Construction of Transition Pieces

What you will need:

• A few hours

• A sound knowledge of Pythagoras

• Ability to substitute into and use formulae

• A protractor, compass, A3 paper, pencil, sharpener and ruler.

• Transition pieces are made to connect two different forms, such as round pipes to square pipes, or square to rectangular.

• They may involve a change in shape and/or size and/or direction.

Transition pieces are used extensively in ductwork of heating and air conditioning systems, and in industrial settings where material has to continuously flow through pipework from one machine or process to another.

Transition pieces are usually made from a single sheet of metal.

The required pattern, called the development, of the transition piece, is accurately drawn on the sheet. It is then cut out and folded along pre-defined lines to form the transition piece.

Think of the development as being the net of the transition piece. Finally the seam edges are welded together to complete the transition piece.

How do you work out the pattern to produce the required transition piece?

The transition piece will be made from a series of triangular surfaces laid side-by-side to form the development. This form of development is known as triangulation. The development of a square to round transition is shown.

Have a think about why we would use triangles rather than quadrilaterals.

The reason is that there is only one possible triangle that can be constructed from 3 given side lengths. This is not the case for quadrilaterals and other polygons. eg a quadrilateral with all sides 80 mm in length could be a square or any rhombus with that side length. To be able to construct a specific quadrilateral we need to know the lengths of all 4 sides and also the length of a diagonal (ie we construct the quadrilateral as two triangles side by side)

Pythagoras’ theorem can be used to find the required lengths before the construction of the development by triangulation.

• In industry, this work would now be done using CAD software. However, we need to be aware of the mechanics behind it or when we hit problems we can’t solve them. We can’t always rely on computers!

• Previously it was done using specialist technical drawing methods and equipment, making use of right angled triangles to construct the ‘true lengths’ to do the triangulation.

Because we are dealing with solid objects we will need to learn a bit about locating points and finding distances in a 3 dimensional coordinate system.

We extend our 2 dimensional (x, y) number plane into a 3 dimensional system by using three axes x, y and z, all at right angles to each other.

We show the y axis as being horizontal, the z axis as being vertical, and the x axis pointing out from the page.

From your seat in a classroom the front wall is the yz plane (blue), the side wall on your left is the xz plane (pink), and the floor is the xy plane (green).

Of course the axes extend in the negative directions also.

• We use the right hand rule to remember the direction of the axes.

• If someone pointed at you with their right hand, their first finger points along the positive x axis, the second finger points along the positive y axis, and the thumb points along the positive z axis.

Practice using 3D coordinates in the classroom with the units in meters: 1. Locate the point (2, 3, 1). 2. Find the coordinates of the point at the centre of the room.

Draw your own neat 3D coordinate system, label the axes and the origin.

On this diagram, show a horizontal square of side 6 units, with the centre of the square at the origin. (It will look like a parallelogram rather than a square).

What are the coordinates of the vertices of the square?

On the same diagram draw a square of side length 4 units at a height of 4 units above the xy plane. The centre of the square is to be on the z axis.

What are the coordinates of the vertices of this square?

We now need to learn how to find the distance between two points in a 3D system.

We will start by finding the distance from the origin to the point (2,3,1)

• The point (2,3,1) is shown.

• Where is the right angle in triangle OAB ?

• Use Pythagoras theorem to find OB.

• The right angle is at A OB2 = 22 + 32 = 4 + 9 = 13 OB = √13

• We now use Pythagoras theorem in triangle OBP to find the

length of OP.

• Where is the right angle in triangle OBP ?

OBP is the right angle

OP2 = OB2 + BP2 = 13 + 1

= 14

The distance from (2,3,1) to the origin is √14 units.

• Can you see that the distance from the origin to any point (x, y,

z) is

222 zyxd

• The distance between the points (x1, y1, z1) and (x2, y2, z2) is given by the formula

We will not prove this, but observe that :

• x1 − x2 is the distance between the points on the x axis

• y1 − y2 is the distance between the points on the y axis

• z1 − z2 is the distance between the points on the z axis

• it is an extension of the formula

221

2

21

2

21 zzyyxxd

222 zyxd

Example: Find the distance between the points (2, 3, 1) and

(5, 1, -2).

Solution: x1 = 2, y1 = 3, z1 = 1,

x2 = 5, y2 = 1, z2 = -2

7.4

22

949

32)3(

)2(11352

222

222

d

Example 1

Follow the next few slides to produce the development for a rectangle to square transition piece.

• The base is to be a rectangle 60 mm by 40 mm.

• The top is to be a square of side 30 mm.

• The top square and bottom rectangle are to be parallel surfaces 40 mm apart, with the centre of the square vertically above one of the vertices of the rectangle. The seam will be on one of the edges.

• Construct the transition piece using a sheet of A4 paper or cardboard.

The transition piece looks like this, watch the video:

Step 1 – Find the coordinates of the points to be used. Watch this video:

Points

A (20, -30, 0)

B (20, 30, 0)

C (-20, 30, 0)

D (-20, -30, 0)

W (-5, 15, 40)

X (-5, 45, 40)

Y (-35, 45, 40)

Z (-35, 15, 40)

We now have these coordinates:

Step 2 - Identify the triangles which will make up the development. There are four surfaces – all quadrilaterals. (not to scale)

Triangles needed are ABW, BWX, BXC, CXY, CYZ, CZD, ZDW and DWA. This makes the seam along AW.

The lengths that need to be calculated are: AW, WB, BX, XC, CY, CZ, ZD, DW

Step 3 - Calculate the lengths using the formula

221

2

21

2

21 zzyyxxd

Rather than doing each individual calculation, excel has been used to calculate these lengths see here. It is always worth checking one or two are correct though, as the formula may have been typed in incorrectly.

Example: Find the distance between the points A (20, -30, 0) and

W (-5, 15, 40).

Solution: x1 = 20, y1 = -30, z1 = 0,

x2 = -5, y2 = 15, z2 = 40

2.65

4250

16002025625

)40()45(25

40015)30()5(20

222

222

d

221

2

21

2

21 zzyyxxd

Finding the length AW

We now have these lengths:

Lengths (mm)

AW 65.2

WB 49.5

BX 49.5

XC 45.3

CY 45.3

CZ 45.3

ZD 62.0

DW 62.0

65.2

49.5 49.5

45.3

45.3 62.0

62.0 45.3

Step 4 – Create the development by triangulation. Watch this video:

Step 5 – Cut out and fold the piece. Watch this video:

There are now four exercises you can try to do on your own using these techniques…

Exercise 1 Produce the development to make a square to square transition piece.

The base is to be a square of length 60 mm.

The top is to be a square of side 40 mm.

The top and bottom squares are to be parallel surfaces 50 mm apart with their centres in a vertical line.

The seam will be on one of the edges. Construct the transition piece using a sheet of A4 paper or cardboard.

Exercise 2 Produce the development to make a rectangle to square transition piece.

The base is to be a rectangle 60 mm by 40 mm.

The top is to be a square of side 30 mm.

The top and bottom squares are to be parallel surfaces 50 mm apart with one vertex of the square directly above a vertex of the rectangle.

The seam will be on one of the edges. Construct the transition piece using a sheet of A4 paper or cardboard.

Exercise 3 Produce the development to make a rectangle to square transition piece to join two sections of ductwork.

One end is to be a vertical rectangle 60 mm wide by 40 mm high; the other is to be a horizontal square of side 60 mm.

The square end is to be 30 mm below the bottom edge of the rectangle and 30 mm out from it.

The seam will be on one of the edges. Construct the transition piece using a sheet of A4 paper or cardboard.

The piece looks like this.

Exercise 4 Produce the development to make an oblique square pyramid, open at the base.

The base is to be a square of side 60 mm.

The height of the pyramid is 80 mm, with the apex of the pyramid directly above one of the vertices of the base.

The seam will be on one of the edges. Construct the pyramid using a sheet of A4 paper or cardboard.

Well done! See your teacher for the extension of this activity…