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Vibrations - Rayleigh-Ritz Method Admissible Functions

  1. Nov 16, 2014 #1
    Hi everyone,

    I'm having a bit of difficulty choosing an admissible function for a fixed-fixed nonuniform bar.

    I chose the function φ(x) = 1 - cos(2πx/L).

    But when solving for the the stiffness and mass coefficients:

    kij = ∫EA(x)φiφjdx

    mij = ∫ρ(x)φiφjdx,

    I am not sure where I should have the "i" and "j" in my function.

    In an example for a fixed-free beam, the function that was given was:

    φ(x) = sin(πx/2L), which was changed to φi(x) = sin[((2i-1)πx)/2L].

    Should I choose my function to be φi(x) = 1 - cos(2πix/L)? And why?
     
  2. jcsd
  3. Nov 16, 2014 #2
    Your choice for phi(x) defined only a single function. Your process requires a whole family of functions (i=1,2,3,...), each of which satisfy the boundary conditions. Try to use your original idea, but extend it to a family.
     
  4. Nov 16, 2014 #3
    Thank you for the quick response Dr. D!

    So in that case, could I choose a function (like before):

    φi(x) = 1 - cos(2πix/L), i = 1,2,3,...
    φj(x) = 1 - cos(2πjx/L), j = 1,2,3,...

    Where they would end up being

    φ1(x) = 1 - cos(2πx/L)
    φ2(x) = 1 - cos(4πx/L)
    φ3(x) = 1 - cos(6πx/L)...

    Thank you!
     
  5. Nov 16, 2014 #4
    I think you've got it!
     
  6. Nov 16, 2014 #5
    Great! Thank you so much!

    If I may ask one more slightly related question:

    If I am using the Assumed Modes method for N=4, for the stiffness and mass coefficients, do the i and j just need to add up to 4?

    For example, I could use:

    mij = m1,3
    or
    mij = m2,2

    I'm not sure if that question is clear, as the book did not explain how to get the number of terms. If this is the correct way, does is matter whether I use 1,3 or 2,2?
     
  7. Nov 16, 2014 #6
    I'm not familiar with your terminology, but N=4 suggests to me that i = 1,2,3,4 and j = 1,2,3,4, so that all the combinations are involved.
     
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