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?