Note: This is a revised and expanded version of my recent posting with the same title. The previous version contained a few confusing typos and tacitly assumed that the inside end of each segment was cut in a circular arc rather than a straight line while, in fact, the latter is the usual procedure. The calculation approach presented below assumes the latter procedure.
There appears to be a growing interest in turning hollow laminated segmented spheres. In addition to the challenging tasks of cutting and gluing up the segments into rings, gluing the rings into a rough sphere and turning the result to obtain a true sphere, there remains the task of determining the required inner and outer sizes of each ring. The direct way to determine the ring sizes is to draw a detailed full-size or accurately-scaled diagram of each ring and make direct measurements from the diagram.
The objective of this note is present a simple way to calculate the outer and inner sizes of the rings for a given desired sphere diameter and desired minimum wall thickness. The method presented works for any number of rings, and the thickness of the various rings need not be the same nor do the various rings need to contain the same number of segments. (Furthermore, the segments in a given ring need not all be the same angular width—more on that later.)
To begin, let us define a few terms.
D = the outer diameter of the sphere that is to be obtained after turning away the corners and edges of the glued up rings.
R = D/2 is the corresponding radius.
r = the outer radius of any particular ring in the stack, measured along the center line of a segment.
W = width of that ring (the length of a segment of the ring measured along its center). (I assume here that the segments will be cut as trapezoids, that is, their inner and outer ends will be parallel straight lines.)
L = the distance measured outward from an imaginary plane that passes through the center of the sphere and is parallel to the rings, to the closer edge of a particular ring of interest.
J = the corresponding distance from that plane to the further edge of the same ring.
N = the number of segments in a ring of interest.
θ = 360/N is the angular width of a segment in degrees.
K = outer width of a particular segment.
T = the desired minimum wall thickness after turning the outside of the sphere.
S = R – T is the inner diameter of the sphere if it were perfectly smoothed inside.
If all the segments in a given ring are to be identical, the outer radius r of that ring and the length W of its segments can be calculated as follows:
As one calculates ring parameters, working outward from the center of the stack, eventually J will be found to be equal to or greater than S. When this happens, that “ring” and subsequent ones are disks (no center opening).
If one wants to use segments of more than one angular size in a given ring, these equations still apply except that now N is interpreted as the number of segments of the particular angular size that would be required to make a complete ring if it were used alone. The computation of W would have to be made for each differently sized segment in the ring. (Optionally, at the expense of added weight, all of the segments in a ring could be cut to the length of the widest one.)
Note that if N is small or the number of rings is small, there will be a lot of waste wood inside the final sphere, adding to its final weight. Much of this can be eliminated by cutting the inside of the segments at an angle so that their trimmed edge matches the adjacent larger disk.
There appears to be a growing interest in turning hollow laminated segmented spheres. In addition to the challenging tasks of cutting and gluing up the segments into rings, gluing the rings into a rough sphere and turning the result to obtain a true sphere, there remains the task of determining the required inner and outer sizes of each ring. The direct way to determine the ring sizes is to draw a detailed full-size or accurately-scaled diagram of each ring and make direct measurements from the diagram.
The objective of this note is present a simple way to calculate the outer and inner sizes of the rings for a given desired sphere diameter and desired minimum wall thickness. The method presented works for any number of rings, and the thickness of the various rings need not be the same nor do the various rings need to contain the same number of segments. (Furthermore, the segments in a given ring need not all be the same angular width—more on that later.)
To begin, let us define a few terms.
D = the outer diameter of the sphere that is to be obtained after turning away the corners and edges of the glued up rings.
R = D/2 is the corresponding radius.
r = the outer radius of any particular ring in the stack, measured along the center line of a segment.
W = width of that ring (the length of a segment of the ring measured along its center). (I assume here that the segments will be cut as trapezoids, that is, their inner and outer ends will be parallel straight lines.)
L = the distance measured outward from an imaginary plane that passes through the center of the sphere and is parallel to the rings, to the closer edge of a particular ring of interest.
J = the corresponding distance from that plane to the further edge of the same ring.
N = the number of segments in a ring of interest.
θ = 360/N is the angular width of a segment in degrees.
K = outer width of a particular segment.
T = the desired minimum wall thickness after turning the outside of the sphere.
S = R – T is the inner diameter of the sphere if it were perfectly smoothed inside.
If all the segments in a given ring are to be identical, the outer radius r of that ring and the length W of its segments can be calculated as follows:
As one calculates ring parameters, working outward from the center of the stack, eventually J will be found to be equal to or greater than S. When this happens, that “ring” and subsequent ones are disks (no center opening).
If one wants to use segments of more than one angular size in a given ring, these equations still apply except that now N is interpreted as the number of segments of the particular angular size that would be required to make a complete ring if it were used alone. The computation of W would have to be made for each differently sized segment in the ring. (Optionally, at the expense of added weight, all of the segments in a ring could be cut to the length of the widest one.)
Note that if N is small or the number of rings is small, there will be a lot of waste wood inside the final sphere, adding to its final weight. Much of this can be eliminated by cutting the inside of the segments at an angle so that their trimmed edge matches the adjacent larger disk.
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