Single Degree of Freedom of Vibration - Undamped and Damped

Statement of the problem

Free vibration is the most fundamental and simplest problem. Here we look at a simple mass-spring system to simulate such motion. As you may expect that displacement of the mass is so called ``harmonic motion'' which can be represented by trigonometry functions sin or cos.

When an viscous damper is added to a free vibration system, it results a decaying oscillation and the system would eventually come to rest. The viscous damper is a dissipation mechanism which is caused by the viscous fluid inside the dashpot. For most dashpots, the rate of extension is proportional to the applied force through a damping constant c. Here we compare the two systems.
 

Exercises

1. By moving the Slider or input a value in the box, set the damping ratio zeta=0.1, Run the program until the system almost stop vibrating then Stop it.
2.  Go to File and Export the Meter Data to a file. Using the Excel to plot the displacement and velocity v.s.. time, measure the time when system almost stop its vibration.

1. By moving the Slider or input a value in the box, set the damping ratioc=1.0, Run the program until the system stops vibrating then Stop it. Now what is the difference between this case and the case of under-damping?

 Go to File and Export the Meter Data to a file. Using the Excel to plot the displacement and velocity v.s. time, measure the time when system almost stop its vibration.

Think Twice

• Drawing `Free Body Diagram' for the undamped system and using the Newton's second law to build up the equations of motion. Then try to solve the equations of motion. Let's assume the initial position and velocity are  and zero respectively. The displacement x is accounted from the equilibrium position, i.e. the position where the spring is unstretched. Notice that we can define a constant call ``natural frequency'' as:

  

which represents the fundamental frequency of a vibrating system.
 Comparing the natural frequency with the frequency measured from your plot. How much is the error?

• Drawing 'Free Body Diagram' for the damped system and using the Newton's second law to set up the equations of motion for the case of damped vibration then try your best to solve the equations of motion. Let's assume the initial position and velocity are  and zeros respectively. The displacement x is accounted from the equilibrium position, i.e. the position where both the spring and damper are unstretched.
•  Can you distinguish the following three different cases:

   

What are differences in system response for these three cases?

•  For the case of damped vibration, the system is experienced a ``exponentially decaying'' process. The rate of decay is  . You can measure this rate from the plot of displacement v.s. time for the case of under-damping and critical damping respectively.