Analysis of Vibration Reduction Mechanism for Variable-pitch End Mills in Hardened Steel

The development of the machining technology is restricted by the problems of serious vibration and low system stability. The variable-pitch mill has certain vibration reduction property due to the unequal pitch angles. In this work, firstly, according to the features of the structure of the variable-pitch end mill, the cutting mechanism is analyzed, and a theoretical model of the dynamic cutting force is developed. Secondly, the cutting force coefficients of the model are determined based on the test of hardened steel for verifying the significance of the theoretical model. Then, the time-frequency characteristics and vibration reduction mechanism of variable-pitch mills are analyzed. The frequency characteristics of different types of pitch angle mills are also explored by the spectrum analysis. Finally, based on the energy and variance methods of the amplitude, the multi-objective optimization of the pitch angle is carried out, where the spectrum lines are found evenly distributed, and the optimal result is 80°-97°-100°-83°. For satisfying the machining requirements, the pitch angle of the variable-pitch end mill is reasonably selected to reduce the amplitude of the forced vibration, which can play an important role in reducing the vibration of the tool and improving its cutting stability.


Introduction
With the increasingly complex structures of products, the requirements of machining accuracy and quality consistency are also being increased continuously, leading to increasing difficulties in machining materials which urgently require high-performance tools. It is difficult to improve the surface quality of a workpiece due to serious vibration. Ma et al. [1] observed that a response surface model based on the acceleration d was more suitable for the prediction and control of the cutting vibration. Xiao et al. [2] proposed a support head of multi-point to adjust the supporting force in realtime. Saini and Yu et al. [3,4] sudied the influence of tool cutting performance on vibration problems for different materials. Zhou et al. [5] presented an approach for systematic singularity analysis of cutting force and vibration in feature extraction. Li et al. [6] developed a single-degree of freedom chatter model by taking the nonlinear hysteretic and exciting force into account. Hu et al. [7,8] analyzed the vibration problem by combining gray correlation degree and light intensity-modulation optical fiber methods. Freyer, et al. [9] combined with piezoelectric transducers to actively detect tools and control vibration. Suyama et al. [10,11] analyzed the tool vibration of steel materials from different processing methods.
In order to reduce the cutting vibration from the point of tool design, Slavicek [12] first proposed the design of the variable-pitch cutter for vibration suppression. Shirase and Tang et al. [13,14] observed that the variable-pitch cutter has better vibration damping performance in comparison to the cutting performance of ordinary cutter. Altintas and Sim et al. [15,16] proposed a model for variablepitch cutting predicting stability in the frequency domain. Budak and Sam et al. [17][18][19] studied the time domain stability of the variable-pitch mill and developed a dynamic model for aluminum alloy.
Olgac et al. [20] explained the phenomenon that the variable-pitch mill can restrain cutting chatter from physical and mathematical perspectives. Sellmeier et al. [21] presented a model of the cutting force and surface machining error of the variable-pitch end mill. Ahmad and Andreas et al. [22,23] explored the stability of variable-pitch cutting and optimized the cutter parameters. Huang et al. [24][25][26] developed another dynamic model of the variable-pitch mill by analyzing the cutting stability for titanium alloy. The superiority of the damping performance of variable-pitch end mills has been widely studied. However, the mechanism reducing the tool vibration still lacks certain analysis in hardened steel materials. By considering the minimization of the vibration amplitude as the objective, the pitch angle is reasonably optimized, so as to improve the cutting stability by suppressing the vibration of the mill due to the cutter angle.
In this paper, the cutting force model of the variable pitch end mill is studied by taking into account the problem of multi-time delay caused by the change in the pitch angle. Considering the properities of the milling hardened steel, the cutting force coefficients are analyzed, and the experimental and simulation results are compared. According to the distribution of different types of pitch angles, the characteristics of the cutting force of the variable pitch end mill are analyzed in the time-frequency domain, and the relationship between the spectrum characteristics of the mill and the mechanism of the forced vibration reduction is established. According to the amplitude diagram of a mill in the frequency domain, the pitch angle of the end mill is optimized by taking the minimum energy and variance as the objective, so that the damping performance of the mill for hardened steel materials can be continuously improved under certain conditions.

Dynamic model of cutting force based on semi-analytical method
As an effective strategy to suppress vibration [27], the variable pitch end mill causes a difference in the cutting time delay due to the unequal angle distribution between teeth. It changes the dynamic cutting force and contact angle during cutting. Accordingly, different teeth bear different chip loads.  According to the force model of the variable pitch end mill (as shown in Fig. 1 Where K tc , K rc and K ac are, respectively, the tangential, radial and axial coefficients of the cutting force. K te , K re and K ae are, respectively, the tangential, radial and axial coefficients of the edge force.
The lag angle of cutter tooth j, relative to the tip of the cutter at the cutting depth z, can be expressed The dynamic cutting force is developed due to the dynamic cutting thickness. Accordingly, the tangential cutting force of a single tooth of the variable pitch end mill can be expressed as: Similarly, the radial and axial cutting forces of single tooth can be expressed as: Where K r and K a are, respectively, the proportional coefficients of the radial and axial cutting forces for different materials, which can be expressed as According to the single-tooth cutting force model, the instantaneous cutting force model of the variable pitch end mill with multiple teeth participating in the cutting process is obtained as follows: Where   g  is the step response function, which indicates whether the cutter teeth participate in the cutting.

 
When the contact angle of the cutter tooth lies between the cut-in angle and cut-out angle, the cutting edge takes part in the cutting. The cut-in angle and cut-out angle during the down milling can be expressed as follows: In the actual processing situation, the cutting force of the end mill is monitored by a dynamometer, which mainly measures the forces in the x, y and z directions of the machine tool. Through coordinate transformation, the cutting forces of the variable pitch end mill in feed, normal and axial direction are transformed into the x-y-z coordinate system as follows: Based on the analysis of the cutter tooth time delay and force in the machining process of the milling cutter, the instantaneous cutting force model of the variable pitch end mill is developed, the cutting mechanism of the mill is determined, and the relationship between the pitch angle and cutting force is established.

Testing of CrMoV hardened steel
The instantaneous cutting force is determined by applying an analytical method to a discrete model of the cutting tool. The accuracy of the predicted cutting force depends largely on its coefficients. A hardened steel has the characteristics of high strength, high hardness, and so on. It is a typically difficult material for machining, and vibration is prominent in its milling process. In order to verify the reliability and improve the theoretical accuracy of the model, the coefficients of cutting force are estimated by using the measured cutting force.
The testing site diagram is shown in Fig. 2, the test tool is carbide, it's geometry is listed in Table1.
The composition and properties of the material are listed in Table 2 and Table 3, respectively. The equipment includes VDL-1000E CNC machining center, Kistler rotary dynamometer, charge amplifier, data acquisition system, acceleration sensor and data processing software.    In this paper, 10 groups of single factor tests were carried out for minimizing the influence of controllable factors on the results of the test. The testing plan is shown in Table 4. Since the material properties of the workpiece are not uniform, the measurement of the cutting force is repeated for 10 times for each group of cutting parameters. The cutting forces of the third, fifth and seventh cutters of each group of parameters are monitored, and their average is taken as the final cutting force of the group of parameter. Based on the theoretical model of the cutting force of the variable pitch end mill, the cutting force coefficients of the model are first measured by a rapid calibration method, then those are related to many factors such as cutting parameters, tool-workpiece material matching, tool geometry and machining system, and the influence of every factor is complex. In 1-5 groups and 6-10 groups of the testing plane, the other three cutting parameters are kept fixed, the feed rate and the cutting depth are increased linearly, respectively. The instantaneous cutting forces F x , F y and F z are measured by the dynamometer. Due to the high axial stiffness of the system, the cutting forces in the x and y directions are discussed, and the regression analysis is conducted (as shown in Fig. 3), where the fitting degrees of the test datas are found to be 99.4%, 99.6%, 99.5% and 99.4%, respectively. The cutting force coefficients are time-varying, the fitting results are averaged. Finally, the cutting force coefficients K tc = 1.21x10 4 N/mm 2 , and K rc = 1.06x10 4 N/mm 2 , and the cutting edge force coefficients K te =   According to the cutting mechanism of the variable pitch end mill, the feed rate of no tooth is the same at different distributions of the pitch angle. The cutting force of each tooth is different, which constantly changes the peak of the force, thus making the cutting law inconsistent. According to Eq.
(10), there is a linear positive correlation between the cutting force and the pitch angle, leading to different loads on cutting edges and premature wear failure on the cutting edge with larger pitch angle.
Therefore, the influence of the difference in the pitch angle on the machining performance of the cutter teeth should be considered in the analysis of the pitch angle. When the difference angle between teeth j+1 and j is   , the relationship between the load difference of the two teeth is expressed as follows:  with equal-tooth end mills. The extreme value of cutting force spectrum in the y direction is not obvious, but scattered. It is found that the extreme cutting force spectrum in the x direction of the variable pitch end mill with asymmetric structure is reduced by more than 10% compared with that of a symmetric structure. Therefore, when the pitch angle of the variable pitch end mill tends to be asymmetrical, the cutting force spectrum of the milling cutter becomes more dispersed and uniform, resulting in smaller tool forced vibration and higher cutting stability under the same cutting conditions.

Optimization of end mills
Moreover the structural rigidity of thin-walled parts is weak, which is easy to be affected by vibration.
Therefore, the pitch angle of the variable pitch end mill is optimized by taking the milling vibration as the index for evaluating the tool and the vibration minimization of the process system as the objective function. When the frequency response function   S Sf and exciting force   e Af of the machine tool-workpiece system are known, the relative vibration amplitude spectrum   R Sf between the tool and the workpiece can be expressed as follows: In the actual milling process, when the cutting tool is suitable for different processing conditions, the response function of the system frequency under different conditions cannot be obtained in real time. In order to make the cutting tool meet all the processing conditions, the response function of the system frequency becomes a horizontal line (default) based on the statistical angle . The cutting force spectrum of the variable pitch end mill needs flatness to minimize its vibration.
The pitch angle of the end mill is taken as the design variable, and the sum of the harmonic energy(energy method Eq.12) and the square of the amplitude deviation (variance method Eq.13) of the cutting force are taken as the objective function. The energy method shows the sum of the vibration amplitudes, and the variance method shows the flatness of the vibration amplitude. The optimal distribution of the angle between the teeth is carried out, so that the amplitude of the forced vibration is reduced and made flat. Let the vibration reduction performance of the variable pitch end mill reaches the theoretical best state. The objective function is obtained from the energy method and the variance method as follows: Where f is the cutting frequency of the cutter teeth, A e (f) is the vibration amplitude. The optimization of the pitch angle is a multi variable and multi constrained parameter optimization problem. In order to ensure the convergence of the optimum results, a global search algorithm is adopted to analyze the problem comprehensively, by combining with multiple sets of initial variable values and constraints (Eq. 14). In order to verify the accuracy of the global search algorithm, the initial values of different pitch angles are optimized with the same constraints (as shown in Table 5).
The results presented in the table show that the convergence of the optimization algorithm is good, and it has higher optimization accuracy.
Where i  is the pitch angle of the i-th cutting edge, min  and max  are the upper and lower bounds of the inter-teeth angle, respectively. The global search method depicts that the optimum result has nothing to do with the initial value.
The pitch angle of the tool in the cutting force test is taken as the initial value in the optimization process. Different groups of constraints are set. The optimization results are shown in Table 6, where it can be seen that the objective value decreases with increasing pitch difference angle(as shown in  Fig. 8 shows the frequency spectrum of the corresponding cutting force, in which the amplitude is greatly reduced and the change trend is relatively gentle. The extreme values of F x and F y amplitude diagrams are reduced by 40% and 15%, respectively, compared with those of the equal-tooth end mill. Therefore, it can be concluded that the optimum pitch angle can reduce the vibration of the milling cutter and effectively improve the cutting stability.

Conclusions
In this paper, the variable pitch end mill is investigated. The time delay characteristics and cutting mechanism of the end mill are analyzed, and a cutting force model of the variable pitch end mill is developed. Based on the test with milling hardened steel(CrMoV), the cutting force coefficients are calculated by using the instantaneous cutting force calibration method, and the significance of the cutting force model is verified. This paper studies the time-frequency characteristics of the cutting force of the end mill, and discusses the frequency spectrum of the cutting force under different types of pitch angles. It is found that the pitch angle is the key factor, which leads to the change of the frequency spectrum characteristics of the milling force. The results show that the end mill with asymmetric structure can reduce the maximum amplitude by more than 10% compared with that with a symmetric structure, which has the function of dispersing harmonic components, indicating that it has better damping performance. At the same time, the pitch angle of the variable pitch end mill is optimized for different objective functions, and the relationship between the optimal distribution of the pitch angle and the objective function is comprehensively analyzed under different constraints. It is found that increasing pitch difference angle leads to the continuous decrease of the objective function. When the pitch difference angle is 20°, the optimization result of the pitch angle is 80°-97°-100°-83°. The extreme values of F x and F y spectrum amplitudes are reduced by about 40% and 15%, respectively, compared with those in the equal-tooth end mill, which shows that the milling cutter structure has better damping performance. In actual machining, the pitch difference angle should be combined with the cutter design criteria, so as to make the cutter meet certain strength requirements.
To sum up, there is a direct relationship between the tooth angle of the variable pitch end mill and the vibration characteristics of the tool. The optimal structure of the tooth angle of the variable pitch end mill has better cutting stability, which is of great significance to prolong the tool life and improve the machining efficiency.
Funding This work was supported in part by the The Central Government for Supporting the Local High Level Talent (number 2020GSP11).

Conflicts of interest
The authors declare no conflict of interest.
Data availability All data generated or analyzed during this study are included in this article.

Compliance with ethical standards
Authors' contributions All authors participate in the analysis and discusse the results and contributed to the final manuscript.  Relationship between pitch angle difference and objective function value Spectrum of end mill with optimal pitch angle (80°-97°-100°-83°)