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Significant Figures, Dimensioning, and Tolerance

The accuracy of any number used in estimations or other technical calculations is specified by the number of significant figures that it contains. A significant figure is any nonzero digit or any leading zero that does not serve to locate the decimal point. A number cannot be interpreted as being any more accurate than its least significant digit, nor should a quantity be specified with any more digits than are justifiable by its measured accuracy. The numbers 128.1, 0.50, and 5.4, for example, imply quantities that have known accuracies of ± 0.1, ± 0.01, and ± 0.1, respectively, but the first is specified to four significant figures, while the second and third are specified to only two. If trailing zeros are placed after the decimal point, they carry the weight of significant figures. Thus, the number 0.50 means 0.5 ± 0.01.

The accuracy of any calculation can only be deemed as accurate as the least accurate number entering into the computation, and the number of significant figures that 0.50 can be claimed for the result should be set accordingly. For example, the product 128.1 × 0.50 × 5.4 entered into a calculator produces the digits 345.87. But because 0.50 and 5.4 are specified to only two significant figures, the rounded-off result of the multiplication must be recorded as 350, also with two significant figures only. Note that a digit is rounded up if the digit to its right is 5 or more. If the digit to its right is less than 5, the digit is rounded down.

When numbers find their way into technical drawings, the number of significant figures takes on a special meaning. No part can ever be made to exact dimensions, because machine tools do not cut perfectly. A cutting tool wanders about its intended position during the machining process. Similarly, changes in temperature, humidity, or vibration during the cutting process can cause the tool to follow a less-than-perfect path. The tolerance of each dimension shown in the drawing specifies the degree of error that will be acceptable for the finished part. As a rule, creating parts with tight tolerances involves the use of more expensive machining equipment and more time, because material cuts must be made more slowly. These features add considerable expense to the finished part. As the designer, you must decide which dimensions are truly critical.

Suppose that you wish to make the vehicle chassis plate shown in Figure 4. The plate, to be made from 0.4-mm-thick aluminum, contains several threaded holes to which axle mounts are to be secured with screws. This fabrication job is a bit complicated for simple hand tools, hence you've decided to have a professional machinist make it. Indeed, such a job requires specialized machining tools, including a milling machine, drill press, and reamer. One issue that you might think about concerns the precision with which the part would need to be built. For a rough prototype, you might be content with an approximate version of the plate that can be produced quickly. For the finished product, you might want the machinist to take the extra time to adhere more closely to the specified dimensions. One way that engineers and machinists communicate on issues of this nature is through the numerical notations on parts drawings. Carefully note the labeled dimensions shown in Figure 4. These numbers communicate to the machinist the acceptable deviation, or tolerance, for each of the plate's various dimensions.

4. Main chassis plate with dimensions and tolerance table.
The numbers on the drawing in Figure 4 have precise meaning for any machinist who reads the tolerance table. In this case, only the hole diameters are especially critical. The length of the chassis plate, for example, is 25 cm. The numbers 25.0, 25.00, and 25.000, though all mathematically equivalent, would mean different things to the machinist. According to the tolerance table, the number 25.0, with one digit after the decimal point, should be interpreted by the machinist to mean 25 ± 0.1 cm. A chassis plate with a finished width diameter anywhere between 25.1 and 24.9 cm would be deemed acceptable. Similarly, the holes are specified as lying 15.00 cm apart, implying a machined tolerance of 15 ± 0.05 cm. The minimum and maximum tolerance limits for the hole centers as machined would be between 15.05 cm and 14.95 cm. According to the tolerance table, the most stringent dimensions of all are those of the hole diameters. Because these holes are intended to hold pins inserted by friction fit, their diameters are specified to three decimal points, implying a strict machining tolerance of 0.200 ± 0.001 cm.

Practice!

Refer to the tolerance table on the logbook page shown in Figure 4. Compute the difference between the maximum and minimum permissible physical values for dimensions specified by the following numbers: 21.0 cm, 8.75 cm, 10 cm, 2.375 cm, 0.003 cm.
Write down the result of the following computation, using only the number of significant figures to which you are entitled: (45 + 8.2) × 91.0 ÷ 12.1.
What is the sum of the numbers 3.00 + 54.0 + 174 + 250?
What is the sum of the integers 3 + 54 + 174 + 250?
Using the tolerance table in Figure 4, write down the following dimensions specified to ±1 mm. 5.1 cm; 954 cm; 573 cm; 15 mm.
If all dimensions on a part are specified to be within ±1 mil (± 0.001 inch) what are the minimum and maximum angles between the sides of a 1-inch square?