Uses for Centre Lines

[studiot]

 

As I said this is always the length of the centre line, which is why engineers use the centre line to measure things.

This is a useful thing to learn, not often taught in maths courses.

 

I was asked to explain this comment I made in another thread, but since discussion would be off topic there I have started this one.

 

I'm sure I have missed many obvious thoughts so comments from Engineers would be welcome.

 

Paradoxically engineers use centre lines both because something balances about the centre line and because things are irregular about the centre line and for other practical reasons as well.

 

Working through my examples

 

1) If we drill a hole or bang a nail in we want to mark the spot where to place the point of our drill or nail. This is at the intersection of two centre lines.

 

2) So let us say we are going to drill these holes to bore out the cylinders in an engine block. It is mechanically important that these are in a straight line to balance the cranking forces, so we reference them to a single line, the centre line.

 

3) Another good reason for using a centre line is that the edges of the workpiece (the block will be a rough casting) so the edges do not form good reference lines. Nor will the corresponding edges of any two blocks be the same.

 

4)Of course the edges may be straight and regular, but they may taper. They are therefore then difficult to measure from. Do you measure at right angles to the edge or what? Measurements are usually made in right angled (rectangular) coordinate systems.

 

5) Say I have a beam that I place on two supports so that as shown in (5a). If I now place a heavy weight on it so it bends as in (5b), the top surface is now shorter than the bottom surface.

 

6) So there must be an intermediate point where there is no change of length as in (6). It can be proved that this point is the centreline of such a beam. So it makes s ense to measure from the line of zero change.

 

7) Because the shortening of the top surface is caused by the ebnding introducing a compressive force and the lengthening of the bottom surface by a corresponding tensile force again there must be a line where there is zero distorting force and this again lies along the centre line. Another name for this is the neutral axis.

 

8)Returning to fig4, if that is actually part of a tapering block or cut in earthworks or whatever and I need the volume, I can calculate the volume as an area times a length. But what area and what length?

Well it turns out that if I take the average of the two end areas and multiply this by (you guessed it) the distance along the centre line the product gives me the volume.

Edited October 15, 2014 by studiot

[Fuzzwood]

One of the more counterintuitive examples of this would be a donut. No matter its 3d shape, as long as the length of the circumference of the center line in the donut matches the one of a straight cylinder with equal diameter of that of the donut, its volume will be the same.

[Curiatron]

Well, in civil engineering we typically dimension off of the centerline for roads, water channels, and other things that are more or less "symetric" about the centerline. However, sometimes things are not dimensioned off of centerline. For example, sometimes roads have additional alignments, for example, along the road curb because the curb alignment deviates substantially in both the horizontal and vertical directions.