Metal Cutting Theory

Based on mapping method and the principle of non-Euclidean geometry, this chapter presents a novel method for expression, analysis and calculation of space geometric angles. According to this method, a space inclined plane is mapped onto a projective plane as an image point, and a space line is mapped on the projective plane as an image line. The problem of calculating the angle between two space inclined planes is thus converted into the problem of calculating the non-Euclidean distance between their image points on the projective plane, and the problem of calculating the angle between two space inclined lines is converted into the problem of calculating non-Euclidean angle of their image lines. The problem of a rigid body rotating in the space is converted into the problem of moving and rotating in the projective plane of its image diagram. Compared with conventional ways, this method is simple and intuitive. It is a more effective approach for designing and calculating the cutting tool angles or angles in machine parts.

In this chapter, the space planes and lines of various cutting tools are mapped onto a plane, so-called projective plane, and then the angles of the cutting tools are analyzed and calculated through the images on the plane by means of the method introduced in last chapter. A series of examples presented in this chapter has shown the validity and visualization of such planar image method. This method explains the profound in straightforward terms, and turns difficult to easy. It can visually synthesize the relationship between the various geometric angles and it’s straightforward to analyze the influence of the rotation of the cutting tool in space on its angles. This method has provided a mathematical model and a theoretical basis for the analysis and calculation of tool working angles and dynamic angles and for their on-line monitoring and optimizing design.

Standard twist drill, under the standard grinding its flank, has a pair of straight major edges. Various new re-sharpening methods, including Qun-drill, however, break the “straight edge” restrictions, re-sharpening the major cutting edges of standard twist drill into a variety of curves to improve its drilling performance. This chapter, with the help of previously introduced method of mapping space angles onto a projective plane, made an in depth study on the cutting edge shape of twist drill, cutting angle distribution along the curved cutting edge and the relationship between cutting edge shape and drilling performance. Study reveals: there is a special cutting edge allowing the rake at each point on it reaches the possible maximum value at the point, thus enabling significantly improved machining ability. This chapter provides the technology for optimizing cutting edge shape to obtain desired rake angle distribution, as well as the technology through the drill flank design and grinding in order to control distribution of drill clearance. Theoretical analysis and experimental research have been performed. Computer software was developed, so that the cutting edge shape realizing required rake angle distribution is calculated.

In the vicinity of original points, the geometrical property of an elliptic plane is very close to that of Euclidian plane, so, when the obliquities of all the planes and lines involved are relatively small we can use Euclidian geometry to approximately replace non-Euclidian geometry and thus greatly simplify the procedure of analyses and calculation or graphic determination of space angle problem with necessary precision; or at least, this approximate approach method can display the varying trend of various angles in mutual effects and it is convenient for optimization tool design by means of error and try.

In this chapter based on experimental and theoretical studies, a principle of minimum energy dissipation is suggested as a basic physical law restricting the process of metal cutting. The principle may be summarized as that the state parameters of a cutting process, under the precondition of satisfying the constraints set by controlling (technological) parameters, always minimize cutting power consumption. On account of this principle, the basic law of cutting mechanics has been deeply studied. The results of theoretical analysis and predictions are verified and supported by experimental facts, and agree with people’s traditional understanding and knowledge about cutting process. Besides giving a general description of the principle of minimum energy dissipation, this chapter focuses on the principle of minimum energy dissipation for the cutting process of an elementary cutting tool, as a basis for further study of the cutting performance of modern complex shaped cutting tools.

Based on the principle of minimum energy dissipation in metal cutting process this chapter has studied the basic characteristics of non-free cutting, probed into the phenomena of chip-ejection interference and chip-ejection compromise widely existing in modern machining processes. A non-linear synthetic method of elementary cutting tools is proposed for establishing the cutting models of complex tools. The general equation governing chip-ejection motion is deduced and its solutions are studied. On the basis of the general equation some examples of non-free cutting are analyzed and the theoretically analyzing and predicting results are verified by experimental data and facts. The sufficient and necessary condition of eliminating chip-ejection interference and realizing free cutting is explored, the basic ideal and the technique route of “Free Cutting Method” are discussed. The research has provided a feasible way for improving and optimizing modern cutting tools.

Based on the Non-linear Synthetic Method of Elementary Cutting Tools described in last chapter, and taking drilling force model as an example, this chapter will talk about the method of establishing a “functional model”. The independent variable of a functional model is some kind of function, which is able to reflect the influence of geometric shape of a cutting tool on its cutting performance. Functional model is a bridge between the basic cutting theory and the cutting processes of modern complex shaped cutting tools, and therefore such kind of model has provide an approach for modeling, evaluating and optimizing advanced cutting tools. Traditional “parameter models” and the modeling method based on parameters are impossible or difficult to do so.

By means of theoretical analysis it is predicted that there must be Bifurcation and Catastrophe phenomena existing in cutting process, and we did find the phenomena in our cutting experiments. The root of the phenomena is the strong non-linearity of non-free cutting process. This finding is beyond people’s traditional knowledge and understanding of metal cutting process, but has been fully verified by experimental facts. The research has revealed the complexity of a cutting process and the difficulty to control it, deepened people’s understanding of the technological process of metal cutting, and required people to renew conception and strategy of machining process design and chip-control.

On the basis of in-depth study of the essence and the law of non-free cutting, in this chapter, a method of free cutting is proposed, and thus a feasible way is found for improving advanced cutting tool design and for optimizing machining process. The key point is properly designing the tool rake face, so as to dredge chip-ejections from different parts of edge line and to eliminate or to reduce the interference in chip-ejection, realizing free-cutting or quasi-free-cutting. The principle and the basic law of free-cutting tool design is proposed and experimental verifications are presented.

In this Chapter, distinguishing experiments were performed, so as to verify which of the rakes: the rake in the direction of chip-ejection γ _ψ , edge normal rake γ _n or orthogonal rake γ _o is the basic parameter determining cutting state and machining effect. From the experimental results, a constraint condition to the designing free-cutting tools will be drawn forth. Together with the basic law of free-cutting tool design presented in last chapter, they will lay foundation for the design and optimization of free-cutting tools. On the basis of these achievements, a new round development of free-cutting cemented carbide inserts was performed and their superiority will be experimentally verified.

During machining process, vibration may be generated, that is, besides nominal cutting motion, there is also a periodical motion superposed between cutter and work-piece. Cutting vibration may lead to a series of undesirable effects, and sometimes, may cause serious results. First of all, the relative motion between the cutter and the work-piece will deteriorate the surface quality of machined parts, affect the performance of the parts, and shorten their life. Secondly, while vibrating, the real instantaneous thickness of cut may fluctuate about the set nominal thickness of cut. And this leads to a periodical alternating dynamic cutting force, the amplitude of it is even bigger than the static cutting force. This alternating dynamic cutting force alternating makes the working portion of the tool fatigued, causes tool tip breaking, and speeds up the wear of the machine tool parts, makes the connections between parts loose, and the machine tool thus loses its precision. In addition, severe vibration also generates big noise, pollutes the environment and hazards operator’s health. The vibration has frequently become one of the main barriers, which limits the productivity. In this Chapter, the outline of linear theory of machine tool chatter are briefly introduced as the foundation necessary for readers to understand the author’s research achievements of non-linear chatter theory, which will be explained in next chapters.

Firstly, a nonlinear chatter model proposed by other researchers was introduced, which reflects the nonlinear stiffness of machine tool structure. The achievement of this model was discussed and the difficulties and problems were pointed out. Then, a new nonlinear chattering model established by the author was introduced. The model is based on a nonlinear factor of tool jumping out of work-piece material, when vibration is severe enough. The characteristics and conclusions of this model, such as amplitude stability and the influence of the cutting parameters on the stabilized amplitude, were discussed in detail.

In this chapter, two basic nonlinear factors in cutting process are taken into consideration at the same time: the pass of the cutting edge partly exceeds the work-piece material and the cutting force nonlinearly dependents on the thickness of cut, and a more perfect theoretical model about self-excited vibration of machine tool is established. This model can explain and predict the two important phenomena of cutting chatter: amplitude stability and the phenomenon of machine tool chatter induced by external shock. The latter is known as “finite amplitude stability”. These phenomena is unexplainable by traditional linear theory. This model leads to some meaningful conclusions, one of the most important is: under certain conditions, increase of the machine load (rather than decrease load), is helpful to suppress chatter, or even completely eliminate the flutter, and the cutting process is stabilized. The theoretical prediction results are in accordance with the experimental data.

Based on the nonlinear theory of machine tool chatter described in last two chapters, this chapter studies the early diagnosis and online control of chatter. First, through the experimental study, the appropriate monitoring signal is selected to monitor the stability of a cutting system. Then, the characteristic changes of the monitoring signal in the transition process from stable to unstable of a cutting system are analyzed, and the characteristics quantities of the system reflecting the degree of tending to instability are selected and their quick algorithm are established. The methods and strategies of the early diagnosis of chatter were discussed. The effectiveness of on line adjusting the cutting parameters to suppress the chatter is experimentally verified and the adjustment strategy established. Finally, an experimental system of on-line cutting chatter monitoring and controlling is established, and good experimental results are obtained.

The modern CNC machine tools have provided a new idea and new possibility for the on-line cutting chatter control—to suppress the chatter with the small continuous periodic disturbance of the spindle speed. This approach is easy to implement and does not require any additional hardware or devices. Moreover, theoretical analysis and cutting test have fully demonstrated the effectiveness of this method.