Thermal Switch Design Project... need help fast!

  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. nvn

    nvn 2,124
    Science Advisor
    Homework Helper

    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. The handout my professor passed out on this project defines Pcr without a negative sign. Why is it suppose to be negative?
     
  5. nvn

    nvn 2,124
    Science Advisor
    Homework Helper

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