CN105824289A - General method for machining complex curved surface of non-spherical cutter in multi-axis-linkage CNC manner - Google Patents

Abstract

The invention discloses a general method for machining the complex curved surface of a non-spherical cutter in the multi-axis-linkage CNC manner. The method comprises the steps of (1) designing the curved-surface generating motion of a cutter relative to a workpiece: firstly, the curved surface of the cutter is described based on a vector function having continuous third-order partial derivatives, and then an envelope surface equation formula for the curved surface of the cutter is obtained with boundary conditions taken into consideration; secondly, a motion optimization functional extremum model formula of the cutter relative to the workpiece and having a minimum machining error, and a motion optimization functional extremum model formula of the cutter relative to the workpiece and having a maximum machining line width are obtained; finally, the above functional extremum model formulas are solved to respectively obtain a motion of the cutter relative to the workpiece at a given machining line width and a minimum curved-surface machining error, and a motion of the cutter relative to the workpiece at a maximum machining line width and a given limit error; (2) realizing the curved-surface generating motion of the cutter relative to the workpiece on a specific machine tool. According to the technical scheme of the invention, the machining accuracy and the machining efficiency of the curved surface are improved through fully developing the machining potential of the multi-axis-linkage CNC.

Description

The universal method that aspheric surface cutter multi-axis NC machining is complex-curved

Technical field

The present invention relates to the universal method that a kind of wide-line processing is complex-curved, particularly relate to the universal method that a kind of aspheric surface cutter multi-axis NC machining is complex-curved.

Background technology

At present, widely used in complex-curved digital control processing is ball head knife.Ball head knife flexibility (adaptability) of operation is strong, tool path scheduling simple, theoretically, only needs three-shaft linkage just can process arbitrarily complicated curved surface.But, ball head knife cannot change machined strip width and machining accuracy by cutter spacing and pose adjustment, thus, machining accuracy and efficiency comparison are low.Five axles and above multi-shaft linkage numerical control machine multifreedom motion function is utilized suitably to adjust cutter spacing and the attitude of aspheric surface cutter, it is possible to obtain optimum machined strip width and machining accuracy.But, owing to tool motion complicated in five axles and above multiaxis NC maching to be lacked the description method of unified standard, and the problem relatively button tool processing such as the tool path scheduling of aspheric surface cutter, precision controlling and interference checking want complicated a lot, at present, the research of this respect is only limitted to for certain types of cutter, the research method using approximation to simplify, is formed without general tool motion and controls optimum theory.

Summary of the invention

The purpose of the present invention is that the universal method providing a kind of aspheric surface cutter multi-axis NC machining complex-curved in order to solve the problems referred to above.

The present invention is achieved through the following technical solutions above-mentioned purpose:

The present invention comprises the following steps:

(1) the curved surface generating motion of design cutter opposite piece:

(1) by the cutter curved surface vector function Σ with continuous print three rank partial derivative_t:r_t=r_t(u_t,υ_t) describe, wherein, u_t,υ_tConstitute Orthogonal Parameter net, by the processed curved surface vector function Σ with continuous print three rank partial derivative_p:r_p=r_p(u_p,υ_p) describe, wherein, u_p,υ_pConstitute Orthogonal Parameter net.By the cutter-contact point trace curve L in processed curved surface_pUse vector function r_pM=r_pM(u_pM(s_p),υ_pM(s_p)) describe, wherein, s_pFor curve L_pArc length parameters；

(2) by the curved surface moving frame of cutter curved surface；

$S_{ft}\[\begin{array}{cccc}M;& e_1=\frac{r_1}{\sqrt{E}}& e_2=\frac{r_2}{\sqrt{G}}& e_3=n_{tM}=e_1\×e_2\end{array}\]r_1=\frac{{\mathrm{dr}}_t}{du},r_2=\frac{{\mathrm{dr}}_t}{d\mathrm{\υ}},E=r_1\·r_1,G=r_2\·r_2;$

Moving frame with cutter-contact point trace curve；

$S_{fp}\[\begin{array}{cccc}M;& \mathrm{\α}& v=n_{pM}\×\mathrm{\α}& n_{pM}\end{array}\]\mathrm{\α}=\frac{{\mathrm{dr}}_p}{{\mathrm{ds}}_p},n_{pM}=(\frac{{\mathrm{dr}}_p}{{\mathrm{du}}_p}\×\frac{{\mathrm{dr}}_p}{{\mathrm{d\υ}}_p})/\left|\frac{{\mathrm{dr}}_p}{{\mathrm{du}}_p}\×\frac{{\mathrm{dr}}_p}{{\mathrm{d\υ}}_p}\right|;$

And consider boundary condition: along cutter-contact point trace curve cutter envelope of surfaces face and processed curved surface, there is second order contact, curved surface generating motion and the movement velocity thereof of cutter opposite piece are expressed as cutter curved surface, processed curved surface and the function of cutter-contact point trace curve intrinsic geometry amount thereof, equation below 1 and formula 2:

$M_{pt}=\left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1y}& -(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z})\\ e_{2x}& e_{2y}& e_{2z}& -(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z})\\ n_{tMx}& n_{tMy}& n_{tMz}& -(x_{tM}{e}_{tx}+y_{tM}{e}_{ty}+z_{tM}{n}_{tz})\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \mathrm{\Δ}& -\sin \mathrm{\Δ}& 0& 0\\ \sin \mathrm{\Δ}& \cos \mathrm{\Δ}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}{\mathrm{\α}}_x& v_x& n_{pMx}& x_{pM}\\ {\mathrm{\α}}_y& v_y& n_{pMy}& y_{pM}\\ {\mathrm{\α}}_z& v_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right];$ Formula 1

Δ is by boundary condition: has second order contact along cutter-contact point trace curve cutter envelope of surfaces face with processed curved surface and determines；

Formula 2:

$V_{tp}(u_{tM},{\mathrm{\υ}}_{tM},s_p,\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},u_t,{\mathrm{\υ}}_t,\mathrm{\Δ})={\mathrm{\ω}}_{tp}\×r_t^{\left(ft\right)}+V_{Mtp};$

${\mathrm{\ω}}_{tp}=-({\mathrm{\ω}}_{23}{e}_1-{\mathrm{\ω}}_{13}{e}_2+{\mathrm{\ω}}_{12}{n}_{tM})+({\mathrm{\τ}}_{g\mathrm{\α}}^{\left(p\right)}\mathrm{\α}-{\mathrm{\κ}}_{n\mathrm{\α}}^{\left(p\right)}v+{\mathrm{\κ}}_{g\mathrm{\α}}^{\left(p\right)}{n}_{pM})\frac{{\mathrm{ds}}_p}{dt}-\frac{d\mathrm{\Δ}}{dt}{n}_{tM};$

$V_{Mtp}=V_{Mp}-V_{Mt}=-(\frac{{\mathrm{du}}_{tM}}{dt}{e}_1+\frac{{\mathrm{d\υ}}_{tM}}{dt}{e}_2)+\frac{{\mathrm{ds}}_p}{dt}\mathrm{\α}=\[-(\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p}{e}_1+\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p}{e}_2)+\mathrm{\α}\]\frac{{\mathrm{ds}}_p}{dt};$

In formula:

$\left\{\begin{array}{c}{\mathrm{\ω}}_{12}=\frac{-E_2{\mathrm{du}}_{tM}/dt+G_1{\mathrm{d\υ}}_{tM}/dt}{2\sqrt{EG}},E_2=\frac{dE}{{\mathrm{d\υ}}_{tM}},G_1=\frac{dG}{{\mathrm{du}}_{tM}}\\ {\mathrm{\ω}}_{23}=\frac{M{\mathrm{du}}_{tM}/dt+N{\mathrm{d\υ}}_{tM}/dt}{\sqrt{G}},M=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂u_{tM}\∂{\mathrm{\υ}}_{tM}},N=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂{{\mathrm{\υ}}_{tM}}^2}\\ {\mathrm{\ω}}_{13}=\frac{L{\mathrm{du}}_{tM}/dt+M{\mathrm{d\υ}}_{tM}/dt}{\sqrt{E}},L=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂{u_{tM}}^2}\end{array}\right.;$

It is curved surface Σ respectively_pAt a M along the Geodesic torsion in α direction and normal curvature,It is curved surface Σ_pAt a M along the short distance curvature in α direction；

(3) by the curved surface mesh equation formula 3 during cutter generated processed curved surface, obtain by cutter curved surface, processed curved surface and the envelope surface equation formulations 4 of the cutter curved surface of cutter-contact point trace curve intrinsic geometry amount description thereof:

Formula 3:N_t·V_tp=0；Formula 4:

(4) by comparing the error between envelope surface and the processed curved surface of cutter curved surface, in the case of obtaining given processing line width, obtain under motion optimization functional extreme model formula 5 and the prescribed limit error condition of the cutter opposite piece of minimum process error, it is thus achieved that the motion optimization functional extreme model formula 6 of the cutter opposite piece of maximum processing line width:

Formula 5:

$F=\min \left(\frac{\underset{L_{p1}}{\∫}{\mathrm{\δ}}_{k\max }^{\left(1\right)}{\mathrm{ds}}_{p1}}{\underset{L_{p1}}{\∫}{\mathrm{ds}}_{p1}}+\frac{\underset{L_{p2}}{\∫}{\mathrm{\δ}}_{k\max }^{\left(2\right)}{\mathrm{ds}}_{p2}}{\underset{L_{p2}}{\∫}{\mathrm{ds}}_{p2}}\right)$

st.

δ_k≥0

${\mathrm{\δ}}_k=r_g(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)-r_p(u_p,{\mathrm{\υ}}_p);$

Formula 6:

$S=max\left({\∫}_{s_{p1}}^{s_{p2}}{\∫}_{{\mathrm{\υ}}_{t1}}^{{\mathrm{\υ}}_{t2}}\sqrt{\[{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\·{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\]\[{\left(r_{gp}\right)}_{s_p}\·{\left(r_{gp}\right)}_{s_p}\]-{\[{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\·{\left(r_{gp}\right)}_{s_p}\]}^2}{\mathrm{d\υ}}_t\right)$

st.

${\mathrm{\δ}}_k(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)=\mathrm{\ϵ}$

δ_k≥0

${\mathrm{\δ}}_k=r_g(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)-r_p(u_p,{\mathrm{\υ}}_p);$

(5) solve the two functional extreme model, in the case of being able to obtain given processing line width, make, under motion and the prescribed limit error condition of the cutter opposite piece of Machining of Curved Surface error minimum, to make the motion of the cutter opposite piece of processing line width maximum.

(2) realizing the curved surface generating motion of cutter opposite piece on concrete lathe, as a example by using certain type 5-shaft linkage numerical control lathe and conical surface wire rod quality processed complex curved surface, it is as follows that this step technique method implements step:

(1) selected 5-shaft linkage numerical control lathe, consolidates a coordinate system in its frame and each kinematic axis, forms the coordinate system describing the motion of cutter opposite piece of this lathe；

(2) by machine coordinates system, cutter is expressed as formula 7 to the curved surface generating motion of workpiece；

(3) press motion design and realize equivalent principle with motion, the design equation formula 1 of the curved surface generating motion of comparison cutter opposite piece realizes equation formulations 7 with the lathe of the curved surface generating motion of cutter opposite piece and obtains equation 8, and solving equation 8 i.e. can determine that the motor control equation of each numerical control axle of machine tooling；

Formula 7:

$M_{pt}=\left[\begin{array}{cccc}{\mathrm{cos\θ}}_B& 0& {\mathrm{sin\θ}}_B& x_m^{\left(h\right)}\\ 0& 1& 0& y_m^{\left(h\right)}\\ -{\mathrm{sin\θ}}_B& 0& {\mathrm{cos\θ}}_B& z_m^{\left(h\right)}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -x_x^{\left(md\right)}\\ 0& {\mathrm{cos\θ}}_A& -{\mathrm{sin\θ}}_A& 0\\ 0& {\mathrm{sin\θ}}_A& {\mathrm{cos\θ}}_A& 0\\ 0& 0& 0& 1\end{array}\right];$

In formula 7, the meaning of each variable is shown in accompanying drawing 4.

Equation 8:

$\begin{array}{c}\left[\begin{array}{cccc}{\mathrm{cos\θ}}_B& 0& {\mathrm{sin\θ}}_B& x_m^{\left(h\right)}\\ 0& 1& 0& y_m^{\left(h\right)}\\ -{\mathrm{sin\θ}}_B& 0& {\mathrm{cos\θ}}_B& z_m^{\left(h\right)}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -x_x^{\left(md\right)}\\ 0& {\mathrm{cos\θ}}_A& -{\mathrm{sin\θ}}_A& 0\\ 0& {\mathrm{sin\θ}}_A& {\mathrm{cos\θ}}_A& 0\\ 0& 0& 0& 1\end{array}\right]=\\ \left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1y}& -(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z})\\ e_{2x}& e_{2y}& e_{2z}& -(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z})\\ n_{tMx}& n_{tMy}& n_{tMz}& -(x_{tM}{e}_{tx}+y_{tM}{e}_{ty}+z_{tM}{n}_{tz})\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \mathrm{\Δ}& -\sin \mathrm{\Δ}& 0& 0\\ \sin \mathrm{\Δ}& \cos \mathrm{\Δ}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}{\mathrm{\α}}_x& v_x& n_{pMx}& x_{pM}\\ {\mathrm{\α}}_y& v_y& n_{pMy}& y_{pM}\\ {\mathrm{\α}}_z& v_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right]\end{array}.$

The beneficial effects of the present invention is:

nullThe present invention is the universal method that a kind of aspheric surface cutter multi-axis NC machining is complex-curved，Compared with prior art，The present invention is when describing and design cutter opposite piece moves，Introduce curved-surface natural moving frame，Describe with the intrinsic geometry amount of cutter curved surface and processed curved surface and design cutter opposite piece motion，Set up the functional extreme model of cutter opposite piece motion optimization design，This motion optimization model can meet the requirement of processed curved surface local characteristics and overall permanence simultaneously，And can guarantee that the motion of obtained cutter opposite piece is continuous and derivable，The lathe equivalence designed with the motion of cutter opposite piece that optimizes moved by cutter opposite piece realizes accounting for respectively，Ensure that the versatility of tool position optimization method and of overall importance，Avoid to cutter curved surface and processed curved surface unnecessary approximation simplification process the mismachining tolerance brought，The potentiality of multi-axis NC machining are given full play to improve precision and the efficiency of Machining of Curved Surface.

Accompanying drawing explanation

Fig. 1 is the relative motion figure of cutter curved surface of the present invention and enveloping surface moving frame thereof；

Fig. 2 is the Error Graph in cutter envelope of surfaces face of the present invention and processed curved surface；

Fig. 3 is the three dimensional structure sketch of 5-shaft linkage numerical control lathe of the present invention；

Fig. 4 is the coordinate system involved by 5-shaft linkage numerical control machine tool motion of the present invention describes；

Fig. 5 is wire rod quality cutter curved surface of the present invention and coordinate system thereof.

In Fig. 3: 1-z axle, 2-x axle, 3-y axle, 4-workpiece column, 5-support, 6-cutter spindle (C axle), 7-rotary table (B axle), 8-work spindle (A axle).

Detailed description of the invention

The invention will be further described below in conjunction with the accompanying drawings:

The present invention comprises the following steps:

(1) the curved surface generating motion of design cutter opposite piece:

(1) by the cutter curved surface vector function Σ with continuous print three rank partial derivative_t:r_t=r_t(u_t,υ_t) describe, wherein, u_t,υ_tConstitute Orthogonal Parameter net, by the processed curved surface vector function Σ with continuous print three rank partial derivative_p:r_p=r_p(u_p,υ_p) describe, wherein, u_p,υ_pConstitute Orthogonal Parameter net.By the cutter-contact point trace curve L in processed curved surface_pUse vector function r_pM=r_pM(u_pM(s_p),υ_pM(s_p)) describe, wherein, s_pFor curve L_pArc length parameters；

(2) by the curved surface moving frame of cutter curved surface；

$S_{ft}\[\begin{array}{cccc}M;& e_1=\frac{r_1}{\sqrt{E}}& e_2=\frac{r_2}{\sqrt{G}}& e_3=n_{tM}=e_1\×e_2\end{array}\]r_1=\frac{{\mathrm{dr}}_t}{du},r_2=\frac{{\mathrm{dr}}_t}{d\mathrm{\υ}},E=r_1\·r_1,G=r_2\·r_2;$

Moving frame with cutter-contact point trace curve；

$S_{fp}\[\begin{array}{cccc}M;& \mathrm{\α}& v=n_{pM}\×\mathrm{\α}& n_{pM}\end{array}\]\mathrm{\α}=\frac{{\mathrm{dr}}_p}{{\mathrm{ds}}_p},n_{pM}=(\frac{{\mathrm{dr}}_p}{{\mathrm{du}}_p}\×\frac{{\mathrm{dr}}_p}{{\mathrm{d\υ}}_p})/\left|\frac{{\mathrm{dr}}_p}{{\mathrm{du}}_p}\×\frac{{\mathrm{dr}}_p}{{\mathrm{d\υ}}_p}\right|;$

And consider boundary condition: along cutter-contact point trace curve cutter envelope of surfaces face and processed curved surface, there is second order contact (as shown in Figure 1), curved surface generating motion and the movement velocity thereof of cutter opposite piece are expressed as cutter curved surface, processed curved surface and the function of cutter-contact point trace curve intrinsic geometry amount thereof, equation below 1 and formula 2:

$M_{pt}=\left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1y}& -(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z})\\ e_{2x}& e_{2y}& e_{2z}& -(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z})\\ n_{tMx}& n_{tMy}& n_{tMz}& -(x_{tM}{e}_{tx}+y_{tM}{e}_{ty}+z_{tM}{n}_{tz})\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \mathrm{\Δ}& -\sin \mathrm{\Δ}& 0& 0\\ \sin \mathrm{\Δ}& \cos \mathrm{\Δ}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}{\mathrm{\α}}_x& v_x& n_{pMx}& x_{pM}\\ {\mathrm{\α}}_y& v_y& n_{pMy}& y_{pM}\\ {\mathrm{\α}}_z& v_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right];$ Formula 1

Δ is by boundary condition: has second order contact along cutter-contact point trace curve cutter envelope of surfaces face with processed curved surface and determines；

Formula 2:

$V_{tp}(u_{tM},{\mathrm{\υ}}_{tM},s_p,\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},u_t,{\mathrm{\υ}}_t,\mathrm{\Δ})={\mathrm{\ω}}_{tp}\×r_t^{\left(ft\right)}+V_{Mtp};$

${\mathrm{\ω}}_{tp}=-({\mathrm{\ω}}_{23}{e}_1-{\mathrm{\ω}}_{13}{e}_2+{\mathrm{\ω}}_{12}{n}_{tM})+({\mathrm{\τ}}_{g\mathrm{\α}}^{\left(p\right)}\mathrm{\α}-{\mathrm{\κ}}_{n\mathrm{\α}}^{\left(p\right)}v+{\mathrm{\κ}}_{g\mathrm{\α}}^{\left(p\right)}{n}_{pM})\frac{{\mathrm{ds}}_p}{dt}-\frac{d\mathrm{\Δ}}{dt}{n}_{tM};$

$V_{Mtp}=V_{Mp}-V_{Mt}=-(\frac{{\mathrm{du}}_{tM}}{dt}{e}_1+\frac{{\mathrm{d\υ}}_{tM}}{dt}{e}_2)+\frac{{\mathrm{ds}}_p}{dt}\mathrm{\α}=\[-(\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p}{e}_1+\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p}{e}_2)+\mathrm{\α}\]\frac{{\mathrm{ds}}_p}{dt};$

In formula

$\left\{\begin{array}{c}{\mathrm{\ω}}_{12}=\frac{-E_2{\mathrm{du}}_{tM}/dt+G_1{\mathrm{d\υ}}_{tM}/dt}{2\sqrt{EG}},E_2=\frac{dE}{{\mathrm{d\υ}}_{tM}},G_1=\frac{dG}{{\mathrm{du}}_{tM}}\\ {\mathrm{\ω}}_{23}=\frac{M{\mathrm{du}}_{tM}/dt+N{\mathrm{d\υ}}_{tM}/dt}{\sqrt{G}},M=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂u_{tM}\∂{\mathrm{\υ}}_{tM}},N=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂{{\mathrm{\υ}}_{tM}}^2}\\ {\mathrm{\ω}}_{13}=\frac{L{\mathrm{du}}_{tM}/dt+M{\mathrm{d\υ}}_{tM}/dt}{\sqrt{E}},L=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂{u_{tM}}^2}\end{array}\right.;$

It is curved surface Σ respectively_pAt a M along the Geodesic torsion in α direction and normal curvature,It is curved surface Σ_pAt a M along the short distance curvature in α direction；

(3) by the curved surface mesh equation formula 3 during cutter generated processed curved surface, obtain by cutter curved surface, processed curved surface and the envelope surface equation formulations 4 of the cutter curved surface of cutter-contact point trace curve intrinsic geometry amount description thereof:

Formula 3:N_t·V_tp=0；Formula 4:

(4) by comparing the error (as shown in Figure 2) between envelope surface and the processed curved surface of cutter curved surface, in the case of obtaining given processing line width, obtain under motion optimization functional extreme model formula 5 and the prescribed limit error condition of the cutter opposite piece of minimum process error, it is thus achieved that the motion optimization functional extreme model formula 6 of the cutter opposite piece of maximum processing line width:

Formula 5:

$F=\min \left(\frac{\underset{L_{p1}}{\∫}{\mathrm{\δ}}_{k\max }^{\left(1\right)}{\mathrm{ds}}_{p1}}{\underset{L_{p1}}{\∫}{\mathrm{ds}}_{p1}}+\frac{\underset{L_{p2}}{\∫}{\mathrm{\δ}}_{k\max }^{\left(2\right)}{\mathrm{ds}}_{p2}}{\underset{L_{p2}}{\∫}{\mathrm{ds}}_{p2}}\right)$

st.

δ_k≥0

${\mathrm{\δ}}_k=r_g(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{*p},\frac{{\mathrm{d\υ}}_{tM}}{*p},s_p,{\mathrm{\υ}}_t)-r_p(u_p,{\mathrm{\υ}}_p);$

Formula 6:

$S=max\left({\∫}_{s_{p1}}^{s_{p2}}{\∫}_{{\mathrm{\υ}}_{t1}}^{{\mathrm{\υ}}_{t2}}\sqrt{\[{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\·{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\]\[{\left(r_{gp}\right)}_{s_p}\·{\left(r_{gp}\right)}_{s_p}\]-{\[{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\·{\left(r_{gp}\right)}_{s_p}\]}^2}{\mathrm{d\υ}}_t\right)$

st.

${\mathrm{\δ}}_k(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)=\mathrm{\ϵ}$

δ_k≥0

${\mathrm{\δ}}_k=r_g(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)-r_p(u_p,{\mathrm{\υ}}_p);$

(5) solve the two functional extreme model, in the case of being able to obtain given processing line width, make, under motion and the prescribed limit error condition of the cutter opposite piece of Machining of Curved Surface error minimum, to make the motion of the cutter opposite piece of processing line width maximum.

(2) realizing the curved surface generating motion of cutter opposite piece on concrete lathe, as a example by using certain type 5-shaft linkage numerical control lathe and conical surface wire rod quality processed complex curved surface, it is as follows that this step technique method implements step:

(1) selected 5-shaft linkage numerical control lathe is (as shown in Figure 3,1-z axle, 2-x axle, 3-y axle, 4-workpiece column, 5-support, 6-cutter spindle (C axle), 7-rotary table (B axle), 8-work spindle (A axle)), its frame and each kinematic axis consolidate a coordinate system, forms the coordinate system (as shown in Figure 4) describing the motion of cutter opposite piece of this lathe；The process tool curved surface used and coordinate system thereof are as shown in Figure 5.

(2) by (as shown in Figure 4) machine coordinates system, cutter is expressed as formula 7 to the curved surface generating motion of workpiece；

(3) press motion design and realize equivalent principle with motion, the design equation formula 1 of the curved surface generating motion of comparison cutter opposite piece realizes equation formulations 7 with the lathe of the curved surface generating motion of cutter opposite piece and obtains equation 8, and solving equation 8 i.e. can determine that the motor control equation of each numerical control axle of machine tooling.

Formula 7:

$M_{pt}=\left[\begin{array}{cccc}{\mathrm{cos\θ}}_B& 0& {\mathrm{sin\θ}}_B& x_m^{\left(h\right)}\\ 0& 1& 0& y_m^{\left(h\right)}\\ -{\mathrm{sin\θ}}_B& 0& {\mathrm{cos\θ}}_B& z_m^{\left(h\right)}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -x_x^{\left(md\right)}\\ 0& {\mathrm{cos\θ}}_A& -{\mathrm{sin\θ}}_A& 0\\ 0& {\mathrm{sin\θ}}_A& {\mathrm{cos\θ}}_A& 0\\ 0& 0& 0& 1\end{array}\right];$

In formula 7, the meaning of each variable is shown in accompanying drawing 4.

Equation 8:

$\begin{array}{c}\left[\begin{array}{cccc}{\mathrm{cos\θ}}_B& 0& {\mathrm{sin\θ}}_B& x_m^{\left(h\right)}\\ 0& 1& 0& y_m^{\left(h\right)}\\ -{\mathrm{sin\θ}}_B& 0& {\mathrm{cos\θ}}_B& z_m^{\left(h\right)}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -x_x^{\left(md\right)}\\ 0& {\mathrm{cos\θ}}_A& -{\mathrm{sin\θ}}_A& 0\\ 0& {\mathrm{sin\θ}}_A& {\mathrm{cos\θ}}_A& 0\\ 0& 0& 0& 1\end{array}\right]=\\ \left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1y}& -(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z})\\ e_{2x}& e_{2y}& e_{2z}& -(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z})\\ n_{tMx}& n_{tMy}& n_{tMz}& -(x_{tM}{e}_{tx}+y_{tM}{e}_{ty}+z_{tM}{n}_{tz})\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \mathrm{\Δ}& -\sin \mathrm{\Δ}& 0& 0\\ \sin \mathrm{\Δ}& \cos \mathrm{\Δ}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}{\mathrm{\α}}_x& v_x& n_{pMx}& x_{pM}\\ {\mathrm{\α}}_y& v_y& n_{pMy}& y_{pM}\\ {\mathrm{\α}}_z& v_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right]\end{array};$

The ultimate principle of the present invention and principal character and advantages of the present invention have more than been shown and described.Skilled person will appreciate that of the industry; the present invention is not restricted to the described embodiments; the principle that the present invention is simply described described in above-described embodiment and description; without departing from the spirit and scope of the present invention; the present invention also has various changes and modifications, and these changes and improvements both fall within scope of the claimed invention.Claimed scope is defined by appending claims and equivalent thereof.

Claims (1)

1. the universal method that an aspheric surface cutter multi-axis NC machining is complex-curved, it is characterised in that comprise the following steps:

(1) the curved surface generating motion of design cutter opposite piece:

(1) by the cutter curved surface vector function Σ with continuous print three rank partial derivative_t:r_t=r_t(u_t,υ_t) describe, wherein, u_t,υ_tConstitute Orthogonal Parameter net, by the processed curved surface vector function Σ with continuous print three rank partial derivative_p:r_p=r_p(u_p,υ_p) describe, wherein, u_p,υ_pConstitute Orthogonal Parameter net.By the cutter-contact point trace curve L in processed curved surface_pUse vector function r_pM=r_pM(u_pM(s_p),υ_pM(s_p)) describe, wherein, s_pFor curve L_pArc length parameters；

(2) by the curved surface moving frame of cutter curved surface:

$S_{ft}\[\begin{array}{cccc}M;& e_1=\frac{r_1}{\sqrt{E}}& e_2=\frac{r_2}{\sqrt{G}}& e_3=n_{tM}=e_1\×e_2\end{array}\]r_1=\frac{{\mathrm{dr}}_t}{du},r_2=\frac{{\mathrm{dr}}_t}{d\mathrm{\υ}},E=r_1\·r_1,G=r_2\·r_2;$

Moving frame with cutter-contact point trace curve:

$S_{fp}\[\begin{array}{cccc}M;& \mathrm{\α}& v=n_{pM}\×\mathrm{\α}& n_{pM}\end{array}\]\mathrm{\α}=\frac{{\mathrm{dr}}_p}{{\mathrm{ds}}_p},n_{pM}=(\frac{{\mathrm{dr}}_p}{{\mathrm{du}}_p}\×\frac{{\mathrm{dr}}_p}{{\mathrm{d\υ}}_p})/\left|\frac{{\mathrm{dr}}_p}{{\mathrm{du}}_p}\×\frac{{\mathrm{dr}}_p}{{\mathrm{d\υ}}_p}\right|;$

And consider boundary condition: along cutter-contact point trace curve cutter envelope of surfaces face and processed curved surface, there is second order contact, curved surface generating motion and the movement velocity thereof of cutter opposite piece are expressed as cutter curved surface, processed curved surface and the function of cutter-contact point trace curve intrinsic geometry amount thereof, equation below 1 and formula 2:

$M_{pt}=\left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1y}& -(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z})\\ e_{2x}& e_{2y}& e_{2z}& -(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z})\\ n_{tMx}& n_{tMy}& n_{tMz}& -(x_{tM}{e}_{tx}+y_{tM}{e}_{ty}+z_{tM}{e}_{tz})\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \mathrm{\Δ}& -\sin \mathrm{\Δ}& 0& 0\\ \sin \mathrm{\Δ}& \cos \mathrm{\Δ}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}{\mathrm{\α}}_x& v_x& n_{pMx}& x_{pM}\\ {\mathrm{\α}}_y& v_y& n_{pMy}& y_{pM}\\ {\mathrm{\α}}_z& v_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right];$ Formula 1

Δ is by boundary condition: has second order contact along cutter-contact point trace curve cutter envelope of surfaces face with processed curved surface and determines；

Formula 2:

$V_{tp}(u_{tM},{\mathrm{\υ}}_{tM},s_p,\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},u_t,{\mathrm{\υ}}_t,\mathrm{\Δ})={\mathrm{\ω}}_{tp}\×r_t^{\left(ft\right)}+V_{Mtp};$

${\mathrm{\ω}}_{tp}=-({\mathrm{\ω}}_{23}{e}_1-{\mathrm{\ω}}_{13}{e}_2+{\mathrm{\ω}}_{12}{n}_{tM})+({\mathrm{\τ}}_{g\mathrm{\α}}^{\left(p\right)}\mathrm{\α}-{\mathrm{\κ}}_{n\mathrm{\α}}^{\left(p\right)}v+{\mathrm{\κ}}_{g\mathrm{\α}}^{\left(p\right)}{n}_{pM})\frac{{\mathrm{ds}}_p}{dt}-\frac{d\mathrm{\Δ}}{dt}{n}_{tM};$

$V_{Mtp}=V_{Mp}-V_{Mt}=-(\frac{{\mathrm{du}}_{tM}}{dt}{e}_1+\frac{{\mathrm{d\υ}}_{tM}}{dt}{e}_2)+\frac{{\mathrm{ds}}_p}{dt}\mathrm{\α}=\[-(\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p}{e}_1+\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p}{e}_2)+\mathrm{\α}\]\frac{{\mathrm{ds}}_p}{dt};$

In formula

$\left\{\begin{array}{c}{\mathrm{\ω}}_{12}=\frac{-E_2{\mathrm{du}}_{tM}/dt+G_1{\mathrm{d\υ}}_{tM}/dt}{2\sqrt{EG}},E_2=\frac{dE}{{\mathrm{d\υ}}_{tM}},G_1=\frac{dG}{{\mathrm{du}}_{tM}}\\ {\mathrm{\ω}}_{23}=\frac{{\mathrm{Mdu}}_{tm}/dt+{\mathrm{Nd\υ}}_{tM}/dt}{\sqrt{G}},M=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂u_{tM}\∂{\mathrm{\υ}}_{tM}},N=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂{{\mathrm{\υ}}_{tM}}^2}\\ {\mathrm{\ω}}_{13}=\frac{{\mathrm{Ldu}}_{tM}/dt+{\mathrm{Md\υ}}_{tM}/dt}{\sqrt{E}},L=n_{tM}\·\frac{{\∂}^2{r}_t}{\∂{u_{tM}}^2}\end{array}\right.;$

It is curved surface Σ respectively_pAt a M along the Geodesic torsion in α direction and normal curvature,It is curved surface Σ_pAt a M along the short distance curvature in α direction；

(3) by the curved surface mesh equation formula 3 during cutter generated processed curved surface, obtain by cutter curved surface, processed curved surface and the envelope surface equation formulations 4 of the cutter curved surface of cutter-contact point trace curve intrinsic geometry amount description thereof:

Formula 3:N_t·V_tp=0；Formula 4:

(4) by comparing the error between envelope surface and the processed curved surface of cutter curved surface, in the case of obtaining given processing line width, obtain under motion optimization functional extreme model formula 5 and the prescribed limit error condition of the cutter opposite piece of minimum process error, it is thus achieved that the motion optimization functional extreme model formula 6 of the cutter opposite piece of maximum processing line width:

Formula 5:

$F=\min (\frac{\underset{L_{p1}}{\∫}{\mathrm{\δ}}_{k\max }^{\left(1\right)}{\mathrm{ds}}_{p1}}{\underset{L_{p1}}{\∫}{\mathrm{ds}}_{p1}}+\frac{\underset{L_{p2}}{\∫}{\mathrm{\δ}}_{k\max }^{\left(2\right)}{\mathrm{ds}}_{p2}}{\underset{L_{p2}}{\∫}{\mathrm{ds}}_{p2}})$

st.

δ_k≥0

${\mathrm{\δ}}_k=r_g(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)-r_p(u_p,{\mathrm{\υ}}_p);$

Formula 6:

$S=\max \left({\∫}_{s_{p1}}^{s_{p2}}{\mathrm{ds}}_p{\∫}_{{\mathrm{\υ}}_{t1}}^{{\mathrm{\υ}}_{t2}}\sqrt{\[{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\·{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\]\[{\left(r_{gp}\right)}_{s_p}\·{\left(r_{gp}\right)}_{s_p}\]-{\[{\left(r_{gp}\right)}_{{\mathrm{\υ}}_t}\·{\left(r_{gp}\right)}_{s_p}\]}^2}d\mathrm{\υ}{t}_{}\right)$

st.

${\mathrm{\δ}}_k(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)=\mathrm{\ϵ}$

δ_k≥0

${\mathrm{\δ}}_k=r_g(u_{tM},{\mathrm{\υ}}_{tM},\frac{{\mathrm{du}}_{tM}}{{\mathrm{ds}}_p},\frac{{\mathrm{d\υ}}_{tM}}{{\mathrm{ds}}_p},s_p,{\mathrm{\υ}}_t)-r_p(u_p,{\mathrm{\υ}}_p);$

(5) solve the two functional extreme model, in the case of being able to obtain given processing line width, make, under motion and the prescribed limit error condition of the cutter opposite piece of Machining of Curved Surface error minimum, to make the motion of the cutter opposite piece of processing line width maximum；

(2) realizing the curved surface generating motion of cutter opposite piece on concrete lathe, as a example by using certain type 5-shaft linkage numerical control lathe and conical surface wire rod quality processed complex curved surface, it is as follows that this step technique method implements step:

(1) selected 5-shaft linkage numerical control lathe, consolidates a coordinate system in its frame and each kinematic axis, forms the coordinate system describing the motion of cutter opposite piece of this lathe；

(2) by machine coordinates system, cutter is expressed as formula 7 to the curved surface generating motion of workpiece；

(3) press motion design and realize equivalent principle with motion, the design equation formula 1 of the curved surface generating motion of comparison cutter opposite piece realizes equation formulations 7 with the lathe of the curved surface generating motion of cutter opposite piece and obtains equation 8, and solving equation 8 i.e. can determine that the motor control equation of each numerical control axle of machine tooling；

Formula 7:

$M_{pt}=\left[\begin{array}{cccc}{\mathrm{cos\θ}}_B& 0& {\mathrm{sin\θ}}_B& x_m^{\left(h\right)}\\ 0& 1& 0& y_m^{\left(h\right)}\\ -{\mathrm{sin\θ}}_B& 0& {\mathrm{cos\θ}}_B& z_m^{\left(h\right)}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -x_x^{\left(md\right)}\\ 0& {\mathrm{cos\θ}}_A& -{\mathrm{sin\θ}}_A& 0\\ 0& {\mathrm{sin\θ}}_A& {\mathrm{cos\θ}}_A& 0\\ 0& 0& 0& 1\end{array}\right];$

Equation 8:

$\begin{array}{c}\left[\begin{array}{cccc}{\mathrm{cos\θ}}_B& 0& {\mathrm{sin\θ}}_B& x_m^{\left(h\right)}\\ 0& 1& 0& y_m^{\left(h\right)}\\ -{\mathrm{sin\θ}}_B& 0& {\mathrm{cos\θ}}_B& z_m^{\left(h\right)}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& -x_x^{\left(md\right)}\\ 0& {\mathrm{cos\θ}}_A& -{\mathrm{sin\θ}}_A& 0\\ 0& {\mathrm{sin\θ}}_A& {\mathrm{cos\θ}}_A& 0\\ 0& 0& 0& 1\end{array}\right]=\\ \left[\begin{array}{cccc}{e}_{1x}& e_{1y}& e_{1y}& -(x_{tM}{e}_{1x}+y_{tM}{e}_{1y}+z_{tM}{e}_{1z})\\ e_{2x}& e_{2y}& e_{2z}& -(x_{tM}{e}_{2x}+y_{tM}{e}_{2y}+z_{tM}{e}_{2z})\\ n_{tMx}& n_{tMy}& n_{tMz}& -(x_{tM}{e}_{tx}+y_{tM}{e}_{ty}+z_{tM}{e}_{tz})\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \mathrm{\Δ}& -\sin \mathrm{\Δ}& 0& 0\\ \sin \mathrm{\Δ}& \cos \mathrm{\Δ}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}{\mathrm{\α}}_x& v_x& n_{pMx}& x_{pM}\\ {\mathrm{\α}}_y& v_y& n_{pMy}& y_{pM}\\ {\mathrm{\α}}_z& v_z& n_{pMz}& z_{pM}\\ 0& 0& 0& 1\end{array}\right]\end{array}.$