Study on the rate of change of thread angle and thread distance considering the interference factor under before and after the dynamic vibration of the fastening system

Precision nuts are widely used in tool spindles and ball screws. Proper locking gives the bearing the right amount of pre-pressure so that the inner balls can roll steadily and inhibit friction and rapid temperature rise. If no pre-pressure is given to the bearing, it will cause the inner ball to slide irregularly and generate extremely high temperatures. In this paper, the precision nuts used are the flank locking precision locked nut and clasp locking precision locked nut for Taguchi method experiments. The control factors are nut type, Fitting clearance, and lubrication oil. Interference factors are surroundings temperature, assembly personnel, and torque wrench. The purpose is to obtain more suitable anti-loosening characteristics by the Taguchi method under the influence of interference factors and to check the thread angle and thread pitch before and after the experiment. The experimental results were analyzed form Taguchi method to find the combination of parameters with the maximum S/N and the variance analysis to obtain the significant factor. Finally, the regression equation of the loosening prevention characteristics was obtained by regression analysis, which shows that the optimized parameters effectively improve the loosening prevention characteristics and explore the interference factor on the loosening prevention characteristics of precision nuts.


Introduction
Precision nuts are mainly used on tool spindles and ball screws.By locking the preload on the bearing, the spindle and ball screw of the machine can run normally and smoothly.Also, it provides good stiffness and indirectly affects the precision of the whole machine operation.Therefore, this study will use the Taguchi method to obtain more suitable anti-loosening characteristics under the influence of the interference factor.Croccolo et al. [1] conducted a failure analysis of bolt joints: a study of the effect of friction coefficient on the moment-preload relationship, which showed that the friction conditions were greatly influenced by surface finish, lubrication, and the number of tightening and loosening cycles.Liu et al. [2] conducted research on the change of friction coefficient during repeated tightening of high-strength bolts.Three cases were considered, namely no lubricant, lubricated thread, and lubricated nut surface.Statistical analysis was carried out by repeating the experiment to study the effect of axial forces on the behaviour of the friction coefficient change.In the absence of lubrication, repeated torsion causes an increase in the coefficient of friction, which leads to a decrease in the axial force.When the lubricant is used, it protects the contact surfaces and the axial force obtained during repeated torsion is more stable.Experimental results show that this method can be used to study the variation of the friction coefficient during repeated locking, and in a study conducted by Zhu et al. [3] to control the preload and estimate the loosening prevention performance of threaded fasteners with an accurate contact model, it was found that the numerical analysis showed that the obtained preload was sensitive to the friction coefficient and the initial preload, and the loosening prevention performance was mainly influenced by the pitch, friction coefficient, and initial preload.Noda et al. [4] conducted a study on the effect of pitch difference between bolt-nut connections on the loosening prevention performance and fatigue life, and the results showed that the pitch difference is very large and the axial force of the bolt becomes worse for a certain tightening moment, but larger values of pitch difference may provide a larger primary torque, thus producing the loosening prevention effect.It was found that considering the proper pitch difference can extend fatigue life.There is a paucity of literature on the effect of interference factors on quality characteristics, as they are usually present to improve the final optimization results.In reality, it becomes an uncontrollable factor, except for the experimental need to design a few levels that can be controlled.In addition, it is a drag on the efficiency of the experiment to investigate the interference factor.In this study, it is assumed that the same type of precision nut has different ambient temperatures depending on different assembly methods, different personnel, different tools, and different working environments.Through Taguchi's analysis, the optimal value of the stability of the precision nut is obtained and the variation of the interference factor is investigated.

Taguchi method
The quality characteristics of this study are flatness and contact surface roughness which mainly affect the important factors of the precision locking nuts.The control factors include the flank locking precision assembly nuts and clasp locking precision locked nuts, the fitting clearance, and the lubricating oil.The interference factors are the ambient temperature before the experiment, the assembler, and the type of torque wrench.The orthogonal array of the internal and external level Table designed by the Taguchi method is L 4 (2 3 ) for the experiment.

Quality loss
The purpose of Taguchi's quality loss function is to quantify and specify the quality so that the relationship between quality loss and quality characteristics can be more easily interpreted.The geometric properties of the precision nuts were studied under different experimental variables and interference factors.The smaller the change in flatness and contact surface roughness during the use of the precision nut, the better it is called smaller-the-better (STB).
where L(y) is the quality loss, y represents the quality characteristic, and k is the quality loss coefficient.

Signal to Noise Ratio (S/N)
The objective of this study is to achieve a small change in flatness and contact surface roughness.Therefore, the quality characteristics are all expected to be small.Thus, the formula of S/N of the desired small characteristic is chosen in the calculation.The calculation is as follows

ANOVA
The experimental factors of the Taguchi method were selected empirically.The S/N calculation of the Taguchi method is used to find the best parameters of the quality characteristics.To find the best combination of parameters and the influence relationship, the use of analysis of variance (ANOVA) is of great importance.The purpose of ANOVA is twofold: the first is the analysis of variance to evaluate the experimental error, and the second is the Significance test.The sum of squares is used to evaluate the amount of deviation from the mean effect of the whole experiment, the mean effect of all control factors, and the analysis of variance features a method to verify whether three or more parent means are equal or whether the control factors have an effect on the variables.Relevant information is shown in Table 1.The standard deviation (S) confidence level of the experimental data shall reach 95%.

Regression analysis
Regression analysis is a research method based on statistical theory.It is used to investigate the relationship between process parameters and quality characteristics and to develop a regression model.Which is finally expressed as a mathematical relationship.Regression analysis investigates the relationship between the independent variables and the dependent variables to predict the value of the dependent variables.The regression model is also used to predict the value of the dependent variable in the case of different independent variables.The regression model of quality characteristics developed in this experiment does not consider the interactions among the factors.
where Y is the response variable (quality characteristic).β 0 and β i are both regression coefficients.i=A, B, and C are control factors.To evaluate the suitability of the regression model after it is built, the size of the coefficient of determination (R 2 ) can be used to evaluate the regression model constructed by the independent variables.The coefficient of determination is between 0 and 1.The closer the R 2 is to 1, the better the explanatory power of the regression model.The calculation results are as follows where SSTotal is the total sum of squares; SSR is the regression sum of squares.

Rate of change to thread angler and rate of change to pitch
In the results and discussion, we will mention the rate of change to thread angle and rate of change to pitch, which is found as follows where θ before is the thread angle (degree) before the experiment and θ after is the thread angle(degree) where d before is the average of the total sum of the first three pitches on the screws before the experiment (mm), d after is the average of the total sum of the first three pitches on the screws after the experiment (mm).

Flank locking precision assembly nut
The experimental sample used in this study is an M30x1.5Pflank locking 30° angle precision assembled nut (Figure 1).The screw nut is made of SCM 440, the hardness of which is HRC 28~32.
The accuracy of the inner thread is precision turned according to ISO 965-3 [5] tolerance class 4H.The set screw is hexagonal locking.The material is CrMo steel SCM 435, hardness HRC 45~53.

Clasp locking precision locked nut
In this study, a clasp locking precision locked nut was used (Figure 2).The screw nut body is made of CrMo alloy steel (SCM 440) with a hardness of HRC 28~32.The accuracy of the inner thread is made by precision turning and grinding according to ISO 965-3 [5], tolerance class 4H.The fastening screw is made of Cr-Mo alloy steel (SCM 435) with a hardness of HRC 45~53, by ISO 898-1 [7].

Standard bolts
The bolts used in this experiment are designed based on commercially available spindles and ball screws.It is mainly used to simulate the fit of the ball screw, precision spindle, and nut.The material is SCM 21, a chromium-molybdenum alloy steel with hardness between HRC 55~58.The accuracy of the outer thread is ISO 965-3 [5].Tolerance class 4h precision grinding process.

Angular contact ball bearings
The bearing used in this study is the Swedish brand SKF 7206 BEP for the experiment.Because of the contact angle of the angled ball bearing, it can withstand both radial load and large unidirectional axial load.The model used in this experiment is 7206 BEP with a contact angle of 40° and a bore diameter of 30 mm, a basic dynamic load of 22.5 kN, a basic static load of 14.3 kN, and a maximum speed of 13,000 rpm with lubricating oil, and a maximum axial force of 29.6 kN.

Contour roughness tester
Japan Mitutoyo formtracer SV-C3200/SV-C4500, measurement of surface roughness and profile shape of the integrated measuring machine.SV-C3200 for surface roughness measurement.SV-C4500 for measuring the profile shape.Surface roughness measurement specifications: the maximum X-axis measurement range is 100 mm, the maximum Z-axis measurement range is 80 μm, the Z-axis resolution: 0.001 μm, measurement speed is 0.02~5 mm/s.Profile measurement specifications: the maximum X-axis measurement range is 100 mm, the maximum Z-axis range is 60 mm, the X-axis resolution is 0.05 μm, the Z-axis resolution is 0.02 μm, and X Axis tilt angle of 45°.The measurement speed is 0.02~5 mm/s.

Vertical dynamic percussion tester
The realistic dynamic knocking test machine was developed by our laboratory (Figure 3).The operation of the spindle is simulated to investigate the anti-loosening characteristics of the precision nut under dynamic conditions.The degree of change in the recession of the nut's shaft force ratio is transmitted to the computer screen through the axial force sensor, and the experimental data can be output for analysis after the experiment.The nominal diameter of the available nuts is between M20 ~ M50.The maximum sensed axial force is 50 kN, the maximum spindle speed is 3000 rpm, and the maximum percussion frequency is 5 Hz.

Experimental process
The experimental samples used in this study were flank locking precision assembly nuts and clasp locking precision locked nuts, which are commonly used in the industry, and the experiments were carried out by a realistic dynamic knocking test machine in our laboratory with Taguchi method design.The control factors are: nut type, fitting clearance, and lubrication oil (Table 2).The interference factors were: ambient temperature, assembly personnel, and torque wrench (Table 3).The effects of control factors and interference factors on the quality characteristics of thread angle and thread pitch were investigated.The original setting combinations of control factors for the Taguchi method in this study were A1 (flank locking), B2 (fitting clearance 0.987 mm), and C1 (Li8002EP).
The original setting combinations of the interference factor are a1 (ambient temperature 20 degrees), b1 (assembler A), and c1 (digital torque wrench).

Assembly nuts and assembly pressure
This experiment uses flank locking precision assembly nuts and clasp locking precision locked nuts of specification M30×1.5P. 5 times of pre-pressing is done by precision digital torque wrench and special sleeve with ISO 2320 and ISO 16047, and then the sixth time is torqued to the desired torque value.The torque applied in the 1st to 5th preloads is 1.5 times the torque value to be assembled.This function is mainly used to eliminate the backlash between the components of the precision fastening system and must be removed and cleaned after each pre-press to remove the iron filings or fine debris generated between the intermediate surfaces during the pre-pressing process.In addition, a preset torque wrench should be used when locking the tightening screw torque.The torque is divided three times according to the rated torque (if the torque value to be assembled is 6 N-m, it is divided into 2 N-m, 4 N-m, and 6 N-m to lock the torque.The assembly sequence is shown in Figure 4 and Figure 5 to tighten the screw no.1→2→3→3→2→1→1→2→3, 1→3→2→4→4→2→3→1→1→3→2→4, respectively, and each time the torque is applied, the torque should be stopped until the preset torque wrench makes a click.This is applied three times until the target torque value is reached.position number.

Thread angle
To understand the effect of each combination of control factors on the thread angle.The thread angle was measured before the experiment and then measured after the experiment.The calculation was done by averaging three teeth before the thread angle of each product.The crest angle was analyzed before and after the experiment.The experimental results of various products were substituted into the S/N formula of smaller-the-better characteristics to find out the S/N and coefficient of variation of each combination of samples (e.g.Table 4 and Table 5).And organize the results into quality characteristic factor response table is shown in Table 6, S/N factor response diagram is shown in Figure 6.S/N is represents the ratio of signal to noise.Therefore, the combination of the maximum value of each factor in the S/N factor response table and response graph is the best parameter level.The best combination of the parameters of the S/N factor response table and the response graph is A1, B1, and C2, i.e., the type of nut is an axially threaded precision nut, the fitting clearance between bolt and nut is 0.528 mm, and the lubricating oil is Lc252.The minimum thread angle before and after the experiment can be obtained under such experimental conditions.Next, the ANOVA analysis table (Table 7) was used to find out the significant factors of thread angle change after the dynamic test.We set the confidence level above 90% as the significant factor.From the analysis, we can see that all of them are greater than 90%, which means that the three control factors are all significant factors in this experiment.

Thread angle benefit analysis
All three factors in this test are significant but can be known by the quality characteristics response table in Table 6.The degree of the fitting clearance is greater than the nut type than the lubricating oil.From Figure 7, Table 8 and combined with the above analysis, the rate of change of thread angle of the T4 combination is only 0.52% compared with the T1 combination, which has a higher negative impact on the thread angle.Due to the size of the fit pitch, the distribution and magnitude of the IOP Publishing doi:10.1088/1742-6596/2631/1/0120147 contact stress between the precision nut and the bolt changes, resulting in an increase in the sliding distance along the thread surface in the radial direction, causing wear on the thread surface.From this, it is clear that the fitting clearance has a greater effect on the variation of the thread angle.

Thread angle regression analysis
The multivariate multiple regressions were analyzed by SPSS, and the primary coefficients in the equations were removed because they could not be estimated after the SPSS analysis.The functional relationship between the target characteristics and the control factors were obtained.Then, the equations for the properties of the control factor and the axial force were established.The coefficient of Determination(R  , B, and C) is the variables of design factor.

Pitch
To understand the effect of each combination of control factors on the pitch.Pitch measurements were performed before the experiment and again after the experiment.The calculation was done by the average of the total sum of three pitches before each product and calculating the rate of change before and after the experiment.After the dynamic test, the data were analyzed by Taguchi method, and the experimental results of various products were substituted into the S/N formula of smaller-the-better characteristics to find out the S/N ratio and the coefficient of variation of each combination of samples (e.g., Table 9 and Table 10).And organize the results into quality characteristic factor response table is shown in Table 11, S/N factor response diagram is shown in Fig. 8. S/N is represents the ratio of signal to noise.Therefore, the combination of the maximum value of each factor in the S/N factor response table and response graph is the best parameter level.The best combination of A1, B2, and C1 for the pitch is obtained from the S/N factor response table and the response graph, i.e., the nut type is flank locking precision assembly nut, the fitting clearance between bolt and nut is 0.987 mm, and the lubrication oil is Li802EP.The minimum pitch variation rate can be obtained under such experimental conditions.The ANOVA analysis table (Table 12) was used to find out the significant factor of contact surface roughness after the dynamic test.We set the confidence level above 90% as the significant factor.From the analysis, we can see that all of them are less than 90%, which means that the three control factors are less significant factors in this experiment.

Pitch analysis
Although the three factors of this test are not significant factors.However, the quality characteristics of the response table in Table 11 show that the degree of influence of the mating gap is greater than that of the nut type than that of the lubricating oil.From Figure 9, Table 12 and combined with the above analysis, the T3 combination pitch change rate is 0.21% compared to the T2 combination has a lower negative impact on pitch change.The fit clearance has a certain influence on the pitch change of the thread.The pitch benefits analysis is shown in Table 13.

Pitch regression analysis
The multivariate multiple regressions were analyzed by SPSS software, and the primary coefficients in the equations were removed because they could not be estimated after the SPSS analysis.The functional relationship between the target characteristic and the control factor is obtained, and then the equation of the control factor and the axial force characteristics is established.The regression coefficient constant is -39.00,-41.502 is for XA, 40.44 is for XB, 44.05 is for XC, and the coefficient of determination (R 2 ) is 1.It can be proved that this regression model has a good ability to explain the rate of change of the pitch of the thread.13.Pitch benefits analysis.

Conclusion
(1) The dynamic test with Taguchi's experimental design shows that the selected control factors (nut type, fitting clearance, and lubricating oil) have a significant impact on the change of thread flank clamping angle and thread pitch.(2) The dynamic test results of thread angle and thread pitch show that the thread surface of the precision nut will be deformed by abrasion after the dynamic knocking test and pre-pressing, which will affect the change of thread angle and thread pitch.However, since this experiment was conducted by heavy pre-pressing, the wear of the thread flank was more intense.As a result, the Thread angle changes significantly compared to the thread pitch.(3) The dynamic test result shows that the rear thread surface is the main stress position in the fastening system.In the fit, the gap is too large, because the nut fastener action, will be accompanied by vibration slip and other phenomena.And cause damage to the thread, and the fitting clearance is small to avoid the rear thread surface from the slide displacement becoming large and affecting the stability of the fastening system.

Figure 6 .
Figure 6.S/N factor response diagram.Figure 7. Comparison of thread angle experimental results.

Figure 7 .
Figure 6.S/N factor response diagram.Figure 7. Comparison of thread angle experimental results.

Figure 8 .
Figure 8. S/N factor response diagram.Figure 9. Comparison of pitch experimental results.

Figure 9 .
Figure 8. S/N factor response diagram.Figure 9. Comparison of pitch experimental results.

Table 2 .
Control factor level.

Table 4 .
Orthogonal array of the thread angle after the dynamic test.

Table 5 .
Data of thread angle experiment after the dynamic test.

Table 6 .
Quality characteristic factor response.

Table 8 .
Thread angle benefits analysis.
2) is 1.It can be proved that this regression model has a good ability to explain the rate of change of thread angle.The prediction equation for the rate of change of thread angle is as

Table 9 .
Orthogonal array of the pitch after the dynamic test.

Table 10 .
Data of pitch experiment after the dynamic test.

Table 11 .
Quality characteristic factor response.