Equation of motion for the mass-spring system using the exact solution and the Abaqus software

Product Overview

In this example, we intend to obtain the equation of motion for the mass-spring system using both the exact analytical solution and the Abaqus software.

The steel bar with a length of 1 meter is hinged about point O. Two springs are connected to both ends of the bar, and the bar can only rotate about point O. The whole bar starts rotating with an initial angular velocity of 10 rad/s.

Mass-Spring System

The spring stiffness of the left spring is 3k, while the spring stiffness of the right spring is k. These springs provide the restoring forces required for the rotational motion of the bar.

The dynamic response of the system is investigated using the Abaqus software and is compared with the corresponding results obtained from the exact analytical solution.

Equation of Motion

The equation of motion of the mass-spring system is derived based on the mechanical and dynamic characteristics of the system.

The exact analytical solution is used to obtain the governing equation of motion and the corresponding dynamic characteristics. The same system is then modeled in Abaqus to obtain its numerical response.

Exact Analytical Solution

The system is solved analytically to obtain the equation of motion, period, and natural angular frequency. These analytical results are used as reference values for evaluating the Abaqus simulation.

Abaqus Simulation

The mass-spring system is modeled in Abaqus and its dynamic response is obtained by applying the initial angular velocity to the steel bar.

The Abaqus simulation is used to obtain the rotational response and determine the dynamic characteristics of the system.

Calculation of the Period

The period is the duration of one complete cycle of oscillation. It can be determined from the Abaqus results by measuring the time interval between two consecutive peaks of the response.

Natural Angular Frequency

The natural angular frequency of the system is calculated using the exact analytical solution and is also obtained from the Abaqus analysis.

The analytical and numerical results are then compared to evaluate the accuracy of the Abaqus model.

Analytical Validation

The problem is solved independently using the exact analytical solution and the Abaqus software. The results obtained from Abaqus are then compared with the analytical results, including the equation of motion, oscillation period, and natural angular frequency.

The comparison between the numerical and analytical results provides a verification of the Abaqus model and demonstrates that the system has been correctly modeled.

Results

The dynamic response obtained from Abaqus is compared with the corresponding results from the exact analytical solution.

The comparison focuses on the equation of motion, oscillation period, and natural angular frequency of the mass-spring system.

What You Will Learn

• How to model a mass-spring system in Abaqus.
• How to model a hinged steel bar with rotational motion.
• How to define springs with different stiffnesses.
• How to apply an initial angular velocity in Abaqus.
• How to derive the equation of motion analytically.
• How to obtain the oscillation period from Abaqus results.
• How to calculate the natural angular frequency.
• How to compare Abaqus results with an exact analytical solution.
• How to verify an Abaqus model using an analytical solution.

Key Features

• Mass-spring system analysis in Abaqus.
• 1-meter steel bar.
• Hinged rotational system.
• Two springs with stiffnesses 3k and k.
• Initial angular velocity of 10 rad/s.
• Equation of motion calculation.
• Exact analytical solution.
• Oscillation period calculation.
• Natural angular frequency calculation.
• Comparison between Abaqus and analytical results.
• Analytical verification of the numerical model.