Research on TA2 dry/wet milling force–vibration characteristics and vibration prediction model

Abstract

To investigate the practical problems such as large cutting force, strong vibration and difficult to control the surface machining quality during TA2 milling process, a dry/wet milling force and milling vibration signal acquisition system was set up, and a three-factor, four-level analytical causality test program was designed to carry out the milling test on TA2, and milling force and milling vibration signals were collected in the process of the test. Force and vibration characteristics under dry and wet milling conditions were extracted and comparatively analyzed, to investigate the influence of dry and wet milling on the cutting performance of TA2. The Spearman correlation coefficients of the milling force and milling vibration signals were calculated to study the correlation between the two signals; the milling process parameters and milling force were used as the continuous input factors to establish the multivariate regression prediction model of milling vibration, and the coupling relationship between the three was further investigated. The results show that: (1) there is a strong correlation between the milling force and vibration signals; (2) the TA2 can obtain better milling performance when the feed rate is v_f = 8, 10 mm/min for dry milling and v_f = 12, 14 mm/min for wet milling; (3) Under dry and wet working conditions, the correlation coefficient R value of the milling vibration prediction model based on the combination of three-way milling force is 0.9184 and 0.9665, respectively, which is higher than that of the prediction model of the combination of the milling process parameters, indicating that the use of the milling force signals can be used to predict the milling vibration in a better way, and the research can provide theoretical references for the actual machining and non-contact measurement of milling vibration in TA2.

Keywords

TA2 pure titanium dry and wet milling milling vibration milling force prediction model

Introduction

Compared with traditional metals like steel and aluminum alloys, titanium alloys offer superior specific strength, excellent low-temperature toughness, and high corrosion resistance, making them the preferred structural materials for advanced applications in aerospace and biomedical engineering.^1,2 Milling is a commonly used conventional machining technique for titanium alloys, yet it presents significant challenges such as high cutting forces, severe vibrations, and difficulties in controlling machining quality. Therefore, the investigation and prediction of the cutting force-vibration characteristics of titanium alloys under various process parameters have become crucial research topics.

In actual machining, the force characteristics of titanium alloy workpieces directly impact the final quality, with process instability leading to machine tool vibrations and even chattering.^3,4 Consequently, the study of cutting force-vibration characteristics has become a focal point among industry researchers. Huang et al.⁵ investigated the vibration process of milling Ti-6Al-4V with a variable pitch end mill and found that the milling force increased significantly when chatter was generated. Zhao et al.⁶ examined the impact of various parameters on the milling force in longitudinal torsional ultrasonic vibratory milling (LTUM) of titanium alloys. They found that milling force positively correlates with cutting speed, depth, and feed per tooth, while inversely with ultrasonic amplitude and tool helix angle. Zhao et al.⁷ investigated how clamping height, feed rate, and radial depth affect the vibration of thin-walled workpieces through single-factor experiments, discovering that feed rate significantly influences vibration, and reducing it can diminish workpiece vibration. Ercetin et al.⁸ used a micro dynamometer to precisely capture cutting force signals, evaluating them by peak and trough analysis. They noted that cutting forces rise with increased feed rate and cutting depth, but decrease as cutting speed increases. Shao et al.⁹ explored the effects of noise, vibration, and force on machining efficiency and quality using an improved particle swarm optimization algorithm, determining that optimal milling conditions under constraints are achieved at a speed of 24.9588 m/min, a feed rate of 9.9324 mm/min, and a depth of 1 mm. Yue et al.¹⁰ analyzed the impact of helix angle on milling force using one-way and orthogonal methods, discovering that an increase in helix angle results in a rise in maximum axial force, a gradual decrease in maximum tangential force, and no significant change in maximum radial force.

In the study of titanium alloy milling processes, the contrasting approaches of dry and wet milling have become key focal points in both industrial and academic research, largely due to their implications for sustainability, tool wear, and cutting force requirements. Numerous scholars have explored these approaches from multiple perspectives, thus establishing a robust theoretical foundation for process optimization. For instance, Al-Falahi et al.¹¹ investigated tool wear patterns during the milling of Hastelloy-C276 under dry and wet conditions, finding reduced adhesive wear in dry machining. Li et al.¹² developed a predictive model for surface roughness under dry and wet milling conditions using the least squares method, noting stronger correlations between roughness, milling force, and milling vibrations in wet milling. Rajeev et al.¹³ assessed the machinability of duplex stainless steel in terms of surface roughness under both dry and wet machining, finding that dry machining produced superior surface quality. Kara et al.¹⁴ studied the unit process energy consumption in dry and wet milling, discovering that dry milling is more energy-efficient than wet milling for the same material removal rate.

Establishing a predictive model is fundamental to optimizing titanium alloy milling parameters; a well-constructed model significantly reduces prediction errors and enhances accuracy. Fuht et al.¹⁵ analyzed data and constructed a prediction model using the Taguchi method and response surface methodology, finding a strong correlation between the experimental and predicted outcomes. Moufki et al.¹⁶ developed a three-dimensional cutting force prediction model for end milling, offering reliable predictions for the typically segmented machining of titanium alloys. Scholars have analyzed the impact of process parameters on titanium alloy milling to develop a prediction model. Sharif et al.¹⁷ created a surface roughness prediction model using factorial experimental design and response surface methodology. Lu et al.¹⁸ established a cutting edge radial runout prediction model through micro-end milling experiments. Wang et al.¹⁹ developed a hybrid prediction model combining finite element analysis with experimental data. Liu et al.²⁰ developed a surface roughness prediction model using an enhanced particle swarm optimization algorithm, which was found to build two orders of magnitude faster and predict with greater accuracy than the backpropagation (BP) model.

In summary, researchers worldwide have studied the machining performance, tool vibration, and cutting deformation of titanium alloy processing through theoretical analyses, numerical simulations, and experimental validation. Researchers have used methods like nonlinear regression to create prediction models that analyze the relationships between milling parameters and forces or vibrations, optimizing milling processes and enhancing model accuracy. However, studies on how dry versus wet milling conditions affect force-vibration characteristics and their inherent connections are scarce. Therefore, in this study a dry and wet milling force-vibration signal collection system is set up to perform experiments, gather data, and conduct comparative analysis of the extracted features. Subsequently, we will analyze the correlation between milling force-vibration signals using correlation analysis. Finally, a multivariate regression model based on the least squares method will be established to further explore the inherent relationships within the data.

Test system and test scheme

Test system

In order to analyze the impact of dry and wet milling conditions on machining outcomes, in this study we combined milling process parameters with vibration and force signals closely related to the machining process to construct a TA2 dry and wet milling force-vibration signal acquisition system, as depicted in Figure 1. This system includes a dry and wet milling system, a milling force measurement system, and a milling vibration measurement system. The dry and wet milling system comprises an XK714 vertical CNC milling machine, a 40 × 100 × 100 mm TA2 pure titanium plate specimen, a GM-4E-D10.0 carbide end milling cutter, and Syntilo 9930c fully synthetic cutting fluid.; The milling force measurement system consists of a piezoelectric triaxial force sensor and an industrial computer. The milling vibration measurement system includes a YD-21 piezoelectric triaxial acceleration sensor, a YE5852 charge amplifier, a high-speed data acquisition device, and WS-AV vibration measurement and analysis software.

Figure 1.

Wet and dry milling force-vibration signal acquisition system.

Test scheme and test steps

To obtain the milling force-vibration characteristics of TA2 under various milling parameter combinations, spindle speed (n), feed rate (v_f), and milling depth (a_p) – as the three key process parameters that directly govern the metal removal mechanics and dynamics during milling – were selected as experimental factors. Additionally, to comprehensively cover actual machining parameters, in this study we utilizes factorial design theory to set four distinct levels for each test factor, based on workpiece material properties, tool geometry, and test conditions. The three-factor, four-level factorial experiment design is outlined in Table 1.

Table 1.

Three-factor, four-level factorial test scheme.

v_f = 8 mm/min			v_f = 10 mm/min			v_f = 12 mm/min			v_f = 14 mm/min
No.	n/rpm	a_p/mm	No.	n/rpm	a_p/mm	No.	n/rpm	a_p/mm	No.	n/rpm	a_p/mm
1	700	4	17	700	4	33	700	4	49	700	4
2	850	4	18	850	4	34	850	4	50	850	4
3	1000	4	19	1000	4	35	1000	4	51	1000	4
4	1150	4	20	1150	4	36	1150	4	52	1150	4
5	700	3	21	700	3	37	700	3	53	700	3
6	850	3	22	850	3	38	850	3	54	850	3
7	1000	3	23	1000	3	39	1000	3	55	1000	3
8	1150	3	24	1150	3	40	1150	3	56	1150	3
9	700	2	25	700	2	41	700	2	57	700	2
10	850	2	26	850	2	42	850	2	58	850	2
11	1000	2	27	1000	2	43	1000	2	59	1000	2
12	1150	2	28	1150	2	44	1150	2	60	1150	2
13	700	1	29	700	1	45	700	1	61	700	1
14	850	1	30	850	1	46	850	1	62	850	1
15	1000	1	31	1000	1	47	1000	1	63	1000	1
16	1150	1	32	1150	1	48	1150	1	64	1150	1

Once all equipment is ready and calibrated, proceed with the test as follows.

(1) Set up the wet and dry milling system. First, to ensure the smoothness and dimensional accuracy of the specimen do not impact the test results, uniformly sand all specimens. Secure the TA2 specimen using the milling machine fixture, and install the milling cutter via the cutter holder. Lastly, fill the pourer with cutting fluid for a test cut, adjusting the position of the pouring tube to ensure precise delivery to the cutting zone.

(2) Connect the milling vibration measurement system. Attach the piezoelectric triaxial acceleration sensor to the milling machine fixture using a strong magnet. Connect the three channels of the sensor to three charge amplifiers via special transmission cables. Transmit vibration signals to the high-speed data acquisition instrument, which summarizes the data and transfers it via USB cable to a computer equipped with WS-AV acoustic vibration measurement and analysis software.

(3) Install the milling force measurement system. Firmly mount the triaxial force transducer under the CNC milling machine table and connect it to the charge amplifier inside the three-way force measuring instrument using a specialized cable, then activate the industrial control machine.

(4) Execute the milling test. Follow the three-factor, four-level factorial design outlined in Table 1 to program and operate the CNC milling machine. Collect milling force and vibration signals synchronously during the process. Upon completing a test set, remove the workpiece and label it with the test serial number. After all tests are complete, save the data and turn off the power.

Test results and analyses

Test results

Utilizing the established test system, dry and wet milling tests were conducted according to the experimental plan in Table 1. To mitigate the impact of operational errors and unforeseen issues on the results, each test with the same process parameters (spindle speed n, feed rate v_f, and milling depth a_p) was performed twice, totaling 256 groups. Upon completion of the tests, root mean square (RMS) values for triaxial milling force and triaxial vibration acceleration were extracted from the sampled force-vibration time-domain signals for further analysis. Extracted test data are presented in Table 2.

Table 2.

TA2 wet and dry milling section test data.

Dry milling							Wet milling
	Tri-axial vibration acceleration/m·s⁻²			Tri-axial milling force/N				Tri-axial vibration acceleration/m·s⁻²			Tri-axial milling force/N
NO.	a _RMSx	a _RMSy	a _RMSz	F _RMSx	F _RMSy	F _RMSz	NO.	a _RMSx	a _RMSy	a _RMSz	F _RMSx	F _RMSy	F _RMSz
1	0.173	0.040	0.076	394.912	230.630	48.993	1	0.156	0.053	0.096	349.625	271.359	53.522
2	0.142	0.027	0.061	291.353	193.826	37.720	2	0.114	0.030	0.071	259.010	196.154	39.376
3	0.129	0.032	0.056	285.100	181.110	35.087	3	0.096	0.024	0.056	224.381	157.373	34.325
4	0.087	0.022	0.035	181.485	136.638	28.886	4	0.083	0.019	0.048	191.394	129.946	33.823
5	0.219	0.044	0.097	495.832	246.797	61.454	5	0.131	0.042	0.083	307.062	169.987	44.661
6	0.191	0.038	0.084	425.870	231.660	52.089	6	0.116	0.025	0.054	286.422	195.428	37.313
7	0.204	0.048	0.088	478.095	257.887	54.709	7	0.108	0.023	0.050	265.369	170.375	35.147
8	0.129	0.033	0.052	275.600	194.062	34.239	8	0.065	0.012	0.030	154.840	104.037	30.325
9	0.006	0.004	0.003	24.505	31.874	10.858	9	0.140	0.032	0.064	333.884	148.708	41.526
10	0.006	0.004	0.003	22.172	28.550	9.938	10	0.043	0.013	0.023	102.634	88.823	21.793
11	0.006	0.004	0.003	20.506	26.269	9.224	11	0.029	0.007	0.015	69.853	72.555	18.984
12	0.008	0.004	0.003	19.808	24.588	8.684	12	0.040	0.008	0.020	101.352	78.754	21.126
13	0.006	0.004	0.003	15.753	19.443	6.365	13	0.005	0.004	0.003	19.390	24.659	8.757
14	0.005	0.003	0.003	13.308	15.706	5.008	14	0.005	0.004	0.003	15.455	19.278	6.690
15	0.006	0.003	0.003	10.040	13.152	3.840	15	0.005	0.003	0.003	30.838	34.531	23.112
16	0.006	0.003	0.003	8.141	10.427	3.640	16	0.005	0.003	0.003	15.728	19.363	6.614
……							……
49	0.146	0.050	0.066	306.006	293.905	51.634	49	0.107	0.034	0.062	257.964	175.626	34.712
50	0.130	0.035	0.059	295.446	231.443	39.153	50	0.074	0.022	0.043	172.408	85.991	22.911
51	0.137	0.062	0.052	259.012	192.528	32.730	51	0.077	0.016	0.034	179.233	98.800	22.085
52	0.154	0.076	0.060	289.103	208.505	38.006	52	0.073	0.018	0.031	165.795	80.151	21.100
53	0.158	0.054	0.073	366.079	283.413	48.372	53	0.072	0.012	0.032	175.644	71.579	21.344
54	0.186	0.061	0.094	385.951	376.137	70.205	54	0.056	0.009	0.026	139.198	34.710	16.832
55	0.165	0.056	0.092	356.849	358.756	62.613	55	0.046	0.018	0.007	30.523	27.594	6.846
56	0.180	0.077	0.108	388.960	412.314	74.305	56	0.036	0.013	0.010	44.062	17.965	8.538
57	0.222	0.091	0.134	504.981	518.565	89.662	57	0.004	0.003	0.003	1.964	1.770	3.299
58	0.114	0.039	0.087	233.240	201.512	43.762	58	0.004	0.003	0.003	2.032	1.935	3.287
59	0.109	0.043	0.076	212.439	145.046	34.784	59	0.050	0.024	0.019	96.205	67.973	13.096
60	0.141	0.062	0.099	267.307	232.218	49.932	60	0.100	0.050	0.021	87.312	32.926	14.057
61	0.154	0.055	0.112	337.894	286.924	53.593	61	0.028	0.008	0.014	65.001	40.416	10.732
62	0.087	0.025	0.049	54.315	217.833	173.162	62	0.026	0.009	0.012	58.634	26.389	8.460
63	0.076	0.034	0.039	22.292	161.156	93.777	63	0.024	0.011	0.007	26.646	22.981	6.309
64	0.091	0.052	0.052	27.660	186.355	116.531	64	0.032	0.013	0.013	52.319	19.404	9.987

Comparison of dry and wet milling force-vibration characteristics

To facilitate a visual comparison of dry and wet milling force-vibration characteristics, integrated milling force F_RMS and integrated milling vibration acceleration a_RMS were computed using equations (1) and (2). These values served as the eigenvalues for comparative analysis, with results displayed in Figure 2.

F_{RMS} = \sqrt{{F_{RMSx}}^{2} + {F_{RMSy}}^{2} + {F_{RMSz}}^{2}}

(1)

a_{RMS} = \sqrt{{a_{RMSx}}^{2} + {a_{RMSy}}^{2} + {a_{RMSz}}^{2}}

(2)

Figure 2.

F_RMS versus a_RMS comparison curve plot.

As depicted in Figure 2, at v_f = 12 and 14 mm/min, the F_RMS and a_RMS values for wet milling are generally lower than those for dry milling. Conversely, at v_f = 8 and 10 mm/min, dry milling yields lower F_RMS and a_RMS values compared to wet milling, although the differences are less pronounced. Therefore, employing dry milling at v_f = 12 and 14 mm/min and wet milling at v_f = 8 and 10 mm/min can optimize the milling performance of TA2 and reduce machining costs.

Considering the use of Syntilo 9930c fully synthetic water-based cutting fluid in this test, which exhibits excellent lubrication and chip removal properties, it can be inferred that in wet milling, the presence of cutting fluid significantly reduces friction between the tool and workpiece, prevents chip accumulation, and thus decreases vibration, effectively ensuring optimal milling performance for TA2. Additionally, the Syntilo 9930c also functions as a coolant, removing heat from both the tool and the workpiece, thereby reducing their temperature. This cooling may lessen thermal deformation of the tool and workpiece, increase the hardness of the workpiece, and elevate the cutting difficulty, which in turn increases milling force and vibration acceleration, resulting in lower F_RMS and a_RMS in wet milling.

Additionally, a noteworthy observation from comparing Figure 2(a) and (b) is that the trends in F_RMS and a_RMS curves for both dry and wet milling are essentially similar, suggesting a correlation between F_RMS and a_RMS.

Correlation analysis

Selecting the appropriate correlation coefficient is a critical step in correlation analysis, which largely depends on whether the data conforms to a normal distribution. Therefore, this study initially employed IBM SPSS Statistics software to conduct a normality test on the experimental data at a 0.05 confidence level. Given the small sample size of 64 in this experiment, the Shapiro-Wilk test was selected. The results are presented in Table 3, where df represents the number of samples, skewness describes the symmetry of the data distribution, kurtosis characterizes the concentration of data distribution, p-value assesses significance, and Z-score is the ratio of the statistical value to the standard error. Typically, in a normality test with a 0.05 confidence interval, the data is considered to follow a normal distribution if the absolute values of skewness and peak Z-score are less than 1.96, and the p-value exceeds 0.05.

Table 3.

Results of normality test.

Processing methods	Parametric	Skewness			Kurtosis			Shapiro-Wilk test
Processing methods	Parametric	Statistics	Standard error	Z-score	Statistics	Standard error	Z-score	Statistics	df	p-value
Dry milling	a _RMSx	−0.139	0.299	−0.465	−0.823	0.590	−1.395	0.964	64	0.058
	a _RMSy	0.345	0.299	1.154	−0.657	0.590	−1.114	0.960	64	0.038
	a _RMSz	0.291	0.299	0.973	−0.770	0.590	−1.305	0.960	64	0.037
	F _RMSx	0.136	0.299	0.455	−0.815	0.590	−1.381	0.957	64	0.026
	F _RMSy	0.735	0.299	2.458	0.376	0.590	0.637	0.941	64	0.004
	F _RMSz	1.917	0.299	6.411	5.807	0.590	9.842	0.848	64	0.000
Wet milling	a _RMSx	0.713	0.299	2.385	−0.376	0.590	−0.637	0.926	64	0.001
	a _RMSy	1.128	0.299	3.773	0.560	0.590	0.949	0.878	64	0.000
	a _RMSz	1.150	0.299	3.846	0.653	0.590	1.107	0.875	64	0.000
	F _RMSx	0.705	0.299	2.358	−0.522	0.590	−0.885	0.923	64	0.001
	F _RMSy	1.121	0.299	3.749	0.519	0.590	0.880	0.880	64	0.000
	F _RMSz	0.889	0.299	2.973	0.287	0.590	0.486	0.929	64	0.001

As Table 3 illustrates, in dry milling, the absolute values of skewness and kurtosis for a_RMSx are less than 1.96 with a p-value greater than 0.05, indicating a normal distribution. However, for a_RMSy, a_RMSz, and F_RMSx, while the absolute values of skewness and kurtosis Z-score are less than 1.96, the p-value are less than 0.05, suggesting these do not follow a strict normal distribution. For F_RMSy and F_RMSz, the absolute values of skewness Z-score exceed 1.96 with p-value less than 0.05. In wet milling, the absolute values of skewness Z-score for both a_RMS and F_RMS exceed 1.96 with p-value less than 0.05, indicating that the data do not conform to a strict normal distribution. Consequently, this study employs the Spearman correlation coefficient for the correlation analysis, due to the non-normal distribution of most data sets. The formula used is:

r h o = \frac{\sum_{i = 1}^{m} (T_{i} - b a r T) (S_{i} - b a r S)}{\sqrt{\sum_{i = 1}^{m} {(T_{i} - b a r T)}^{2} \cdot \sum_{i = 1}^{m} {(S_{i} - b a r S)}^{2}}}

(3)

In the formula, rho represents the Spearman correlation coefficient, m is the sample size, T_i and S_i are the ranks of the I observation in the two variables respectively, barT and barS are the average ranks of these variables. The results of the Spearman correlation analysis are presented in Table 4.

Table 4.

Results of rho calculation.

Processing method	Parametric	a _RMSx	a _RMSy	a _RMSz	df
Dry milling	F _RMSx	0.933	0.657	0.906	64
	F _RMSy	0.896	0.700	0.946	64
	F _RMSz	0.828	0.694	0.897	64
Wet milling	F _RMSx	0.966	0.849	0.988	64
	F _RMSy	0.896	0.769	0.933	64
	F _RMSz	0.886	0.746	0.919	64

Typically, a rho value between 0.7 and 1 indicates an extremely strong correlation; between 0.3 and 0.7 suggests moderate to weak correlation; and between 0 and 0.3 signifies very weak or no correlation. According to Table 4, the correlation coefficients between a_RMSy and F_RMSx, F_RMSy in dry milling fall within the 0.6–0.7 range, indicating a moderate correlation. The rho values between the remaining a_RMS and F_RMSx, F_RMSy, F_RMSz are within the 0.7–1 range, demonstrating an extremely strong correlation. This further validates the existence of a significant correlation between F_RMS and a_RMS.

Predictive modelling and comparative analysis

To further validate the correlation between milling force and vibration signals, this study utilizes dry milling test data from Table 2 to develop a multiple regression predictive model. Using the response surface methodology, milling process parameters and triaxial milling force eigenvalues are treated as continuous factors, with vibration acceleration (a_RMS values) as the response variable. Sensitivity analyses test the sensitivity of the model to the input parameters and are generally used more often in forecasting in the humanities and social sciences. The parameters selected in our experiments are process parameters and relevant sensor data, which are less in need of sensitivity analysis. In addition, we performed correlation analysis on the selected eigenvalues to further verify the validity of the selected data.

The predictive model, Model-1, incorporates spindle speed (n), feed rate (v_f), and milling depth (a_p) as continuous factors, and is defined as follows:

\begin{array}{l} a_{RMS} = - 0.404 - 0.001298 n + 0.0086 v_{f} + 0.1997 a_{p} + 0.000001 n^{2} - 0.00174 {v_{f}}^{2} \\ - 0.01000 {a_{p}}^{2} + 0.000003 n \cdot v_{f} - 0.000031 n \cdot a_{p} + 0.00799 v_{f} \cdot a_{p} \end{array}

(4)

The predictive model, Model-2, is developed using F_RMSx, F_RMSy, and F_RMSz as continuous factors, as outlined below:

\begin{array}{l} a_{RMS} = - 0.00670 + 0.000443 F_{RMSx} - 0.000436 F_{RMSy} + 0.00368 F_{RMSz} - 0.000001 {F_{RMSx}}^{2} + 0.000001 {F_{RMSy}}^{2} \\ - 0.000004 {F_{RMSz}}^{2} + 0.000001 F_{RMSx} \cdot F_{RMSy} + 0.000004 F_{RMSx} \cdot F_{RMSz} - 0.000012 F_{RMSy} \cdot F_{RMSz} \end{array}

(5)

This study employs the correlation coefficient R to evaluate the goodness of fit of the predictive model, which assesses the accuracy with which the model fits the data. For m samples (x₁, y₁), (x₂, y₂), …, (x_m, y_m), the model predicts (x₁, Y₁), (x₂, Y₂), …, (x_m, Y_m). Then the formula for R is as follows:

R = 1 - \frac{S_{r e s}}{S_{t o t}}

(6)

In the formula, the value of R ranges from 0 to 1, and the closer it is to 1 means that the model fits the data better. S_reg is the sum of the squares of the differences between the sample value y_i and the sample mean

\bar{y}

, and S_tot is the sum of the squares of the residuals of the predicted value Y_i and the true value y_i as shown in equations (7) and (8):

S_{r e g} = \sum_{i}^{m} {(y_{i} - \bar{y})}^{2}

(7)

S_{r e g} = \sum_{i}^{m} {(y_{i} - \bar{y})}^{2}

(8)

Using equation (6), the R-values for Model-1 and Model-2 are calculated to be 0.6122 and 0.9182, respectively, indicating that Model-2 significantly outperforms Model-1 in prediction accuracy. To facilitate a clearer visual comparison, comparison curves of the predicted values from Model-1 and Model-2 against the actual values are illustrated in Figure 3.

Figure 3.

Comparison curve between real and predicted values for dry milling.

As illustrated in Figure 3, there is a low degree of overlap between the predicted and actual values for Model-1, whereas Model-2 shows a high degree of overlap, reinforcing the existence of a correlation between milling force and vibration signals. This suggests that milling force signals provide a more accurate prediction of milling vibrations than cutting parameters alone.

Furthermore, to further substantiate this conclusion, this study applies the same modeling and comparative analysis steps to the wet milling test results presented in Table 2. The model, Model-wet1, constructed with n, v_f and a_p as continuous factors, is outlined as follows:

\begin{array}{l} a_{RMS} = 0.366 - 0.000867 n - 0.0082 v_{f} + 0.1401 a_{p} + 0.000000 n^{2} - 0.000517 {v_{f}}^{2} \\ - 0.00087 {a_{p}}^{2} + 0.000028 n \cdot v_{f} - 0.000052 n \cdot a_{p} + 0.00484 v_{f} \cdot a_{p} \end{array}

(9)

The predictive model, Model-wet2, developed using F_RMSx, F_RMSy, and F_RMSz as continuous factors, is described as follows:

\begin{array}{l} a_{RMS} = 0.00267 + 0.000425 F_{RMSx} - 0.000540 F_{RMSy} + 0.00261 F_{RMSz} - 0.000005 {F_{RMSx}}^{2} - 0.000001 {F_{RMSy}}^{2} \\ - 0.000133 {F_{RMSz}}^{2} + 0.000006 F_{RMSx} \cdot F_{RMSy} + 0.000043 F_{RMSx} \cdot F_{RMSz} - 0.000018 F_{RMSy} \cdot F_{RMSz} \end{array}

(10)

Similar to dry milling, the R-values for Model-wet1 and Model-wet2, calculated using equation (6), are 0.7262 and 0.9665, respectively. Model-wet2 demonstrates significantly higher prediction accuracy than Model-wet1. Subsequently, the predictive performance of Model-1 and Model-2 is illustrated in Figure 4, where Model-wet2 shows a significantly better fit to the actual values compared to Model-wet1, indicating that Model-wet2’s prediction accuracy surpasses that of Model-wet1. This substantiates the paper’s conclusion that the observed correlation between milling force and vibration signals possesses a degree of universality and is not merely coincidental.

Figure 4.

Comparison curve between real and predicted values for wet milling.

In summary, Combining the prediction models based on wet and dry milling data, it can be found that the prediction models based on both vibration and force data have very high accuracy, verifying the stability of the proposed method.

Conclusion

To investigate the influence patterns of dry and wet milling on force-vibration characteristics and the intrinsic relationships of force-vibration signals, in this study we first established a TA2 dry and wet milling force-vibration signal acquisition system and designed a three-factor, four-level factorial experimental scheme. Subsequently, a series of TA2 milling tests were conducted under both dry and wet conditions, where milling force-vibration signals were collected and their relevant eigenvalues were extracted for comparative analysis. Finally, the Spearman correlation coefficient was employed to analyze the correlations among the test data, and multiple regression prediction models of milling vibration were established under both dry and wet conditions using milling process parameters and milling force as continuous factors, followed by model comparisons.

(1) At v_f = 12 and 14 mm/min, the F_RMS and a_RMS values for wet milling were generally lower than those for dry milling; at v_f = 8 and 10 mm/min, dry milling resulted in lower F_RMS and a_RMS compared to wet milling, though the differences were less pronounced. By applying dry milling at v_f = 8 and 10 mm/min, and wet milling at v_f = 12 and 14 mm/min, the milling performance of TA2 can be enhanced, providing guidance for practical machining and reducing economic costs.

(2) The trends in dry and wet milling force-vibration signals were largely consistent, exhibiting a strong correlation. The predictive model of milling vibration, developed from the triaxial milling force signals, demonstrated significantly greater accuracy than that based on cutting parameters, providing a theoretical foundation for the non-contact measurement of TA2 milling vibrations.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: In this paper, the research was sponsored by Jiangsu Natural Science Foundation Project (Grant No. BK20231173), Xuzhou Science and Technology Project (Grant No. KC23054), Key Project of Jiangsu University Students’ Innovation and Entrepreneurship Programme in 2023 (Grant No. 202310320010Z).

ORCID iDs

Xin Wang

Shuncai Li

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