Find Node Points on a Beam: Analytical Solution

[SEG9585]

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.

Homework Equations

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?

 

[SEG9585]

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]

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.