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Question about free damped vibration in an SD

  1. Mar 26, 2016 #1
    1. The problem statement, all variables and given/known data
    "If a horizontal force of F = 299kN is applied to the right at the girder level by a hydraulic jack and then the force is removed suddenly, find the equation of motion of the girder. Consider 5% critical damping. Find the
    displacement and velocity of the girder at 0.2 seconds after release of the force."

    This is of a braced single storey building. (bottom left is braced to top right etc)
    I = 7.6*10^6mm^4
    E = 200 GPa
    Length of Girder = 6m
    Height of Columns = 3.5m
    The braced bars are 25mm diameter. This meant that the Area of them came to 490.874mm^2 as Area = (pi*d^2) / 4

    mass of frame = 4000kg.

    stiffness (girder and two columns) = 11404.775N/mm

    natural frequency = 53.4 rad/s

    2. Relevant equations
    m*(acceleration) + c*velocity +k*displacement = F(x) -> (Inertia Force + Viscous force + Stiffness force = F(x))

    critical damping constant = 2*mass*omega

    damped natural frequency = natural frequency * sqrt (1-ξ^2)

    natural frequency = sqrt (stiffness / mass)

    3. The attempt at a solution

    I calculated Stiffness in both the columns and the rigid girder. This came to 11404.775. I solved for the columns by using (12EI/L^3) multiplied by 2 as there are two columns. For the rigid girder, I had to take into account theta.

    I then calculated natural frequency of vibration after finding stiffness and dividing it out by the mass. After square rooting the fraction, it came to 53.4 rad/sec.

    I then calculated the critical damping constant, this came to 427200kg/sec as critical damping constant Cc = 2*mass*natural frequency.

    C actually was less than Cc, meaning it is under critically damped. ξ = 0.05.

    I then went back to m*(acceleration) + c*velocity +k*displacement = F(x)

    I then made all of it equal to zero, deriving from a characteristic equation and then solving for the roots using the quadratic formula. All this did was bring me back to an equation I already knew from my notes, as shown with attached image.

    I am wondering, since it is applied instantaneously, does m*acceleration + c*velocity + k*displacement still equal to 0? or does it equal to 299kN?? The 299 is throwing me off because I was told if the frame is free moving, then it should equal to zero.

    Also I'm not sure about writing the equation of motion when I am not given the initial velocity or displacement constraints.

    Apologies if this question sounds a bit silly - I haven't come across one of these questions in my tutes!
     

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  2. jcsd
  3. Mar 31, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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