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Homework Help: Kinematics of a linkage system of 4 bars

  1. Sep 8, 2010 #1
    1. The problem statement, all variables and given/known data


    Tabulate and plot the angular position, velocity and acceleration of θ4 for t=0 to t=10 in increments of 0.1

    r1= 30 mm
    r2 = 12 mm
    r3 = 39 mm
    r4 = 36 mm

    θ2 = 0.1t (radians)

    3. The attempt at a solution

    Well first I wrote this down:

    r1 = r2cos(θ2) + r3cos(θ3) - r4cos(θ4)

    It didn't really get me anywhere so after some research I found Freudenstein's Eqn:

    K 1 cos θ 2 + K2 cos θ 5 + K 3 = cos ( θ 2 - θ 5 )

    K1 = l1 / l4 K2 = l 1 / l 2 K3 = ( l 32 - l 12 - l 22 - l 2 4 ) / 2 l 2 l 4

    Inputting the values I ended up with this:

    (30/36)cosθ2 + (30/12)cosθ5 - (91/96) = cos(θ2 - θ5)

    I have no idea how to solve this equation in terms of θ5 though!

    (once I find θ5 I'll just use θ5 = 360 - θ4 to get θ4)

    Am I doing this right at all? Any help would be soooooooo much appreciated!!!

    Let me know if you need any more info or if I posted wrong or anything!
    Thanks again!
  2. jcsd
  3. Sep 8, 2010 #2
    r2 and r4 have fixed points of rotation, but r3 is floating. Can you find a relationship concerning that the rotating ends of r2 and r4 have a fixed distance r3 between them?
    Another approach is to let t=0.1 Are you then able to calculate the new geometry? Obviously, you could take days to slog through this 0.1 seconds at a time, but the process may indicate to you what you have to do to make it more general, and easier.
  4. Sep 8, 2010 #3
    Yeah, well what I ended up doing is just throwing Freudenstein's eqn into Mathematica, have it solve it for θ5 and then just use that... it wasn't a pretty equation, but it seemed to have gotten the job done:

    y = -cos^(-1)((160 cos^2(x)-582 cos(x)-2 sqrt(9216 sin^4(x)+49319 sin^2(x)+2816 sin^2(x) cos^2(x)-31520 sin^2(x) cos(x))+455)/(48 (4 sin^2(x)+4 cos^2(x)-20 cos(x)+25)))

    (y is θ5, x is θ2)

    Still this solution is obviously not how they wanted me to do it...

    If anyone knows a more elegant way of solving this problem, I've already handed it in but I'd love to know how to do it!
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