Analysis of Cutting Forces From Conventional to High-Speed Milling with Straight and Helical-Flute Tools for Flat-End Milling Incorporating the Effect of Tool Runout

: This paper presents a systematic study to analyze the dependence of cutting forces on tool geometry, workpiece material and cutting parameters such as spindle speed, tool engagement and cutting direction in flat-end milling with tool runout. The cutting forces are determined according to a mechanistic force model considering the trochoidal flute path to calculate the undeformed chip thickness, and average cutting force and linear regression model are applied for identifying the coefficients of the force model. A series of milling processes are conducted on AZ31 Magnesium (Mg) alloy and titanium alloy (Ti6Al4V) to analyze the instantaneous cutting force curves, amplitudes of cutting forces and peak forces over a wide range of spindle speeds from conventional to high-speed milling. It is demonstrated that the values of the cutting force coefficients are higher at conventional spindle speed and decrease with an increase in spindle speed, especially when machining Ti6Al4V alloy. For the edge force coefficients, it is observed a slight variation when using cutting tools with different helix angles. Besides, the cutting force amplitudes strongly depend upon the workpiece material. The helix angle has a significant influence on the transverse force amplitude at conventional speed. The forces obtained mechanistically are also substantiated by comparison with measurements.


Introduction
End milling is one of the extensively utilized complex manufacturing methods in metalworking industries. Tool runout occurs from several sources such as tool axis offset, tool axis tilt and setting of inserts in end milling operations, which affects the instantaneous undeformed chip thickness and cutting forces depending on the cutting conditions and cutting geometry [1]. A large number of predictive cutting force models have been developed to examine the milling process, which focus on physical aspects. However, some fundamental physical phenomena are still difficult to understand in a wider range of cutting conditions. Thus, it seems interesting to analyze the physical phenomena with the help of cutting force modeling in many conditions.
Modeling of cutting forces in end milling constitutes the basis for predicting process stability [2], identifying surface location errors [3] and analyzing surface generation [4]. There are three different approaches to predict cutting forces in milling processes: analytical, numerical and mechanistic [5], where the situations of tool runout are considered. The analytical models relate cutting forces to process variables and mechanical aspects. Amarego and Deshpande proposed a force model using oblique cutting analyses [6]. In this model, they discussed the effects of deflection and tool eccentricity. Li et al. [7] developed another model including the effects of runout to predict milling forces applying Oxley's machining theory. In their model, the action of an individual flute segment was described as oblique cutting. Wang and Liang [8] proposed an analytical formulation to compute the chip load in milling processes with the effect of runout. In this formulation, the cutting geometry was modelled based on the circular flute trajectory. While this approximation greatly simplifies the analysis, it causes significant errors. Advances in computers have led to the development of numerical methods, where the relationship between tool geometry and workpiece is studied with the finite element-based models [9].
In mechanistic model, cutting forces are proportional to undeformed chip area [10]. Kline et al. [11] presented a mechanistic model to predict milling forces for the cutting geometry with tool runout. They calibrated the empirical coefficients with the measured average forces and an undeformed chip thickness approach, which are referred to as force coefficients. Based on a similar mechanistic cutting force model considering the runout and tool deflection, Yun and Chu [12] determined the cutting forces for flat-end milling. A general approach was established by Wan and Zhang [13] for cutting force prediction in milling operations, who included the influences of the tool and workpiece and immersion angle variation. Montgomery and Altintas [14] proposed a theoretical cutting force method which describes the undeformed chip thickness considering trochoidal motion of the tool. Recently, Wojciechowski [15] investigated cutting forces in ball-end milling of inclined surfaces by including the effect of surface inclination and tool runout. However, in die and mold making operations, where cutting conditions change, a variety of cut tests are necessary to determine the cutting forces. Considering the cutting tool, workpiece and cutting parameters, a systematic investigation of the influences of tool runout on the milling process based upon trochoidal flute trajectories should be performed.
There are mainly two approaches for identifying the force coefficients, i.e., mechanistic approach [16][17][18], oblique transformation method [19][20][21]. These approaches require a linear regression of the measured cutting forces over a range of feed rates using a constant cutting speed and depth of cut. Another method [22] employs a nonlinear optimization approach to carry out a least-squares fitting of predicted cutting forces to measured ones, which allows to study the effects of machining parameters on dynamic cutting forces. By decomposing the cutting forces into nominal and perturbation components, Wan et al. [23,24] also suggested an approach which calibrates simultaneously the force coefficients and tool runout by solving nonlinear equations for flat-end milling.
This paper presents a systematic study to determine the cutting forces for flat-end milling process with tool runout by considering the effects of tool geometry, workpiece material and cutting parameters such as spindle speed, tool engagement and cutting direction. Numerous milling processes are also conducted to accurately determine the instantaneous cutting force curves, amplitudes of cutting forces and peak forces over a wide range of spindle speeds from conventional to high-speed machining. A mechanistic force model which determines the cutting forces based on the undeformed chip thickness calculated from the trochoidal flute path is used, and the force coefficients are derived using average cutting force and linear regression model.

Mechanistic cutting force model
An end milling with a straight-flute end mill is depicted in Fig. 1. The motion of cutting edge is a combination of rotation and translation. The orientation and magnitude of cutting forces change because this relative motion causes a chip geometry that varies with time. Thus, it is required to consider the trajectory of cutting edges to determine forces accurately.
To model the cutting forces for straight end milling, a mechanistic force model is applied, which is proven in the literature [25,26]. This model relates the cutting force components to the undeformed chip thickness. Tangential (Ft) and radial (Fr) forces applied on the kth flute of a rigid end mill at time t are described by where Ktc and Krc denotes the tangential and radial cutting force coefficients, respectively. Kte and Kre are the corresponding edge force coefficients. da is the axial depth and wk(t) is the undeformed chip thickness.
The tangential and radial forces are then resolved into feed (Fx) and transverse (Fy) force components as is the rotation angle of the kth flute.
The contributions of entire flutes are computed, and added to achieve the total instantaneous forces on the tool at the current time as follows where Nf is the flute number. To account for the effect of helix angle (), the end mill is discretized into L number of disk elements along its axis, whose elementary length is , as illustrated in Fig. 2. Considering the ith disk of the kth flute at the jth angular position of the tool, the immersion angle is expressed as is the position angle of flute tip, and is calculated with respect to the initial position angle () by indexing the amount of angular increment (d).
is the pitch angle. The lag angle )) ( ( i  is calculated using where r is the tool radius.  [27] In helical end milling, the elemental tangential (dFt) and radial (dFr) forces are defined as Subsequently, the elemental forces are expressed in the Cartesian coordinate system as given below The total forces applied on the kth flute at time t are evaluated by integrating Eq. (7) along the axial depth where z is the axial coordinate of the ith disk of the kth flute.
Finally, the total instantaneous forces at the current time are calculated by summing the forces generated by all flutes.
The flowchart for determining the cutting forces in helical end milling simulation is illustrated in Fig. 3. The simulation begins by defining the cutting parameters (spindle speed (Ω), axial depth (da), feed rate (f), entry and exit angles (θSt, θEx), tool radius (r), number of flutes (Nf), force coefficients etc.) and numerical parameters (initial time and angle values, numbers of time steps (T), angular steps (K) and axial steps (L), time interval (dt), integration angle (d) etc.). Then, the program runs the time loop for discretized disks. The program checks for each time interval (dt) whether the angular position of kth flute of the ith disk is in cut or out of cut. If cutting edge is not in cut, the forces are adjusted to zero. Otherwise, the forces in feed and transverse directions (Fx, Fy) are evaluated using Eq. (7). The main loop continuous until the number of simulated time steps equals the predefined one. At the final stage, the program plots the forces.

Calibration of force model coefficients
It is necessary to calibrate the force model coefficients to make cutting force predictions. There are various methods to calibrate the model coefficients. Here, the force coefficients were derived from the average force components measured during slot cutting experiments which were conducted over a range of feeds per tooth (fz) with a constant spindle speed and axial depth. Such a method was also used by Dikshit et al. [28]. A single-flute end mill was used to eliminate the effects of runout. Projecting the tangential and radial force components into the Cartesian coordinate system and integrating over one revolution yield the following expressions for average cutting forces in feed (X) and transverse (Y) directions [22].
where St  and Ex  represent the entry and exit angles, respectively. Eqs. (9) and (10) can be arranged in matrix form as where F and K are the force and coefficient vectors, respectively. G is the geometry matrix which is related to the cutting geometry. The subscript m represents the number of cutting conditions for the calibration. In this study, calibration was performed using four different feed rates. Thus, the sizes of the F vector, G matrix and K vector are 8x1, 8x4 and 4x1, respectively.
Applying least-squares fitting method to the force vector and geometry matrix, the coefficients are calibrated as follows

Instantaneous chip thickness model
There are two types of tool runout, radial and axial. The axial runout, whose influence on the cutting forces is disregarded, is very small compared to the axial depth. However, the radial runout has the same order as feed per tooth [7]. Therefore, the analysis is focused on radial runout, which is described as the offset of tool from the axis of rotation.
The trajectories of cutting edges with respect to the workpiece in milling are trochoidal [29], as illustrated in Fig. 4a. The undeformed chip thickness can be modeled by considering flute trajectories incorporating tool runout [30].
The trajectory of the cutting point at the tip of the kth flute is given as The position of the tool center in the coordinate system XOY can also be described as follows where xC and yC are the coordinates of tool center in feed and transverse directions, respectively. f is the feed rate. t is the time.  is the runout offset, and  is the location angle. The trochoidal motion, which occurs in feed direction, is accounted for through the expression . t f The undeformed chip thickness is the length between two consecutive workpiece surfaces as illustrated in Fig. 4b. Assuming that the kth flute removes the surface generated by a previously passing one. The line PC intersects the path curve left by the tip of the previous flute at point R. When the tip of the previous flute is at point R, the coordinates of point R (xR, yR) can be found from Eq. (21). At this moment, the center of tool is at point C, and its coordinates (xC, yC) can be obtained using Eq. (22).
The length of the line segment PR corresponds the instantaneous undeformed chip thickness (wk), and can be determined as follows

Determination of runout parameters
The radius of each flute can be measured when the end mill is attached to the spindle. Assuming that on an end mill with clockwise rotation, there are Nf flutes labelled counterclockwise as No. 1, 2, , Nf, and the flutes are uniformly distributed. The runout offset of the kth flute is erk, where k = 1, 2, , Nf. As presented in Ref. [31], the actual radius of the kth flute can be regarded as rk = r +erk. The runout perpendicular to the cutting edge is neglected as in Eq. (24). Fig.  5 shows the convention employed for runout.

Fig. 5 Runout offset and location angle definitions
Thus, the edge radius equations for a four-flute end mill can be defined as  From the above equations, the runout offset and location angle can be found as 6 Milling experiment procedure

Milling machine
The entire milling experiments were performed on 3-axis Quaser MV154C vertical milling machine. This machine allows to cut materials at high spindle speeds. It has a maximum spindle speed of 10,000 rev/min. The machine table moves in feed and transverse directions, and the spindle moves vertically on the column.

Experimental setup
The cutting forces were measured by using a dynamometer (Kistler 9257B) which was fixed on the machine table. The dynamometer was connected to the computer through a charge amplifier (Kistler 5070) and National Instruments data acquisition card. Post-processing of data was treated through DynoWare software. Before the cutting experiments, the tool runout was adjusted by placing a thin aluminum alloy film with the end mill in the tool chuck. Then, the radii of all flutes at the free ends of the end mills were measured through a dial indicator with a precision of 0.001 mm to quantify the runout values for each cutting tool. To minimize the influence of runout measurement deviations, each measurement was repeated two times to establish a statistical mean. Table 1 lists the calculated runout parameters for each tool. Fig. 6 shows the experimental setups for cutting force and tool runout measurements.

Workpiece materials
Cutting forces were analyzed under dry machining for two different workpiece materials with high strength: AZ31 Mg-alloy and Ti6Al4V alloy. All the machined specimens had dimensions of 170  100  12 mm, which were rigidly clamped on the dynamometer. The workpiece was fed towards the tool so that there was a relative movement between the rotating tool and workpiece during experiments.

Cutting tools
The cutting tools were uncoated tungsten carbide flat-end mills with 12 mm diameter held in a tool holder mounted on the spindle, which had either straight or helical flutes. The helix angle for helical-flute tools was 30°. For calibration, experiments were performed with single-flute tools to simplify the experimental work. Then, four-flute ones were used to investigate the effects of tool engagement, tool geometry and workpiece material. The cutting tools are shown in Fig. 7.

Cutting conditions
An extensive range of spindle speeds ranging from conventional to high-speed cutting was considered to obtain completely data about machinability of AZ31 Mgalloy and Ti6Al4V alloy with the straight and helical-flute end mills. To evaluate the effect of spindle speed on the force coefficients, three sets of calibration experiments were performed with three different spindle speeds of 600, 1200 and 3000 rev/min. For each set, four different feedrate were tested by changing the feed rate (fz) from 0.05 to 0.30 mm/flute in steps of 0.05 mm/flute while other parameters constant. Table 2 depicts the experimental design for calculating the force coefficients. In addition to calibration experiments, cutting experiments were performed to evaluate the effects of tool engagement, tool geometry and workpiece material on cutting forces using straight and helical-flute end mills with four flutes. It was focused on slot cutting and shoulder milling in down and up cutting configurations to study different tool engagement conditions. The tool engagement was defined with a radial depth (dr) of 6 mm for shoulder milling. The axial depth was kept constant at 2 mm for the entire experiments.

Force coefficients for different tool/workpiece materials
All the cutting force curves obtained from milling experiments have the characteristic shape presented in Fig. 8a. Unstable regions appear at the beginning and end of the curves, corresponding to entry and exit of the tool. To calibrate the force coefficients, average cutting forces were calculated from graphical data obtained in the steady region at the center of the curves for 10 spindle revolutions, as given in Fig. 8b. alloy. Fig. 9 shows the relationship between feed rates and average cutting forces for carbide/AZ31 Mg-alloy pair. It can be observed that the average cutting forces change linearly with the feed per tooth. Besides, when the spindle speed increases from 600 to 3000 rev/min, these linear relationships have the similar gradient. The relationship between feed rates and average cutting forces for carbide/Ti6Al4V alloy pair is illustrated in Fig. 10. The average cutting forces increase linearly as the feed rate increases as reported by other researchers [22], and the gradients of the linear relationships of average cutting forces in feed and transverse directions vary with increasing spindle speed. In addition to, the magnitudes of the feed and transverse forces decrease as the spindle speed increases. The cutting and edge force coefficients calibrated from the least-squares method by using the average forces for two different tool/workpiece pairs are listed in Table  3. As can be seen, the force coefficients for each tool/workpiece combination vary when cutting with the same milling parameters. The tangential and radial cutting force coefficients calculated when machining Ti6Al4V alloy are higher than those obtained in cutting AZ31 Mg-alloy. Further, it is found that when the speed is changed from 600 to 3000 rev/min, the tangential and radial cutting force coefficients decrease slightly by 2.44 and 11.57% for AZ31-Mg alloy, respectively. However, there is a larger decrease of 33.71 and 71.61% in tangential and radial cutting force coefficients for Ti6Al4V alloy, respectively. These phenomena reveal that the cutting force coefficients calculated for AZ31-Mg alloy are not very sensitive to the variation of spindle speed. However, due to the variation of shearing behavior during cutting Ti6Al4V alloy at different spindle speeds, the cutting force coefficients depend upon the speed as seen in Fig. 11a. Also, the resulting trend is in agreement with results in Ref. [32].
It is well known that the edge force mainly results from the ploughing mechanism at the cutting edge, which is associated with the elastic-plastic property of workpiece material [33]. For AZ31 Mg-alloy, there is no significant difference in tangential and radial edge force coefficients with an increase in spindle speed. On the other hand, as can be also observed in Fig. 11b, there is a decrease of 50.48 and 59.74% in tangential and radial edge force coefficients with increasing spindle speed from 600 to 3000 rev/min for Ti6Al4V alloy, respectively. Consequently, the edge force coefficients are noticeably dependent on the properties of workpiece materials. The decrease in coefficients can be also associated with the softening of the workpiece material because of the tool-chip interface temperature increasing with spindle speed.

Force coefficients for cutting tools with different helix angle
To identify the force coefficients for two different types of end mills with straight and helical-flute, the average cutting forces in feed and transverse directions were obtained under different spindle speeds during milling of AZ31 Mg-alloy.
The calibrated cutting and edge force coefficients are listed in Table 4. It is found that when the spindle speed is changed from 600 to 3000 rev/min, the tangential cutting force coefficients decrease by 5.56 and 2.44% for straight and helical-flute end mills, respectively. Moreover, a comparison of radial cutting force coefficients reveals that there is a decrease of 9.85 and 11.57% with increasing spindle speed from 600 to 3000 rev/min for straight and helical-flute tools, respectively. It is evident from the above results that the magnitudes of the tangential and radial cutting force coefficients are influenced from the variation of the spindle speed.
By comparing the cutting force coefficients, it can be also seen that the cutting coefficients determined using straight end mill are higher than those obtained by the helical tool. It is noteworthy here that increasing helix angle serves to decrease the cutting force coefficients. For the tangential and radial edge force coefficients, it is observed a slight variation when using cutting tools with different helix angle or different spindle speeds. It can be concluded that the edge coefficients are independent of tool geometry.

Tool engagement conditions in milling
To more carefully investigate the effect of runout on the instantaneous force profile within the envelope, the tool engagement conditions including slot cutting and shoulder milling in down and up cutting configurations were analyzed for spindle speeds ranging from conventional to high-speed cutting. The cutting tool was a four-flute helical end mill. The workpiece material was AZ31 Mg-alloy. Fig. 12 illustrates the slot cutting configuration and the cutting force curve in feed direction with considering the tool runout obtained mechanistically at conventional spindle speed of 600 rev/min and feed rate of 0.025 mm/flute. It can be observed that the force curve has different peak values in time domain of one tool rotation since the runout causes each flute to experience a different chip thickness. Besides, its peaks and shape are easily identifiable. Thus, the predicted feed force was adopted to analyze cutting force curves. For four-flute k = 1, 2, 3, 4, the mean radial runouts (erk) are 0.008, 0.0055, 0.002, 0.0025 mm, respectively. The difference of the radius of the actual kth flute with respect to the previous (k+1)th one is denoted as δerk = erk  erk+1. In this case, δerk is −0.0025, −0.0075, 0.0045, −0.0055 mm for k = 1, 2, 3, 4, respectively. Since the absolute radius variation for a flute with respect to the previous one is smaller than the feed per tooth, i.e., , z rk f e   every flute engages in cutting.
To reveal the presence of runout, four stages are considered as a function of the cutting tool rotation angle, corresponding to the entry and exit cut of the four flutes. Each one describes a characteristic evolution of the cutting force curve: Slot cutting begins when the initial flute gradually engages in cutting. A unique force peak occurs and the force decreases quickly to valley value. During the second stage, the cut geometry makes the force peak slightly different. In the third stage of slot cutting, the curve immediately begins another identical cycle but a higher force peak is produced. It is clear that this step corresponds to the entry of the high side of the tool into the workpiece. It can be also concluded that when the tool enters the cut, its low side is engaged with the workpiece. The fourth stage corresponding to the end of the cutting cycle produces a slightly higher force peak but lower force valley compared with the third stage. As a result, the magnitude of force peak becomes different at each stage since the chip load for each flute varies due to the presence of runout. In case of slot cutting, the comparisons of cutting force curves obtained from the experiment and predicted from the time domain simulation are also presented in Fig. 13. It can be observed that the shape and amplitudes of the predicted cutting force profiles are in good agreement with the measured ones. Due to the existence of tool runout, the amplitudes of the measured and predicted cutting forces vary with the cutting tool rotation. The flutes on the high side of the tool are engaged with the workpiece more than those on its low side, and cause higher cutting force peaks. The differences of feed and transverse force peaks between the low and high sides of cutting tool are ratios of about 50 and 25%, respectively.
For a better comparison of the results obtained from the mechanistic model, the varying amplitudes of feed and transverse force curves are analyzed, and the average of the cutting force amplitudes is obtained considering two successive revolutions, i.e., eight peaks and valleys in the curves. The peaks (Fx,P1, , Fx,P4) and valleys (Fx,V1, , Fx,V4) for one tool revolution are marked in Fig. 13. The average amplitude is calculated as given below In this study, the mean squared errors of average amplitudes (MSE) are taken as a measure of performance of the mechanistic model, and calculated as follows  Measured Predicted increase as the spindle speed increases in slot cutting conditions with the same axial depth and feed rate. The best correlation is achieved under low spindle speed. The force model performs well for the feed force with a mean squared error of 5.2 while a slightly higher prediction error of 9.38 is calculated for transverse force. The total of the mean squared errors is also found as 14.58. This can be attributed to considering only peripheral cutting edges of end mill in the model. In practice, the end cutting edge introduces additional cutting forces. Up cutting configuration is illustrated in Fig. 14a, which corresponds to a radial immersion of 50%. Fig. 14b plots the predicted cutting force curve at conventional spindle speed of 600 rev/min and feed rate of 0.05 mm/flute, and four main stages are described: Up cutting begins with a chip thickness equal to zero. As the tool rotates, the cutting force gradually increases and a force peak occurs. At the end of the cut, the chip is removed, and the initial flute comes out of the workpiece. In the second stage, a slightly varying force peak and valley reveal compared with the first stage. In the third stage, the force peak is greater than those in the preceding stages. This means the entry of the high side of the tool into the workpiece. Many phenomena such as wear, vibration can occur due to increasing chip load. The fourth stage corresponds the largest force peak. It should be noted here that there are nickpoints on the curve, which correspond to the quickly entry of the flutes into the workpiece.
For the case of up cutting that is illustrated in Fig. 15, the mechanistic model gives a good description of feed and transverse forces with slight deviations in the peak and valley regions. The cutting force curves indicate a behavior close to sinusoidal, and the tool runout vary the distribution of cutting forces between each flute of the cutting tool. Also, it can be observed from the curves that the cutting process is continuous.  It is interesting to compare the average amplitudes of the experimental cutting forces with those calculated mechanistically for three different cutting speeds. For up milling, Table 6  found as 17.53 and 65.35 for feed and transverse forces, respectively. The total of the mean squared errors is as high as 82.88. There is a lower accuracy for the forces derived from the mechanistic model with the rigid tool assumption at the highest spindle speed. Thus, the vibration phenomenon and flexible behavior of the tool should be considered when modelling the cutting forces for high spindle speed. Down cutting configuration with a radial immersion of 50% and the cutting force curve predicted at spindle speed of 600 rev/min and feed rate of 0.075 mm/flute are shown in Fig. 16. Down cutting begins with the entry of the initial flute from the straight workpiece surface. Negative magnitudes are observed in the curve because of mainly the projection of forces on the relief surface of the tool [34], which coincide with the findings of Kline and DeVor [11]. In the second stage, the tool produces a slightly lower force peak than the first stage. The third stage occurs a higher force peak than the other stages due to the entry of the high side of the tool into the workpiece. The nickpoint on the curve causes the cutting force to increase quickly in the positive direction. The fourth stage is characterized by the largest negative value, and is a critical place for the cutting force.
In the down cutting case, the experimentally observed force curves and the predicted force curves are also presented in Fig. 17. It can be observed that the model is able to predict the variations of cutting forces with rotation angle while small deviations are available in the peak and valley regions of cutting force curves. In case of four-flute, each flute path has its own profile shape and magnitude in one tool revolution since the tool runout exists. The variations in the forces are symmetric to those in up cutting. In order to compute the deviation between the force amplitudes for down cutting configuration with runout, the experimentally measured force amplitudes and the amplitudes derived from the mechanistic force model are presented in Table 7. The predicted feed force from the force model integrated with the chip thickness model has a mean squared error of 27.71 while the prediction error is high up to 52.59 for transverse force. Further, the total of the mean squared errors is calculated as 80.3. A better accuracy is obtained under the condition with low spindle speed.  As presented above, the accuracy of force predictions is sensitive to the tool engagement condition. The predicted cutting forces are more accurate in the slot cutting configuration for the tested cutting speed range. For the cases of up and down cutting, the feed and transverse forces are predicted with a lower accuracy.

Influence of different tool geometries on cutting forces
To study the influences of tool geometries on cutting forces for milling process with runout, the straight and helical end mills having four flutes were analyzed at constant feed rate of 0.1 mm/flute for a wide range of speeds, namely 600, 1200 and 3000 rev/min. For each spindle speed, slot cutting processes were performed on AZ31 Mg-alloy.
As can be seen from the predicted force curves in Fig. 18, the straight-flute end mill gives greater force peaks than helical-flute tool at conventional spindle speed of 600 rev/min. There is a sharper transition at the peak regions of feed force for end mill that has helical flutes compared with straight-flute end mill. On the other hand, a smoother transition is observed at peak regions of feed force for straight-flute end mill. For transverse forces, both straight and helical-flute end mills exhibit a sharp transition at the peak regions of each flute pass period. fact that tool wear is not considered in the model. The amplitudes of the measured and predicted feed and transverse forces tend to decrease as the spindle speed increases, which can be explained by the following: (i) It is generally observed that a decrease in spindle speed decreases the shear angle, and a small shear angle gives a large shear-plane area. Thus, the shear forces needed for deformation increase; (ii) At low spindle speed, the cutting forces increase due to high friction coefficient. At conventional spindle speed of 600 rev/min, the helical end mill leads to a relatively small decrease of the feed force compared with the straight-flute tool. However, the measured and predicted transverse forces by the helical end mill are quite lower than those obtained using straight-flute tool at the same spindle speed. From these explanations, it is clearly seen that helical end mill greatly decreases the transverse force at low speed. For the highest spindle speed of 3000 rev/min, the feed and transverse forces increase slightly with the helix angle. In addition, the prediction errors for straight and helical-flute end mills are presented in Table 8. The prediction errors in feed and transverse forces for helicalflute end mill are 7.79 and 6.97, respectively. The prediction error in feed force is identified as 6.3 in the case of straight-flute end mill, but the prediction error in transverse force is as high as 28.87. For these tools, the errors of the predicted cutting forces in feed direction are similar. However, the helical-flute end mill provides more accurate predictions for transverse force in contrast to the straightflute tool. It is obvious that for the case of helical-flute end mill, the mechanistic force model can predict the cutting forces in both feed and transverse direction with a better accuracy.

Influence of different workpiece materials on cutting forces
To reveal the influences of workpiece materials on cutting forces for milling process with runout, two different workpiece materials were considered, AZ31 Mg and Ti6Al4V alloys. Slot cutting processes were performed at three different spindle speeds of 600, 1200 and 3000 rev/min with constant feed rate of 0.05 mm/flute using four-flute helical end mill.
Based on the force coefficients identified at conventional spindle speed of 600 rev/min for two different workpiece materials, the predicted cutting force curves are shown in Fig. 20. Ti6Al4V alloy gives substantially greater force amplitudes than AZ31 Mg-alloy under the same cutting condition. From this result, it can be found that the machinability of Ti6Al4V alloy seems to be more critical because of the high value of amplitudes compared with AZ31 Mg-alloy. This phenomenon can be very important in high-speed cutting owing to the dynamic effects appearing with the increase of spindle speed. Fig. 21 presents the trends of the measured and predicted cutting force amplitudes changing with spindle speed in machining of AZ31 Mg-alloy and Ti6Al4V alloy. It can be observed that the amplitudes of the measured and predicted cutting force components in feed and transverse directions for Ti6Al4V alloy are higher at conventional speed, and show a decrease with increasing spindle speed. Also, AZ31 Mg-alloy results in slightly varying amplitudes with spindle speed. In the case of slot cutting of Ti6Al4V alloy, the measured and predicted feed and transverse forces are considerably higher than those predicted in slot milling of AZ31 Mg-alloy over the spindle speed range.  Table 9 gives the prediction errors for AZ31 Mg and Ti6Al4V alloys. As can be seen from the table, the prediction accuracy of feed and transverse forces is dependent on the workpiece material. In machining of AZ31 Mg-alloy, the prediction errors in feed and transverse forces are 9.7 and 7.69, respectively. The prediction error in feed force is identified as 580.28 in machining of Ti6Al4V alloy; moreover, the error in transverse force prediction is as high as 860.2. That is, Ti6Al4V alloy results in more accurate predictions for feed force in contrast to the transverse force. Finally, it appears that for machining case of AZ31 Mg-alloy, the mechanistic model including the effect of runout can predict the cutting forces with a better accuracy.

Conclusions
In this research, a systematic study to model the cutting forces for flat-end milling with tool runout has been presented by considering the effects of tool geometry, workpiece material and process parameters such as tool engagement and cutting direction for a wide range of spindle speeds ranging from conventional to highspeed milling. A mechanistic force model which predicts the cutting forces based on the undeformed chip thickness calculated from the trochoidal flute trajectories was used, and the average force method and linear regression model were considered to evaluate the force coefficients. Based on this research, the following observations are drawn: 1. The force coefficients can be determined as constant values with respect to the variation of spindle speed to improve the accuracy of the predicted forces for a particular tool-workpiece couple. Since the shearing mechanism changes with increasing spindle speed, there is a tendency to decrease especially in cutting force coefficients. Also, the edge coefficients are independent to the tool geometry. 2. The conventional spindle speed produces larger cutting forces than high speed, and the amplitudes of the predicted cutting forces under conventional spindle speed agree well with the measured values. However, in the case of high-speed end milling, the predicted cutting forces deviate from those obtained experimentally since dynamic phenomena appear with increasing spindle speed. Accordingly, the rigid tool assumption drastically limits the predictive capacity of the model to the conventional spindle speed. 3. The accuracy of the predicted cutting forces highly depends on the workpiece material. For spindle speeds ranging from conventional to highspeed machining, AZ31 Mg-alloy provides higher cutting force prediction accuracy than Ti6Al4V alloy. Also, the helix angle has a significant effect on the cutting forces, particularly the transverse force at conventional speed. 4. For a rigid end milling process, the tool runout causes a considerable variation in the cutting force curve over the rotation of a multi-flute tool. Besides, the flutes on the high side of cutting tool with runout encounter larger force peaks than those on the other side. Thus, the runout effect should be taken into account to analyze the cutting forces in end milling.
Author contribution All the authors have equally involved in preparing the manuscript.

Data Availability
The data and materials supporting the results of this study are included in the article.

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Competing interests
The authors declare no competing interests