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The Modelling and Analysis of Micro-Milling Forces for Fabricating Thin-Walled Micro-Parts Considering Machining Dynamics

In the fabrication process of thin-walled micro-parts, both micro-cutting tools and thinwalled micro-parts have the characteristics of small size and low stiffness. Therefore, the regenerative chatter during the machining process cannot be ignored. The influence of the tool runout error and actual trochoidal trajectories of the cutting edge on micro-milling forces should also be considered comprehensively. In this paper, the tool runout error in the micro-milling process is first analysed, and an instantaneous undeformed chip thickness model is established considering the runout error. On this basis, the dynamic deformation of the micro-cutting tool and thin-walled micro-part is studied, and an instantaneous, undeformed, chip-thickness model is proposed with the consideration of both the runout error and dynamic deformation. The dynamic parameters of the machining system are obtained using the receptance coupling method. Finally, thin-walled micro-part machining experiments are carried out, and the obtained results of micro-milling force simulation based on the proposed model are compared with the experimental results. The results indicate that the micromilling force modelling, by taking the influence of machining dynamics into account, has better prediction accuracy, and the difference between the predicted resultant forces and the experimental results is less than 11%.


Introduction
Thin-walled micro-parts generally refer to thin-walled structures with a thickness less than 100 µm and a height-to-thickness ratio greater than 5. They are widely used in the medical, aviation, and aerospace industries [1]. Among them, Ti-6Al-4V alloy is widely used, considering its high specific strength, specific stiffness, and good biocompatibility [2]. Micro-milling technology has superior characteristics, including high processing efficiency, high machining accuracy, and a wide range of processing materials, so it is used as an effective means of thin-walled micro-part fabrication. The diameter of the micro-milling tool is typically between 100 µm and 1 mm, and usually, it has to bear more stress than the traditional cutting tool and is more prone to tool wear and breakage [3]. The stiffness of the thin-walled micro-structure is also weak, which makes the regenerative chatters nonnegligible during the machining process.
Cutting force is an important index for the machining quality and cutting tool status. Therefore, it is crucial to establish a modelling and analysis approach that can be used for predicting the cutting forces and consequently the process optimisation in thin-walled micro-part fabrication in particular. The mechanistic cutting force modelling combines the advantages of analytical and empirical methods. Therefore, it has been widely studied.

Modelling of IUCT
The cutting force is proportional to the IUCT in the mechanistic micro-milling force model. Therefore, the calculation of the IUCT is the premise for the micro-milling forces model. In addition to the tool runout error, the influence of the tool-workpiece dynamic deformation error on the IUCT and micro-milling force cannot be ignored, as shown in Figure 1. On the basis of the traditional micro-milling force model, the influence of dynamic characteristics is taken into account. In this section, a model for the accurate IUCT calculation is presented based on the real cutting-edge trajectories of the micro-milling cutter. It can be observed that the cutting force prediction is a feedback process.

Modelling of IUCT
The cutting force is proportional to the IUCT in the mechanistic micro-milling force model. Therefore, the calculation of the IUCT is the premise for the micro-milling forces model. In addition to the tool runout error, the influence of the tool-workpiece dynamic deformation error on the IUCT and micro-milling force cannot be ignored, as shown in Figure 1. On the basis of the traditional micro-milling force model, the influence of dynamic characteristics is taken into account. In this section, a model for the accurate IUCT calculation is presented based on the real cutting-edge trajectories of the micro-milling cutter. It can be observed that the cutting force prediction is a feedback process.

Modelling of the Cutting-Edge Trajectories
As shown in Figure 2, the micro-milling cutter is equally divided into K micro-elements along the z-axis direction. The corresponding element cutting force is acted on to each micro-element. The element cutting force has three components: radial, normal, and axial, in which the axial component is relatively small [25]. β ω (1) where β is the helix angle of the cutting tool, R is the tool radius, ω is the angular velocity of the spindle rotation, is the axial depth of cut, and K is the number of micro-elements. Therefore, the angular position of the jth cutting edge at any time can be expressed as follows:

Modelling of the Cutting-Edge Trajectories
As shown in Figure 2, the micro-milling cutter is equally divided into K micro-elements along the z-axis direction. The corresponding element cutting force is acted on to each micro-element. The element cutting force has three components: radial, normal, and axial, in which the axial component is relatively small [25].

Modelling of IUCT
The cutting force is proportional to the IUCT in the mechanistic micro-milling force model. Therefore, the calculation of the IUCT is the premise for the micro-milling forces model. In addition to the tool runout error, the influence of the tool-workpiece dynamic deformation error on the IUCT and micro-milling force cannot be ignored, as shown in Figure 1. On the basis of the traditional micro-milling force model, the influence of dynamic characteristics is taken into account. In this section, a model for the accurate IUCT calculation is presented based on the real cutting-edge trajectories of the micro-milling cutter. It can be observed that the cutting force prediction is a feedback process.

Modelling of the Cutting-Edge Trajectories
As shown in Figure 2, the micro-milling cutter is equally divided into K micro-elements along the z-axis direction. The corresponding element cutting force is acted on to each micro-element. The element cutting force has three components: radial, normal, and axial, in which the axial component is relatively small [25]. Due to the influence of the helix angle, the corresponding time of each micro-element at height z can be expressed as Equation (1).
where β is the helix angle of the cutting tool, R is the tool radius, ω is the angular velocity of the spindle rotation, is the axial depth of cut, and K is the number of micro-elements. Therefore, the angular position of the jth cutting edge at any time can be expressed as follows: Due to the influence of the helix angle, the corresponding time of each micro-element at height z can be expressed as Equation (1).
where β is the helix angle of the cutting tool, R is the tool radius, ω is the angular velocity of the spindle rotation, a p is the axial depth of cut, and K is the number of micro-elements. Therefore, the angular position of the jth cutting edge at any time can be expressed as follows: where N t denotes the tooth number. The central position of the ith tool micro-element can be expressed as Equation (3).
where x a [t i (z)] and y a [t i (z)], respectively, represent the actual coordinates of the cutting-edge centre at height z; x n [t i (z)] and y n [t i (z)], respectively, represent the nominal coordinates of the cutting-edge centre at height z. It can be expressed as follows: , respectively, denote the position deviation caused by the tool runout error, as shown in Figure 3.
where Nt denotes the tooth number.
The central position of the ith tool micro-element can be expressed as Equation (3).
where [ ] and [ ], respectively, represent the actual coordinates of the cutting-edge centre at height z; [ ] and [ ], respectively, represent the nominal coordinates of the cutting-edge centre at height z. It can be expressed as follows: where [ ] and [ ], respectively, denote the position deviation caused by the tool runout error, as shown in Figure 3.  Additionally, they can be expressed as Equation (5).
[ ] where ρ and α represent the runout error value and position angle of the cutting tool, respectively. and represent the position deviation caused by the dynamic deformation between the cutting tool and workpiece in the x and y direction, respectively.
According to the machining system dynamics, the dynamic deformation of the micro-milling tool and thin-walled micro-parts can be obtained from Equation (6).
where Fx and Fy are the cutting force in the x and y directions; gpx (p = t, w) are the impulse responses of the cutting tool and thin-walled micro-part in the x direction, respectively; gpy (p = t, w) is the impulse responses of the cutting tool and thin-walled micro-part in the y direction, respectively. Therefore, the relative displacement between the tool and workpiece can be obtained as follows: Additionally, they can be expressed as Equation (5).
where ρ and α represent the runout error value and position angle of the cutting tool, respectively. x d [t i (z)] and y d [t i (z)] represent the position deviation caused by the dynamic deformation between the cutting tool and workpiece in the x and y direction, respectively. According to the machining system dynamics, the dynamic deformation of the micromilling tool and thin-walled micro-parts can be obtained from Equation (6).
where F x and F y are the cutting force in the x and y directions; g px (p = t, w) are the impulse responses of the cutting tool and thin-walled micro-part in the x direction, respectively; g py (p = t, w) is the impulse responses of the cutting tool and thin-walled micro-part in the y direction, respectively. Therefore, the relative displacement between the tool and workpiece can be obtained as follows: The dynamic deformation and the relative displacement of the previous cutting cycle can be expressed as Equations (8) and (9), respectively.
On this basis, the position deviation caused by the dynamic deformation can be expressed as follows:

IUCT Modelling Considering the Tool Runout
Based on the cutting tool edge trajectory modelling above, the IUCT can be obtained. In the micro-milling process, there are three different forms of material removal mechanisms, as shown in Figure 4.
On this basis, the position deviation caused by the dynamic deformation can be expressed as follows:

IUCT Modelling Considering the Tool Runout
Based on the cutting tool edge trajectory modelling above, the IUCT can be obtained. In the micro-milling process, there are three different forms of material removal mechanisms, as shown in Figure 4. When the IUCT is less than the critical thickness hcr, no chips will be formed, and only elastic deform occurs. Conversely, when the IUCT is greater than the critical thickness, the material is removed by shearing. Additionally, when the IUCT approaches the critical thickness value hcr, elastic and plastic deformation occurs simultaneously. The minimum cutting thickness is determined by the method proposed by Johnson [26] and further supported by the published research results [27].
The IUCT is the distance between the current toolpath and the workpiece surface. As shown in Figure 5, Oi,j−1 and Oi,j represent the central positions of the (j − 1)th and the jth cutting edge trajectory at the height z, respectively. I2 is the intersection point of the straight line Oi,j − I1 and the (j − 1)th cutting edge trajectory. Therefore, the IUCT at φi,j is the distance between point I1 and point I2. When the IUCT is less than the critical thickness h cr , no chips will be formed, and only elastic deform occurs. Conversely, when the IUCT is greater than the critical thickness, the material is removed by shearing. Additionally, when the IUCT approaches the critical thickness value h cr , elastic and plastic deformation occurs simultaneously. The minimum cutting thickness is determined by the method proposed by Johnson [26] and further supported by the published research results [27].
The IUCT is the distance between the current toolpath and the workpiece surface. As shown in Figure 5, O i,j−1 and O i,j represent the central positions of the (j − 1)th and the jth cutting edge trajectory at the height z, respectively. I 2 is the intersection point of the straight line O i,j − I 1 and the (j − 1)th cutting edge trajectory. Therefore, the IUCT at ϕ i,j is the distance between point I 1 and point I 2 . In order to calculate the IUCT, it is necessary to acquire the coordinate values of points I1 and I2. The coordinates of point I1 can be expressed as Equation (11). In order to calculate the IUCT, it is necessary to acquire the coordinate values of points I 1 and I 2 . The coordinates of point I 1 can be expressed as Equation (11).
where x O i,j and y O i,j can be obtained as follows: The coordinate of point I 2 is expressed as Equation (12).
where x O i,j and y O i,j can be obtained as follows: As the three points O i, j , I 1 , and I 2 are on the same straight line, their relationship can be expressed as Equation (13).
The distance between the points I 1 and I 2 is the IUCT that only considers the tool runout at any position angle ϕ i,j .
Based on Equation (12), as long as the value of t i is determined, the coordinate value of point I 2 can be obtained. The Newton iteration method is employed to solve Equation (13). Additionally, the initial value of the iteration t i0 (z) is selected as follows: Once t i is obtained, the IUCT that considers the tool runout error can be obtained by Equation (14).
The simulation flowchart for determining the IUCT is shown in Figure 6. The tool runout error can cause variations in chip thickness, and therefore, some cutting edges may be not engaged in machining. In order to obtain the real IUCT values, the tooth skipping in the cutting process is taken into account. Once is obtained, the IUCT that considers the tool runout error can be obtained by Equation (14).
The simulation flowchart for determining the IUCT is shown in Figure 6. The tool runout error can cause variations in chip thickness, and therefore, some cutting edges may be not engaged in machining. In order to obtain the real IUCT values, the tooth skipping in the cutting process is taken into account.

Micro-Milling Forces Modelling
As the IUCT is greater than the critical value, the material removal mechanism is the same as that of traditional milling. The tangential, radial, and axial forces acting on different cutter micro-elements during micro-milling are expressed in Equation (16).
The cutting force elements consist of a shear force term and a friction force term, where Kte, Kre, and Kae represent the tangential, radial, and axial cutting-edge coefficients, respectively. Ktc, Krc, and Kac represent the tangential, radial, and axial cutting force coefficients, respectively. hj[ti(z)] is the IUCT, and dz is the height of the micro-element.
The tangential and radial cutting forces elements at the angular position φ can be expressed in Cartesian coordinates as Equation (17).
After integrating the element cutting force, the total cutting force can be obtained.

Micro-Milling Forces Modelling
As the IUCT is greater than the critical value, the material removal mechanism is the same as that of traditional milling. The tangential, radial, and axial forces acting on different cutter micro-elements during micro-milling are expressed in Equation (16).
The cutting force elements consist of a shear force term and a friction force term, where K te , K re , and K ae represent the tangential, radial, and axial cutting-edge coefficients, respectively. K tc , K rc , and K ac represent the tangential, radial, and axial cutting force coefficients, respectively. h j [t i (z)] is the IUCT, and dz is the height of the micro-element.
The tangential and radial cutting forces elements at the angular position ϕ can be expressed in Cartesian coordinates as Equation (17).
After integrating the element cutting force, the total cutting force can be obtained.
Due to the low stiffness of the micro-cutting tool and the thin-walled micro-structure, regenerative chatter is unavoidable under the action of periodically milling force, which can cause the change in IUCT, as shown in Figure 7.
Due to the low stiffness of the micro-cutting tool and the thin-walled micro-structure, regenerative chatter is unavoidable under the action of periodically milling force, which can cause the change in IUCT, as shown in Figure 7. Combined with the analysis in Section 2.1, the IUCT considering both the tool runout error and the dynamic regeneration effect can be expressed as Equation (19).

Cutting Force Experiments and Runout Error Measurement
In order to verify the accuracy of the present micro-milling force model, micro-milling thin-walled experiments were carried out on a Kern five-axis machining centre, as shown in Figure 8.  Combined with the analysis in Section 2.1, the IUCT considering both the tool runout error and the dynamic regeneration effect can be expressed as Equation (19).

Cutting Force Experiments and Runout Error Measurement
In order to verify the accuracy of the present micro-milling force model, micro-milling thin-walled experiments were carried out on a Kern five-axis machining centre, as shown in Figure 8.
Due to the low stiffness of the micro-cutting tool and the thin-walled micro-structure, regenerative chatter is unavoidable under the action of periodically milling force, which can cause the change in IUCT, as shown in Figure 7. Combined with the analysis in Section 2.1, the IUCT considering both the tool runout error and the dynamic regeneration effect can be expressed as Equation (19)

Cutting Force Experiments and Runout Error Measurement
In order to verify the accuracy of the present micro-milling force model, micro-milling thin-walled experiments were carried out on a Kern five-axis machining centre, as shown in Figure 8.  The positioning accuracy of the Kern Evo machine in x, y, and z directions can reach 0.1 microns, and the spindle speed can reach up to 50,000 rpm. The workpiece was mounted on the dynamometer by means of a fixture. Additionally, its dimensions are illustrated in Figure 8, where W = 1 mm, B = 100 µm, L = 3 mm, and the height of the thin walls ws 600 µm. Table 1 lists the essential tooling parameters of the carbide micro-milling cutters used in experiments; the type of micro-milling tool was SD AFK40. A Kistler 9256C piezoelectric dynamometer was used to measure the cutting force in x, y, and z directions. It was installed on the machine tool table. The sampling rate for force measurements was 10 kHz. The cutting force data were obtained after processing, as shown in Figure 9. A Kistler 9256C piezoelectric dynamometer was used to measure the cutting force in x, y, and z directions. It was installed on the machine tool table. The sampling rate for force measurements was 10 kHz. The cutting force data were obtained after processing, as shown in Figure 9. The cutting-edge radius was measured by scanning electron microscope, as shown in Figure 10. The workpiece materials were Ti-6Al-4V and Al 6061 in test 1 and test 2, respectively. To validate the proposed mechanistic model, the processing parameters used in test 1 and test 2 are shown in Table 2. The cutting parameters were selected according to the stability prediction model; chatter can be avoided using these parameters [28]. The runout error of the cutting tool was measured with a micrometre, as shown in Figure 11. The micrometre was installed on the machine table and rotated manually. During the rotation process, the result was recorded every 45°, the maximum deviation of the results was the tool runout value, and the corresponding direction angle was the runout direction angle, as shown in Figure 12 [6]. The cutting-edge radius was measured by scanning electron microscope, as shown in Figure 10. The workpiece materials were Ti-6Al-4V and Al 6061 in test 1 and test 2, respectively. To validate the proposed mechanistic model, the processing parameters used in test 1 and test 2 are shown in Table 2. The cutting parameters were selected according to the stability prediction model; chatter can be avoided using these parameters [28].    The runout error of the cutting tool was measured with a micrometre, as shown in Figure 11. The micrometre was installed on the machine table and rotated manually. During the rotation process, the result was recorded every 45 • , the maximum deviation of the results was the tool runout value, and the corresponding direction angle was the runout direction angle, as shown in Figure 12 [6].

Identification of Dynamic Parameters
The cutting force model proposed in this paper requires the dynamic parameters of the micro-milling system. Since the micro-milling tool and thin-walled micro-parts are small in size and inconvenient to install sensors, the receptance coupling method was employed to determine the corresponding dynamic parameters.
The response of a structure refers to the relationship between the force exerted on the structure and the deformation of the structure. The receptance coupling method refers to dividing the structure into several substructures; then, the dynamic parameters of each substructure were obtained individually. Finally, the dynamic parameters were assembled. In the micro-milling system, the dynamic parameters of the spindle were obtained by the modal experiment, and the dynamic parameters of the micro-tool and the thin-

Identification of Dynamic Parameters
The cutting force model proposed in this paper requires the dynamic parameters of the micro-milling system. Since the micro-milling tool and thin-walled micro-parts are small in size and inconvenient to install sensors, the receptance coupling method was employed to determine the corresponding dynamic parameters.
The response of a structure refers to the relationship between the force exerted on the structure and the deformation of the structure. The receptance coupling method refers to dividing the structure into several substructures; then, the dynamic parameters of each substructure were obtained individually. Finally, the dynamic parameters were assembled. In the micro-milling system, the dynamic parameters of the spindle were obtained by the modal experiment, and the dynamic parameters of the micro-tool and the thin-

Identification of Dynamic Parameters
The cutting force model proposed in this paper requires the dynamic parameters of the micro-milling system. Since the micro-milling tool and thin-walled micro-parts are small in size and inconvenient to install sensors, the receptance coupling method was employed to determine the corresponding dynamic parameters.
The response of a structure refers to the relationship between the force exerted on the structure and the deformation of the structure. The receptance coupling method refers to dividing the structure into several substructures; then, the dynamic parameters of each substructure were obtained individually. Finally, the dynamic parameters were assembled. In the micro-milling system, the dynamic parameters of the spindle were obtained by the modal experiment, and the dynamic parameters of the micro-tool and the thin-walled micro-parts were obtained by the finite element method; then, the frequency response function of the machining system could be obtained.
As shown in Figure 13, substructure A is the spindle, and substructure B is the micromilling cutter. walled micro-parts were obtained by the finite element method; then, the frequency response function of the machining system could be obtained. As shown in Figure 13, substructure A is the spindle, and substructure B is the micromilling cutter.
The transfer function of the whole system can be expressed as Equation (20).  (20) where G and H are the transfer functions of the assembled structure and substructures, respectively [24]. X and F are the deformation and the cutting force, respectively.
In the modal experiment, the Brüel and Kjaer 8206-001 impact hammer was used to generate the input excitation signal, and the Brüel and Kjaer 4508-B-001 accelerometer was used to gain the output signal. The acquired signals were processed by a dynamic signal analyser, as shown in Figure 14.  The transfer function of the whole system can be expressed as Equation (20).
where G and H are the transfer functions of the assembled structure and substructures, respectively [24]. X and F are the deformation and the cutting force, respectively. In the modal experiment, the Brüel and Kjaer 8206-001 impact hammer was used to generate the input excitation signal, and the Brüel and Kjaer 4508-B-001 accelerometer was used to gain the output signal. The acquired signals were processed by a dynamic signal analyser, as shown in Figure 14. used to gain the output signal. The acquired signals were processed by a dynamic signal analyser, as shown in Figure 14.  The obtained modal parameters in various directions, including natural frequency ω n , stiffness K, and damping coefficients are listed in Table 3. The subscripts of x and y, t, and w denote the tool and workpiece, respectively.  The obtained modal parameters in various directions, including natural frequency ω n , stiffness K, and damping coefficients c are listed in Table 3. The subscripts of x and y, t, and w denote the tool and workpiece, respectively. After obtaining the modal parameters, the transfer function can be expressed as follows: where H denotes transfer function, K denotes dynamic stiffness, F denotes dynamic cutting force, and X denotes dynamic deformation. All the above variables are in the frequency domain. Moreover, vibration isolation was performed around the machine tool. Therefore, the vibration of other equipment could not be transmitted to the machine tool, and only the vibrations caused by its inner excitation were considered in the article.

Results and Discussion
The cutting force is periodic in the process of micro-milling. Therefore, only two cycles of the simulation results were selected to compare with the experimental results. The comparison results are shown in Figures 15 and 16. The cutting force model that considered the tool runout error and dynamic deformation error simultaneously is denoted as Model 1, and the cutting force model that only considers the tool runout error is denoted as Model 2, and the cutting force under the ideal condition in which there is no tool runout and chattering is denoted as Model 3. Before the cutting tests, the clamping force acting on the workpiece was measured, while no cutting was performed. It can be seen that the amplitudes of clamping forces in the x, y, and z directions are less than 0.08 N in Figure 17, which is smaller than 4.1% and 5.3% of the cutting forces amplitudes in tests 1 and 2, respectively. Therefore, the influence of the clamping force can be neglected. ered the tool runout error and dynamic deformation error simultaneously is denoted as Model 1, and the cutting force model that only considers the tool runout error is denoted as Model 2, and the cutting force under the ideal condition in which there is no tool runout and chattering is denoted as Model 3. Before the cutting tests, the clamping force acting on the workpiece was measured, while no cutting was performed. It can be seen that the amplitudes of clamping forces in the x, y, and z directions are less than 0.08 N in Figure  17, which is smaller than 4.1% and 5.3% of the cutting forces amplitudes in tests 1 and 2, respectively. Therefore, the influence of the clamping force can be neglected.    It can be observed from the results that the cutting force Model 1 and the cutting force Model 2 are in good agreement with the experimental results in trend. Moreover, it can be concluded that the prediction accuracy of cutting force Model 1 is higher than that of Model 2, which means the dynamics of the machining system has a considerable influence It can be observed from the results that the cutting force Model 1 and the cutting force Model 2 are in good agreement with the experimental results in trend. Moreover, it can be concluded that the prediction accuracy of cutting force Model 1 is higher than that of Model 2, which means the dynamics of the machining system has a considerable influence on the cutting force. In addition, at the peaks and troughs of the cutting force, the errors between the predicted values and experimental values are relatively large, which can be caused by tool overhang measurement errors and tool geometry manufacturing errors. Due to the difficulty in identifying the bottom end of the small-diameter tool, the distance from the tool clamping position to the bottom end is difficult to measure, which results in errors during the simulation process. Additionally, the cutting forces at the peaks and troughs are relatively larger. Therefore, the deformation of the cutting tool and workpiece is greater, so are the cutting force errors.
A significant difference is observed between the experimental results and the models at the F z force. The factors such as noise and random vibration in the micro-milling process have a considerable influence on the z-direction cutting force component due to its small amplitude. Therefore, the measurement signal fluctuates constantly. In comparison, the force signal is significantly attenuated when a low-pass filter is used for noise reduction. It can be seen from the comparison results that the measured force signal and the simulated force signal have good consistency in the overall trend. Figure 18 shows the root mean square error between the predicted cutting force and the experimental value of test 1. From Figure 18, Model 1 is found to have smaller RMS values in the x, y, and z direction, and resultant force, generally. on the cutting force. In addition, at the peaks and troughs of the cutting force, the errors between the predicted values and experimental values are relatively large, which can be caused by tool overhang measurement errors and tool geometry manufacturing errors. Due to the difficulty in identifying the bottom end of the small-diameter tool, the distance from the tool clamping position to the bottom end is difficult to measure, which results in errors during the simulation process. Additionally, the cutting forces at the peaks and troughs are relatively larger. Therefore, the deformation of the cutting tool and workpiece is greater, so are the cutting force errors. A significant difference is observed between the experimental results and the models at the Fz force. The factors such as noise and random vibration in the micro-milling process have a considerable influence on the z-direction cutting force component due to its small amplitude. Therefore, the measurement signal fluctuates constantly. In comparison, the force signal is significantly attenuated when a low-pass filter is used for noise reduction. It can be seen from the comparison results that the measured force signal and the simulated force signal have good consistency in the overall trend. Figure 18 shows the root mean square error between the predicted cutting force and the experimental value of test 1. From Figure 18, Model 1 is found to have smaller RMS values in the x, y, and z direction, and resultant force, generally.

Conclusions
In this paper, a mechanistic micro-milling forces modeling method was presented, particularly applicable for thin-walled micro-part fabrication. The modelling considers both the influence of the tool runout error and the dynamic regenerative effect. The proposed model was verified by two groups of micro-milling experiments. The cutting force

Conclusions
In this paper, a mechanistic micro-milling forces modeling method was presented, particularly applicable for thin-walled micro-part fabrication. The modelling considers both the influence of the tool runout error and the dynamic regenerative effect. The proposed model was verified by two groups of micro-milling experiments. The cutting force model considering only the influence of the tool runout and the cutting force model under the ideal conditions were also simulated as references. The proposed cutting force model would be helpful for processing precision and machining monitoring in the process of micromachining thin-walled parts. Based on the obtained results, the following conclusions can be drawn: (1) In the micro-milling of thin-walled parts, the actual trajectories of the cutting edge were analysed by considering the tool runout error and the dynamic deformation. On this basis, the instantaneous undeformed chip thickness model was established, which also considers the "tooth skipping" phenomenon.