Three Methods of Unfolding Expandable Surfaces of Sheet Metal Components

Sheet metal components, despite their complex and varied shapes, are mostly made up of basic geometries and their combinations. The basic geometry can be divided into two types: planar and curved. The common planar three-dimensional (mainly quadrangular prisms, truncated prisms, oblique parallel surfaces, quadrangular cones, etc.) and their planar assemblies are shown in figure (a) below, while the common curved three-dimensional (mainly cylinders, spheres, orthocones, oblique cones, etc.) and their curved assemblies are shown in figure (b) below. As can be seen from the basic curved three-dimensional sheet metal components shown in (b) below, there is a rotating body formed by a bus bar (plain line: straight or curved) rotating around a fixed axis. The surface on the outside of the rotating body is called the rotating surface. Cylinders, spheres and cones are all rotating bodies and their surfaces are rotating surfaces, whereas oblique cones and irregularly curved bodies are not rotating bodies. Obviously, a cylinder is a straight line (bus) rotating around another straight line that is always parallel and equidistant. A cone is a straight line (bus) intersecting an axis at a point and always rotating at a certain angle. A sphere is a semi-circular arc with the diameter as the axis of rotation.

There are two types of surface: expandable and non-expandable. To determine whether a surface or part of a surface is spreading, use a ruler against an object, rotate the ruler and see if the ruler fits all the way around the surface of the object in a certain direction, and if it does, write down the position and choose a new position near any point. The surface of the measured part of the object is extensible. In other words, any surface where two adjacent lines can form a plane (i.e. where two lines are parallel or intersect) is expandable. This type of surface is the plane of three dimensions, column surface, cone surface, etc.; where the parent line is a curve or two adjacent lines is the intersection of the surface, are not scalable surface, such as the sphere, ring, spiral surface and other irregular surface, etc.. For non-expandable surfaces, only approximate expansion is possible.

There are three main methods of unfolding expandable surfaces, namely: the parallel line method, the radial line method and the triangle method. The method of unfolding operation is as follows.

Parallel line method

In accordance with the prism of the prism or cylinder of the line, the prism or cylinder surface into a number of quadrilateral, and then spread out in turn, to make the expansion of the map, this method is called parallel line method. The principle of the parallel line method of unfolding is: because the surface of the form by a set of numerous parallel to each other straight lines, so the two adjacent lines and their upper and lower ends of the tiny area enclosed by the line, as an approximate plane trapezoid (or rectangle), when divided into an infinite number of tiny area, then the sum of the small plane area, is equal to the surface area of the form; when all the tiny plane area in accordance with the original The surface of the truncated body is unfolded when all the tiny planes are laid out in their original order and relative to each other, without omission or overlap. Of course, it is not possible to divide the surface of a truncated body into an infinite number of small planes, but it is possible to divide it into dozens or even several small planes.

Any geometry where the cords or prisms are parallel to each other, such as rectangular tubes, round tubes, etc., can be surface unfolded by the parallel line method. The diagram below shows the unfolding of the prismatic surface.

The steps to make an unfolding diagram are as follows.

1. to make the main view and top view.

2. make the base line of the unfolding diagram, i.e. the extension line of 1'-4' in the main view.

3. record the perpendicular distances 1-2, 2-3, 3-4, 4-1 from the top view and move them to the datum line to obtain points 10, 20, 30, 40, 10 and draw perpendicular lines through these points.

4. drawing parallel lines to the right from points 1', 21', 31' and 41' in the main view, intersecting the corresponding perpendiculars to give points 10, 20, 30, 40 and 10

5. Connect the points with straight lines to obtain the unfolding diagram.

The diagram below shows the unfolding of a diagonally cut cylinder.

The steps to make an unfolding diagram are as follows.

1. make the main view and top view of the oblique truncated cylinder.

2. Divide the horizontal projection into a number of equal parts, here into 12 equal parts, the half circle is 6 equal parts, from each equal point up to the vertical line, in the main view of the corresponding line, and cross the oblique section circumference at 1', ... , 7' points. The points of the circle are the same.

3. Expand the cylindrical base circle into a straight line (the length of which can be calculated using πD) and use it as a reference line.

4. Draw a vertical line from the equidistant point upwards, i.e. the plain line on the surface of the cylinder.

5. Draw parallel lines from the main view at 1', 2', ... , 7' respectively, and intersect the corresponding prime lines at 1", 2", ... The endpoints of the lines on the unfolded surface.

6. Connect the endpoints of all the plain lines into a smooth curve to obtain a diagonal cut of the cylinder 1/2. The other half of the unfolding is drawn in the same way to obtain the desired unfolding.

From this, it is clear that the parallel line method of expansion has the following characteristics.

1. The parallel line method can only be applied if the straight lines on the surface of the form are parallel to each other and if the real lengths are shown on the projection diagram.

2. using the parallel line method of solid expansion of the specific steps are: any equal (or arbitrary division) of the top view, from each equal point to the main view of the projection ray, in the main view of a series of intersection points (which is actually the surface of the form into a number of small parts); in the direction perpendicular to the (main view) straight line intercept a line segment, so that it is equal to the section (perimeter), and photographed on the top view of the points, over this line segment The vertical line of this line is drawn through the points on the line and the vertical line of the line drawn from the intersection point in the first step of the main view, and then the intersection points are connected in turn (this is actually a number of small parts divided by the first step in order to spread out), then the unfolding diagram can be obtained.

Radiometric method

On the surface of the cone, there are clusters of lines or prisms, which are concentrated at the top of the cone, using the top of the cone and the radiating lines or prisms to draw the expansion method, called the radiometric method.

Radial method of unfolding the principle is: the shape of any adjacent two lines and its bottom line, as an approximate small plane triangle, when the small triangle bottom infinitely short, small triangle infinite, then the small triangle area and the original truncated side area is equal, and when all small triangles are not missing, not overlapping, not creased according to the original left and right relative order and position When all the small triangles are laid out in their original relative order and position, the surface of the original form is also expanded.

The radial method is the method of unfolding the surface of all kinds of cones, whether they are orthocones, oblique cones or prisms, as long as they have a common cone top, they can be unfolded by the radial method. The diagram below shows the unfolding of the oblique truncation of the top of a cone.

The steps to make an unfolding diagram are as follows.

1. Draw the main view and fill in the top truncation to form a complete cone.

2. Make a cone surface line by dividing the base circle into a number of equal parts, in this case 12 equal parts, to obtain 1, 2, ..., 7 points, from these points to draw a vertical line upwards, and intersect the base circle orthographic projection line, and then connect the intersection point with the top of the cone O, and intersect the oblique surface at 1', 2', ..., 7' points. The lines 2', 3', ..., 6' are not real lengths.

3. Draw a sector with O as the centre and Oa as the radius. The arc of the sector is equal to the circumference of the base circle. Divide the sector into 12 equal parts, intercepting equal points 1, 2, ..., 7. The arc lengths of the equal points are equal to the arc lengths of the circumference of the base circle. Using O as the centre of the circle, make leads (radial lines) to each of the equal points.

4. From the points 2', 3',..., 7' make leads parallel to ab, intersecting Oa, i.e. O2', O3',... O7' are the real lengths.

5. Using O as the centre of the circle and the perpendicular distance from O to each of the intersection points of Oa as the radius of the arc, intersect the corresponding prime lines of O1, O2, ..., O7, to obtain the points of intersection 1'', 2'', ..., 7''.

6. Connect the points with a smooth curve to obtain a diagonal intercept of the top of the conical tube. The radiometric method is a very important method of expansion and is applicable to all cone and cone truncated components. Although the cone or truncated body is unfolded in a variety of ways, the unfolding method is similar and can be summarised as follows.

In the second view (or only in one view) the entire cone is expanded by extending the edges (prisms) and other formalities, although this step is not necessary for truncated bodies with vertices.

By dividing the perimeter of the top view equally (or arbitrarily, without dividing it equally), the line over the top of the cone (including the lines over the vertices of the lateral ribs and sides of the prism) corresponding to each of the equal points is made, the point of this step being to divide the surface of the cone or truncated body into smaller parts.

By applying the method of finding the real lengths (the rotation method is commonly used), all the lines that do not reflect the real lengths, the prisms, and the lines associated with the expansion diagram are found without missing the real lengths.

Using the real lengths as a guide, the entire side surface of the cone is drawn, together with all the radiating lines.

On the basis of the whole cone side surface, draw the truncated body on the basis of the real lengths.

Triangulation method

If there are no parallel lines or prisms on the surface of the part, and if there is no cone top where all the lines or prisms intersect at one point, the triangle method can be used. The triangle method is applicable to any geometry.

The triangle method is to divide the surface of the part into one or more groups of triangles, and then find out the real length of each side of each group of triangles, and then these triangles in accordance with certain rules according to the real shape flattened to the plane and get unfolded, this method of drawing unfolded diagrams is called the triangle method. Although the radial method also divides the surface of a sheet metal product into a number of triangles, the main difference between this method and the triangular method is that the triangles are arranged differently. The radial method is a series of triangles arranged in a sector around a common centre (cone top) to make an unfolding diagram, whereas the triangular method divides the triangles according to the surface shape characteristics of the sheet metal product, and these triangles are not necessarily arranged around a common centre, but in many cases are arranged in a W-shape. In addition, the radial method is only applicable to cones, whereas the triangular method can be applied to any shape.

Although the triangle method can be applied to any shape, it is only used when necessary because it is tedious. For example, when the surface of the part without parallel lines or prisms, can not use the parallel line method to expand, and no concentration of all lines or prisms of the vertex, can not use the radial method to expand, only when the triangle method for the surface expansion. The diagram below shows the unfolding of a convex pentagram.

The steps of the triangle method for the expansion diagram are as follows.

1. Draw a top view of the convex pentagram using the method of a positive pentagon within a circle.

2. Draw the main view of the convex pentagram. In the diagram, O'A' and O'B' are the real lengths of the OA and OB lines, and CE is the real length of the bottom edge of the convex pentagram.

3. Use O'A' as the major radius R and O'B' as the minor radius r to make the concentric circles of the diagram.

4. Measure the lengths of the circles in order of m 10 times on the major and minor arcs to obtain 10 intersections of A"... and B"... on the major and minor circles respectively.

5. Connect these 10 points of intersection, resulting in 10 small triangles (e.g. △A "O "C" in the diagram), which is the expansion of the convex pentagram.

The 'sky is round' component shown below can be seen as a combination of the surfaces of four cones and four flat triangles. If you apply the parallel line method or the radial line method, it is possible, but it is more troublesome to do so.

The steps of the triangle method are as follows.

1. will be 12 equal parts of the circumference of the plan, will be equal parts of the points 1, 2, 2, 1 and similar angle point A or B connected, and then from the equal points up for the vertical line intersection of the main view of the upper mouth in 1', 2', 2', 1' points, and then connected with A' or B'. The significance of this step is that the side surface of the sky is divided into a number of small triangles, in this case into sixteen small triangles.

2. From the symmetrical relationship between the front and back of the two views, the lower right corner of the plan 1/4, the same as the remaining three parts, the upper and lower ports in the plan reflect the real shape and real length, because GH is the horizontal line, and thus the corresponding line projection 1'H' in the main view reflects the real length; while B1, B2 but in any projection map does not reflect the real length, which must be applied to find the real length of the line method to find the real length, here The right triangle method is used (note: A1 equals B1, A2 equals B2). Next to the main view, two right-angled triangles are made so that one right-angled side CQ equals h and the other - right-angled sides A2 and A1 - are the hypotenuse QM and QN, the real length line. The significance of this step is to find out the length of all the small triangle sides, and then analyse whether the projection of each side reflects the real length, if not, then the real length must be found one by one using the real length method.

3. Make an expansion diagram. Make the line AxBx so that it is equal to a, with Ax and Bx respectively as the centre of the circle, the real length of the line QN (i.e. l1) as the radius of the arc intersected by 1x, which makes a plane diagram of the small triangle △AB1; with 1x as the centre of the circle, the plane diagram of S arc length as the radius of the arc, and Ax as the centre of the circle, the real length of QM (i.e. l2) as the radius of the arc intersected by 2x, which makes a plane diagram of the small triangle △A12 This gives the expansion of the triangle ΔA12 in the plan. Ex is obtained by intersecting an arc drawn with Ax as the centre and a/2 as the radius, and an arc drawn with 1x as the centre and 1'B' (i.e. l3) as the radius. Only half of the full spread is shown in the spread diagram.

The significance of choosing FE as the seam in this example is that all the small triangles divided on the surface of the form (truncated body) are laid out on the same plane, in their actual size, without interruption, omission, overlap or crease, in their original left and right adjacent positions, thus unfolding the entire surface of the form (truncated body).

From this, it is clear that the triangular method of unfolding omits the relationship between the original two plain lines of the form (parallel, intersecting, dissimilar) and replaces it with a new triangular relationship, thus it is an approximate method of unfolding.

1. Correctly dividing the surface of the sheet metal component into a number of small triangles, correctly dividing the surface of the form is the key to the unfolding of the triangle method, in general, the division should have the following four conditions to be the correct division, otherwise it is the wrong division: all the vertices of all small triangles must be located on the upper and lower edges of the component; all small triangles must not cross the internal space of the component, but can only be attached to the All two adjacent minor triangles have and can have only one common side; two minor triangles separated by one minor triangle can have only one common vertex; two minor triangles separated by two or more minor triangles either have a common vertex or no common vertex.

2. Consider the sides of all the small triangles to see which reflect the real length and which do not. Any that do not reflect the real length must be found one by one according to the method of finding the real length.

3. Using the adjacent positions of the small triangles in the diagram as a basis, draw all the small triangles in turn, using the known or found real lengths as radii, and finally connect all the intersections, depending on the specific shape of the component, with a curve or with a dash, to obtain an unfolding diagram.

Comparison of the three methods

According to the above analysis can be seen: triangle unfolding method can unfold the surface of all expandable forms, while the radial method is limited to unfolding the intersection of lines at a point of composition, parallel line method is also limited to unfolding the elements parallel to each other's components. Radial method and parallel method can be seen as a special case of the triangle method, from the simplicity of drawing, the triangle method to unfold the steps more cumbersome. Generally speaking, the three methods of unfolding are chosen according to the following conditions.

1. If the component of a plane or surface (regardless of its cross-section closed or not), on the projection of all the lines on a projection surface, are parallel to each other's solid long lines, and in another projection surface, the projection of only a straight line or curve, then you can apply the parallel line method to expand.

2. If a cone (or part of a cone) is projected on a projection plane, its axis reflects the real length, and the base of the cone is perpendicular to the projection plane, then the most favourable conditions for the application of the radiometric method are available ("most favourable conditions" does not mean the necessary conditions, because the radiometric method has a real length step, so regardless of the cone (in what kind of projection position, always can find out all the necessary elements line real length, and then expand the side of the cone).

3. When a plane or a surface of a component is polygonal in all three views, that is, when a plane or a surface is neither parallel nor perpendicular to any projection, the triangle method is applied. The triangle method is particularly effective when drawing irregular shapes.