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Vibration of a cantilevered beam with a Tip mass

  1. Apr 10, 2012 #1
    Hey guys,

    I was wondering if anyone could point me in the next, correct direction for this problem.
    I understand how to determine the mode shapes and the natural frequencies of a cantilevered beam without a tip-mass, but adding the tip-mass baffles me a little bit.

    The boundary conditions are as follows:
    Assume that the beam is clamped and rigid at the wall, therefore the deflection and the slope are zero at the wall.
    I also understand that the bending moment at the tip would be zero (this assumes that the tip mass can be modelled as a point-load with no dimensions).
    The shear force at the tip baffles me a little bit. But from what I can gather it should be the negative product mass of the tip mass and the acceleration of the beam at the tip.

    So, I take the general equation for the motion of a continuous system such as this

    y(x) = Asin(beta*x)+Bcos(beta*x)+Csinh(beta*x)+Dcosh(beta*x)

    I integrate through three times giving -
    First Integration
    y'(x) = A*beta*cos(beta*x)- B*beta*sin(beta*x)+C*beta*cosh(beta*x)

    Second Integration
    y''(x) = -A*beta^2*sin(beta*x)- B*beta^2*cos(beta*x)+C*beta^2*sinh(beta*x)

    Third Integration
    y'''(x) = -A*beta^3*cos(beta*x)+ B*beta^3*sin(beta*x)+C*beta^3*cosh(beta*x)

    The first two boundary conditions at the wall give
    A+C = 0 & B+D = 0

    The last two equations can be rearranged to:
    y''(x) = -A*beta^2(sin(beta*x)+sinh(beta*x)) - B*beta^2(cos(beta*x)+cosh(beta*x))

    y'''(x) = -A*beta^3(cos(beta*x)+cosh(beta*x))+ B*beta^3(sin(beta*x)-sinh(beta*x))

    The first boundary condition at the tip gives:

    0= -A*beta^2(sin(beta*x)+sinh(beta*x)) - B*beta^2(cos(beta*x)+cosh(beta*x))

    and the second boundary condition at the tip gives:

    -mδ^2y/δt^2 = -A*beta^3(cos(beta*x)+cosh(beta*x))+ B*beta^3(sin(beta*x)-sinh(beta*x))

    This is the part which confuses me slightly.

    The equation for y(x,t) is given by:

    y(x,t) = Y(x)*e^iwt

    Does this mean, in order to determine δ^2y/δt^2, I should differential this twice with respect to t?
    This would yeild:

    δ^2y/δt^2 = -w^2*Y(x)*e^iwt

    Am I doing this correctly? If so, what would I do AFTER this point? If I recall correctly, the solution for an un-loaded cantilever allows you to eventually determine points at which cos(bL) and cosh(bL) cross, giving the natural frequencies. How would this proceed for a cantilever with a tip mass?

    Any help would be greatly appreciated.


  2. jcsd
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