Natural frequencies of two degree of freedom system using the Abaqus software and Mechanical Vibrations book

Project Overview

In this Abaqus tutorial, the natural frequencies of a two-degree-of-freedom system are obtained using the Abaqus software and compared with the results presented in the Mechanical Vibrations book by Rao.

The main objective of this example is to model the two-degree-of-freedom mass-spring system in Abaqus, obtain its natural frequencies, and compare the numerical results with the corresponding results reported in the Mechanical Vibrations book.

Two-Degree-of-Freedom System

In this problem, the system consists of two masses and springs. The parameters of the system are defined as follows: m₁ = 10 kg, m₂ = 1 kg, k₁ = 30 N/m, k₂ = 5 N/m, and k₃ = 0.

The damping coefficients c₁, c₂, and c₃ are all equal to zero. Therefore, there is no damping in the system, and the model consists only of masses and springs.

Mechanical Vibrations Book

This problem has been selected from the Mechanical Vibrations book by S. S. Rao.

The results presented in the book are used as a reference for evaluating the results obtained from the Abaqus simulation.

Analytical Solution

The natural frequencies of the two-degree-of-freedom system are first determined using the analytical solution. The system properties, including the masses and spring stiffnesses, are used to determine the two natural frequencies of the system.

Based on the analytical solution, the squared natural frequencies are obtained as ω₁² = 2.5 and ω₂² = 6.

These analytical results are used as reference values for comparison with the natural frequencies obtained from the Abaqus analysis.

Abaqus Simulation

The two-degree-of-freedom system is modeled in the Abaqus software, and the natural frequencies of the system are obtained from the numerical analysis.

The Abaqus results are then compared with the corresponding results reported in the Mechanical Vibrations book.

Comparison of Results

The natural frequencies obtained from the Abaqus software are compared with the results presented in the Mechanical Vibrations book.

As observed from the results, the results obtained from the Abaqus software are exactly concordant with those presented in the Mechanical Vibrations book.

What You Will Learn

• How to model a two-degree-of-freedom system in Abaqus.
• How to define masses and springs in Abaqus.
• How to model a system without damping.
• How to obtain the natural frequencies of a two-degree-of-freedom system.
• How to compare Abaqus results with results from a Mechanical Vibrations textbook.
• How to verify a numerical vibration model using reference results.

Key Features

• Two-degree-of-freedom mass-spring system.
• Natural frequency analysis.
• No damping in the system.
• Abaqus numerical simulation.
• Comparison with the Mechanical Vibrations book by Rao.
• Results comparison between Abaqus and the reference solution.
• Free Abaqus tutorial.