# Eigenfreuency of beam

How would you calculate the eigenfrequency of such a cantilever?

• The eigenfrequency of which mode are you interested in? The frequency of the fundamental oscillation in the up-down, left-right, back-front or any equivalent torque/rotational modes? And do we have to assume that the green plate is rigid? Dec 5, 2017 at 13:48
• Is the length $\gg$ than the width? If so we can assume both are beams and you can find the combined mass $\mathbf{M}$ and stiffness matrix $\mathbf{K}$ to form the eigenvalue problem $$\left(\mathbf{M}^{-1}\mathbf{K}-\omega^2 1\right) \mathbf{X} = 0$$ Dec 5, 2017 at 18:01
• @ja72 Yes the width is longer than the length. Then how do I solve this equation for omega ? Dec 6, 2017 at 13:57
• Are you interested in the twisting mode? Dec 6, 2017 at 15:45
• See "DYNAMIC ANALYSIS OF COMPOSITE MEMBERS WITH INTERLAYER SLIP" by Girhammer and Pan, International J. Solids and Structures, 30(6), 1993, p. 797-823 for the equations and some solutions. That approach works if you assume that the members can be approximated as beams. Things get more complicated if these are plates. Dec 6, 2017 at 22:13

Are you looking for a single number for one particular beam, or a closed form solution for any possible combination of dimensions?

If the former, the easiest will be ANSYS (or Abaqus or NASTRAN or any other commercial FEA software). Model the two different beams as shell elements. With shell elements you can use a single layer for each sub-beam. With brick elements, you'll need at least 7 layers for each sub-beam, which means a longer computation time.

If the later, you'll probably want to pick up a graduate level textbook on the vibration of continuous systems. You'll need to set up and solve partial differential equations. It is probably easiest to set up two differential equations that are linked by a boundary condition that the motion at the interface is the same.

Material 0 would have to be modeled as a plate (https://en.wikipedia.org/wiki/Bending_of_plates). If $w1<<w0$,then Material 1 could potentially be modeled as an Euler-Bernoulli beam (https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory), which would be a little easier. But otherwise they'd both have to be plates.

Calculating the eigenfunctions in closed form would be pretty hard. Instead, if $h1<h0$ and $w1<w0$, you could probably use the Raleigh-Ritz method (https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Ritz_method) using the eigenfunctions of the Material 0 plate by itself as the basis functions.