Single Degree of Freedom (SDOF) Dynamic System

A single degree of freedom (SDOF) system is the simplest model which may be used to idealize a complex engineering dynamic system. One of such examples is a structure resting on base isolators.

The idealization uses mass (lumped mass), stiffness (spring) and damping (dashpot) components to characterize the response of system.

Depending on the consideration, we may have either an undamped or damped system.

If a dynamic system is subjected to only initial displacement or velocity but without any external forces, then the system is under free vibration.

On the other hand, when force is exerted to the system, then the system would be under a forced vibration.

Using SDOF model, we can derive several important parameters for dynamic study. Most notable ones are natural frequency, damping ratio and critical damping.

Here we encounter a single degree of freedom (SDOF) system with the mass of 5-ton, stiffness of 32.15kN/m and damping ratio of 5%.
From these properties, we know this is a damped system which subjected to underdamping. Apart from that, with the absence of external force and presence of initial displacement, the system will undergo free vibration.
After we have categorized the system, we can identify suitable equations to solve for its dynamic responses i.e. displacement, velocity and acceleration over time.
First, we need to calculate the natural frequency and angular frequency of this damped system. With these parameters, we can simplify our equations.
After that, we just substitute the value of time instance, corresponding displacement and velocity into the equations, and solve for the dynamic responses.
Lastly, we can plot the graphs of responses against time for the time period we are interested in.

Download your spreadsheet here

We are now facing a single degree of freedom (SDOF) system with the mass of 5-ton, stiffness of 32.15kN/m and damping ratio of 5%. The system is subjected to a dynamic load with magnitude of 2kN. This loading is a sine function of time.

From these properties, we know this is a damped system which subjected to underdamping. The system will undergo forced vibration as external load is anticipated.

From categorization, we successfully identify the equations that can help us to solve for the dynamic responses i.e. displacement, velocity and acceleration over time.

First, we need to calculate several parameters, namely natural frequency, angular frequency of damped system, frequency ratio and theta. Then, we may proceed with simplifying the equations.

First, we solve for transient responses of the system. We substitute the value of time instance and solve for displacement. Subsequently, we provide necessary parameters and solve for the remaining dynamic responses.

Next, we solve for the steady-state responses of the SDOF system.

After both sets of responses are obtained, we can simply sum them up and determine the total dynamic responses of the system.

Lastly, we can plot the graphs of responses against time for the time period we are interested in.

Download your spreadsheet here