Method of Finding The Real Length of a Component
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Method of Finding The Real Length of a Component

Views: 280     Author: Site Editor     Publish Time: 2023-12-19      Origin: Site

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In the processing of sheet metal parts, workpieces of various shapes are often encountered, such as ventilation pipes, deformed joints, etc. To complete their processing, the sheet metal must first be unfolded, the surface of the object is spread out on a plane according to its actual shape and size. The unfolding of sheet metal is a preparatory process for the sheet metal material, and is also a prerequisite for the correct processing of the sheet metal parts. In order to draw a sheet metal unfolding diagram correctly, it is necessary to know the actual dimensions of the unfolding diagram or the actual dimensions of the relevant components of the unfolding diagram. When the three-dimensional surface of the line and the projection surface is not parallel, the design drawings in the projection is not reflected in its real length, so before the unfolding must be used as a graphical method to find out the real length of the line segment.


The methods of solving for the real length of a line segment include the rotation method, the right triangle method, the right trapezoid method and the auxiliary projection plane method. The mastery and application of these methods of finding the real length of a line segment is a prerequisite and basis for the acquisition of sheet metal unfolding skills.


The Rotation Method

The rotation method involves rotating a tilted line around an axis perpendicular to a projection plane to a position parallel to another projection plane, where the projected line segment on that projection plane is the real length of the tilted line. For graphical convenience, the axis generally passes over one of the end points of the inclined line, the end point is the centre of the circle and the inclined line is the radius of the rotation.


The principle of rotation for real length: the diagram below shows the principle of rotation for real length. ab is a general position line, which is inclined to any projection plane. ab's projection a'b' on the V-plane and ab's projection on the H-plane are both shorter than the real length. Assuming that the axis AO is perpendicular to the H-plane at one end of AB, when AB is rotated around the AO axis to a position AB1 parallel to the V-plane, its projection a'b1' on the V-plane (the dashed line in the diagram indicates the real length) will reflect its real length.

Right Triangle Method

Rotation method for real lengths: The diagram below shows the specific method of using the rotation method for real lengths. In the diagram below (a), the horizontal projection ab is rotated so that it is parallel to the orthographic projection, resulting in points a1 and b1, connecting a1b' or a'b1, which is the real length of the line segment AB; in the diagram below (b), the orthographic projection a'b' is rotated so that it is parallel to the horizontal projection, resulting in a1 and b1, connecting a1b or ab1, which is the real length of the line segment AB.

Right Triangle Method

Example: The diagram below shows a diagram of the real length of the prism of an oblique prism using the rotation method. As can be seen from the projection, the base of the oblique prism is parallel to the horizontal plane and its horizontal projection reflects its solid form and real length. The remaining four faces (sides) are two sets of triangles, whose projections do not reflect the real form. To obtain the real form of the two sets of triangles, the real length of their prisms must be found. As the shape is symmetrical from front to back, only the real lengths of the two lateral prisms are required to draw the diagram.

Right Triangle Method

The specific steps in making an unfolding diagram are

1. Use the rotation method to find the real lengths of the lateral ribs Oc and Od. As shown in the diagram below, take O as the centre of the circle, respectively Oc, Od as the radius of the rotation, cross the horizontal line in c1, d1. c1, d1 from c1, d1 up the vertical line, and orthographic projection c'd' extension line intersected in c1'd1', connecting O'c1', O'd1' is the real length of the side prism Oc and Od.

2. Make a line AD of length equal to ad at the appropriate position on the diagram, and then draw △AOD with A and D as the centre of the circle and Od' as the radius of the arc, intersecting at O; then make an arc with O as the centre of the circle and Oc1' as the radius, intersecting with the arc made with D as the centre and dc as the radius at C. Connect OC and DC to obtain △DOC. Draw the remaining two sides of △COB and △BOA in the same way to obtain a trigonal cone with the sides expanded.


The figure below is a truncated cone, the real length of the cone and the expansion, should first draw the top of the cone, become a complete cone, and then make a series of cone surface, and use the rotation method to find these lines were truncated part of the real length of the line (also available to leave part of the real length of the line), you can make the expansion of the figure.

Right Triangle Method

To find the real length of the truncated part of the line, the diagramming steps are as follows.

1. extend the shape line 1'1" and 7'7" to intersect, resulting in the top of the cone O'.

2. Make the base circle of the cone, and divide the circumference of the base circle into a number of equal parts (here 1/2 the circumference of the base circle is divided into 6 equal parts), to obtain equal parts 1, 2, ..., 7, from each equal point to the main view of the vertical lead, and the base circle orthogonal projection intersected at 1', 2', ..., 7' points, and then from each point and the top of the cone O' for the line, to obtain the cone the lines of the conic surface.

3. Among the lines of the cone, only the outline lines 1"1' and 7"7' are parallel to the orthogonal projection and reflect its length, while the rest do not reflect the real length. The method is to make a parallel line of 7'1' from 7", 6"..., 2" and intersect the O'1' contour line at 7°, 6°,..., 2°, O'6°, O'5°,..., O'2° for O'6", O'5",..., O' 2" respectively. 2" of real length.

Right Triangle Method

The diagram above shows the real length of the skew cone by rotation. The steps are as follows.

1. first make 1/2 the base circle, the circumference of the base circle into a number of equal parts (in the diagram into 6 equal parts).

2. with the vertical foot O as the centre of the circle, O1, O2, ..., O6 for the radius of the arc, and 1 ~ 7 line intersection at 2 "and so on each point.

3. Make a line from the points 2" etc. to O', O'2' etc. being the real length of the line through the equinoxes. In other words, O'2' is the orthogonal projection of the O2 line and O'2" is the real length of the O2 line.


The diagram below shows the real lengths of the prisms of a square joint using the rotation method and expanding them.

Right Triangle Method

The steps for drawing the real lengths of the prisms are

1. draw the main view and the top view, equate the top view circle opening and connect the corresponding plain lines.

2. rotate the plain lines a1, (a4), a2, (a3) and draw vertical lines upwards to derive their real lengths a-1, (a-4) and a-2, (a-3) on the right side of the main view.

3. Using the plain line real lengths, the square mouth edge lengths and the round mouth equivalent arc spread lengths, draw the 1/4 spreads in turn.


Where the transition part of the square tube is opposite to the round tube, there must be a square-round joint. The square mouth can be a square mouth or a rectangular mouth, the round mouth can be in the centre or to one side or to one corner, therefore, the form of such joints can be varied, but the method of seeking the real length of the square and round joints is basically the same.


Right Triangle Method

The right triangle method is a commonly used method for finding the real length.


The principle of the right triangle method and the method of drawing: the following diagram (a) is the principle diagram of the right triangle method for real length. The line segment AB is not parallel to the projection plane, and its projection ab and a'b' do not reflect the real length. In the ABba plane, a line is made parallel to ab through point A and intersects Bb at point B1, giving the right triangle ABB1. In this triangle, the real length of the hypotenuse AB of the right triangle can be found by knowing the lengths of the two right-angled sides AB1 and BB1. And the lengths of AB1 and BB1 are found on the projection diagram as AB1 = ab, BB1 = b'b1', or BB1 = b'bx - a'ax. Knowing such two right-angled sides uniquely draws the right triangle sought.

Right Triangle Method

Figure (b) above shows the use of the right triangle method to find the real length. The projection of the AB line is known as ab and a'b', to find the AB real length, you can first make a horizontal line through the point a', cross bb' line in the point b1', bb1' that is, the length of a right-angle side of the request. Then the top view of the ab for another right-angle edge, over the point b cited vertical line and intercept bB0 = b'b1', connected to aB0, that is, the real length of the line segment.


Example: The diagram below shows a small and large square mouth joint, try to find the real length of its prime line AC and auxiliary line BC.

Right Triangle Method

It can be seen from the diagram that the real length AC can be found in a right triangle with aC and Aa as the two right-angled sides, while the real length BC can be found in the right triangle BbC. In both triangles, Aa= Bb= h, which is equal to the height of the joint. The other two right angled sides aC and bC are equal to the projections ac and bc of AC and BC in the top view respectively. In this way, the real lengths of AC and BC can be found as follows.

1. make a right angle B0OC0.

2. intercept OA0 and OB0 on the horizontal side of that right angle respectively equal to ac and bc in the top view, and intercept OC0 on the vertical side equal to the height h in the main view.

3. connect C0A0 and C0B0, then the hypotenuse C0A0 and C0B0 are the real lengths of the requested AC and BC.


The Right-angle Trapezoid Method

The right-angle trapezoid method is also a common method of finding real lengths.


The principle of the right-angle trapezoid method for real length and the method of drawing: the following diagram shows the principle of using the right-angle trapezoid method for real length. The general location of the line AB in the V surface and H surface can not reflect the real length, but the two endpoints of the line AB and the distance between the V surface can be obtained on the H surface, that is, Aa and Bb, the same, A, B two points and the distance between the H surface can also be obtained on the V surface, that is, Aa 'and Bb'. Based on this principle, the real length of the line AB can be found using the right-angle trapezoid method. There are two specific methods of graphing the real lengths.

1. using the orthographic projection of the real length of the line AB: the orthographic projection of AB a'b' as the bottom edge of the right-angled trapezoid, from a', b' two points respectively up the vertical line, intercept the length of Aa', Bb', connected to AB, that is, for the requested.

2. is the use of the horizontal projection of the real length of the line segment AB: the horizontal projection of AB as the bottom edge of a right-angled trapezoid, from a, b two points respectively up the vertical line, intercept the length of Aa, Bb, connect AB that is the requested.

Right Triangle Method

Example: The following figure shows a horseshoe deformation joint, its upper and lower mouth are circles, but the two circles are not parallel and not equal in diameter, try to make a right-angle trapezoid method of its line length and expansion diagram.

Right Triangle Method

From the above figure (a) can be seen, because its surface is not a conical surface, in order to make its expansion diagram, can only use the line to and from the surface into a number of triangles, and one by one to find the real shape of these triangles. The specific graphing steps are as follows.

1. Make 12 equal parts of the upper and lower mouths, and divide the surface into 24 triangles as shown in the diagram.

2. Find the real lengths of the lines Ⅰ-Ⅱ, Ⅱ-Ⅲ, ..., Ⅵ-VII, and then make the real shape of the series of triangles.


For such examples, if the rotation method or the right triangle method is used to find the real length, the projection of the line segment on the top view must be made. As the top surface of the horseshoe deformation joint and the horizontal projection plane inclined, so the top surface in the top view is reflected as an ellipse, obviously, these two methods for the expansion of the map, are more trouble, at this time, it is appropriate to use the right-angle trapezoidal method.


Such as the above figure (b) in the Ⅰ-1-Ⅱ-2-Ⅲ-3...XII-12 folded surface stretch spread into the figure shown below, then the figure above the fold line Ⅰ-Ⅱ-Ⅲ...XII, that is, the real length Ⅰ-Ⅱ, Ⅱ-Ⅲ, ..., Ⅵ-VII and so on the line. This method of finding the real lengths is the right-angle trapezoid method.

Right Triangle Method

As can be seen from the diagramming method, the right-angle trapezoid method is also based on a projection of an inclined line as the base, with the distance of the two end points of the inclined line from the same projection plane as the two right-angle sides, after forming a right-angle trapezoid, then the hypotenuse of the right-angle trapezoid, that is, the real length of the requested line. The right triangle can be seen as a special case of the right-angled trapezoid method where the length of the right-angled side is equal to zero.

The above method is used to obtain the two side lines of each triangle on the surface of the horseshoe deformation joint, the other side of which is the length of the upper and lower circular opening equal to the unfolded arc. In this way, a series of triangles can be made by the method of triangles with three known sides, which are arranged in order to obtain the following diagram of the horseshoe deformation joint.


Face Change Method

In addition to the above methods of finding the real length of the line, there is also the common method of changing the surface.

Right Triangle Method

The principle of the method of changing the surface for the real length and the method of drawing: the principle of the method of changing the surface is to keep the space segment unchanged, another new projection surface to make it parallel to the requested segment, and perpendicular to the original one, the projection of the segment on the new projection surface will reflect its true length. The diagram above shows a schematic diagram of the real length of a line segment.

Right Triangle Method

As can be seen from the diagram above (a), the line segment AB is not parallel to both the H and V projection planes and its projection does not reflect the real length. The new projection a1'b1' reflects the real length of AB. Further analysis of the space shown in figure (a) above reveals the following projection relationships for the surface change method.


1. Since the new projection surface P is parallel to AB and perpendicular to the H-plane, then the line of intersection between the new projection surface P and the H-plane, O1X1 (called the new projection axis), is necessarily parallel to the H-plane projection ab of the line AB, O1X1 // ab, as reflected in the H-plane projection.


2. Since the P and V surfaces are simultaneously perpendicular to the H surface, the distance from the projection a1'b1' of the P surface to O1X1 and the distance from the projection a'b' of the V surface to OX must simultaneously reflect the perpendicular distances from the two endpoints A and B of the spatial line to the H surface, and they are equal to each other, a1ax1 = a'ax = Aa and b1'bx1 = Bb. For ease of designation, the newly made projection parallel to AB The projection a1'b1' which reflects the real length is called the new projection, the projection a'b' which originally did not reflect the real length is called the old or replacement projection, and the projection of the H-plane which is perpendicular to them at the same time is called the invariant projection. In this way, this projection relationship for the replacement surface method can be expressed as the distance from the new projection to the new axis being equal to the distance from the old projection to the old axis.


3. Since both the P and V surfaces are perpendicular to the H surface, the connection between the P projection and the H projection at any point on the line must be perpendicular to the new projection axis O1X1, the line between the invariant projection and the old and new projections is perpendicular to the old and new projection axes respectively, after unfolding.


In accordance with the above projection relationship of the permutation method, the graphing steps should be

1. as shown in (b) above, make the new projection axis O1X1 parallel to ab.

2. Draw a perpendicular line through points a and b to the O1X1 axis and intersect O1X1 at points ax1 and bx1.

3. Move the projections a' and b' of the V-plane to the OX-axis to the new projection plane, measure ax1a1'=axa' and bx1b1'=bxb' on the vertical lines.

4. Connect the points a1' and b1', the new projection a1'b1' of the AB line, which reflects the real length of AB.


Example: The diagram below shows the use of the auxiliary projection plane method to find the real shape of a cylindrical section.

Right Triangle Method

The steps in the drawing are as follows.

1. make a main and top view, dividing the top view by 1/2 the circumference of the circle in 6 equal parts.

2. draw a vertical line upwards through the equidistant point to give the position of the prime line in the main view.

3. drawing perpendiculars downwards from the equidistant points to intersect the bottom centre line, the width between the plain lines of the section

4. drawing perpendicular lines through the intersection of the lines on the oblique opening of the section to the long axis parallel to the oblique opening of the section, and then drawing the distance between the equidistant points in the top view and the centre line of the bottom circle, in turn, to the points in the secondary view, in accordance with the rule of "equal width".

5. Connect the points in order to create a solid ellipse of the section.


The diagram below shows the use of the auxiliary projection plane method to find the real shape of the orthocone section. The diagrams ①, ②, ... (7) indicate the order of drawing and connecting lines.

Right Triangle Method

In general, it is not necessary to draw lines on the cone surface to make the real shape of the conic section, but it is better to use the weft circle method, as shown in the figure above. In order to make the lines clear, the three steps of the diagram will be drawn separately in this example, the actual diagram does not need to be separated. The steps are as follows.


1. Weft circles: the projection line of the section is divided into 6 equal parts; the horizontal line of the above equal points is intersected with the contour line; the vertical line is drawn downwards from each intersection point on the contour line and intersected at the bottom of the cone; the weft circles are drawn in turn with the centre of the O circle, see figure (a) above.


2. Top view of the cross-section: by drawing a vertical line down through each equivocation of the cross-section lines in the main view, intersecting with the corresponding latitude circle, a series of intersection points is obtained; by connecting the intersection points, the top view projection of the cross-section can be obtained, see figure (b) above.


3. To find the real shape of the section: make an ellipse parallel to the long axis of the section 1"7"; draw perpendicular lines from each equal point of the section 1~7 to the long axis 1"7"; in accordance with the principle of equal widths, draw a series of widths a, b, c, d and e of the section in the top view to the auxiliary projection, resulting in 2", 3", 4", 5" and 6" points; connect the points, that is, the real shape of the conical section, see diagram (b) above. Figure (c) above.


The diagram below shows the use of the auxiliary projection surface method to find the real shape of the oblique conic section.

Right Triangle Method

The use of the auxiliary view for the real shape of the oblique conic section is similar to that of the real shape of the orthogonal conic section. However, the oblique cone has the characteristic that the top of the cone is inclined to one side and its axis is also inclined, so that the centre of a series of weft circles does not lie at the same point on the same axis. Therefore, instead of making concentric circles, a cone is made with one centre for each weft circle. This feature can be mastered by following the three steps described above to draw out the auxiliary view of a solid section.


The specific drawing steps are as follows.


1. For the weft circle: the section line 4 equal parts; for equal points of the horizontal line, intersecting with the contour line; from the contour line on the points down to the vertical line, intersecting with the bottom circle; equal points of the horizontal line and the axis intersection of the points for the weft circle of the centre, the centre of the circle to the bottom circle; respectively, the centre of the weft circle and the corresponding radius for the weft circle.


2. The top view of the section: through the main view of the section lines of each equivocation, downward lead vertical lines, and the corresponding latitude circle intersection, resulting in a series of intersection points; along with the intersection points, you can get the top view of the section projection.


3. To make the real shape of the section: according to the width of the section shape found in the top view, make 1/2 auxiliary view to draw the 1/2 real shape of the oblique conical section.


Comparison of Real Length Methods

Based on the above analysis, a simple comparison can be made between the four methods of finding the real length of a real line.


The rotation method solves for the real length by changing the position of the figure in space, without changing the position of the projection plane.


The permutation method solves for the real length by changing the position of the projection plane without changing the position of the figure.


The right triangle method and the right-angled trapezoid method (the right triangle method can be seen as a special case of the right-angled trapezoid method) solve for the real length line by changing neither the position of the space figure nor the position of the projection plane.

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