A bolted joint assembly process optimization method of a fusion virtual variable response surface model

Abstract

This invention discloses a bolt assembly process optimization method based on a fused dummy variable response surface. First, by comparing and analyzing the impact of performance degradation of different bolt joint surfaces on the vibration response of the bolt tip, weak joint surfaces in the entire machine are identified. A full-factor experiment is designed using the tightening sequence and the preload ratio α in the first step of the two-step tightening method as optimization variables. Then, a binary dummy variable is introduced with cross-tightening as the reference group, and α is centered to establish a cubic polynomial response surface model containing both dummy and continuous variables. The fitting accuracy is evaluated using the coefficient of determination and root mean square error. Finally, with the goal of minimizing the vibration response amplitude of the bolt tip, a genetic algorithm is used for global optimization to obtain the optimal assembly process parameters, which are then compared and verified with similar processes. This method can provide technical guidance for the assembly optimization of weak bolt joint surfaces in engineering.

Description

Technical Field

[0001] This invention relates to the fields of bolted connection surface performance analysis and machine tool dynamics research, specifically to a method for optimizing the assembly process of bolted joint surfaces by integrating a dummy variable response surface model. Background Technology

[0002] Heavy-duty gantry milling machines contain numerous bolted connections, such as the slide-column joint surface and the connecting beam-column joint surface. Due to their discontinuous structural characteristics and susceptibility to creep and plastic deformation, these bolted joint surfaces inevitably become weak points in the overall machine. Related research indicates that the stiffness of bolted joint surfaces accounts for 40% to 60% of the total stiffness of the machine, demonstrating that the assembly quality of these joints has a decisive impact on the dynamic response and even machining accuracy of the entire machine. Therefore, optimizing the assembly process for these weak bolted joint surfaces is of significant engineering importance for improving the dynamic performance and accuracy retention of the machine tool.

[0003] Currently, research on bolt assembly process optimization primarily employs orthogonal experimental design to investigate the effects of factors such as tightening sequence and number of tightening steps on the distribution of bolt preload or the contact stiffness of the mating surfaces after assembly. However, this method has two shortcomings: first, it can only find the optimal combination in discrete experimental schemes, making it difficult to achieve precise optimization in continuous parameter space; second, existing studies mostly use the consistency of preload or the stiffness of the mating surfaces after assembly as evaluation indicators, failing to establish a direct mapping relationship between assembly process parameters and the dynamic performance of the entire machine. Furthermore, assembly process parameters often include both discrete variables (such as tightening sequence) and continuous variables (such as tightening ratio and tightening speed), making it difficult for traditional response surface methodology to uniformly handle both types of variables, leading to challenges in modeling.

[0004] Therefore, this invention proposes a bolt assembly process optimization method based on fused dummy variable response surface. By introducing a dummy variable encoding strategy, a unified modeling of discrete tightening sequence and continuous tightening ratio is achieved. A genetic algorithm is then used to perform global optimization in the continuous parameter space, ultimately obtaining the assembly process parameters that optimize the dynamic performance of the whole machine. Summary of the Invention

[0005] The objective of this invention is to propose a bolt assembly process optimization method based on a fused dummy variable response surface. A response surface model with fused dummy variable encoding is established to uniformly handle two types of variables: discrete tightening sequence and continuous tightening ratio. A genetic algorithm is then used for global optimization, aiming to minimize the vibration response at the tool tip to obtain the optimal assembly process parameters. This provides technical guidance for optimizing the assembly of weak bolt joint surfaces in engineering. The specific solution adopted in this invention is as follows:

[0006] Step 1: By comparing and analyzing the impact of performance degradation of different bolted joint surfaces on the vibration response of the tool tip, the weak joint surfaces of the entire machine are identified. Based on the bolt layout characteristics of the weak joint surfaces, the optimization variables and their value ranges are determined. The optimization variables include the tightening sequence (discrete variable) and the first-step preload ratio α in the two-step tightening method (continuous variable). The tightening sequence has three typical schemes, and the first-step preload ratio α is divided into multiple levels within a specified range with a certain step distance. A full factorial test scheme is designed.

[0007] Step 2: Based on the full factorial testing scheme, simulation calculations were performed on the assembly process parameters for each group to obtain the corresponding tool tip vibration response amplitude A as the optimization target. It should be noted that the tool tip vibration response amplitude A was calculated based on the fractal-finite element joint modeling method and the multibody system transfer matrix method dynamic model. Both methods have been verified through modal tests on different components and can reliably reflect the mapping relationship between assembly process parameters and the overall dynamic performance of the machine.

[0008] Step 3: Using the tightening sequence and the pre-tightening ratio α from the first step as input variables, and the vibration response amplitude A at the blade tip as the output variable, a response surface model is established using a fusion dummy variable encoding method. Specifically: for the discrete tightening sequence variable, one tightening scheme is selected as a reference group, and binary dummy variables are introduced to encode various tightening schemes; the continuous variable α is centered; a polynomial response surface model containing dummy and continuous variables is established, and the model is fitted and its accuracy is evaluated.

[0009] Step 4: With the minimum vibration response amplitude A at the tool tip as the optimization objective, and the binary value constraints of the dummy variables and the range of α as constraints, the genetic algorithm is used to perform global optimization on the response surface model to obtain the optimal assembly process parameters.

[0010] Step 5: Substitute the optimal assembly process parameters into the original simulation model for verification calculation and compare them with the predicted values ​​of the response surface model; at the same time, set up multiple control schemes and compare the tool tip response of the optimal scheme with similar assembly process schemes to verify the effectiveness and engineering applicability of the optimization method.

[0011] Step one specifically involves:

[0012] Based on the bolt layout characteristics of the weak joint surfaces of the gantry milling machine, a two-step tightening strategy is determined. The first step involves tightening each bolt sequentially to α times the target preload F according to the selected tightening sequence; the second step involves tightening each bolt sequentially to the target preload F according to the same tightening sequence.

[0013] Tightening sequence is treated as a discrete variable, and typical tightening schemes are set. The first-step pre-tightening ratio α is treated as a continuous variable, and its value range is defined. The step size is set according to the number of tightening schemes and the first-step tightening ratio α, and a full factorial experiment is designed.

[0014] Step two specifically involves:

[0015] For each test scheme, the tightening sequence and the first-step pre-tightening ratio α are set, and the load is applied to each bolt in sequence according to the two-step tightening method using finite element simulation software to extract the contact stress distribution of the mating surface.

[0016] Based on previous research findings, the contact stiffness values ​​of the mating surfaces under various assembly processes were calculated based on the contact stress distribution at the mating surfaces. These mating surface stiffness values ​​were then substituted into the overall machine dynamics model to calculate the vibration response at the tool tip for each scheme, and the response amplitude was extracted as the optimization target value.

[0017] The tightening sequence, the first-step pre-tightening ratio α, and the corresponding blade tip acceleration response amplitude of all full-factor test schemes are summarized to form a sample dataset.

[0018] Step three specifically involves:

[0019] The discrete tightening order variables are encoded using dummy variables. Taking a specific tightening order as a reference group, two binary dummy variables, D1 and D2, are introduced. When the tightening order is another tightening order, D1=1, D2=0; when the tightening order is a third tightening order, D1=0, D2=1; when the tightening order is the reference tightening order, D1=0, D2=0.

[0020] The continuous variable α is centered, the mean of α is calculated for all test schemes, and the centered tightening ratio is defined. .

[0021] A cubic polynomial response surface model incorporating dummy and continuous variables is established, and its expression is:

[0022]

[0023] In the formula, A is the vibration response amplitude at the tool tip, and β0 to β9 are undetermined coefficients.

[0024] The response surface model was fitted with least squares using the sample dataset to obtain the estimated values ​​of each undetermined coefficient, thus completing the establishment of the response surface model.

[0025] Step four specifically involves:

[0026] An assembly process optimization model is established, with the minimum vibration response amplitude A at the tool tip as the optimization objective and D1×D2=0 and α within the constraint range as the constraint conditions.

[0027] A genetic algorithm is used to solve the above optimization model. The algorithm's parameters, such as crossover and mutation probabilities, are set, and a global search is performed in the parameter space through selection, crossover, and mutation operations. The optimal individual is output when the convergence condition is met.

[0028] Decoding the optimal individual yields the optimal assembly process parameters: the optimal tightening sequence and the optimal first-step pre-tightening ratio α.

[0029] Step five specifically involves:

[0030] Substitute the optimal assembly process parameters obtained in step four into the finite element simulation model and the whole machine dynamics model described in step two, recalculate the vibration response amplitude of the tool tip, and compare the simulation calculation value with the optimal target value predicted by the response surface model in step three to verify the prediction accuracy of the response surface model.

[0031] Multiple control schemes were set up, including: changing the tightening sequence while keeping α constant, and keeping the tightening sequence but changing the α value. The blade tip vibration response of each control scheme was calculated and compared with the response of the optimal scheme.

[0032] If the vibration response amplitude of the optimal solution at the tool tip is lower than that of all control solutions, then the effectiveness and engineering applicability of the assembly process optimization method proposed in this invention are verified. Attached Figure Description

[0033] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings, wherein:

[0034] Figure 1 This is a flowchart of the technical solution of the present invention.

[0035] Figure 2 These are schematic diagrams of three bolt tightening sequence schemes, where (a) is sequential tightening S1, (b) is cross tightening S2, and (c) is cross tightening S3.

[0036] Figure 3 It is a 3D surface plot and contour plot of the response surface that integrates dummy variable encoding. (a) 3D surface plot; (b) contour plot;

[0037] Figure 4 This is a flowchart of a genetic algorithm.

[0038] Figure 5 The diagram shows a comparison of the tool tip acceleration response between the optimal assembly process and similar processes, where (a) is a time domain response comparison and (b) is a frequency domain response comparison. Detailed Implementation

[0039] The present invention will be described in detail below with reference to the accompanying drawings and embodiments:

[0040] A finite element method for calculating the dynamic characteristics of machine tools based on parametric modeling of bolted joint surfaces, the specific implementation of which includes the following steps:

[0041] A bolt assembly process optimization method based on fused dummy variable response surface, specifically including the following steps:

[0042] Step 1: By comparing and analyzing the impact of performance degradation of different bolted joint surfaces on the vibration response of the tool tip, the weak joint surfaces of the entire machine are identified. Based on the bolt layout characteristics of the weak joint surfaces, the optimization variables and their value ranges are determined, and a full factorial test scheme is designed.

[0043] Step 2: Based on the full factorial test scheme, simulate and calculate the assembly process parameters of each group to obtain the vibration response amplitude A of the tool tip for each group.

[0044] Step 3: Using the tightening sequence and the pre-tightening ratio α from the first step as input variables, and the vibration response amplitude A at the blade tip as the output variable, a response surface model is established using the fusion dummy variable coding method, and the fitting accuracy is evaluated.

[0045] Step 4: Using the minimum vibration response amplitude A at the tool tip as the optimization objective, a genetic algorithm is used to perform global optimization to obtain the optimal assembly process parameters.

[0046] Step 5: Perform model accuracy verification and similar process comparison verification.

[0047] The specific process is as follows: Figure 1 As shown.

[0048] Step one specifically includes the following steps:

[0049] Based on our previous research, quantitative comparative analysis has identified the slide-column mating surface as a weak bolted connection in a heavy-duty gantry milling machine. Specifically, the bolt preload at both the slide-column and connecting beam-column mating surfaces was relaxed by 10% to 30%, and the displacement and acceleration response at the tool tip were calculated using the overall machine dynamics model. The results show that with the same 30% relaxation, the tool tip acceleration response at the slide-column mating surface increases by 5%, while that at the connecting beam-column mating surface only increases by 0.6%. Therefore, this study selects the slide-column mating surface as the object for assembly process optimization.

[0050] The slide-column mating surface consists of 32 M33 bolts with a performance grade of 10.9 and a target preload of 326 kN per bolt. A two-step tightening method is selected for assembly. The specific process is as follows: First, tighten each bolt sequentially to α times the target preload F according to the selected tightening sequence; second, tighten each bolt sequentially to the target preload F according to the same tightening sequence.

[0051] The tightening order is treated as a discrete variable, and three typical schemes are set. For example... Figure 2 As shown in (a), sequential tightening of S1 means tightening the bolts in the order they are arranged in the space. Figure 2 As shown in (b), cross-tightening S2 involves tightening in a cross sequence. Figure 2 As shown in (c), the cross-tightening S3 is performed by tightening from the inside out and alternating diagonally.

[0052] The first-step pre-tightening ratio α is treated as a continuous variable, with a value range of 30% to 80%. The continuous variable is discretized into 11 levels with a step size of 5%. The full factorial experimental design includes all combinations of 3 tightening sequences and 11 ratio levels, totaling 33 experimental schemes.

[0053] Step two specifically includes the following steps:

[0054] For each test scheme, the tightening sequence and the first-step pre-tightening ratio α were set, and the load was applied to each bolt in sequence according to the two-step tightening method using Abaqus finite element simulation software to extract the contact stress distribution of the mating surface.

[0055] Based on the contact stress distribution at the interface, the normal stiffness, tangential stiffness, and torsional stiffness of the interface under each assembly process were calculated using a fractal-finite element joint modeling method. This method has been validated through modal testing of double-T specimens, with the natural frequency error within 8%.

[0056] Substituting the stiffness value of the bonding surface into the overall machine dynamics model based on the multibody system transfer matrix method, the acceleration response at the tool tip corresponding to each scheme is calculated, and the acceleration response amplitude A is extracted as the optimization target value. This dynamic model simplifies each component of the gantry milling machine into EB beams, concentrated mass blocks, and spring hinge elements, constructs the overall machine transfer equation, solves for the natural frequencies and mode shapes, and then calculates the time-domain signals of the tool tip displacement and acceleration response under cutting conditions using the modal superposition method and the Runge-Kutta method. This method has been verified through modal tests on a scaled model, with the natural frequency error within 5%.

[0057] The tightening sequence number, the first-step pre-tightening ratio α, and the corresponding blade tip acceleration response amplitude A of all 33 test schemes were summarized to form a sample dataset.

[0058] Step three includes the following steps:

[0059] The discrete tightening sequence variables are encoded using dummy variables. Taking cross-tightening S3 as the reference group, two binary dummy variables D1 and D2 are introduced. The conversion relationship between tightening sequence and dummy variables is as follows: sequential tightening S1 corresponds to D1=1, D2=0; cross-tightening S2 corresponds to D1=0, D2=1; and cross-tightening S3 corresponds to D1=0, D2=0. A constraint condition D1×D2=0 is applied to ensure that the three tightening sequences are mutually exclusive in actual operation.

[0060] Center the continuous variable α. Calculate the mean of α for the 33 experimental schemes. The tightening ratio after centralization is defined as R = α - 0.55.

[0061] A cubic polynomial response surface model incorporating dummy and continuous variables is established, and its expression is:

[0062]

[0063] In the formula, A is the vibration response amplitude at the tool tip, and β0 to β9 are undetermined coefficients.

[0064] The response surface model was fitted with least squares using 33 sets of data from the sample dataset to obtain the estimated values ​​of each undetermined coefficient.

[0065] The specific response surface model obtained by fitting is as follows:

[0066]

[0067] The fitting accuracy of the response surface model was evaluated. The coefficient of determination R² = 0.9995, and the root mean square error RMSE = 2.44 × 10⁻⁶. -4 This indicates that the model fits very well. Leave-one-out cross-validation was used, with a cross-validation determination coefficient R²-CV = 0.9987 and a root mean square error RMSE-CV = 3.75 × 10⁻⁶. -4 This indicates that the model has good generalization ability and no overfitting phenomenon has occurred.

[0068] The fitted response surface three-dimensional surface plot and contour plot are as follows: Figure 3 As shown in the figure, as the tightening sequence changes from sequential tightening to cross tightening and then to crisscross tightening, the acceleration response amplitude A shows a decreasing trend; when the pre-tightening ratio α in the first step increases from 30% to 80%, the acceleration response amplitude A shows a trend of first decreasing and then increasing, indicating a clear optimal point.

[0069] Step four specifically includes the following steps:

[0070] An assembly process optimization model is established. The optimization objective is to minimize the acceleration response amplitude A at the tool tip. The optimization variables are dummy variables D1, D2, and the first-step preload ratio α. The constraints include: D1, D2 ∈ {0, 1}, D1 × D2 = 0 to ensure that the three tightening sequences are mutually exclusive in actual operation, and 0.3 ≤ α ≤ 0.8.

[0071] The above optimization model is solved using a genetic algorithm. The genetic algorithm process is as follows: Figure 4 As shown. The genetic algorithm's parameters are set as follows: crossover probability Pc = 0.8, mutation probability Pm = 0.15. The algorithm starts from a randomly generated initial population, calculates the fitness of each individual, and iteratively evolves through selection, crossover, and mutation operations until the convergence condition is met, at which point the optimal individual is output.

[0072] Decoding the optimal individual component yields the optimal assembly process parameters. The optimal tightening sequence is cross-tightening S3; the optimal first-step preload ratio α = 64.3%; and the corresponding predicted value of the tool tip acceleration response amplitude is 2.542058 m / s².

[0073] The rationality of the above optimal result can be analyzed from a mechanistic perspective: First, the cross-tightening, which is applied from the inside out and diagonally alternately, can effectively avoid the unidirectional accumulation of assembly errors and macroscopic warping deformation, making the contact of the mating surfaces more uniform and the overall contact stiffness higher; Second, α=64.3% achieves the optimal balance between too large and too small. If the first-step pre-tightening force is too large (α is too high), the micro-protrusions on the mating surfaces are prone to plastic crushing, leading to stress concentration. If the proportion is too small (α is too low), good pre-contact cannot be formed, and elastic interaction is aggravated.

[0074] Step five specifically includes the following steps:

[0075] Substituting the optimal assembly process parameters obtained in step four—the cross-tightening S3 fit with the first-step preload ratio α = 64.3%—into the overall machine dynamics model previously established by the research group in step two, the acceleration response at the tool tip was re-simulated and calculated, yielding a simulation value of 2.542006 m / s². Comparing this simulation value with the optimal target value of 2.542058 m / s² predicted by the response surface model in step three, the error between the two is only 0.002%, verifying that the response surface model has extremely high prediction accuracy.

[0076] Multiple control schemes were set up to compare and verify similar processes. The control schemes included: changing the tightening sequence (such as cross-tightening S2 or sequential tightening S1) while keeping α=64.3% unchanged, and keeping cross-tightening S3 but changing the α value (such as 60.3% or 68.3%), etc.

[0077] Calculate the tip acceleration response of each control scheme separately, and plot the time-domain response curve and frequency-domain response curve as follows: Figure 5 As shown in Figure (a) and Figure (b).

[0078] from Figure 5 As can be seen from Figures (a) and (b), the optimal assembly process scheme—cross-tightening S3 fit α=64.3%—has a lower time-domain response amplitude of the tool tip acceleration than all the control schemes, and the resonance peak value corresponding to the first three natural frequencies in the frequency domain response is also lower than all the control schemes.

[0079] In summary, the optimal assembly process can effectively reduce the vibration of the tool tip, thereby improving the overall dynamic performance of the machine. This verifies the effectiveness and engineering applicability of the bolt assembly process optimization method based on the fused dummy variable response surface proposed in this invention.

Claims

1. A bolt assembly process optimization method based on fused dummy variable response surface, characterized in that, include: Determine the optimization variables and their value ranges for the weak joint surfaces. The optimization variables include the tightening sequence and the first-step pre-tightening ratio α in the two-step tightening method. The tightening sequence is a discrete variable, and the first-step pre-tightening ratio α is a continuous variable. Design a full factorial test scheme. Based on the full factorial test scheme, the assembly process parameters of each group were simulated and calculated to obtain the vibration response amplitude of the tool tip point of each group as the optimization target. Using the tightening sequence and the first-step pre-tightening ratio α as input variables, and the vibration response amplitude at the blade tip as the output variable, a response surface model is established using a fusion dummy variable encoding method. Binary dummy variables are introduced for discrete variables, and continuous variables are centered to establish a polynomial response surface model containing dummy variables and continuous variables. With the minimum vibration response amplitude at the tool tip as the optimization objective, a genetic algorithm is used to globally optimize the response surface model to obtain the optimal assembly process parameters. The optimal assembly process parameters were substituted into the original simulation model for verification calculation, and the tool tip response was compared with similar assembly process schemes to verify the effectiveness of the optimization method.

2. The method according to claim 1, characterized in that, The method of establishing a response surface model using fused dummy variable encoding includes: Select one tightening scheme as a reference group, introduce binary virtual variables D1 and D2 to encode the three tightening schemes, and apply the constraint D1×D2=0 to ensure that the schemes are mutually exclusive. Calculate the mean of α across all experimental schemes. Define the centralized tightening ratio ; Construct a cubic polynomial response surface model that includes dummy variables and continuous variables: ； In the formula, A is the vibration response amplitude at the tool tip, and β0 to β9 are undetermined coefficients.

3. The method according to claim 2, characterized in that, The establishment of the response surface model also includes: The response surface model was fitted using a sample dataset with least squares. The fitting accuracy was evaluated by the coefficient of determination R² and the root mean square error RMSE. The generalization ability of the model was evaluated by leave-one-out cross-validation.

4. The method according to claim 1, characterized in that, The method of using a genetic algorithm to globally optimize the response surface model includes: An optimization model is established with the goal of minimizing the vibration response amplitude at the blade tip, and with the constraints of the binary values ​​of the dummy variables and the range of α as the conditions. Set the crossover probability and mutation probability, and iterate through selection, crossover and mutation operations. After the convergence condition is met, output the optimal individual. Decode the optimal tightening order and the optimal α value.

5. The method according to claim 1, characterized in that, The step of substituting the optimal assembly process parameters into the original simulation model for verification calculation includes: Substitute the optimal assembly process parameters into the simulation model to recalculate the vibration response at the tool tip, and compare it with the predicted value of the response surface model to verify the accuracy of the model. Multiple control schemes with different tightening sequences or α values ​​were set up. The response at the tool tip was calculated and compared with the optimal scheme to verify the effectiveness of the optimization method.

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