Eigenvalue Computation for a cantilever beam

Author: Philipp Bucher

Kratos version: 9.0

Source files: Beam Eigenvalues

Case Specification

This simple example shows how the eigenvalues (Eigenfrequencies and Eigenmodes) of a cantilever beam can be computed.

The properties of the beam can be seen in the Materials file

Here we are only interested in the first three bending modes, for which we can calculate the analytical solution with (source):

f_i = λ_i/(2 π L²) (EI/ρA)^1/2

with lambda for the first eigenfrequencies being:

• λ₁ = 1.875
• λ₂ = 4.694
• λ₃ = 7.855

Using the specified properties we obtain for the first three bending modes:

• f₁ = 0.5595 Hz
• f₂ = 3.5068 Hz
• f₃ = 9.82 Hz

The results obtained with Kratos are :

• f₁ = 0.5596 Hz
• f₂ = 3.5063 Hz
• f₃ = 9.81685 Hz

The corresponding mode-shapes are shown below:

First Bending Mode

Second Bending Mode

Third Bending Mode