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Homework Help: Node Points on a Beam?

  1. Apr 17, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the node points (zero displacement) for a clamped-pinned and clamped-spring beam.
    I am trying to figure out how to find these points on a beam of length L (constant E*I,m) for the first 3 natural frequencies.

    2. Relevant equations
    3. The attempt at a solution

    [tex]\upsilon(x,t) = \sum \phi(x)*\zeta(t)[/tex]
    ODE:
    [tex]\phi''(x) - \omega\phi(x) = F[/tex]

    General solution:
    [tex]\phi(x) = A*sinh(\alpha*x) + B*cos(\alpha*x) - C*sin(\alpha*x) - D*cosh*(\alpha*x)[/tex]

    A,B,C,D to be found using boundary conditions (in this case clamped-pinned):
    [tex]\phi(0)=0[/tex]
    [tex]\phi'(0)=0[/tex]
    [tex]\phi(L)=0[/tex]
    [tex]\phi''(L)=0[/tex]


    My question is: the mode shape can be found by solving for the constants, etc. But how can you analytically find the zeros for this? The solution is non-linear. Is there another method I can use to find the node points?
     
  2. jcsd
  3. Apr 20, 2008 #2
    Anyone there??
    One thought I had was to use Newton's method but I've tried this and the radius of convergence seems to be too small for many of my cases.
     
  4. Apr 20, 2008 #3

    Redbelly98

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    Staff Emeritus
    Science Advisor
    Homework Helper

    Unless A=D=0, I'm pretty sure you need to solve numerically.
    An analytic solution would be possible when A=D=0, since it's just trig terms then.
     
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