Two-Layer Cantilever  Beam with One End Fixed and One End Free

System Overview:

You are analyzing a composite beam with the following configuration:

• Beam Configuration: One end fixed, one end free.
  
  

• Material Layers:
  
  

  • Bottom Layer: Polycrystalline Silicon
    
    

  • Top Layer: Zinc Oxide
    
    

Analysis Type:

This analysis is focused on the natural frequency (Eigenfrequency) analysis of a composite beam. The purpose is to calculate the beam's natural frequencies and mode shapes under the given boundary conditions.

Tool Used:

The analysis is performed using COMSOL Multiphysics, specifically version 5.2a, which provides the Structural Mechanics Module for analyzing vibrations and natural frequencies in beams.

Analysis Objective:

The primary goal is to determine the natural frequencies of the beam (how it vibrates) for different mode shapes and to understand how the two-layer structure (Polycrystalline Silicon at the bottom and Zinc Oxide at the top) influences the vibrational behavior.

The beam is modeled with the boundary conditions:

• One end fixed: At this point, both displacement and rotation are zero.
  
  

• One end free: At this point, the beam is free to vibrate.
  
  

Natural Frequency Results:

The natural frequencies you obtained from the COMSOL model are:

1. 16245 Hz
   
   

2. 82584 Hz
   
   

3. 96466 Hz
   
   

4. 100,030 Hz (1.0003E5 Hz)
   
   

5. 259,020 Hz (2.5902E5 Hz)
   
   

6. 237,300 Hz (2.373E5 Hz)
   
   

These frequencies represent the vibration modes of the beam, where the first frequency corresponds to the fundamental mode, and the higher frequencies correspond to higher-order vibration modes.

Theoretical Background:

For beams with one fixed end and one free end, the natural frequencies can be approximated using the Euler-Bernoulli beam theory, which is based on the beam’s material properties (such as Young’s Modulus, density, and Poisson’s ratio) and geometry (length, cross-sectional area, and moment of inertia).

For a fixed-free beam, the natural frequencies can be estimated by the formula given below

Where:

• fn is the natural frequency,
  
  

• L  is the length of the beam,
  
  

• E is the Young’s Modulus,
  
  

• I is the second moment of area (area moment of inertia),
  
  

• ρ  is the density of the material,
  
  

• A is the cross-sectional area of the beam,
  
  

• αn ​ are the mode shape constants for a fixed-free beam.
  
  

For a fixed-free beam, the first few values of αn\alpha_nαn​ are approximately:

• α1≈1.875 (for the first mode),
  
  

• α2≈4.694 (for the second mode),
  
  

• α3​≈7.855 (for the third mode), and so on.
  
  

COMSOL Multiphysics Setup:

In COMSOL, you would set up the model by defining:

• Boundary Conditions: One end fixed (displacement and rotation are zero) and one end free.
  
  

• Material Properties: Polycrystalline Silicon for the bottom layer and Zinc Oxide for the top layer.
  
  

• Geometry: Define the beam’s length and cross-sectional shape.
  
  

• Meshing: Apply a mesh to accurately capture the vibrational modes.
  
  

The Eigenfrequency Analysis in COMSOL will compute the natural frequencies based on the beam's material properties, geometry, and boundary conditions.

Summary:

• Name of the system: Two-Layer Composite Beam with One End Fixed and One End Free.
  
  

• Material Properties: Polycrystalline Silicon (bottom layer), Zinc Oxide (top layer).
  
  

• Analysis Type: Natural Frequency (Eigenfrequency) Analysis.
  
  

• Tool Used: COMSOL Multiphysics 5.2a.
  
  

The results you obtained (frequencies: 16245 Hz, 82584 Hz, 96466 Hz, 100,030 Hz, 259,020 Hz, and 237,300 Hz) represent the vibrational modes of the composite beam. These can be compared with theoretical estimates based on the beam's material and geometric properties.