The Blerg

Virtual Work

So We’ve talked about mdc equations before in math monkey. They are usually in the form  m\: \ddot{x} + d\: \dot{x} + cx\: = 0\:

m Being the Mass, d the dissapation (dampening) and c the spring constants.

I won’t go into how one builds this equation up, there are many and I covered in math monkey pretty in depth on how to do it with the basic math pendel. One of these days I may get into a more indepth analysis of maybe a car or a real double pendle. Anyways, a part that wasn’t explained at all was the “Virtual Work” of the system, or namely the Q\: part of the Lagrange equation

\mathrm{Q}=\frac{d}{dt}\frac{\delta\mathrm{L}}{\delta\mathrm{\dot{\phi}}}-\frac{\delta\mathrm{L}}{\delta\mathrm{\phi}}

So whats Q? Well according to wiki:

Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements.

Or whatever that means. In mathemagics, it means this:

Q\:= \displaystyle\sum_{i=1}^{n}\mathrm{M_i(t)}\frac{\delta\phi}{\delta\mathrm{x}}+\mathrm{F_i(t)}\frac{\delta\mathrm{r}}{\delta\mathrm{x}}+\mathrm{F_{di}(t)}\frac{\delta\mathrm{r_d}}{\delta\mathrm{x}}+\mathrm{R_i(t)}\frac{\delta\mathrm{r}}{\delta\mathrm{x}}

Namely the Sum of the Moment, external force, friction and dampening times their directional vector components first derivative. Or Well thats what it looks like when I read it anyways. I still haven’t quite got the hang/understanding of it. But I’ve never claimed to know what I’m doing. Just that I look like I know what I’m doing 😉

Well to calculate all this out theres quite a bit of work to do. I recently had to do such a thing in a system I had to describe and while digging through my notes I happened to fall on a trick I seemed to write to myself in a exam cheatsheet. Normally, you would calculate the vectoral force (for dampening in this example) and multiple it with its directional derivative like so:

\mathrm{F_{di}(t)}\frac{\delta\mathrm{r_d}}{\delta\mathrm{x}}

F could look like this (or rather, does):

\mathrm{\underline{F}_{d}(t)} = -d \dot{x} \vec{e_x}

and r would look like maybe this:

\delta\mathrm{r} = x\vec{e_x} +y \vec{e_y}

So you can imagine there are a bunch of steps to get this done, each partial div. blah blah multiplying, blah blah.

Well good news everyone!

I guess it’s not a surprise cause I already said I found a cheat. Well here it is:

\mathrm{\underline{F}_{d}(t)} = -d \Delta\Delta\dot{x} = -d \Delta\dot{x}^2

So if you’ve already kinematically described your system, You can take the derivatives you already know, square them, add them together and multiply by -d and you have your answer immediately.

Yay! I’ve tested this conjecture exactly once with 100% success rate. So that’s good math in my books. I wish I could credit where I found this from…but since it’s on my cheat sheet. I guess I’ll have to claim it as my own.

\mathrm{e}^{\sqrt{2}}

Dokumenta14 Updates #2

It’s the third week since the beginning of the exhibit I’ve been sharing with Valeria and some things have changed, been updated and the like.

First of the likes is the promised dipped skull and it’s development.

that was after 12 dips. I expected a particular kind of development, mostly that after enough layers some kind of face would be developed. Even surprisingly a nose! Though some more needs to be done…the mouth and the horn shouldn’t really be there at all.

Second of likes:

Maybe it’s part of the first likes. Either way it’s the parts that were dipped last week.

This is after the first cycle. They don’t really look spectacular in the first dippings, although you can see the skull then compared to the skull now.

Tomorrow, I will make a mother post about the upgrade to the animatronics.

 

e^{\sqrt{2}}

Documenta14 Updates

We had our official exhibition this past friday. Everything went pretty well, my robots didn’t explode, or burn anything down…Not that that should happen…but well, old machines can just…give up sometimes. I updated the dokumenta14 page with some photos…but here I’d like to post a video and a photo of what my robot actually does. There is also a little preview of whats to come in the following weeks.

 

The videos came from facebook, and Jan Hendrik Neumann, But they’re some nice videos. I may replace them later.

 

And for the small preview, for next week.

Cheap or cleaver?

Clever.

There is something called the “Dokumente” in the city where I live…it’s some giant art festival that is surprisingly world wide. Once every 5 years it shows up and my tiny city of 250k turns into a million person city for 3 months during the summer. Interestingly I was approached by a new friend of mine Valeria if I was interested in doing an exhibit with her for the duration of the Dokumente. I gladly said yes of course. I would never turn down the chance to do something arty, and be able to use the chance to make something.

Continue reading “Cheap or cleaver?”

Who Framed Roger Rabbit

One of my favourite movies of all time. Who Framed Roger Rabbit it’s a film that was absolute truth and evidence that cartoons were real live, living things.

Welllll not quite. But it sure as hell was a film that tests the sands of time and definitely holds up. If you get the chance check it out. But for those who already have seen or atleast know the film, I found an interesting tidbit about some of it’s animation tricks and general laws of the films universe.

Continue reading “Who Framed Roger Rabbit”