# Cantilever bending

Is a well studied problem. I am a textiles engineer and am doing some work on modelling the interaction of the yarn with a brush ring (circular brush with hairs slowing the yarn down). You can easily calculate the deformation of a hair caused by a force perpendicular to the curve of the hair in 2D by dividing the hair in finite elements or finite volumes - as the deformation is rather big - and apply the classical formulas for a beam in bending to the elements. Piece of cake.

A bit more difficult: the yarn moving through the brush is a dynamic phenomenon, I also want to model inertia. When you move a rigid object with mass m an acceleration a in a certain direction, you know the force F = ma. However, when applying this on a hair in bending, the base of the hair is fixed and the yarn bends in the process, which corresponds to a translation and rotation of the different elements. Anyone an idea of how to calculate the inertia force caused by an acceleration of a point on the hair perpendicular to the curve of the hair in 2D, or of the inertia force of a beam in bending or something? An idea how and where to start?

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It's been a long time since physics class, I've been wondering something.

Say you have a rod with length L and mass M fixed on one end, divided in n elements (*) with mass m = M/L and accelerate the free end downwards with an acceleration a, than I would like to know the inertia force F. I tend to think that the acceleration of a point at a distance r from the fixed end will experience an acceleration a*r/L, so I would have to make the sum of m*r/L for all elements obtaining M/2.

So for the inertia force F when experiencing an acceleration a at its free end of the rod I would obtain M*a/2. Is this correct, as the inertia moment for a rod at an axis through one end is M*L²/3, or am I confusing two different things here?

Code:
                          F   ^
|
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-

Fixed end  <-> free end

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arildno
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Okay, am I to understand this as a segmented non-rigid rod composed of rigid elements?

arildno said:
Okay, am I to understand this as a segmented non-rigid rod composed of rigid elements?
Basically, it is a model for a hair. Simply put, the question is if the tip (or another point) is accelerated by an object, what is the inertia force the object faces?

I modelled the hair as a number of small elements as the theory of bending would otherwise no longer be valid as the deformation can be pretty big. Dividing in small elements allows to apply the known bending formulas validly on the small elements.

arildno
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All right, so I got that right (i.e. the hair is allowed to deform during the process, whereas each element remains a rigid rod).

First of all:
It is incorrect to use the moment of inertia formula for a straight rod for the hair, since the hair is non-rigid (basically, that means it's moment of inertia about the fixed end will be a function of time, and not a constant)

Here's what I suggest you must do:
This analysis does not explicitly include the specification of the force F or the acceleration, that specification will be an additional equation to those I derive.

Let's therefore, for the moment disregard it, and look at the general picture:

1) Counting unknowns:
Elementwise, we must have 2n scalars specifying the centres of mass for each element
To each rigid element, there is associated an unknown angular velocity, so we have n of these also.

Using Newton's 3.law, we essentially have 2n distinct force components
(these act at the joinings between elements)
(Clearly, for the hair system, these are "internal forces", but they are "external" to the element we consider)

So, we have a total of 5n unknowns.
(In addition, we might have your force F, in which case you'll need the accceleration specification to close your system)

2) Equations:
F=ma applied to each center of mass yields 2n equations

In addition, you have n independent angular momentum equations.

In order to close your system, you must require that the velocities at the joinings are equal for the 2 elements combined (or at the wall, that the velocity must be zero for the contact point of the first element)

These relations give you the last 2n equations.

3) Conclusion:
At the face of it,solving a 5n (+1) equation system seems rather formidable,
but possibly, there is a distinct pattern to the solutions to be found here.
I suggest that you try out the 2-element case first to get a "feel" of the problem

I have a finite volume scheme to calculate the movement of a yarn. This involves dividing the yarn into a number of finite volumes, the center of which is called a node. My integration scheme involves the following steps:

1. Calculate contact : project the position p normally on the surface, p'. Take p = p'
2. Calculate the sum of tension force, air friction force, bending force in the yarn, bending force exerted by the hairs of the brush ring F acting on a node
3. Calculate the surface reaction. Adjust speed v(old) and force F as below, with (n) the normal and (t) the transversal component:

v(old) = v(old)(t) - Beta*v(old)(n),
F = F(t) + F(n) + FrictionForce with F(n) = 0 if F(n) is directed into the surface
4. Calculate the new speed of a node : v(new) = v(old) + F*dt/m;
5. Calculate the new position of a node : p(new) = p(old) + v(new)*dt

All good and well, except that when a node of my yarn is moving through the brush and deforming the hairs, the acceleration of that node will not be F*dt/m but will be smaller due to the fact that the hairs in the brush ring will also have to be accelerated. The easiest would be just to suppose the acceleration would be F*dt/(m+M) with M the sum of all of the weights of all the hairs making contact, but this is not exactly pretty accurate.

arildno
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Your scheme is decidedly more complex (and realistic!) than what I had in mind!

I will study it in detail, and hopefully, be able to come up with some constructive suggestions, but your scheme deserves more than a casual glance from my part.

I would just add to what I wrote earlier, and that I believe, has some bearing on the problem:

Clearly, if you are to model the yarn with the use of the tensile force, this is clearly related to the material's elastic properties.
(My model is unrealistic in this respect, by considering each element as completely rigid; each element should at least be able to expand/contract in the length direction, according to the material's elastic properties)

At the moment, I believe that simplifying measures are necessary to find a practical solution, possibly along the lines you suggest.
I'll have to think about it..

I don't need any improvement to the scheme (except perhaps for modelling collision, but that's another question I intend to post here) but it is useful in thinking how you would fit in the inertia force caused by moving through the brush. I'm writing a paper but haven't quite figured out whether I should make a simplification to correct F*dt/m to F*dt/(m+M) with M as said earlier or something more advanced. It would be nice to have something more advanced in order to publish it.

Okay, so what if I choose to displace the end point with a certain distance a*dt*dt/2 with dt and a chosen arbitrarily, and again displace it with a*(2*dt)*(2*dt)/2.

If I calculate which form of the bent hair is necessary to displace the end point like that, I could calculate the acceleration of all the other point by using:

a = ( p(t) - 2*p(t+dt) + p(t+2*dt) ) / ( dt * dt )

And hence calculate the inertia force of not only the end point, but also each other point, and by totalling the inertia of the entire hair. Any faulty assumption?

arildno