A numerical control machine tool electric spindle vibration reliability sensitivity verification method

Abstract

This invention provides a method for verifying the vibration reliability sensitivity of CNC machine tool electric spindles. First, the CNC machine tool electric spindle system is simplified using the lumped mass method, considering rotational unbalance forces and bearing nonlinear restoring forces. Based on the Lagrange fundamental equations, the dynamic equations are derived. Then, the Runge-Kutta method is used to solve the dynamic equations, obtaining the theoretical values ​​of the spindle-bearing system dynamic equations with respect to vibration displacement under steady-state conditions. The effectiveness of the spindle-bearing system dynamic model is verified. Limit state equations are established based on the allowable radial runout of the spindle in engineering practice. Finally, adaptive Kriging analysis is used to analyze the vibration reliability of the spindle-bearing system. This method fully considers the random factors present in the manufacturing of ball bearings and proposes a reliability assessment method using spindle end vibration displacement as the evaluation index. This is of great significance for preventing excessive spindle vibration displacement from affecting machining quality and improving machining accuracy and reliability.

Description

Technical Field

[0001] This invention belongs to the field of mechanical processing technology, specifically relating to a method for verifying the vibration reliability sensitivity of CNC machine tool electric spindles. Background Technology

[0002] With the rapid development of the manufacturing industry, higher requirements are being placed on the machining accuracy of machine tools. During rotation, the machine tool spindle is subjected to loads, which causes vibration. In most cases, this vibration has negative effects, reducing the machining accuracy of the machine tool and shortening its service life. Therefore, studying machine tool spindle vibration is of great significance.

[0003] To analyze the vibration mechanism of machine tool spindles during service, many scholars at home and abroad have conducted extensive research on the dynamic characteristics of machine tool spindle models. Cheng et al. (Cheng MK, Gao F, Li Y. Vibration Detection and Experiment of PMSM High Speed ​​Grinding Motorized Spindle Based on Frequency Domain Technology[J]. Measurement Science Review, 2019, 19(3): 109-125.) established radial vibration, tilting vibration and axial vibration models, and verified the established vibration models using experimental modal analysis. Zhang et al. (Zhang SJ, Yu JJ, To S, et al. A theoretical and experimental study of spindle imbalance induced forced vibration and its effect on surface generation indiamond turning[J]. International Journal of Machine Tools & Manufacture, 2018, 133: 61-71.) considered the influence of unbalanced magnetic force and air buoyancy pressure, established a five-degree-of-freedom model of spindle forced vibration, and studied the influence of unbalanced magnetic force on vibration. Xi et al. (Xi ST, Cao HR, Chen XF, et al. Dynamic modeling of machine tool spindle bearing system and model based diagnosis of bearing fault caused by collision[J]. Procedia CIRP, 2018, 77: 614-617.) proposed a dynamic modeling method for machine tool spindle bearing system supported by angular contact ball bearings and floating displacement bearings, which simultaneously considers bearing defects, and verified the accuracy of the modeling through experiments. In terms of reliability analysis, since the dynamic model of the spindle system is a strongly nonlinear system, the moment method or digital simulation method will encounter the problem of excessive computational load, which is difficult to be accepted by practical engineering. Zhang Feng et al. (Zhang Feng, Tang Zhangchun, Liu Yongshou, Yue Zhufeng. Reliability sensitivity analysis based on stratified sampling method[J]. Journal of Computational Mechanics, 2012, 29(06): 841-846.) proposed a stratified importance sampling method for reliability analysis.Chen Chuanhai et al. (Chen Chuanhai, Yang Zhaojun, Chen Fei, Hao Qingbo et al. Reliability Modeling of Machining Center Spindle Based on Bootstrap-Bayes [J]. Journal of Jilin University (Engineering Science), 2014, 44(01): 95-100.) proposed a reliability modeling method based on Bootstrap-Bayes, but this method may result in low reliability. While the above reliability analysis method can reduce the number of samples to some extent, it may not meet the needs of strongly nonlinear systems in practical engineering.

[0004] The aforementioned studies have laid the foundation for the dynamic characteristics and reliability analysis of machine tool spindles, and their basic principles are quite mature. However, these studies did not consider the randomness of bearing and spindle structural parameters caused by manufacturing and assembly processes. This randomness causes changes in the vibration displacement of the spindle end during service, leading to discrepancies between theoretical analysis and actual conditions. Therefore, it is necessary to conduct reliability analysis of machine tool spindles considering the randomness of bearing and spindle structural parameters. Summary of the Invention

[0005] Based on the above problems, this invention adopts the Kriging surrogate model method, which uses the fitted explicit function to replace the original complex nonlinear dynamic equations of the spindle system, thereby realizing the reliability analysis of machine tool spindle vibration.

[0006] This invention proposes a method for verifying the vibration reliability sensitivity of CNC machine tool electric spindles, comprising:

[0007] Step 1: Simplify the CNC machine tool electric spindle system using the lumped mass method, considering rotational unbalance forces and nonlinear restoring forces of the bearings, and establish the dynamic equations of the machine tool spindle-bearing system based on the fundamental Lagrange equations;

[0008] The dynamic equation of the spindle-bearing system in step 1 is expressed as follows:

[0009]

[0010] In the formula, M is the system's generalized mass matrix, K is the system's generalized stiffness matrix, C is the system's damping matrix, G is the gyro matrix, and F is the system's excitation force matrix; the system's excitation force mainly consists of two parts: the nonlinear restoring force of the rolling bearing and the unbalanced force caused by the unbalanced eccentricity of the spindle system; q represents the generalized coordinate vector of the spindle-bearing system relative to the base platform, q=(x1,θ y1 ,x b1 ,x2,x b2 ,x3,θ y3 ,y1,θ x1 ,y b1 ,y2,y b2,y3,θ x3 ), where x i y i (i = 1, 2, 3) represent the displacement degrees of freedom of each lumped mass block in the x and y directions, respectively, θ x1 θ y1 θ y3 θ x3 Represents the torsional degrees of freedom of lumped mass blocks m1 and m3 about the x and y axes, respectively. bi y bi (i = 1, 2) represents the bearing's degrees of freedom in the x and y directions.

[0011] The machine tool spindle-bearing system is simplified to a 14-DOF spindle-bearing system model using the lumped mass method. The derivation process of the dynamic equations of the spindle-bearing system is as follows:

[0012] 1) The Lagrange equation is established as follows:

[0013]

[0014] In the formula, T represents the total kinetic energy of the system, V represents the total potential energy of the system, U represents the damping loss of the system, and F represents the total potential energy of the system. i q represents external incentives. i Represents a generalized coordinate vector;

[0015] 2) Mass blocks m1 and m3 are dynamically symmetric rigid bodies undergoing steady motion. Their kinetic energy can be calculated using the following formula:

[0016]

[0017] In the formula, m represents the mass of the lumped mass block m1 or m3, and J d J represents the moment of inertia. p Represents the polar moment of inertia, θ x θ represents the torsion angle about x, x and y represent the displacements in the x and y directions, respectively. y ω represents the torsion angle about the y-axis, and ω is the angular velocity of rotation around the principal axis.

[0018] 3) For lumped masses m2, m b1 m b2 The formula for calculating kinetic energy is:

[0019]

[0020] In the formula, m represents the lumped mass blocks m2 and m b1 or m b2 The quality;

[0021] 4) The formula for calculating the potential energy of each concentrated mass block is:

[0022]

[0023] In the formula, V x V y Let k represent the potential energy in the x and y directions, respectively. rr k φr k φφ k rφ The stiffness of a massless shaft is represented by x. b This represents the displacement of the bearing in the x-direction, y-direction. b This indicates the displacement of the bearing in the y-direction;

[0024]

[0025] In the formula, E represents the elastic modulus of the material, l represents the length of the axis of rotation, and I represents the moment of inertia of the cross section;

[0026] 5) The formula for calculating the damping loss of each concentrated mass block is:

[0027]

[0028] In the formula, c represents the damping loss coefficient;

[0029] The rolling elements are evenly distributed at equal intervals between the inner and outer raceways of the bearing, and the motion between the rolling elements and the inner and outer raceways is pure rolling, satisfying the Hertz contact condition. The rotational speed ω of the cage in pure rolling is... c for:

[0030]

[0031] In the formula, R is the outer diameter of the bearing, and r is the inner diameter of the bearing;

[0032] 6) The angle of the j-th rolling element is represented by φ. j :

[0033]

[0034] In the formula, Z is the number of balls in the rolling bearing, and t is the time variable;

[0035] 7) The inner and outer raceways of the bearing are in contact with the rolling elements. At any given moment, the deformation r of the j-th rolling element at the contact point is... j Represented as:

[0036]

[0037] In the formula, γ0 is the initial clearance of the bearing;

[0038] 8) Under the condition of satisfying the Hertz contact condition, the rolling elements and the inner and outer raceways will generate a nonlinear restoring force f.j :

[0039]

[0040] In the formula, k b The Hertzian contact stiffness is related to the material properties and geometry of the rolling bearing; n b Contact index;

[0041] 9) Because the spindle rotation system cannot be perfectly aligned, an unbalanced force will be generated during rotation. This unbalanced force is expressed as:

[0042]

[0043] In the formula, e represents the spindle eccentricity, and F x F represents the external excitation in the x-direction. y This indicates an external excitation in the y-direction;

[0044] 10) Combining equations (1) to (12), we obtain the vibration differential equation of the fourteen-degree-of-freedom spindle-bearing system as follows:

[0045]

[0046] In the formula, m1 and m b1 m2, m b2 m3 and m3 represent the masses of each simplified lumped mass block, respectively, and k 1b1 k 2b1 k 2b2 k 3b2 k φφ1 k rφ1 k φφ3 k rφ3 k represents the stiffness coefficient of the shaft between each lumped mass. b1y k b1x k b2y k b2x These represent the contact stiffness of each bearing, c 1b1 c 2b1 c 2b2 c 3b2 c represents the damping coefficient of the shaft between each lumped mass. b1y c b1x c b2y c b2x These represent the damping coefficients of each bearing.

[0047] Step 2: Solve the dynamic equation (i.e., formula (13)) using the Runge-Kutta method to obtain the theoretical value of the dynamic equation of the spindle-bearing system with respect to the vibration displacement response under steady state.

[0048] Step 3: Verify the validity of the spindle-bearing system dynamic model, build a test bench, conduct vibration tests on the CNC machine tool spindle, and determine the limit state equation for vibration failure of the spindle-bearing system; including:

[0049] Step 3.1: Build a test bench for studying the vibration displacement response of the electric spindle end, and use an eddy current displacement sensor to sample the vibration signal at the spindle end to obtain the experimental value of the vibration displacement response;

[0050] Step 3.2: Compare and verify the experimental results with the theoretical model results; when the relative deviation between the theoretical value and the experimental value of the vibration displacement response at each corresponding point is within the set threshold δ', the dynamic modeling of the spindle-bearing system is considered to be effective; otherwise, the dynamic modeling of the spindle-bearing system is considered to be invalid, and it is necessary to return to step 1 to re-establish the vibration differential equation of the spindle-bearing system.

[0051] Step 3.3: Establish the limit state equation Z based on the allowable radial runout of the spindle in the actual engineering project:

[0052] Z = g(X) = S(X) - x * (14)

[0053] In the formula, S(X) is the theoretical value of the vibration displacement response; x* is the maximum allowable radial runout of the machine tool shaft, taken as 75μm; and X is a random variable of bearing parameters.

[0054] Step 4: Establish a surrogate model using the adaptive Kriging method, verify the accuracy of the adaptive Kriging using Monte Carlo, calculate the reliability and sensitivity, and verify the reliability; including:

[0055] Step 4.1: Generate the initial sample pool S in the input variable space using the Monte Carlo method;

[0056] Step 4.2: Use Latin hypercube sampling to extract initial training sample points X0, and substitute them into Step 2 to solve for the corresponding vibration response numerical solution Y0, forming the initial training sample set S0.

[0057] Step 4.3: Construct the Kriging proxy model using the initial training sample set S0;

[0058] Step 4.4: Calculate the learning function value for each sample in the initial sample pool and select the next sample point to be updated. Determine whether the Kriging self-learning process stopping criterion is met. If the learning criterion is met, stop the learning process and proceed to Step 4.5; if the learning stopping criterion is not met, expand the sample pool and return to Step 4.2 to continue building the Kriging model.

[0059] Step 4.5: Calculate the reliability using the current Kriging model, and verify the accuracy of the Kriging model by calculating the reliability using the Monte Carlo method. Calculate the error between the reliability obtained by the Kriging model and the Monte Carlo method. If the error is within the set threshold, it proves that the reliability result calculated by the Kriging model is accurate; otherwise, it is considered that the reliability calculated by the Kriging model is invalid, and the Kriging model is rebuilt.

[0060] Step 4.6: Calculate the failure probability and sensitivity using the current Kriging model;

[0061] Step 4.7: Calculate the coefficient of variation of the failure probability estimate. If the coefficient of variation is less than the set threshold δ, the estimate of the reliability local sensitivity calculated in Step 4.5 is considered acceptable; otherwise, return to Step 4.4.

[0062] The beneficial effects of this invention are:

[0063] This invention proposes a method for verifying the reliability sensitivity of CNC machine tool electric spindle vibration. It takes into account the random factors present in the manufacturing of ball bearings, making it more consistent with real working conditions. This invention uses the vibration displacement of the spindle end as the evaluation index for reliability assessment, which is of great significance for preventing excessive spindle vibration displacement from affecting machining quality and improving machining accuracy and reliability. Attached Figure Description

[0064] Figure 1 This is a flowchart of the vibration reliability sensitivity verification method for CNC machine tool electric spindles in this invention;

[0065] Figure 2 This is a model diagram of the machine tool spindle structure in this invention;

[0066] Figure 3 This is a simplified diagram of the machine tool spindle structure model in this invention;

[0067] Figure 4 This is a cross-sectional view of the bearing used for stress analysis in this invention;

[0068] Figure 5 The figures show a comparison between simulation results and experimental results at different rotational speeds in this invention. (a) represents the relationship between vibration displacement and time at a rotational speed of 27000 r / min, (b) represents the relationship between vibration displacement and time at a rotational speed of 37200 r / min, (c) represents the relationship between vibration displacement and time at a rotational speed of 43500 r / min, and (d) represents the relationship between vibration displacement and time at a rotational speed of 49800 r / min.

[0069] Figure 6This is a graph showing the sensitivity results of machine tool spindle vibration reliability in this invention. Detailed Implementation

[0070] The invention will be further described below with reference to the accompanying drawings and specific implementation examples. To study the reliability of machine tool spindle vibration, this invention proposes a method for verifying the reliability sensitivity of CNC machine tool electric spindle vibration. This method uses the vibration displacement at the spindle end during spindle rotation as a metric, considers the uncertainty of bearing structural parameters, calculates the corresponding reliability and sensitivity of each structural parameter, verifies the accuracy of the dynamic modeling through experiments, and verifies the accuracy of the adaptive Kriging surrogate model through the Monte Carlo method. This demonstrates that the dynamic spindle-bearing system modeling and reliability analysis method proposed in this invention has high accuracy.

[0071] This invention proposes a method for verifying the vibration reliability sensitivity of CNC machine tool electric spindles, such as... Figure 1 As shown, it includes:

[0072] like Figure 2 As shown, the electric spindle mainly consists of a housing, front and rear bearing assemblies, stator and rotor, cooling water channels, shaft retaining rings, and a rotating shaft. Considering the complexity of the spindle system itself, the lumped mass method is used to simplify the spindle unit into a fourteen-degree-of-freedom system, such as... Figure 3 As shown in the figure. m1, m b1 m2, m b2 m3 and m3 represent the masses of each simplified lumped mass block, respectively, and k 1b1 k 2b1 k 2b2 k 3b2 and represent the stiffness coefficients of the shafts between the lumped masses, respectively, k b1y k b1x k b2y k b2x These represent the contact stiffness of each bearing, c 1b1 c 2b1 c 2b2 c 3b2 c represents the damping coefficient of the shaft between each lumped mass. b1y c b1x c b2y c b2x These represent the damping coefficients of each bearing.

[0073] Step 1: Taking the simplified 14-DOF spindle system model as the research object, considering the rotational unbalance force and the nonlinear restoring force of the bearing, the dynamic equation of the machine tool spindle-bearing system is established based on the Lagrange fundamental equation.

[0074]

[0075] In equation (1), M is the system's generalized mass matrix, K is the system's generalized stiffness matrix, C is the system's damping matrix, G is the gyro matrix, and F is the system's excitation force matrix. The system's excitation force mainly consists of two parts: the nonlinear restoring force of the rolling bearing and the unbalanced force caused by the unbalanced eccentricity of the spindle system. q represents the generalized coordinate vector of the spindle-bearing system relative to the base platform, q=(x1,θ) y1 ,x b1 ,x2,x b2 ,x3,θ y3 ,y1,θ x1 ,y b1 ,y2,y b2 ,y3,θ x3 ), where x i y i (i = 1, 2, 3) represent the displacement degrees of freedom of each lumped mass block in the x and y directions, respectively, θ x1 θ y1 θ y3 θ x3 Represents the torsional degrees of freedom of lumped mass blocks m1 and m3 about the x and y axes, respectively. bi y bi (i = 1, 2) represents the bearing's degrees of freedom in the x and y directions.

[0076] The Lagrange equation used to derive formula (1) is as follows:

[0077]

[0078] In the formula, T represents the total kinetic energy of the system, V represents the total potential energy of the system, U represents the damping loss of the system, and F represents the total potential energy of the system. i q represents external incentives. i Represents a generalized coordinate vector;

[0079] The vibration differential equation of the spindle-bearing system is established using the Lagrange equation, and the energy of each simplified mass block and the external excitation force of the system need to be solved.

[0080] 1) Mass blocks m1 and m3 are dynamically symmetric rigid bodies undergoing steady motion. Their kinetic energy can be calculated using the following formula:

[0081]

[0082] In the formula, m represents the mass of the lumped mass block m1 or m3, and J d J represents the moment of inertia. p Represents the polar moment of inertia, θ x θ represents the torsion angle about x, x and y represent the displacements in the x and y directions, respectively. yω represents the torsion angle about the y-axis, and ω is the angular velocity of rotation around the principal axis.

[0083] 2) For lumped masses m2, m b1 m b2 The formula for calculating kinetic energy is:

[0084]

[0085] In the formula, m represents the lumped mass blocks m2 and m b1 and m b2 quality

[0086] 3) The formula for calculating the potential energy of each concentrated mass block is:

[0087]

[0088] In the formula, V x V y Let k represent the potential energy in the x and y directions, respectively. rr k φr k φφ k rφ The stiffness of a massless shaft is represented by x. b This represents the displacement of the bearing in the x-direction, y-direction. b This indicates the displacement of the bearing in the y-direction;

[0089]

[0090] In the formula, E represents the elastic modulus of the material, l represents the length of the axis of rotation, and I represents the moment of inertia of the cross section.

[0091] 4) The formula for calculating the damping loss of each concentrated mass block is:

[0092]

[0093] In the formula, c represents the damping loss coefficient.

[0094] 5) The generalized excitation force consists of rotational unbalance force and bearing nonlinear restoring force. For the bearing nonlinear restoring force: for the convenience of subsequent research, the following assumptions are made: the rolling elements are evenly distributed at equal intervals between the inner and outer raceways of the bearing, and the motion between the rolling elements and the inner and outer raceways is pure rolling, and the Hertz contact condition is satisfied. Figure 4 For a simplified model of a pure rolling bearing, the rotational speed ω of the cage is... c for:

[0095]

[0096] In the formula, R is the outer diameter of the bearing, r is the inner diameter of the bearing, and ω is the angular velocity of the spindle rotation.

[0097] 6) By Figure 4 The angle of the j-th rolling element can be obtained as φ. j :

[0098]

[0099] In the formula, Z is the number of balls in the rolling bearing, and t is the time variable.

[0100] 7) The inner and outer raceways of the bearing are in contact with the rolling elements. At any given moment, the deformation r of the j-th rolling element at the contact point is... j It can be expressed by the following formula:

[0101]

[0102] In the formula, γ0 is the initial clearance of the bearing.

[0103] 8) Under the condition of satisfying the Hertz contact condition, the rolling elements and the inner and outer raceways will generate a nonlinear restoring force f. j :

[0104]

[0105] In the formula, k b The Hertzian contact stiffness is related to the material properties and geometry of the rolling bearing; n b The contact index is n = 1.5 for rolling bearings.

[0106] 9) Regarding rotational unbalanced forces: During rotation, because the spindle rotation system cannot be perfectly aligned, an unbalanced force will be generated during rotation, namely:

[0107]

[0108] In the formula, e represents the spindle eccentricity F. x F represents the external excitation in the x-direction. y This indicates an external excitation in the y-direction.

[0109] 10) Taking all the above factors into account, the system vibration differential equation is established according to equation (2):

[0110]

[0111] In the formula, m1 and m b1 m2, m b2 m3 and m3 represent the masses of each simplified lumped mass block, respectively, and k 1b1 k 2b1 k 2b2 k 3b2 k φφ1 k rφ1 k φφ3 krφ3 k represents the stiffness coefficient of the shaft between each lumped mass. b1y k b1x k b2y k b2x These represent the contact stiffness of each bearing, c 1b1 c 2b1 c 2b2 c 3b2 c represents the damping coefficient of the shaft between each lumped mass. b1y c b1x c b2y c b2x These represent the damping coefficients of each bearing.

[0112] Step 2: Solve the dynamic equations using the Runge-Kutta method to obtain the numerical solution of the dynamic equations of the spindle-bearing system under steady-state conditions;

[0113] The various material and structural parameters used for simulation are shown in Tables 1, 2, and 3:

[0114] Table 1 Bearing Parameters

[0115]

[0116] Table 2 Quality Parameters

[0117]

[0118] Table 3 Stiffness and Damping Parameters

[0119]

[0120] Step 3: Verify the validity of the spindle-bearing system dynamic model, build a test bench, conduct vibration tests on the CNC machine tool spindle, and determine the limit state equation for vibration failure of the spindle-bearing system; including:

[0121] Step 3.1: Set up a spindle-bearing system test bench, use an eddy current displacement sensor to sample the vibration signal at the end of the spindle, and obtain the experimental value of the vibration displacement at the end of the spindle.

[0122] A test bench for the vibration displacement response of the electric spindle end was constructed, with the spindle end as the test object. An eddy current displacement sensor with a sampling frequency of 10kHz (Sinocera) was used. The CWY-DO-500 sensor was used to collect vibration displacement response data at the spindle end. The vibration displacement parameters were converted into electrical signals, which were then amplified and transmitted to a computer via a data acquisition system (Sinocera YE6251) for data processing. The experiment selected four common operating speeds of the machine tool spindle: 27000 r / min, 37200 r / min, 43500 r / min, and 49800 r / min for measurement. The experimental procedure was as follows: First, the sensor was zeroed by placing it 1500 μm from the spindle end. At this point, the data acquisition system displayed a displacement of 0. Then, the machine tool was started and idled at 27000 r / min. The displacement-time signal was transmitted back to the computer via the data acquisition system (Sinocera YE6251). The computer displayed the vibration displacement versus time curve, and the data was saved. The above experimental steps were repeated for the other speeds.

[0123] After the experimental test was completed, the time-domain response results were obtained, and the acquired signals were processed. During the experiment, due to inherent errors in the measurement system and interference from external factors, some questionable data were generated. We used the method of repeating the experiment to reduce the impact of errors. In addition, some data with large deviations appeared. The Rheinda criterion was used to discard some data. For some questionable data that could not be discarded, the mechanism was analyzed to determine its cause. After completing the experiment and data processing, the vibration displacement versus time curves at four rotational speeds were obtained.

[0124] Step 3.2: Compare and verify the experimental results with the theoretical model results. If the relative deviation between the theoretical value and the experimental value of the vibration displacement response at each corresponding point is within the set threshold, the dynamic modeling of the spindle-bearing system is considered valid. Otherwise, the dynamic modeling of the spindle-bearing system is considered invalid, and it is necessary to return to step 1, adjust the lumped mass division and the stiffness coefficient of each massless shaft, and re-establish the vibration differential equation of the spindle-bearing system.

[0125] The experimental results at four operating speeds of 27000 r / min, 37200 r / min, 43500 r / min, and 49800 r / min were compared with the theoretical modeling and simulation results. The results are as follows: Figure 5 As shown. From Figure 5 The results show that the theoretical model and the experimental results have a high degree of consistency, and the relative error is within the set threshold.

[0126] Step 3.3: Establish the limit state equation Z based on the allowable radial runout of the spindle in the actual engineering project:

[0127] Z = g(X) = S(X) - x * (14)

[0128] In equation (14), S(X) is the theoretical value of the vibration displacement response calculated by formula (13); x* is the maximum allowable radial runout of the machine tool spindle, taken as 75 μm; X is a random variable of bearing parameters. According to the spindle-bearing system dynamic model established in this paper, the bearing structural parameters will be reflected in the stiffness coefficient k during the change process. b Regarding the bearing clearance, the stiffness and clearance of the front and rear bearings are selected as random variables.

[0129] When Z>0, it means that the dynamic response of the spindle end vibration displacement exceeds the allowable radial runout value of the spindle, and the spindle vibration fails; when Z<0, it means that the dynamic response of the spindle end vibration displacement does not exceed the allowable radial runout value of the spindle, and the spindle vibration is reliable; when Z=0, it means that the spindle-bearing system is in the limit dynamic response state.

[0130] Step 4: Establish a surrogate model using the adaptive Kriging method, verify the accuracy of the adaptive Kriging using Monte Carlo, and calculate the reliability and sensitivity; including:

[0131] Step 4.1: Generate the initial sample pool S in the input variable space using the Monte Carlo method;

[0132] Using bearing stiffness and radial clearance as design variables, 10,000 data points were extracted as the initial sample pool. The random parameters and distributions of each design variable are shown in Table 4.

[0133] Table 4 Random parameters of bearing clearance and contact stiffness

[0134]

[0135]

[0136] Step 4.2: Use Latin hypercube sampling to extract initial training sample points X0, and substitute them into Step 2 to solve for the corresponding vibration response numerical solution Y0, forming the initial training sample set S0.

[0137] Using Latin hypercube sampling, 20 sets of data were extracted from the initial sample pool as initial training sample points. These were then substituted into the differential equation of the spindle-bearing system dynamics in step 2 to obtain numerical solutions, thus forming 20 sets of initial training sample sets.

[0138] Step 4.3: Construct the Kriging proxy model using the initial training sample set S0;

[0139] Step 4.4: Calculate the learning function value for each sample in the initial sample pool and select the next sample point to be updated. Determine if the Kriging self-learning process stopping criterion is met. If the learning criterion is met, stop the learning process and proceed to Step 4.5; if the learning stopping criterion is not met, expand the sample pool, add the next sample point to the sample pool, return to Step 4.2, and continue building the Kriging model.

[0140] Step 4.5: Calculate the reliability using the current Kriging model, and verify the accuracy and precision of the Kriging model by calculating the reliability using the Monte Carlo method. Calculate the error between the two methods. If the error is within the set threshold, it proves that the reliability result calculated by the Kriging model is accurate; otherwise, it is considered that the reliability calculated by the Kriging model is invalid, and the Kriging model is rebuilt.

[0141] Reliability calculations were performed using both adaptive Kriging and Monte Carlo methods, and the results are shown in Table 5.

[0142] Table 5 Reliability Calculation Results

[0143]

[0144] The results in the table show that the error compared to the Monte Carlo method is within the acceptable range, proving that the adaptive Kriging method has high accuracy. Based on this, sensitivity analysis is performed using a surrogate model.

[0145] Step 4.6: Calculate the failure probability and sensitivity using the current Kriging model;

[0146] Step 4.7: Calculate the coefficient of variation of the failure probability estimate. If the coefficient of variation is less than 5%, the estimate of the reliability local sensitivity calculated in Step 4.5 is considered acceptable; otherwise, return to Step 4.4.

[0147] In order to obtain the degree of influence of bearing and spindle structural parameters on reliability, the sensitivity coefficient of failure probability on the mean and standard deviation of random variables is used to represent the reliability, and the sensitivity coefficient of each design variable parameter is obtained.

[0148] Depend on Figure 6 It can be seen that as the clearance between the front and rear bearing assemblies increases, the reliability decreases; conversely, as the contact stiffness of the front and rear bearing assemblies increases, the reliability increases. The clearance and stiffness of the rear bearing assembly have a greater impact on reliability. Considering that bearing stiffness cannot be increased indefinitely, the precision requirements and assembly requirements for the rear bearing assembly are more stringent during manufacturing, processing, and assembly.

[0149] This invention comprehensively considers the balancing forces and nonlinear restoring forces of the bearings during spindle rotation, establishing a vibration dynamic model of the spindle-bearing system using the Lagrange equation. The effectiveness of the spindle system dynamic model is verified through simulation and experimentation. Limit state equations are established based on the allowable radial runout of the spindle in engineering practice, and design variables for the limit state equations are determined according to the manufacturing and assembly processes. Adaptive Kriging analysis is then used to analyze the vibration reliability of the spindle-bearing system. This method fully considers the random factors present in ball bearing manufacturing and proposes a reliability assessment method using spindle end vibration displacement as the evaluation index. This is of great significance for preventing excessive spindle vibration displacement from affecting machining quality and improving machining accuracy and reliability.

Claims

1. A method for verifying the reliability and sensitivity of vibration of an electric spindle in a CNC machine tool, characterized in that, include: Step 1: The machine tool spindle-bearing system is simplified to a fourteen-degree-of-freedom spindle-bearing system model using the lumped mass method. Considering the rotational unbalance force and the nonlinear restoring force of the bearing, the dynamic equations of the machine tool spindle-bearing system are established based on the Lagrange fundamental equations. The dynamic equation of the machine tool spindle-bearing system in step 1 is expressed as follows: (1) In the formula, M For the system's generalized mass matrix, K The system's generalized stiffness matrix. C Here is the system damping matrix. G For gyroscope matrices, F Here is the system excitation force matrix; The system excitation force consists of two parts: the nonlinear restoring force of the rolling bearing and the unbalanced force caused by the unbalanced eccentricity of the spindle system. q This represents the generalized coordinate vector of the spindle-bearing system relative to the base platform. ,in x i'' 、y i'' Each represents a lumped mass block in x and y Degrees of freedom of displacement in the direction, i'' =1, 2, 3, θ x1 θ y1 θ y3 θ x3 Represents lumped mass block m 1 or m 3 wraps x, y Torsional degree of freedom of the shaft x bi' 、y bi' Represents the bearing in x and y Degrees of freedom of displacement in the direction, i' =1, 2; The vibration differential equation of the fourteen-degree-of-freedom spindle-bearing system is: In the formula, m 1 、m b1 、m 2 、m b2 、m 3 represents the mass of each of the simplified mass blocks. k 1b1 、k 2b1 、k 2b2 、k 3b2 、 k φφ1 、 、k φφ3 、 These represent the stiffness coefficients of the rotational shafts between each concentrated mass block. k b1y 、k b1x 、k b2y 、k b2x These represent the contact stiffness of each bearing. c 1b1 、c 2b1 、c 2b2 、c 3b2 These represent the damping coefficients of the rotational shafts between each lumped mass block. c b1y 、c b1x 、c b2y 、c b2x These represent the damping coefficients of each bearing. Indicates the moment of inertia. Represents the polar moment of inertia. ω The angular velocity of the main shaft rotation; Step 2: Solve the dynamic equations using the Runge-Kutta method to obtain the theoretical values ​​of the dynamic equations of the spindle-bearing system under steady-state conditions regarding the vibration displacement response; Step 3: Verify the effectiveness of the spindle-bearing system dynamic model, build a test bench, conduct vibration tests on the CNC machine tool spindle, and determine the limit state equation for vibration failure of the spindle-bearing system; Step 4: Establish a surrogate model using the adaptive Kriging method, verify the accuracy of the adaptive Kriging using Monte Carlo, calculate the reliability and sensitivity, and verify the reliability.

2. The method for verifying the vibration reliability sensitivity of a CNC machine tool electric spindle according to claim 1, characterized in that, In step 1, the machine tool spindle-bearing system is simplified to a fourteen-degree-of-freedom spindle-bearing system model using the lumped mass method. The derivation process of the dynamic equations of the spindle-bearing system is as follows: 1) Establish the Lagrange equation as follows: In the formula, T Represents the total kinetic energy of the system. V Represents the total potential energy of the system. U Represents the damping loss of the system. F i Represents external incentives. Represents a generalized coordinate vector; 2) Mass block m 1. m 3 is a dynamically symmetric rigid body undergoing steady motion; its kinetic energy is calculated using the following formula: (3) In the formula, m This represents the mass of the lumped mass block. Indicates the moment of inertia. Represents the polar moment of inertia. Indicates circling x The angle of twist, x , y express x , y Displacement in the direction, Indicates circling y The angle of twist, ω The angular velocity of the main shaft rotation; 3) For centralized quality m 2 、m b1 , m b2 The formula for calculating kinetic energy is: (4) 4) The formula for calculating the potential energy of each concentrated mass block is: In the formula, V x , V y They represent x and y Potential energy in the direction, k rr , k φr 、k φφ 、k rφ Indicates the stiffness of a massless rotating shaft. Indicates that the bearing is in x Displacement in the direction, y b Indicates that the bearing is in y Displacement in the direction; In the formula, E Represents the elastic modulus of the material. l Represents the length of the shaft. I Represents the moment of inertia of the cross section; 5) The formula for calculating the damping loss of each concentrated mass block is: (7) In the formula, c Indicates the damping loss coefficient; The rolling elements are evenly distributed at equal intervals between the inner and outer raceways of the bearing, and the motion between the rolling elements and the inner and outer raceways is pure rolling, satisfying the Hertz contact condition. The rotational speed of the cage in pure rolling is... ω c for: (8) In the formula, R The outer diameter of the bearing. r This refers to the inner diameter of the bearing. 6) No. j The angle of each rolling element is expressed as: φ j : (9) In the formula, Z This refers to the number of balls in a rolling bearing. t It is a time variable; 7) The inner and outer raceways of the bearing are in contact with the rolling elements. At any given moment, the first... j Deformation of the rolling element at the contact point r j Represented as: In the formula, γ 0 represents the initial clearance of the bearing; 8) Under the condition of satisfying the Hertz contact condition, the rolling elements and the inner and outer raceways will generate a nonlinear restoring force. f j : In the formula, k b The Hertzian contact stiffness is related to the material properties and geometry of the rolling bearing. n b Contact index; 9) Because the spindle rotation system cannot be perfectly aligned, an unbalanced force will be generated during rotation. This unbalanced force is expressed as: (12) In the formula, e Indicates the spindle eccentricity. express x External excitation in direction express y External stimulus in the direction; 10) Combining formulas (1) to (12) yields the vibration differential equation of the fourteen-degree-of-freedom spindle-bearing system.

3. The method for verifying the vibration reliability sensitivity of a CNC machine tool electric spindle according to claim 1, characterized in that, Step 3 includes: Step 3.1: Build a test bench for studying the vibration displacement response of the electric spindle end, and use an eddy current displacement sensor to sample the vibration signal at the spindle end to obtain the experimental value of the vibration displacement response; Step 3.2: Compare and verify the experimental results with the theoretical model results; when the relative deviation between the theoretical and experimental values ​​of the vibration displacement response at corresponding points is within the set threshold... If the model is valid, the spindle-bearing system dynamics model is considered valid; otherwise, the model is considered invalid and the process needs to return to step 1 to re-establish the vibration differential equation of the spindle-bearing system. Step 3.3: Establish the limit state equations based on the allowable radial runout of the spindle in the actual engineering project. Z : Z=g(X)=S(X)-x * (14) In the formula, S( X ) This represents the theoretical value of the vibration displacement response; x* X represents the maximum permissible radial runout of the machine tool spindle, and X is a random variable representing bearing parameters.

4. The method for verifying the vibration reliability sensitivity of a CNC machine tool electric spindle according to claim 1, characterized in that, Step 4 includes: Step 4.1: Generate the initial sample pool S in the input variable space using the Monte Carlo method; Step 4.2: Use Latin hypercube sampling to extract initial training sample points X0, and substitute them into Step 2 to solve for the corresponding vibration response numerical solution Y0, forming the initial training sample set S0. Step 4.3: Construct the Kriging proxy model using the initial training sample set S0; Step 4.4: Calculate the learning function value for each sample in the initial sample pool, select the next sample point to be updated, and determine whether the Kriging self-learning process stops. If the stopping criterion is met, stop the learning process and proceed to Step 4.

5. If the stopping criterion is not met, expand the sample pool and return to Step 4.2 to continue building the Kriging model. Step 4.5: Calculate the reliability using the current Kriging model, and verify the accuracy of the Kriging model by calculating the reliability using the Monte Carlo method. Calculate the error between the reliability obtained by the Kriging model and the Monte Carlo method. If the error is within the set threshold, it proves that the reliability result calculated by the Kriging model is accurate; otherwise, it is considered that the reliability calculated by the Kriging model is invalid, and the Kriging model is rebuilt. Step 4.6: Calculate the failure probability and sensitivity using the current Kriging model; Step 4.7: Calculate the coefficient of variation of the failure probability estimate. When the coefficient of variation is less than a set threshold... If the estimate of the reliability local sensitivity calculated in step 4.5 is deemed acceptable, otherwise return to step 4.4.

Images