Solved: Cantilever Experiment: Deriving Formulae

[ash1098]

[SOLVED] Cantilever experiment

Hi,

I am writing a report on cantilever oscillations, my experiment involves fixing different cantilvers e.g. a ruler to the end of a table then measuring the period and height of oscillations while varing the mass attached to the cantilever, and other varients e.g. length of cantilever.

I have come across these two formulae:

T= 2(pi)*[(4ML^3)/(bd^3E)]^1/2

and:

h= 4MgL^3/Ebd^3

where: b= width of cantilever
d= thickness of cantilever
E= Youngs Modulus
M=Mass
L=Lenght of cantilever
T=period of oscillations
h=height of oscillation

I have looked at eqn's involving Hooke's and simple harmonic motion but cannot work out how these formulae have been derived.

Does anyone know how these formulae where derived, or where I can find information on this in general?

Thanks.

ash.

p.s. I have written out the formulae using math open office and attached them in pdf if it helps make them easier to read.

 

[Gokul43201]

Have you studied how to calculate bending moments and shear stresses in beams? That's the starting point. You will finding it covered in any standard Engineering Mechanics/Solid Mechanics/Strength of Materials textbook (e.g., Timoshenko) or online by Googling the above terms (also try deflection of beams).

If you don't want to start from first principles, you can start with the general deflection equation for a cantilevered beam:

$$h=\frac{PL^3}{3EI}$$

which is essentially, the cross-section independent form of your second equation. The equation for the time period of small oscillations comes from plugging the shear stress into the restoring force and linearizing to first order. Or if you'll settle for a quick wave of the hands, notice that the equation for T is exactly what you would get if you take the expression for h and plug this into the place of the length of a simple pendulum.

 

[ash1098]

Hi Gokul, Sorry I took so long to reply, I forgot I had this post running. I understand my investigation now! Thank you for your help.