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}}

Math Monkey

φI study mechanical engineering, as anyone who knows me know. I’m pretty bad at it. Hence this website….One thing I’ve noticed, and discussed with others is the entire lack of…speciality in subjects, or specifically. explanations of things. Any Google-fu’er can tell you, the more indepth you get into a subject the harder it can be to find real answers. A Recent subject for me, while looking into/researching vibration engineering for a 5 axis laser cutter I’ve been working on was “reactionary forces“. Often described by Newtons impulsesatz as m\: \ddot{x} = \displaystyle\sum_{i=1}^{n}\,F_i + Z_i Where Z_i is the reactionary force. I’m the last person in the world…or well maybe second last who should be asked what its all about. But basically to conserve energy in a system, you need it…but it doesn’t really exist. ‘Cause it’s virtual or some bullshit.

Anyways, it’s something that I’ve only ever had to calculate once, for a single assignment, and never had to do again. So I figured I’d write a little about it. Mostly because it was frustrating, but interesting at the same time. This is math heavy post btw. Continue reading “Math Monkey”