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Math Help - Rearrange equation to find crank angle theta?

  1. #1
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    Rearrange equation to find crank angle theta?

    The following equation gives an offset piston's displacement (Y) for a known crankshaft angle (theta).

    Given a known piston displacement (Y) and that r,b & d are also known, I would like to be able to calculate the crankshaft angle - to rearrange equation to solve for (theta)?

    Any help greatly appreciated.

    Thanks,

    Russ

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    Re: Rearrange equation to find crank angle theta?

    I caught an error in this post. I shall try again.
    Last edited by JeffM; July 14th 2014 at 09:49 AM.
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  3. #3
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    Re: Rearrange equation to find crank angle theta?

    $y = rcos( \theta ) + \sqrt{b^2 - (rsin( \theta ) - d)^2} \implies r\sqrt{1 - sin^2( \theta )} = y - \sqrt{b^2 - (rsin( \theta ) - d)^2}.$

    $u = sin( \theta ) \implies r\sqrt{1 - u^2} = y - \sqrt{b^2 - (ru - d)^2} \implies r^2 - r^2u^2 = y^2 - 2y\sqrt{b^2 - (ru - d)^2} + b^2 - (ru - d)^2 \implies$

    $r^2 - r^2u^2 = y^2 - 2y\sqrt{b^2 - (ru - d)^2} + b^2 - r^2u^2 + 2dru - d^2 \implies 2y\sqrt{b^2 - (ru - d)^2} =y^2 + b^2 + 2dru - d^2 - r^2.$

    $v = y^2 + b^2 - d^2 - r^2 \implies 2y\sqrt{b^2 - (ru - d)^2} = v + 2dru \implies$

    $4y^2 \{b^2 - (r^2u^2 - 2dru + d^2)\} = v^2 + 4druv + 4d^2r^2u^2 \implies$

    $4b^2y^2 - 4r^2u^2y^2 + 8druy^2 - 4d^2y^2 = v^2 + 4druv + 4d^2r^2u^2\implies$

    $u^2\{4r^2(d^2 + y^2)\} + u\{4dr(v -2y^2) + v^2 - 4y^2b^2 + 4d^2y^2 = 0.$

    Assuming I have not screwed up again

    $a = 4r^2(d^2 + y^2),\ b = 4dr(v -2y^2),\ and\ c = v^2 - 4y^2b^2 + 4d^2y^2 \implies u = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \implies \theta = arcsin(u).$
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    Re: Rearrange equation to find crank angle theta?

    Jeff,

    Many thanks for working this through. A lot more involved than I thought, but I can follow your steps above. Time to put it to the test with some numbers.
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    Re: Rearrange equation to find crank angle theta?

    Quote Originally Posted by russiver View Post
    Jeff,

    Many thanks for working this through. A lot more involved than I thought, but I can follow your steps above. Time to put it to the test with some numbers.
    If it doesn't work with numbers, then it is because I blew the algebra. The fundamental idea is right: convert the cos to sin, substitute u for sin, solve for u, and then take the arcsin.
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