The following content has been provided by the University of Erlang and Nuremberg.

Okay, so good morning everyone. So let's start with where we left last time.

So we were discussing configuration of a rod which is uniformly strained.

Right?

By VK. Okay? So it takes some shape and what I mean is that it has same VK everywhere.

That's why I say it is uniform, uniformly along the arc length only.

Okay? And we found out the analytical formula for the center line which was

S equals magnitude of V times cosine of phi times S times K hat plus sine of phi

into 1 over magnitude of K into identity minus e to the power SK K hat cross K hat perp.

Okay? And what we saw was that this term generates a straight line whereas this term generates a circle.

And the combination of these two then leads to a helix. Right? So this generates a helix.

And similarly the rotation big R of S we had found it to be e to the power SK.

So these two are such that at S equal to 0 the cross section at S equal to 0 has its center at the origin

and the orientation of the cross section at S equal to 0 is along the axis of the coordinate system. Right?

Big R of S at S equal to 0 is identity. Alright? So let's look at some particular cases.

For example, if V is parallel to K then what happens? So if V is parallel to K, sorry, parallel to K,

remember that phi was the angle between V and K. So if V is parallel to K, phi is 0, sine phi becomes 0.

Right? So only this term survives. So this center line then degenerates into a straight line.

And can you think of some of the simplest situations where V and K are parallel?

So if V is equal to 0, 0, V3 and K is equal to 0, 0, K3. So you can see this is a situation where V and K are parallel. Right?

And what does this correspond to? So it has got only stretching, no shear and here you got only twist.

So this corresponds to combined extension twist situation. So this is, so one example is combined extension torsion.

Isn't it? So you have got a beam, you stretch it and twist it, the center line remains straight.

And that's why that formula also tells you that center line will remain straight. Right?

So that was one particular situation. You think of another situation where V is perpendicular to K.

So what's going to happen in this case? This term drops out, but only this survives.

So the helix degenerates into a circle.

And can you think of a deformation where this can happen? When are V and K perpendicular?

What about pure bending? Right? In case of pure bending, the center line is a circle. Right? It's an arc of a circle.

So V, 0, 0, 1, pure bending, no shear, no stretch. Whereas K could be K, 1, 0, 0. Right?

Or you could also have K1, K2 together. Still V and K are perpendicular to each other.

So that's the case of pure bending. So one example here is your pure bending.

And of course, if these two situations are not there, then you have the full helix. Right?

And last time we also wrote down this formula slightly differently, which was,

we wrote it as little r of s equal to s times tau K hat plus identity minus e to the power sk into little xf.

And then we had defined tau as V times cos phi. Right? And this little x of f,

so we had defined our tau as magnitude of V times cos phi.

Whereas this little x of f, you could see that it's same as V cross K divided by magnitude of K squared.

And this is the formula for a helix in a standard form.

This is called fixed point of the helix. Fixed point.

And this tau is related to pitch of the helix. You know, pitch meaning,

how much does the helix move along the axis when you make a full circle?

So, this tau is related to that. Can you, for example, find out what is the pitch of the helix in terms of tau?

So, maybe I should go to the next board.

What's the pitch of the helix? So, how much s you have to move so that the helix takes a full turn? 2 pi.

See, the rotation is given by this formula. So, how much should s be so that the angle here becomes 2 pi?

For a given s, how much is the angle here? So, this is a rotation about K axis.

And how much is the angle of rotation for a given s?

So, you have to say s times magnitude of K. That's the angle of rotation of this rotation matrix.

And this has to be equated to 2 pi. So, that means s is equal to 2 pi over magnitude of K.