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Best answer by mnori

Analytical.png

5 replies

L
LucMeekes23-Emerald IV
23-Emerald IV
August 6, 2018

Valery,

Try this.

LM_20180806_SymbolicSolution.png

Success!
Luc

    W
    Werner_E25-Diamond I
    25-Diamond I
    August 6, 2018

    You can easily calculate the integral by hand if you remember that 1+sinh^2(a) = cosh^2 (a). Then, using real arguments, the root "cancels" with the square and as X and arsinh(tan(alpha)) are constants, simple linear substiution gives you the integral. Then you can quickly solve for X and if yo do it manually you will arrive at the very same solution which Luc had provided.

     

    But you can also let Mathcad's symbolics di the work. As usual when it gets a little more complicated Mathcad/Mupad needs a lot of help with substitution and the placement of the right commands in the correct order.

    muPad seems to prefer ln instead of arsinh and asinh is not a valid argument for "rewrite", so it looks that there is no way to convince muPad to give the solution using arsinh instead of ln.

    B.png

      V
      24-Ruby IV
      ValeryOchkov24-Ruby IVAuthor
      24-Ruby IV
      August 6, 2018

      Thanks, Luc and Werner!

      But If you are so smart, solve please this problem symbolically until the end!

      Closed-Chain-Alpha-PE-symbolic.png

      W
      Werner_E25-Diamond I
      25-Diamond I
      August 6, 2018
      @ValeryOchkov wrote:

      Thanks, Luc and Werner!

      But If you are so smart, solve please this problem symbolically until the end!

      But if we do, there is no fun left for you ! 😉

       

      Here's my attempt.

      As you can see that in the last phase muPad switches to numeric mode. You may try your luck with something more capable like Maple or Mathematica, but chances are that there is not closed symbolic solution for that equation.

      At least we see that the angle is constant and independent of the parameter L and the symbolics arrives at the same result as the numeric "root" function. At least sometimes. Sometimes, as I show, muPad would return results which are simply wrong! Sure a no-go - this should never happen!.

      B.png

      Sheet in MC11 format attached - maybe Luc is about to give it a try with Maple.

        M
        mnori1-VisitorAnswer
        1-Visitor
        August 7, 2018

        Analytical.png

          V
          24-Ruby IV
          ValeryOchkov24-Ruby IVAuthor
          24-Ruby IV
          August 7, 2018

          I think it is one new Physical and Math constant!

          Steven Finch! Where are you?

          Closed-Chain-Alpha-close-formula.png

          V
          24-Ruby IV
          ValeryOchkov24-Ruby IVAuthor
          24-Ruby IV
          August 7, 2018

          More Physic less Math - a numerical solution of the system of 3 physical equations!

          Fig-16.png

          A
          AlanStevens19-Tanzanite
          19-Tanzanite
          August 8, 2018

          Just for fun I thought I'd see how far I could get on Valery's problem with a combination of Mathcad symbolics plus "Hand" symbolics.  Like others, I found it's easy (if tedious) until trying to find where PE = -L^2/4.  At this point one needs to resort to numerics.  Here's my approach (in MC15):

           

          Valery1.PNG

          Valery2.PNG

          Valery3.PNG

          Valery4.PNG

          Alan

            V
            24-Ruby IV
            ValeryOchkov24-Ruby IVAuthor
            24-Ruby IV
            August 8, 2018

            Thanks, Alan!

            But solution without animation is not solution

            Closed-Chain-Alpha-close-formula.gif

              F
              Fred_Kohlhepp23-Emerald I
              23-Emerald I
              August 8, 2018
              @ValeryOchkov wrote:

              Thanks, Alan!

              But solution without animation is not solution

              Closed-Chain-Alpha-close-formula.gif

              Disagree!  Alan did the hard work, the animation is window dressing.

                V
                24-Ruby IV
                ValeryOchkov24-Ruby IVAuthor
                24-Ruby IV
                August 9, 2018

                Ease to remember this constant: 50+20+20=90

                Fig-16-90.png

                 

                M
                mnori1-Visitor
                1-Visitor
                August 10, 2018

                I made some mathematical tourism, I read that this Gudermannian.pngis Gudermannian function, here there is a discussion with several links:

                http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1993229&messageID=6864360

                V
                24-Ruby IV
                ValeryOchkov24-Ruby IVAuthor
                24-Ruby IV
                August 10, 2018

                We have created the MOSNE function

                (Meekes, Ochkov, Stevens, Nori and Exinger)

                MOSNE.png

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