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Mathematical modeling of the deflection of a rod

  1. Aug 20, 2015 #1
    A rod is radially mounted on a motor shaft and provided with a flat end plate. By turning movement, the end plate will touch a pointed object, and the angle turned is measured.By the torque of the motor during acceleration and deceleration, a vibration and deflection happens on the rod. The model shows the deflection depending on temporal torque and angle.

    This deflection will affect the angle measurement . So the idea is to find the mathematical formulae for the deflection and vibration, account it as the error and minimize it using controller(Simulink model or so).
    This system is a higher order system(parameters like acceleration, length, torque, angle, etc) and for me it looks a very complicated task but interesting.

    Can someone guide me to start the modeling. Can an FEM analysis help me with understanding the behaviour of vibration and deflection?
    Even after FEM how do/ where do I start the modeling task?
    Can anyone suggest me/ tip me some similar mathematical modeling examples or even a suitable literature so that I can learn more?
  2. jcsd
  3. Aug 20, 2015 #2
    For me the first step is always to draw a picture and include all the forces and torques.

    I don't see yet why FEM would be required. It seems like certain progress can likely be made with pencil and paper and fundamental principles.
  4. Aug 20, 2015 #3
    Thank you. My idea was to change the values of Torque and note down the simulated values. So that later when the mathematical model is made, i could compare the deflection values. Do you think this will work.

    I am a Control system guy not Mechanical. Could you guide me on the topics that I should know to do this? I have been studying Free body diagrams. Could you please tip me on other such topics, or any literature or texts that I can look into? My major task is control system part. But this is where I should start, unfortunately I have very little knowledge in this.
  5. Aug 21, 2015 #4


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    You are looking at this problem backwards .

    The motor characteristics and the way motor is controlled dominate the response of the overall system . The rod and target are just springs with a bit of inertia .

    If this is a problem being looked at for a real application then choose a motor with suitable characteristics and then use electronic control and feed back to set up a system which will do exactly what you want under programmable control .
  6. Aug 22, 2015 #5

    I am sorry , i didnt get that. This is how typically I have seen such a problem is worked out. kindly correct me if I am wrong.
    1. Find the dynamics of the system. This is basically mathematical modelling or simply finding the governing equation.
    2. Find the Transfer function of the system with the above equation using Laplace transform.
    3. Model creation , optimisation etc in controller.(PID or watever we use).

    I have littte knowlege in the first step. Which all topics of mechanics(eg. deflection, vibration, some governing principles etc.) should I know well to derive the mathematical equation. Please someone tip me on any good text books or worksheets so that I can learn it.
  7. Aug 25, 2015 #6
    Hai again,
    I have read through various papers and I think this is a Centrifugal spring - like system. Can anyone help me confirm if I am right? Any other suggestions...
  8. Aug 26, 2015 #7
    Please see the attachments for the mechanism explained Basic mechanism.jpg When motor starts.jpg

    Attached Files:

  9. Aug 26, 2015 #8


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    How to model this system depends upon what your concern is.

    Are you really concerned with the angle-time relation at every instant, or only with how long it takes the tool to arrive at the work piece?

    Are you really concerned about the flexure of the rod, or simply the impact velocity of the tool at the work piece?]

    What is it you are really after in modeling this system?
  10. Aug 27, 2015 #9
    Yes, flexing of the rod is a major concern. Due to vibration and deflection the reading ( angle value ) is disturbed. So a generic formula to find the behavior of the rod, irrespective of the length will be first found out. This equation will depend on various parameters(torque, etc etc.).
    So we can control the motor later with this equation in mind to reduce the error.
    Hope its clear
  11. Aug 27, 2015 #10


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    You cannot model the motion of the rod without describing the torque applied by the motor. The motor torque is a major input to the rod motion. You will have to model the rod, the motor, and the control system all together, because they interact.

    Suppose for a first case that you apply an exceedingly small torque to the motor. The rod will rotate, but it will be very, very slow and with very little dynamic deflection.

    Now at the other extreme, suppose you apply a very large but brief torque pulse from the motor. The rod will rotate, but it will not be uniform. Rather, there will be significant bending as well as rotation, so a bending wave is transmitted down the length of the rod.

    These two cases are nothing alike, which points to why you must model the entire system together. In such a model, you can represent the rod by the Euler-Bernoulli beam bending model with distributed mass included, or you can use the more elaborate Timoshenko beam model. Either way, the beam will be represented by a PDE in space and time. For either model, the end moment resulting from the motor torque is a critical input. Since you propose a control system, the motor torque will be dependent upon the control system inputs to the motor, and the control system will be dependent upon the feedback from the angular position sensor. It is a SYSTEM, have fun with it!
  12. Sep 10, 2015 #11
    Hello again.
    can someone help me with how to find these two terms. A solved example is appreciated. because I am stuck without knowing how to integrate this. φi(x) is defined as the normal mode of a beam. And I am interested only in the first mode. I cant figure out how to solve it. Please help me out. equations.JPG
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