Unit 3 - Oscillation

Lab 3C: Damped Oscillations

UCLA Physics Department

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University of California, Los Angeles

Department of Physics and Astronomy

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Physics 4AL

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Outline of Lab 3C

• 3C In-Lab
• Overview of damped oscillations
• Obtain data for a damped harmonic oscillator using the Arduino
• Fit this data to the model for an underdamped harmonic oscillator
• Analyze the energy of the oscillator over time

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Damped Oscillations

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Damped Oscillations

• While our mass-spring system worked well for a simple harmonic oscillator, many systems involve a form of damping
• Oscillating systems with damping
• Can oscillate within a decaying “envelope” that bounds these oscillations.
• Underdamped regime of the oscillator

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Damping term

Differential equation for SHM

Differential equation for damped oscillations

Underdamped Regime

• The oscillator’s position (a function of time):

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• Similar to the simple oscillator scenario
• The amplitude of the sinusoid decays with time

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• A new parameter: 𝛕 (the lifetime of the decay)

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Damping with Eddy Currents

• In this lab, to create damping, we will use eddy currents
• Eddy currents
• Created in metals when a magnet moves past the metal surface
• Currents create damping by�creating opposing magnetic fields,�which push back against the magnet

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• Energy is transferred out of the motion�and into heat from the current

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Image source: Wikipedia - Eddy Currents

Experimental Setup

• Use the same Arduino setup as from lab 3A
• String up the Arduino like you did in lab 3A
• We will be working with the same springs.
• Make sure that you don’t stretch them out too far, or they WILL break!

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Experimental Setup

1. Weigh your Arduino setup with the magnetized mass
2. Attach the magnetized mass to the spring
3. Your Arduino will be attached as shown underneath the metal tubing

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Step o

Experimental Setup

4. Remove the Arduino from the spring and replace it with a few masses to weigh down the spring (so when you hang your arduino, there will be room for the arduino to oscillate)

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5. Add the metal tube by sliding it up from below and screwing it into the holder

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6. Attach the Arduino to the spring and remove the masses. Ensure that the magnetized mass and the string don’t directly touch the metal tube

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TA Checkpoint 1 - Damped Oscillation Data

• Pull down your Arduino setup by a few centimeters and release
• Collect data for 10-15 periods of oscillation.
• No Accelerometer used: No calibration needed but can still have it on the Arduino.
• Save your data to a text file and import this data into python. You can use the notebook in Unit 3 to import and analyze your data
• Physics_4AL_2023-24/Unit 3/Unit 3 - Lab 3C.ipynb
• Plot your distance vs. time data

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Fitting a Damped Oscillator

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Fitting to a Damped Harmonic Oscillator

• Recall how to fit data to a function:
• scipy.curve_fit
• Define a function to fit to (new function!)
• A function has new parameters
• Guess parameters

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• Refer to Lab 3B

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Guessing the Lifetime

• Guessing the lifetime ()
• Analyze the envelope of the amplitude. If we select just the peaks of the sinusoid, we can get points for our guesses
• Guess the heights of the first two peaks, y(t₀) and y(t₁)

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• Subtract the offset from these peaks and divide them from each other

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Guessing the Lifetime

• By doing this, we eliminate the component of the sinusoid, the amplitude, and the offset

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• We can then take the natural log of this expression and divide appropriately to obtain the lifetime:

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TA Checkpoint 2 - Damped Oscillation Fit

• Obtain a good guess for the lifetime of the oscillator

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• Fit your position vs. time data to a damped harmonic oscillator.
• Things to note: Including good guess parameters
• Things to note: Order of parameters in the guesses and in the fitted coefficients

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• Plot your distance vs. time with the best-fit.
• Your fit should match your data well!
• If not a good fit, what would you tweak? (Refer to Lab 3B)

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Energy of Harmonic Oscillations

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Energy of a Harmonic Oscillator

• In a harmonic oscillator, the energy switches between kinetic energy and potential energy.
• What is the potential energy for mass-spring system?

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offset/mean value

Potential energy: elastic potential energy of the spring.

x₀ (the equilibrium position) can be found by finding the mean of displacement which is the offset in your best fit parameters

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Energy of a Harmonic Oscillator

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All energy is in potential energy

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Spring is unstretched, all energy is in kinetic energy.

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TA Checkpoint 3 - Energy vs. Time Plots

• Using the spring constant you found in lab 3A, determine the potential energy of your system from your position data

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• Use theoretical differentiation by taking the derivative of the position fit function to obtain your velocity. Determine the kinetic energy of your system from your velocity data
• Create a plot that includes 1. your potential energy, 2. kinetic energy, 3. total energy (potential + kinetic) as a function of time. Be sure to include a title, axis labels, and a legend

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Post-Lab Requirements for lab 3C

• Plot your distance vs. time for the damped oscillator.
• Plot your energies vs. time (kinetic, potential, and total).
• Describe the shape of your total energy over time. Explain whether this result makes sense based on the model for the damped harmonic oscillator

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