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Unit 3 - Oscillation

Lab 3C: Damped Oscillations

UCLA Physics Department

University of California, Los Angeles

Department of Physics and Astronomy

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. This is called the underdamped regime of the oscillator

Damping term

Differential equation for SHM

Differential equation for damped oscillations

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Underdamped Regime

In the underdamped regime, the oscillator’s position as a function of time has the form:

This is similar to the simple oscillator scenario, except that the amplitude of the sinusoid decays with time

In addition, we have a new parameter for our function given by 𝛕 called 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 are created in metals when a magnet moves past the surface of the metal

These currents create damping by�creating opposing magnetic fields,�which push back against the magnet

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

Image source: Wikipedia - Eddy Currents

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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

Attach the magnetized mass to the spring

Your Arduino will be attached as shown underneath the metal tubing

Weigh your Arduino setup with the magnetized mass

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Experimental Setup

Remove the Arduino from the spring and replace it with a few masses to weigh down the spring

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

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. Since we will not be working with the accelerometer output in this lab, you don’t need to worry about calibration

Save your data to a text file and import this data into python. You can use this notebook to import and analyze your data

Plot your distance vs. time data

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Fitting a Damped Oscillator

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Fitting to a Damped Harmonic Oscillator

Since our damped harmonic oscillator has a different form from the simple harmonic oscillator, we have to define a new function with an additional fit parameter

We can input our data into curve_fit like we did with the simple harmonic oscillator, but we must first obtain a good guess for the lifetime

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Guessing the Lifetime

To guess the lifetime, we need to 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)

Subtract the offset from these peaks and divide them

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Guessing the Lifetime

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

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

Using curve_fit, fit your position vs. time data to a damped harmonic oscillator. Be sure to include your lifetime guess in the right position in the array according to your fit function parameters

Plot your distance vs. time with the best-fit. Your fit should match your data well

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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. For a mass spring system, the potential energy is the elastic potential energy.

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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offset/mean value

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Energy of a Harmonic Oscillator

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All energy is in potential energy

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

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 with your potential energy, kinetic energy, and 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. Be sure to include a title, axis labels, the best-fit, and a legend

Plot your energies vs. time (kinetic, potential, and total). Be sure to include a title, axis labels, the best-fit, and a legend

Describe the shape of your total energy over time. Explain whether this result makes sense based on the model for the damped harmonic oscillator