Bifurcation and Catastrophe in Metal Cutting Process

Abstract

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.

Appendix 9.1

Deduction of Eq. (9.1).

A non-free cutting model with double edged cutting tool was presented in this chapter, as shown in Fig. 9.1, and its mathematical presentation is Eq. (9.1). Here, we will give a brief deduction of Eq. (9.1).

The unit vector in the direction of resultant force on rake face is

$${\mathbf{R}}_{{\mathbf{1}}} = - \sin \beta \,\sin \psi_{\lambda } \, {\mathbf{i}} - \sin \beta \,\cos \psi_{\lambda } {\mathbf{j}} + \cos \beta \, {\mathbf{k}}\text{,}$$

(9.6)

where, β is the friction angle on the interface between the rake and the chips, and i, j, k are the unit vectors in the directions of axes x, y, z, respectively (they are not shown in Fig. 9.1). The resultant force imposed on chip by the rake is;

$$\varvec{R} = R{\mathbf{R}}_{{\mathbf{1}}} ,$$

(9.7)

The vector of cutting velocity is,

$$\varvec{ V} = V{\mathbf{k}}\text{,}$$

(9.8)

The vector of chip-ejection velocity is,

$$\varvec{U} = r\varvec{V}(\sin \psi_{\lambda } {\mathbf{i}} + \cos \psi_{\lambda } {\mathbf{j}}),$$

(9.9)

where, r is cutting ratio. The vector of shear velocity is,

$$\varvec{W} = \varvec{U} + \varvec{V} = V(r\,\sin \psi_{\lambda } {\mathbf{i}} + r\,\cos \psi_{\lambda } {\mathbf{j}} + {\mathbf{k}}).$$

(9.10)

Shear speed, i.e., the module of vector W is,

$$\left| \varvec{W} \right| = V\sqrt {1 + r^{2} }$$

(9.11)

The unit vector in the shear direction is,

$${\mathbf{W}}_{{\mathbf{1}}} = \frac{{r\,\sin \psi_{\lambda } {\mathbf{i}} + r\,\cos \psi_{\lambda } {\mathbf{j}} + {\mathbf{k}}}}{{\sqrt {1 + r^{2} } }}$$

(9.12)

The shear force, i.e., the projection of R on the shear direction W ₁ ,

$$\varvec{R} \cdot {\mathbf{W}}_{{\mathbf{1}}} = R({\mathbf{R}}_{{\mathbf{1}}} \cdot {\mathbf{W}}_{{\mathbf{1}}} ) = R\frac{(\cos \beta - r\,\sin \beta )}{{\sqrt {1 + r^{2} } }} = \tau S,$$

(9.13)

where, τ is shear stress of the work-piece material, and S is the area of shear plane. So,

$$R = \frac{{\tau S\sqrt {1 + r^{2} } }}{(\cos \beta - r\,\sin \beta )}.$$

(9.14)

The cutting power is,

$$A = R \cdot \,\cos \beta \cdot V = \frac{V \cdot \tau \cdot \,\cos \beta }{(\cos \beta - r\,\sin \beta )}\sqrt {1 + r^{2} } \cdot S.$$

(9.15)

Shear plane is divided into two parts: the shear plane of major edge and the shear plane of minor edge, the areas of them are denoted as S _I and S _II , respectively. The calculation of S _I and S _II are divided into three cases:

The first case: tan ψ _λ < a _w/ a _c , in this case, vector W intersects with the top surface AFCE of the work-piece, and the intersection point is P. What is shown in Fig. 9.1 is just this case. The coordinates of point P In the coordinate system xyz are $P_{x} = a_{c} \tan \psi_{\lambda } \,$, $P_{y} = \, a_{\text{c}} \,$ and $P_{z} = \, {{a_{\text{c}} } \mathord{\left/ {\vphantom {{a_{\text{c}} } {(r\,{ \cos }\,\psi_{\lambda } )}}} \right. \kern-0pt} {(r\,{ \cos }\,\psi_{\lambda } )}}$. In this case, S _I and S _II are shown as Fig. 9.14a, b and the total area of the two shear plane

Fig. 9.14

The shear planes (tan ψ _λ  < a _w /a _c ). a Major shear plane. b Minor shear plane

$$\begin{aligned} S & = S_{\text{I}} + S_{\text{II}} = \, \square BCEO - \Delta OPE + \Delta OAP \\ & = a_{c} (a_{w} - \frac{1}{2}a_{c} \tan \psi_{\lambda } )\sqrt {1 + \frac{1}{{r^{2} \cos^{2} \psi_{\lambda } }}} + \frac{1}{2}\frac{{a_{c}^{2} }}{{r\,\cos \psi_{\lambda } }}\sqrt {1 + r^{2} \sin^{2} \psi_{\lambda } } . \\ \end{aligned}$$

(9.16)

The second case: tan ψ _λ > a _w/ a _c , in this case, vector W intersects with the side surface FCDB of the work-piece, and the intersection point is P. The coordinates of point P In the coordinate system xyz are $P_{x} = a_{w}$, $P_{y} = \, {{a_{w} } \mathord{\left/ {\vphantom {{a_{w} } {{ \tan }\psi_{\lambda } }}} \right. \kern-0pt} {{ \tan }\psi_{\lambda } }} \,$ and $P_{z} \, = {{a_{w} } \mathord{\left/ {\vphantom {{a_{w} } {\text{(}r\,{ \sin }\psi_{\lambda } }}} \right. \kern-0pt} {\text{(}r\,{ \sin }\psi_{\lambda } }})$. In this case, S _I and S _II are shown as Fig. 9.15b and c, and the total area of the two shear plane is,

Fig. 9.15

The area of cut and shear planes (tan ψ _λ  > a _w /a _c ). a Chip sections. b Major shear plane. c Minor shear plane

$$\begin{aligned} S & = S_{\text{I}} + S_{\text{II}} = \Delta OBP +\square ACDO - \Delta OPD \\ & = \frac{{a_{w} }}{{r\,\sin \psi_{\lambda } }}\left[ {a_{c} \sqrt {1 + r^{2} \sin^{2} \psi_{\lambda } } + \frac{1}{2}a_{w} \left( {\sqrt {1 + r^{2} \cos^{2} \psi_{\lambda } } - \sqrt {\cot^{2} \psi_{\lambda } + r^{2} \cos^{2} \psi_{\lambda } } } \right)} \right]. \\ \end{aligned}$$

(9.17)

The third case, tan ψ _λ  = a _w/ a _c , it is a critical case between the above two cases, the vector W intersects with the side ridge EC of the work-piece, and the intersection point is P, which coincides with the corner point C as shown in Fig. 9.16a. The coordinates of point P in the coordinate system xyz are $P_{x} = a_{w}$, $P_{y} = \, a_{c} \,$ and $P_{z} \, = {{a_{w} } \mathord{\left/ {\vphantom {{a_{w} } {\text{(}r\,{ \sin }\psi_{\lambda } }}} \right. \kern-0pt} {\text{(}r\,{ \sin }\psi_{\lambda } }}) = {{a_{c} } \mathord{\left/ {\vphantom {{a_{c} } {\text{(}r\,{ \sin }\psi_{\lambda } }}} \right. \kern-0pt} {\text{(}r\,{ \sin }\psi_{\lambda } }})$. In this case, S _I and S _II are shown as Fig. 9.15b, c, and the total area of the two shear plane is,

Fig. 9.16

The area of cut and shear planes (tan ψ _λ  = a _w /a _c ). a Chip sections. b Major shear plane. c Minor shear plane

$$S = S_{\text{I}} + S_{\text{II}} = \Delta BPO + \Delta PAO = \frac{1}{2r}\left[ {a_{c} \sqrt {a_{w}^{2} (1 + r^{2} ) + a_{c}^{2} } + a_{w} \sqrt {a_{c}^{2} (1 + r^{2} ) + a_{w}^{2} } } \right],$$

(9.18)

Substituting Eqs. (9.6)–(9.8) into Eq. (9.5), respectively, and considering the physical unit conversion result in the Eq. (9.1), i.e., the expression of cutting power .