Trigonometry Homework for the Maths Gurus: Please help :)

[awemawson]

In the attached drawing I need to calculate the length 'L' (line A-D) that runs from the centre of circle radius R to intercept chord B-C, where the chord subtends angle B-A-C and the unknown line is at angle B-A-D. Note I need this for any angle B-A-D, not just for the line intercepting at right angles


Why do I need this? Think Traub CNC Lathe : The circle represents a part in the chuck in the main spindle. The line L is in the direction of the X axis of the lathe - ie tool in or out, and the tool will be a milling cutter rotating with its axis parallel to the main spindle. The main spindle will progress round the circle using it's precision 'C' axis for positioning, and the X axis will move in and out producing a flat on the work represented by the chord B-C.

I want to produce a general purpose algorithm for work of varying diameter and varying chord lengths.

Any help from the less mathematically challenged than I would be much appreciated 

[mklotz]

angle ABC = (180 - BAC)/2

angle ADB = 180 - ABC - BAD

Then:

L/sin(ABD) = R/sin(ADB)

so:

L = R*sin(ABD)/sin(ADB)

A remark on notation:  Labeling angles as you have done can lead easily to confusion.  Mathematicians generally label angles with a single (often Greek) letter to simplify identification.

[awemawson]

Thanks Marv. It's a quick autocad sketch and I couldn't find alpha beta gamma etc

I actually need L as a function of angles BAC and BAD. BAD will be the control variable, ie the rotation of the main spindle as the cut progresses, BAC being part of the set up defining the part.

[mklotz]

BAC should have appeared in the first equation.  Damn notation caught me.  I've fixed it in my post.

[awemawson]

Thanks Marv, but I'm obviously missing something here Your equation is in terms of angles ABD and ADB, neither of which are known values

My known values are:

R = radius of circle
Angle BAC = angle subtended by the chord
Angle BAD = angle of line L intersecting the chord whose length I want.

(I suspect that as angle BAD sweeps from zero to ultimately being equal to angle BAC, the point of intersection of the line L and the chord describes a segment of a circle but I've not been able to prove it. It is this point of intersection that is the tangential point of cutting of the milling cutter.)

[mklotz]

I've given you an equation to solve for ADB and ABD = ABC.

We need both those angles in order to use the law of sines to solve for L.

[philf]

Andrew,

Maths was always my favourite subject and it's nice for someone to set such a brainteaser.

Try:

L = (R x sin(90-BAC/2))/(sin(90+BAC/2-BAD))

My first effort had one too many "/2" in. I have now checked it with a spreadsheet and an Autocad drawing.

Hope that's what you needed.

Phil.

[Lew_Merrick_PE]

Andrew -- Is there a reason yoiu cannot make the line "AD" perpendicular to the chord "BC"?  If you are coming in with an endmill to mill that face, would it not need to be perpendicular to cut the chord line smoothly?

If that is the case, then angle "BAD" = angle "CAD" and the only "issue" is deciding the subtended angle "BAC."

[awemawson]

Marv, I'll try and do another drawing later that shows what is actually happening, but in words:

The X axis tool slide holds a tool turret with driven tooling. The selected tool will be a spiral end mill cutting on it's flank, not its end, and it's axis of rotation will be parallel to the main lathe spindle.

As this tool can only move in towards the lathe axis, or away from it (ie not up and down) it cannot cut a flat, unless the main spindle rotates slowly as the tool slide slides in and out. It is the mathematical relationship between the work rotation (represented by the angle of the line L) and the movement of the tool slide (represented by the length of line L) that I am trying to build into my algorithm.

When there are less grandchildren crawling over me I'll try and get a sketch drawn up

[Manxmodder]

Andrew,I don't know how how helpful this is but by coincidence I was sent an email offer from a metrology seller yesterday. They are offering for sale a specialised hand held machinists calculator which does trig,sin,tan,arctan and a great deal of other math specifically associated with shop calculations.

 I am going to ask other posters opinions/experiences of this calculator in another thread.

Link to site advertising: http://www.machine-dro.co.uk/engineers-calculator-machining-reference-tool.html?___SID=U   

.....OZ

[mklotz]

 It is the mathematical relationship between the work rotation (represented by the angle of the line L) and the movement of the tool slide (represented by the length of line L) that I am trying to build into my algorithm.

When there are less grandchildren crawling over me I'll try and get a sketch drawn up

Rather than do any more drawings I think you should spend some time studying what I wrote.  I've given you a solution for L in terms of BAD and BAC which is what you asked for.  If you can't understand it  I'll be glad to answer specific questions you might have.

[awemawson]

Andrew,I don't know how how helpful this is but by coincidence I was sent an email offer from a metrology seller yesterday. They are offering for sale a specialised hand held machinists calculator which does trig,sin,tan,arctan and a great deal of other math specifically associated with shop calculations.

 I am going to ask other posters opinions/experiences of this calculator in another thread.

Link to site advertising: http://www.machine-dro.co.uk/engineers-calculator-machining-reference-tool.html?___SID=U   

.....OZ

Thanks Oz, but the intention is to write a canned cycle that I can call up using the inbuilt trig functions of the controller. It is to be a generalised sub routine to mill a flat on round work - be it one or many facets.

[awemawson]

Andrew,

Maths was always my favourite subject and it's nice for someone to set such a brainteaser.

Try:

L = (R x sin(90-BAC/2))/(sin(90+BAC/2-BAD))

My first effort had one too many "/2" in. I have now checked it with a spreadsheet and an Autocad drawing.

Hope that's what you needed.

Phil.

Thanks Phil,

You are working on the lines I need - no pun intended 

[philf]

Thanks Phil,

You are working on the lines I need - no pun intended 

Andrew - I thought a diagram might help. Phil

[philf]

Andrew,I don't know how how helpful this is but by coincidence I was sent an email offer from a metrology seller yesterday. They are offering for sale a specialised hand held machinists calculator which does trig,sin,tan,arctan and a great deal of other math specifically associated with shop calculations.

 I am going to ask other posters opinions/experiences of this calculator in another thread.

Link to site advertising: http://www.machine-dro.co.uk/engineers-calculator-machining-reference-tool.html?___SID=U   

.....OZ

Hi Oz,

I've seen the calculator before. It may help you working out some trig but I doubt it would give you a formula to build into a program or spreadsheet which is what I believe Andrew needed.

The speeds/feeds feature would possibly be the feature of most use to me.

A couple of days ago I found a free online speeds/feeds calculator called FSWizard: http://zero-divide.net/index.php?page=fswizard . From the same author there is a paid-for program called HSMAdvisor which has geometry, threads etc as well as speeds and feeds.

I tried the online calculator yesterday working out feeds and speeds for a CNC milled loco component and it worked fine.

There are free Android/iOS versions with very few materials in the database but I will be buying an Android version which has many materials. It's only $29.99 (£20) which is about the cost of a carbide cutter so, if I don't break any cutters, will very soon pay for itself.

Phil.

[awemawson]

Phil you've come up trumps again, many thanks 

Phil sent me his spread sheet, and sure enough when plotted, running the rotation angle against motion ie length 'L', it plots out as the segment of a circle as I expected.

Many thanks for all the support that you all have given me