Experiment 2: Damped Simple Harmonic Motion.

Brief introduction:

 

             The simple harmonic oscillator is one of the central problems in physics. It is useful in understanding springs, small amplitude pendulums, electronic circuits, quantum mechanics, and even cars that shake at 53 MPH. Furthermore, many problems can be considered the sum of a large number, or infinite number, of harmonic oscillators. (In a later experiment of coupled harmonic oscillator you will investigate the behavior of coupled oscillators)

 

             Almost everyone has an intuitive understanding of the playground swing, and so it is a good first example. If the person in the swing is neither “pumping” nor being pushed, and if frictional losses are small, one has a simple harmonic oscillator, at least for small amplitudes. If the rider drags his or her feet then there is damping.

 

 

The Damped Harmonic Oscillator:

 

If the damping force, f_D  , is proportional to the velocity, v, with a damping constant, b, then

 

                                                 f_D= -bv

 

The equation of motion for this system is:

 

                                                md²x/dt² + b dx/dt  + kx = 0            

 

 

There are three cases depending on the degree of damping discussed below.

 

 

 

 

Case I.   If b is small enough such that (b/2m) < √((k/m), then this is called under damped case.

 

The equation becomes:

 

x= A e^-(b/2m)t cos (ω' t + φ) 

 

Where, ω' =√((k/m) - (b/2m)² )

         Or, ω' =√(ω₀² - d² ) with d = b/2m, called damping factor.

 

 

 

Case II.   If (b/2m) = √((k/m), then it is critically damped case.

 

The equation becomes:

 

x =A(1+ω₀t)*e^-ω₀^t

 

 

 

Case III.   If (b/2m) > √((k/m), it is called over damped case.

 

The equation becomes:

 

x = [(e^ωt ₊ e^-ωt)+(d/ω)*(e^ωt - e^-ωt)]/2

where ω =√(d²-ω₀² )

 

 

 

 

 

 

 

 

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Experiments can be done using this virtual experiments:

 

· Under damped case

 

1. Set a value of damping factor and measure the time period of oscillation and hence ω'.

2. Calculate the ω' from ω₀ and damping factor d.

3. Taly the calculated ω' with the measured value.

4. Repeat this and calculate ststistical error.

 

 

Questions:

 

Under construction

 

 

 

Please feel free to send your feedback, suggestions or queries regarding the experiment to: oscillations.vlab@gmail.com

In your email, please mention the experiment no. and name of the experiment.

 

 

 

 

 

 

 

 

 

 

 

For detail theory, follow the link below:

Developed and maintained by: Satyajit Banerjee, Pabitra Mandal and Gorky Shaw

Click on the links below to download the video instructions for running the programmes:

 

1. Underdamped Simple Harmonic Motion

2. Critically Damped Simple Harmonic Motion

3. Overdamped Simple Harmonic Motion

There are three programmes for three different damping conditions (see ‘Brief introduction’ section) for the Damped case. To run the programs on your own PC offline, click on the following links below to download them (recommended):

 

1. Underdamped Simple Harmonic Motion

2. Critically Damped Simple Harmonic Motion

3. Overdamped Simple Harmonic Motion

Procedure to download and run the programs offline (follow the same procedure for all three):

 

· To download the programs, right click on the links above and choose ‘save target as’, or, ‘save link as’ depending on the browser.

· Save the ‘.zip’ file to any directory on your PC.

· Extract the ALL contents of the ‘.zip’ file to the SAME folder.

· Double click on the file with extension ‘.exe’, for example, “underdamped_SHM.exe”, or “criticallydamped_SHM.exe” or “Overdamped_SHM.exe” to execute the program.

· Now, perform the experiment as instructed in the video demonstration provided in the link below: