by Jeff Anderson and Michael McCusker
Last updated: December 10, 2018

Introduction

Are you curious to learn how to use eigenvalues to solve applied problems in our observable world? Have you ever wanted to build your own eigenvalue problems to check if the theory you learn in math class is actually useful in practice? In this project, we present a learning activity that enables you to apply eigenvalue theory to investigate a useful modeling problem. We demonstrate how to build a spring-coupled pair of pendula and how students can analyze this system using eigenvalues. This activity is designed to enhance student motivation in the classroom and to prepare students to apply eigenvalues to real-world problems. To learn more, please visit the support website for the eigenvalue project. This math is very important for mechanical, civil, electrical, and aerospace engineers. In fact, the example we highlight above sets the foundation for so many fields that rely on vibrations analysis.

Welcome

Welcome to the companion website for the paper Make the eigenvalue problem resonate with our students. Before we begin, let’s get the basics out of the way. Interested readers can download a pdf copy of this article by clicking the link below:

Make the eigenvalue problem resonate with our students Accepted Manuscript (pdf)

This is the Accepted Manuscript of an article published online on June 14, 2018 by Taylor & Francis in the journal PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies. For more information, please note that this article is also referenced online here: https://doi.org/10.1080/10511970.2018.1484400. We owe a big thank you Taylor & Francis for their friendly policies that allow us to share our work in this form.

This webpage is designed to help college mathematics and engineering instructors enhance eigenvalue curriculum with an authentic modeling exercise. The goal of our work is to empower students create and solve their own eigenvalue problem within the context of a useful mathematical modeling paradigm. To achieve this goal, we show you how to design and assemble a spring-coupled pair of pendula. We also provide a modeling scheme that supports you in analyzing the dynamics of this system using eigenvalue theory. The entire apparatus and data acquisition system is easy-to-build, inexpensive, simple-to-assemble, safe, and of appropriate size for in-class demonstrations and student laboratory explorations. This support website includes links to online videos that introduce relevant curriculum, a laboratory project prompt that can be assigned to students, sample experiment videos that can be analyzed using image processing software, example spreadsheets that provide analysis for the modeling process, and many other resources to support you in applying eigenvalue theory to an authentic, real-world problem.

Our Pedagogical Approach

One major reason that most introductory linear algebra courses cover eigenavalue theory is that these tools can be extremely useful for transforming hard problems in differential equations into much easier problems in linear algebra. The entire field of vibration analysis depends on this theory. In fact, eigenvalues are one of the tools that allow us to move back and forth between finite and infinite dimensional vector spaces. Eigenvalues enable us to transform complex dynamical systems problems into much simpler algebraic problems. Moreover, these problems show up in diverse fields within industry, government, and academia.

Because eigenvalue problems are so popular in STEM fields, we believe there is a need to invite students to experience a more authentic treatment of eigenvalues in their introduction to linear algebra. This belief was born during of our first experience teaching an introductory linear algebra course. Before our class began, we sampled a wide variety of introductory textbooks on the subject. In the more than 30 books that we perused, we found a curious pattern in the choices each author made in presenting eigenvalue theory.

Most introductory linear algebra textbooks do NOT develop eigenvalue theory within the context of application problems that are directly relevant to phenomena that students can observe in their daily life. On the contrary, popular linear algebra textbooks dedicate many sections to developing algorithms to compute eigenvalues by hand and to interpreting the eigenvalue problem geometrically in terms of eigenbasis. There are, of course, a limited number of introductory linear algebra textbooks that devote multiple sections to the relationships between eigenvalue theory and dynamical systems. However, even in this best-case scenario, the reader is asked to take a huge leap of faith if she is to believe that eigenvalues are useful tools in modeling observable phenomenon that exist in her world. Almost nowhere in the introductory literature to linear algebra are students invited to verify eigenvalue-based models of dynamical systems by comparing measured and modeled data. This presents a barrier for our students that prohibits their ability to transfer their knowledge of eigenvalue theory out of the classroom and into the working world.

If we want to teach young students to use math to understand their world, we need to give them practice with the entire mathematical modeling process in their mathematics courses. When it comes to eigenvalues, there are many students (eg. mechanical, civil, and electrical engineers) who deserve the opportunity to transform a physical vibrations problem into a mathematical model, analyze that model using eigenvalues, and then compare the dynamics of the physical system with those of the model. If we’ve done our work well, these students should feel empowered to make their own conclusions about the accuracy of eigenvalue-based models in describing dynamical systems. Moreover, after they complete such an authentic modeling problem, these students will likely be able to recognize other phenomena in their observable surroundings that might be modeled by eigenvalue analysis. It is with this perspective in mind that we develop the work below.

Design physical apparatus

We began with a relatively simple description of a design problem: can we build a system to enable students to study and model a useful vibrations problem, use eigenvalue theory to analyze said model, and then compare the resulting measured and modeled data to make conclusions about the utility of eigenvalues as modeling tools. After many months of thinking, conversation, and iteration, we designed the apparatus described in this paper. Below we’ve included a link to documents that guide you on how to build one version of such an apparatus.

McCusker’s Guide to Design and Construct the Apparatus, Version 1 Design

The design highlighted above was the first working design for the apparatus we used in our paper. Michael McCusker built this apparatus. Dr. McCusker is an experimental physicist by training and has lots of experience working with this hands. Thus, this first design is great for people who feel comfortable working with their hands and can get creative. As you’ll notice, this design is inexpensive. Below is a video of Dr. McCusker describing his work and highlighting relevant features of his apparatus.

In order to test to make sure that this design is accessible to other instructors, Jeff built his own version without any help from Dr. McCusker. Jeff is a mathematician by training and has very little experience working with his hands. The entire purpose of building a second version of this apparatus was to prove that even a novice maker (Jeff) can build a functional system for a low cost. This second design might be helpful for people who feel less comfortable working with their hands, though the price is slightly higher compared the original version. Below we link to a video showing the completed design and the entire set up process.

Measuring the Spring Constant

The spring constant can be measured with a low-cost method. Rather than purchasing calibrated masses, we use US treasury coins, which have uniform masses and, even in the smallest budget schools, are readily available.

A digital kitchen scale can be used to measure the mass of a coin (e.g. a US quarter) as follows:  Place several quarters on the scale (perhaps 30) and measure the average value. Then hang a selected spring from a frame and attach a small plastic bag that also has been weighed from the lower end of the spring. Measure the spring extension. Successively add coins to the bag and measure the spring extension as each successive quarter is added to the bag. Record the data in a spreadsheet with a column for weight added and one for spring extension. Use a least-squares fit utility (such as LINEST or a trend line) available within the spreadsheet to compute the slope of the plot of extension vs added weight (The force in newtons is m* g). The slope will yield the spring constant in N/m. Hooke’s Law formula is F (Newtons) = k x where x is measured in meters. The spring constant k will have the units of Newtons/meter. Below we've provided a link to a handout that includes this overview and sample data for the spring Michael McCusker used in his physical system.

Measuring the Spring Constant Handout (.pdf)

Capture Experiment Videos

Once we’ve built our apparatus, the next step is to film various eigenvalue experiments. We begin this process by attaching a digital camera to the top of our physical frame and pointing the camera downward to capture overhead videos of the masses. After pressing the record button, we place the left and right masses in different initial positions and capture a birds-eye-view video of the dynamics of this system. Below we provide six different videos captured on our two separate apparatus.

Set up data analysis tools

The third step in our process is to transform our raw videos into data files that represent measurements made on our system. To accomplish this step, we import each file into an open-source video analysis and modeling software called Tracker. Tracker is a free video analysis tool that conforms to the Open Source Physics Java framework. This software uses image processing algorithms to transform the input video file into time, position, velocity, and acceleration data. rely on the open-source, image processing software. In the video below, we give you a quick overview of how to use Tracker software to analyze an eigenvalue experiment video file. This video is a brief overview of how to use Tracker in this modeling activity. This introduction should not be considered a substitute for the Tracker tutorials available on YouTube (links provided below). 

After building the physical apparatus and filming our eigenvalue experiments, we use an open source video analysis and modeling software called Tracker to analyze the dynamics of our physical system. This software is available for download, free-of-charge, at the Tracker website. At the top of the page, you will find links to "Download Tracker installer for" your computer. Click the link corresponding to the operating system on the computer(s) that you will to use for the data analysis portion of this laboratory exercise. If you need extra support getting this software to run properly, Tracker's webpage has a number of useful links including Help and Installer Help.

Once you successfully download and install this software, we recommend that you learn how to properly utilize the video analysis features to support this laboratory exercise. Tracker's author, Douglas Brown, has created a number of useful Youtube videos for this purpose. We've provided links to these tutorials below for ease of reference:

Tracker Quick Start Tutorial (2 min, 30 sec)
Getting Started with Tracker (15 min, 04 sec)
Tracker Autotracker Tutorial (17 min, 31 sec)

Film and Import Eigenvalue Videos

Now that Tracker is successfully installed, you can use this software to analyze your eigenvalue experiments. For readers who would like to recreate the results shown our paper, below we provide the videos we used to produce all of our results. 

First Set of Experiments: Michael McCusker's Apparatus

All data from this paper were gathered from author Michael McCusker's eigenvalue apparatus. Below, we provide links to the videos and data we used to create the graphs presented in this paper. The physical apparatus seen in these videos has the following measurements:

Readers who are interested in verifying the results we present can simply download the .mp4 files from the table below, import these files into Tracker (as discussed above) and reproduce the desired position and velocity data associated with these videos. For convenience, we have also provided the raw data files (.xlsx) produced by tracker and the manipulated data files that we used to present our analysis in this paper. 

Second Set of Experiments: Jeff Anderson's Apparatus

As part of the process of developing this laboratory exercise, the authors created multiple eigenvalue apparatus. In particular, while Michael McCusker is trained experimental physicist, Jeff Anderson is not. Thus, in order to convince himself that a mathematician with no experience building experiments could build such a system, Jeff designed and built his own eigenvalue apparatus with Dr. McCusker's prototype in mind. Then, Jeff used his couple, mass-spring chain to produce and analyze data in all three modes (normal mode 1, normal mode 2, and the mixed mode). This was a useful exercise to prove that even a highly unskilled technician (Jeff) can produce a useful, inexpensive apparatus for classroom use.

In the first column of the table below, we provide a set of three videos (one for each mode) associated with Jeff's apparatus. In the second column, we provide three useful files including:

A. Raw exported data associated with the left mass (as a .csv file) exported from Tracker

B. Raw exported data associated with the right mass (as a .csv file) exported from Tracker

C. The Tracker project file (as a .trk file to be opened in Tracker) used to produce the raw exported data for each mass. 

Finally, in the third column of this table, we provide a refined data analysis (as a .xlsx file) for both the left and right masses of this system. This refined analysis includes modeled displacement and velocity data and charts that compare the modeled behavior against the collected data. These refined files also include guidance on how to manipulate the raw data exported from Tracker in order to compare this data against the modeled behavior.

This third column is useful because the data we collect in Tracker represents raw position measurements in the frame of reference we defined in Tracker. However, the models that we construct all depend on displacement measurements. In order to create said displacement measurements from the raw position data, we must execute some nontrivial calculations. For example, we must find a valid amplitude measurement for our displacement function and we need an approximate equilibrium positions of each mass at rest with zero velocity and zero net force. The refined data files give some insight into how the authors dealt with these issues. 

Jeff Anderson's physical apparatus seen in these videos has the following measurements:

Create a mathematical modeling framework

One of the major goals of this project is to empower students to create a meaningful eigenvalue problem from an observable phenomenon. In so doing, we hope students experience the challenge and joy of mathematical modeling. As part of this process, students will need to mathematize the physical system by creating the appropriate modeling framework. To support students in this goal, we have created some introductory videos to help students work through this process. Below is a link to twelve videos that explore the entire modeling process of translating the coupled pendula problem into a standard eigenvalue problem.

How do we use this activity in our introductory linear algebra classes?

Below our drafted copies of resources that the authors use when teaching linear algebra. The first link includes a sample calendar for a quarter-long class in linear algebra. Portions of this activity are assigned during the last week of class. The authors plan to make videos on lecture content for this introductory linear algebra class. In this case, this laboratory exercise will happen live in class during the final week of the quarter and student's written results would be due on the day of the final exam. 

Sample quarter-long calendar for introductory linear algebra

How can you use these resources?

One of the really cool features of this project is that we’ve designed resources that can be integrated into the classroom in many forms. Of course, we encourage you to build your own apparatus and conduct the experiments from scratch using data that you generate with your students. However, we realize that taking on such a DIY schedule project might not be feasible upon first consideration. With this in mind, we outline some of the ways that you might integrate this project into your classroom.

Activity 1: Incorporate background material into your lecture notes and course curriculum

List of applications that students study in-class prior to the eigenvalue activity

Activity 2: Give students tracker data (above) and have them generate the model using our system paramters

Activity 3: Have students download the .mp4 videos and create their own Tracker data and then proceed with Activity 2.

Activity 4: Instructor build eigenvalue system and students film their own eigenvalue videos. Then proceed with activity 3 and then activity 2.