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.

Model Analysis and Convergence Study in ANSYS

A host structure such as a cantilever or simply supported beam and its surface bonded with piezoelectric patches acting as sensor and actuator is called as an ‘Integrated Structure’. The first step in understanding the working of smart structures is to analyze the host structure using finite element analysis. The smart structure can then be controlled accordingly.

Finite Element Modal is created in ANSYS© using the various tools present. Size of the beam selected is 500x25.4x0.8 mm.

Meshed Model of beam

The ends of the beam are constrained in the UZ direction. Modal Analysis is performed in ANSYS© using the Block Lanczos Method. The number of modes of vibration to be extracted and the frequency range is specified. To determine the mesh size of the beam it is important to perform a convergence study. The first three natural frequencies of the vibration of the beam are calculated for the various mesh sizes by performing modal analysis and are compared with the theoretical values as given below:

In order to compromise between accuracy and computational time the mesh size of (60*8*1) is selected whose natural frequency is found to be 7.155 Hz which is very close to 7.151 Hz.