Fig.1 Equal-error linear approximation of non-circular curves
1 INTRODUCTION Since most CNC machine tools do not have interpolation commands for tool nose trajectories of non-circular curves, they are usually replaced by straight-line sections or arc sections in the preparation of CNC programs for such curves. Because straight line replacement is simple and intuitive, it is used more often. The method of replacing non-circular curves with straight lines is shown in Figure 1. Under the condition of meeting the accuracy requirements, a polyline can be used instead of a non-circular curve. In the figure, a, b, c, d, etc. are called nodes. The key to achieving NC programming of the tool nose trajectory is to determine these nodes. In order to simplify the calculation, the nodes are usually determined by the equal spacing method and the equal step method. The equal interval method is to make the pitch ∆x of the projections of each node on the x-axis equal when the maximum deviation between the theoretical curve and the straight line is less than the allowable deviation (d max ≤ d allow). The equal step method is to make the straight line length ∆L equal to each other under the condition that the maximum deviation between the theoretical curve and the straight line is less than the allowable deviation (d max ≤ d allow). Their common characteristic is that the calculation is relatively simple. However, when the curvature of the curve between the nodes changes greatly, since ∆x and ∆L are fixed values, the surface roughness of the machined part will change greatly, which will affect the surface processing quality of the workpiece; at the same time, the curvature of the curve The change also changes the machining error d of the workpiece. On the other hand, the spacing of the equal spacing method and the step size of the equal-step method are determined by the minimum radius of curvature of the non-circular curve according to the machining accuracy, so the two methods will generate many nodes in the entire non-circular curve, making Computation and programming are quite tedious. If the equal error linear approximation method is adopted, the above problems can be effectively avoided. 2 The theoretical calculation of equal error linear approximation is shown in Fig. 1. The characteristic of equal error linear approximation method is to make the error d equal to the non-circular curve between the nodes and the straight line. The specific solution steps are as follows: Take the starting point a (xa, ya) as the center of the circle and d as the radius as the circle, and determine the circle equation of the allowable error as
(x-xa)2+(y-ya)2=d2 (1) The slope of the common tangent PT of the circle and the curve
K= yT-yp xT-xp (2) where xT, yT, xp, and yp are obtained by solving the following simultaneous equation: { yT-yp=f1'( xp)( xT-xp) yp=f1( xp yT-yp=f2'(xT)(xT-xp) yT=f2(xT) (3) where f1(x) - error circle function f2(x) - the function of the machining curve is known to be parallel to PT The slope of chord ab is K, then the linear equation of chord ab is y-ya=K(x-xa) (4) The simultaneous curve equation and chord ab equation, the coordinate of b-point can be obtained as {y=f2(x) y -ya=k(x-xa) (5) Repeat the above steps to sequentially find the coordinates of points c, d, and e. 3 Node calculation of equal error linear approximation method In NC machining, the theoretical curve of the tool nose trajectory is generally taken as a parabola y=ax2 (a>0, x>0), then there is y'=2ax. According to the tolerance circle equation (1), {y=ya-[d2-(x-xa)2]1â„2 y'=- x-xa y-ya (6) is available. Therefore, the equation group (3) can be rewritten as {yT -yp=- xp-xT (xT-xp) yp-yT yp=ya-[d2-(xp-xT)2]1â„2 yT-yp=2axT(xT-xp) yT=axT2 (7) Simultaneous equations (7) Available 4au3-4au2ya-t3+4aut3+4autxa=0 (8) Where: t=xp-xa u=(d2-t2)1â„2
Figure 2 program flow chart
Known conditions are known: 0 ≤ t ≤ d. To solve the value of t, we can use a step-by-step search method or a dichotomy method on the computer to find the root (the stepwise search method is used in this paper). The equation can be solved by solving t from the equation (8). According to equation (5), {y=ax2 y-ya=k(x-xa) (9) where k=(yT-yp)/(xT-xp) solves the system of equations (9). The node coordinates are x = k + del 2a (10) where del = [k2 - 4a (kxa - ya)] 1â„2 Then take this node as a new starting point and solve it repeatedly to find all the nodes sequentially. 4 Compilation program development In AutoCAD embedded VBA (Visual Basic for Applications) development environment to develop computing programs, program flow shown in Figure 2. The ideal curve of the NC tool nose trajectory is a parabola. The trajectory of the simulated curve of the polygonal line drawn at the node d=0.05 is shown in FIG. 3 . The calculated node data is stored in the "c:/data.txt" file, as shown in Figure 4.
Figure 3 Simulation curve trace (d=0.05)
Figure 4 Save the node coordinate data file (d = 0.01)
The subroutine for solving the equation is as follows: // Solve the equation For tana=delta To 0 Step -0.0001 u=Sq(r delta ^ 2-tana ^ 2) texpr=4 * a * u ^ 3 + 4 * a * u * tana ^ 2 - 4 * a * u ^ 2 * ya-tana ^ 2 + 4 * a * u * tana * xa If texpr < 0.00001 And texpr > 0 Then Exit For Next xp=xa + tana:yp=ya-u xt =tana / (2 * a * u): yt = a * xt ^ 2 k = (yp-yt) / (xp-xt) del = Sq (rk ^ 2-4 * a * (k * xa-ya) Xb = (k + del) / (2 * a) yb = a * xb ^ 2 // Solution The end error value d can be set by "variable setting". 5 Equal-error straight-line approximation of other non-circular curves For equal-error linear approximation of other non-circular curves, it is only necessary to change the expression of t in the calculation program. Take the hyperbolic curve as an example. Just use the expression "tpr=4*a*u^3 + 4*a*u*tana^2-4*a*u^2*ya-tana^2 + in the program. 4*a*u*tana*x" is replaced by "texpr=t^2 + t*xa-u*ya-2*sq(ru*t)". 6 Conclusion In the NC programming of the tool nose trajectory of non-circle curves, the equal-error linear approximation method is the method with the least number of nodes and the least number of numerical control blocks, and the machining surface accuracy and dimensional accuracy of the parts are high. However, the calculation of equal-error linear approximation method is cumbersome. It is difficult to obtain the exact solution by solving the quadratic quadratic equation manually. The calculation program compiled in the AutoCAD embedded VBA development environment introduced in this article can effectively realize the node calculation of the error linear approximation method such as non-circular curve, and it is portable, and it can be conveniently used for the calculation of other non-circular curve nodes. It can also simulate the difference between the tool trajectory and the actual machining curve, so it has high practical value.
Forward-out Baler Machine
Ecohydraulic forward-out baler has been used by many customers to press non-ferrous metals, such as used beverage cans (UBC), scrap aluminum chips, scrap copper chips etc. So it is also called tin Cans Baler, aluminum chips baler, metal chips baler, aluminum powder baler etc. Forward-out aluminum Balers are widely used by recycling centers due to their efficient and proficient working mechanism. The larger aluminum can baler models can be customized to compress metal leftover, steel shavings and steel scraps. Furthermore, the optional manual and automatic control systems of the cans baler enable the automation of machine and allow the manual use as well.
Forward-out Baler, Forward-out Baling Machine, Forward-out Baling Press, Forward-out Baler Equipment, Forward-out Metal Compactor
Jiangyin Metallurgy Hydraulic Machinery Factory , https://www.eco-metalsrecycle.com
![<?echo $_SERVER['SERVER_NAME'];?>](/template/twentyseventeen/skin/images/header.jpg)