Method and equipment for determining the order of distributed dynamic loads based on the golden section method

Abstract

This invention discloses a method and device for determining the order of distributed dynamic loads based on the golden section method. The method comprises: establishing a relationship expression between response and load based on frequency-domain load identification theory; transforming the solution to the distributed load into the solution of the generalized Fourier series expansion order and its coefficient using a generalized Fourier series expansion; using the multiplication method to determine the interval range of the generalized Fourier series expansion order, and then applying the golden section method to obtain the optimal order within this interval. This method can quickly find the optimal fitting order for the distributed load, has high computational efficiency, and effectively improves the accuracy of distributed load identification.

Description

Technical Field

[0001] The present invention belongs to the field of dynamic load identification, and in particular relates to a method and device for determining the order of distributed dynamic loads based on the golden section method. Background Art

[0002] Distributed dynamic loads, a common load type in engineering practice, play a crucial role in engineering applications. Examples include bridges and buildings subjected to wind loads, and ships and dams subjected to wind and waves. The identification of distributed dynamic loads requires dynamic calibration, and the quality of this calibration directly impacts the accuracy of load identification. In load identification, distributed loads are continuously distributed functions. The limitations of infinite-dimensional dynamic calibration make it difficult to implement in engineering practice. Therefore, converting infinite-dimensional dynamic calibration in distributed dynamic load identification to finite-dimensional dynamic calibration is a highly research-oriented issue in both theoretical and engineering applications. A common approach is to expand the loads using orthogonal polynomials, establish a relationship between the polynomial coefficients and the response, and determine the loads using the polynomial coefficients. When using a generalized Fourier series expansion, the order of the expansion is infinite. However, in numerical calculations, the error between the fitting function and the fitted function must be kept within a certain error range. Furthermore, some studies have shown that a higher fitting order is not necessarily better. For example, when solving inverse problems, a higher fitting order can lead to more severe ill-conditioning. Therefore, it is necessary to quickly determine the reasonable fitting order. Reasonable determination of the truncation order can improve the calculation efficiency and control the error range. Summary of the Invention

[0003] Purpose of the invention: The purpose of the present invention is to provide a method and device for determining the order of distributed dynamic loads based on the golden section method, which has high computational efficiency, can quickly find the best fitting order of the distributed load, and effectively improve the accuracy of distributed load identification.

[0004] Technical solution: To achieve the above-mentioned purpose, the present invention adopts the following technical solution:

[0005] A method for determining the order of distributed dynamic loads based on the golden section method comprises the following steps:

[0006] (1) Based on the frequency domain load identification theory, the relationship between the response and the load is established. Combined with the generalized Fourier series expansion, the solution of the distributed load is transformed into the solution of the generalized Fourier series expansion order and its coefficients;

[0007] (2) The doubling method is used to determine the appropriate interval range [j / t, j] of the generalized Fourier series expansion order, where j is the Fourier series expansion order and t is the doubling step size. Based on the appropriate interval range [j / t, j], the optimal order is located using the golden section method.

[0008] Furthermore, the appropriate interval range for determining the order of the generalized Fourier series expansion using the doubling method in step (2) includes:

[0009] (2.1) Calculate the maximum relative error ε of load identification for the jth order of distributed load expansion j ;

[0010] (2.2) Expand j by t times, determine point c: c = ceil(j / t + 0.382(jj / t)), and calculate ε c ,ε j , judge ε c <ε j / t And ε c <ε j Is it true? If not, loop through step (2.2) until it is true, then terminate the loop and get a single peak interval [j / t,j], where point c is the point at 0.382 of the golden section method in the interval [j / t,j], ε j / t ,ε c ,ε j The maximum relative errors of load identification for the j / t, c, and j order distributed load expansions are shown in Figure 1. Ceil() indicates rounding up.

[0011] Furthermore, the maximum relative error of load identification is defined as: in is the relative error in identifying the real part of the distributed load, The relative error of the imaginary part of the distributed load is calculated as follows:

[0012]

[0013] Where, f r (x),f i (x) are the real and imaginary parts of the measured load, are the real and imaginary parts of the calculated load of the j-th order generalized Fourier series expansion of the distributed load, and abs() means finding the absolute value.

[0014] Furthermore, in step (2), locating the optimal order by the golden section method includes:

[0015] (2.a) Let m = j / t, d = m + jc, and calculate ε d , where d is the point at 0.618 of the golden section method in the interval [m, j], ε d The maximum relative error of load identification for the d-order expansion of the distributed load;

[0016] (2.b) Determine whether c = d and jd ≠ 1. If so, set d = d + 1 and proceed to the next step; otherwise, proceed directly to the next step;

[0017] (2.c) Determine whether jm>1. If so, end the loop and output the optimal expansion order j; otherwise, proceed to the next step.

[0018] (2.d) Determine ε c <ε d Is it true? If so, assign j = d, d = c, c = m + jd; and determine whether c = d and cm ≠ 1 is true. If so, set c = c-1 and calculate ε c , return to step (2.c), otherwise directly calculate ε c And return to step (2.c), if ε c <ε d If not, proceed to the next step;

[0019] (2.e) Assign m = c, c = d, and d = m + jc; and determine whether c = d and jd ≠ 1 holds. If so, set d = d + 1 and calculate ε. d , return to step (2.c), otherwise directly calculate ε d And return to step (2.c).

[0020] The present invention also provides a computer device comprising: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and are configured to be executed by the one or more processors, and when the programs are executed by the processors, the steps of the distributed dynamic load order determination method based on the golden section method as described above are implemented.

[0021] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the distributed dynamic load order determination method based on the golden section method as described above.

[0022] Beneficial effects: The present invention establishes the relationship between the response and the load based on the frequency domain load identification theory, and combines the generalized Fourier series expansion to convert the solution of the distributed load into the solution of the generalized Fourier series expansion order and its coefficients, so that the identified load can be obtained by fitting. The expansion order of the distributed load seriously affects the accuracy of load identification, so it is necessary to determine the optimal expansion order. The present invention first determines the interval range of the reasonable expansion order of the distributed load through the multiplication method, and then obtains the optimal expansion order through the golden section method. The example verifies that this method has high computational efficiency, can quickly find the optimal fitting order of the distributed load, effectively improves the distributed load identification accuracy, and the method is easy to implement and has broad application prospects in the field of load identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 This is a flow chart of a method for determining the order of a distributed load based on the golden section method according to an embodiment of the present invention;

[0024] Figure 2 2. This is a schematic diagram of distributed load on a simply supported beam according to an embodiment of the present invention;

[0025] Figure 3 This is a flow chart of the fast golden section calculation method according to an embodiment of the present invention;

[0026] Figure 4 1 is a graph showing the maximum relative error in load identification corresponding to the 1st to 10th order expansion of the generalized Fourier series of the distributed load according to an embodiment of the present invention;

[0027] Figure 5 This is a comparison of the identification results of the real and imaginary parts of the distributed load of the 7th order, the best order of distributed load expansion, according to an embodiment of the present invention;

[0028] Figure 6 This is a graph showing the absolute error between the identification result and the true value of the optimal order 7 of the distributed load expansion according to an embodiment of the present invention;

[0029] Figure 7 This is a diagram showing the relative error between the identification result and the true value of the 7th order, the optimal order of distributed load expansion, according to an embodiment of the present invention. DETAILED DESCRIPTION

[0030] The technical solution of the present invention will be further described below with reference to the accompanying drawings. For the purpose of simplicity and explanation, the principles of the present invention are mainly described with reference to embodiments. The embodiments of the present invention provide an implementation description of the present method using a simply supported beam model, but it should be understood that the present method is also applicable to the identification of distributed loads of other structures. In the following description, many specific details are presented to provide a thorough understanding of the embodiments. However, it is obvious to those skilled in the art that these embodiments may not be limited to these specific details in practice.

[0031] The generalized Fourier series fitting function of the distributed load in the present invention uses Legendre polynomials. In order to have a clearer and more comprehensive understanding of the technical solution of the invention, the generalized Fourier series fitting function is first theoretically derived.

[0032] Legendre polynomials are orthogonal polynomials with weighting function ρ(x)=1 on the interval [-1,1], and are expressed as follows:

[0033]

[0034] The expanded results are as follows:

[0035]

[0036] The Legendre polynomials have the following properties:

[0037] Recursion: (n+1)L n+2 (x)=(2n+1)xLn+1 (x)-nL n (x)(n≥2)

[0038] Orthogonality:

[0039] Parity: When the order n is an odd number, L n (x) is an even function; when the order n is an even number, L n (x) is an odd function, that is:

[0040] L n (-x) = (-1) n-1 L n (x)(n≥2)

[0041] When x∈[a,b], do variable substitution Then t∈[-1,1], the transformed Legendre polynomial is:

[0042]

[0043] in, Solve back substitution

[0044] The distributed load of the beam structure is fitted using a one-dimensional Legendre polynomial, while the distributed load of the plate structure is fitted using a two-dimensional Legendre polynomial, whose expression is L mn (x,y)=L m (x)L n (y), by the same token, it can be proved that the two-dimensional Legendre polynomials also satisfy the recursion, orthogonality and parity properties.

[0045] If the function f(x) is weighted square integrable on the interval [a, b], then f(x) can be expanded using the generalized Fourier series:

[0046]

[0047] where a n is the coefficient after expansion, is the normalized form of Legendre polynomial, calculated by the following formula:

[0048]

[0049] Reference Figure 1 The distributed dynamic load order determination method of the embodiment of the present invention includes the following steps:

[0050] Step 1: Based on the frequency domain load identification theory, the relationship expression between the response and the load is established. Combined with the generalized Fourier series expansion, the solution of the distributed load is transformed into the solution of the generalized Fourier series expansion order and its coefficients.

[0051] First, a simply supported beam model is established. The simply supported beam has a length of l = 1 m, a width of a = 0.04 m, a height of h = 0.008 m, and an elastic modulus of E = 2.1 × 10 11 Pa, density ρ = 7800 kg / m 3 , Poisson's ratio μ = 0.3, divided into 50 beam elements. Figure 2 The distributed load diagram of a simply supported beam is shown. The distributed dynamic load acting on the beam is:

[0052]

[0053] Derive the Legendre polynomial expression, x∈[0,1], and make variable substitution x=2x-1, then the x interval becomes x∈[-1,1], and the expression is as follows:

[0054]

[0055] In the frequency domain, the relationship between the single-point excitation and response of the structure is:

[0056] X(ω)=H(ω)F(ω)

[0057] Where X(ω) is the displacement response function in the frequency domain; H(ω) is the corresponding frequency response function; and F(ω) is the force in the frequency domain.

[0058] For continuous structures, the above formula can be written as:

[0059]

[0060] Where x and k are the positions of the excitation point and the response point, respectively.

[0061] For continuous structures, load identification can be expressed in the complex domain as:

[0062]

[0063] Natural frequency ω=20Hz, distributed load F r (ω,x),F i (ω,x) can be expressed as follows through the generalized Fourier series expansion:

[0064]

[0065] Where, is the normalized form of Legendre polynomial, a j ,b j (j=1,2,...∞) are the coefficients of Legendre polynomial. r (x),F i Substitute (x) into the complex domain expression of the load:

[0066]

[0067] Where j is the truncation order of the Legendre polynomial for the distributed load, are the real and imaginary parts of the frequency response function of point k under the excitation of point x, is the normalized form of the j-th Legendre polynomial, a j ,b j are the real and imaginary coefficients of the Legendre polynomial, are the real and imaginary parts of the displacement response at point k on the beam.

[0068] Select 50 equally spaced measuring points, and the above formula can be simplified as follows:

[0069] QA=X

[0070] Where,

[0071] Where k = 50 and j is to be determined. The coefficients of the Legendre polynomials can be obtained by taking the generalized inverse of the above equation:

[0072] A=[Q T Q] -1 Q T X=Q + X

[0073] Substitute into the fitting polynomial to obtain the distributed load expression:

[0074]

[0075] Among them F r (x) is the real part of the distributed load, F i (x) is the imaginary part of the distributed load.

[0076] Step 2: Use the doubling method to determine the appropriate interval range of the generalized Fourier series expansion order, and based on the appropriate interval range, locate the optimal order using the golden section method.

[0077] The distributed load is expanded using the generalized Fourier series. If the expansion order is low, the load fitting accuracy will be low. If the expansion order is high, the load identification accuracy will be low due to the ill-posed problem. Therefore, it is necessary to select the optimal fitting order to obtain a higher identification effect.

[0078] Define the relative error of load identification:

[0079]

[0080] Where, is the relative error in identifying the real part of the distributed load, is the relative error of the imaginary part of the distributed load, abs() means to find the absolute value, f r(x),f i (x) are the real and imaginary parts of the measured load, are the real and imaginary parts of the calculated load of the j-th order generalized Fourier series expansion of the distributed load, respectively.

[0081] Define the maximum relative error for load identification:

[0082]

[0083] Where, ε j is the maximum relative error of load identification of the j-th order generalized Fourier series expansion of the distributed load.

[0084] In order to obtain the optimal generalized Fourier series expansion order of the load, the present invention uses the accelerated golden section method. First, the reasonable order range is determined by the multiplication method, that is, a multiplication step t is preset, and the search intervals [1, t], [t, t 2 ]、[t 2 ,t 3 ]...dynamically change the interval until the appropriate interval range is found, and then use the golden section method to locate the optimal order.

[0085] Reference Figure 3 The following are the specific steps for accelerating the golden section method using t=3 as an example:

[0086] (1) Set the initial parameters, let j = 1, t = 3;

[0087] (2) Calculate the maximum relative error ε of load identification for the jth order of distributed load expansion j ;

[0088] (3) Expand j by t times, determine point c: c = ceil(j / t + 0.382(jj / t)), and calculate ε c ,ε j , judge ε c <ε j / t And ε c <ε j Is it true? If it is true, execute the next step. Otherwise, execute step (3) repeatedly. Where ceil() means rounding up.

[0089] The third step is to find the solution interval by the doubling method. Since the order of the generalized Fourier series expansion of the distributed load affects the load identification accuracy, the maximum relative error of load identification ε j As the expansion order increases, it will first become smaller until the optimal expansion order is reached. At this time, the maximum relative error of load identification ε j The trend of dropping to the lowest point and then suddenly increasing is shown in Figure 4 . Figure 4The maximum relative error ε of the load identification corresponding to the j-order (j=1, 2, ..., 10) expansion of the generalized Fourier series of the distributed load is shown. j It can be seen that the order of expansion of the generalized Fourier series of the distributed load is not the larger the better, so it is necessary to determine the optimal expansion order to reduce the load identification error. It can be seen that the optimal expansion order j is the minimum point of the single peak interval. The single peak interval is the maximum relative error ε of the load identification j The interval changes according to the "high-low-high" rule, so it satisfies ε c <ε j / t And ε c <ε j Then the loop is terminated and a single peak interval [j / t,j] is obtained. Point c is the point at 0.382 of the interval [j / t,j] using the golden section method, ε j / t ,ε c ,ε j The maximum relative errors of load identification for the j / t, c, and j order distributed load expansions are shown respectively.

[0090] (4) Let m = j / t, d = m + jc, and calculate ε d ;

[0091] The single peak interval [m, j] is obtained by the doubling method in step 3. The golden section method is used in this interval to gradually reduce the single peak interval, where d is the point at 0.618 of the golden section method in the interval [m, j]. d Maximum relative error in load identification for the d-th order of distributed load expansion.

[0092] (5) Determine whether c = d and jd ≠ 1. If so, set d = d + 1 and proceed to the next step; otherwise, proceed directly to the next step;

[0093] Since point c is the golden section 0.382 rounded up, there is a situation where c=d. The purpose here is to make c<d to improve the efficiency of the golden section.

[0094] (6) Determine whether jm>1 is true. If not, end the loop and output the optimal expansion order j; otherwise, proceed to the next step;

[0095] For the obtained interval [m,j], if jm≤1, the loop is terminated and j is the optimal expansion order.

[0096] (7) Determine ε c <ε d Is it true? If so, assign j = d, d = c, c = m + jd; and determine whether c = d and cm ≠ 1 is true. If so, set c = c-1 and calculate ε c , return to step (6), otherwise directly calculate ε cAnd return to step (6). If ε c <ε d If not, proceed to the next step;

[0097] The purpose of this step is to narrow the interval according to the golden section method. c <ε d If holds, then the interval is

[0098] [m,d], and update the 0.382 and 0.618 points of the golden section method in the interval [m,d].

[0099] (8) Assign m = c, c = d, d = m + jc; and determine whether c = d and jd ≠ 1 holds. If so, set d = d + 1 and calculate ε d , return to step (6), otherwise directly calculate ε d And return to step (6).

[0100] If ε c >ε d , then the interval is [c,j], and the points 0.382 and 0.618 of the golden section method are updated within the interval [c,j].

[0101] Steps 2-3 of the above steps involve the doubling method, which quickly determines the range of reasonable fitting orders. Steps 4-8 involve the golden section method, which is used to find the extreme value of the fitting error and locate the optimal order of the generalized Fourier series expansion of the distributed load. Once the optimal Fourier series order is determined, the final fitted load identification result is obtained based on the distributed load expression expressed using the Legendre polynomial expansion, which is obtained by inverting QA = X.

[0102] In the examples, a Matlab program was written to perform load identification and optimal expansion order analysis. Table 1 shows the process of finding the optimal expansion order for distributed loads and the load identification error for the example. As can be seen from the table, only five calculations are required to find the optimal expansion order for the distributed loads, and as the optimal expansion order for the distributed loads is higher, the computational efficiency is significantly improved.

[0103] Table 1 Expanded order and identification error records

[0104] Calculation times Expansion order Error ε 1 1 0.873743 2 3 0.559030 3 9 1.051320 4 6 0.110588 5 7 0.005290

[0105] In the embodiment, the above steps return the optimal expansion order j=7 of the generalized Fourier series of the distributed load.

[0106] Figure 5 、 Figure 6 、 Figure 7The results of the real and imaginary parts of the distributed load, as well as the absolute and relative errors from the true values, are shown for the optimal order of the distributed load expansion, 7. The results show that the distributed load order determination method based on the golden section method can quickly find the optimal fitting order of the distributed load, effectively improving the accuracy of the distributed load identification.

[0107] The present invention also provides a computer device comprising: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and are configured to be executed by the one or more processors, and when the programs are executed by the processors, the steps of the distributed dynamic load order determination method based on the golden section method as described above are implemented.

[0108] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the distributed dynamic load order determination method based on the golden section method as described above.

Claims

1. A method for determining the order of distributed dynamic loads based on the golden section method, characterized in that: The following steps are involved: (1) Based on the frequency domain load identification theory, the relationship expression between the response and the load is established. Combined with the generalized Fourier series expansion, the solution of the distributed load is transformed into the solution of the generalized Fourier series expansion order and its coefficient. The relationship expression between the response and the load is as follows: In the frequency domain, the relationship between the single-point excitation and response of the structure is: X(ω)=H(ω)F(ω) Where X(ω) is the displacement response function in the frequency domain; H(ω) is the corresponding frequency response function; F(ω) is the frequency domain force; For a continuous structure, the above formula is expressed as: Where x and k are the positions of the excitation point and the response point, respectively, which can be expressed in the complex domain as: l is the target structure length, ω is the natural frequency; (2) Use the doubling method to determine the appropriate interval range [j / t, j] of the generalized Fourier series expansion order, where j is the Fourier series expansion order and t is the doubling step size. Based on the appropriate interval range [j / t, j], the optimal order is located by the golden section method, where the golden section method for locating the optimal order includes: (2.a) Let m = j / t, d = m + jc, and calculate ε d , where d is the point at 0.618 of the golden section method in the interval [m, j], point c is the point at 0.382 of the golden section method in the interval [j / t, j], and ε d The maximum relative error of load identification for the d-order expansion of the distributed load; (2.b) Determine whether c = d and jd ≠ 1. If so, set d = d + 1 and proceed to the next step; otherwise, proceed directly to the next step; (2.c) Determine whether jm>1. If so, end the loop and output the optimal expansion order j; otherwise, proceed to the next step. (2.d) Determine ε c <ε d Is it established? c The maximum relative error of load identification for the c-order distributed load expansion is given. If it is established, assign j = d, d = c, c = m + jd; and determine whether c = d and cm ≠ 1 is established. If it is established, set c = c-1 and calculate ε c , return to step (2.c), otherwise directly calculate ε c And return to step (2.c), if ε c <ε d If not, proceed to the next step; (2.e) Assign m = c, c = d, and d = m + jc; and determine whether c = d and jd ≠ 1 holds. If so, set d = d + 1 and calculate ε. d , return to step (2.c), otherwise directly calculate ε d And return to step (2.c).

2. The method for determining the order of distributed dynamic loads based on the golden section method according to claim 1, characterized in that: In the step (1), combined with the generalized Fourier series expansion, the solution of the distributed load is converted into the solution of the generalized Fourier series expansion order and its coefficients, including: When the natural frequency ω is known, the distributed load F r (ω,x),F i (ω,x) is expressed by the generalized Fourier series expansion as: Where, is the normalized form of the j-th Legendre polynomial, a j ,b j are the real and imaginary coefficients of Legendre polynomials, j is the truncation order of Legendre polynomials for distributed loads, j=1,2,...∞, and F r (x),F i Substitute (x) into the complex field expression: Where, are the real and imaginary parts of the frequency response function of point k under the excitation of point x, are the real and imaginary parts of the k-point displacement response on the target structure.

3. The method for determining the order of distributed dynamic loads based on the golden section method according to claim 1, characterized in that: In step (2), the appropriate interval range for determining the order of the generalized Fourier series expansion using the doubling method includes: (2.1) Calculate the maximum relative error ε of load identification for the jth order of distributed load expansion j ; (2.2) Expand j by t times, determine point c: c = ceil(j / t + 0.382(jj / t)), and calculate ε c ,ε j , judge ε c <ε j / t And ε c <ε j Is it true? If not, loop through step (2.2) until it is true, then terminate the loop and get the single peak interval [j / t,j], where point c is the point at 0.382 of the golden section method in the interval [j / t,j], ε j / t ,ε c ,ε j The maximum relative errors of load identification for the j / t, c, and j order distributed load expansions are shown in Figure 1. Ceil() indicates rounding up.

4. The method for determining the order of distributed dynamic loads based on the golden section method according to claim 3 is characterized in that: The maximum relative error of load identification is defined as: in is the relative error in identifying the real part of the distributed load, The relative error of the imaginary part of the distributed load is calculated as follows: Where, f r (x),f i (x) are the real and imaginary parts of the measured load, are the real and imaginary parts of the calculated load of the j-th order generalized Fourier series expansion of the distributed load, and abs() means finding the absolute value.

5. A computer device, characterized in that: include: one or more processors; Memory; as well as One or more programs, wherein the one or more programs are stored in the memory and are configured to be executed by the one or more processors, and when the programs are executed by the processors, the steps of the distributed dynamic load order determination method based on the golden section method as described in any one of claims 1 to 4 are implemented.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method for determining the order of distributed dynamic loads based on the golden section method according to any one of claims 1 to 4 are implemented.

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