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Engineering Quarterly VOL.II. . . No.1

University of Mary

FREE

FALL 2019

W ELCOME ! Welcome to the first issue of Volume 2 of the Engineering Quarterly, a newsletter published four times a year by the University of Mary School of Engineering. In it we include articles from each of the four engineering majors that we currently offer on campus: Electrical Engineering, Mechanical Engineering, Civil Engineering, and Construction Management. We report on items and events of interest to engineering students, engineering companies, and alumni of our engineering program. You will read about our service and outreach activities in the community, the scholarships our students have been awarded, descriptions of the internships our students have worked at over the summer, guest speakers that have come in and enlightened us, and even assignment problems and projects that our students have completed as part of their course work. We hope that you will enjoy revisiting some of the problems that you had to solve back when you were studying for your own engineering degree – or at least you will enjoy the fact that you don’t have to solve them! The intended audience of the Engineering Quarterly is engineers, engineering students, engineering alumni, future engineering students, and people working in fields closely related to engineering such as science, computing, and mathematics. Enjoy! Terry Pilling

On Thursday, August 1, this year’s Go Baby Go service was completed and the cars were presented to the kids. Dr. Heather Lundeen and Rodrigo da Costa Aparecido are the faculty members overseeing the project. The students raised $1500 and five children from two to six years old benefited with cars activated with a push button or a joystick.

first design, this service group, under Dr. Lundeen and Mr. Aparecido, have added more electronic features and modifications to the frame. In 2017, one of the cars had a special compartment for to hold a backpack with the child’s feeding equipment. In 2018, more children were served and some received cars with a button and others with a joystick to control the car. Now, in 2019, one car was had a double seater for twin brothers, in which either of them could drive the car.

In 2006, Cole Galloway initiated an inclusion movement for children with motor impairments, called Go Baby Go, at the University of Delaware’s Department of Physical Therapy. This program includes modifying an electric toy car in any way necessary for a child with motor or developmental delays to access their environment and interact with their peers. After taking the first measurements, and a few weeks of design and build, we had the great pleasure of giving the cars to the children. Watching the children learn how to use the controls to accelerate, and move on their own along with or after their siblings was a beautiful experience.

DAY OF S ERVICE By ANTHONY WALDENMAIER

G O BABY G O ! By RODRIGO DA COSTA APARECIDO

The Go Baby Go project was initiated three years ago as a partnership between Physical Therapy and Engineering where Engineering faculty and students work side-byside with Physical Therapy doctoral students to adapt ride-on toys that help with the inclusion and development of children with special needs.

For the 2019 iteration of the project, 13 Physical Therapy doctorate students identified children in the Bismarck/Mandan area that would benefit from those adapter cars and then collaborated with the Engineering School to design a project that would best fit each individual child, and then build and modify the cars.

Every year, the University of Mary has a Day of Service where students and faculty go out into the community and perform service activities to help people and businesses in Bismarck-Mandan. Service activities range from seasonal work like raking leaves or picking up garbage to reading stories to young school students and everything in between.

We in the Engineering school also arrange for our own service events which have The PT students would decide how to best a uniquely engineering flavor to them. This accommodate the needs of the child and the The students worked over several weeks year, we traveled to Apple Creek Elemenengineering students would design the frame to design the modifications that would benefit tary School where we conducted two separate and electronics to meet those needs. the children and their families. From Cole’s projects.

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was selected for the outdoor exposure of the the student body and the broader community) boxes and the eventual wet conditions of daily and sponsors for the trip tentatively scheduled watering. to run from April 27th to May 7th .

First, students stained and performed minor maintenance on the Gaga Ball Pit which The bottom of each box was constructed was constructed on-site during last yearâ€™s day using six planks of wood which provide supof service event. port to plastic netting. The boards will carry the bulk of the load of the soil, while the netting will allow for drainage. In the spring, students will install landscaping fabric to the box before filling with soil and the various plants.

We were excited to see that the pit was loved by the elementary school students over the past year. A red stain was selected to bring school spirit to the Apple Creek yard. We were also very happy to see that the pit was in very good condition after a full year in the North Dakota weather!

We will be working with the Ayaviri Prelature and Salesian Sisters, to serve the needs of the Sisters, the children in a special needs school the Sisters operate, and/or the broader community of Ayaviri. The trip will conclude with the opportunity to hike to Machu Picchu (one of the seven wonders of the world!).

Another element of the design was mobility. The school did not want to secure these to the ground until further groundskeeping and landscaping plans are made. The boxes are light enough to be moved, but sturdy enough to last for many years.

We really enjoyed these projects and are glad to bring some engineering skill to the Day of Service. We hope to continue providing Engineering projects to the community for years to come!

J OIN OUR P ERU M ISSION T RIP FOR E NGINEERS !

The plan is to identify appropriate projects which may include work on water heaters, water purification, construction projects, and/or other needs and we will also conduct an on-site needs assessment for future trips. During the spring semester we will plan/prototype the project and then execute/deploy it when we arrive. Please note that it is not necessary for you to participate in the planning/prototyping stage in order to go on the trip.

By JAMES CARRICO AND PAULA

Second, we constructed above-ground KITZENBERG planter boxes that the school can use for gardens and plant science projects in the coming spring and summer. The final design was This coming summer, if there is enough based on several do-it-yourself designs, but participation, the School of Engineering will catered to the needs of the school. formally participate in the Prelature Support Mission (formerly known as the Medical Mission) in Peru, for the first time.

Six of these boxes were constructed onsite to allow each grade level to have its own set of planting projects. The boxes stand at 2ft high to allow younger students to plant, and each box is 4ft by 6ft. Green treated wood

The all inclusive cost of the trip will be We are seeking participants (both within between $2,000 and $3,000 and so we are cur-

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rently seeking sponsorship and fund-raising opportunities for participants from interested companies and community members. Participants: We need at least 21 student participants total (at least six from engineering). Please inform your friends and classmates!

the 2019-2020 academic year. Mark worked mainly with AutoCAD, ArcGIS, and Map Wise designing Fiber Optic broadband internet and 911 address layout for the South Dakota communities of Arlington, DeSmet, and Lake Preston. He also worked fixing and redesigning the existing and newly built aerial Contact: For more information or if you are and buried Fiber Optic internet in Glasgow, interested in participating in the trip, sponsor- MT along with other projects across North ing students, or if you have information about Dakota, South Dakota, and Montana. potential fund-raising opportunities we could Carter Trefz is a pursue please contact: Sophomore MechanDr.Jim Carrico, jdcarrico@umary.edu ical Engineering stuDr.Paula Kitzenberg, phkitzenberg@umary.edu dent who worked for

S UMMER P ROJECTS AND I NTERNSHIPS Many of our engineering students who are Sophomore level and above have internships each summer and they return with many great stories of the fun and interesting things they did while working at companies and firms. Here are some of their stories. Braden Benz is a Freshman Engineering student who worked for Advanced Mechanical this summer. Advanced Mechanical is an HVAC and plumbing company based out of Bismarck. However, many jobs require travel to Dickinson, Williston, and sometimes Montana or South Dakota. “This summer I mainly worked in Bismarck, but worked two weeks in Dickinson,” Braden said, “and my job consisted of a variety of things including installing new air conditioners and furnaces in residential houses as well as commercial job sites, installing ductwork in a variety of places (one including the new engineering center for the University of Mary) as well as demolition of old ductwork and heating and air conditioning systems also at the University of Mary. I worked in Dickinson for two full weeks. One week consisted of adding zoning to a school to provide a more comfortable atmosphere for each classroom. The next week I helped in the construction of a new electric company building.” “I also followed service techs and did service work on HVAC equipment at places like Steffes Manufacturing in Dickinson, ND, other housing HVAC equipment, and many other places around Bismarck. I learned a lot about how HVAC systems work and how to service them as well as knowledge from each job site about construction in general. The company was impressed by my work ethic and how easily I caught on to everything.” Mark Sammons is a Sophomore Mechanical Engineering student who started his first internship during the spring semester of his Freshman year with Finley Engineering in Bismarck, ND. He is continuing it through

Common Sense MFG under the Solar division. “The shop where I worked is located near Faulkton SD, however most of the time I was on the road doing installations all across North and South Dakota. I built and installed solar powered water tanks for cattle which involved welding, wiring, plumbing, heavy equipment operating, equipment repair, and truck driving. Some fun things I learned includes how to talk to customers about our product and troubleshooting broken systems. My boss was impressed with how diverse my skill set was stepping into this position and how eager I was to learn more.” Evan Sekerak is a sophomore Mechanical Engineering student who did an internship at Curation Foods, Inc. in Bowling Green, OH over the summer. Much of his work was using AutoCAD LT to lay out equipment on the production floor and to design new equipment. Some of his projects even came to fruition before coming back to school, including new conveyors and equipment stands. Through working in the fresh food industry, specifically green beans and salads, he learned a lot about sanitary design, refrigeration (38◦ F) and stainless steel, both essential to keeping products safe from harmful contamination. Working under the engineer there, he could see into the life of the company and corporate structure through online town hall meetings and visits with salesmen. He also got to travel to another plant in Hanover, PA and observe their operations, especially automation robots and systems. “I’ve always joked I could do CAD all day,” Evan says, “but during this internship, I really did and loved it!” Snædı́s Danı́elsdóttir is a junior Electrical Engineering student who worked for an engineering firm in Iceland. “This summer was my second summer interning at Verkı́s Consulting Engineers.” she said, “It is one of the largest consulting engineering companies in Iceland. I worked at their headquarters in Reykjavı́k as an assistant in Electrical Engineering projects in the Energy division. I spent most of my time there us-

ing DIgSILENT: PowerFactory to model a geothermal power plant, Svartsengi, which not only produces hot water and energy, but also produces the lagoon in the known Blue Lagoon. Some of my other projects were to draw both miscellaneous electrical and general schematics and even electrical floor plans for summer houses as a side project from another division. I did the above projects using AutoCAD. While working this summer, I learned a lot about high voltage substations and geothermal power plants. But, my coworkers really enjoyed that little boost of confidence they got when I asked them even the simplest questions related to our projects or electrical engineering in general. Lastly, not only were my coworkers impressed with my background as a journeyman electrician, but they were also particularly impressed with my quick learning skills with all the different programs they used.” Logan Mahoney is a junior Civil Engineering student who worked for Hopfauf Custom Builders outside of Mandan this summer. “The type of work that I did for Hopfauf included framing carpentry and a bit of finish carpentry. I usually worked on high-end residential and light commercial buildings. The sites themselves were scattered around the Bismarck-Mandan area, including the new Runnings in Mandan and then Zorell’s Jewelry Store on 9th in Bismarck. The main thing that I learned this summer was the fundamentals of framing carpentry and structure building. This experience gave me a different perspective on the construction process. It allowed me to see the implication of a design and see it go from design to construction. I also got to see the logical part of the building process as well.” Logan also worked with CHI St. Alexius on the design of a memorial for the employees they lost in an airplane accident last year. Logan’s team competed in a design contest for the memorial and his team’s design was one of the ones chosen for the memorial space. Logan will write an article about the design and the November 18 memorial service in the next Engineering Quarterly. Peter Collart is a Junior Mechanical Engineering student who worked at Steffes headquarters in Dickinson, ND. Steffes is a manufacturing company providing a lot of products for the oil industry. Additionally, Steffes creates Electric Thermal Storage products used in several states and countries, and contracts with companies such as Bobcat. Peter’s work consisted mostly of drafting and 3D design using Solidworks, Creo Parametric, and AutoCAD. “ I started the job with very little 3D design experience; however, now I would consider myself extremely proficient at 3D modeling and draft-

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ing! I was assigned a variety of projects but I will mention two of them: I worked with Bobcat and Steffes locations in Grand Forks to improve the SG 60 stump grinder which Steffes manufactures for Bobcat. This required many phone calls, 3D design, and updating drawing dimensions and welds. I also worked with another intern to perform Flow Simulations in Solidworks. We were assigned the task of delving into this new software to determine whether it is something that could be of use to the company. We simulated airflow and heat transfer in heating units manufactured by Steffes. It was a joy to work with Steffes this summer and I am pleased to announce the company has made the list of 50 Best Places to Work for the fifth year in a row! I would highly recommend the company to anyone looking for a job or manufacturer.” Michael Storick is a junior Electrical Engineering student who worked for Killdeer Mountain Manufacturing in Dickinson this summer. “I started out with a few miscellaneous projects, mostly prep work for a new airplane contract KMM had recently acquired. This involved a lot of excel entry, so over time I taught myself some VBA to speed up the repetitive parts of my job. I eventually worked on a VBA project to automate the prep work for future contracts, saving the company work-hours which would normally be spent on that.” The company was extremely happy with Michael’s work and they have contracted him to continue to work on the project remotely over the school year. William Blewett is a Senior Civil Engineering student who interned for Advanced Engineering and Environmental Sciences (AE2S) where he worked on civil municipal projects. His summer location was in New Salem and his job dealt with new utilities and roads. William was the resident project representative (RPR) on the job site and his hard work ethic and attention to detail impressed his bosses. Aaron Bales is a senior Electrical Engineering student who worked for Bobcat this summer and continues to work with them through their student development program this semester. “My initial project was to build up three environmental test stations for electrical testing. The test stations include a PC, a programmable logic controller, a breakout to six connectors for units (logic control system), among other things. I also performed testing of electrical equipment which includes vibration and thermal shock. My time at Bobcat has been extended to complete and assist in projects such as improving certain test equipment. These improvements include writing programs in C# to communicate with such equipment, performance, and

Engineering Quarterly FALL 2019

basically making something complex easy for low co-workers. This internship has decided everyone to use. I have learned more than I for me which area of Mechanical Engineering could imagine with my time at Bobcat. I’ve that I would like to pursue after graduation.” learned project management skills, ladder Ida Narbuvoll is a selogic, different programming languages, and nior Civil Engineermore about industry circuits. As I continue to ing student who had work at Bobcat I hope that I can continue to an internship at the improve the test environment and reduce time Bridge Division at the that is required to perform such tasks.” ND DOT this summer. Richard Bendish is a senior Electrical Engineering student who worked for Prairie Engineering this summer in their Bismarck Office and on-site at various job locations. His job included working in CAD and Revit doing modeling. He also performed circuit and load calculations and created spec sheets. Richard enjoyed learning load calculations for circuits in buildings and leaning more about the circuitry of larger scale power systems. The company was impressed with Richard’s hard work and his eagerness to learn.

She says, “I spent most of my time at the central office in Bismarck, but I got to go out on several field trips to various structures in North Dakota. The internship position was set up so that I split time between the Hydraulics Section and the Bridge Management Section, and my main tasks included preparing field review documents, issuing overload permits, and reviewing bridge inspections. It was very interesting to get a better insight in all the work and challenges that goes into documenting data, inspecting, maintaining, and improving the current bridges and culverts in the state. Particularly, I enjoyed learning about bridge inspections and how the process Levi Clagett is a se- of storing and analyzing the data from all the nior Civil Engineering structures is done.” Caleb Weisgarber is student who worked a senior Civil Engifor the City of Bisneering student who marck Engineering worked for Holly Beck Department as a reSurveying and Engiturning intern. Levi neering based out of says, “I worked all Bismarck this summer. over the City of BisHe says “I worked marck on all the roads, sidewalks, and new on surveying the Bisdevelopments. My job was to inspect the quality of the construction work and ensure marck airport, Minot bypass, and multiple it was up to specifications. The main thing I property surveys around the Bismarck area. I learned was how to stay very organized, and was able to further my education in the area that it’s a challenge when it rains after pour- of what happens on a construction site, espeing concrete. I’ve been impressed with how cially in the surveying area. They were very hard construction crews work everyday. It’s pleased on how I was able to hit the ground been a great experience and I’ve learned an running because of what I learned in our surincredible number of practical skills for this veying class and the prior experience I have had.” career field.” Jarad Zittleman is a senior Mechanical Engineering student who worked for Bartlett & West this summer in Bismarck as was part of the MEP (Mechanical, Electrical, Plumbing) group. Jarad mentioned that “Bartlett & West are the mechanical engineering contractor for the University of Mary’s new engineering school, so I was lucky enough to do a lot of work on the project. I was able to attend weekly on site job meetings and see how the school was coming along. I also played roles in jobs in Kansas and Missouri. I did building heat load calcs, submittal reviews, lighting calcs, mechanical and pluming design, and any extra drafting that needed to be done throughout the summer. I was able to learn how consulting engineers interact with clients and achieve deadlines along with understanding any new engineering software. I also traveled down to Topeka to spend a week with the other interns throughout the company. My ability to learn new things, past work experience in plumbing and HVAC, and time management and people skills were all things that impressed my fel-

Evan Anderson is a Senior Electrical Engineering student who went to Phoenix Arizona to work for Central Arizona Project this summer. Central Arizona Project (CAP) is Arizona’s single largest resource for renewable water supplies. CAP is designed to bring about 1.5 million acre-feet of water from the Colorado River to Central and Southern Arizona every year. More than 5 million people, or more than 80% of the state’s population, live in Maricopa, Pima and Pinal counties, where CAP water is delivered. CAP works to bring water from the Colorado River to three counties throughout Arizona. The canal stretches about 340 miles from Lake Havasu to south of Tucson. Along the canal there are 15 pumping stations where the water is lifted and then descends along the canal. CAP is also the largest power consumer in the state of Arizona. Evan said, “what I did was create a document on the SEL-421 relay settings used at one of the main pumping stations as well as working on the Human Machine Interfaces (HMI) at several plants

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and working on alarms placed on each motor of each pump using PLCs. I was able to learn a lot about the power industry as well as substation design and the future of the power industry by attending conferences hosted by GE and Schweitzer Engineering Laboratories. While in my position I noticed quite a large age gap in the field of power systems. Many of my co-workers were approaching retirement age and although experts in their field were not technologically inclined so most of the work I did with PLC’s or on the HMI’s or with the relay software was pretty impressive to them. Being able to navigate software, use terminal commands, and use ladder logic/function blocks were some of the skills they liked.” Noah Krebs is a Senior Civil Engineering student who worked with BARR Engineering this summer in Bismarck. His projects included Civil 3D, Pipeline routing and planning, estimation costs for projects, CCR Removal Inspection, Construction Observation, SPCC Plans, Microsoft Office, and Risa. Noah says “my summer was mainly focused around Civil 3D, which I had no experience in before I started working at Barr. I quickly became one of the Engineering Techs that had Civil 3D sheets to work on every day. I learned a lot about the consulting world of engineering and really enjoyed what I did this summer. My different knowledge of a variety of software impressed them. It showed that I was versatile at whatever they put in front of me I would find a way to get it done. They also liked how personable I am, which is a very important attribute when being in a consulting company. My internship may have ended but I am planning on staying on so I am in the process of interviewing for the Civil Engineer-Entry Level position in the Bismarck location.” Josef Sollmann is a senior Electrical Engineering student who worked for Killdeer Mountain Manufacturing (KMM) at their Dickinson site. “I spent the summer designing components and processes for machines which KMM wanted to use to automate a process in their manufacturing line. Pretty much I was given the rundown of what they envisioned it would look like to implement it in their facility and what they waned it to be capable of and then I was given free range to make it work however I could. By the time I left I had completed my project to the point where it could start to be used. I learned a great deal of 3D design and modeling using Auto-CAD, as well as programming in Visual Basic for data manipulation and organization. I also learned about creating processes for operators while adhering to outside specification. I think the company was impressed with the ideas and solutions I came up with to make everything work. They also were impressed with a code

Engineering Quarterly FALL 2019

I worked on with another intern which compiled specific data pertaining to our project into readable format for the automated machines, saving them about 1-2 months of work had it been done by hand. Overall, I think they were pleased with the work that I put in and the progress I made on my project.”

all kinds of medical equipment.” He impressed his coworkers and supervisors with how quickly he learned each modality. He enjoyed working in the medical engineering field and says that he may be interested in a career in the area upon graduation but is going to keep his options open since he is interested Alex Davis is a senior in a wide range of subjects. Civil Engineering student who completed P ROBLEMS TO T RY his 2nd year internship with Braun Intertec this past summer R EINFORCED C ONCRETE P ROBLEM out of the Dickinson, By ERIC GARCIA ND, office. Alex says, “most of my work was around Watford City ND. The services I provided included soils and concrete testing and the work varied from a new high school being developed, to an old city receiving brand new roads, or even performing epoxy grout tests for a gas plant on its final stages of completion. I was all over the place this past summer learning from engineers, contractors and even fellow coworkers. The thing that most impressed me the most was how much work is available west of Bismarck and how much North Dakota is growing as a whole. I am excited to see where we are at in the next 10 years and am very Determine the value φMn for the section thankful for the opportunity given to me this shown. past summer.” Austin Konschak is a senior Civil Engineering student who worked for Northern Technologies, LLC out of Bismarck as an Engineering Technician Intern. Austin says, “I worked primarily in Bismarck, ND and New Town, ND. I would take my work truck to various construction sites to perform plasticity tests on concrete or nuclear density gauge tests on compacted fill. Additionally, I would perform proctor and gradation tests on soil samples brought into the lab. I learned more about the construction process and how to handle confrontational situations on job sites. I cannot speak for my boss, but I felt that I was a hard worker and that I handled my unpredictable and extensive work schedule very well.” Phillip Springsteen is a senior Electrical Engineering student who worked at GE Healthcare this summer. He worked as a field engineering apprentice in Bismarck and Minot. His job involved repairing imaging equipment such as MRI, CT scanners, ultrasound imagers, etc. Later in the summer he worked as a biomedical engineering apprentice where his job entailed repairing biomedical equipment such as electro-surgical units, IV pumps, blood warmers, etc. Phillip says, “I traveled around the state to various hospitals and repaired

C IRCUITS I P ROBLEM By TERRY PILLING

In the following circuit each of the resistors is 1.8 kΩ, find the resistance between points a and b. Note: you may want to use a ∆ − Y transformation.

Check your answer using NGSPICE.

DYNAMICS P ROBLEM By ERIC GARCIA

A robot arm moves so that P travels in a circle of radius 0.8m about Point B, which is not moving. Knowing that P starts from rest, and its speed increases at a constant rate of 10mm/s2 , determine

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(a) the magnitude of the acceleration when by far one of the most phenomenal experit = 4s ences that faculty members at the University of Mary encounter. (b) the time for the magnitude of the accelerThis retreat took place this year from Friation to be 80mm/s2 day October 18th until Sunday October 21st. In this retreat that starts from 8:00 AM until around 9:00 PM everyday, newly hired, faculty members from Engineering, Theology, P OWER S YSTEMS P ROBLEM Communications, Biology, Arts and other disBy KHALID J. OWEIS ciplines throughout the university come together under the inspiring guidance of the University President Monsignor Shea, the deep support of the VP for Academic affairs Dr. Diane Fladeland, and the intellectual care of the head of Catholic Studies Dr. Scott Cleveland, to talk about how to work together and incorporate Catholic and Benedictine values. This beautiful and mind lifting dialogue is immersed in an environment of friendship, fine dining and great entertainment. At the end of the retreat, new faculty members pray The RLC circuit is driven by a sinusoidal the Sunday evening Mass together with the voltage source, vs (t) = 100 cos (500t + 30◦ ). students and the faithful at the Annunciation Find the current, i(t), as a function of time. chapel at the University, and they are commissioned after Mass by Monsignor Shea in a delightful ceremony where they are given icons that carry the words of the Virgin Mary after N EW FACULTY F ORMATION AT M ARY the annunciation that say: “Ecce Ancilla DoBy KHALID J. OWEIS mini” and mean: “Behold I am the handmaid of the Lord” This overwhelming experience allows facThere are numerous aspects of the Uni- ulty members to talk with each other openly, versity of Mary that makes it different than agree and disagree respectfully with each any other university in the United States. The other, and laugh and share their emotions tomost important of all these aspects is its cul- gether. The result of this remarkable experiture. The culture of the University of Mary is ence is a group of faculty members that feel a culture of unity. All of the employees, staff very close to each other and to the adminand faculty members work in unity for the istration. They share together a clear vision same mission, purpose and goals. The ques- on what the purpose of the university is. A tion then arises, how is this grand teamwork servant leadership vision that started with the being achieved? Who is behind that, and how Sisters of the Annunciation Monastery and is is it being lead? There are several answers being carried out to this day by the previous to this important question. However, one of and current administrations. They feel enthe most prevalent answers is a very special trusted with the souls and futures of their stuevent that every faculty member at the Uni- dents. They go about it in the beautiful, warm, versity goes through at the time of their hire. and embracing North-Dakota culture. Where This event is called: New Faculty Formation does an experience like this take place anyRetreat. where in Higher Education except at the UniPrior to this wonderful event, newly hired versity of Mary? faculty members are given a packet that inIn a place like no-other, the University of cludes a number of select passages to be read Mary Faculty members and departments do thoroughly prior to the event and discussed not co-exist. Quite the contrary, they work in detail in this three-day retreat. The se- hand-in-hand in union under a parenting adlected texts are from the Catholic Intellectual ministration that guides them and helps them and Benedictine Wisdom Tradition. They in- achieve their ultimate mission, the formation clude: “The University and The Church” by of our students according to the Christian, the late founder of the Catholic Studies pro- Catholic, and Benedictine tradition. gram Dr. Don Briel, “Ex Corde Ecclesiae” by Saint John Paul II, excerpts from “The idea of a University” by the newly canonized St. John Henry Newman, “Catholic Universities: Dangers, Hopes and Choices” by Alasdair MacS OLUTIONS TO P ROBLEMS Intyre, “The Decline and Fall of the Christian College” by James Burychael, two addresses of Monsignor James Shea, the first took place on his inauguration as President at • Reinforced Concrete problem: the University of Mary, and the second is en- φMn = 379.1kf t titled: “Ex Corde Ecclesiae at the University of Mary”, and numerous other readings that • Circuits I Problem: discuss the farm culture of German-Russian The resistance between points a and b is immigrants of North Dakota, the Sports and R = 2 kΩ which we check by hooking up Catholic Studies program, as well as the Mis- a 2 kV battery between a and b and verifying sion and Identity Statement at the University, that 1 A of current flows from it. Here is an along with several other insightful readings. NGSPICE netlist that simulates the circuit: While this sounds like a tedious task, it is

6 circuit.net ∗ Pyramid r e s i s t o r network V1 R1 R2 R3 R4 R5 R6 R7 R8 R9

a a a d c c e d d f

0 c d c e 0 0 e f e

2k 1.8 k 1.8 k 1.8 k 1.8 k 1.8 k 1.8 k 1.8 k 1.8 k 1.8 k

. control op p r i n t i ( V1 ) . endc . end

We run this code using: ngspice < circuit.net The output is: No. of Data Rows : 1 i(v1) = -1.00000e+00 which says that 1 A of current flows from the battery. Hence we have verified that the total resistance between a and b in the circuit is 2 kΩ. • Dynamics problem: a) a = 10.20mm/s2 , b) t=25.2 sec • Power Systems problem: i(t) = 0.707 cos(500t − 15◦ ) A

T ESSERACTS , H YPERCUBES , AND I NFINITE D IMENSIONAL R ESISTOR N ETWORKS By TERRY PILLING

A Tesseract is a 4-dimensional ‘hypercube’ which is what happens to a 3dimensional cube when you add an additional direction at each vertex. In the same way that adding a 3rd direction to the vertices of a square give you a cube, adding a 4th direction to the vertices of a cube give you a Tesseract. Unfortunately, one cannot draw a Tesseract on a piece of paper like we can for a square and a cube. However, the projection of it into 3-dimensional space looks sort of like the above picture. One can continue in this fashion to create ‘hypercubes’ of higher and higher dimensionality by simply starting with a cube of dimension d and adding another direction to space and then attaching another edge to each of the vertices to produce a cube of dimension d + 1.

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This article comes as the result of a guest lecture that we had in my EEL 206 Circuits I, course. As we often do in our engineering courses here at the University of Mary Engineering School, I invited an engineer from industry to come and speak to the students. For this course, I invited Clyde Eisenbeis, a retired electrical engineer, who worked in electronics for many years with the company 3M, to come and speak to the class. He spoke about his career as an electrical engineer and he imparted words of wisdom and advice to the students. During his talk he challenged the students with the following problem. Find the resistance between points a and b in this cube made out of 1Ω resistors:

then three, and so on, until you start to see the general pattern. In the case of a resistor cube, the zero dimensional case is just a single point with a total equivalent resistance of zero. This is not very interesting since there are no resistors, so we will instead begin with the next case which is a one-dimensional cube. This is just a line, containing a resistor, connecting two points. The resistance is just that of the resistor – which in our case is 1 Ω. A twodimensional cube is a square: b

The way that Clyde solved this in class was to explain that, because of the symmetry of the situation, the vertices which are one step away from a must be at the same voltage since, if you rotate the entire diagram by 120◦ around the vertex a, you would get exactly the same diagram. Hence those voltages could not be different from each other. Since the voltages one step away from a are the same, then if you hooked up a battery across a and b, 1/3 of the current would flow down each edge away from a. Next, each of the edges which continue from a split into two and so half of the current, 1/6 of the total, flows each way. The exact same argument from the opposite corner tells us that 1/3 of the total current must flow toward b through each edge meeting b. The entire cube is therefore equivalent to a circuit with three parallel resistors in series with six parallel resistors in series with three parallel resistors. The total equivalent resistance is thus: 1 1 1 5 Req = + + = Ω 3 6 3 6 at this point in the class I interrupted with a grin, “I am going to assign the students to find the equivalent resistance of a 4-dimensional Tesseract!” to which one of the students, Margaret Kuhar, said with a smirk, “If you can draw it on the board then we will find the resistance.” Haha. Clever girl. She knows it is impossible to draw a four dimensional cube in three dimensional space. After the class I couldn’t help but continue to think about the problem of the equivalent resistance of a Tesseract. I figured, why not see if I can do it? Except, instead of just the Tesseract, why not try to find a general formula for the equivalent resistance of an ndimensional hypercube for any value of n? There are a number of ways to attempt a solution to problems like this, but the one I like to try first is to start with zero dimensions, then move to one dimension, then two,

So for n = 4 we have R4 =

3 X k!(4 − k − 1)!

k=0

1 1 1 1 + + + 4 12 12 4

which is the correct expression that we found before. The reader may wish to challenge themselves to show that, for n = 5 and n = 6 the formula gives: 1 1 1 1 1 + + + + 5 20 30 20 5 1 1 1 1 1 1 R6 = + + + + + 6 30 60 60 30 6 R5 =

Finally, let us see what is happening to the resistances as the dimension of space increases. So far we have:

R1 R2 R3 R4 R5 R6

= 1.000 = 1.000 = 0.833 = 0.667 = 0.533 = 0.433

We see that an imaginary line connecting a and b has a symmetry under reflection. I.e. if we think of the line as a mirror on side is the mirror image of the other – meaning there is a symmetry under reflection, or 180◦ rotation about the axis coming out of the page. The We can write a C program which produces circuit is equivalent to two resistors in series, these numbers: in parallel with another two resistors in series. hypercube.c The equivalent resistance is therefore: Req =

1 1 + = 1Ω 2 2

Next we move on to three dimensions, and find the answer we found above for the cube. The case of the four-dimensional cube, or Tesseract, is just a vertex a with four edges leaving it in parallel. The other side of the Tesseract has vertex b with four edges leaving it in parallel. Each of these edges must end on a vertex and each vertex must have four edges attached to it. Hence each of the four lines must then branch into three more lines for a total of 12 edges in parallel. The total resistance is: Req =

1 1 1 2 1 + + + = 4 12 12 4 3

So we have found that the equivalent resistance of a Tesseract. Now we want to see if we can go even further! Can we distinguish a general pattern here? Let’s write out the sums we have found so far: 1 1 1 1 n=2: + 2 2 1 1 1 n=3: + + 3 6 3 1 1 1 1 n=4: + + + 4 12 12 4 n=1:

# i n c l u d e < s t d i o . h> # i n c l u d e < s t d l i b . h> # d e f i n e N 30 f l o a t Cube ( i n t k , i n t n ) { float i ; f l o a t R1 = 1 . 0 ; f l o a t R2 = 1 . 0 ; f l o a t R3 = 1 . 0 ; f o r ( i = 1 . 0 ; i <= n−k −1; i ++) R1 =R1∗ i ; f o r ( i = 1 . 0 ; i <= k ; i ++) R2=R2∗ i; f o r ( i = 1 . 0 ; i <= n ; i ++) R3=R3∗ i; r e t u r n ( R1∗R2 ) / R3 ; } i n t main ( i n t a r g c , char ∗∗ a r g v ) { f l o a t R; int n ; f o r ( n = 1 ; n<=N ; n ++) { R = 0.0; f o r ( i n t k = 0 ; k<n ; k ++) R = R + Cube ( k , n ) ; p r i n t f ( ”%d %f \ n ” , n , R ) ; } return 0; }

Let’s graph the output:

this is actually a very interesting pattern of numbers called a Harmonic Triangle. Each entry is equal to the sum of the two entries directly below it and below to the right. For example 1/2 = 1/3 + 1/6, or 1/6 = 1/12 + 1/12, etc. The general formula in n dimen- As n → ∞ the graph approaches the curve sions is: R = 2/n. So an infinite dimensional hypercube will have resistance R → 2/∞ = 0. n−1 X k!(n − k − 1)! For those who are interested in exploring this Rn = (n − 1)! problem further here is a puzzle for you to try: k=0

VOL.II. . . No.1

Problem: We have given the formula for the resistance from one corner to the opposite corner of an n-dimensional hypercube. Can you find the resistance between any two points a distance of k resistors apart?

Engineering Quarterly FALL 2019

and also in the preprint by Prof. Nicholas Pippenger from Harvey Mudd College entitled The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, which can be found at https://arxiv. org/abs/0904.1757. Prof. Pippenger also gives the detailed derviation of the The answer is given in D. Singmaster, Prob- asymptotic limit, R = 2/n, which we have lem 79-16, SIAM Review, 21:4 (1978) 559, found experimentally.

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