Tower crane damage identification method based on vehicle-tower coupling finite element model

Abstract

A tower crane damage identification method based on a vehicle-tower coupling finite element model comprises the following steps that 1, a vehicle-tower coupling finite element unit is constructed, and the vehicle-tower coupling finite element unit is provided with a horizontal beam unit and a variable amplitude trolley; 2, according to the motion equation of the variable-amplitude trolley and the motion equation of the beam unit, an expression of a trolley-tower coupling finite unit is obtained; 3, coding the degrees of freedom of the n beam units, and assembling the n beam units into a beam part of the vehicle-tower coupling finite element model; 4, assembling the variable amplitude trolley and the beam part to obtain a trolley-tower coupling finite element model; and 5, inputting related parameters of the tower crane and the variable-amplitude trolley in the entity vehicle-tower coupling system into the vehicle-tower coupling finite element model, and calculating to obtain acceleration response data of the variable-amplitude trolley. The vehicle-tower coupling finite unit has the advantages that the vehicle-tower coupling finite unit can be used for subsequent damage mode research of the tower crane and can also be used for identifying the frequency and the vibration mode of the tower crane.

Description

Technical Field

[0001] The present invention relates to the technical field of damage detection of tower cranes, and particularly to a method for identifying damage of tower cranes based on a vehicle-tower coupled finite element model. Background Art

[0002] At present, we usually adopt lossy and non-destructive detection methods including visual inspection, ray detection, ultrasonic detection, stress testing, magnetic particle detection, acoustic emission detection, etc. to conduct health and safety monitoring on tower cranes. The first step of most of the above-listed methods is to detect the damage location or degree of the tower crane by identifying the modal parameters of the tower crane. It can be seen that in the health and safety monitoring of tower cranes, the modal parameters of the structure are very important.

[0003] In reality, the modal parameters of all objects are similar to human fingerprints and are all different, consisting of countless unique data. The modal parameters of each structure itself usually do not change, such as natural frequency, modal shape, and structural damping, etc. However, when its own structure is damaged, its modal parameters will also change. Theoretically, this is the reason why damage can be identified by analyzing the modal parameters. The main content of modal analysis is actually to perform coordinate transformation on data. It transforms the response values originally in the physical coordinates to be described in the defined modal coordinates. Each basis vector in the modal coordinates is the eigenvector of the vibration response in the original physical coordinates. That is to say, the relationship between the response vectors can be simply described by using the correlation between each basis vector in this coordinate system.

[0004] However, the computational modal analysis method cannot consider the coupling relationship between each response vector. Therefore, the experimental modal analysis method is often combined with it to identify the damage of the structure. The experimental modal analysis method, as the name implies, is to obtain the response values through on-site tests or model tests and then identify the actual modal parameters of the structure through numerical processing. In recent years, people usually apply external excitation to the structure to pick up the vibration response values of the structure, and then obtain the modal parameters of the structure through the fast Fourier transform FFT. However, this method has relatively strict environmental requirements, and when using this method to measure large machinery, a large number of test instruments such as sensors are required, the amount of data collected at the measuring points is large, and the data processing is rather cumbersome. Summary of the Invention

[0005] A method for identifying damage of a tower crane based on a vehicle-tower coupled finite element model provided by the present invention can accurately detect the damage result of the tower crane.

[0006] To achieve the above object, the key of a method for identifying damage of a tower crane based on a vehicle-tower coupled finite element model provided by the present invention is to include the following steps:

[0007] Step 1: Construct a vehicle-tower coupled finite element, where the vehicle-tower coupled finite element is provided with 1 horizontal beam element and 1 luffing trolley, and the mass of the luffing trolley is m v , with a stiffness of k v , and a damping of c v , the mass of the wheel is m w , and the luffing trolley moves on the beam element;

[0008] Step 2: According to the motion equation of the luffing trolley and the motion equation of the beam element, obtain the expression of the vehicle-tower coupled finite element;

[0009] Step 3: Encode the degrees of freedom of n beam elements, and assemble the n beam elements into the beam part of the vehicle-tower coupled finite element model according to the degrees of freedom encoding of the beam elements;

[0010] Step 4: Assemble the luffing trolley and the beam part to obtain the vehicle-tower coupled finite element model;

[0011] Step 5: Input the boom length l, unit length mass m q , elastic modulus E, and cross-sectional moment of inertia I of the tower crane in the physical vehicle-tower coupled system, as well as the mass m v , stiffness k v and speed v of the luffing trolley into the vehicle-tower coupled finite element model, and the vehicle-tower coupled finite element model calculates the acceleration response data of the luffing trolley according to the above data;

[0012] Step 6: Perform a fast Fourier transform (FFT) on the acceleration response data to obtain an acceleration spectrum;

[0013] Step 7: Identify the modal parameters of the tower crane according to the acceleration spectrum;

[0014] Step 8: Detect the damage condition of the tower crane in the physical vehicle-tower coupled system according to the modal parameters or acceleration response data.

[0015] Through the above design, by constructing a vehicle-tower coupling finite element and assembling to obtain a vehicle-tower coupling finite element model, the coupling relationship between the tower crane and the luffing trolley in the physical vehicle-tower coupling system is simulated. Furthermore, by inputting various parameter information of the tower crane and the luffing trolley in the physical vehicle-tower coupling system, the acceleration response data of the luffing trolley is calculated. The acceleration response data is converted into the acceleration frequency spectrum diagram of the luffing trolley through the fast Fourier transform (FFT). Through the acceleration frequency spectrum diagram, modal parameters such as the frequency and vibration mode of the tower crane can be accurately obtained, and then the damage result of the tower crane can be detected and identified. The damage result of the tower crane includes whether the tower crane is damaged, as well as the damage location and damage degree.

[0016] Preferably, in step 2, the process of obtaining the expression of the vehicle-tower coupling finite element according to the motion equation of the luffing trolley and the motion equation of the beam element is as follows:

[0017] The vehicle-tower coupling finite element is composed of a luffing trolley and a beam element. The mass of the luffing trolley is m v , the stiffness is k v , the damping is c v , the mass of the wheel is m w , the position where the luffing trolley contacts the beam element is at a distance x c from the left end of the beam element, and this position changes with the moving speed and time of the luffing trolley. The motion equation of the luffing trolley is expressed as:

[0018]

[0019] where q v is the vertical displacement generated by the luffing trolley itself during operation. Since there is no spring in the simplified calculation model of the luffing trolley, the vertical displacements q w and q v generated by the wheels are the same;

[0020] The luffing trolley and the tower crane are connected and move through steel wheels, and the contact surface is a point contact. Since the wheels are rigid, there is no elastic restoring force between the top of the wheels and the luffing trolley, and only the damping force F1 acts; at the same time, due to the interaction relationship between the boom and the luffing trolley, the wheels will also receive the reaction force p1 from the boom; then, the expression of the motion equation of the wheels of the luffing trolley is:

[0021]

[0022] where the expression of F1 is:

[0023] The motion equation of the vehicle-tower coupling finite element is as follows:

[0024]

[0025] Among them, g represents the acceleration due to gravity, and m b represents the consistent mass matrix of the beam element, and c b represents the damping matrix of the beam element, and k b represents the stiffness matrix of the beam element, and {u b} represents the displacement column matrix of the ordinary beam element, and its value is:

[0026] {u b} = {u A θ A u B θ B} (5)

[0027] Among them, u A , θ A represent the degrees of freedom of the left end node of the beam element, and u B , θ B represent the degrees of freedom of the right end node of the beam element;

[0028] The value of {N} is the Hermite interpolation polynomial, and the result after cubic interpolation is:

[0029]

[0030] Here, {N} is the distribution coefficient of the force on the wheel in the vertical displacement and rotational displacement at both end nodes, and is related to the position x c of the wheel and the length L of the beam element;

[0031] For the vehicle-tower coupled finite element, its consistent mass matrix can be obtained in the following way:

[0032] Assume that the left end of the ordinary beam is subjected to a unit angular acceleration , then the acceleration distribution along the beam length is:

[0033]

[0034] Among them, ψ(x) is the displacement shape function at the node of the ordinary beam. According to D'Alembert's principle, the calculation result of the inertial force resisting this acceleration is:

[0035]

[0036] Among them, m(x) represents the mass of the beam at position x;

[0037] The nodal inertial force generated by the acceleration is calculated from the distributed inertial force in Equation (8) through the principle of virtual displacement, which is called the mass influence coefficient associated with the acceleration. Introduce a vertical virtual displacement and set the nodal external force p a The work done is equal to the work done by the distributed inertial force f I (x), that is, using the following formula:

[0038]

[0039] where δv a represents the virtual displacement of the nodal external force p a , and δv(x) represents the virtual displacement of the distributed inertial force f I (x);

[0040] Then, represent the internal virtual displacement with an interpolation function and substitute it into Equation (8), and finally derive the mass influence coefficient formula as:

[0041]

[0042] where m ij is any one of the mass influence coefficients of any beam segment, i is the number of the beam segment, and j represents the jth displacement degree of freedom at the beam end node;

[0043] Since the vehicle-tower coupling finite element is a homogeneous ordinary beam structure, its consistent mass matrix is:

[0044]

[0045] where the represents the mass per unit length of the vehicle-tower coupling finite element;

[0046] The stiffness matrix k of the beam element b is calculated by a method similar to that for analyzing the element mass coefficient. Any stiffness coefficient corresponding to the beam bending is expressed by the following formula:

[0047]

[0048] where ψ″(x) represents the virtual curvature, EI is the elastic stiffness of the beam element, and through the interpolation function, the stiffness matrix of the vehicle-tower coupling finite element is obtained:

[0049]

[0050] The damping matrix c of the beam element is calculated using Rayleigh damping b :

[0051] c b = a0m b + a1k b (14)

[0052] Among them, α0 and α1 represent damping proportional constants, the unit of α0 is s, and the unit of α1 is s -1 ;

[0053] The two coefficients in the above formula are obtained by solving a pair of simultaneous equations:

[0054]

[0055] Among them, ω m and ω n are two specific frequencies of the known tower crane, ξ m and ξ n are the damping ratios corresponding to the first two frequency values. The relationship between the damping ratio and the frequency is obtained from formula (15) as:

[0056]

[0057] Assume that the damping ratios applied to the control frequencies of the two tower cranes are the same, that is, ξ m = ξ n = ξ. Formula (15) is simplified into the following expression:

[0058]

[0059] In the trolley-tower coupling finite element, the luffing trolley will not accidentally fall outside the boom slide rail, and at the same time, it will not rush out of the slide rail or jump off the slide rail due to excessive speed. Then, there is the following formula for the luffing trolley:

[0060] q w = u c = {N} T {u b} (18)

[0061] Taking the first derivative of the above formula gives:

[0062]

[0063] Taking the second derivative gives:

[0064]

[0065] The following relationships are used in the differentiation of the above two formulas:

[0066]

[0067] Among them, dx represents the displacement change, dt represents the time change, and v represents the luffing trolley movement speed;

[0068] Substitute equations (19) and (20) into equation (1) to obtain the motion equation of the car body, and the expression is as follows:

[0069]

[0070] Then, substitute equations (3), (19), (20), and (22) into equation (2) to obtain the contact force between the luffing trolley and the tower crane as:

[0071]

[0072] Substitute equation (23) into equation (4) to express the motion equation of the car-tower coupled finite element as:

[0073]

[0074] Combine equations (22) and (24) into a matrix, and the expression of the car-tower coupled finite element is as follows:

[0075]

[0076] The above car-tower coupled finite element can be used for subsequent research on the damage mode of tower cranes, and can also be used for luffing trolleys with different structures to identify the frequencies and vibration modes of tower cranes. Ignoring the wheel mass, simplify the expression of the car-tower coupled finite element as:

[0077]

[0078] Preferably: In equation (26), the expression of the car-tower coupled finite element includes two parts, the beam element and the luffing trolley. The car-tower coupled finite element has 5 degrees of freedom, and the 5 degrees of freedom are the vertical displacement and rotational displacement of the left end node of the beam element, the vertical displacement and rotational displacement of the right end node of the beam element, and the vertical displacement of the luffing trolley.

[0079] The car-tower coupled finite element well simulates the coupling effect between the car and the tower.

[0080] Preferably: According to equation (26), rewrite the expression of the car-tower coupled finite element as:

[0081]

[0082] Or rewrite the expression of the car-tower coupled finite element into two elements, as follows:

[0083]

[0084] Among them, formula (28) represents the beam element expression, and formula (29) represents the luffing trolley expression.

[0085] Preferably: in the said step 3, the degrees of freedom of n beam elements are encoded. The degrees of freedom of the fixed end are 0, the degrees of freedom of the free end are 2, the degrees of freedom of n beam elements are 2n, and adding one degree of freedom of the luffing trolley, the number of degrees of freedom of the trolley-tower coupled finite element model is 2n + 1.

[0086] The said fixed end is the end where the first beam element is fixed on the tower crane, that is, the end where the boom is fixedly connected to the tower crane; the free end is the joint end of two adjacent beam elements and the suspended end of the nth beam element.

[0087] Preferably: according to the said formula (28), n beam elements are assembled to obtain the beam part; on the basis of formula (28), formula (29) is added to assemble the luffing trolley and the beam part.

[0088] The trolley-tower coupled finite element model consists of two parts, one part is the beam part and the other part is the luffing trolley. Therefore, when assembling the trolley-tower coupled finite element model, the two parts are assembled first, and then the luffing trolley is assembled on the beam part to complete the overall assembly.

[0089] Preferably: the assembly of the beam part includes the assembly of the mass matrix, the stiffness matrix and the damping matrix, and the assembly methods of the mass matrix, the stiffness matrix and the damping matrix are the same;

[0090] Among them, the assembly method of the stiffness matrix is as follows:

[0091] First, according to the order of the degrees of freedom encoding, the stiffness coefficients of the right node of the s-th beam element and the left node of the s + 1-th beam element are superimposed and calculated, that is, the stiffness coefficients at the intersection nodes of the two connected beam elements are superimposed and assembled to complete the process of converting from the element stiffness matrix to the global stiffness matrix;

[0092] Secondly, in order to facilitate the assembly of the stiffness matrix of the luffing trolley, the degrees of freedom q v of the luffing trolley and the beam degrees of freedom {u b} are swapped in order, and the following formula is obtained after adjustment:

[0093]

[0094] Among them, the vertical displacement of the luffing trolley is differentiated twice to obtain the acceleration response of the luffing trolley

[0095] According to the moving position of the luffing trolley on the beam part over time, the stiffness matrix of the luffing trolley is added to the element stiffness matrix of the beam element corresponding to the position where it is located, and the element stiffness matrix of this beam element is updated on the overall stiffness matrix, thereby completing the update of the overall stiffness matrix.

[0096] Preferably: in the step 8, the method for detecting and identifying the damage condition of the tower crane is either a damage identification method based on the change of natural frequency; or a damage identification method based on the change of vibration mode; or a damage identification method based on the change of flexibility; or a damage identification method based on wavelet transform.

[0097] The damage identification method based on the change of natural frequency is as follows:

[0098] Build a vehicle-tower coupled finite element model. Before the tower crane is damaged, input various initial parameters of the tower crane and the luffing trolley into the vehicle-tower coupled finite element model, calculate the initial acceleration response data of the luffing trolley, perform a fast Fourier transform to obtain the acceleration spectrum diagram before damage, and identify the natural frequency of the tower crane before damage through the acceleration spectrum diagram before damage;

[0099] After the tower crane is damaged, input various parameters of the damaged tower crane into the vehicle-tower coupled finite element model, calculate the acceleration response data of the luffing trolley after damage, perform a fast Fourier transform to obtain the acceleration spectrum diagram after damage, and identify the natural frequency of the tower crane after damage through the acceleration spectrum diagram after damage;

[0100] Compare the natural frequency of the tower crane before damage and the natural frequency of the tower crane after damage to obtain the damage result of the tower crane.

[0101] The natural frequency is one of the modal parameters that is most easily obtained and has a high identification accuracy. There are many methods for damage identification based on the change of natural frequency. Its characteristics are: when the structure is damaged, only the stiffness of the structure decreases while ignoring the change of the structure mass, and a modified theoretical model is established before the early damage of the structure. Theoretically, the ratio of the change in any second-order frequency after damage is only a function of the damage position and has nothing to do with the damage size.

[0102] The damage identification method based on the change of vibration mode is as follows:

[0103] (1) Build a vehicle-tower coupled finite element model, input various parameters of the tower crane and the luffing trolley into the vehicle-tower coupled finite element model, and calculate the acceleration response data of the luffing trolley;

[0104] (2) Perform a fast Fourier transform on the acceleration response data of the luffing trolley to obtain an acceleration spectrogram and identify the frequency of the tower crane;

[0105] (3) Adopt the band-pass filtering method (BPS) to extract the response data related to the frequency of the tower crane from the acceleration response data;

[0106] (4) Use the Hilbert transform to obtain the instantaneous amplitude of the response data;

[0107] (5) Obtain the vibration mode of the tower crane from the instantaneous amplitude;

[0108] (6) Regularize the vibration mode of the tower crane;

[0109] (7) In the physical vehicle-tower coupling system, the vibration mode of the tower crane is obtained by the vibration mode curvature method or the vibration mode change graph method

[0110] The vibration mode is a basic modal parameter. Although its test accuracy is low, it contains more information. Therefore, there are also many damage identification and diagnosis techniques based on the change of the vibration mode. (1) Vibration mode curvature method. If there is damage in the structure, the stiffness at the damaged location will decrease and the curvature will increase. The change of the vibration mode curvature increases with the increase of the curvature. Therefore, the location of damage can be determined according to the change of the vibration mode curvature. (2) Vibration mode change graph method. The relative change of the vibration mode is used as the positioning parameter, that is, the ratio of the difference between the vibration modes before and after damage to the vibration mode before damage. When damage occurs, a relatively large value will appear in the relative change of the vibration mode on the degrees of freedom affected in the damaged area. Therefore, the location of damage can be identified by using the relative change graph of the vibration mode.

[0111] The damage identification method based on the change of flexibility is as follows:

[0112] Under the condition that the mode satisfies normalization, the flexibility matrix is a function of the reciprocal of the frequency and the vibration mode. As the frequency increases, the influence of the reciprocal of the high frequency in the flexibility matrix can be ignored. In this way, as long as the first few low-order modal parameters and frequencies are measured, a matrix with better accuracy can be obtained. The maximum element in each column of the difference matrix is obtained according to the difference matrix of the two flexibility matrices before and after damage, and the location of damage can be found by checking the maximum element in each column.

[0113] The damage identification method based on wavelet transform is as follows:

[0114] Filter and perform double integration on the acceleration response data of the luffing trolley to obtain the displacement time history signal of the sampling point. Perform wavelet analysis on the acceleration time history signal and the displacement time history signal respectively to realize the detection of structural damage.

[0115] Advantages of the present invention: By constructing a vehicle-tower coupled finite element and deriving the expression of the vehicle-tower coupled finite element, and then realizing the overall assembly of the vehicle-tower coupled finite element model according to the expression, calculating the acceleration response data of the luffing trolley based on the parameter information of the tower crane and the luffing trolley in the physical vehicle-tower coupled system, and then identifying the modal parameters of the tower crane through the acceleration response data, and further obtaining the damage condition of the tower crane. Description of the Drawings

[0116] Figure 1 is a schematic flow chart of the present invention;

[0117] Figure 2 is a schematic structural diagram of the vehicle-tower coupled finite element in the embodiment;

[0118] Figure 3 is a schematic diagram of a rigid wheel in the embodiment;

[0119] Figure 4 is a schematic diagram of the degree-of-freedom coding of the first part of the vehicle-tower coupled finite element model in the embodiment;

[0120] Figure 5 is a schematic diagram of the degree-of-freedom coding of the second part of the vehicle-tower coupled finite element model in the embodiment;

[0121] Figure 6 is a schematic diagram of the assembly of a common beam in the first stage of the embodiment;

[0122] Figure 7 is a schematic diagram of the assembly of the luffing trolley in the second stage of the embodiment;

[0123] Figure 8 is a comparison diagram of the acceleration response of the luffing trolley in the embodiment;

[0124] Figure 9 is a comparison diagram of the acceleration response of a flat-top tower crane in the embodiment. Detailed Embodiments

[0125] The present invention will be further described in detail below with reference to the drawings and specific examples. The following embodiments or drawings are used to illustrate the present invention, but not to limit the scope of the present invention.

[0126] As Figure 1 shown: A method for identifying the damage of a tower crane based on a vehicle-tower coupled finite element model includes the following steps:

[0127] Step 1: Construct a vehicle-tower coupled finite element. The vehicle-tower coupled finite element is provided with 1 horizontal beam element and 1 luffing trolley, and the mass of the luffing trolley is m v , the stiffness is k v , and the damping is cv , the mass of the wheel is m w , the luffing trolley moves on the beam element;

[0128] Step 2: According to the motion equation of the luffing trolley and the motion equation of the beam element, obtain the expression of the vehicle-tower coupled finite element;

[0129] Step 3: Encode the degrees of freedom of n beam elements, and assemble the n beam elements into the beam part of the vehicle-tower coupled finite element model according to the degrees of freedom encoding of the beam element;

[0130] Step 4: Assemble the luffing trolley and the beam part to obtain the vehicle-tower coupled finite element model;

[0131] Step 5: Input the boom length l, mass per unit length m q , elastic modulus E and section moment of inertia I of the tower crane in the solid vehicle-tower coupled system, as well as the mass m v , stiffness k v and velocity v of the luffing trolley into the vehicle-tower coupled finite element model, and the vehicle-tower coupled finite element model calculates the acceleration response data of the luffing trolley according to the above data;

[0132] Step 6: Perform fast Fourier transform (FFT) on the acceleration response data to obtain the acceleration spectrum diagram;

[0133] Step 7: Identify the modal parameters of the tower crane according to the acceleration spectrum diagram;

[0134] Step 8: Detect the damage condition of the tower crane in the solid vehicle-tower coupled system according to the modal parameters or acceleration response data.

[0135] In order to be able to analyze the modal parameters of the whole tower crane more accurately, the tower crane is discretized into multiple units, and it is assumed that the luffing trolley only acts on one unit. In order to more realistically simulate the coupling situation between the tower crane and the luffing trolley in reality, the stiffness and damping of the luffing trolley will be considered. Since the luffing trolley and the boom are in contact through steel wheels in real life, the stiffness of the luffing trolley will be taken as approaching infinity.

[0136] The vehicle-tower coupled finite element reflects well the coupling relationship between the luffing trolley and the boom. In order to facilitate subsequent finite element simulation, the beam part of the vehicle-tower coupled finite element is assumed to be a beam element. The degrees of freedom of the entire vehicle-tower coupled finite element can be from Figure 2It can be seen that there are 5 degrees of freedom. This is because the lateral displacement degrees of freedom at the nodes are not considered in this embodiment. So, for two nodes, there are 4 degrees of freedom. At the same time, the luffing trolley also belongs to a separate structural part, and its uniform motion will also cause vertical displacements of itself and the boom, which is the 5th degree of freedom.

[0137] As Figure 2 shown: The vehicle-tower coupled finite element is composed of a luffing trolley and a beam element. The luffing trolley is simplified into a calculation model with a mass of m v , a stiffness of k v , a damping of c v , and a wheel mass of m w . The position where the luffing trolley contacts the beam element is at a distance of x c from the left end of the beam element. This position changes with the moving speed and time of the luffing trolley. Then, the motion equation of the luffing trolley can be expressed as:

[0138]

[0139] where q v is the vertical displacement generated by the luffing trolley during its own operation. Since there is no spring in the simplified calculation model of the luffing trolley, the vertical displacements q w and q v are the same;

[0140] In reality, the luffing trolley and the tower crane are connected and move through steel wheels, and the contact surface is a point contact, which is simulated as a schematic diagram of a rigid wheel as Figure 3 shown. Since the wheel is rigid, there is no elastic restoring force between the top of the wheel and the luffing trolley, and it is only subjected to the damping force F1; at the same time, due to the interaction relationship between the boom and the luffing trolley, the wheel will also be subjected to the reaction force p1 from the boom; then, the motion equation expression of the wheel of the luffing trolley is:

[0141]

[0142] where F1 can be expressed as:

[0143] The motion equation of the vehicle-tower coupled finite element is as follows:

[0144]

[0145] where m b represents the consistent mass matrix of the beam element, c b represents the damping matrix of the beam element, k b represents the stiffness matrix of the beam element, {u b} represents the displacement array of the ordinary unit beam, and its value is:

[0146] {u b} = {u A θ A u B θ B} (5)

[0147] Among them, u A , θ A represent the degrees of freedom of the left node of the ordinary unit beam, and u B , θ B represent the degrees of freedom of the right node of the ordinary unit beam (right endpoint degrees of freedom).

[0148] The value of {N} is the Hermite interpolation polynomial, and the result after cubic interpolation is:

[0149]

[0150] Here, {N} is the distribution coefficient of the force on the wheel at the vertical displacement and rotational displacement of the two end nodes, and is related to the position x c of the wheel and the length L of the beam element;

[0151] For the vehicle-tower coupled finite element, its consistent mass matrix can be obtained in the following way:

[0152] Assume that the left end of the ordinary beam is subjected to a unit angular acceleration The acceleration distribution along the beam length is:

[0153]

[0154] Among them, ψ(x) is the displacement shape function at the nodes of the ordinary beam. According to D'Alembert's principle, the calculation result of the inertial force resisting this acceleration is:

[0155]

[0156] By using the principle of virtual displacement, the nodal inertial force generated by this acceleration is calculated from the distributed inertial force in formula (8), which is called the mass influence coefficient associated with the acceleration. Introduce a vertical virtual displacement, and make the work done by the nodal external force p a equal to the work done by the distributed inertial force f I (x), that is, use the following formula:

[0157]

[0158] Then represent the internal virtual displacement with the interpolation function and substitute it into formula (8), and finally derive the mass influence coefficient formula as:

[0159]

[0160] Among them, m ij is any mass influence coefficient of any beam segment, i is the number of the beam segment, and j represents the j-th displacement degree of freedom at the beam end node.

[0161] Since the vehicle-tower coupled finite element is a homogeneous ordinary beam structure, its consistent mass matrix is:

[0162]

[0163] Among them, represents the mass per unit length of the vehicle-tower coupled finite element;

[0164] The stiffness matrix k of the beam element b is calculated by a method similar to that for analyzing the element mass coefficient. Any stiffness coefficient corresponding to the beam bending can be expressed by the following formula:

[0165]

[0166] Among them, ψ″(x) represents the virtual curvature, EI is the elastic stiffness of the beam element. Through the interpolation function, the stiffness matrix of the vehicle-tower coupled finite element is obtained:

[0167]

[0168] Assuming that the damping is proportional to the combination of the consistent mass matrix and the stiffness matrix, then a simple damping matrix formula can be obtained. This method is called Rayleigh damping. Using this method to calculate, the damping matrix c of the beam element is b :

[0169] c = a0m + a1k (14)

[0170] The two coefficients in the above formula are obtained by solving a pair of simultaneous equations:

[0171]

[0172] Among them, ω m and ω n are two specific frequencies of the tower crane that are known, ξ m and ξ n are the damping ratios corresponding to the first two frequency values. The relationship between the damping ratio and the frequency is obtained from formula (15) as:

[0173]

[0174] Since detailed information on the damping ratio varying with frequency is rarely available, it is usually assumed that the damping ratios for the control frequencies applied to the two tower cranes are the same, i.e., ξ m = ξ n = ξ, and Equation (15) is simplified to the following expression:

[0175]

[0176] In the vehicle-tower coupled finite element, the luffing trolley will not accidentally fall outside the boom slide rail, and at the same time, it will not rush out of the slide rail or jump off the slide rail due to excessive speed. Then, for the luffing trolley, there is the following formula:

[0177] q w = u c = {N} T {u b} (18)

[0178] Taking the first derivative of the above formula gives:

[0179]

[0180] Taking the second derivative gives:

[0181]

[0182] The following relationships are used in taking the derivatives of the above two formulas:

[0183]

[0184] Substituting Equations (19) and (20) into Equation (1), the motion equation of the vehicle body can be obtained and expressed as follows:

[0185]

[0186] Then, substituting Equations (3), (19), (20), and (22) into Equation (2), the contact force between the luffing trolley and the tower crane can be obtained as:

[0187]

[0188] Next, substituting Equation (23) into Equation (4), the motion equation of the vehicle-tower coupled finite element can be expressed as:

[0189]

[0190] Combining Equation (22) and Equation (24) into a matrix, the expression of the vehicle-tower coupled finite element is as follows:

[0191]

[0192] The above vehicle-tower coupling finite element can be used for subsequent research on the damage mode of tower cranes, and can also be used for luffing trolleys with different structures to identify the frequencies and vibration modes of tower cranes. In most cases, the mass of the wheels is much smaller than the overall mass of the tower crane, so it can be ignored. Then the above expression of the vehicle-tower coupling finite element can be simplified as:

[0193]

[0194] It can be seen from the above formula (26) that the expression of the vehicle-tower coupling finite element contains two parts, and all five degrees of freedom of the vehicle-tower coupling finite element are taken into account. The five degrees of freedom are the vertical displacement, rotational displacement of the beam end node, and the vertical displacement of the luffing trolley. The vehicle-tower coupling finite element well simulates the coupling effect between the vehicle and the tower.

[0195] The expression of the vehicle-tower coupling finite element is derived above. Next, the MATLAB software will be used to realize the overall simulation of the vehicle-tower coupling finite element model by combining multiple vehicle-tower coupling finite elements, so as to identify the modal parameters of the tower crane.

[0196] Since a series of programming is required for the MATLAB software, in order to more simply realize the simulation of the vehicle-tower coupling finite element model, especially the simulation of the uniform motion of the luffing trolley, the expression of the vehicle-tower coupling finite element is decomposed in this embodiment. According to formula (26), the general formula of the vehicle-tower coupling finite element can be rewritten as:

[0197]

[0198] Or the expression of the vehicle-tower coupling finite element can also be directly rewritten as two elements:

[0199]

[0200] Formulas (28) and (29) are the expressions of the two parts of the vehicle-tower coupling finite element respectively. It can be clearly seen that the expression of the luffing trolley is more complex, because the coupling effect between the luffing trolley and the boom is expressed through the expression of the luffing trolley in this embodiment.

[0201] The expression of a single vehicle-tower coupling finite element was derived above. Next, the overall model assembly of the vehicle-tower coupling finite element model will be carried out. However, before assembly, the degree-of-freedom encoding of the entire model needs to be listed as a whole. The degree-of-freedom encoding required in this paper is divided into two major parts. The first part is to first consider the vehicle-tower coupling finite element model as composed of countless ordinary beams. At this time, the boundary conditions of the structure are the boundary conditions of a simply supported beam, ignoring the lateral displacement degrees of freedom at the nodes. Then the number of degrees of freedom of the entire vehicle-tower coupling finite element model is 2N + 3, and the (2N + 3)-th degree of freedom is the vertical displacement degree of freedom generated by the luffing trolley itself, as Figure 4 shown. The reason for placing the degrees of freedom of the luffing trolley in the degree-of-freedom encoding is, firstly, that it can be better divided into two major parts, namely the ordinary beam part and the moving mass block part, and secondly, it is to better realize the coupling effect between the vehicle and the tower and be able to code the overall structure more quickly and accurately.

[0202] As Figure 5 shown, it is a schematic diagram of the change in the overall degrees of freedom after considering the boundary conditions of the vehicle-tower coupling finite element model. This is the second part of the degree-of-freedom encoding. Since it is a cantilever beam structure, the two ends are respectively a fixed end and a free end. The degrees of freedom at the fixed end are 0, and the degrees of freedom at the free end are 3. Also, because the lateral displacement degrees of freedom are not considered, the degrees of freedom at the free end are 2, and the degrees of freedom of each intermediate node do not change and remain 2. Then the total number of degrees of freedom of the beam elements is 2N, plus one degree of freedom carried by the luffing trolley, and the total number of degrees of freedom of the overall VTI system is 2N + 1. The difference between the two is very important and is an important prerequisite for realizing the assembly of the vehicle-tower coupling system through MATLAB in the following text.

[0203] Next, it is the most important step in realizing vehicle-tower coupling in MATLAB programming, that is, how to assemble the two major elements mentioned above. The significance of this step is to form the stiffness matrix, mass matrix, and damping matrix of the vehicle-tower coupling system. However, since the assembly process is relatively complex and to avoid confusion in understanding, the entire assembly process will be divided into two stages to gradually complete the assembly of the vehicle-tower coupling finite element model. First, like the degree-of-freedom encoding, the assembly of the ordinary beam structure is carried out first, and the formula used in this part is (28); then, considering the luffing trolley on the structure of the ordinary beam, formula (29) is added.

[0204] Next, the specific processes of the two major assembly stages will be described in detail with reference to the diagrams. First, the first assembly stage is relatively simple, which is to assemble N individual identical beam elements. For beam elements, the overall mass matrix, overall stiffness matrix, and overall damping matrix all adopt the same assembly method. To avoid repeated explanations, only the assembly method of the overall stiffness matrix will be listed in this article. When the luffing trolley moves uniformly to the (n = 2) beam element, the degree-of-freedom stiffness coefficients of the left and right nodes of this beam element are shown in Table 1:

[0205] Table 1

[0206]

[0207]

[0208] As Figure 6 shown, it is a simple schematic diagram of the first assembly stage. The horizontal and vertical coordinates of the whole diagram are the numbers of the first block of degree-of-freedom codes. Square blocks are used to represent ordinary beam structures. Since the number of degrees of freedom of an ordinary beam is 4, the side length of this block is also 4×4. That is, the degree-of-freedom codes of the ordinary unit beam with n = 1 are 1, 2, 3, and 4, where 1 and 2 are the degrees of freedom at the left end, and 3 and 4 are the degrees of freedom at the right end. It can be clearly seen from the figure that there is an overlap between the first block and the second block, which means that the stiffness coefficients of the right node of the first block and the left node of the second block need to be superimposed. For the first stage, it is actually to superimpose and assemble the stiffness coefficients at the intersecting nodes of ordinary beams in pairs, completing the process of converting from the element stiffness matrix to the overall stiffness matrix. Through Figure 6 it can be seen that the rows and columns of the overall stiffness matrix formed after the completion of the entire first-stage assembly are both 2N + 2. The assembly of the luffing trolley will be carried out in the second stage.

[0209] First, before the second-stage assembly, in order to more conveniently assemble the stiffness matrix of the luffing trolley, in formula (29), the degrees of freedom q v of the luffing trolley and the degrees of freedom {u b} of the beam need to be swapped in order. After adjustment, the following formula can be obtained:

[0210]

[0211] For the example given in the first stage, the luffing trolley element needs to be added to the beam element with n = 2, which means that the stiffness matrix of the luffing trolley needs to be added to the original element matrix of the ordinary beam, and the stiffness matrix of the overall structure of the trolley-tower coupled finite element model is complete. It should be noted that N and N′ in formula (30) change with the position x cIt changes with movement. For the vehicle-tower coupling finite element proposed in the present invention, how to achieve the movement of the luffing trolley unit on the beam unit is the most crucial step, as Figure 7 shown. First, the position of the luffing trolley unit needs to be obtained. Then, the element matrix of this beam unit will be superimposed with the element matrix of the luffing trolley next. Due to the external force of the luffing trolley acting on this beam unit, the {N} of this beam unit is different from that in the first stage. Recalculate to obtain the latest element matrix, and finally superimpose the element matrix on the overall matrix obtained in the first stage.

[0212] The above text describes how to superimpose the luffing trolley unit on the beam unit. Next, how to achieve real-time superposition of the luffing trolley will be explained. Assume that the luffing trolley is moving at a constant speed. Therefore, the {N} of the ordinary beam also changes continuously with time, and it is necessary to update {N} in real time according to the time transformation. Assume that the update rate is the same as the trolley moving speed, with a value of Δt. Next, the description of the assembly path of the luffing trolley matrix in the second stage will continue with the example in the above text. When the luffing trolley runs to the beam unit with n = 2, at this time, it is necessary to update the element matrices of the degrees of freedom numbered 3, 4, 5, and 6 of this beam unit. At the same time, the degree of freedom 2N + 3 of the luffing trolley is added to the element matrix of the original beam unit along with the corresponding element matrix, and the result is shown in Table 2. At the same time, the degrees of freedom of all other beam units do not change. When the luffing trolley moves to the next unit, such as n = 3, the corresponding degrees of freedom at this time are 5, 6, 7, 8, and 2N + 3, and then update the matrix state at this moment.

[0213] Table 2

[0214]

[0215] Finally, the matrix of the vehicle-tower coupling finite element model is obtained by using the assembly of each step, and then the system matrix within the entire time range can be obtained completely. At this time, the boundary conditions of the cantilever beam are not considered in the matrix of the vehicle-tower coupling finite element model. Next, it is to follow Figure 5As shown, considering the boundary conditions, that is, the degrees of freedom of the beam element when n = 1 are only the vertical and rotational degrees of freedom of the right - hand node, and the degrees of freedom of the beam element when n = N are 4 degrees of freedom. Therefore, the degrees of freedom of the entire vehicle - tower coupled finite - element model are 2N + 1. Through accurate boundary conditions, the correct VTI system matrix is extracted from the total matrix. Up to this step, the overall mass matrix, overall stiffness matrix, and overall damping matrix of the final VTI system are calculated. Finally, by using appropriate numerical solution methods, the responses of any degree of freedom of the entire system at any time can be obtained. Commonly used numerical solution methods include the central difference method, fourth - order R - k method, and newmark - β. In this paper, the commonly used newmark - β method is adopted to solve the VTI system. In the newmark - β method, the calculation of integral constants is required, as shown below:

[0216] a0 = 1 / (β×dt 2 ) (31)

[0217] a1 = γ / (β×dt) (32)

[0218] a2 = 1 / (β×dt) (33)

[0219] a3 = 1 / (2×β)-1 (34)

[0220] a4 = γ / β - 1 (35)

[0221] a5 = dt / 2(γ / β - 2) (36)

[0222] a6 = dt×(1 - γ) (37)

[0223] a7 = γ×dt (38)

[0224] Among them, γ represents the coefficient of the weight of the linear change between the influence of the initial and final accelerations on the velocity change, γ = 0.25; β represents the coefficient of the weight of the contribution of these initial and final accelerations to the displacement change, β = 0.5; a0 - a7 represent the calculation integral constants, and dt represents the time change amount.

[0225] Next, the various parameters of different parts of the adopted vehicle - tower coupled finite - element model are shown in Table 1 below:

[0226] Table 1

[0227]

[0228] According to the above parameter settings, the acceleration response of the luffing trolley is used to obtain the modal parameters of the tower crane through two schemes: theoretical derivation and numerical simulation for verification.

[0229] In numerical simulation, the MATLAB programming software was used to realize the assembly of the finite element model of the vehicle-tower coupling. The frequency of the tower crane was successfully identified by the response of the luffing trolley, and the acceleration response values of the vehicle body obtained by theoretical derivation and numerical simulation were compared with those of the flat-top tower crane, as Figure 8 , Figure 9 shown.

[0230] As Figure 8 shown, it includes the acceleration response value of the luffing trolley obtained by theoretical derivation and the acceleration response value of the luffing trolley obtained by numerical simulation. It can be clearly seen from Figure 8 that there are small differences between the theoretical value and the numerical simulation value: the peak values of the response values are roughly equal, and the overall vibration period and vibration waveform are relatively consistent, but there are slight deviations in the response values generated at the underestimated parts of the vibration waveform. The main reason for the above-described deviation may be that many real external factors are ignored in the theoretical derivation process, including the damping of the luffing trolley itself. At the same time, in the theoretical derivation, the stiffness of the luffing trolley is considered to be 0, while in the numerical simulation, the stiffness of the luffing trolley is considered to be close to infinity, and the numerical simulation is closer to the coupling effect between the tower crane and the luffing trolley in reality.

[0231] As Figure 9 shown, it includes the acceleration response value of the tower crane obtained by theoretical derivation and the response value of the tower crane obtained by numerical simulation. It can be clearly seen from Figure 9 that the vibration modes and periods between the two are roughly the same, but compared with the acceleration response comparison diagram of the luffing trolley, there is a deviation in the response value between the simulation value and the theoretical value. Whether at the trough or peak of the vibration waveform of the response value, the response values differ by a certain value, and moreover, the difference between the two is continuously increasing with the increase of time.

[0232] Figure 8 and Figure 9 both verify the accuracy between the theoretical derivation and the numerical simulation through the comparison of the simulation value and the theoretical value. In addition to the difference in the stiffness value of the luffing trolley, another reason for the above difference is the difference in the methods used. In the theoretical derivation, the modal superposition method is used to represent the displacement response value, while in the numerical simulation, the Hermite interpolation method is used to obtain the displacement response value distribution coefficient at the node of the unit beam. It can be seen that it is inevitable to have a difference between the theoretical value and the simulation value, and the difference is small and can be ignored. Comparing Figure 8 and Figure 9 it can be seen that the acceleration response of the luffing trolley oscillates more obviously than that of the flat-top tower crane. This is because the luffing trolley directly absorbs energy to generate vibration, and secondly, it is affected by the vibration wave, which makes the amplitude of the high-frequency component in the acceleration of the luffing trolley larger.

[0233] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, various modifications and variations can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for identifying damage of tower cranes based on a vehicle-tower coupled finite element model, characterized in that It includes the following steps: Step 1: Construct a vehicle-tower coupled finite element. The vehicle-tower coupled finite element is provided with 1 horizontal beam element and 1 luffing trolley, and the mass of the luffing trolley is m v , the stiffness is k v , the damping is c v , the wheel mass is m w , and the luffing trolley moves on the beam element; Step 2: According to the motion equations of the luffing trolley and the beam element, obtain the expression of the vehicle-tower coupled finite element; Step 3: Encode the degrees of freedom of n beam elements, and assemble the n beam elements into the beam part of the vehicle-tower coupled finite element model according to the degrees of freedom encoding of the beam elements; Step 4: Assemble the luffing trolley and the beam part to obtain the vehicle-tower coupled finite element model; Step 5: Input the boom length l, mass per unit length m q , elastic modulus E, and moment of inertia I of the tower crane in the entity vehicle-tower coupling system, as well as the mass m v , stiffness k v and speed v of the luffing trolley into the vehicle-tower coupling finite element model. The vehicle-tower coupling finite element model calculates the acceleration response data of the luffing trolley according to the above data; Step 6: Perform fast Fourier transform (FFT) on the acceleration response data to obtain an acceleration spectrum diagram; Step 7: Identify the modal parameters of the tower crane according to the acceleration spectrum diagram; Step 8: Detect the damage condition of the tower crane in the physical vehicle-tower coupled system according to the modal parameters or acceleration response data.

2. The method for identifying the damage of a tower crane based on a vehicle-tower coupled finite element model according to claim 1, wherein: In Step 2, the process of obtaining the expression of the vehicle-tower coupled finite element according to the motion equations of the luffing trolley and the beam element is as follows: The vehicle-tower coupling finite element is composed of a luffing trolley and a beam element. The mass of the luffing trolley is m v , the stiffness is k v , the damping is c v , the mass of the wheel is m w , the position where the luffing trolley contacts the beam element is at x from the left end of the beam element c . This position changes with the moving speed and time of the luffing trolley. The motion equation of the luffing trolley is expressed as: Among them, q v is the vertical displacement generated by the luffing trolley during its own operation. Since there is no spring in the simplified calculation model of the luffing trolley, the vertical displacement q w and q v are the same; The luffing trolley and the tower crane are connected and move through steel wheels, and the contact surface is a point contact. Since the wheels are rigid, there is no elastic restoring force between the top of the wheel and the luffing trolley, and only the damping force F1 acts; at the same time, due to the interaction relationship between the boom and the luffing trolley, the wheel will also receive the reaction force p1 from the boom; then, the expression of the motion equation of the wheel of the luffing trolley is: Among them, the F1 expression is: The motion equation of the vehicle-tower coupled finite element is as follows: where, g represents the acceleration due to gravity, m b represents the consistent mass matrix of the beam element, c b represents the damping matrix of the beam element, k b represents the stiffness matrix of the beam element, {u b} represents the displacement column matrix of the ordinary beam element, and its value is: {u b} = {u A θ A u B θ B} (5) Among them, u A , θ A represent the degrees of freedom of the left end node of the beam element, and u B , θ B represent the degrees of freedom of the right end node of the beam element; The value of {N} is the Hermite interpolation polynomial, and the result after cubic interpolation is: Here, {N} is the distribution coefficient of the vertical displacement and rotational displacement of the force on the wheel at both end nodes, which is related to the position x of the wheel c and the length L of the beam element; For the vehicle-tower coupled finite element, its consistent mass matrix can be obtained in the following way: Assume that the left end of a common beam is subjected to a unit angular acceleration , then the acceleration distribution along the beam length is as follows: Among them, ψ(x) is the displacement shape function at the ordinary beam node. According to D'Alembert's principle, the calculation result of the inertial force resisting this acceleration is: Among them, m(x) represents the mass of the beam at position x; The nodal inertial force generated by this acceleration is calculated from the distributed inertial force in Equation (8) through the principle of virtual displacement, and is called the mass influence coefficient associated with the acceleration. Introduce a vertical virtual displacement, and let the external force p of the node a The work done is equal to the work done by the distributed inertial force f I (x), that is, use the following formula: Among them, δv a represents the virtual displacement of the external force p of the node a , and δv(x) represents the virtual displacement of the distributed inertial force f I (x); Then, use the interpolation function to represent the internal virtual displacement and substitute it into formula (8), and finally derive the mass influence coefficient formula as: where m ij is any mass influence coefficient of any beam segment, j is the number of the beam segment, and j represents the type of displacement degree of freedom at the beam end node; Since the vehicle-tower coupled finite element is a homogeneous ordinary beam structure, its consistent mass matrix is: Among them, represents the mass of the vehicle-tower coupling finite element length; The stiffness matrix k of the beam element b Calculated in a similar way to the analysis of the element mass coefficient, any stiffness coefficient corresponding to the beam bending is expressed by the following formula: Among them, ψ″(x) represents the virtual curvature, EI is the elastic stiffness of the beam element, and then through the interpolation function, the stiffness matrix of the vehicle-tower coupled finite element is obtained: The damping matrix c of the beam element is calculated using Rayleigh damping b : c b = a0m b + a1k b (14) wherein, α0 and α1 represent damping proportional constants, the unit of α0 is s, and the unit of α1 is s -1 ; The two coefficients in the above formula are obtained by solving a pair of simultaneous equations: Among them, ω m and ω n are two specific frequencies of a known tower crane, ξ m and ξ n are damping ratios corresponding to the first two frequency values. The relationship between the damping ratio and the frequency is obtained from formula (15) as follows: Assume that the damping ratios of the control frequencies applied to two tower cranes are the same, i.e., ξ m = ξ n = ξ, and formula (15) is simplified into the following expression: In the vehicle-tower coupled finite element, the luffing trolley will not accidentally fall outside the boom slide rail, and at the same time will not rush out of the slide rail or jump off the slide rail due to too high speed. Then, the luffing trolley has the following formula: q w = u c = {N} T {u b} (18) where u c represents the vertical displacement of the beam element; Take the first derivative of the above formula to get: Take the second derivative to get: The following relationship is used in the derivation of the above two formulas: Among them, dx represents the displacement change, dt represents the time change, and v represents the moving speed of the luffing trolley; Substitute formulas (19) and (20) into formula (1) to obtain the motion equation of the vehicle body, and the expression is as follows: Then, substitute formulas (3), (19), (20), and (22) into formula (2) to obtain the contact force between the luffing trolley and the tower crane as: Substitute Equation (23) into Equation (4) to express the motion equation of the vehicle-tower coupled finite element as follows: Combine Equation (22) and Equation (24) into a matrix, and the expression of the vehicle-tower coupled finite element is as follows: Neglect the wheel mass and simplify the expression of the vehicle-tower coupled finite element to:

3. The method for identifying the damage of a tower crane based on a vehicle-tower coupling finite element model according to claim 2, wherein: In Equation (26), the expression of the vehicle-tower coupled finite element includes two parts: the beam element and the luffing trolley. The vehicle-tower coupled finite element has 5 degrees of freedom, which are the vertical displacement and rotational displacement of the left end node of the beam element, the vertical displacement and rotational displacement of the right end node of the beam element, and the vertical displacement of the luffing trolley.

4. The method for identifying the damage of a tower crane based on a vehicle-tower coupled finite element model according to claim 2, characterized in that: According to Equation (26), rewrite the expression of the vehicle-tower coupled finite element as: Alternatively, rewrite the expression of the vehicle-tower coupled finite element as two elements, as follows: Among them, Equation (28) represents the expression of the beam element, and Equation (29) represents the expression of the luffing trolley.

5. The tower crane damage identification method based on the vehicle-tower coupling finite element model according to claim 4, characterized in that: In Step 3, encode the degrees of freedom of n beam elements. The number of degrees of freedom of the fixed end is 0, the number of degrees of freedom of the free end is 2, and the number of degrees of freedom of n beam elements is 2n. Adding one degree of freedom of the luffing trolley, the number of degrees of freedom of the vehicle-tower coupled finite element model is 2n + 1.

6. The tower crane damage identification method based on the vehicle-tower coupling finite element model according to claim 4, characterized in that: According to Equation (28), assemble n beam elements to obtain the beam part; based on Equation (28), add Equation (29) to assemble the luffing trolley and the beam part.

7. The method for identifying damage of tower crane based on vehicle-tower coupling finite element model according to claim 6, characterized in that: The assembly of the beam part includes the assembly of the mass matrix, the stiffness matrix, and the damping matrix. The assembly methods of the mass matrix, the stiffness matrix, and the damping matrix are the same; Among them, the assembly method of the stiffness matrix is as follows: First, according to the degree-of-freedom encoding order, perform a superposition operation on the stiffness coefficients of the right node of the s-th beam element and the left node of the (s + 1)-th beam element, that is, perform a superposition assembly on the stiffness coefficients at the intersection node of the two connected beam elements to complete the process of converting from the element stiffness matrix to the global stiffness matrix; Secondly, to facilitate the assembly of the stiffness matrix of the luffing trolley, the degrees of freedom q of the luffing trolley in formula (29) v and the degrees of freedom {u b} of the beam are swapped in order, and the following formula is obtained after adjustment: Among them, the vertical displacement of the luffing trolley is differentiated twice to obtain the acceleration response of the luffing trolley According to the moving position of the luffing trolley on the beam part over time, add the stiffness matrix of the luffing trolley to the element stiffness matrix of the beam element corresponding to the position where it is located, and update the element stiffness matrix of this beam element on the global stiffness matrix, thereby completing the update of the global stiffness matrix.

8. The method for identifying damage of tower cranes based on the vehicle-tower coupled finite element model according to claim 1, wherein: In Step 8, the method for detecting and identifying the damage condition of the tower crane is either a damage identification method based on the change in natural frequency; or a damage identification method based on the change in vibration mode; or a damage identification method based on the change in flexibility; or a damage identification method based on wavelet transform.

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