# The relationship between geometric tolerances and alternative applications

The national standard GB1182～1184 “Shape and Position Tolerance” includes shape tolerance-straightness, flatness, roundness, cylindricity, line profile, surface profile; orientation position tolerance-parallelism, perpendicularity, inclination; Positioning position tolerance-concentricity, symmetry, position; runout-radial, oblique, circular end face runout, radial and end face full runout. Although some of these items have different concepts, they are closely related. Some items are similar or controlled by other items, some are single tolerances, and some are comprehensive tolerances, which can replace each other under certain conditions. But often fail to pay attention to this problem. Sometimes the designer draws the geometric shape and size of the part, but the labeling of the shape and position tolerance is sloppy, and the phenomenon of improper labeling or repeated labeling often occurs. Sometimes due to the different understanding of it by technicians, it causes confusion in the application and brings difficulties to the manufacture and inspection of parts. Therefore, it is necessary to have a deep understanding of the relationship between shape and position tolerances, and to master their various replacement usages. In this way, when marking the shape and position tolerances of parts, it is the most concise, clearest, most practical, the most economical for processing, and the most convenient for testing when the requirements are met. 1. Shape tolerance 1. Cylindricity, straightness, roundness and cylindricity is an index that limits the amount of change of the actual cylinder to the ideal cylinder surface. Its tolerance zone is the area between two coaxial cylindrical surfaces with the tolerance value t as the radius difference. It controls the various shape tolerances in the cross section of the cylinder and the shaft section, such as roundness, axis straightness, straightness of plain lines, etc. When in use, it is not necessary to mark the roundness and straightness if the cylindricity is generally marked. If the roundness and straightness must be marked separately, the tolerance value must be less than the cylindricity tolerance value (see Figure 1), to indicate that the design requires further requirements for radial or axial shape tolerances.
Figure 1 Cylindricity and roundness or straightness are marked at the same time. Usually, the cylindricity error is detected by a roundness meter or a three-coordinate measuring device equipped with a computer. If these devices are not available, it is best not to use cylindricity. In this case, roundness can be used separately. The parallelism with the cylindrical surface is used instead (see Figure 2).
Figure 2 Combination of roundness and parallelism instead of cylindricity When roundness and parallelism are used instead of cylindricity, the roundness tolerance value and parallelism tolerance value should be determined according to the aspect ratio of the cylinder. ·When the length of the cylinder is greater than its diameter, the parallelism tolerance value of the plain line must be correspondingly greater than the roundness tolerance value (see Figure 3a). ·When the length of the cylinder is equal to its diameter, the parallelism tolerance value of the plain line and its roundness tolerance value should also be equal (see Figure 3b). ·When the length of the cylinder is less than its diameter, the parallelism tolerance value of the plain line must be correspondingly smaller than the roundness tolerance value (see Figure 3c).
a)L＞D b)L=D c)L＜D Figure 3 Determine the roundness tolerance and parallelism tolerance according to the length-diameter ratio of the cylinder 2. Roundness, line profile roundness is to limit the variation of the actual circle to the ideal circle An index of the tolerance zone is the area between two concentric circles with the tolerance value t as the radius difference. The line profile is an index that limits the amount of change of the actual curve to the ideal curve. The tolerance zone is the area between the two envelopes of a series of circles with a diameter of tolerance t. The centers of the circles should be located on the ideal profile. . From the line profile tolerance zone (see Figure 4b), it can be seen that the line profile not only requires its contour shape to be correct, but also has certain size requirements, that is, its ideal shape is related to the size, similar to the size deviation. The roundness is not the case. It only limits the difference between the radii of the two concentric circles. As for the diameter of the two concentric circles, there is no requirement for the diameter of the two concentric circles, and the position of the two concentric circles is uncertain. Therefore, marking the line profile can get an effect similar to the use of the principle of inclusion (as shown in Figure 4c, the actual curve must be located between two concentric circles with a diameter of 79.9mm and 80.1mm). The effects marked in Figure 4a and Figure 4c are actually the same.
Figure 4 Line profile and the principle of tolerance are well known. When the principle of tolerance is applied to a single element, it can comprehensively control the various shape errors of the longitudinal and cross-section of a cylindrical hole or shaft, including roundness errors. Therefore, the roundness error can be completely controlled by marking the line profile without marking the roundness, that is, the line profile can be used instead of the roundness. Generally, it is more intuitive and clear to use roundness for circular curves. Especially in actual production, it is very convenient to use two-point and three-point methods to measure roundness. The line profile is dedicated to non-circular curves. 2. Position tolerance and shape tolerance The actual position and direction of the measured element of a part are always closely related to its actual shape. Therefore, the ideal boundary of the related element controls the actual position and direction of the element, and it inevitably controls the shape error of the element. For the convenience of operation, whether to use a comprehensive gauge or to measure with an indicating gauge, it is generally carried out directly on the contour surface of the measured element. Therefore, the position error is the combined effect of the actual position and the actual shape, that is, the measured position error includes the shape error. So usually the shape tolerance value given by the same element should be smaller than the position tolerance value (see Figure 5).
Figure 5 Simultaneous labeling of shape tolerance and position tolerance 3. Orientation position tolerance and positioning position tolerance The relationship between orientation tolerance and positioning tolerance is the same as the relationship between position tolerance and shape tolerance, usually positioning tolerance can control the orientation requirements, because the actual element being measured is in the positioning tolerance Not only is the position tolerance change (translation) within the belt controlled, but the direction change (angular displacement) is also controlled. 1. Coaxiality and parallelism. The coaxiality tolerance of the axis of the two holes in Figure 6 can completely control the parallelism requirements of the two axes. Because it controls the translation, tilt or bending of the measured axis to the reference, it is not necessary to mark the two Parallelism of the hole axis.
Figure 6 Concentricity comprehensive control parallelism 2. Position degree and perpendicularity position degree is a comprehensive tolerance. As shown in Figure 7, the straightness of the axis of the two holes and the perpendicularity of the axis of the two holes to the reference plane can be comprehensively controlled by the position degree, and there is no need to repeat the marking.
Figure 7 Position degree comprehensive control verticality and straightness 3. Position tolerance (position degree, coaxiality, symmetry) All positioning tolerance items can be replaced by position degree (see Figure 8, Figure 9).
Figure 8 Position degree comprehensive control coaxial degree
Fig. 9 Position degree comprehensive control Symmetry degree a) and b) in Fig. 8 and Fig. 9 have the same control effect, and the tolerance zone shape and detection method are the same. Therefore, the degree of position can be used to replace the degree of concentricity and symmetry. Since marking the coaxiality and symmetry of the above conditions in production is more intuitive and clear than marking the position degree, it is more appropriate to mark the coaxiality and symmetry on the drawing, and the position degree is usually used to limit the position error of points and lines. 4. Various runouts 1. Radial circle runout and radial full runout. The tolerance zone of radial circle runout is any measurement plane that is perpendicular to the reference axis. The difference in radius is the tolerance value t, and the center of the circle is two points on the reference axis. The area between two concentric circles (see Figure 10a), the tolerance zone is limited to two coordinates (plane coordinates).
Figure 11 End face circle runout and end face full runout The tolerance zone of end face full runout is the area between two parallel planes perpendicular to the reference axis and the distance is the tolerance value t (see Figure 11b). Obviously, the end face circular runout is only a part of the end face full runout, and the effects of the two are different. It should be determined according to the functional requirements whether to mark the end face full runout or the end face round runout. Generally, only when the flatness of the end face is small enough, can the end face circular runout be used instead of the end face full runout. For example, for the shaft shoulder where the bearing is installed, because its radial dimension (d1-d2) is small, the end face circle runout error can be controlled to achieve the purpose of controlling the end face full runout (see Figure 12).
Figure 12 Use the end face circular runout to control the end face full runout 3. Radial circular runout and oblique circular runout Generally, the oblique circular runout should be marked on the conical surface and the forming surface of the symmetrical axis of rotation. Only when the cone angle of the conical surface is small (such as a≤10°) can the radial circular runout be marked instead of the oblique circular runout to facilitate detection. As shown in Figure 13, suppose the radial circle runout error is H and the oblique circle runout error is h, then: h=Hcosa.
Figure 13 Oblique circular runout 5. Runout tolerance and other geometric tolerances 1. Radial circular runout, roundness, coaxiality Radial circular runout is a comprehensive tolerance, which not only controls the concentricity error, but also Contains roundness error. When the axis of the measured cylindrical surface is coaxial with the reference line, because the measured element has a roundness error, there will be a radial circle runout error; when the measured element is an ideal circle but there is a coaxiality error, it will also A radial circle runout error occurs. It can be seen that as long as there is a concentricity or roundness error, there must be a radial circle runout error, and vice versa. Since it is more convenient to detect the radial runout error, the radial runout is often used instead of the coaxiality tolerance in production. For the same measured element, after the radial circle runout is marked, there is no need to mark the concentricity or roundness (see Figure 14), otherwise, the concentricity tolerance value must be less than the runout tolerance value.
Figure 14 Circle runout comprehensive control coaxiality 2. End face circle runout, end face full runout, end face perpendicularity, flatness a. End face circle runout and end face perpendicularity End face perpendicularity limits the verticality of the entire end face to the reference axis. The tolerance zone is the area between two parallel planes perpendicular to the reference axis. It not only limits the perpendicularity error of the entire measured end to the reference axis, but also limits the flatness error of the entire measured end surface. The end face circle runout only limits the position error of each point on the measured circumference and the shape error along the axial direction on the circumference, and does not control the flatness error and perpendicularity error of the entire end face. When the measured end faces the reference axis and there is an end face circle run-out error, the measured end face must have a perpendicularity error. Conversely, when the end face has a perpendicularity error, the end face circle run-out error may be zero (see Figure 15). There is an error in the flatness of the end face.
Figure 15 End face perpendicularity and end face circle runout Therefore, marking the end face perpendicularity tolerance can control the end face circle runout and end face flatness error. In the design, for the end faces that generally play a fixed connection role, the end face circular runout tolerance should be used first, because it is convenient to detect, for example, the shoulder of the rolling bearing, the end face of the gear blank, etc. When the end face is more important for processing and positioning, the perpendicularity tolerance should be adopted to control the flatness error at the same time. Such as the face of the face of the lathe, the work surface of the vertical lathe, etc. b. End face full runout and end face perpendicularity The end face full runout and end face perpendicularity tolerance have exactly the same control on the measured elements, and the two can be substituted for each other, or the same detection method can be used. In production, the end face full runout is used for workpieces that can (conveniently) rotate around the reference centerline, such as general shaft parts. The end faces of box parts and the center line of the hole are usually marked with perpendicularity tolerances. 3. Radial total runout, cylindricity, coaxiality a. Radial total runout tolerance is a comprehensive control index. The radial total runout of a single element is the cylindricity. However, the total radial runout of the related elements can control the cylindricity error and the concentricity error at the same time. Therefore, the total radial runout cannot be simply equated with the cylindricity. A cylindricity error will inevitably lead to a radial full runout error, and a coaxiality error will inevitably lead to a radial full runout error (see Figure 16).
Figure 16 Radial total runout and cylindricity, coaxiality b. Replacement usage·To test the total runout error of a single element and cylindrical surface, if limited by the part structure or testing equipment, the parallelism and roundness of the plain line can be used Instead (the labels in Figures 17a and 17b are equivalent).
Figure 17 Parallelism and roundness comprehensively replace the full run-out of a single element. For the full run-out of related elements, parallelism, roundness and coaxiality of the prime line can be used instead of control (the labels in Figure 18a and 18b are equivalent).
Figure 18 Parallelism, roundness, and concentricity comprehensively replace the full runout of the related elements. When the full radial runout cannot be detected, if the cylindricity detection method is mature or advanced measuring instruments are available, the full radial runout of the related elements can also be used Cylindricity and concentricity are substituted.

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