Method for identifying weak rigidity link of cantilever beam structure under working condition of sensors distributed at non-equal intervals

Abstract

The invention provides a method for identifying a weak rigidity link of a cantilever beam structure under the working condition of sensors distributed at non-equal intervals, many structures on a machine tool can be equivalent to relatively simple structures, but in the equivalent process, due to the fact that differences exist among all substructures, the rigidity of the cantilever beam structure cannot be identified, and the rigidity of the cantilever beam structure cannot be identified. On the basis of the method, a finite element numerical model of a model is established, and a flexibility curvature matrix and a flexibility curvature change rate are combined for various working conditions that sensors are distributed at unequal intervals; under the condition that the sensors cannot be distributed at equal intervals and the stress concentration position is refined, damage identification can be realized. And adjusting the element stiffness matrix and the element mass matrix for solving the flexibility curvature matrix according to the actual sensor distance, and carrying out two-time difference calculation on the obtained flexibility matrix to obtain a weak identification index. And a more accurate identification index is provided for weak link identification of the complex structure of the machine tool.

Description

Technical Field

[0001] The invention proposes a new method for identifying weak links in rigidity of non-equidistant structures based on a flexibility curvature matrix and using a new weak link identification index of a dynamic testing method, and belongs to the field of mechanical vibration testing. Background Art

[0002] Studies have shown that the stiffness of machine tools directly affects the processing performance and processing accuracy of machine tools. Improving the stiffness of machine tool structures is of great significance to improving the overall performance of machine tools. In many fields such as aerospace and machine tool processing, complex structures are equivalent to simple structures such as beam structures. However, due to the existence of "joint surfaces" between structures and structural damage, the equivalent beam structure will have weak links. These weak links seriously affect the stiffness of the equivalent structure, and then affect the overall structural stiffness. If you want to improve the stiffness of the weak parts of the machine tool, you must be able to accurately identify the location of the stiffness weak links of the beam structure.

[0003] In actual applications, the arrangement of sensors faces the limitation of unequal spacing distribution, which has great limitations. When the span of the structure to be detected is large and the structure is complex, the sensors cannot be evenly distributed. If the measured data is obtained by using unequal spacing of sensors, the actual measurement error will often be mixed into the identification index, affecting the damage identification effect; in the actual working process of the cantilever beam, the location of the damage often appears at the maximum stress. Placing the sensor closely at the vulnerable position can obtain a more accurate damage location; in actual damage detection, due to the complex structure of the actual structure, the sensors often cannot be evenly spaced. However, the commonly used damage identification indicators cannot accurately identify the damage location and degree of damage when the sensors cannot be evenly spaced. Therefore, a damage identification indicator that does not depend on the sensor spacing is needed. This damage identification indicator is more universal, does not depend on the spacing of the sensor distribution, and can be applied to a variety of scenarios. Summary of the invention

[0004] Step 1: Establish a cantilever beam numerical model with non-uniformly spaced units

[0005] A machine tool is a continuous and complex structure with infinite degrees of freedom. When identifying its weak links in stiffness, the structure to be tested is discretized into several subsystems with several masses. Different subsystems are connected by equivalent springs and equivalent dampers to form a simple dynamic model. In the equivalent numerical model of the structure, the joint surfaces between the machine tool structures are regarded as the weak links in stiffness of the equivalent structure, and a finite element numerical model of the cantilever beam is established according to the actual situation of the structure to be tested.

[0006] According to the sensor distribution spacing, confirm the actual length of each unit. For beam units with non-uniformly spaced sensor distribution, the non-uniformly spaced sensor distribution often causes differences in the unit stiffness matrix and the unit mass matrix. Both the unit stiffness and unit mass matrices involve unit lengths, and often fail to obtain damage identification effects under unequally spaced unit division conditions. The unit stiffness matrix and unit mass matrix need to be processed.

[0007]

[0008]

[0009] Where: k e --Element stiffness matrix, m e --Element mass matrix, E--elastic modulus, I--section inertia moment, A--cross-sectional area, l--unit length, ρ--material density

[0010] The actual sensor distribution spacing is used to replace the unit length in the unit stiffness matrix and the unit mass matrix. Assuming that δ is the length of the unit cell and k is the number of units containing the unit cell length, the length of the element can be expressed as k*δ. The actual sensor distribution spacing is used to replace the unit length in the unit stiffness matrix and the unit mass matrix.

[0011]

[0012]

[0013] Where: k el --Optimized unit stiffness matrix, E--elastic modulus, I--section inertia moment, k--the number of unit unit lengths, δ--unit unit length, m el --Optimized unit stiffness matrix, ρ--material density, A--cross-sectional area.

[0014] The beam element has two nodes, and each node has two degrees of freedom. Using the matrix elements of the same node, the stiffness matrix K and the mass matrix M of the prefabricated beam are:

[0015]

[0017]

[0018] Where: K-- overall stiffness matrix, --unit stiffness matrix value, n--unit number, M--overall mass matrix, --Element mass matrix value.

[0019] Step 2: Establish the identification index of the weak structural unit under the condition of unequally spaced distribution sensors. The unit stiffness matrix and unit mass matrix in the above formula are different in different units. Assemble the element matrices into an overall matrix.

[0020] The flexibility matrix contains a lot of important information, including natural frequencies and mode shapes. When a structure is damaged, the stiffness of the system decreases, the flexibility increases, and the mass matrix is ​​less affected. Therefore, the change in flexibility can be used to detect structural damage.

[0021]

[0022] Where: F--structural flexibility matrix, φ--mode matrix, Ω-diagonal matrix, m-mode order, i--natural frequency order, w i - the i-th order circular frequency, --i-th order vibration mode, n-number of vibration mode measurement points, f yy --Flexibility matrix values ​​at nodes.

[0023] From the above formula, it can be concluded that the higher the modal order, the smaller the contribution to the flexibility matrix. Therefore, when using the flexibility matrix for damage detection, only low-order modes need to be considered.

[0024] By using the center difference of each column of the flexibility matrix to determine the damage location, the flexibility curvature change rate (FCCR) can be obtained, thereby obtaining the damage identification index.

[0025]

[0026] Where: Δx j+1 --The distance between the node and the next node, F j+1 --Flexibility matrix value at node j+1, F j --Flexibility matrix value at node j, Δx j-1 --The distance between the previous node and the next node, F j-1 --Flexibility matrix value at node j-1, Δx j --The distance between the node and the previous node.

[0027] FCCR is the curvature of the displacement influence line under the unit force at each measuring point. Since the unit lengths considered are not equal, the unit length in the identification index should be omitted, and the flexibility matrix before and after damage should be solved separately.

[0028]

[0029]

[0030] Where: UFCCR-flexibility curvature matrix in the undamaged case, F ju+1 --The value of the damage-free flexibility matrix at the j+1 node, F ju --The value of the damage-free flexibility matrix at node j, F ju-1 --The value of the damage-free flexibility matrix at the j-1 node, DFCCR - the flexibility curvature matrix in the case of damage, F jd+1 --The value of the damaged flexibility matrix at the j+1 node, F jd --The value of the damaged flexibility matrix at node j, F jd-1 --The value of the damaged flexibility matrix at the j-1 node.

[0031] The curvature change rate matrix IDI of the first-order differential flexibility before and after structural damage is:

[0032]

[0033] The above method is very accurate in the case of equal spacing, but not so accurate in the case of unequal spacing. Now, by performing two differential operations on the flexibility curvature matrices UFCCR and DFCCR, we can obtain UFCCR′ and DFCCR′ before and after damage, which reflects the change of flexibility curvature of adjacent measuring points under unit force.

[0034]

[0035]

[0036] Where: UFCCR′ – the second-order difference flexibility matrix in the undamaged case, DFCCR′ – the second-order difference flexibility matrix in the damaged case, UFCCR j+1 --The value of the secondary difference flexibility matrix at the j+1 node in the undamaged case, UFCCR j --The value of the secondary difference flexibility matrix at the j node in the undamaged case, UFCCR j-1 --The value of the secondary difference flexibility matrix at the j-1 node in the undamaged case, DFCCR j+1 --The value of the secondary difference flexibility matrix at the j+1 node in the case of damage, DFCCR j --The value of the secondary difference flexibility matrix at node j in the case of damage, DFCCR j-1 --The value of the secondary difference flexibility matrix at the j-1 node in the case of damage.

[0037] When a structure is damaged, it usually causes changes in the stiffness and flexibility of the structure. The UFCCR′ and DFCCR′ before and after damage are obtained respectively. The difference matrix before and after damage is compared with the difference matrix before damage to obtain two differential flexibility curvature change rate matrices IDII:

[0038]

[0039] NI=diag(IDII)(15)

[0040] Where: IDII - twice differential flexibility curvature change rate matrix, NI - new weak identification index, diag - diagonal matrix. The place where the change between the two adjacent points in NI is greater than 50% corresponds to the position where the cantilever beam structure is weak in stiffness. At the same time, the value of NI is used to evaluate the relative size of the stiffness weakness.

[0041] The present invention has the following advantages due to the adoption of the above technical solution:

[0042] (1) The identification of structural stiffness weak links is deduced through theoretical calculations of structural damage detection, and the sensitivity of the flexibility curvature matrix to the weak links is used to optimize the weak identification index under the distribution of unequally spaced sensors. This provides more accurate guidance for the application of stiffness weak identification index on machine tools.

[0043] (2) This weak identification index is established from the foundation. In the overall stiffness matrix and the overall mass matrix of the structure, the unequal spacing units are refined. According to the sensor spacing under actual working conditions, the refined units are allocated to each unit, so this method has higher applicability.

[0044] (3) By studying the identification effect of the damaged weak link of the equivalent beam structure, we can explore the identification effect of the stiffness weakness index on the stiffness weak link of the machine tool structure, which makes the establishment of the identification index more convenient. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 Schematic diagram of the equivalent beam structure of the machine tool

[0046] Figure 2 Sensor distribution diagram for equivalent beam structure

[0047] Figure 3a The frequency domain diagram of each node under the distribution of undamaged unequally spaced sensors without noise

[0048] Figure 3b The frequency domain diagram of each node under the distribution of damaged unequally spaced sensors without noise

[0049] Figure 4 Time domain displacement map of each node after reconstruction

[0050] Figure 5a This is the identification effect diagram of the IDII indicator without noise and single damage.

[0051] Figure 5bThe identification effect diagram of the IDII with single damage in the presence of noise

[0052] Figure 6a This is the identification effect diagram of the IDII indicator with multiple damages due to noise.

[0053] Figure 6b This is the identification effect diagram of the IDII indicator with multiple damages due to noise. DETAILED DESCRIPTION

[0054] The present invention is described in detail below with reference to the accompanying drawings.

[0055] 1 Equivalent cantilever beam model

[0056] In such Figure 1 In the model shown, the cantilever beam has a length of L = 1.2 m and a cross-sectional area of ​​A = 0.025 m 2 There are 12 units with unequal spacing to verify the identification effect of the new identification index. Each node of each beam unit is assumed to have only two degrees of freedom, so the unit stiffness matrix and unit mass matrix obtained in this way have rotation angle and lateral displacement variables. The elastic modulus and mass density of each unit are E=3*10 11 Pa and p = 7850 kg / m 3 Due to the existence of the fixed end, the displacement angle variables of the left end of the unit node 1 are all 0. During the damage process of the structure, its overall mass matrix often does not change with the structural damage, and the main impact is on the overall stiffness matrix of the structure. By weakening the stiffness of the unit to simulate the damage of the structure, the identification effect of the identification index on the working condition in noisy and non-noise environments is studied.

[0057] 2 Test Data

[0058] Through numerical simulation and simulation experiments, the spectrum diagrams of numerical simulation and simulation experiments are as follows Figure 3a As shown in the figure, different degrees of noise are added to the input signal to verify the recognition effect of the identification algorithm under different damage degrees and different noises. Since the signal collected by the test is interfered by noise, when processing frequency domain data and time domain data, it is necessary to use wavelet decomposition denoising and low-pass filtering to reduce noise, and obtain the processed data, so as to further improve the accuracy of the identification index.

[0059] 3. Weak identification effect

[0060] The cantilever beam model is systematically reconstructed using the state space method, and the cantilever beam before and after damage is reconstructed to form a weak link in stiffness. The process of testing indicators is divided into two cases: one is the case where there is noise or no noise in a single damage, and the other is the case where there is noise or no noise in multiple damages. The identification effect of the identification indicator is determined through multi-angle and multi-dimensional detection. The displacement time domain diagram of each node after Simulink reconstruction is shown in the figure below. Figure 4 shown.

[0061] 3.1 Identification effect of single damage on cantilever beam with and without noise

[0062] Considering that the structure is damaged at only one location and that it is impossible to evenly distribute the sensors during the actual test, the damage to the cantilever beam structure occurs at unit node 4, and the damage degrees are 5%, 10%, 15%, and 20%, respectively. Figure 5a To identify the effect of the identification index IDII in the absence of noise, it can be clearly seen that the index value at the damage position has a sudden change (the mutation occurs at the two nodes of the damaged unit). When 15dB noise is added to the input signal, Figure 5b The identification index IDII shows the identification effect under noise interference. It can be clearly seen that when there is noise, the identification index can still mutate at the damaged part, and the other indicators fluctuate around 0 when no damage occurs, which has no substantial impact on the identification effect. The identification index has good noise resistance.

[0063] 3.2 Identification effect of multiple damages on cantilever beam with and without noise

[0064] Since the machine tool is in complex processing conditions for a long time, when damage occurs, the damage often occurs in multiple places. Considering that the cantilever beam structure is damaged in different units, it is assumed that multiple damages occur at unit nodes 4 and 9 of the cantilever beam. The damage degrees are set to 5%, 10%, 15%, and 20% respectively. The unit node is represented by E-, the noise degree is represented by EN-, and the damage identification index effect is as follows: Figure 6a When the damage occurs at different nodes, the damage index changes suddenly at the damaged node even under the condition of unevenly distributed sensor spacing, and the damage index value increases with the increase of damage degree. Add white noise signal to the input signal, input 15dB white noise, and the recognition effect is as follows: Figure 6b As shown, it is obvious that the index at the non-damaged node fluctuates to varying degrees, but the mutation values ​​at the two nodes of the damaged unit are sufficient to identify the damage location of the structure, which can prove that the damage identification index has an identification effect for multiple losses under this working condition.

[0065] The simulation results of the above examples show that the applicability of this identification index has great advantages over other proposed identification indexes. In the implementation process of the identification index, the state space used to reconstruct the modal data of the original structure can greatly reduce the impact of noise and test errors. When applied to the identification index, it can maximize the accuracy of the identification effect of the new identification index, that is, the location where the mutation value appears is the weak location of the structural stiffness.

Claims

1. A method for identifying weak links in the stiffness of a cantilever beam structure under the condition of non-equidistantly distributed sensors, characterized in that: The implementation process is as follows: Step 1: Establish a cantilever beam numerical model with non-uniformly spaced units When identifying the weak links in the stiffness of a machine tool, the structure to be tested is discretized into several subsystems with several masses. Different subsystems are connected by equivalent springs and equivalent dampers to form a simple dynamic model. In the equivalent numerical model of the structure, the joint surfaces between the machine tool structures are regarded as the weak links in the stiffness of the equivalent structure, and a finite element numerical model of the cantilever beam is established according to the actual situation of the structure to be tested. According to the sensor distribution spacing, confirm the actual length of each unit; For beam structures under the condition of non-uniformly spaced sensor distribution, there will be a difference between the unit stiffness matrix and the unit mass matrix. It is often impossible to obtain the damage identification effect under the condition of unequally spaced unit division. It is necessary to process the unit stiffness matrix and the unit mass matrix. Where: k e --Element stiffness matrix, m e --unit mass matrix, E--elastic modulus, I--section inertia moment, A--cross-sectional area, l--unit length, ρ--material density; The unit length in the unit stiffness matrix and the unit mass matrix is ​​replaced by the actual sensor distribution spacing; assuming that δ is the length of the unit cell and k is the number of units containing the unit cell length, the length of the element is expressed as k*δ. The unit length in the unit stiffness matrix and the unit mass matrix is ​​replaced by the actual sensor distribution spacing; Where: k e1 --Optimized unit stiffness matrix, E--elastic modulus, I--section inertia moment, k--the number of unit unit lengths, δ--unit unit length, m e1 --Optimized unit stiffness matrix, ρ--material density, A--cross-sectional area; The beam element has two nodes, and each node has two degrees of freedom. If the matrix elements of the same node are used, the stiffness matrix K and the mass matrix M of the assembled beam are respectively: Where: K-- overall stiffness matrix, --The value of the element stiffness matrix at the corresponding point, n--element number, M--overall mass matrix, --The value of the unit mass matrix at the corresponding point; Step 2: Establish the identification index of the structural weak unit under the condition of unequally spaced distribution sensors. The unit stiffness matrix and unit mass matrix in the above formula are not the same in different units. Assemble the element matrices into an overall matrix. The flexibility matrix contains a lot of important information, including natural frequency and mode vibration shape; when the structure is damaged, the stiffness of the system will decrease and the flexibility will increase, so the change in flexibility is used to detect structural damage; Where: F--structural flexibility matrix, φ--mode matrix, Ω-diagonal matrix, m-mode order, i--natural frequency order, w i - the i-th order circular frequency, --i-th order vibration mode, y--the number of vibration mode measurement points to be extracted, f yy --Flexibility matrix value at the node; From the above formula, it can be concluded that the higher the modal order, the smaller the contribution to the flexibility matrix; therefore, when using the flexibility matrix for damage detection, only the first three modes need to be considered; The damage location is determined by using the center difference of each column of the flexibility matrix to obtain the flexibility curvature matrix FCCR (Flexibility curvature change rate), thus obtaining the damage identification index; Where: Δx j+1 --The distance between the node and the next node, F j+1 --Flexibility matrix value at node j+1, F j --Flexibility matrix value at node j, Δx j-1 --The distance between the previous node and the next node, F j-1 --Flexibility matrix value at node j-1, Δx j --The distance between the node and the previous node; FCCR is the curvature of the displacement influence line under the unit force at each measuring point; Solve the flexibility matrix before and after damage respectively; Where: UFCCR-flexibility curvature matrix in the undamaged case, F ju+1 --The value of the damage-free flexibility matrix at the j+1 node, F ju --The value of the damage-free flexibility matrix at node j, F ju-1 --The value of the damage-free flexibility matrix at the j-1 node, DFCCR-the flexibility curvature matrix in the case of damage, F jd+1 --The value of the damaged flexibility matrix at the j+1 node, F jd --The value of the damaged flexibility matrix at node j, F jd-1 --The value of the damaged flexibility matrix at the j-1 node; The curvature change rate matrix IDI of the first-order differential flexibility before and after structural damage is: The above method is very accurate in the case of damage identification with equal spacing, but is less accurate in the case of identification of unequally spaced units; In the case of unequal spacing element identification, the flexibility curvature matrices UFCCR and DFCCR are differentially calculated twice to obtain UFCCR′ and DFCCR′ before and after damage, which reflects the change of flexibility curvature of adjacent measuring points under unit force. Where: UFCCR′--the second-order difference flexibility matrix in the undamaged case, DFCCR′--the second-order difference flexibility matrix in the damaged case, UFCCR j+1 --The value of the secondary difference flexibility matrix at the j+1 node in the undamaged case, UFCCR j --The value of the secondary difference flexibility matrix at the j node in the undamaged case, UFCCR j-1 --The value of the secondary difference flexibility matrix at the j-1 node in the undamaged case, DFCCR j+1 --The value of the secondary difference flexibility matrix at the j+1 node in the case of damage, DFCCR j --The value of the secondary difference flexibility matrix at node j in the case of damage, DFCCR j-1 --The value of the secondary difference flexibility matrix at the j-1 node in the case of damage; The UFCCR′ and DFCCR′ before and after damage are obtained respectively; the difference matrix before and after damage is compared with the difference matrix before damage to obtain two differential flexibility curvature change rate matrices IDII: NI=diag(IDII) (15) Where: IDII - twice differential flexibility curvature change rate matrix, NI - new weak identification index, diag - diagonal matrix; The place where the change between the data of two adjacent points in NI is greater than 50% corresponds to the location where the stiffness of the cantilever beam structure is weak. At the same time, the value of NI is used to evaluate the relative size of the stiffness weakness.

Images