# Node Points on a Beam?

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

$$\upsilon(x,t) = \sum \phi(x)*\zeta(t)$$
ODE:
$$\phi''(x) - \omega\phi(x) = F$$

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

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

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?

## Answers and Replies

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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.

Redbelly98
Staff Emeritus
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.