Beam structure frequency order determination method and application thereof

Abstract

This invention discloses a method for determining the frequency order of beam structures and its application, applicable to simply supported beams, cantilever beams, or other beams with defined boundary conditions. The method includes the following steps: (1) establishing the ratio relationship of the beam's vibration frequencies at different orders; (2) when the beam's boundary conditions are determined, determining the solution k of its frequency characteristic equation. n L is determined, and n takes a positive integer; based on the first n orders k n The L value can be used to solve for the ratio γ of adjacent order frequencies. n,n‑1 The ratio γ of each frequency order to the fundamental frequency n,1 (3) Using the order n as the independent variable and k as the independent variable. n L, γ n,n‑1 γ n,1 One or more of them are used as dependent variables to fit a function, and the fitted function is used as the regular formula; (4) Based on the fitted function and the k obtained from the experimental test, the function is fitted to the dependent variable and the regular formula is used to fit the function. n L, γ n,n -1, γ n,1 Value, determine k n L, γ n,n‑1 γ n,1 The frequency order n corresponding to the value can be used to determine the frequency order n of the test beam. Based on the above frequency order determination method, this invention can quickly and accurately determine the frequency order, or calculate other frequencies based on known frequencies and their orders, and can also determine the true frequency of the beam from numerous noise spectra.

Description

Technical Field

[0001] This invention relates to a method for determining the frequency order of beam structures, and more particularly to a method for determining the frequency order of beam structures and its application. Background Technology

[0002] Methods for determining the order of frequencies using measured measurements often involve comparing the results with theoretical or finite element method (FEM) calculations, or determining the order by the number of intersections between the measured mode shape and the equilibrium position. However, theoretical and FEM calculations often contain errors compared to the true frequency values; measured frequencies and mode shapes can also be affected by noise, errors, and other factors, leading to incorrect order determination and ultimately, erroneous modal analysis results. Furthermore, while measured frequency determination is feasible for low-order frequencies, higher-order frequencies require deploying more vibration sensors to test mode shapes, significantly increasing the testing and analysis workload. Additionally, a large number of sensors can affect the dynamic characteristics of the beam. Therefore, experimental methods for determining frequency order have limited engineering practicality. Summary of the Invention

[0003] Purpose of the invention: To address the problems existing in the prior art, this invention provides a frequency order determination method for beam structures based on ratios, and also provides applications of this frequency order determination method for beam structures in order identification, frequency calculation, and true / false frequency identification.

[0004] Technical solution: A frequency order determination method for beam structures, applicable to simply supported beams, cantilever beams, and beams with other boundary conditions (such as steel-connected at both ends, hinged at one end, steel-connected at one end, and hinged at the other end, etc.).

[0005] Specifically, the steps include the following:

[0006] (1) Construct the ratio relationship of the vibration frequencies of the beam at different orders;

[0007] A beam has length L, width b, and height h, respectively. Its mass per unit length and bending stiffness are respectively... And EI(u), where u is the axial coordinate of the beam and t is the vibration time;

[0008] According to the beam's vibration equation and the method of separation of variables, the beam's vibration frequency ω n satisfy

[0009]

[0010] Where, k n is a characteristic parameter; n represents the modal order of the beam, hereinafter referred to as the order, and n takes a positive integer;

[0011] Define γ i,j Let ω be the i-th order frequency of the beam. i and the j-th order frequency ω jIf the ratio is given, then the ratio of the vibration frequencies of the beam at different orders is:

[0012]

[0013] (2) Once the boundary conditions of the beam are determined, its frequency characteristic equation can be determined. Therefore, the solution k of the frequency characteristic equation of the beam with such boundary conditions is... n L can be determined, and k can be solved. n L is independent of the beam's geometric and physical parameters;

[0014] Given the frequency characteristic equation of a beam, solve for the first n k-th orders of the frequency characteristic equation. n The value of L, where n can theoretically take any positive integer, but in practical engineering, a value between 1 and 20 is generally sufficient; based on the first n orders k n L-value calculation Adjacent frequency ratio γ n,n-1 The ratio γ of each frequency order to the fundamental frequency n,1 ;

[0015] (3) Based on the first n order k n L, Adjacent frequency ratio γ n,n-1 The ratio γ of each frequency order to the fundamental frequency n,1 With order n as the independent variable and k as the independent variable n L, γ n,n-1 γ n,1 One or more of them are used as dependent variables to fit a function, and the fitted function is used as a regular formula to characterize k. n L, γ n,n-1 γ n,1 The relationship with the order n;

[0016] (4) Based on the fitting function and the k obtained from the experiment n L, γ n,n-1 γ n,1 Value, determine k n L, γ n,n-1 γ n,1 The order n corresponding to the value.

[0017] Optionally, the beam is a simply supported beam; the frequency characteristic equation of a simply supported beam is:

[0018] sink n L=0

[0019] Where, k n L = nπ, which is independent of the beam's geometric and physical parameters;

[0020] The frequency ratio of a simply supported beam is

[0021]

[0022] When i > j, γ i,j >1; when i < j, γ i,j <1;

[0023] Through data fitting, the γ n,n-1 It is a decreasing function of order n, and The γ i,1 It is a quadratic increasing function of order 1, and its formula is as follows:

[0024] .

[0025] Optionally, the beam is a cantilever beam; the frequency characteristic equation of the cantilever beam is:

[0026] 1+cosk n Lcoshk n L=0

[0027] Where, k n L is a solution to the frequency characteristic equation, which is independent of the beam's geometric and physical parameters;

[0028] The frequency ratio of the cantilever beam is

[0029]

[0030] Based on the frequency characteristic equation of the cantilever beam, solve the first n k-th order frequency characteristic equation. n The value of L, where n is a positive integer; based on the first n orders k n L-value calculation Adjacent frequency ratio γ n,n-1 The ratio γ of each frequency order to the fundamental frequency n,1 ;

[0031] Based on the first n order k n L, Adjacent frequency ratio γ n,n-1 The ratio γ of each frequency order to the fundamental frequency n,1 With order n as the independent variable and k as the independent variable n L, γ n,n-1 γ n,1 One or more of them are used as dependent variables to fit a function, and the fitted function is used as a regular formula to characterize k. n L, γ n,n-1 γ n,1 The relationship with the order n;

[0032] Based on the fitting function and the k obtained from the experiment n L, γ n,n-1 γ n,1 Value, determine k n L,

[0033] γ n,n-1 γ n,1 The order n corresponding to the value.

[0034] Specifically, through data fitting, with the order n as the independent variable, k n L is the dependent variable, and the linear fitting function is obtained.

[0035] k n L = an-b

[0036] Among them, a and b are related to the boundary conditions of the beam. Once the boundary conditions are determined, a and b can be determined.

[0037] With order n as the independent variable, The value is the dependent variable, resulting in a linear fitting function.

[0038]

[0039] in, It is related to the boundary conditions of the beam. Once the boundary conditions are determined, c can be determined.

[0040] With order n as the independent variable, γ n,n-1 The value is the dependent variable, resulting in a univariate quadratic fitting function.

[0041]

[0042] Same as above. It is related to the boundary conditions of the beam. Once the boundary conditions are determined, c can be determined.

[0043] With order n as the independent variable, γ n,1 The value is the dependent variable, resulting in a univariate quadratic fitting function.

[0044]

[0045] in, It is related to the boundary conditions of the beam. Once the boundary conditions are determined, d and f can be determined.

[0046] A method for order identification based on the frequency order determination method for beam structures includes the following steps:

[0047] When the geometric and physical parameters and boundary conditions of the beam are determined, and the k-th and (k+1)-th order frequencies of the beam are known, respectively... k f k+1 ;

[0048] Calculate f k f k+1 frequency ratio γ k+1,k =f k+1 / f k ;

[0049] Based on the frequency characteristic equation of the beam, the frequency ratio γ between any two adjacent orders can be calculated. i,i-1 As a theoretical value;

[0050] γ k+1,k and theoretical value γ i,i-1 For comparison, when k is a positive integer, the measured value γ k+1,k and theoretical value γ i,i-1 If there is a minimum error, then the positive integer k is f. k The order of.

[0051] A frequency determination method based on the aforementioned beam structure, including the following steps for calculating the beam's frequency:

[0052] When the geometric and physical parameters and boundary conditions of a beam are determined, and the k-th order frequency of the beam is known to be f... k ;

[0053] Based on the frequency characteristic equation of the beam, the arbitrary frequency ratio γ can be calculated. i,j ;

[0054] Calculate the nth order frequency f n =f k / γ k,n .

[0055] A method for identifying true and false frequencies based on the frequency order determination method for beam structures includes the following steps:

[0056] When the geometric and physical parameters and boundary conditions of a beam are determined, and the k-th order frequency of the beam is known to be f... k ;

[0057] Another frequency obtained through actual testing Through analysis, if the If the frequency is the frequency of the beam, then it should be the i-th order frequency;

[0058] Calculate f k , The frequency ratio

[0059] Based on the frequency characteristic equation of the beam, calculate the theoretical value γ of the ratio of these two frequencies. k,i ,Will and γ k,i Compare them and calculate the difference.

[0060] If the difference Δ is within the preset threshold e, then a judgment is made. The value is the i-th order frequency of the beam;

[0061] When the difference Δ exceeds the preset threshold e, then a judgment is made. The value is not the i-th order frequency of the beam.

[0062] Beneficial effects

[0063] Compared with existing technologies, this invention can quickly and accurately determine the order of a frequency, and can also calculate other orders of frequencies based on known frequencies and their orders, and can also determine the true frequency from a large number of noise spectra. Attached Figure Description

[0064] Figure 1 The first 20 γ stages of the simply supported beam in Embodiment 1 of the present invention n,n-1 γ n,1 As the order changes;

[0065] Figure 2 The first 20 steps of the cantilever beam in Embodiment 2 of the present invention are k n L, γ n,n-1 γ n,1 As the order changes;

[0066] Figure 3 A simplified calculation diagram for a cantilever beam;

[0067] Figure 4 Finite element mesh for cantilever beam;

[0068] Figure 5 To obtain the frequency spectrum curve of vibration acceleration after vibration simulation of a finite element model of a cantilever beam;

[0069] Figure 6 This is a spectrum diagram of a cantilever beam when the present invention is applied to the identification of true and false frequencies;

[0070] Figure 7 This is the spectrum diagram of a cantilever beam when the present invention is applied to calculate other order frequencies;

[0071] Figure 8 This is a spectrum diagram of a cantilever beam when the present invention is applied to identify the frequency order. Detailed Implementation

[0072] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0073] Example 1 (Simply Supported Beam)

[0074] The frequency characteristic equation of a simply supported beam is denoted as...

[0075] sink n L=0(1)

[0076] Therefore, we can obtain k n L = nπ or n is the order of the beam frequency, which is a positive integer.

[0077] Define γ i,j Let be the ratio of the i-th frequency to the j-th frequency (where i and j are positive integers).

[0078]

[0079] For a simply supported beam, there is

[0080]

[0081] Substituting equation (3) into equation (2) yields

[0082]

[0083] For a simply supported beam with any parameters, the frequency ratio γ i,j It depends only on the order of the frequency and is independent of the specific geometric and physical parameters of the beam. Therefore, for any simply supported beam, as long as the two orders are determined, their ratio γ is constant. i,j It is also definite and independent of the specific frequency. Table 1 gives the frequency ratio γ of the first 20 simply supported beams. n，1 and γ n,n-1 .

[0084] Table 1 Frequency ratios of simply supported beams

[0085]

[0086] As shown in Table 1, the frequency ratio of a simply supported beam to γ i,j Both are constant, and when i > j, γ is satisfied. i,j >1, when i < j, γ is satisfied i,j <1.

[0087] Based on the data in Table 1 and equation (4), we can obtain γ i,i-1 The graphs showing the relationship between the order and the degree of change are as follows: Figure 1 .from Figure 1 It can be seen that γ ii-1 It is a decreasing function of i, and γ i,1 It is a quadratic increasing function of i.

[0088] Example 2 (Cantilever Beam)

[0089] The frequency characteristic equation of a cantilever beam is

[0090] 1+cosk n Lcoshk n L=0(5)

[0091] Where n is the order of the beam frequency, which is a positive integer. For a cantilever beam with any parameters, the corresponding parameter k n L all satisfy equation (5), and the solution k of equation (5) n L always has a specific solution.

[0092] According to the definition in equation (2), we have

[0093]

[0094] For a cantilever beam with any parameter, k n L is constant; as long as the orders i and j are determined, its frequency ratio is equal to γ. i,j It is also definite, namely γ i,j Regardless of the specific frequency value, Table 2 gives the frequency ratios γ for the first 20 orders. n,1 and γ n,n-1 .

[0095] Table 2 Frequency Ratio of Cantilever Beams

[0096]

[0097] As shown in Table 2, once the order of the cantilever beam is determined, then γ i,j All are definite; the frequency of the cantilever beam is greater than γ. i,j The frequency ratio of a simply supported beam to γ i,j They have different numerical values, but their properties and patterns are basically the same.

[0098] Based on the data in Table 2, we can obtain k. n L, γ n,n-1 γ n,1 The graphs showing the relationship between the order and the degree of change are as follows: Figure 2 (a) Figure 2 (b) Figure 2 (c) Figure 2 As shown in (d).

[0099] According to the k of the cantilever beam n L value and Figure 2 (a) Data fitting was performed using MATLAB, with the order as the independent variable and k. n L is the dependent variable, and the linear fitting function is...

[0100] k n L=3.137 47n-1.513 8 (7)

[0101] According to cantilever beam Value and Figure 2 (b) The fitted function is

[0102]

[0103] Based on the adjacent frequency ratio γ of the cantilever beam n,n-1 and Figure 2 (c) The fitted function is

[0104]

[0105] The errors of the three fitting functions are given in Table 3.

[0106] Table 3 Cantilever Beam k n L, and γ n,n-1 Fitting function error

[0107]

[0108]

[0109] As can be seen from Table 3, the errors of these three fitting functions are very small. Therefore, the fitting functions can be regarded as regular formulas, which makes it easy to calculate the corresponding order k. n L, and γ n,n-1 The value can also be determined based on k obtained from experiments. n L, and γ n,n-1 The value is used to deduce the order of the value.

[0110] Example 3 (Beam with Other Boundary Conditions)

[0111] Beams have various boundary conditions, and different boundary conditions result in different frequency equations. In existing technology, Zhuo Shujun and Ge Yujun simulated the boundary conditions of beams as support springs and bending springs, unifying the beam's vibration equations. The beam ends were simulated with vertical stiffness springs and bending springs to represent the elastic boundary conditions at the beam ends, with vertical spring stiffness coefficients K1 and K2 respectively. The vertical displacement ranges from free to fixed, so the values ​​of the vertical spring stiffness coefficients are 0 to ∞, and the bending spring stiffness coefficients are... and The bending angle changes from free to fixed, and the spring constant takes values ​​from 0 to ∞. The frequency equation of the beam under elastic boundary conditions is given by equation (10).

[0112]

[0113] Among them, A=2sinkLsinhkL, B=sinkLcoshkL-coskLsinhkL, C=coskLcoshkL-1

[0114] As can be seen from equation (10), when the geometric physical parameters and boundary conditions of a beam are determined, equation (10) is only a function of kL, and equation (10) will degenerate into a polynomial consisting entirely of trigonometric functions, hyperbolic functions, and constants. The solution of kL will also be deterministic and invariant, and can be used with k n L represents n, where n = 1, 2, 3, ... Therefore, for other beams with defined boundary conditions, their frequency ratios still satisfy equation (6), meaning that the frequencies of these beams can also be determined using the frequency order determination method based on the ratio, and other frequency values ​​can be calculated. Equation (6) can be used for frequency order determination of beams with arbitrary boundary conditions.

[0115] To verify the frequency order determination method based on ratios proposed in this invention, a finite element model was used to validate the theoretical method. This simulation employed the large-scale general-purpose finite element software Abaqus.

[0116] The basic data of the model are as follows: a cantilever beam model, using a rectangular continuous beam with uniform cross-section, a cantilever beam length (L) of 0.5m, a thickness (h) of 0.019m, a width (b) of 0.012m, and an elastic modulus (E) of 1.84 × 10⁻⁶. 11 Pa, shear modulus (G)

[0117] 0.86×10 11 Pa, density (ρ) 7.758 × 10 3 Kg / m 3 A cantilever beam model was created in Abaqus using plane stress elements (CPS4). The dynamic modal superposition method was employed for analysis. A simplified model diagram is shown below. Figure 3 As shown, the mesh division is shown in the image. Figure 4 As shown.

[0118] Vibration simulation of a cantilever beam using a finite element model was conducted to obtain vibration acceleration, followed by spectral analysis. Figure 5 The spectrum curves shown can be used to obtain the first four frequencies, and the specific values ​​are shown in Table 4. The ratio of adjacent frequencies of these four frequencies is calculated and compared with the theoretical values ​​of Example 2. It can be seen that the maximum error of the ratio of three adjacent frequencies of the first four frequencies does not exceed 1.5%, indicating that the method is consistent with the actual structural characteristics.

[0119] Table 4. Cantilever beam frequencies calculated using finite element analysis.

[0120]

[0121] To verify whether the adjacent frequency ratio holds true at higher frequencies, a cantilever beam with a length (L) of 1.2m, a thickness (h) of 0.02m, a width (b) of 0.02m, and an elastic modulus (E) of 2.1×10⁻⁶ was used. 11 Pa, density (ρ) 7.8 × 10 3 Kg / m 3 Finite element analysis was performed on the case. Due to the large slenderness ratio of the beam, more frequencies could be calculated. The dynamic characteristic analysis function of the finite element model can theoretically calculate any number of frequencies. However, since the calculation error of higher frequencies gradually increases and the testing of higher frequencies of the beam is difficult, only the first 20 frequencies were selected, as shown in Table 5. The ratios and relative errors of each frequency are also given in Table 5. It can be seen that compared with the finite element results, the error of the ratio of the first 20 frequencies based on the frequency order determination method is within 1.3%, and the following trend exists: the lower the frequency order, the smaller the error. The error of the first 12 frequencies is within 1%, which can meet the accuracy requirements of laboratory and actual engineering.

[0122] Table 5. Frequency and its ratio in numerical simulation of cantilever beam, along with error.

[0123]

[0124]

[0125] The following examples, in conjunction with Examples 4-6, explain in detail the application of the above-mentioned frequency order determination method in distinguishing true and false frequencies, calculating frequency values, and identifying order.

[0126] Example 4 (Distinguishing between true and false frequencies)

[0127] Based on the known frequencies and their orders, it can be determined whether other frequencies on the spectrum are the true frequencies of the beam or interference frequencies from other excitation sources.

[0128] For example Figure 6The image shows the frequency spectrum of a cantilever beam. From the graph, we know that f1 = 59.96 Hz and f2 = 385.4 Hz. Therefore, the frequency ratio can be calculated as follows:

[0129]

[0130] By referring to Table 2, we can see that the frequency ratio of the cantilever beam is γ. 2,1 The theoretical value is 6.2669, so the error in this test is 2.56%, which is relatively small. According to Table 2, γ... 3,2 =2.8, then the theoretical value of the third-order frequency of the cantilever beam should be:

[0131]

[0132] As can be seen from the figure, the frequency of 808.2Hz after the second-order frequency f2 = 385.4Hz is different from the theoretically calculated third-order frequency. The difference is 25.1%, which is significant. Furthermore, the frequency after 808.2 Hz in the spectrum is 1049 Hz, which differs from the calculated value. The difference is 2.78%, which is relatively small, so it can be determined that the third frequency of the cantilever beam is f3 = 1049Hz.

[0133] Similarly, we can determine that f4 = 2001Hz, f5 = 3233Hz, and f6 = 4749Hz. Furthermore, the frequency of 1616Hz does not conform to the frequency order determination principle based on ratios, therefore it is not the frequency of the cantilever beam.

[0134] This case study can be used to verify and demonstrate that the frequency order determination method based on ratios can determine whether the frequency values ​​in the spectrum are the frequencies of the corresponding cantilever beams.

[0135] Example 5 (Calculating other frequency values)

[0136] Suppose that a certain order frequency of a beam is known under certain boundary conditions, where the order is k and the frequency is f. k =f is a known quantity. Then, according to equation (4) or (6), other frequencies and their orders of the simply supported beam or cantilever beam can be calculated. The frequencies and their orders of beams with other boundary conditions can be calculated using the methods in equation (10) and (6). Based on the frequency equation of the beam, the frequency ratio γ can be calculated. i,j Where i and j are positive integers, the method for calculating other order frequencies is as follows: Calculate the nth order frequency: f n =f k / γ k,n , where n is any positive integer.

[0137] For example Figure 7As shown in the spectrum, f1 = 59.96 Hz, f2 = 385.2 Hz, f3 = 1048 Hz, and f4 = 2000 Hz are clearly visible. However, the 5th and 6th order frequencies do not show significant excitation. Therefore, by referring to Table 2 and the known first four order frequencies, the 5th and 6th order frequencies can be calculated as follows:

[0138]

[0139]

[0140] The calculated 5th and 6th order frequencies are... Figure 6 In the frequency comparison (different tests on the same beam), the calculation errors of the 5th and 6th frequencies in this case are 2.26% and 3.99%, respectively. This shows that the frequency order determination method based on ratios can accurately calculate other frequencies based on known frequencies and frequency comparisons.

[0141] Example 6 (Identifying Frequency Order)

[0142] Suppose we know two adjacent frequencies of a beam under certain boundary conditions, and their frequencies are fi and fj respectively. k and f k+1 Since γ is a known quantity, its ratio is calculated according to equation (6) to obtain the ratio of adjacent frequencies γ. k+1,k Then, using the frequency equation of the beam, the arbitrary frequency ratio γ can be calculated. i,j Through γ k+1,k and γ i,j Compare or calculate all γ i,j List them in a table, and find γ by looking up the table. k+1,k The specific value of k can be obtained by determining the corresponding frequency order. Then, using the specific value of k and the known frequency order f... k Therefore, the method in Example 5 can be used to calculate the frequency value of any order.

[0143] For example Figure 8 As shown, assume that frequencies below 2003Hz are unknown, and that the three adjacent frequencies above 2003Hz are known to be fi. k =2003Hz, f k+1 =3234Hz and f k+2 =4749Hz, then the ratio of adjacent frequencies can be calculated:

[0144]

[0145]

[0146] By comparing with the adjacent frequency ratios in Table 2, it can be determined that when k = 4, With γ 5,4=1.6531 has a minimum error of 2.33%, and With γ 6,5 =1.4938 has a minimum error of 1.69%, which means that 2003Hz is the 4th order frequency, 3234Hz is the 5th order frequency, and 4749Hz is the 6th order frequency. This is very consistent with the actual situation, which shows that the order of adjacent frequencies can be identified by the frequency order determination method based on the ratio.

[0147] In summary, by solving the vibration frequency equation of a beam, this invention discovers that for beams with defined boundary conditions, the frequency ratios are a series of fixed values, independent of the beam's geometric and physical parameters. Based on this principle, this invention proposes a frequency order determination method based on ratios. Taking common simply supported beams and cantilever beams as examples, it provides a series of ratios between adjacent frequencies and the ratio between each frequency and the fundamental frequency, and fits a ratio variation function. Finite element models and physical experimental models verify the correctness and accuracy of this method. This invention also proposes three application scenarios for the frequency order determination method: (1) when some frequencies are known, determining whether other frequencies are the true frequencies of the beam; (2) if a certain frequency and its order are known, other frequencies can be calculated using this method; (3) if two frequencies are known to be adjacent, the order of these two frequencies can be identified using this method, and other frequency values ​​can be calculated accordingly. Since the frequency of a beam is relatively insensitive to damage, the frequency order determination method of this invention can also be used for beams with minor damage, identifying the frequency order and calculating other frequencies through a frequency ratio table or ratio fitting function.

[0148] The embodiments described above are merely illustrative of several implementations of the present invention and are not intended to limit the scope of protection of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A method for determining the frequency order of beam structures, applicable to simply supported beams, cantilever beams, or other beams with defined boundary conditions, characterized in that, Includes the following steps: (1) Construct the ratio relationship of the vibration frequencies of the beam at different orders; The length, width, and height of a beam are respectively , , The beam has a mass per unit length and a bending stiffness of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ... and ,in It is the axial coordinate of the beam; Based on the beam's vibration equation and the method of separation of variables, the beam's vibration frequency is... n satisfy: ； Where, k n For feature parameters; Indicates the order of the beam. Take a positive integer; definition For Liang's First frequency i and the First frequency j If the ratio is given, then the ratio relationship of the vibration frequencies of the beam at different orders is: ； (2) When the boundary conditions of a beam are determined, its frequency characteristic equation is also determined. Therefore, the solution to the frequency characteristic equation of a beam with such boundary conditions is... Determined, i.e., solution It is independent of the beam's geometric and physical parameters; Given the frequency characteristic equation of a beam, solve the frequency characteristic equation before... Step Value, of which Take a positive integer; based on the previous... Step Value before calculation Step Adjacent frequency ratio The ratio of each frequency order to the fundamental frequency ; (3) Based on the previous Step , Adjacent frequency ratio The ratio of each frequency order to the fundamental frequency In order of order As the independent variable, with , , , One or more of these variables are used as dependent variables to fit a function, and the fitted function serves as a formula to characterize the pattern. , , , sum order The relationship; specifically including the following: (3-1) In order of order As the independent variable, The value is the dependent variable, and the linear fitting function is obtained. ; in, and It depends on the boundary conditions of the beam, but not on the beam's geometric and physical parameters. When the boundary conditions are determined, then... and That's it; (3-2) In order of order As the independent variable, The value is the dependent variable, resulting in a linear fitting function. ; in, This is related to the boundary conditions of the beam; (3-3) In order of order As the independent variable, The value is the dependent variable, resulting in a univariate quadratic fitting function. ; (3-4) in order of As the independent variable, The value is the dependent variable, resulting in a univariate quadratic fitting function. ; in, , This is related to the boundary conditions of the beam; (4) Based on the fitting function and the experimental results , , , Value, determined , , , The order corresponding to the value .

2. The frequency determination method for beam structures according to claim 1, characterized in that, The beam is a simply supported beam; The frequency characteristic equation of a simply supported beam is: ； in, It is independent of the beam's geometric and physical parameters; The frequency ratio of a simply supported beam is: ； when hour, ;when hour, ; Through data fitting, the It is the order. The decreasing function, and The It is a quadratic increasing function of order 1, and its formula is as follows: 。 3. The method for determining the frequency order of beam structures according to claim 1, characterized in that, The beam is a cantilever beam; The frequency characteristic equation of a cantilever beam is: ； in, The solution to the frequency characteristic equation is independent of the beam's geometric and physical parameters; The frequency ratio of the cantilever beam is: ； Based on the frequency characteristic equation of the cantilever beam, before solving the frequency characteristic equation... Step Value, of which Take a positive integer; based on the previous... Step Value Calculation Adjacent frequency ratio The ratio of each frequency order to the fundamental frequency ; According to the previous Step , Adjacent frequency ratio The ratio of each frequency order to the fundamental frequency In order of order As the independent variable, with , , , One or more of these variables are used as dependent variables to fit a function, and the fitted function serves as a formula to characterize the pattern. , , , sum order Relationship; Based on the fitting function and experimental results , , , Value, determined , , , The order corresponding to the value .

4. A method for order identification based on the frequency determination method for beam structures according to any one of claims 1 to 3, characterized in that, Includes the following steps: When the geometric and physical parameters and boundary conditions of a beam are determined, and the first... Rank, The frequency values ​​of the order are respectively , ; calculate , The frequency ratio = / ; Based on the frequency characteristic equation of the beam, the ratio of any adjacent frequency order can be calculated. As a theoretical value; Will and theoretical value To make a comparison, when When it is a certain positive integer, the measured value and theoretical value If there is a minimum error, then the positive integer for The order of.

5. A frequency calculation method based on the frequency order determination method for beam structures according to any one of claims 1 to 3, characterized in that, Includes the following steps: When the geometric and physical parameters and boundary conditions of a beam are determined, and the beam's... The first frequency is ; Based on the frequency characteristic equation of the beam, the arbitrary frequency ratio can be calculated. ; Calculate the first First frequency .

6. A method for identifying true and false frequencies based on the frequency order determination method for beam structures according to any one of claims 1 to 3, characterized in that, Includes the following steps: When the geometric and physical parameters and boundary conditions of a beam are determined, and the first... The order frequency value is ; Another frequency obtained through actual testing Through analysis, if the If the frequency is the frequency of the beam, then it is the first... First frequency; calculate , The frequency ratio ; Calculate the theoretical value of the ratio of these two frequencies based on the frequency characteristic equation of the beam. ,Will and Compare them and calculate the difference. ; When the difference At the preset threshold If it is inside, then judge. The value is the first of the beams. First frequency; When the difference Exceeding the preset threshold Then judge The value is not the first of the beams. First frequency.

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