Theoritical Model Analysis of a Simply Supported Beam

The theoretical analysis of a simply supported beam is necessary to find out the exact values of the natural frequencies. The exact frequencies would be used further to validate the results obtained by the analysis software ANSYS©.

As cross sectional dimensions of our beam are small as compared to its length, it can be treated as Euler Bernoulli Beam [10]. The reference figure for the ANSYS© is shown below.

Continuous beam Analysis

Let
A = Cross-sectional area of the beam
E = Modulus of elasticity of the beam material
ρ = Density

Consider an element dx of the beam subjected to shear force Q and bending moment M.

Assumptions
1. No axial forces are acting on the beam.
2. Effects of shear deflection are neglected.
3. The deformation of the beam is assumed due to moment and shear force.