## Description

### Calculation of production deviation

The processing process of the stone spiral staircase railing is basically as described above, but this processing method is still not perfect in some directions. This is because the spiral staircase is formed on the basis of the spiral surface, and the modeling of the three-dimensional spiral surface is the most difficult link in processing. Therefore, in order to reduce processing difficulties, this processing method processes a single piece of the spiral surface into a cylindrical surface that is similar to it, thereby forming a certain error with the spiral surface. In order to reduce this error, the author of this article has derived a series of approximations with CNC bead wire saws A convenient and practical calculation formula for processing spiral noodles, and an example calculation that illustrates the application of this new method.

The spiral staircase can not only meet practical requirements but also embodies the modern art style of decorative design. Therefore, the application of spiral stairs is becoming more and more extensive. However, many companies are not good at making spiral staircase guardrails. This article introduces the manufacturing process of the spiral staircase guardrails and provides a calculation formula for processing the spiral surfaces.

### (1) Establishment of the equation

The surface of the cover of the spiral stair railing is a spiral surface, as shown in Figure 1. It can be viewed geometrically as a line segment perpendicular to the Xyz axis and moving in an equal-angle circular motion around the z-axis, and at the same time along the direction of the straight line. Space curved surfaces formed by linear motion at a constant speed.

The mathematical formula of space analytic geometry is expressed as:

{x=rcosθ; y=rsinθ; z=kθ}(r1≤r ≤r2) ①

Where θ is the angle between the line between the projection point of a point on the cylindrical helix plane and the center of the bottom surface and the x-axis, r1, r2” are the radii of the inner and outer circles of the bottom surface, and k is a constant. The spiral expansion of the spiral surface is a right triangle

r0 = 0.5 (r1 + r2);

K=r1 tgλ1=r2 tgλ2=r0 tgλ0 ②

If a generatrix coincides with the generatrix of this helix on the x-axis and passes through the midpoint A0 cylindrical surface of another generatrix on the helix to approach the helix block A1A2B2B1, its radius is R.

(“+” means the error between the cylindrical surface and the helix that intersects the normal line farther from point A1 on the helix at point A1, which is much greater than the difference between the cylindrical surface and the helix closer to point A1 at point A1 Error, so it needs to be discarded)

### (2) Deviation analysis

It can be seen from the error formula that △1 and △2 are obviously related to factors such as β and θ. The smaller θ is, the smaller △1 and △2 will be. However, the more stone slabs are divided, the worse the decorative effect will be. Therefore, θ should be taken to an appropriate value. Generally, △1 and △2 can be less than 3mm. It is known that β<θ, and to make the error smaller, the best value of β can be obtained from the error formula so that the error between the cylindrical surface and the spiral surface can be made smaller.

### (3) Examples

Take the curved surface of the spiral staircase cover as an example to analyze the error between the spiral surface and the cylindrical surface that replaces it. For a spiral surface with a chord length of 860mm, the base circle radius of its inner and outer spirals is: r1=6442mm, r2=6542mm, get the rotation angle θ=7.54°, r2=17.47°, substitute it into the formula ② to get k=2059mm, and then substitute into the formula ③ to get R=648612mm. Take β = r0 = tg-1k/ r0, then substitute it into the error formula, and the result is as follows:

In summary, if you want to make the spiral staircase railing have a small error in the modeling process, you can use the CNC bead wire saw to process the cylindrical surface radius formula obtained above.

### (4) Conclusion

From the perspective of future development, whether it is in terms of the practical value or artistic value of the spiral staircase, it will be appreciated and adopted by more people, and more and more companies will also have a strong influence on its technical processing. Interest, so the processing of the spiral staircase guardrail will have a broader development prospect.