α is the peak acceleration, x, y, z are the coordinates of the measuring point, and X, Y, Z are the coordinates of the excitation source.

6. The application of the vibration wave propagation calculation method based on the WPA method as described in any one of claims 1 to 5, characterized in that: It is used to determine the response status of the target location.

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