First Longitudinal Natural Frequency of a Cantilever Beam in Abaqus

Project Overview

In this training, we first calculate the first longitudinal natural frequency of a cantilever beam using the Abaqus software. Then, the result is compared with the Analytical Solution based on axial vibration theory to evaluate the accuracy of the numerical results.

Model Description

The model consists of a steel cantilever beam with a length of 0.8 m, a width of 0.2 m, and a thickness of 0.02 m.

The material properties are defined as follows: Young's modulus is 200 GPa, Poisson's ratio is 0.3, and the material density is 7850 kg/m³.

One end of the beam is fixed, while the other end is free to vibrate. The model is used to determine the first longitudinal natural frequency and to verify the numerical results obtained from the Abaqus simulation.

Abaqus Simulation

First, the cantilever beam is simulated using the Abaqus software to obtain its first longitudinal natural frequency.

A frequency analysis is performed to calculate the natural frequencies and corresponding mode shapes. The longitudinal mode is identified by examining the deformation pattern along the length of the beam.

The longitudinal natural frequency obtained from the Abaqus software is equal to 1582.1 Hertz.

Identification of the First Longitudinal Mode

In the Abaqus results, the longitudinal mode is identified from its mode shape. In this mode, the beam vibrates primarily along its longitudinal axis, while the fixed end remains constrained.

The mode shape helps identify longitudinal vibration and distinguish it from bending and torsional vibration modes.

Analytical Solution

Next, the first longitudinal natural frequency of the cantilever beam is calculated using the Analytical Solution based on axial vibration theory.

For a uniform bar with one end fixed and the other end free, the first longitudinal natural frequency can be calculated using the following equation:

f₁ = (1 / 4L) √(E / ρ)

In this equation, E is Young's modulus, ρ is the material density, and L is the length of the bar.

The first longitudinal natural frequency obtained from the Analytical Solution is approximately 1577.3 Hertz.

Comparison of Results

The results obtained from the Abaqus software are compared with those obtained from the Analytical Solution based on axial vibration theory.

The first longitudinal natural frequency obtained from Abaqus is 1582.1 Hz, while the value obtained from the Analytical Solution is 1577.3 Hz.

The difference between the Abaqus result and the Analytical Solution can be calculated once the numerical result is entered.

What You Will Learn

• How to model a cantilever beam in Abaqus.
• How to define material properties and density in Abaqus.
• How to apply boundary conditions to a cantilever beam.
• How to perform a natural frequency analysis.
• How to calculate the first longitudinal natural frequency using axial vibration theory.
• How to obtain natural frequencies and mode shapes using Abaqus.
• How to identify the first longitudinal mode from the mode shape.
• How to compare Abaqus results with the Analytical Solution.
• How to evaluate the accuracy of a numerical vibration model.

Key Features

• Longitudinal vibration analysis of a cantilever beam.
• Three-dimensional solid finite element model.
• Steel material properties.
• First longitudinal natural frequency calculation.
• Analytical Solution based on axial vibration theory.
• Abaqus numerical simulation.
• Identification of the first longitudinal mode shape.
• Comparison between Abaqus and Analytical Solution.
• Free Abaqus tutorial.