Thread: Rearrange equation to find crank angle theta?

1. 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

2. Re: Rearrange equation to find crank angle theta?

I caught an error in this post. I shall try again.

3. 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

$a = 4r^2(d^2 + y^2),\ b = 4dr(v -2y^2),\ and\ c = v^2 - 4y^2b^2 + 4d^2y^2 \implies u = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \implies \theta = arcsin(u).$

4. 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.

5. Re: Rearrange equation to find crank angle theta?

Originally Posted by russiver
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.