International Journal of Machine Tools and Manufacture
Simulation and experimental investigation of the end milling process considering the cutter flexibility
Introduction
The milling process is widely used in many industrial departments, such as automotive, aerospace and die/mould manufacturing etc., thus it is very important to study how to improve the throughput and part quality. Surface quality directly influences the working performance of the tooling; therefore, it has been the focus of much research in recent years.
Precise and reasonable modelling of the milling process is the key to studying milled surface topography. The milling process models that are used for predicting the milled surface topography can be categorized into three types. The first one is the “Instantaneous Rigid Force, Static Deflection Model” [1], which does not consider the dynamic properties of the “machine/tool/workpiece” system or the effects of the deflection of the cutter and workpiece on the instantaneous undeformed chip thickness. Based on it, Kline et al. [2] predicted the surface error by calculating the deflection of the cutter and workpiece at the tool point where the new surface was generated. The second one is the “Instantaneous Force with Static Deflection Feedback” model [1]. Based on this model, Sutherland and DeVor [3] developed an improved method for cutting force and surface error prediction in flexible end milling systems, which only considered the effects of static deflection of the cutter on the instantaneous chip thickness. In pursuit of the calculation of the flexible system chip loads, an iterative procedure was used to balance the forces and deflections generated during the milling process [3]. The third one is the “Regenerative Force, Dynamic Deflection Model” [1], which considered the cutting dynamics and simplified the milling system as a planar 2-degree of freedom system. Based on this model, Smith and Tlusty [4], and Montgomery and Altintas [5] predicted the milled surface roughness and wave while Ismail et al. [6] considered the effects of the tool dynamics and wear on the milled surface, and presented a mechanistic model for surface generation. However, this type of milling process model can’t be used to evaluate the geometric shape error caused by the static deflection of the cutter. As a matter of fact, with a long slender mill, the static and the dynamic deflection of the cutter are both very important and non-negligible when considering surface generation.
Due to the high rigidity of modern numerically controlled machining centers and when the ratio of the length to the diameter of the cutter is large, the main factors influencing the milled surface topography are the static and dynamic deflections of the cutter. Thus, in order to overcome the shortcomings stated above, the regenerative effect of the static as well as the dynamic deflections of the cutter on the transient undeformed chip thickness should be taken into consideration. If the workpiece is thin, its dynamics also should be taken into consideration. Therefore, in this paper, the “cutter” is simplified and regarded as a flexible cantilever beam acted on by a dynamically distributed cutting force, which vibrates near its statically balanced position in two mutually perpendicular directions.
In section 2, according to the basic assumption for a linear elastic body, the force acting on the cutter and its deflection are both divided into two parts: the static and the dynamic. In section 3, applying differential geometry theory, and considering the regenerative feedback in the practical milling process, the dynamic milling force acting on a unit length of the cutter is derived. In section 4, a new kind of dynamic deflection model of a cutter is established, and a new set of efficient numerical simulation algorithms are presented. The equations for the static deflection of a cutter are formulated in section 5. By considering the cutter flexibility, a surface generation model is formulated in section 6. The comparative assessment of simulation and experimental results is given in section 7. And last, the conclusions are given in section 8.
Section snippets
Analysis of the cutter deflection
Fig. 1 (a) shows an end mill with an effective length L. The section of the cutter that engages the workpiece is acted on by two mutually perpendicular distributed milling forces and . Oz represents the position of the cutter axis without deflection, Oz′ represents the static balanced position of the cutter axis under the action of the static milling force. Let the fixed end of the cutter (O) be the coordinate origin, then establish a left-handed system Oxyz, in order to be
Entrance and exit angles
As shown in Fig. 2, the entrance and exit angles can be determined as follows. [7]
Intensity of dynamic milling force
Suppose that the cutter vibrates near its statically balanced position. Take a cross-section at the elevation h
Governing equation of a small element of the cutter under transverse vibration
In expression (9), letwhere and are the unit vectors in the positive directions along the three coordinate axes in the inertial frame Oxyz; Fx,Fy and Fz are the and components of the left-hand resultant vector.
Then, the expression (9) can be written in the form of components. That is to say, the intensity of the dynamic milling forces (Fd,u and Fd,v) acting on the cutter in the xOz and yOz planes can be expressed as
Calculation of the static deflection of the cutter
As stated above, in a practical milling process, the overall deflection of the cutter can be regarded as the sum of the static and the dynamic deflection. The sections above have dealt with the dynamic milling force and dynamic deflection of the cutter. In this section, the calculation of the static deflection of the cutter will be derived.
In an ideal milling process without vibration, suppose that the static milling forces in the x and y directions are uniformly distributed on the engaging
A surface generation model by considering the cutter flexibility
In order to derive the trajectory equation of the cutting edge relative to the workpiece in the peripheral milling process considering the kinematic characteristics of the peripheral milling process, it is necessary to establish five sets of Cartesian coordinate systems as shown in Fig. 5. [7], [11] and [12]. These are
- 1.
O0x0y0z0 system: The inertial reference system fixed on the workpiece in working space.
- 2.
O1x1y1z1 system: The coordinate system attached to the spindle, which translates relative to
Validation of the model
Tests and simulations were carried out to validate the model presented above. To show the deflection of the cutter and the surface topography clearly, an up-milling process with a small-diameter end-mill with a larger feed than normal was considered. Aluminum was used as the workpiece. The parameters used are listed in Table 1.
From [7], the effective cross-sectional coefficient of the cutter (ζ) was 8.728. The effective length of the cutter was 80 mm, and the first order inherent frequency of
Conclusions
The main conclusions that can be drawn from this work are as follows.
- 1.
Under stable cutting conditions, the static deflection of the cutter is the most significant deflection. Therefore, if only the static deflection is taken into account, the error of the predicted result will not be too great.
- 2.
Under unstable cutting conditions, the errors of the predicted surface profiles relative to the measured values at the bottom, middle and top elevations were all less than 20 percent.
- 3.
In an up-milling
Acknowledgements
This work was funded by the National Natural Science Foundation of China, the Natural Science Foundation of Hebei Province grant No. 500046 and the Doctoral Foundation of Education Committee of Hebei Province.
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