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

Product Overview

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

The steel bar has a length of 1 meter and is hinged about point O. Two springs are connected to both ends of the bar, while a damper is attached to the middle of the bar. The bar can only rotate about point O.

The whole bar starts rotating with an initial angular velocity of 10 rad/s.

Mass-Spring-Damper System

The mechanical system consists of a rigidly connected steel bar, two springs attached to its ends, and a damper connected to the middle of the bar.

The bar is hinged at point O and is allowed to rotate about this point. An initial angular velocity of 10 rad/s is applied to initiate the motion.

Equation of Motion

The equation of motion of the system is derived based on the mechanical properties of the bar, springs, and damper.

The analytical formulation is used to determine the dynamic response of the system and to calculate its period and natural angular frequency.

Exact Analytical Solution

The mass-spring-damper system is solved using an exact analytical solution. The analytical solution provides a reference for evaluating the numerical results obtained from Abaqus.

The equation of motion, oscillation period, and natural angular frequency are obtained from the analytical formulation.

Abaqus Simulation

The same mass-spring-damper system is modeled in Abaqus to obtain its numerical dynamic response.

The springs and damper are defined according to the properties of the analytical model, and the initial angular velocity is applied to the steel bar.

Calculation of the Period

The oscillation period is obtained from the dynamic response of the system. The numerical period is determined from the Abaqus results and compared with the value obtained from the exact analytical solution.

Natural Angular Frequency

The natural angular frequency of the system is calculated using the analytical solution and is also evaluated from the Abaqus results.

The numerical and analytical values are compared to assess the accuracy of the Abaqus model.

Equation of Motion from Abaqus

The equation of motion of the mass-spring system is obtained from the Abaqus analysis. The resulting equation is then used to describe the rotational response of the hinged steel bar.

Results

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

The comparison focuses on the equation of motion, oscillation period, and natural angular frequency. Good agreement between the two approaches verifies the accuracy of the Abaqus model.

The comparison demonstrates good agreement between the numerical and analytical solutions and confirms that the mass-spring-damper system has been correctly modeled in Abaqus.

What You Will Learn

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

Key Features

• Mass-spring-damper system analysis in Abaqus.
• 1-meter steel bar.
• Hinged rotational system.
• Two springs connected to the ends of the bar.
• One damper connected to the middle of the bar.
• 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 validation of the numerical model.