In my work on sand casting foundry processes, I have often encountered the challenge of producing high-quality molds for protective covers. These covers, commonly used for dust prevention and collision protection, are typical cast parts requiring robust and accurate molds. For high-volume production, I select aluminum alloy for the sand casting foundry mold due to its excellent wear resistance and machinability. The geometry of the protective cover is composed of straight segments, circular arcs, and elliptical arcs, connected by four screw holes for assembly. In the sand casting foundry, the numerical control (NC) machining of such molds demands precise calculation of coordinate values at critical points, especially the tangency points between different curve segments. Additionally, the thermal behavior during solidification and cooling is vital: the mold cavity must be maintained at the correct temperature to ensure the iron casting solidifies properly before opening, yet not so long that productivity suffers. Therefore, I set out to develop a systematic approach combining numerical computation, NC programming, and finite element analysis to optimize both the machining and the cooling schedule for this sand casting foundry mold.
This article presents my methodology in detail. I first derive the nonlinear equations based on the analytical geometry of the profile, then solve them using the bisection iterative method implemented in a custom calculator. The calculated coordinates enable me to write a macro program for the CNC lathe. Next, I perform a transient thermal finite element simulation of the entire sand casting foundry assembly (mold plus sand), considering material properties of cast iron and silica sand. The resulting temperature field provides the necessary cooling time before mold opening, ensuring casting quality while maximizing productivity.
Numerical Calculation for NC Machining
Because the protective cover is symmetric, I take half of the sand casting foundry mold as my study object. The key geometric features include a straight line tangent to a small circular arc at point A, and the same straight line tangent to an outer elliptical arc at point B. Similarly, another straight segment is tangent to a larger circular arc at point C and to an inner elliptical arc at point D. The mold is machined from an aluminum alloy bar. Before turning the inner surface, drilling is required. The drill diameter must be chosen so that the drill tip (with a 118° point angle) does not interfere with the elliptical inner profile. I define point E as the intersection of a line through point F (the far right end of the mold) with the inner ellipse, and the drill radius must be less than the y-coordinate of E.
I now establish the mathematical models. Let the small arc have center (10, d) and radius 5. The outer ellipse has semi-axes a1 and b1, with center (e, 0). The tangency conditions at A (x1, y1) and B (x3, y3) require that the straight line AB shares the same slope as the tangent to the small arc at A and also the same as the tangent to the outer ellipse at B. This yields the following system of nonlinear equations:
$$(x_1 – 10)^2 + (y_1 – d)^2 = 25$$
$$\frac{(x_3 – e)^2}{a_1^2} + \frac{y_3^2}{b_1^2} = 1$$
$$\frac{10 – x_1}{y_1 – d} = \frac{y_3 – y_1}{x_3 – x_1}$$
$$\frac{b_1^2 (e – x_3)}{a_1^2 y_3} = \frac{y_3 – y_1}{x_3 – x_1}$$
To solve these equations, I apply the bisection method, which is simple, robust, and easy to implement in a computer program. The idea is to repeatedly halve an interval that contains the root until the interval length is sufficiently small, and then take the midpoint as the approximate root. For this problem, I can determine a reasonable interval for x1 from geometry: (5, 10). From the first equation, I express y1 and its derivative:
$$y_1 = d – \sqrt{25 – (x_1 – 10)^2}, \quad y_1′ = \frac{10 – x_1}{y_1 – d}$$
Using the slope equality, I then compute x3 and y3 in terms of x1:
$$x_3 = \frac{b_1^2 e – a_1^2 y_1′(y_1 – x_1 y_1′)}{a_1^2 y_1’^2 + b_1^2}$$
$$y_3 = \frac{b_1^2 (e – x_3)}{a_1^2 y_1′}$$
The bisection loop continues until the residual of the ellipse equation at (x3, y3) is less than \(10^{-10}\). I store the final x1, y1, x3, y3 values.
Similarly, for the second tangency pair (points C and D), I denote the larger circular arc (center (8, d), radius 10) and the inner ellipse with semi-axes a2, b2 (center (e, 0)). The system is analogous, with the same bisection procedure applied for x2 in the interval (5, 10). The formulas become:
$$y_2 = d – \sqrt{100 – (x_2 – 8)^2}, \quad y_2′ = \frac{10 – x_2}{y_2 – d}$$
$$x_4 = \frac{b_2^2 e – a_2^2 y_2′(y_2 – x_2 y_2′)}{a_2^2 y_2’^2 + b_2^2}$$
$$y_4 = \frac{b_2^2 (e – x_4)}{a_2^2 y_2′}$$
Again, the iteration stops when the inner ellipse equation at (x4, y4) is sufficiently satisfied.
For the drill point E, I define line EF through point F (197, 0) at an angle of 121° from the positive x-axis (since the drill point angle is 118°, the included half-angle is 59°, so the line makes 180° – 59° = 121° with the x-axis). Thus:
$$y_5 = (x_5 – 197) \tan 121^\circ$$
And E lies on the inner ellipse:
$$\frac{(x_5 – e)^2}{a_2^2} + \frac{y_5^2}{b_2^2} = 1$$
I solve this with bisection for x5 in the interval (x4, 197). The maximum permissible drill radius is then y5. In my practical parameters: a1=50, b1=60, a2=45, b2=55, d=80, e=152. The results are listed in the table below.
| Point | x (mm) | y (mm) | Iterations |
|---|---|---|---|
| A | 9.483 | 75.027 | 97 (for first pair) |
| B | 156.312 | 59.777 | |
| C | 8.967 | 70.054 | |
| D | 155.810 | 54.802 | |
| E | — | 52.479 | — |
The drill diameter must therefore be less than \(2 \times 52.479 = 104.958\) mm. I choose a drill diameter of 104 mm to be safe.
Calculator Development for the Sand Casting Foundry Mold
To avoid manual computation, I developed a simple calculator in a programming environment (e.g., Visual Basic or Python with a GUI). The interface has six input fields for the ellipse semi-axes (a1, b1, a2, b2) and the parameters d and e. The bisection algorithm is implemented using a Do loop with termination condition. Six output fields display the four tangency coordinates, the y-coordinate of point E, and the iteration count. The tool drastically reduces calculation time and eliminates human error; it is particularly useful when the sand casting foundry mold design is adjusted frequently.
I tested the calculator with the same input values. The results matched those in the table above, confirming correctness. The number of iterations (97) indicates that the bisection method, though simple, converges slowly; however, for this application the speed is acceptable because the calculations are performed only once per design iteration.
NC Programming of the Sand Casting Foundry Mold
With the coordinates known, I proceed to generate the NC program. The mold is to be machined from an aluminum alloy bar (φ210 mm). I use a FANUC CNC lathe. The turning of both internal and external surfaces is accomplished with macro programming, which provides flexibility and portability. I select tools: T0101 for external turning, T0202 for internal turning, and T0303 (4 mm wide cut-off tool, left edge as reference). The clamping and drilling steps are performed manually: the bar is held in a three-jaw chuck with about 240 mm overhang; the right face is faced; then holes of φ20 mm and φ50 mm are drilled to a depth of 226 mm to facilitate internal turning. The extra 30 mm length is used for clamping during second setup.
The program for the internal surface (O1234) is as follows:
O1234;
M03 S800 M07 T0101;
G00 X212.0 Z2.0;
G90 X208.0 Z-236.0 F0.1;
M05;
T0202;
M03 S500;
G41 G00 X48.0 Z2.0;
G71 U2.0 R1.0;
G71 P100 Q200 U-0.4 W0.2 F0.1;
N100 G01 X80.0 F0.1;
Z-30.0;
G02 X140.108 Z-38.967 R10.0;
G01 X109.604 Z-185.810;
#1=-185.810;
N150 #2=110*SQRT(1-[#1+182]^2/2025);
G01 X#2 Z#1 F0.1;
#1=#1-0.001;
IF [#1 GE -227.0] GOTO150;
N200 G70 P100 Q200;
G40 G00 X220.0 Z100.0;
M05;
T0303;
M03 S400;
G00 X212.0 Z-232.0;
G01 X18.0 F0.1;
X212.0;
G00 X220.0 Z100.0;
M09;
M05;
M30;
In this program, the G71 cycle roughs the inner contour, including the circular arc and the elliptical region approximated with a macro loop. The step size is 0.001 mm, which gives a smooth surface. After finishing with G70, the part is cut off at Z-232 mm, leaving the chucking portion.
The external surface program (O1235) is similar but uses the outer ellipse parameters. After cutting off, the total length is 202 mm. The program ensures that all tangency points are accurately machined, producing a mold ready for the sand casting foundry process.
O1235;
M03 S800 M07 T0101;
G42 G00 X212.0 Z2.0;
G71 U2.0 R1.0;
G71 P300 Q400 U0.4 W0.2 F0.1;
N300 G01 X0 F0.1;
Z0;
#3=0;
N350 #4=120*SQRT(1-[#3+50]^2/3600);
G01 X#4 Z#3 F0.1;
#3=#3-0.001;
IF [#3 GE -45.688] GOTO350;
G01 X150.054 Z-192.517;
G02 X160.0 Z-197.0 R10.0;
G01 X200.0;
N400 Z-206.0;
G70 P300 Q400;
G40 G00 X220.0 Z100.0;
M05;
T0303;
M03 S400;
G00 X212.0 Z-206.0;
G01 X18.0 F0.1;
X212.0;
G00 X220.0 Z100.0;
M09;
M05;
M30;
Transient Temperature Field Finite Element Analysis of the Sand Casting Foundry Assembly
In a sand casting foundry, the mold is placed upside down in the lower flask, surrounded by silica sand. After ramming, the upper flask is placed and the sprue and riser patterns are positioned. The sand casting foundry mold is then removed, leaving cavities for the cover and gating system. The entire assembly is shown schematically in the process layout. To analyze the solidification and cooling behavior, I create a finite element model using ANSYS. Because of symmetry, I use a planar model, ignoring the sprue and riser to reduce computational cost. The materials involved are cast iron (the molten metal) and silica sand (the mold). Their thermal properties are given in the table below.
| Material | Density (kg/m³) | Specific Heat (J/(kg·K)) | Thermal Conductivity (W/(m·K)) |
|---|---|---|---|
| Cast iron (melt) | 7800 | 455 | 50 |
| Silica sand | 1600 | 1100 | 0.27 |
I use the PLANE55 2D thermal element. The initial temperature of the iron nodes is set to 1534 °C (typical pouring temperature), and the sand nodes are initially at room temperature (25 °C). A transient analysis is performed for a total time of 1370 seconds, with 100 time steps. The time integration is activated, and all substeps are saved.
After solving, I examine the temperature distribution. The overall model temperature at 1370 seconds is shown in the contour plot (the actual image is omitted here, but I describe its content). The highest temperature is 146.112 °C, located in the sand region that is enclosed by the concave part of the iron mold. This is expected because the sand in that region traps heat. The iron casting itself reaches a maximum temperature of 145.496 °C at the same time, which is below the recommended mold opening temperature of 150 °C. Therefore, the mold can be safely opened after 1370 seconds of cooling. In colder or hotter workshop environments, the cooling time might need adjustment accordingly.
Conclusion
Through this work, I have demonstrated a systematic methodology for the design and production of a sand casting foundry mold for a protective cover. By leveraging the analytical geometry of the profile, I derived nonlinear equations that describe the tangency points. The bisection method, implemented in a custom calculator, efficiently solved these equations, yielding exact coordinates for CNC programming. The macro programs I wrote for the FANUC lathe accurately machine both the internal and external contours, including the elliptical sections. Furthermore, I performed a transient thermal finite element analysis of the entire sand casting foundry system, obtaining the temperature field over 1370 seconds. The results confirmed that the casting cools below the safe opening temperature within that period, which provides a reliable basis for process planning. The combination of numerical computation, NC programming, and thermal simulation provides a robust framework for any sand casting foundry mold requiring complex geometries and controlled cooling. Future work could extend this approach to include optimization of cooling channel design or the use of different mold materials to further improve productivity in the sand casting foundry environment.