Mechanical-Structural Engineering: Forces/Moments on Complex Beam

[Graham1874]

Homework Statement

To give a bit of context, I am doing my final year university project on micro-mechanical interactions between an AFM probe and a sample surface. I do not have notes for a system this complicated, as we always considered our systems to be rigid bodies. I was always relatively clueless at mechanics so this particular small part of my project is a nightmare for me!

I have labelled the Free Body Diagram attached, but the following information should help to explain further:

Section A has a downwards-vertical displacement being applied, and can be considered a rigid body, fixed in all other DOFs.
A key feature of this system is the bending cantilever beam (Section B as marked in FBD).
It begins with an angle of 13 degrees from the horizontal axis, the inclined beam can be seen on the image FBD.
Section C (probe tip) can also be assumed to be a rigid body.
Section D is the sample surface which can be assumed rigid and fixed in all DOFs.

How do I get to the solution for finding Fz/Fy components and moments on the sample surface from the probe tip?

I have not given values for the system because I want to use the help provided to work through it myself.

Homework Equations

I know there are moments relating to a bending with a stiffness. Not used to a system where a displacement is producing the force vector on the other end of the system.

The Attempt at a Solution

As I say, this is a relative nightmare for me and I don't really know where to start, so all help will be greatly appreciated.

I'm sure I'll need to provide more information to helpers, so these will be answered in EDITS below.

 

[Nidum]

Doesn't seem too difficult .

Just to clarify :

Your system is something like this : Link[/PLAIN]

The components are very small .

You want to estimate the varying load on the probe tip caused by the varying displacement of the tip as it traverses the undulations of the sample .

There is some preload on the tip .

Question : The beam you show is steeply inclined relative to the sample surface . Shouldn't it be more nearly parallel ?

 

[Graham1874]

Doesn't seem too difficult .

Just to clarify :

Your system is something like this : Link[/PLAIN]

The components are very small .

You want to estimate the varying load on the probe tip caused by the varying displacement of the tip as it traverses the undulations of the sample . There is some preload on the tip .

Question : The beam you show is steeply inclined relative to the sample surface . Shouldn't it be more nearly parallel ?

Hi Nidum, thanks for your quick reply.

Yes, the AFM probe is being used in that manner. For this stage of the project, I am wanting to assume initial contact between probe/surface on an assumed-flat surface to see the force components and moments which are acting at the point of contact.

To answer your question simply: the beam is inclined at 13 degrees from the hoizontal for the inspection process.

 

[Graham1874]

I think I may have to consider the support at the probe tip as Pinned, and I'm not sure what, if any, to consider as the support for Section A? It is fixed in all DOFs apart from the z-direction.

Any further help with this? I haven't managed to get very far, but this attached file shows the way I'm trying to go with this. Let me know if I'm identifying this problem wrongly please!

 

[Nidum]

The beam itself can be analysed by standard methods .

Problem is deciding what the conditions of loading and possible fixation at the probe tip are .

There are at least two different basic cases - static and dynamic .

For the static case I think that you could reasonably assume a simple vertical reaction load and no fixation .

To make a more informed decision you need to look at the physics and decide whether the tip digs into the sample surface enough to stop it sliding .

For the dynamic case the situation is more difficult to define but basically comes down to getting answers to questions :

(a) Is there any significant drag force as the tip is traversed across the sample ?
(b) Is the motion stable ?

 

[Graham1874]

Beam is just a cantilever which can be analysed by standard methods .

Problem is deciding what the conditions of loading and possible fixation at the probe tip are .

There are at least two different basic cases - static and dynamic .

For the static case I think that you could reasonably assume a simple vertical load and no fixation .

For the dynamic case the situation is more difficult to define but basically comes down to getting answers to questions :

(a) Is there any significant drag force as the tip is traversed across the sample ?
(b) Is the motion stable ?

For my study, I am primarily looking at the force vectors in a static state - although the full scenario is actually quasi-static I believe, as time does pass but none of the parameters I'm measuring are time-dependent.

With this I am trying to decipher whether the common assumption that the resultant force at the tip is not in fact normal to the surface at the instance of initial contact i.e. even before applying any 'lateral' displacement there is a non-normal force vector applied at the tip from the vertical displacement.

Basically my issue with this calculation comes down to the fact that the beam is inclined and has both rigid and deformable (elastic) elements.

When reading textbooks about static mechanics of beams, I just get the sense that the examples are too simplified to apply to my scenario.

Any further thoughts to help me progress? I really appreciate your help so far.

 

[Nidum]

I'll come back on this subject in a day or two .

 

[Nidum]

Easiest way to tackle this problem is just to derive an expression relating force to displacement at the probe tip . We should be able to do this by standard means .

Before starting though we need to have a clearer idea of what that beam looks like . Is it a plain rectangular strip or something more complicated ?

 

[Graham1874]

Thanks Nidum, yes it's a plain rectangular strip.

 

[Graham1874]

Easiest way to tackle this problem is just to derive an expression relating force to displacement at the probe tip . We should be able to do this by standard means .

Before starting though we need to have a clearer idea of what that beam looks like . Is it a plain rectangular strip or something more complicated ?

Also, I have realized that friction will need to be considered because it will be opposing the horizontal movement of the tip caused by the horizontal component of the force vector at the tip. Hence this will create the static equilibrium condition.

Let me know if you need any further information about the system. I appreciate your time with this

 

[Nidum]

 

[Nidum]

Deleted

 

[Nidum]

Deleted

 

[haruspex]

[is there] a non-normal force vector applied at the tip from the vertical displacement.

Is that important in itself, or is the important question the consequence of such a force for the beam shape? Or, perhaps, the resulting additional resistance to the movement of the beam support?
For the moment, think of the beam as rigid but with a hinge at the support, and ignore friction at the probe. If the support gets a small distance $\delta x$ closer to the substrate, how far along it would the probe slide? You say the beam angle is 13^o, so it would be $\delta x \tan(13^o)$, right? Now consider the beam as being end-loaded enough so that its buckling shortens it by that amount. What end load is required? Could the frictional force be that great? If it could be, what would be the component of that force on the vertical direction?

 

[Graham1874]

Thanks for your reply, haruspex.

First of all, the important question for me - at this stage - is 'What are the y and z components of the force vector at the position of probe-substrate contact?'. I understand that this can be taken further but I purely want to analyse whether there is a y (horizontal) component acting on the surface.

Secondly, can you explain how it is δxtan(13deg)? When Section A displaces downwards, would the 13 degrees not reduce too? Are you proposing this as a consideration to find out how to bring the shortening of the beam into static equilibrium with the 'lengthening' of the beam caused by the positive y-component of the force vector at probe-substrate contact?

My feeling is that I need to consider the probe tip to be in contact with the sample surface 'freely' - i.e. without a support, have the Section A being fixed like a cantilever, but allowing vertical displacement and derive an expression in a state of equilibrium where the negative y-displacement due to buckling cancels out the lengthening due to horizontal component of the force vector at contact. This is why I think friction will come into play because, without a support at the contact position, there should be free movement along the surface leading to an expression including friction which would help to bring the system to static equilibrium.

I appreciate you giving me these thoughts but, in all honesty and with total respect, I feel more confused than I did before!

If you think my statements above are indicating some confusion of the system then let me know because I don't feel I fully understand some of your questions.

Thanks again for your time and I look forward to any further thoughts.

 

[Nidum]

All we actually have to analyse is a simple mechanism the same as an old type gramophone pickup .

Back soon .

 

[Nidum]

 

[Nidum]

Whoever chose that 13 deg angle either knew what they were doing or they got lucky - with that angle the deflected shape is almost the same for both pinned and sliding cases at the probe tip .

 

[Nidum]

Is seeing the deflected shape enough for now or do you want to start exploring deflection v load calculations ?

 

[Graham1874]

Whoever chose that 13 deg angle either knew what they were doing or they got lucky - with that angle the deflected shape is almost the same for both pinned and sliding cases at the probe tip .

Hi Nidum,

That's very interesting! I am modelling the AFM in Abaqus FEA software but I'm a complete beginner so it's taken me 3 weeks to create the model! Can I ask:
1. Did you apply the displacement to the whole of Section A?
2. Did you make the Section A as a rigid body?
3. What material property in your model defines the stiffness of the beam? (I believe the answer to this would be Young's modulus, but let me know)

I want this thread to discuss the mathematical side of this problem but I'm interested in those ^ question for the FEA. See my next reply regarding your question in your follow-up post.

 

[Graham1874]

Is seeing the deflected shape enough for now or do you want to start exploring deflection v load calculations ?

For my project I have been advised that, to get the best mark, I need to back up my Abaqus results with mathematical calculations to prove that the Abaqus model can be considered legitimate. So yes, I will need to back up the visualisations with mathematical proof of the behaviour.

I want to be able to describe the behaviour mathematically so that I don't necessarily need to have the whole model simulated, and so that I can simplify the Abaqus model down to the tip and substrate contact, to save computational time.

Any help with the maths behind the force vector at contact would be super-helpful because, as I say, that's the only way of proving the legitimacy of my software model.

 

[Graham1874]

I hate to bump my thread but I still haven't managed to derive an expression for the force vector. Anyone able to help?

 

[Graham1874]

FAO Nidum:

The attached conversion sheet explains the conversions from standard units to microscale units.

Examples:
15 nanometres is equal to 15 x 10^9 metres. Then look at chart to convert to micro-units, giving 0.015 units to be input for FE model.
304GPa is equal to 304 x 10^9 Pascals. Then look at chart to convert to micro-units, giving 304000 units for FE input.

 

[Nidum]

OK