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Solution to differential equations of piezoelectric vibration

  1. May 22, 2014 #1
    dear all, I am working on a problem about the vibration of a piezoelectric ring with electrodes on the upper and lower surface. the coordinate system is cylinder coordinate system. the structure is shown below. This is an axisymmetry problem.

    attachment.php?attachmentid=69968&d=1400765791.png

    the polarization of the piezoelectric ring is along thickness direction.

    constitutive equations are
    attachment.php?attachmentid=69969&d=1400765791.png

    the equation of motion are
    attachment.php?attachmentid=69970&d=1400765791.png

    The geometric relations are
    attachment.php?attachmentid=69971&d=1400765791.png

    substituting constitutive relation and geometric relation into the equation of motion , I obtain
    attachment.php?attachmentid=69972&d=1400765791.png


    I want to find a solustion to the above equ. (4). Any one who is familiar with such equations could provide some references for me to read ? Thanks a lot.
     

    Attached Files:

  2. jcsd
  3. May 22, 2014 #2

    UltrafastPED

    User Avatar
    Science Advisor
    Gold Member

    Do you expect an analytic solution?

    You might determine some appropriate boundary conditions and carry out a series of numerical solutions; this will help you locate the most sensitive parameters, and guide you in some appropriate approximations which will simplify the equations.
     
  4. May 23, 2014 #3
    I just want to find a form for the solution. For example, in Cartesian coordiantes, the vibration equation is
    attachment.php?attachmentid=70014&d=1400853516.png

    the solution for the equations is
    attachment.php?attachmentid=70016&d=1400853785.png
    Then I can continue the discussions.
     

    Attached Files:

  5. May 24, 2014 #4
    I guess you want to say that there is one of the possibile solutions for the equations you wrote!
     
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