Laboratory 5 - Procedure¶

First, please take note of three points regarding Lab 5:

It is very likely you will need two full lab sessions to complete this lab. Please budget your time accordingly.
When you arrive for the second session, you are expected to have written a proposed method for measuring the unknown mass. This needn’t be in-depth - a list of a few bullet points will be sufficient. Your TAs will come by to check on your proposal at the beginning of lab session 2.
For the first session, the TAs have been instructed to provide limited help. You have done everything in this lab before. Here, you are simply combining the skills in a new way.

Introduction¶

In this lab, you will use FEM simulation to produce a predictive model that describes the variation of the natural frequencies of a cantilevered beam in response to an applied point mass. Using this model, you will determine the mass of an object attached to your real-world beam by observing the changes in the beam’s natural frequencies.

This lab is an open-ended application of concepts you have learned up until this point in the semester. There are multiple valid approaches to solve this problem, and it is up to you and your group to find a procedure you think will produce an accurate estimate of the unknown mass. The most accurate group and the most precise section will win prizes, as explained in the contest rules.

Broadly speaking, you will:

Measure the first three natural frequencies of your beam by performing FFT on accelerometer data
Refine your FEM model to match the experimental frequencies
Adjust your FEM modal analysis by adding point masses to the beam
Perform a regression analysis to determine the relationship between mass and frequency change
Estimate the unknown mass, and determine the uncertainty in the estimate.

Note: You aren’t required to use the values for natural frequencies and Young’s modulus that you obtained from earlier labs. This means that if you previously saw unusual results, they won’t affect your performance for this lab.

Task 0 - Background¶

Read the Background and Resources section for this lab. Make sure you and your partner(s) have a good idea of the goal and a general understanding of how you might apply the concepts from previous labs in order to achieve it.

Task 1 - Unloaded Beam¶

With your beam unloaded, measure the acceleration vs. time signal that the accelerometer outputs in response to an impulse.
Perform a fast Fourier transform on the time-domain data to determine the first three natural frequencies.

Task 2 - Model Validation¶

In Ansys, confirm that the geometry and material properties of your model match your beam.
Run the modal analysis, and vary your model in Ansys until the FEM modal analysis agrees with the frequencies discovered via FFT. It’s not necessary that your measured frequencies match with previous labs - the added accelerometer will have changed them slightly. The main goal is seeing agreement between measured frequencies and FEM frequencies.
- Note - for this lab, it’s perfectly acceptable to vary parameters (e.g., beam geometry and topography, Young’s Modulus, Poisson’s ratio, etc.) until your frequency values “look close.” Briefly justify your variations in your report.

Task 3 - Apply Masses to Your FEM Modal Analysis¶

In Ansys, you can expand the “Geometry” tab of the main project tree to see the solid that makes up your model. Right-clicking on the solid will let you add a “point mass.” After clicking on a face of your model, you will be able to specifiy the exact location and mass of the point mass.
Apply various masses at various locations, recording the resulting natural frequencies. It is important to approach this with purpose. You will be performing a regression on data taken from this analysis - decide in advance how and where you will apply the masses to generate the most useful data for a regression.

Here are some points to consider:

What is your rough estimate of the unknown mass from simply holding it in your hand? Taking this into account, what should be the magnitudes of the point masses you introduce to your model?
Where are the three mounting holes in your beam? Will you use all three? Where should you apply the point masses?

Task 4 - Regression Analysis¶

Using data from Task 3, perform a regression analysis to relate the mass to natural frequency.

Here are some points to consider:

Do you want to take into consideration only one natural frequency, or all three (using the curve_fit implentation from the Background and Resources section for multiple independent variables)?
Do you want to use the frequency values themselves, or the change between the unloaded and loaded frequency? How would this affect the y-intercept of your regression? Would you need to add another coefficient if the intercept is non-zero?
Does your data look like a linear fit, or should you consider a quadratic?

Task 5 - Measure the Frequencies¶

Attach the mass to the beam, and measure the resulting natural frequencies using FFT of the acceleration vs time signal. Depending on your plan, you may only need to use one mounting hole. You may need to record data for all three.

Task 6 - Determine the Mass¶

Using the regression from Task 4 and the measurements from Task 5, determine the mass of your object.
Calculate the uncertainty in your mass estimate, using either of the two methods described in the Background and Resources section - the standard error of the fit and Student’s t-tables, or the covariance diagonals and the sensitivity indices.

Your Report¶

For this assignment, you are limited to 6 pages and 5 figures. In addition to the common themes we have discussed this semester, your report should also include details of your experimental design (i.e., your method for determining the mass and the reasoning behind it).

Mass Measurement Contest¶

Each group’s results will be entered into a contest, with the most accurate group and the most precise section receiving prizes.

Rules of Contest

The masses must not leave the lab.
You may not mount other known masses to the cantilever, or use any mounting point other than the three pre-dilled holes.
You must report your uncertainty in your mass measurement to enter the competition. In other words, your estimate of your mass must include a confidence interval.
You must report the serial number of your mass (i.e., “TJM 01-TJM 12”) to enter the competition.
You may use the following tools and software: accelerometer, calipers, ruler/tape, Ansys, SolidWorks, Labview, Python, Matlab, and Excel.
It should go without saying that you may not weigh your mass.

Note: This is intended to be a friendly competition on top of a normal lab assignment. Poking around to find loopholes will disqualify you, and could also earn you a poor grade for not following the lab procedure.

Winners of the contest

There will be two sets of winners for the contest:

Lab group with the most accurate mass measurement calculated with $A=|m_{reported}-m_{actual}|$
Lab section with the most precise mass measurement calculated with $P=\frac{\sum_{i=1}^{N}(m_{reported}-m_{actual})^2}{N}$

Where $A$ is the accuracy, $P$ is the precision, $m_{reported}$ is the reported mass from your experiment, $m_{actual}$ is the actual mass of the object, and $N$ is the total number of lab groups in a section. The group and section with smallest $A$ and $P$, respectively, will win prizes. The prizes are:

Each member of the winning lab group will receive a $25 gift card of their choice.
Donuts will be brought to the winning section.

Note - due to smaller section sizes, sections 004L and 008L will be grouped as one section for the purpose of calculating $P$, as will sections 001L, 011L, and 014L.

Introduction to Sensors and Data