What are these equations representing?

[optics.tech]

Does anyone know what equations are these (please see attached image)?

Also, does anyone know these complete equations?

Thank you

 

[Mech_Engineer]

They look like they could be related to rotational dynamics, as terms for angular speed ( $$\omega$$ ) radius ( $$r$$ ) and time ( $$t$$ ) are in them. It's also possible they are related to some form of harmonic oscillation, based on the $$sin(\omega t)$$ terms I see in there.

However they look like they are probably based a specific geometry, as they don't look generalized to me. Where did you see these equations, and what was the context of their presentation?

 

[minger]

The lambda $$\Lambda$$ and what Mech_ said makes me think they rotordynamic oriented as well.

 

[FredGarvin]

The r/L term is usually a slenderness ratio in shaft dynamics. I'll have to look through my rotor dynamics handbook. I agree with Mech in that I bet this is a derivation for a specific condition/geometry. It definitely is taking me back to the days of harmonic functions and Fourier transforms...

 

[Mapes]

The r/L term is usually a slenderness ratio in shaft dynamics.

I don't disagree with this, but on the far right of the photo it looks like L is the length of a bar connected to a rotating radial link with length r. That would make

$$\sqrt{L^2-r^2\sin^2\omega t}=L\sqrt{1-\left(\frac{r}{L}\sin\omega t\right)^2}$$,

which appears in one of the equations, the y coordinate of the end of the bar.

 

[optics.tech]

However they look like they are probably based a specific geometry, as they don't look generalized to me. Where did you see these equations, and what was the context of their presentation?

How do you know that the equation is dedicated to specific geometry?

If so, what kind of geometry is it? 2 or 3 dimensional?

 

[FredGarvin]

I don't disagree with this, but on the far right of the photo it looks like L is the length of a bar connected to a rotating radial link with length r. That would make

$$\sqrt{L^2-r^2\sin^2\omega t}=L\sqrt{1-\left(\frac{r}{L}\sin\omega t\right)^2}$$,

which appears in one of the equations, the y coordinate of the end of the bar.

I think you're right on that. I saw the "a" on the circumference of the circle and thought that r may be the radius of the bar with length L. Your slant is more probable. I wonder if it's just a kinematics equation for a linkage, like you mentioned...