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General Cantilever Equations for Bending Aluminum

  1. Feb 14, 2007 #1
    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:
    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 gauge

    My 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.
    Last edited: Feb 14, 2007
  2. jcsd
  3. Feb 14, 2007 #2


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    Homework Helper

    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.
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