Twisted Transformer Functions first created 05/02/09 - last modified 03/10/09 Page Author: Ty Harness
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The Twisted Transformers (TT) application is capable of producing curved sides as shown in figure 1 where the green curve is a quadratic. TT has 3 built in shape functions which describe the mitring between the segments:
  • Linear
  • Quadratic
  • Elliptical
    Figure 1 defines the axes t and Z(t) which are shown red and blue respectively. The length of the t axis is indirectly defined by the chosen geometry parameters:Base Inscribed radius and the top inscribed radius. The axis is then divided into N equal divisions. Z(t) is the vertical ordinate where h is a proportianal factor.

    $ Z(t) = h/(t_1 - t_0) * t $ .............The linear function.

    $ Z(t) = h/(t_1 - t_0)^2 * t^2 $..........The quadratic function.

    In the elliptical shape function then h becomes the semi major axis and t1-t0 becomes the semi minor axis.

    $t = (t_1 - t_0)*sin(a)$ and $ Z(t) = h*(1-cos(a))$

    All 3 equations have boundary conditions that force the curve to go through 0,0 and t1,h. Of course, the curve does not need to pass through points and these are just 3 equations out of an infinite choice, but I've chosen them for simplicity and to suit a wide variety of sheet metal transformers and ornamental finials.

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    Figure 1 - Curved sides

    If you want to define your own curve then it's possible to import Z(t) from a text file. The shape function Z(t) value is specified as a text file (or sometimes called a comma separated, CSV file). In order for the TT application to load the data correctly you must format the file using the following simple rules:

  • At least 13 items of data are required
  • No more than 121 items of data
  • The data must be a numeric form (e.g. 1, 10.223 or -2.3 etc) or an exponential form (e.g. 10E-2 etc.)
  • Data items must be separated by a carriage return and line feed (It's not comma separated as implied by the term csv)
  • A carriage return and line feed must follow the last data item.
  • The t axis is divided into equal divisions. e.g. 13items of data then 12 equal divisions.

    Use Notepad(TM) from the Windows(TM) menu and just use carriage return( enter) between the data items (and the line feed is appended automatically in notepad), See figure 2. One function I should have maybe built in by standard is a circular dome as quite a few people have asked how can TT produce such a shape. Figure 2 shows that you determine the z heights of a dome requied by TT can be with a graphical method. Simply divide the t axis up equally and measure the vertical ordinate. Enter the vertical ordinates into a notepad document.

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    Figure 2a - Determination of the z heights by graphical construction
    b) Specifying Z(t) using MS Notepad(TM)
    c) Twisted Transformers DXF export of the data shown in figure 2a.

    You will see h is automatically set to 1 and be thought of as a multiplying factor for Z(t):

    $Z(t_i) = h*Z(t_i)$

    where you may wish to tweak the design to fit the transformer to revised dimensions without the need to re calculate all the values again. For example by letting h=2 we have created and elliptical dome without the need redraw or calculate the explicit z values. Figure 3b shows a screen shot of TT where h=2.

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    Figure 3a - Screen shot of TT using the above z values
    3b) Setting h = 2.

    Once you decide you need more than 12 divisions then typing in all the data is going to be tedious. Rather than use a graphical construction method you may which to derive the z height using algebra.

    For example, using the equation of a circle

    $(x-x_0)^2 + (y-y_0)^2 = a^2$

    where a is the radius and $x_0$ and $y_0$ are the circle's centre coordinate. In our case $y_0 = 0$, $z = y$ and $x_0 = a = 55 $. Therefore z as a function of x:

    $z(x) = sqrt( 55^2 - (x-55)^2)

    You can then use a spreadsheet like MS Excel(TM) or OpenOffice Calc(TM) to enter data quickly. Figure 4 shows a screen shot from OpenOffice Calc(TM) to calculate z(t) with 12 divisions as per the previous graphical method example. Note the cell formula used C10 = SQRT(55^2 - (B10-55)^2). When using a spreadsheet always use cell formulae to do the calculation for you.

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    Figure 4 - Screen shot of Open Office Calc(TM)

    If you need 120 divisions to describe a smoother pattern you can be quite creative with cell formulae. The table below needs only cell A4 and B4 to be changed to calculate 120 z(t) values.

    A B C D E
    4 =120 =55 =0 =B4*(C4/A4) =SQRT(B4^2 - (D4-B4)^2)
    5 =A4 =B4 =C4+1 =B4*(C5/A5) =SQRT(B5^2 - (D5-B5)^2)

    With only a few cell formulae and then copying row 5 into row 6 Open Office will auto increment the row and calculate the cells. I could then archive this spreadsheet in case I need any circular domes in the future.

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    Figure 5a - Using a spreadsheet to calculate 120 heights.
    b)Import into the TT application to produce a smoother pattern plot.

    Finally just copy the data from column E and paste into a notepad document and save. Use custom function in TT and then load the csv file.

    text here text here
    Figure 6a and b - Theodolite bombshell type case

    ASCII to MathML used in this page: ASCIItoMathML homepage