# Damped Simple Harmonic Oscillator and the Driven Damped Harmonic

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```					Eastern Nazarene College                                                   PY 312 Applied Mechanics

Damped Simple Harmonic Oscillator and the
Driven Damped Harmonic Oscillator
This is a two week lab. We will do the Harmonic Oscillator and Damped Harmonic Oscillator parts in
the first week and the Driven Harmonic Oscillator part in the second week.

Harmonic Oscillator and Spring Constant
1. Read Appendix A: Experimental Uncertainty and Appendix B: Propagation of Experimental
Uncertainties. You can find these on the General Physics course web site under Resources.
2. Get two springs from the instructor and measure the spring constant of both springs. Make sure you
do not stretch the spring beyond 2 or 3 times the unstretched length. We would like to measure the
spring constant as accurately as possible so think about different ways to make the measurements
and how you might reduce the systematic and random uncertainties.
3. Read Appendix C: Plotting and Interpreting Graphs.
4. Plot the data (using Excel, Maple, MATLAB, or DataStudio) and use a linear least-square fit to find
the spring constant. Use a program that will give you the uncertainty for the slope and intercept.
Your uncertainty should be only a few percent.
5. Calculate the effective spring constant for two springs connected in series. See problem 3.?? in your
textbook. Again calculate the uncertainty for the effective spring constant.
6. Connect the two springs in series and measure the effective spring constant for the two connected in
series.
7. We will use these same spring in the next section and in next week’s lab so get a baggie from your
lab instructor and use a marker to identify your springs.

Damped Harmonic Oscillator
Underdamped
1. Set up a Pasco track and cart with your two springs holding the cart near the center of the track. Do
not use a cart with a plunger on the end. The two springs should be stretched about 2 or 3 times their
unstretched lengths. Set up a sonic motion detector to measure the displacement of the cart. You
may want to use a small cardboard square to reflect the sonic signal. Make sure it is not so large as
to cause damping.
2. Displace the cart about 15 cm and release the cart. Record the motion of the cart until it comes to
rest with as little damping as possible. You may want to clean the track grooves and check the cart
wheels. You should get 15 to 20 oscillations without magnetic damping.
3. Use your recorded data and any other measurements needed to calculate the spring constant and
compare it to the values found in procedure 1. You should have an uncertainty or root-mean-
squared value for your spring constant so that you can make a statement if they are equal or not.
Procedure 1 was a static measurement whereas the measurement you just made was a dynamical
one. They are not the same, but the difference is very small.
4. Next we will add damping to the oscillator (cart). It will be convenient to use magnetic damping
because it is much easier to control. The magnetic damping device is shown below and is placed on
the front of the cart using the magnets on the cart to hold it on. You can increase or decrease the
damping by moving the device up or down so it is closer or farther away from the track.

5. Again displace the cart about 15 cm and release the cart. Record the motion until the cart comes to
rest. Adjust the magnetic damping so that you get about 10 oscillations. Do this for three or four
different damping parameters. The largest damping parameter should give about 3 oscillations.

6. Analyze the recorded motion in procedure 5 to find the damping parameter for each case. Use
DataStudio, Maple, or MATLAB to do a nonlinear fit to the motion data to find the damping
parameter. Calculate the uncertainty.

Overdamped
8. Now increase the damping so that the oscillator (cart) is overdamped and record the motion for three
or four different damping parameters.
9. Analyze the motion to find the damping parameter.

Critically Damped
10. Can you find the critically damped case? It should be the motion with the shortest settling time.
Calculate the critical value for the damping parameter in terms of the natural frequency using the
measured spring constant from the first section. Does this value fall between the values you used for
the under- and over-damped parts?
11. Play around with the damping parameter between these two values and find the critical damping
parameter by find the shortest decay time.

Driven Damped Harmonic Oscillator
Underdamped (the only case we will study)
1. Use the same set-up but add the motor to drive the spring-cart system. Adjust the bracket that holds
the string to the spring so that it goes in a circle about 1 cm in diameter. Add damping so that the
motion damps out in about 10 to 15 oscillations with no drive. Calculate the resonance frequency
(approximately) and turn on the motor and adjust the voltage until the drive frequency is about one-
half the resonance frequency. After 20 to 30 oscillations the motion should settle down into the
2. For this section of the lab we want to record the position of the cart and the position of the drive.
Think of a way to measure the position of the drive as a function of time. We will get the phase
difference between the drive and cart from this data.
3. Now we want to record the motion of the cart and drive as the transient dies out and steady-state
grows. The way to do this is to hold the card steady at its equilibrium position, start the motor, and
release the cart. After about 20 or 30 oscillations start recording the motion of the cart and drive
with Science Workshop. Record until the transient dies out and you can see the steady-state for 10
or 15 oscillations.
4. Repeat procedures 1 through 3 for eight or 10 different drive frequencies (some above the resonance
frequency and some below).

Resonance Curve
5. Next we would like to plot the resonance curve. From the recorded data above find the amplitude
and frequency and plot a resonance curve.
6. Next use Science Workshop to do a nonlinear least-squared fit of the amplitude equation to the data
and see if the fitting parameters agree with your measurements.
7. Repeat procedures 1 through 6 for a different damping constant.

Phase Plot
8. Finally we would like to plot phase difference verses frequency. From the data above plot the phase
difference verses frequency for the two different damping constants.

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