- 34

- 0

**1. Homework Statement**

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. Homework 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?