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Homework Help: Lagrangian problem of a cylinder on inclined plane and two springs

  1. Oct 28, 2014 #1
    1. The problem statement, all variables and given/known data
    A cylinder on a inclined plane is rolling without slipping. Inclined plane is connected to wall with a spring and cylinder is connected to wall with a spring too. All frictions will be neglected, and all the given data has shown on the image below.

    As seen above, k2 spring and cylinder is only moving at vertical axis which can be thought as y.

    First of all, i have doubts on generalized coordinates. I think only one generalized coordinate should be used and it should be y. I need to solve this problem with D'Alembert Principle and Lagrangian Dynamics.

    Thank you in advance for your help.

    2. The attempt at a solution

    in order to apply Lagrangian dynamics,


    Does it look right or am i missing something about rotational kinetic energy?
  2. jcsd
  3. Oct 28, 2014 #2
    Kinetic energy;

    for x=x1 and y=x2

    is second one right?
  4. Oct 28, 2014 #3


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    You are correct in saying that this system has only one degree of freedom.

    It would be helpful if you would very carefully define your coordinates on the figure. Where is x = 0? Where is y = 0? You may also find it useful to define an "up-slope coordinate, s" as a secondary variable.

    The biggest part of your difficulty right now is that you have not adequately dealt with the kinematics of this problem. This is very common, because everyone says, "oh, kinematics ... that's trivial," but it is not. It is what is causing most of your difficulty right now.
  5. Oct 29, 2014 #4

    I can define our generalized coordinates like this. This graphic is designed for D!alembert principle.

    In this situation our Lagrange eqn is;


    we can write
    then we can solve this problem,isnt it?
  6. Oct 29, 2014 #5
    I think it is true.
  7. Oct 29, 2014 #6


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    Your equation says that x1 = 0 implies x2 = 0 also. Why is this true?

    Exactly what does x1 measure? You show the left end of the arrow at the wall, but the right end is to what?

    x2 reference is how far above the bottom of the ramp?

    What is the relation between rotation of the disk and x1?

    I think you still have not really dealt with the kinematics of the problem in a complete way. It will make your life much, much simpler if you do so.
  8. Oct 29, 2014 #7
    Ohh.Sorry.. T is not equal the zero. You re right.

    -x2 reference is how far above the bottom of the ramp?

    I think it is not important. It defines cylinder's motion on vertival axis. If I say x2 ,It starts from O.

    -What is the relation between rotation of the disk and x1?

    upload_2014-10-29_18-30-45.png and upload_2014-10-29_18-31-16.png

    and my answer is;

  9. Oct 29, 2014 #8
    I think Lagrange equation is more simle method than others(D'alembert, Hamilton...etc) to solve this problem.
  10. Oct 29, 2014 #9


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    Actually, since there is only a single degree of freedom in this system, the Eksergian's equation is available and it is usually more simple than Lagrange, although they give the same result.
  11. Oct 29, 2014 #10
    I think my result is exactly true.
    Now I try to solve this problem with d'Alembert principle.
    I hope that I will find same result.

    Then I want to find natural frequency of the system. Finally when alpha=0, what is meaning?

    Attached Files:

  12. Oct 29, 2014 #11
    Ohhh. Nooo. My result is a bit wrong. True result is;

    upload_2014-10-29_20-37-37.png İS WRONG.
  13. Oct 29, 2014 #12


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    I think you have an algebraic error in your original work; I have done a very similar analysis but got you last equation (sort of!).

    This problem is not well posed in this respect: Nothing is given about the free length of the springs. For what values of x and y are the springs relaxed? You assumed (in your form for V) that both displacements were measured from the relaxed position, but his may violate the kinematic relations.

    To be properly posed, the problem should specify the free length conditions (ie, values of x and y that make the springs relaxed).
  14. Oct 29, 2014 #13
    Thanks for your helping.

    But I dont understand your saying. What is your last solution? Can you write it here?
  15. Oct 29, 2014 #14


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    Oh, that would not be fair! You will learn a lot more if you puzzle this out yourself.

    I suggest that you accept my observation that there is not enough information given about the relaxed state of the springs, and simply assume that there are values xo and yo that apply at the same system configuration for which both springs are relaxed. Based on that, re-formulate your potential energy expression.

    In order to do this, you will have to do a more complete kinematic analysis than you have done thus far (as it appears to me). I suggest that you assume that the length of the base of the wedge is B and that the horizontal distance from the wall at left to the centerline of the vertical spring is C.

    To do the kinematics, I strongly recommend that you employ vector loop equations, or better, their scalar equivalents.
  16. Oct 29, 2014 #15
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