Plate structure displacement detection method based on polynomial basis function

Abstract

The invention relates to the technical field of mechanical test and structure detection, in particular to a plate structure displacement detection method based on a polynomial basis function, which comprises the following steps: arranging a plurality of strain sensors on the plate surface of a cantilever plate in a dispersed manner; applying an external load to the free end of the cantilever slab; the strain sensor measures the strain of the position where the strain sensor is located, and the position and the strain measurement value of a strain measurement point are obtained; based on a Schmidt orthogonalization method, obtaining a cantilever slab displacement polynomial basis function group and a cantilever slab displacement model; obtaining a strain polynomial basis function group of the cantilever slab, and determining a strain model of the cantilever slab; based on the position of the strain measurement point and the strain measurement value, performing parameter estimation on the cantilever slab strain model to obtain an estimated parameter; obtaining a displacement value of each strain measuring point based on the estimation parameters and the cantilever slab displacement model; the displacement of the plate structure can be efficiently detected based on the strain data of the plate structure.

Description

Technical Field

[0001] The invention relates to the technical field of mechanical testing and structural detection, and in particular to a plate structure displacement detection method based on polynomial basis functions. Background Art

[0002] In the field of engineering structures, plate structures (such as bridges, building floors, and aerospace wings) are widely used in various engineering projects. Since plate structures are affected by various loads and environmental factors during use, their displacement and deformation are important indicators for evaluating the state of the structure. Therefore, accurate detection of the displacement of plate structures is very important for the systematic identification of structures.

[0003] Structural modal parameters are inherent characteristics of structures. Changes in structures such as reduced stiffness will cause changes in structural parameters, so changes in structural modal parameters can be used to detect whether the structure is damaged. However, obtaining structural modal parameters through experiments requires system identification of the structure. Existing system identification methods obtain structural modal parameters by identifying the response of the structure, and the data used for identification are usually response data such as displacement of the structure. Directly using velocity sensors to measure structural responses brings more additional mass to the structure than strain sensors, which has a greater negative impact on the structural response. To address this issue, obtaining structural displacement data through structural strain data obtained by strain sensors is of great significance for structural system identification. Summary of the invention

[0004] In view of the above problems, the present invention provides a plate structure displacement detection method based on polynomial basis functions, which solves the problem in the prior art that the structural system identification effect is poor directly based on the strain data of the cantilever plate structure.

[0005] The present invention provides a plate structure displacement detection method based on polynomial basis functions, comprising the following steps:

[0006] Step S1, arranging a plurality of strain measuring points on the surface of the cantilever plate in a dispersed manner, and laying a plurality of strain sensors at the positions of the plurality of strain measuring points; applying an external load to the free end of the cantilever plate; the strain sensor measures the strain at the position, and obtains the position and strain measurement value of the strain measuring point;

[0007] Step S2, based on the Schmidt orthogonalization method, obtain the cantilever plate displacement polynomial basis function group and determine the cantilever plate displacement model;

[0008] Step S3, obtaining a cantilever plate strain polynomial basis function group based on the cantilever plate displacement polynomial basis function group, and determining a cantilever plate strain model based on the cantilever plate strain polynomial basis function group;

[0009] Step S4, based on the position of the strain measuring point and the strain measurement value, perform parameter estimation on the cantilever plate strain model to obtain estimated parameters;

[0010] Step S5: obtaining the displacement value of each strain measuring point based on the estimated parameters and the cantilever plate displacement model.

[0011] Preferably, step S1 specifically includes: for a homogeneous cantilever plate with a length of L and a width of H, N strain sensors are evenly distributed on the surface of the cantilever plate, a random external load is applied to the cantilever plate structure near the free end, and the position coordinates (x i ,z i ) and strain measurement value ε i (i=1,2,3LN), where i is the strain measurement point number and N is the total number of strain measurement points on the cantilever plate.

[0012] Preferably, step S2 specifically includes:

[0013] One end of the cantilever plate in the z direction is fixed, the other end is free, and both ends in the x direction are free; the initial polynomial in the z direction is determined as c 1 z 2 ; Determine the initial polynomial in the x direction as a 1 ;

[0014] Based on the Schmidt orthogonalization method, the initial polynomial in the z direction and the initial polynomial in the x direction are orthogonalized to obtain the displacement polynomial basis function group Z in the z direction. n The polynomial basis function set X in the z and x directions m (x), where n represents the basis function number in the z direction and m represents the basis function number in the x direction;

[0015] The expression of the cantilever plate displacement model is:

[0016]

[0017] Where d(x,z) is the displacement of the cantilever plate at coordinate (x,z), A mn is the coefficient to be determined, q is the number of basis functions in the z direction, and p is the number of basis functions in the x direction.

[0018] Preferably, step S3 specifically includes:

[0019] Calculate the polynomial basis function set Z in the z direction n The second-order derivative of (z) is used to obtain the polynomial basis function group of the strain in this direction. The x-direction shift polynomial basis function group X m (x) is the strain polynomial basis function set in the x direction;

[0020] The strain polynomial basis function set X m (x) and Determine the z-direction strain expression of the cantilever plate strain model:

[0021]

[0022] Where e(x,z) is the strain in the z direction at the cantilever plate coordinate (x,z).

[0023] Preferably, step S4 specifically includes:

[0024] The position coordinates (x i ,z i ) is normalized to obtain the normalized position coordinates (x i ′,z i ′);

[0025] The normalized position coordinates (x i ′,z i ′) and the corresponding strain measurement value ε i Substitute into the expression e(x,z) of the cantilever plate strain model to obtain the coefficient A in the expression: mn .

[0026] Preferably, step S5 specifically includes:

[0027] The coefficient A to be determined mn and the normalized position coordinates (x i ′,z i ′) is substituted into the expression d(x, z) of the cantilever plate displacement model to obtain the displacement value of each strain measuring point.

[0028] Preferably, the cantilever plate has a length L = 0.255 m, a width H = 0.0762 m, and an elastic modulus E = 7×10 10 Pa, the number of strain sensors N=20 evenly distributed on the cantilever plate surface.

[0029] Compared with the prior art, the present invention has at least the following beneficial effects:

[0030] (1) The present invention converts the strain data of multiple measuring points on the cantilever plate structure into displacement data through the method of polynomial basis function fitting, thereby improving the accuracy of displacement detection and the success rate of structural system identification, and effectively solving the problem of poor system identification effect directly based on the cantilever plate strain data.

[0031] (2) The present invention obtains strain data by using simple sensors such as strain gauges instead of directly using displacement or acceleration sensors, thereby avoiding the influence of the additional mass introduced by the sensor on the structural response, while ensuring data accuracy and reducing interference with the structure itself.

[0032] (3) The strain sensor of the present invention is simpler and more economical than displacement or acceleration sensors. Accurate structural displacement data can be obtained through this method, which reduces costs and provides a more economical and efficient solution for structural system identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The drawings are only for the purpose of illustrating particular embodiments and are not to be construed as limiting the invention.

[0034] Figure 1 This is a flow chart of the plate structure displacement detection method based on polynomial basis functions provided by the present invention.

[0035] Figure 2 This is a model diagram of the cantilever plate structure provided by the present invention.

[0036] Figure 3 A comparison diagram of the displacement fitting results of the cantilever plate structure provided by the present invention. DETAILED DESCRIPTION

[0037] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein, and therefore, the protection scope of the present invention is not limited by the specific embodiments disclosed below.

[0038] In order to solve the problem that the structural system identification effect is not good directly based on the strain data of the cantilever plate structure, the present invention provides a plate structure displacement detection method based on polynomial basis functions, which obtains the displacement values ​​of the structural strain measuring points through polynomial fitting based on the strain values ​​of multiple points on the structure, and then performs system identification on the structure, thereby increasing the success rate of system identification.

[0039] In order to illustrate the effectiveness of the method proposed by the present invention, the above technical solution of the present invention is described in detail below through a specific embodiment. Figure 1 As shown, a plate structure displacement detection method based on polynomial basis functions is disclosed, and the specific implementation steps are as follows:

[0040] Step S1, a plurality of strain measuring points are dispersedly arranged on the surface of the cantilever plate, and a plurality of strain sensors are laid corresponding to the positions of the plurality of strain measuring points; an external load is applied to the free end of the cantilever plate; the strain sensor measures the strain at the position, and obtains the position and strain measurement value of the strain measuring point.

[0041] like Figure 2 As shown, the plate structure in the embodiment of the present invention is a homogeneous cantilever plate with a length of L = 0.255m, a width of H = 0.0762m, and a cross-sectional size of 3.175*10 -3 *0.0762m 2 , elastic modulus E = 7 × 10 10 Pa, 20 strain sensors are evenly distributed on the cantilever plate surface, a random external load is applied to the cantilever plate structure near the free end, and the position coordinates (x i ,z i ) and the strain measurement value ε of each strain measuring point i (i=1,2,3L20), where i is the strain measurement point index.

[0042] Step S2: Based on the Schmidt orthogonalization method, a polynomial basis function group of the cantilever plate displacement is obtained to determine the cantilever plate displacement model.

[0043] In this step, the Schmidt orthogonalization method is used to obtain a normalized orthogonal polynomial basis function group in each direction of the plate displacement based on the initial polynomial of the existing cantilever plate boundary conditions in the length and width directions. The two groups of polynomial basis functions are multiplied and then summed to form a displacement expression based on the polynomial basis function.

[0044] For the cantilever plate, assume that one end in the z direction is fixed, the other end is free, and both ends in the x direction are free. Select the initial polynomial that satisfies the boundary conditions. Based on the condition that one end in the z direction is fixed and the other end is free, the initial polynomial that satisfies the boundary conditions in the z direction is selected as c 1 z 2 ; Based on the condition that both ends are free in the x-direction, the initial polynomial that satisfies the boundary conditions in the x-direction is selected as a 1 .

[0045] Based on the boundary conditions in the length and width directions of the cantilever plate, the Schmidt orthogonalization method is used to determine a 1 =1.

[0046] The Schmidt orthogonalization method is a method for transforming a set of linearly independent polynomials into a set of orthogonal polynomials. The present invention uses the Schmidt orthogonalization method to orthogonalize the initial polynomials, obtain the coefficients and subsequent high-order polynomials according to the properties of the orthogonal polynomials, and obtain the basis function group Z n (z) and Xm (x), where Z n (z) is the displacement polynomial basis function group in the z direction, n is the basis function number in the z direction; X m (x) is the displacement polynomial basis function group in the x direction, and m represents the basis function number in the x direction. The product of these mutually orthogonal polynomial basis functions is linearly combined to form the expression of the cantilever plate displacement model:

[0047]

[0048] Where d(x,z) is the displacement of the cantilever plate at coordinate (x,z), A mn is the coefficient to be determined, q is the number of basis functions in the z direction, and p is the number of basis functions in the x direction.

[0049] In some embodiments, the step of orthogonalizing the initial polynomial using the Schmidt orthogonalization method may be: for the z direction, the initial polynomial is c 1 z 2 , in order to generate the subsequent high-order polynomial basis function group Z n (z), we can introduce more polynomials. For example, we can choose the polynomial set {z 2 ,z 3 ,z 4 ,...}.

[0050] The Schmidt orthogonalization process is as follows: Let P a (z) = α 1 z 2 , P b (z) = α 2 z 2 +α 3 z 3 , where α 1 ,α 2 ,α 3 is the coefficient to be determined, then the coefficient to be determined should satisfy the Schmidt orthogonalization condition:

[0051]

[0052] Solve α by calculating the above integral equation 1 ,α 2 ,α 3 , and get P a (z), P b (z). In a similar way, more high-order polynomials orthogonal to the known polynomials are calculated, so that each new polynomial is orthogonal to all previous polynomials, and an orthogonal normalized polynomial basis function group Z is obtained. n (z). The subsequent high-order polynomial basis function group X in the x direction is calculated in a similar way m(x).

[0053] The above example illustrates a method in which the present invention uses the Schmidt orthogonalization method to orthogonalize the initial polynomial to obtain a subsequent high-order polynomial. It can be understood that there are other ways to achieve Schmidt orthogonalization, and the present invention does not limit the specific method of achieving Schmidt orthogonalization.

[0054] Obtain X by Schmidt orthogonalization method m (x) and Z n (z), where m = 5, n = 4, indicating that there are five polynomials in the x direction and four polynomials in the z direction. The displacement model of the cantilever plate is expressed as follows:

[0055]

[0056] Step S3, obtaining a cantilever plate strain polynomial basis function group based on the cantilever plate displacement polynomial basis function group, and determining a cantilever plate strain model based on the cantilever plate strain polynomial basis function group.

[0057] In this step, according to the relationship between displacement and strain in elastic mechanics, the polynomial basis function of strain is obtained on the basis of the displacement polynomial basis function, thereby obtaining the strain expression based on the strain polynomial basis function.

[0058] Specifically, according to the relationship between displacement and strain in elastic mechanics, the strain of the cantilever plate structure is the second-order derivative of the displacement. Calculate the polynomial basis function group Z of the displacement in the z direction n The second-order derivative of (z) obtains the polynomial basis function set of the strain in this direction The x-direction shift polynomial basis function group X m (x) is the strain polynomial basis function set in the x direction.

[0059] The polynomial basis function set X m (x) and The product of the polynomial basis function group is linearly combined to obtain the strain expression in the z direction of the cantilever plate strain model:

[0060]

[0061] Where e(x,z) is the strain in the z direction at the cantilever plate coordinate (x,z), A mn is the coefficient to be determined, q is the number of basis functions in the z direction, and p is the number of basis functions in the x direction.

[0062] For the expression of the cantilever plate strain model, the cantilever plate of the present invention has one end fixed in the z direction and the other end free, and both ends in the x direction are free, so the main strain occurs in the z direction. According to the relationship between the displacement and strain of the plate in elastic mechanics, the strain in a certain direction is the second-order derivative of the displacement. Therefore, when only the strain in the z direction is considered and the strain in the x direction is ignored, the strain expression of the cantilever plate can be expressed as the second-order derivative of the displacement expression of the cantilever plate with respect to z. When the displacement expression of the cantilever plate is derived, due to X m (x) depends only on x, and the derivative of z will not affect X m (x), so it can be regarded as a constant. So the expression of the cantilever plate strain model has Z n The second derivative of (z) And X m (x) remains unchanged.

[0063] The present invention obtains the second-order derivatives of the four polynomial bases in the z direction respectively, and there are five polynomial bases in the x direction that remain unchanged, forming a fitting expression for the positive strain in the z direction of the plate:

[0064]

[0065] Step S4: Based on the position of the strain measuring point and the strain measurement value, the cantilever plate strain model is estimated to obtain estimated parameters.

[0066] Based on the known position coordinates of each strain measuring point on the cantilever plate under external load (x i ,z i ) and the measured strain value ε i , normalize the position coordinates of each strain measuring point on the plate to obtain the normalized position coordinates (x i ′,z i ′). The normalized coordinate (x i ′,z i ′) and the corresponding strain measurement value ε i Substitute into the expression e(x,z) of the cantilever plate strain model to obtain the coefficient A in the expression: mn .

[0067] In some embodiments, the normalized coordinates and the corresponding strain measurement values ​​are substituted into the expression e(x, z) of the cantilever plate strain model to obtain a linear equation system, and the least square method or other methods can be used to solve the linear equation system to achieve parameter estimation and obtain the coefficient A to be determined. mn The present invention does not limit the specific method of solving the coefficients to be determined.

[0068] Step S5: obtaining the displacement value of each strain measuring point based on the estimated parameters and the cantilever plate displacement model.

[0069] The coefficient A to be determined mn and the normalized position coordinates (x i ′,z i ′) is substituted into the expression d(x, z) of the cantilever plate displacement model to obtain the displacement fitting response value at the corresponding position.

[0070] The displacement fitting response value obtained by MATLAB and the displacement value obtained by Abaqus simulation are compared. Figure 3 As shown, it can be seen that the displacement fitting response value obtained by strain conversion in the present invention is consistent with the simulation displacement response result, and the calculation result of the cantilever plate structure displacement detection method provided by the present invention has a high accuracy.

[0071] The present invention adopts the above technical scheme, based on the measured strain values ​​of multiple points on the cantilever plate structure, obtains the displacement values ​​of the structural strain measuring points through polynomial fitting, and then performs system identification on the structure, thereby increasing the success rate of system identification.

[0072] Although the specific embodiments of the present invention depict various actions or steps in a specific order, this should be understood as requiring such actions or steps to be performed in the specific order shown or in a sequential order, or requiring all illustrated actions or steps to be performed to obtain the desired results. Under certain circumstances, multitasking and parallel processing may be advantageous. Similarly, although several specific implementation details are included in the above discussion, these should not be interpreted as limiting the scope of the present disclosure. Certain features described in the context of a separate embodiment can also be implemented in a single implementation in combination. On the contrary, the various features described in the context of a single implementation can also be implemented in multiple implementations individually or in any suitable sub-combination. The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or replacements that can be easily thought of by any technician familiar with the technical field within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

[0073] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by any technician familiar with the technical field within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

Claims

1. A plate structure displacement detection method based on polynomial basis functions, characterized in that: The following steps are involved: Step S1, arranging a plurality of strain measuring points on the surface of the cantilever plate in a dispersed manner, and laying a plurality of strain sensors at the positions of the plurality of strain measuring points; applying an external load to the free end of the cantilever plate; the strain sensor measures the strain at the position, and obtains the position and strain measurement value of the strain measuring point; Step S2, based on the Schmidt orthogonalization method, obtain the cantilever plate displacement polynomial basis function group and determine the cantilever plate displacement model; Step S3, obtaining a cantilever plate strain polynomial basis function group based on the cantilever plate displacement polynomial basis function group, and determining a cantilever plate strain model based on the cantilever plate strain polynomial basis function group; Step S4, based on the position of the strain measuring point and the strain measurement value, perform parameter estimation on the cantilever plate strain model to obtain estimated parameters; Step S5: obtaining the displacement value of each strain measuring point based on the estimated parameters and the cantilever plate displacement model.

2. The plate structure displacement detection method based on polynomial basis functions according to claim 1 is characterized in that: Step S1 specifically includes: For a homogeneous cantilever plate with a length of L and a width of H, N strain sensors are evenly distributed on the cantilever plate surface. A random external load is applied to the cantilever plate structure near the free end. The position coordinates (x i ,z i ) and strain measurement value ε i (i=1,2,3L N), where i is the strain measurement point number and N is the total number of strain measurement points on the cantilever plate.

3. The plate structure displacement detection method based on polynomial basis functions according to claim 2 is characterized in that: Step S2 specifically includes: One end of the cantilever plate in the z direction is fixed, the other end is free, and both ends in the x direction are free; the initial polynomial in the z direction is determined as c1z 2 ; Determine the initial polynomial in the x direction as a1; Based on the Schmidt orthogonalization method, the initial polynomial in the z direction and the initial polynomial in the x direction are orthogonalized to obtain the displacement polynomial basis function group Z in the z direction. n The polynomial basis function set X in the z and x directions m (x), where n represents the basis function number in the z direction and m represents the basis function number in the x direction; The expression of the cantilever plate displacement model is: Where d(x,z) is the displacement of the cantilever plate at coordinate (x,z), A mn is the coefficient to be determined, q is the number of basis functions in the z direction, and p is the number of basis functions in the x direction.

4. The plate structure displacement detection method based on polynomial basis functions according to claim 3 is characterized in that: Step S3 specifically includes: Calculate the polynomial basis function set Z in the z direction n The second-order derivative of (z) is used to obtain the polynomial basis function group of the strain in this direction. The x-direction shift polynomial basis function group X m (x) is the strain polynomial basis function set in the x direction; The strain polynomial basis function set X m (x) and Determine the z-direction strain expression of the cantilever plate strain model: Where e(x,z) is the strain in the z direction at the cantilever plate coordinate (x,z).

5. The plate structure displacement detection method based on polynomial basis functions according to claim 4 is characterized in that: Step S4 specifically includes: The position coordinates (x i ,z i ) is normalized to obtain the normalized position coordinates (x i ′,z i ′); The normalized position coordinates (x i ′,z i ′) and the corresponding strain measurement value ε i Substitute into the expression e(x,z) of the cantilever plate strain model to obtain the coefficient A in the expression: mn .

6. The plate structure displacement detection method based on polynomial basis functions according to claim 5 is characterized in that: Step S5 specifically includes: The coefficient A to be determined mn and the normalized position coordinates (x i ′,z i ′) is substituted into the expression d(x, z) of the cantilever plate displacement model to obtain the displacement value of each strain measuring point.

7. The plate structure displacement detection method based on polynomial basis functions according to any one of claims 2 to 6, characterized in that: The cantilever plate has a length of L = 0.255 m, a width of H = 0.0762 m, and an elastic modulus of E = 7 × 10 10 Pa, the number of strain sensors N=20 evenly distributed on the cantilever plate surface.

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