Dave Martin is a Creo, Windchill, and PTC Mathcad instructor and consultant. He is the author of the books “Top Down Design in Creo Parametric,” “Design Intent in Creo Parametric,” and “Configuring Creo Parametric,” all available at amazon.com. He can be reached at [email protected].
Dave currently works as the configuration manager for Elroy Air, which develops autonomous aerial vehicles for middle-mile delivery. Previous employers include Blue Origin, Amazon Prime Air, Amazon Lab126, and PTC. He holds a degree in Mechanical Engineering from MIT and is a former armor officer in the United States Army Reserves.
This collection covers the six Mathcad Community Challenges that took place throughout 2023. The challenges are bimonthly community events where a challenge is posted at the beginning of the month, and throughout that month, community members post their Mathcad Prime solutions to those problems and can collaborate or iterate off earlier submissions. Participants also win exclusive Community badges for their profiles.
Mathcad Community Challenges 2022 Collection
Write-ups and solutions of the first-ever six Mathcad Community Challenges on diverse topics.
In 2023, we had six challenges covering these wide range of topics:
Black Holes
Multiple Voltage Sources
Optimize Horizontal Distance
Steel I-beam (Civil Engineering)
Art Gallery
Crankshaft & Key (Mechanical Engineering)
Be sure to use the links provided for each challenge to download participants’ Mathcad Prime worksheets and read along with the commentary. That’s the best way to learn from your peers’ work.
Mathcad Community Challenge January 2023: Black Holes
The Mathcad Community Challenge January 2023 was on black holes, because let’s face it, black holes are cool! As a massive star dies, it can end up collapsing upon its own weight. It becomes so dense that it basically punches a hole in the space-time continuum. Here was the challenge:
Create a worksheet that:
Calculates the event horizon (Schwarzschild radius), last photon orbit, last stable particle orbit, and temperature if the sun, moon, and planets of our solar system were to become black holes. (You can choose whether to include Pluto.)
Uses the Chart Component to depict the event horizon and temperature as a function of mass up to the size of UY Scuti, the largest known star in the Milky Way galaxy (which may become a black hole “soon” on a cosmic timescale).
It turns out that the equations are actually not that complicated. I really wanted people to get a sense of the extraordinary. For example, the Earth has a diameter of 7,917.5 miles (12,756 km). That corresponds to a radius of 3,958.8 miles (6,378 km). If the Earth were a star, it would have to collapse to a Schwarzschild radius of 0.349 inches (8.87 mm). That is nuts!
I also asked for temperature because I was surprised that black holes have a temperature. They are so massive that not even light can escape a black hole, but Professor Stephen Hawking figured out that black holes emit radiation and therefore have a temperature. A very, very low temperature, but a temperature, nonetheless. Absolute zero is -459.67 degrees Fahrenheit (-273.15 degrees Celsius). If the Earth were a black hole, its temperature would be -459.633 degrees Fahrenheit (-273.129 degrees Celsius).
The submissions
The first entry was made by user PGrist. I had provided a NASA reference for the equations, but I mistakenly directed people towards the equation that calculates Schwarzschild radius as a function of the mass of the sun. I prefer the “purer” equation that calculates the event horizon as a function of object mass. PGrist calculated the event horizon, last photon orbit, and last stable particle orbit, but left the values in kilometers. Expressing the values between 10 to the -7 power and 10 to the -10 power diminishes the impact of the results.
The second submission was by frequent contributor Fred Kohlhepp. He created some nice XY plots depicting event horizon radius and temperature as a function of mass. Fred also used a table to compile the masses of the sun, moon, and planets. By vectorizing the mass column, Fred placed a second trace for the solar system objects onto the plot. Nice.
The third and final submission was by another frequent contributor, ppal. Like PGrist, ppal calculated the temperatures at the event horizon, last photon orbit, and last stable particle orbit. (It turns out there are multiple temperature calculations for black holes.) ppal created nice XY Plots as well. I like the plotting of the names of celestial objects on the X-axis in some plots as well as logarithmic scales on both axes in others.
Chart Component
I was disappointed that none of the submissions depicted the Schwarzschild radius or temperature using the Chart Component. Everyone used the old XY 2D plot. I understand that people might be more familiar with the standard XY plot, and it has functionality you can’t find elsewhere. But for creating publication-ready images, the Chart Component is the way to go. Here is an example of the Chart Component I made depicting the event horizon and temperature as a function of mass:
Mathcad graphing options: XY plots and chart components
Learn the differences between Mathcad Prime’s XY Plots versus the Chart Component feature.
There’s an old saying that math is the language of physics. Mathcad’s documentation and graphing tools can help you convey the real meaning behind phenomena and its meaning. Download the worksheets from this challenge here.
Mathcad Community Challenge March 2023: Multiple Voltage Sources
Scenario 1. We have a circuit with a motor (modeled as a 100 Ohm resistor), a 6V voltage source, and a 100 Ohm resistor. Calculate the current in the motor and the resistor. Scenario 2. We add another loop, with an additional 8V voltage source and a 120 Ohm resistor. Calculate the current in the motor and the two resistors. Scenario 3. We add another loop, with an additional 10V voltage source and a 140 Ohm resistor. Calculate the current in the motor and the three resistors. Scenario 4. We add another loop, with an additional 12V voltage source and a 160 Ohm resistor. Calculate the current in the motor and the four resistors.
By having multiple EMF sources in the circuits, I was looking for users to apply a network theorem like superposition. This states that each voltage source acts independently of each other. You can remove other EMF sources, solve for the circuit, and add the results. However, I am not an electrical engineer, and it appears our brilliant challengers were able to solve without superposition.
Let’s look at the solutions provided.
The six submissions
Werner E. created a system of equations for the currents based on Kirchhoff’s laws. After getting a complicated result when solving symbolically, he wrote a program involving summation and product operators and the stack function to construct a vector of the currents. Werner also wins the award for fastest solution – two hours for his first post and four hours for a Mathcad Prime worksheet.
ppal, another frequent contributor, submitted two worksheets with different approaches. The first had good use of images and matrix math. However, as someone who is not an electrical engineer, I would have liked a little more documentation explaining the rationale behind the solution.
ppal’s second submission used Solve Blocks, which I find much easier to understand even without additional documentation. If you haven’t explored the power of Solve Block functionality, I strongly recommend you do so. It’s my favorite Mathcad tool and will expand your capabilities for engineering calculations immensely.
I love the way that Fred Kohlhepp uses different tools to explain his approach. In addition to text, he documents the functions via the Comparison Equal To operator (I’m a big fan of that technique). Fred also used a collapsed area to hide an image. The equations he constructs to generate matrices for the motor voltages, resistor currents, and motor currents are complex yet elegant, making use of index notation, summations, and the if function.
Alan Stevens made use of the deliberate increments to the voltage and resistance in each scenario to construct vectors for the inputs. Then he constructs a resistance matrix and calculates the resistor and motor currents. It’s an economical approach to the answer. But once again, a little more documentation will help outsiders understand the problem and solution better.
TTokoro. Once again, just a beautiful worksheet, using color and highlighting. Their solution was based on Millman’s Theorem and Ohm’s Law with the results checked by Kirchhoff’s current law. Like the others, Ttokoro used vectors, matrices, built-in functions, and custom functions. Here are some of the unique aspects of this worksheet:
Use of Draft View to hide some in-depth programs
A Combo Box input to select the scenario! Neat stuff!
TTokoro is the master of creating images using 2D XY plots. If you change the scenario from the Combo Box, the XY plot updates to show the scenario!
Mathcad 15?
The rules to the contest state that the submissions must be in PTC Mathcad Prime. User MFra submitted two Mathcad 15 worksheets. Fortunately, Mathcad Prime can convert legacy worksheets to Mathcad Prime. Unfortunately, the documentation for the lengthy (~11 pages) worksheets was in Italian, so I couldn’t understand the network diagram approach.
What can we learn?
I’m going to repeat what I say almost every month because it’s true: It’s amazing how many different approaches our community finds to solve problems in Mathcad. In these worksheets I’ve seen a combination of functions, programs, matrices, summation operators, product operators, symbolic solving, Solve Blocks, Combo Boxes, and more. The work was documented via text, images, plots, and formatting tools.
If you want to learn advanced techniques for Mathcad, check out the submissions from this month’s and previous months’ challenges. Also, the challenges are meant to be collaborative. In future months, if you can’t come up with an initial solution, maybe you can build on someone else’s work with different Mathcad tools. Are you up to the challenge?
Mathcad for Electrical Engineers webinar replay
Watch a Sr. Hardware Design Engineer’s examples on why Mathcad is used by electrical engineers
A ball with a mass of 1 kilogram is at the top of a frictionless ramp 10 meters above the ground. The ball rolls down the incline and launches from a height of 2 meters and an angle of 30 degrees above the ground.
The problems to be solved are:
Create a function that calculates the horizontal distance as a function of initial height, launch height, and launch angle.
Calculate the horizontal distance the ball will land from the end of the ramp.
Solve for the angle that will optimize the horizontal distance.
How will the horizontal distance change if this were performed on the Moon instead of on the Earth’s surface? Assume the acceleration due to gravity on the Moon’s surface is 1/6 that of Earth.
Use the Chart Component to depict how the horizontal landing distance changes as a function of angle.
Use a 3D Plot to show how the horizontal landing distance changes as a function of ramp height and launch angle. Assume the ball starts at a height of 10 meters.
The submissions
Frequent contributor ppal submitted the first and multiple worksheets. I like that they build on previous worksheets. I hadn’t intended people to take moment of inertia into account, but ppal did. Functionality in the solutions included user-defined functions, symbolic evaluation, the polyroots function, XY plots, and 3D plots. The worksheets incorporated multiple aspects to make the worksheet friendly to others, including text with different fonts, images, and highlighting.
Fred Kohlhepp, another loyal contributor, submitted an attractive Mathcad Express worksheet with numerous XY plots that greatly help to convey information to the reader. Fred also uses another technique I like: documenting the thinking behind your equations using the equality comparison operator (the thick equals sign). Fred also figured out the trick behind question 4. Since the acceleration due to gravity falls out of the equations, the horizontal distance is the same on the Earth, Moon, or any planetary body.
Jeff Henning’s submission wins the award for the most publication-ready worksheet. It could be used to teach physics to high school and college students. Jeff uses collapsible areas to organize the worksheet into assumptions, derivations, inputs, function definitions, and the answers to each question. Jeff gets extra points for being the first person since the challenge started to use the Chart Component. He plots the trajectory of the ball as well as the distance versus launch ramp angle.
The 3D Plot for question 6 is also beautifully formatted. Another small thing that makes a huge difference: Jeff uses the document footer including page numbers. Just a beautiful work of art.
6 Ways to make your Mathcad worksheets publication ready
How can you make your own worksheets beautiful and easy for anyone to understand?
As usual, TTokoro submitted an inviting worksheet complemented with text, highlighting, XY Plots, 3D Plots, and headers and footers. Draft View was used to hide some calculations, which helps keep worksheets neat. The math made heavy use of symbolic evaluation. A nice, compact piece of work. If you want to take your XY Plots to the next level, I recommend checking out TTokoro’s challenge worksheets.
Alan Stevens posted an economical Mathcad Express worksheet. (Since the Express version lacks the Chart Component and 3D plots, he and Fred were not able to complete questions 5 and 6.) As I had suggested in a subsequent comment, Alan considered conservation of potential energy into kinetic energy, resulting in compact equations and functions. The optimum angle was solved by defining a custom function and applying the root function.
Jan Claeys’s worksheet started off with a table of contents consisting of hyperlinks to other parts of the worksheet. The calculations were documented well with text, highlighting, and use of the equality comparison operator (like Fred and others). The calculations included symbolic evaluation, the root function, an XY Plot, and even a Solve Block (my personal favorite Mathcad functionality). Jan completed questions 5 and 6 with a very nice Chart Component and 3D Plot, then concluded the worksheet with hyperlinks to external references. Like Jeff’s, this worksheet makes a very nice teaching tool.
The outcome
There were two sets of answers, depending on whether rotational inertia was taken into account. The people who submitted worksheets found those results. The more time I have worked in industry, the more I have realized the importance of documentation. It doesn’t matter how great your results are if you fail to communicate them effectively. We’re also not able to explain our results personally (especially years after the fact), so it’s important that the work stands on its own. Like previous months, we see a variety of solution tools, but also a variety of different tools to document that work. I recommend checking out the submissions to see which communication methods resonate with you to include in your own work. You can also check out all of the previous challenges and solutions that have covered a wide range of topics so far.
Mathcad Community Challenge July 2023: Civil Engineering
Scenario: You have a standard 300mm steel I-beam with a length of 12 meters. It is simply supported at one end and at 8 meters from that end. There is a load of 4500 Newtons in the middle and 2500 Newtons at the unsupported end.
Calculate the cross-sectional area of the I-beam.
Calculate the cross-sectional moment of inertia about the midpoint of the beam. (You may choose to ignore the contribution of the fillets. If you want the additional challenge, you may include them.)
Use the Chart Component to display shear and bending moment diagrams.
Calculate the deflection at the end of the I-beam.
The submissions
Frequent contributor Fred Kohlhepp submitted the first worksheet, and later submitted a second worksheet that took the I-beam fillets into account. He calculated the cross-sectional area and moment of inertia via integration, requiring a tightening of Mathcad’s convergence tolerance. He used vectors and matrices to calculate the reactions at the supports and the shear as a function of position along the beam. He plotted the bending moment by integrating the shear function. Then he used the slopes of the moment lines to calculate and plot the deflection.
Visually, I like Fred’s use of highlighting to draw attention to the important parts.
Terry Hendicott’s approach was quite ambitious, consisting of eight (!) worksheets linked together using the Include functionality. He acknowledges that the problem can be solved in a much shorter format, but uses this longer technique to help users gain confidence working on more complex problems in Mathcad.
Terry’s approach is essentially a Finite Element Analysis (FEA) method using an open-source mesh generator called Gmsh. The solutions are quite involved with several complex programs. Anyone interested in performing advanced structural analysis using Mathcad should check it out.
As usual, Jan Claeys created a beautiful worksheet that can be used as a teaching tool. He made great use of Mathcad’s documentation functionality, including text, images, headers, footers, external and in-worksheet hyperlinks, areas, and even programs. (I’ve never seen programs used for documentation. Interesting.) Some unique aspects of his worksheet include Tables and the SIUnitsOf function. The results were presented beautifully with Chart Components.
(By the way, you can see more of Jan in this Mathcad video.)
Robert B submitted a worksheet with original images for the beam length and section as reference for his calculations. He calculated the cross section and moment of inertia by breaking up the beam into three sections. He calculated a function for the load; integrated that to get the shear; integrated the shear to get the moment; integrated the moment to get the slope; and integrated the slope to get the deflection. He then used a Solve Block (my personal favorite Mathcad functionality) to solve for the reactions.
Robert plotted the various quantities using both XY Plots and Chart Components. I recommend you view his worksheet to decide for yourself which tool communicates the results better.
Previous contributor Dennis Fallon, the former dean of the School of Engineering at The Citadel, has returned for this month’s challenge with an exemplary worksheet that could be used in a college classroom. It has great use of images with transparent backgrounds, fonts, color, borders, and internal hyperlinks. Professor Fallon used two different methods for calculating the properties of the beam including the fillets, a classical approach and a second using Solve Blocks and programs. The Chart Components for shear and moment take full advantage of formatting options, especially color.
Special thanks
Dr. Pat Heffernan provided great help by vetting this problem during the creation phase. His worksheet is posted in the challenge replies. You can catch his presentation at the Mathcad for Civil Engineers webinar here. He also has a YouTube channel that covers many Mathcad topics.
Mathcad for Civil Engineers webinar replay
Watch why the software is a perfect fit for the day-to-day work of civil and structural engineers.
As a mechanical engineer and Creo Parametric user, I’m used to thinking of Mathcad as a tool for product development and mechanical engineering related work. We have had previous challenges that have shown Mathcad’s utility for electrical engineering, and now we can see it is just as well suited for structural analysis and civil engineering problems. And, of course, Mathcad can be used for more than just engineering calculations, including statistics, image processing, and finance.
If you have not looked at the submissions for yourself, please do so. I guarantee you will learn something.
Mathcad Community Challenge September 2023: Art Gallery
“Painting is a science and all sciences are based on mathematics.”
Leonardo da Vinci
September’s Mathcad challenge was to use the Chart Component, XY Plot, Polar Plot, Contour Plot, or 3D Plot, create some kind of image that evokes the beauty of math. There were no other guidelines regarding simplicity or complexity. Let’s dive into the submissions!
The challengers
The first submission was a smiley face from the Mathcad Community Challenge organizer, David Newman. Normally, people involved in the challenge don’t submit to it. But I’m citing it because even though I consider myself highly proficient at the Chart Component, I learned a lot from this. Most of the image was created by applying a gradient fill to the background and – something I’ve never even considered – using a black chart border with huge values for the size of the corners. Ingenious. The worksheet itself uses a collapsed area to hide some range variables. There are seven traces in the Chart Component: a parabola, two ellipses, and five points. Changing the weight values for the traces helps achieve the smiley face image.
You might not think this art is particularly aesthetic; beauty is in the eye of the beholder. But you can apply these techniques to your charts.
As I remarked in the community site comments, the submission by frequent contributor Alan Stevens reminded me of the repeating patterns of a Spirograph toy from my youth. The build-up of the math is quite interesting. A range variable is created from 0 to 360 for degrees. But since Mathcad uses radians, he constructs a vector using the range variable to define radian values. Six different variables (vectors) created using the sine function, modulus function, and a constant value. These variables are stacked, and then graphed on an X-Y plot using the vectorize operator. A lot of good stuff here.
A subsequent image appears to have eliminated the stacking so that different colors can be applied to each of the variables. As noted in the discussion on the community site, the NaN (Not a Number) value can be used to avoid lines being drawn from the end of one value in the stack to the first value in the next variable in the stack.
(Side note: the image was also helped by turning off the grid display on the worksheet.)
Frequent contributor TTokoro submitted two worksheets. The first displays a 3D version of the Lissajous curve, which is also known as the Bowditch curve. You have seen this curve on oscilloscopes, television images, and old science fiction movies because of the beauty of its harmonic motion. The first worksheet makes nice use of the Combo Box input-output control to change the values of the coefficients and phase angles. A Lissajous curve is normally 2D, but TTokoro made it 3D by also defining a parametric equation for z-value, controlled by its own Combo Box. I recommend downloading the worksheet and both playing around with the different choices as well as seeing how the Combo Box was constructed.
There are four X-Y plots and one 3D plot. The X-Y plots depict the traditional 2D Lissajous and the projections of the 3D Lissajous onto the XY, YZ, and ZX planes. The 3D plot shows the 3D version of the Lissajous.
TTokoro’s second worksheet has a couple long, complex programs that I won’t even pretend to understand. They are used to create some cool 3D plots that I recommend you play with. They consist of repeating 5- and 6-sided 3D polygons in structures of connected nodes that remind me of finite element meshes and additive manufacturing beam structures. They would also be at home in a Marvel movie. Check them out and spin them around because the image changes depending on the angle.
The final submissions came from PTC’s own Tetiana "Tanya" Pakhomova. She is not even a Mathcad user, and she was able to graph a simple and sweet 2D heart in the Chart Component. My only advice would be to turn off the plot border, axes, and tick marks and turn on a user-defined range for the Y-axis. Nice use of color and thickness for the trace.
The challenge then inspired Tetiana to learn more about Mathcad. She said that initially she was apprehensive about the Chart Component but quickly found it easy to grasp. She then submitted a dark pink cosmos flower, reminiscent of her childhood home in Ukraine, by using the parametric form of the rose curve for the petals. The pistil was formed via a point in the middle with weight and opacity. The frame to the picture was made by using gradient colors for the chart background and plot area, and a thick centerline dash line style. Very creative use of the formatting tools in the Chart Component.
What can we learn?
Even with a handful of submissions, users created art with the Chart Component, X-Y Plots, and 3D Plots. They used functions, range variables, vectors, programs, the vectorize operator, and Combo Boxes. There’s a breadth of tools and techniques available to create art and communicate your results to an audience.
This challenge reminds us that math is fun and it’s beautiful. Download the submissions and then make art in Mathcad for yourself. If you want even more ideas for generating 2D/3D curves and 3D surfaces in Mathcad, I recommend checking out Alfred Gray’s book, Modern Differential Geometry of Curves and Surfaces.
Mathcad Community Challenge November 2023: Mechanical Engineering
In a surprise twist of events, the winner of this month’s challenge is… me. For some reason, this challenge did not appeal to the audience. In the last third of the month, I began posting steps in the progress of my solution, hoping to inspire someone to build on my work. Alas, there were no takers.
The challenge
This month's challenge was a mechanical engineering problem. A key is a mechanical component used to transfer power between a rotating shaft and a machine component (in this example, a flywheel).
The challenge was to create a worksheet that performs the following:
Calculate the torque that can be transmitted by the reduced shaft diameter (the diameter not including the keyway).
Calculate the shear stress and compressive stress on the key and shaft.
(Optional) Using the technique presented by Dr. Heffernan in this video, include a simple image of the shaft and key with a transparent background. The image does not have to be to scale. The transparent background is the important part.
(Optional) Create a Chart Component or X-Y Plot of the shear stress as function of the breadth/width of the key. (Or for more intuitive communication, plot the ratio of shear stress relative to the yield strength in shear as a function of the key. Assume that the material fails in shear at half the nominal yield strength.)
(Optional) Use Combo Boxes, Tables, Matrices, or other methods to turn this into a “working” worksheet, where users can select different materials, design factors, torque cycles, or other factors of their choosing.
The solution
The main part of the challenge was fairly straightforward: calculate the torque capacity of the shaft, the compressive stress of the key onto the crankshaft, and the shear stress in the key. The torque capacity required an intermediate step of calculating the moment of inertia. But other than that, I solved for these values with just variables and math expressions.
When tackling later steps in the challenge, I realized that I had been shortsighted. I had to turn the math expressions into functions, which I should have done from the start. Functions provide much more flexibility.
One of Mathcad’s greatest strengths for engineering calculations, especially when compared to tools like Microsoft Excel, is the ability to document your worksheet. Being inspired by previous submissions, I used text fonts, text color, and Areas to help readability and organization.
The first optional challenge was to add an image using the technique demonstrated by Dr. Heffernan in the Mathcad for Civil Engineers webinar. It involves creating an image manually in PowerPoint, saving that image to a picture file, and then inserting the image into a worksheet.
This imported image has a transparent background and looks simply fantastic against Mathcad’s graph paper interface. I recommend everyone learn this technique and use it whenever possible.
Communicating results to your audience
The second optional challenge was to graph the shear stress versus key width using either the X-Y Plot or Chart Component. I prefer Chart Components because they offer so many more formatting options for generating publication-ready graphs and can be exported to standard image file formats.
It doesn’t matter how good your results are if your audience doesn’t understand them. That’s why I prefer normalizing stress results relative to the yield strength. In the second Chart Component, I added a second Y-axis expression with a value of 1 and changed its formatting to a dashed red line. This helps convey to people with less of a technical background what’s safe and what isn’t.
Working worksheets
Generally, I find that worksheets fall into three categories: a solution to a specific problem, teaching tools, and “working worksheets.” The latter means that the worksheet behaves like a form where people can vary the inputs for a standardized problem to drive problem development. Key sizing is a task that certain engineers may need to perform over and over again.
Combo Boxes from the Input/Output tab can be used for this purpose. However, there are other techniques you can use, like tables and matrices with lookup functions. “Working worksheets” can be embedded into Creo Parametric parts and assemblies to drive 3D CAD geometry. When you create a worksheet, think about how you can make it reusable for yourself and others and how you can leverage it for future problems.
Mathcad for Mechanical Engineers webinar replay
Learn from Dave Martin how Mathcad couples with 3D CAD programs as an engineering notebook.
The first thing I realized is how much I have learned from the Mathcad community and those who have participated in previous challenges. If you look at my worksheets, there are techniques that came from previous submissions. I highly encourage everyone to download previous challenge entries to see how you can improve your own work.
Secondly, I was reminded of something I used to tell all my Introduction to Pro/ENGINEER and Introduction to Creo students: The more thinking you do upfront, the fewer problems you’ll have later. I jumped in and created my worksheets essentially on the fly. Just a few minutes of planning would have generated better results.
Up Next
Check out the challenges from 2024 next.
The Mathcad Community Challenge series continued to 2024, with brand new topics to cover.
Learn From Your Peers
Dave Martin
Dave Martin is a Creo, Windchill, and PTC Mathcad instructor and consultant. He is the author of the books “Top Down Design in Creo Parametric,” “Design Intent in Creo Parametric,” and “Configuring Creo Parametric,” all available at amazon.com. He can be reached at [email protected].
Dave currently works as the configuration manager for Elroy Air, which develops autonomous aerial vehicles for middle-mile delivery. Previous employers include Blue Origin, Amazon Prime Air, Amazon Lab126, and PTC. He holds a degree in Mechanical Engineering from MIT and is a former armor officer in the United States Army Reserves.
