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# Validation Case: Modal Analysis of a Cantilever Beam

In this case, modal analysis is performed on a cantilever beam which is fixed at one end and free at the other. The aim of this FEA analysis is to find the first 5 natural frequencies of the beam.

The FEA model and the resulting comparison of the simulation results against the analytical calculations validates the use of following conditions for modal analysis in Solve:

• Restraint

## Geometry: • Geometry Dimensions
• Cross Sectional Area: 0.04 m by 0.04 m
• Length of Part (l): 1 m

### Material:

• Steel
• Density (ρ) = 7700 kg/m3
• Young’s Modulus (E) = 480 GPa
• Poisson’s Ratio (ν) = 1e-6

### Physics:

• Restraint (Fixture)
• Face1 ( ABCD) ## Meshing

OnScale Solve generated the mesh automatically and the meshing statistics, at the time of writing this document, are as follows:

Mesh Quality: Medium

## Analysis

The analysis type can be selected in the Simulator section.

Mechanical Analysis Type: Modal

Definition: Modes

Number of Modes: 5

## Reference Solution:

The natural frequency (in cycles per second) of a cantilever beam is given by: where Kn is the constant where n refers to the mode of vibration, g is the gravitational acceleration, E is the Young’s Modulus, I is the area moment of inertia, w is the weight of the beam and l is the length of the beam.

For this case these values are as follows:

g = 9.81 m/s2

E = 480 GPa

I = 2.1333e-7 m4

w = 120.8592 kgm/s2

l = 1 m

The mode vibration constants for the first five modes are given by:

 Mode Kn 1 3.51601527 2 22.0344915 3 61.6972144 4 120.901916 5 200

## Results Comparison

The table below compares the values obtained using the analytical methods described above and the FEA analysis performed using OnScale Solve.

 Outputs Analytical Method (Hz) OnScale Solve (Hz) Natural Frequency 1 (f1) 50.964 50.965 Natural Frequency 2 (f2) 317.406 317.476 Natural Frequency 3 (f3) 880.068 880.554 Natural Frequency 4 (f4) 1,286.704 1,311.113 Natural Frequency 5 (f5) 1,700.772 1,702.683 