Thermal Switch Design Project need help fast!

  1. Oct 16, 2010 #1
    1. The problem statement, all variables and given/known data

    There are three vertical metal strips (Steel, Aluminum, Steel) between two horizontal rigid bodies

    alphaa = 12.5 E-6 /degF -- coefficient of thermal expansion
    alphas = 6.6 E-6 /degF
    Ea = 10 E6 psi -- Young's Modulus
    Es = 30 E6 psi

    The original dimensions of the switch are
    ta = .0625 in -- thickness of aluminum
    wa = .25 in -- width of aluminum
    ts = .0625 in
    ws = .125 in
    L = 4 in -- length of all metals

    With these dimensions, the switch will activate with a 180 degF temperature increase

    By only changing the dimensions of the aluminum strip, I have to modify the switch to activate with 100 degF temperature increase


    2. Relevant equations

    δ = P*L/(E*A) + alpha*ΔT*L

    Pcr = 4*pi^2*Ea*Ia / L^2 -- critical axial compressive load for the aluminum to buckle
    Ia = wa*ta^3/12 -- minimum second moment of inertia


    3. The attempt at a solution

    I tried setting the deformation of the aluminum equal to that of the steel

    δa = δs
    Pa*L/(Ea*Aa) + alphaa*ΔT*L = Ps*L/(Es*As) + alphas*ΔT*L

    from a free body diagram, I got Ps = -.5*Pa

    Pa*L/(Ea*Aa) + alphaa*ΔT*L = -Pa*L/(2*Es*As) + alphas*ΔT*L
    Pa*(1/(Ea*Aa) + 1/(2*Es*As)) = ΔT*(alphas - alphaa)

    solving for Pa and simplifying using (1/a + 1/b)^-1 = ab/(a+b)

    Pa = ΔT*(alphas - alphaa)*(2*Ea*Aa*Es*As / (Ea*Aa + 2*Es*As))

    substituting Pcr = Pa

    pi^2*Ea*wa*ta^3/(3*L^2) = ΔT*(alphas - alphaa)*(2*Ea*Aa*Es*As / (Ea*Aa + 2*Es*As))

    now solving this for the temperature..

    ΔT = pi^2*ta^2/(3*L^2) * (Ea*Aa + 2*Es*As)/(2*Es*As(alphas - alphaa))

    when I plug in the variables for the 180 degF switch from above, I am getting
    ΔT = -181.5 degF and I do not see anything wrong in my algebra; my units still come out to be degF.

    Can someone tell me what I am doing wrong? Am I approaching this the right way?
     
    Last edited: Oct 16, 2010
  2. jcsd
  3. Oct 17, 2010 #2

    nvn

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    naevitar77: Excellent work, so far, except for one minor mistake. Pcr should be, Pcr = -4*Ea*Ia*(pi/L)^2. You erroneously omitted the negative sign here. Try it again.

    By the way, numbers less than 1 should always have a zero before the decimal point. E.g., 0.25, not .25. See the international standard for writing units (ISO 31-0).
     
  4. Oct 17, 2010 #3
    The handout my professor passed out on this project defines Pcr without a negative sign. Why is it suppose to be negative?
     
  5. Oct 17, 2010 #4

    nvn

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    Pcr is negative because it is compressive force. Your professor listed the magnitude.
     
  6. Oct 17, 2010 #5
    That makes sense. Thank you so much nvn. You're a life saver!
     
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