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Homework Help: Mass-Spring Model?

  1. Jan 2, 2015 #1
    1. The problem statement, all variables and given/known data
    A half-full water tank mounted on the top of a building is modelled as shown below. Find the shear force it can produce?
    m = 500 kg
    k = 1000 N/m
    model: https://app.box.com/s/qke2kbwag2mp9a23kkqq

    2. Relevant equations
    Transfer Matrix Method:
    Point & Field Matrices: https://app.box.com/s/0gmyi1hqaecp1hfjd3f3
    Rearranged model: https://app.box.com/s/nycu9csciavufusnlnfz

    3. The attempt at a solution
    I'm unsure if I should use this method; find T (period of oscillation), use T to find force 'F'.
    working 1: https://app.box.com/s/he5ji206c1c8s76lrf09

    Or if I should use 'Transfer Matrix Method'. If I should use this method, how should I start?
  2. jcsd
  3. Jan 2, 2015 #2


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    Your first figure looks nothing like a tank of water. I don't see the connection at all. Maybe you could elaborate?
  4. Jan 2, 2015 #3
    It's not 100% clear to me either, but if you look at page two of the attached .pdf below. It seems that what I've been given is a simplified model where the mass of the water (m) is 500 kg & the equivalent stiffness of the building is (k) 1000 N/m. It then asks me to find the shear force?

    pdf: https://app.box.com/s/ftdnwffvx3y0wf9kl44f

    The method I've been studying recently is 'Transfer Matrix Method' but this requires an applied force over a period of time (sinusoidal excitation force). But as this problem doesn't have one, the only way I can think of to resolve it is just to use T = SQRT(k/m), F = 2*pi/T. The problem comes from an old Engineering Analysis Paper MEng/BEng (Hons) but it's only worth 4 marks. Just not sure where to start with the limited info??
  5. Jan 2, 2015 #4


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    Ah, now we are getting somewhere! Your first post said exactly nothing about horizontal ground motion. You really need to help by telling us what the problem of interest is here.

    If you read the Housner paper, I think you need to consider most Fig. 3. You need to use your given data to evaluate M0, M1, k, k1 etc., according to what Housner has described in his paper. This will enable you to write an approximate equation of motion for the system. Do you know if your tank is circular or rectangular in plan section?

    Basically the question is asking you to apply (and slightly extend) Housner's ideas. This is a very good problem, and an opportunity to learn some very practical engineering analysis.
  6. Jan 2, 2015 #5


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    By the way, T is the period, not the natural frequency!!
  7. Jan 3, 2015 #6
    Yes you're correct, T (period of oscillation) = 2*π√m/k. Therefore F (max force produced) = (2π/T)*m. I'll have to σo through the paper properly & post once I have a more thorough understanding. Thanks for your input, back asap...
  8. Jan 4, 2015 #7
    Stiil pretty much stuck with this one? Using Housner paper the max shear force that can be produced by the half filled tank is; T = 2*π√m/k ∴ T = 2*π√500/1000 = 4.443 seconds. This is the period of oscillation therefore shear force F = (2*π/T)/m = 500/(√500/1000) = 707.1 N. The answer given in the exam paper is F = 2000 N.

    From post #4, I can't evaluate M0, M1, k, k1 as there is no further info given i.e. no dimensions, displacement, shape of tank etc. Only mass (m) & equivalent stiffness (k).

    I've thought about turning this into an eigenvalue problem & using Lagrange's eq's but with the limited info I'm not sure how to proceed?

    • If anyone has any ideas please post...
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