General Cantilever Equations for Bending Aluminum

[X1088LoD]

I am trying to determine some general cantilever equations.

I have an aluminum beam extending out 235 mm (L) from an aluminum
block. The beam is 25 mm wide (W) and 3 mm thick (H). A force is
applied at a point approx. 200 mm from the block and I am interested
at a point 25 mm away from the block. I know the exact measurement of
vertical deflection at a point 100 mm from the block.

I am assuming the modulus of elasticity E is 10*10^6 psi.

In general, I have the following diagram:
http://www.brentless.com/Images/station2.jpg
In the diagram

A: a driving rod that moves up and down from a loudspeaker setup not
shown, this guides the cantilever on this end
B: a measuring caliper to measure the vertical displacement at point
alpha measured from the aluminum block
C: a mounted strain gauge, the point of interest, centered at delta
from the aluminum block
D: an aluminum block mounted the cantilever on one end

Z1: a known, measurable displacement at alpha distance
Z2: a displacement not known, at the end of the beam

alpha: the measurement from the block to the measuring caliper
beta: the measurement from the caliper to the end of the beam
gamma: the measurement from the caliper to the driving rod
delta: the measurement from the block to the strain gaugeMy problems to this point, most formulas I have found assume that the
measurement of deflection is actually taken at the end of the beam.
So how can I use the measurement at the 100 mm point above. I don't
specifically know the value of the force being applied (my cantilever
is being driven up and down by a loudspeaker, so if possible I would
like to leave out the force value and determine an equation based on
the measurement of deflection. Thoughts?

I am trying to determine a general equation of which I can relate to
stress and strain the values and measurements I have stated above for
a testing model. I am not a mechanical engineer, so I don't really
understand this stuff, so I appreciate any advice or help anyone can
give me.

 

[radou]

If I understood the problem correctly, you need to find the deflection of the end point, and all you know is the measured deflection at point B. Well, using a dummy force method, you can find a relation between the applied end force, let's say F, in a form F = F(d*), where d* is the deflection at B. Further on, you can again use a dummy force method to find the relation between the deflection d at the end point (i.e. point A) and the force F, i.e. d = d(F). After plugging the force in terms of d* into that equation, you should be able to retrieve a relation of form d = d(d*), which is what you need to find.

 

[ravenprp]

Thank you for providing the details of your cantilever setup. The general equation for bending of an aluminum beam can be written as:

𝛿 = (𝐹𝑙^3)/(3𝐸𝐼)

where 𝛿 is the deflection at a point, 𝐹 is the applied force, 𝑙 is the length of the beam, 𝐸 is the modulus of elasticity, and 𝐼 is the moment of inertia of the beam.

In your case, the length of the beam (𝑙) is 235 mm, the width (𝑊) is 25 mm, and the thickness (𝐻) is 3 mm. The moment of inertia (𝐼) for a rectangular beam is given by:

𝐼 = (𝑊𝐻^3)/12

Substituting these values in the general equation, we can write the equation for deflection at a point 25 mm from the block as:

𝛿 = (𝐹(235)^3)/(3(10*10^6)(25*3^3)/12)

𝛿 = (𝐹(235)^3)/(7.2*10^9)

Now, since you have the measurement of deflection (𝛿) at a point 100 mm from the block (𝛿𝟏𝟎𝟎), we can write the following equation:

𝛿𝟏𝟎𝟎 = (𝐹(100)^3)/(7.2*10^9)

We can solve for the force (𝐹) in this equation, which will give us an equation for force in terms of the deflection at 100 mm:

𝐹 = (𝛿𝟏𝟎𝟎 * 7.2*10^9)/(100)^3

Now, we can substitute this equation for force (𝐹) in the first equation for deflection at 25 mm, which will give us an equation in terms of the deflection at 100 mm:

𝛿 = ((𝛿𝟏𝟎𝟎 * 7.2*10^9)/(100)^3 * (235)^3)/(7.2*10^9)

𝛿 = (𝛿𝟏𝟎