Transfer of dimensions: a simple example
Suppose that the task at hand is
to manufacture a lot of several thousand parts, as pictured below in figure 1. In
that figure, only the dimensions of interest for this exposition are indicated.
Those dimensions have been carefully determined by a designer, based on his
best knowledge, applying state of the art technology. Those dimensions happen
to be all 10 mm, with a tolerance of ±0.1
mm. This means that the part is expected to function properly if those
dimensions are kept between 9.9 and 10.1 mm.
Fig. 1. A design
showing the functional dimensions of a part.
Suppose that, for the purpose of
manufacturing those parts, the manufacturing people in the factory request that
the print, as used in the process, be modified so that all the dimensions are
taken from the left end of the part, as shown in figure 2. They request that a drawing be prepared for the manufacturing process, based on the original functional design, so that their work can be facilitated. For example, they want to know how long the part should be, if their initial manufacturing operation is to cut the part and machine it to the final length.
It is clear that the nominal
dimensions have to be 10, 20 and 30 mm. The question that is unclear is how the values
of tolerances are affected by the change in dimensioning. The new tolerances
are represented by the variables x, y and z, as indicated in figure 2. It is the purpose of this article to
show how the new values for the tolerances can be calculated.
Fig. 2. The design is changed, to show all the
dimensions measured from the left end of the part.
The two dimensions that have
been replaced still must be met, as they were in the original design. The
tolerances in the modified drawing will have to be such that conditions A y B,
as shown in figure 3, have to be the result of the calculations for the new
tolerances.
Let’s suppose that –in a first
approach– we decide that the tolerance for 10 mm (variable x) will be half of that for 20 mm (variable y), and then it would be fair to give the dimension of 30 mm a
tolerance z that is three times the
value of x. Somehow it makes sense to
assign tolerance values that are directly proportional to the magnitude of the
nominal dimensions. A better approach would be to assign tolerances following
the practices of the ISO sistem of limits and fits (ISO 286-1 and ISO 286-2), but let’s assume that we
are satisfied now by making the tolerances directly proportional to the nominal
dimensions, in this first approach to the problem. It will be shown that this
will lead to an overconstrained problem. Thus, we would have the following
equations:
y = 2 x
|
[1]
|
z = 3 x
|
[2]
|
In figure 3 shown below, the two functional conditions A and B are established, being those that were originally specified in the original design. Their values are as follows:
Condition A (maximum value) = AMAX = 10.1
Condition A
(minimum value) = Amin = 9.9
and the same values apply in this case for condition B, by the original design.
That is, BMAX = 11.1, and Bmin
= 9.9 mm.
Fig. 3. The dimensions that have been replaced in the
original design still have to be met, and are now functional conditions A and B, indicated with the wide block arrows in this figure.
It must be kept clear that
conditions A and B will no longer appear in the design that must be produced in the
shop floor, and inspected for approval. The manufacturing can be done according
the the instructions in the shop floor, but the inspection must comply with
what is specified in figure 1.
From figure 3, we can see that
satisfying condition A implies the
following expressions:
(20 + y) – (10 – x) ≤ AMAX = 10.1
|
[3]
|
|
|
|
(20 – y) – (10 + x) ≥ Amin =
9.9
|
[4]
|
|
|
|
And then, in order to satisfy condition B, we write the next two equations:
(30 + z) – (20 –
y) ≤ BMAX = 10.1
|
[5]
|
(30 – z) – (20 +
y) ≥ Bmin = 9.9
|
[6]
|
The resulting expressions can be
presented in matrix form, as a system of four equations with three unknowns, as
follows:
2 x
|
- y
|
= 0
|
[7]
|
|
3 x
|
- z
|
= 0
|
[8]
|
|
x
|
+ y
|
≤ 0.1
|
[9]
|
|
y
|
+z
|
≤ 0.1
|
[10]
|
This system is
overconstrained. There are more mathematical constraints than variables. Sound
judgement has to be applied to solve it. There are standard mathematical
procedures to deal with this kind of problems, but fortunately this one is easy
enough to be solved without resource to advanced knowledge.
Now it is only necessary to
realize that constraints [9] and [10] are more important, for the purpose of
complying with the design specifications shown in figure 1, than equations [7]
and [8], since these latter equations were only suggested by common sense.
An easy way to work around the
problem is to replace equations [7] and [8] by the following much simpler requirements:
x ≤ y
|
[11]
|
y ≤ z
|
[12]
|
In the following section it will
be shown that a valid solution could be given by x = 0.04, y = 0.04 and z = 0.06 mm.
Results
Figure 4 shows a result that is
consistent with the previous calculations. It can be easily verified that
conditions A and B are correctly met.
Fig. 4. The results of the calculations for transfer of dimensions with the new tolerances.
The figure 4 above complies with the
original design specifications that were shown in figure 1. However, figure 1 is not
equivalent to figure 4. The relationship of equivalence or compliance works
only in one direction. That is, figure 4 complies with the functional requirements of figure 1; those requierements were the functional conditions A and B.
It must also be noticed that sizable
portions of the tolerances originally specified in figure 1 are no longer
available for the process, after the original dimensions have been transferred. This is an
unavoidable result of transfering dimensions with tolerances. The smallest
dimension, which originally was 10 ±0.1
mm, has now been transferred to become 10 ±0.04
mm, thus giving up 60% of the tolerance.
Another valid solution, different
from that shown above in figure 4, could have been x = 0.05, y = 0.05 and z = 0.05 mm. However, it is better to
assign a slightly larger tolerance to the larger dimension of 30 mm, and
tighter tolerances to the smaller 10 mm and 20 mm dimensions. There are multiple solutions, but once one solution is chosen, the manufacturing must proceed according to that particular solution.
However, regarding the inspection procedures and gages used to measure the parts in the production process, those must be strictly based on the original functional drawing, which is the design shown in figure 1.
Conclusion
Transferring dimensions with
tolerances is sometimes unavoidable, when the manufacturing process requires
the transfer to be done to make some process possible. Sometimes the transfer
is only convenient for purposes of inspection. The least expensive
manufacturing process is often that in which the transfer of design
specifications can be avoided.
The motivation to write this
article came from multiple occasions in which the author, when inquiring
practicing engineers in industry, professors in academia or engineering
students, has presented the problem of transfering the design specifications
shown in figure 1 to the form of figure 2. Unfortunately, and so far without
exception, the quick answer has been as shown in figure 5, which is incorrect,
as the reader can now see.
Fig. 5. An incorrect solution to the problem, which is often encountered in practice.
The author of this article has
often encountered quality problems in manufacturing industries, problems that
can be traced back to inappropriate knowledge of the little known –albeit
simple– subject of transfer of dimensions.
References:
1. Jiménez, Pierre. Acotación Funcional. Ed. Limusa, México, 1985.
3. Chevalier, A. Tecnología del diseño y fabricación de piezas metálicas. Ed. Limusa, 1998.
