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Who would have thought: algebra is useful (on topic)

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  • Who would have thought: algebra is useful (on topic)

    I needed a cam-lever like this:

    Click image for larger version

Name:	cam.png
Views:	249
Size:	30.8 KB
ID:	1849651
    But I couldn't find one with enough "throw" - what McM-C calls clamping distance.

    Taking the opportunity to have some fun, I decided to make it myself. The cam would be just be an eccentric piece of round - i.e., the center of rotation offset from the stock center:
    Click image for larger version

Name:	cam2.jpg
Views:	241
Size:	28.8 KB
ID:	1849652

    The clamping distance is the difference between d1 (unclamped) and d2 (clamped) when rotated CCW 90°. The tricky part is determining the offset (o) required. Which is where the algebra comes in. We know:
    o = r (radius) - d1
    d1 = d2 - clamping distance,
    d2 = SQRT(r^2 - o^2) (thanks, Pythagoras)
    The radius and clamping distance are given, leaving 3 unknowns and 3 equations. Documenting the algebra involves a lot of squares & square roots, which is ugly in text-only, so I'll skip that & get to it:

    o = r - d1
    d1 = (-b + SQRT(b^2 - 4*a*c) ) / 2 [quadratic formula] where a = 1, b = r - clamping distance, c = 0.5*(clamping distance^2)

    for 1.25 round & a clamping distance of 0.125, the required offset is 0.141.

    Note - this is for 90° rotation, which gives the right-triangle & its Pythagorean relationship. The offset can be figured for other than 90°, but it would involve the law of cosines & much messier algebra.





  • #2
    Originally posted by Bob Engelhardt View Post
    I needed a cam-lever like this:

    Click image for larger version

Name:	cam.png
Views:	249
Size:	30.8 KB
ID:	1849651
    But I couldn't find one with enough "throw" - what McM-C calls clamping distance.

    Taking the opportunity to have some fun, I decided to make it myself. The cam would be just be an eccentric piece of round - i.e., the center of rotation offset from the stock center:
    Click image for larger version

Name:	cam2.jpg
Views:	241
Size:	28.8 KB
ID:	1849652

    The clamping distance is the difference between d1 (unclamped) and d2 (clamped) when rotated CCW 90°. The tricky part is determining the offset (o) required. Which is where the algebra comes in. We know:
    o = r (radius) - d1
    d1 = d2 - clamping distance,
    d2 = SQRT(r^2 - o^2) (thanks, Pythagoras)
    The radius and clamping distance are given, leaving 3 unknowns and 3 equations. Documenting the algebra involves a lot of squares & square roots, which is ugly in text-only, so I'll skip that & get to it:

    o = r - d1
    d1 = (-b + SQRT(b^2 - 4*a*c) ) / 2 [quadratic formula] where a = 1, b = r - clamping distance, c = 0.5*(clamping distance^2)

    for 1.25 round & a clamping distance of 0.125, the required offset is 0.141.

    Note - this is for 90° rotation, which gives the right-triangle & its Pythagorean relationship. The offset can be figured for other than 90°, but it would involve the law of cosines & much messier algebra.



    Whatda you mean - "we know" That's the first time I have seen algebra used since high school, not that I was there.....

    Comment


    • #3
      That looks like a simple trig problem to me. One triangle.

      Comment


      • #4
        I have no use for algebra.

        -D
        DZER

        Comment


        • #5
          Originally posted by Doozer View Post
          I have no use for algebra.

          -D
          Unless...
          Attached Files

          Comment


          • #6
            The "throw" is twice the pivot offset at 180 degrees of rotation, neither algebra nor trig is required for a simple round cam.
            If a cam is needed that has a uniform lift per degree of rotation more complex calculations are used.

            I turn several of these parts per year, 5 1/2" Dia. round cams, the center is offset by 3/4" giving a lift of 1 1/2".

            Comment


            • #7
              I once tugged on Algesbra, but she turned around and slapped me in the face.

              Comment


              • #8
                As I recall from all those years back, Jane, fresh from Teacher college had a very nice azz, wore black skirts and had a habit of chalklining her azz. Jane couldn't teach a dam thing. Her students who passed had older siblings who taught them. 82% of Jane's class failed the final. I went to Summer School and got a B from a man who could and did teach.
                The School said Jane was licensed and equal to any other Algebra teacher.
                Jane was really there to find a husband.

                Jane had ideas about Salvitore the social studies teacher and they had a couch in the unused janitor closet near their respective rooms. Someone unknown slipped a speaker into that closet and connected it to the Strombeg Carlson school PA system. The PA had a listen feature so school admins could monitor and evaluate individual classrooms. Sal and Jane were very entertaining when some dastardly doer broadcast their activities schoolwide + outside speakers. Many students learned new material, girls in Speedwriting class made notes. The sane employee, school secretary hit the Drill switch on the Fire Alarm not knowing the Sal & Jane Show was on outside the building. It took a while for the PA to get shut down.

                Salvitore got served with Divorce papers in 3rd period a few weeks later. Jane went on leave, and the couch was hauled out of the unofficial lounge. The muscular male teacher who learned school solid core doors don't shoulder open wore a sling for a couple months. Nobody was ever caught or convicted. The Mimeograph copied of the Sal & Jane show were on sale on the busses for a while.

                Comment


                • #9
                  Originally posted by Toolguy View Post
                  That looks like a simple trig problem to me. One triangle.
                  It took me nearly sixty years to discover that trigon is just another name for a triangle, so trigon-ometry just means measuring triangles.

                  George

                  Comment


                  • #10
                    Yes, I did the same thing a few weeks ago when I was 3d printing glass scale mounts for my CNC mill for mapping the ballscrew. Didn't feel like placing the mill on a surface plate and using a height gauge to find the actual reference heights, so I printed my mounts with a rough ballpark figure. Of course the glass scale wasn't sitting level, so I used trig(let the CAD program do the numbers, sketched out a triangle and found the height I needed to add to one of the mounts to get the scale to sit level.

                    Comment


                    • #11
                      Trigon is the abbreviation of triangle and polygon. A trigon is sort of a 6 sided triangle. At least, that's what the trigon carbide inserts are.
                      Last edited by Toolguy; 01-18-2020, 01:43 PM.

                      Comment


                      • #12
                        Originally posted by Bented View Post
                        The "throw" is twice the pivot offset at 180 degrees of rotation, neither algebra nor trig is required for a simple round cam.
                        ...
                        Well, 180° is the "gimme" case. 90° is much harder, even with a simple round cam, and if neither 90° nor 180°, it is very much harder.

                        Comment


                        • #13
                          I can't say I've used a lot of algebra in my shop. But I've used a LOT of trigonometry over the years.

                          Joe Pie on his YT channel has some videos where there's no machining done. Just a really nasty measurement issue which he breaks down into a series of right triangles and comes up with clever yet in the end simple ways of solving. Some great use of trig to arrive at a solution to measure some feature we machine.
                          Chilliwack BC, Canada

                          Comment


                          • #14
                            Originally posted by Toolguy View Post
                            That looks like a simple trig problem to me. One triangle.
                            Not quite: the base of that triangle (d2) depends upon d1, which depends upon o, which depends upon d2, ....

                            Comment


                            • #15
                              Originally posted by Georgineer View Post

                              It took me nearly sixty years to discover that trigon is just another name for a triangle, so trigon-ometry just means measuring triangles.

                              George
                              The 'gon' in trigonometry comes from the Greek root 'gonia' meaning angle. So trigonometry is the study of things with three (tri-) angles, i.e. triangles.

                              The 'gonia' root is also evident in "goniometer", a device for measuring angles and in the words describing the regular polygons - pentagon, hexagon, heptagon, etc.

                              Regards, Marv

                              Home Shop Freeware - Tools for People Who Build Things
                              http://www.myvirtualnetwork.com/mklotz

                              Location: LA, CA, USA

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