Theoretical spring constant calculation for a tube of pipe

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SUMMARY

The discussion focuses on calculating the theoretical spring constant of a hollow glass pipette intended for use as a cantilever tip in atomic force microscopy experiments. The target spring constant is 100 Newtons/metre, and the user seeks to determine the appropriate diameter of the pipette while considering the effects of tapering and bending. Key equations provided include the moment of inertia for a cylinder and the deflection formula for beams, which are essential for deriving the spring constant. The conversation highlights the importance of material choice, noting that glass may be too brittle for this application.

PREREQUISITES
  • Understanding of spring constants and their calculations
  • Familiarity with moment of inertia for cylindrical shapes
  • Knowledge of beam deflection principles
  • Basic concepts of atomic force microscopy and cantilever design
NEXT STEPS
  • Research "Moment of Inertia for Tapered Beams" to understand how tapering affects calculations
  • Study "Beam Deflection in Mechanics of Materials" for more in-depth equations and applications
  • Explore "Spring Constant Calculations for Non-Uniform Cross Sections" to address complex geometries
  • Investigate alternative materials for cantilevers in atomic force microscopy applications
USEFUL FOR

Researchers and engineers in the fields of materials science, mechanical engineering, and nanotechnology, particularly those involved in atomic force microscopy and cantilever design.

ReliableSin
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Homework Statement



I require the a means to theoretically calculate the spring constant of a hollow tube.
Essentially, I need to find the diameter of the glass pipette which will have a spring constant of 100 Newtons/metre. If someone could point me towards relevant information on calculating, or using software to find the solution it would be greatly appreciated.

Homework Equations



This is what I need. I've been searching, but all I've been able to find are solutions based on spring geometry.

The Attempt at a Solution



No idea.
 
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Welcome to PF!

Hi ReliableSin! Welcome to PF! :smile:

I don't understand … what are you intending to do with this glass pipette? :confused:

Are you using it to support something, as a building column?
 
I'm using it as a cantilever tip in some atomic force microscopy experiments. Usually we fabricate our own cantilevers using optical fiber with a spring constant of ~100 Newtons/metre. Now we want to use a pipette as the cantilever tip. As such, we need the pipette to have a spring constant of about the same value. We can adjust the spring constant of the pipette by tapering or etching. However, we also need to make a bend in this pipette, and the thinner we make the pipette before bending, the higher the chance that our bend will stop fluid from flowing through the pipette.
To sum up, I need to know how to get the spring constant of a tube so I calculate what is the largest pipette diameter I can use which will have an appropriate spring constant.

I've been reading through the pages on wikipedia on spring constants and deflection. I think I need to find the appropriate moment of inertia for a cylinder, which is simple, and sub this into the equations given for a normal cantilever. However, I'm having a hard time following their math, specifically how they eliminate their force term.
 
ReliableSin said:

Homework Statement



I require the a means to theoretically calculate the spring constant of a hollow tube.
Essentially, I need to find the diameter of the glass pipette which will have a spring constant of 100 Newtons/metre. If someone could point me towards relevant information on calculating, or using software to find the solution it would be greatly appreciated.

Homework Equations



This is what I need. I've been searching, but all I've been able to find are solutions based on spring geometry.

The Attempt at a Solution



No idea.

ReliableSin said:
I'm using it as a cantilever tip in some atomic force microscopy experiments. Usually we fabricate our own cantilevers using optical fiber with a spring constant of ~100 Newtons/metre. Now we want to use a pipette as the cantilever tip. As such, we need the pipette to have a spring constant of about the same value. We can adjust the spring constant of the pipette by tapering or etching. However, we also need to make a bend in this pipette, and the thinner we make the pipette before bending, the higher the chance that our bend will stop fluid from flowing through the pipette.
To sum up, I need to know how to get the spring constant of a tube so I calculate what is the largest pipette diameter I can use which will have an appropriate spring constant.

I've been reading through the pages on wikipedia on spring constants and deflection. I think I need to find the appropriate moment of inertia for a cylinder, which is simple, and sub this into the equations given for a normal cantilever. However, I'm having a hard time following their math, specifically how they eliminate their force term.

It does sound like glass will be too brittle for this application. Have you considered making the bending pipette out of some other material?
 
We use glass for these applications all the time. The issue is that the stiffness of a non-tapered glass pipette is too great to perform contact-mode measurements.
 
I'm assuming you have a cantilevered tube made of glass, but then you imply that the tube is actually tapered. If it isn't tapered, you can use straight beam equations:

I = 3.14159 (Do4-Di4) / 64
Where:
Do = OD
Di = ID

I is moment of inertia in units of length raised to the 4'th power.

Deflection of this beam is:

d = F L3 / (3 E I)
Where:
d = deflection (units of length)
F = Force
L = beam length
E = bending modulus (units of force per length squared)

To get spring constant:

k = F / d

k = 1 / ( L3 / (3 E I) )

This assumes the beam is tubular and straight and of uniform cross section and the force is perpendicular to the axis of the tube.

If the tube is actually tapered, curved, nonuniform cross section, etc... it will get a bit more messy...
 
Thank you so much! That's incredibly helpful. Could you point me towards your source for further reading on the topic?
And as you said, the tapering will modify these calculations. Will it essentially be a change of the moment of inertia, or is it more complex than that?
 

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