Engineering Quarterly VOL.II. . . No.2
Welcome to the second issue of Volume 2 of the Engineering Quarterly, a newsletter published four times a year by the University of Mary School of Engineering, which includes articles from each of the four engineering majors that we currently offer on campus: Electrical Engineering, Mechanical Engineering, Civil Engineering, and Construction Management. Hands-on design constitutes a large part of our engineering degree program in all disciplines and in all years from freshman through senior. In the fall semester of the senior year, our students do a senior design project within their engineering discipline and in the spring semester of their senior year they do a crossdisciplinary design project. This fall the senior design projects were presented to the public during the week of December 9 to 13. We will outline below some of those projects as well as other design projects done by our sophomore and junior level students in their courses. In this issue of EQ, we will outline the first semester design projects as well as other items and events of interest to engineering students, engineering companies, and alumni of our engineering program such as service and outreach activities in the community, scholarships, internships, guest speakers, and even assignment and exam problems. Enjoy! Terry Pilling
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The project was broken into two phases due to timing. Phase 1 was the concrete pad and monument/benches, and phase 2 was the ramp going from the monument to the hospital. The concrete pad for the monument was poured on September 26, and the monument along with the benches was shipped November 1 and installed shortly thereafter. Waldos Construction donated the concrete work at cost minus 10% and Coldspring donated Over the course of several months, I the benches and monument also at cost minus worked with Kilee Harmon, the Development 10%. The CHI St. Alexius Health Auxiliary Director for CHI St. Alexius, to design and ladies funded the rest of the project. plan the construction of the memorial space. The project began as a design competition that the engineering students undertook during the spring semester last year. We broke up into groups and each group designed a memorial for the space at the hospital.
We then held a poster session where the staff from CHI St. Alexius studied each of the student designs and decided on the winning project. During the summer, professor Waldenmaier and I met with representatives from St. Alexius hospital on several occasions to work out final design plans.
CHI S T. A LEXIUS M EMORIAL By LOGAN MAHONEY
On November 18, 2018, a Bismarck Air Medical plane crashed, killing medical staff members. Todd Lasky, a 48-year-old pilot, died in the crash, along with nurse Bonnie Cook, 63, and Metro Area Ambulance paramedic Chris Iverson, 47. They were en route from Bismarck to Williston to pick up a neonatal patient when the crash occurred. Shortly after the crash, more than 1,400 people, including hundreds of emergency responders from throughout North Dakota, attended a civic memorial service for the three. Dozens of ambulances participated in a procession through Bismarck. This year, on the first anniversary of the crash, Bismarck Air Medical and CHI St. Alexius Health each held a private memorial services and the hospital unveiled and blessed a memorial space outside its South Broadway entrance. The space is a sitting area designated for reflection and was designed by our University of Mary School of Engineering students.
University of Mary
I started the construction of a module for the final design and presented it to the Benedictine Sisters of Annunciation Monastery. Following the presentation, I began work on the architectural plans for bidding. St. Alexius wanted a simple but elegant design. The main platform was made in the shape of an arch. The arch shape is a symbol of the CHI hospital. It also creates a great design flow for the memorial as it establishes a very welcoming feel, which will also draw people in to look at it. The next major elements were the benches and the main centerpiece. The centerpiece is a three-sided stone slab, with three benches surrounding it. Each of the three victims will have a side of the slab devoted to them. Then three benches will then sit perpendicular to each face. This will allow people to come in, pay their respects and stay for a little while if they so choose. There will also be short 2’ bollards to light up the memorial at night.
Logan Mahoney
FE E XAM P REPARATION We are offering an FE Exam Preparation course this semester which runs on Tuesday and Thursday evenings from 6pm to 8pm at our Front Avenue location north of the Bismarck Event Center. The course is open to anyone who would like to take it. So if you are an engineer who is currently working in industry and would like to take the FE exam this spring, or if you plan to take your PE exam and would like a refresher on the material, you can sign up for the preparation course and then come to whichever sessions cover topics that are on your specific exam. The course covers all of the topics on the Fundamentals of Engineering exams for civil, electrical, and mechanical engineering. The format of the class is quite informal. We simply meet at 6pm and go through problems from old FE exams and from the various study guides, reminding students of the key elements to look for and explaining the fastest ways to solve each type of problem. The course began on January 9 and will continue through to March 17 and it is our hope that most, if not all, the students will then immediately take the FE exam. We offer this course every spring semester and it will remain open to anyone who would
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creative young scientists to choose a career in engineering. Of course, we at the University of Mary School of Engineering jumped at the chance to attend the event, to tell our own individual stories, and to encourage these students to choose the University of Mary for their undergraduate engineering degree. I told my story to the students and disN ORTH DAKOTA S OCIETY OF cussed our engineering program and how it P ROFESSIONAL E NGINEERS differs from other schools. I followed this by handing out issues of the Engineering Quarterly along with applications materials and inThe NDSPE Chapter 3 held its November formation packets for the University of Mary meeting at our Engineering School’s tempo- School of Engineering.
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like a refresher on the FE exam subjects. For more information and to enroll in this course please see the University of Mary’s continuing education site here: https:// continuinged.umary.edu/
rary location on Front Avenue. They brought pizza for the students and professor James Carrico gave a presentation about our upcoming engineering service trip to Peru that he is organizing for spring 2020 (see the article about the Peru trip in the Fall 2019 issue of Engineering Quarterly). The students got to see what the NDSPE has been doing lately and the hope is that they will be encouraged to join this and other engineering service organizations and remain members throughout their careers.
N ORTH DAKOTA P ETROLEUM C OUNCIL
A PP C ONTROLLED V EHICLE By AARON BALES, RICHARD BENDISH, AND PHILLIP SPRINGSTEEN
They call it a Mobile App Controlled Articulated Vehicle (MACAV). Springsteen designed the system, and developed the code for the microcontroller boards, and the cellphone app so that operator can control the vehicle from the smartphone. Based on articles from the manufacturer on how to run the motors and how to control the motor controller boards using command line from a computer, he put together a stand-alone system controlled by microcontroller development boards, which is able to handle all the information to run the motors, calibration of encoders, feedback of the system, WiFi connection, etc. and pair it with an app that has an operator-friendly user interface.
The first group of senior electrical engineering students built an articulated electric vehicle controlled by a smart phone. The project was sponsored by GenesysGates Technology of Bismarck.
Science Olympiads, math competitions, and science bowls are very important events
for encouraging students to excel in STEM fields and to choose STEM careers. We love to attend and host these events and we also love to congratulate and encourage the winning teams. In the evening of Nov. 20, 2019, the North Dakota Petroleum Council hosted the winning team of the Legacy Science Olympiad to their location to eat pizza and listen to engineers from various branches of engineering tell how they ended up with careers as engineers, as well as stories about the amazing projects they have worked on up to now.
The group also developed the testing equipment used to control and collect data from the cooling system and the actuator control tests in order to choose the final design for the cooling system and the position for the linear actuator.
The sponsor gave the students four 3-phase induction motors, two controller boards, two linear actuators, and batteries, with the goal of making them into an electric vehicle. Aaron Bales (Bismarck, ND), Richard Bendish (Mandan, ND), and Phillip Springsteen (Graham, WA) spent the semester working on the code that would run the controller boards and the motors; reliability tests for the linear actuator control, and its positioning and the cooling system design of the controller Microcontrollers were programmed to run The goal was to encourage these smart and boards compartment. a predetermined sequence and variables were
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observed, such as current and temperature. The result of these tests were used to choose the final design for this 3-credit design class. “I couldn’t have done this project without the rest of my team members,” said Springsteen, who is also an Academic AllConference and All-American grappler for the University of Mary wrestling program. “This was a group project and we all contributed. Everyone’s part in this project was necessary.” Springsteen adds that mechanically, this project is nowhere near complete. They hope to make adjustments and be able design it so that the MACAV is able to remotely complete tasks around the home – as if the operator was playing a video game on their phone.
W IRELESS P OWER T RANSFER By JOHN SANTIAGO AND JOSEF SOLLMANN
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the students had a limited budget and were not of the project. These students were assigned allowed to go over it: $200 including every- the following tasks: 1) The projection of the thing. required text to the LED matrix, 2) Calibrating the temperature sensor, and getting a good reading from it to be projected to the LED matrix, 3) and finally the students added an intermission display between each phase of the required text one of these intermediate displays was a clip of Conways’ Game of Life. After assembling the LEDs in the box, the power supply and the rest of the components, the front side of the box was covered with a semi-transparent display glass to diffuse the high illumination of the LEDs and make the text being displayed comfortably legible. The LED sign was a great success, and The first thing that the students thought it will be displayed in the new Engineering of: Are a hundred LEDs enough to make the writing legible? To answer this question, they building once it is done. To take a look at opened a spreadsheet and tried to write their a time-lapse showing the steps of the project names by filling the cells in the spreadsheet. using this link: https://tinyurl.com/ They realized that they needed more LEDs to UmaryLED make the writing on the sign readable from a distance. Consequently, after studying the size of the box that housed the LEDs, and studying the proper minimum distance between two adjacent LEDs for correct illumination, the students decided that there should be 720 LEDs in that box to properly display the text. They were to be put in a matrix of 48 × 15 LEDs.
Senior students Josef Sollmann and John Santiago studied an off-the-shelf smart phone charging station in order to discover how the various properties such as phone case thickness and phone position on top of the charging pad affect the station’s ability to charge the phone. These variables were chosen based their perception as consumers of issues when charging their devices. They did numerous trials and statistically analyzed the data to The next step was to build the box itself. come up with an optimal configuration and tested the charging capacity of the wireless They got the materials from the local hardcharged and wireless charger receiver used in ware store, and assembled the box of the digital sign. The box was then painted black the project. to give it a professional feel. The box had two main levels and compartments: The back compartment housed the DC power supply, the Arduino microcontroller that controlled D IGITAL LED D ISPLAY the LEDs and the wires. The wires went By KHALID OWEIS out from the back compartment very neatly to the front compartment. The front compartDuring the fall 2019 semester, four of ment had the LEDs and the plastic mesh grid the Electrical Engineering junior students in that separated the LEDs for clear illumination. the Electronics I Lab: Gabriel Zimmer, Josh They glued the LEDs with even spacing to Reiner, Snædı́s Danı́elsdóttir, and Jeff Strong, the wall that separated the compartments in were assigned an interesting term project. The the box. Then they soldered the wires to the project consisted of a 30 by 40 inch electronic LEDs. sign with a colored LED matrix that included at least 100 LEDs. The sign should present a welcome to the visitors of the School of Engineering. It should also state the six Benedictine values that the University of Mary holds very deeply (Community, Hospitality, Moderation, Prayer, Respect for Persons, Service) with optional explanation for each one. These values can be found here: https://tinyurl. com/benvalues. Furthermore, this sign has to display the current indoor temperature. While the hardware design and building On top of that, they had to build everything from scratch, including the box that was being done by two students, the other two housed the LEDs and the equipment. Finally, students were working on the software side
E QUINE V IBRATION T HERAPY P LATE By JARAD ZITTLEMAN, BAILEY MEYER, REBEKKAH KELLER, AND DYLAN LARSON
Our senior Mechanical Engineering students presented their project of designing a therapeutic vibration plate for horses, particularly those that compete in rodeo competitions. This project was requested by Professor Alanna Dennison in the Athletic Training and Kinesiology Department. The requirements of the design were that it needed to be low cost, control multiple vibration settings, operate on uneven terrain, and portable. The students used CAD, initial prototyping, and performed engineering analysis in coming up with a final design for a vibration platform. Furthermore, they also developed a dynamic simulation model on Octave to determine the forces and stresses potentially occurring at the joints when a horse experiences the vibrations generated by the plate.
DYNAMICS P ROJECT By ERIC GARCIA
Our sophomore Engineering students had
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a catapult competition in their Dynamics class snow removal, and traffic flow issues, and dein which they were tasked to design and con- sign the campuses stormwater distribution and struct a catapult which would launch a projec- drainage system. tile to hit their instructor, Dr. Eric Garcia, in the face at a distance at least 10 meters away. The students were limited to only two shots to ensure that their design was based on their calculations and not just trial and error. There were four teams of three students and each team was only provided $20 worth of wood materials and any scrap wood in the machine shop. Each team came up with a unique design that used either ropes or bungee cords. One team even designed and built their own spring using our Lathe in the machine shop. Student were also tasked to analyze the campus’ infrastructure, identify any functional and/or aesthetic issue and design an engineering solution to the problem.
One team successfully hit Dr. Garcia in the face and was the first team to do so in the two years this project has been assigned. The team (pictured) that succeeding in hitting their target was Karsen Meyer, Daniel Minor, and Peter Smith. In addition to a built catapult, students were tasked to put together design reports documenting their design process and calculations.
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knee joint is not planar because the amount of rolling and slip is not identical on the lateral and medial sides of the joint. To simplify the problem, students in design of machinery focused on the lateral side, which exhibits more rolling than slip and thus chose to model the knee as pure rolling contact. In so doing, the students were able to approximate the instantaneous center of rotation (ICR) between the femur and tibia to be always be at the point of contact between the two and thus follow the curve of the femoral head. This simplified the design of a knee-brace mechanism to being simply the design of a four-bar mechanism with an ICR that followed the trajectory described by the femoral head and the appropriate joint angle for any given point along that trajectory.
Both teams came up with creative engineering solutions and designs to solve various issues they identified on campus such as indoor walkways between campus buildings and building expansions.
G IVING T REE
The school of Engineering took part in the university-wide giving tree competition for By ERIC GARCIA the first time this year. Each team’s design reports totaled close to The trees are located in each of the schools 100 pages of calculations, technical drawings, and they contain ornaments which list items The senior Civil Engineering students cost estimates, and other design details! were separated into two groups for their design projects. The first group consisted of Austin Konschak, Alex Davis, Colin GudD ESIGN OF M ACHINERY ereit, and Caleb Weisgarber, and the second By JAMES CARRICO group consisted of Levi Clagett, Jackson Penner, and Robert Spady. C IVIL E NGINEERING S ENIOR D ESIGN
Their project was to develop their own ‘landscape masterplan’ for Phase II of the University of Mary’s Vision 2030 plan to improve the campus’ layout, functionality, and aesthetics. The overarching goal of the project was to improve the infrastructure of the University of Mary campus. The specific tasks assigned were to design a reinforced concrete parking structure, redesign the roads/pavements of campus to address campus entrance, parking,
In design of machinery, students are tasked with simulating the motion of physical systems and synthesizing their own mechanisms ideally with some practical end (to give authenticity and purposefulness to projects). One learning outcome of these design projects is how fundamentally different devices can exhibit identical motion to each other. This is a powerful insight for design of mechanisms because it illustrates the wide is the solution space for design problems. If desired outcomes can’t be achieved with a cam mechanism, what about a linkage, or non-circular gearing? This equivalence between different mechanisms suggests an approach to design. For instance, one method to design of a knee brace might be summarily described as identifying the type of joint a knee is and identifying mechanisms that produce an identical motion. In truth, a knee joint is not a planar joint and exhibits both rolling and slip. The
that are needed by various non-profit organizations in the Bismarck-Mandan community. The idea is that the faculty and staff will select an ornament from the tree, purchase the item, then return the item to the collection box beside the tree. We got our tree on the Dec. 3 and we found to our delight that all of the ornaments
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Engineering Quarterly WINTER 2020 played games, ate, and discussed how the semester went. It was a successful semester even though it has been a difficult one given that our
had been completed and the items returned to the collection box by December 4! As a result, our faculty and students enjoyed a celebration of our giving on Decemcourses were split between campus and downtown Bismarck. Our students and our faculty haven’t complained, and in fact, I think we all feel we have plenty to be thankful for this year.
F UN S TUFF Here are some problems taken from our various engineering courses that our students had to solve. The solutions can be found at the end of this newsletter. C IRCUITS II P ROBLEM ber 17th which consisted of a donut and hot chocolate buffet! The buffet lasted all day and the students were able to have a cup of hot chocolate and a donut between their final exams. Thanks to Melinda Seidling for helping the Engineering School get involved in the Giving Tree this year.
P ROFESSOR A NTHONY G ARCIA N OMINATED FOR AWARD
By NANCY KELLY
Write an NGSPICE netlist which will plot the transient response of and RLC circuit between 0.5µs and 500µs for an inductor of L = 100mH with an initial current of 1mA, a capacitor of C = 200pF , and a resistor of R = 1kΩ. Show the plot of the voltage across the resistor and the currents through the capacitor and the inductor. E LECTRONICS II PROBLEM By KHALID OWEIS
The basketball players nominated profesProblem: For the amplifier in the figure, sor Anthony Garcia as outstanding faculty this determine: (a) R in(base) , (b) Rin , (c) Av , (d) semester and on Saturday, December 21, the A , and (e) A i p men’s basketball team hosted a home game were they announced professor Garcia’s name before the lineup. Professor Garcia even got a T-shirt from the team! Thanks to all our faculty for having such a great impact on our University of Mary student athletes.
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and end with a blizzard. Winter started particularly early in 2019, with our first substantial snowfall occurring in early October. This left inches of snow on the ground. ECI 313 General Surveying is a course covering the basic elements of a land survey; locating and measuring precise locations, angles, elevations, and distances of large land areas and properties. This begs the question; how can students learn to conduct an accurate survey on icy, slippery, inaccessible, and snow-covered land areas? Due to the initial snowfall and the regular inclement weather that occurred on survey lab days, the fall 2019 ECI 313 had to adapt. Surveying was brought indoors with a combination of inside instrument use and the use of Google Earth. The class was lucky enough to conduct initial leveling, profile, and basic traversing surveys using traditional leveling instruments, staking, and taping methods outdoors. As for advanced survey methods involved in traversing, routes were established within connected buildings on campus. The combination of hands-on indoor/outdoor applications allowed the class to understand the basic principles involved in field techniques. The final lab projects for the course were assigned using strictly computer-based methods. Students were to conduct a series of surveys utilizing the measuring tools available on Google Earth; distance, elevation, and angles. The first assignment was conducting a long open-link traverse spanning several miles or a plot survey of a very large structure (such as a sports stadium). Students were to record measurements consistent with field notes for all points observed – treating the survey as if is were on-site. The final assignment was to conduct a thorough survey of roughly onethird of the UMary campus. This included profiles of all roads and walking paths, grid surveys of fields and parking lots, and perimeters of buildings within the section. The goal was to combine the efforts of the three groups into an all-inclusive campus survey. Students were also directed to tie in known elevations from field surveys to calibrate the elevation estimation from Google Earth. Conducting a hands-on lab via computer was never the plan for the course. However, it is a relief to know that technology can help us adapt to cope with Mother Natures everchanging plans for North Dakota. Problem to try on your own: Using Google Earth (https://earth. google.com/web/) conduct a closed-loop traverse of a location of your choosing. Measure distances between points and record elevations. Recommendations for locations: Highway sections of Bismarck, professional sports stadiums, or a city block from your hometown.
C HRISTMAS PARTY
The faculty of the University of Mary School of Engineering held a Christmas party
S URVEYING I NDOORS : U SING M ODERN T ECHNOLOGY TO ACCOUNT FOR W INTER IN N ORTH DAKOTA By ANTHONY WALDENMAIER
For the reader who lives in North Dakota, at the temporary building on Front Avenue or really anywhere in the Midwest, winter can just before finals week, where the full time make an appearance as early as September. A professors as well as the adjunct professors week may start with temperatures in the 90s
• The measuring tool is located on the left border of the app (lowest option) • Select a node and note the elevation which is found in the lower right corner • Select the next node with the measuring tool, record the distance, and note the elevation of the node • Proceed to the next node and continue the process until the end is reached (a
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closed loop should end on the location and thus of the initial node) z = x + jy = r cos θ + jr sin θ A sample Survey of Wrigley Field (Chicago, = r (cos θ + j sin θ) IL) is provided below. = rejθ
C OMPLEX N UMBERS By TERRY PILLING
A complex number, z, is a number which has both a real part and and imaginary part and can be written as z = x + jy √ where x and y are both real numbers and j = −1. These numbers are very familiar to both electrical engineering students and mechanical engineering students, since they appear any time one deals with oscillating systems such as AC circuits or springs, pendulums, or even satellites orbiting the Earth. Complex numbers are called algebraically complete because the roots of any polynomial equation will be complex numbers. On the other hand, the real numbers are not algebraically complete. For example, the equation z 2 +1 = 0 is a polynomial equation which has no solution in the real numbers but has two solutions in the complex numbers: z = ±j are the solutions and both are pure imaginary numbers. One of the most amazing properties of complex numbers was discovered by the famous mathematician Leonhard Euler1 in the mid 1700’s. This is the famous Euler Formula: ejθ = cos θ + j sin θ
where we used Euler’s formula in the last step. One cool thing that you can do with Euler’s formula is use it as a way of deriving trigonometric identities if you happen to be stuck on a deserted island with no internet and a hostile tribe of cannibals demands to know the trig identities or you will be eaten. Exponents have the following nice property under system with the function showing up as a surface graph. multiplication: Now what about complex functions? X a X b = X a+b What if we want to graph a function f (z) from the complex numbers to the complex numand so bers? In other words both z is a complex numjθ jφ j(θ+φ) ber and f (z) is also a complex number. This e e =e is a bit problematic since the complex numand if we insert Euler’s formula in for these bers are equivalent to 2-dimensional vectors. exponentials we have So we would need 2 dimensions for the domain – just like our f (x, y) plot above, but we cos(θ + φ) + j sin(θ + φ) = would also need 2 dimensions for the range [cos(θ) + j sin(θ)] + [cos(φ) + j sin(φ)] since f (z) is also two dimensional. Thus the actual graph would be a 3-dimensional “surwhich we can multiply out, using the fact that face” in a 4-dimensional space! It is imposj 2 = −1, to get sible to graph this in our 3-dimensional universe. Hence we need a different way to visucos(θ + φ) + j sin(θ + φ) = alize what is going on here. [cos(θ) cos(φ) − j sin(θ) sin(φ)] Here is a method that we can use: We + j [cos(θ) sin(φ) + sin(θ) cos(φ)] pick a random point and plot it in a certain color and then we plot the value that the point and equating the real parts and the imaginary maps to under the function. So if the comparts gives finally: plex function was f (z) = z 2 for example, we would choose a random point, say z = cos(θ + φ) = cos(θ) cos(φ) − sin(θ) sin(φ) 0.54 + 0.32j = (0.54, 0.32) and we plot it, sin(θ + φ) = cos(θ) sin(φ) + sin(θ) cos(φ) then we also plot Notice that, if θ = φ these become cos(2θ) = cos2 (θ) − sin2 (θ) sin(2θ) = 2 cos(θ) sin(θ)
and so on. We have seen how to visualize complex numbers as vectors, or points, on a 2-dimensional plane. What about complex functions? Recall that we usually visualize a function using a graph. For example if we have a function f (x) of a single real number x then we can draw This is a useful formula in a multitude of ways a horizontal axis to represent the values of x First, notice that if you put θ = π in Euler’s and then draw a vertical axis to represent the values of f (x) and the function will then be a formula you get the remarkable equation
f (z) = z 2 = (0.54 + 0.32j) ∗ (0.54 + 0.32j) = 0.1892 + 0.3456j = (0.1892, 0.3456) in a different color. Now if we choose a whole bunch of random numbers from a region of the xy-plane such as a square between x = ±1 and y = ±1 and map each of the points we should get a feel for what the complex map is doing to the points. Here is a G NUPLOT script that chooses 5000 random numbers between zero and one, then shifts them so they are between −1 and +1 and plots both the original square region and the region that it maps to under the complex function f (z) = z 2
ejπ = −1 which relates e = √2.71828 · · · , π = 3.141592 · · · , and j = −1. Complex numbers can be written as vectors in 2-dimensional Cartesian coordinates by identifying the vertical axis as the ‘imaginary axis’ and writing z = x + jy −→ (x, y) and if you transform to polar coordinates curve drawn on this graph. The graph shows that the original square What about a function of two variables x where the vectors have radius r2 = x2 + y 2 −1 and y? Well in that case the function is f (x, y) region gets warped into a sort of oval region and angle θ = tan (y/x) then you have and so we use two axes for the x and the y where the density of points increases toward x = r cos θ variables and we use a third one for f (x, y) the center. which will give us a 3-dimensional coordinate y = r sin θ 1
Note that the German name ‘Euler’ is pronounced ‘Oiler’ as in the Edmonton Oilers hockey team.
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complex-map.gnuplot
might like to try.
set t i t l e ” f ( z ) = z ˆ2 ” s e t xrange [ − 2 : 2 ] s e t yrange [ − 2 : 2 ] u n s e t key
Assignment: Modular forms are functions of complex numbers that obey the following symmetry relation: az + b f = (cz + d)k f (z) cz + d
set offset 0.1 ,0.1 ,0.1 ,0.1 s e t s a m p l e s 5000
where
f x ( x , y ) = x∗x−y∗ y f y ( x , y ) = 2∗ x ∗ y p l o t ’ + ’ u s i n g ( x=rand ( 0 ) , y= rand ( 0 ) , 2∗ x −1) : ( 2 ∗ y −1) p t 7 ps 0 . 2 , \ ’ + ’ u s i n g ( x=rand ( 0 ) , y=rand ( 0 ) , f x ( 2 ∗ x −1 ,2∗ y −1) ) : ( f y ( 2 ∗ x −1 ,2∗ y −1) ) p t 7 p s 0 . 3
So the map f (z) = z 2 is pushing points toward the center and it is also stretching them out vertically. This can be seen more easily by recalling Euler’s formula:
f (z) = z 2 ⇒ f (rejθ ) = r2 ej2θ
we see by the formula that the angles are all doubled and the radiuses are squared. Hence radii which are less than 1 will be compressed (if you square a number that is less than 1 it gets even smaller) and radii which are greater than 1 are stretched. This is why we see the accumulation at the origin. Also the doubling of the angles is why the oval shape appears. Let’s take a look at another complex function for fun. This time we will look at f (z) = 1/(z + 1). We start by just looking at random points inside a square between 0 and 1 on both the x and the y coordinates. Here is the G NUPLOT code that generates it
a b c d
to prove this using Euler’s formula or some other method? Let’s shift the square region so that it surrounds the origin like we did in our first plot. Here is the code that does it:
is a matrix in the group SL(2, Z). I.e. it is a 2 × 2 matrix with integer entries which has determinant equal to 1. There are many unsolved problems and open research questions dealing with modular forms. They are used in Cryptography, Number Theory, and complex-map3.gnuplot in Quantum Physics, High Energy Quantum Field Theory, Quantum Gravity, Superstring s e t t i t l e ” f ( z ) = 1 / ( z +1) ” Theory, and they were famously used by Pros e t xrange [ − 1 : 1 . 5 ] s e t yrange [ − 1 . 5 : 1 . 5 ] fessor Andrew Wiles in his proof of the 400 u n s e t key year old problem called Fermat’s Last Theoset offset 0.1 ,0.1 ,0.1 ,0.1 rem2 . We would like to try to play with modular s e t s a m p l e s 10000 forms using our plotting method to try and get f x ( x , y ) = ( x + 1 ) / ( x∗ x +2∗ x+1+y ∗y ) a more intuitive feel for them, however, usf y ( x , y ) = −y / ( x∗ x +2∗ x+1+y ∗y ) ing gnuplot is a bit too tedious when rapidly changing parameters and trying new things on p l o t ’ + ’ u s i n g ( x=rand ( 0 ) , y= the fly. Better is to write a program that acrand ( 0 ) , 2∗ x −1) : ( 2 ∗ y −1) p t 7 ps 0 . 2 , \ cepts parameters from the command line or ’ + ’ u s i n g ( x=rand ( 0 ) , y=rand ( 0 ) from a menu so you can use it to play and , f x ( 2 ∗ x −1 ,2∗ y −1) ) : ( f y ( 2 ∗ x explore ideas quickly. To this end, write a −1 ,2∗ y −1) ) p t 7 p s 0 . 3 program in C++ using the OpenGL graphics library which takes integers a, b, c, d as input Now we see that corners don’t seem to parameters and outputs the transformation of map to corners after all do they? There seems a square region of the complex plane under to be more than four corners on the image and the linear fractional transformation: more than four edges also. It is a cool map az + b though. z −→ cz + d in a way similar to the method we used with G NUPLOT.
complex-map2.gnuplot s e t t i t l e ” f ( z ) = 1 / ( z + 1) ” s e t xrange [ − 1 : 1 ] s e t yrange [ − 1 : 1 ] set offset 0.1 ,0.1 ,0.1 ,0.1 u n s e t key s e t s a m p l e s 2000 f x ( x , y ) = ( x + 1 ) / ( x∗ x +2∗ x+1+y ∗y ) f y ( x , y ) = −y / ( x∗ x +2∗ x+1+y ∗y ) p l o t ’ + ’ u s i n g ( x=rand ( 0 ) , y= rand ( 0 ) , x ) : ( y ) p t 7 p s 0.2 ,\ ’ + ’ u s i n g ( x=rand ( 0 ) , y=rand ( 0 ) , fx ( x , y ) ) : ( fy ( x , y ) ) p t 7 ps 0.3
Interestingly, we see that the original square gets shrunken down, rotated and has some cusps. It actually still has four corners which leads us to believe that corners map to corners and edges to edges. Is there a way 2
Complex maps are the first step toward generating fractals as you may have guesses by the images that we are seeing. The second Above is a screenshot of the output of the ingredient that you need to generate fractals 2 is repeated iterations of a complex map. One desired C++ program under the z → z transthing you might want to try for fun is the fol- formation we studied before using G NUPLOT. The problem is to modify it to output the relowing technique: sult of a linear fractional transformation instead of z 2 . z → f (z) → f (f (z)) → f (f (f (z))) One possible solution to this problem, usand so on iteratively coloring each iteration ing OpenGL along with the (now mostly depby a different color. I wonder what cool recated) GLUT windowing library is given at images you would find? If you are inter- the end of this newsletter. ested by this stuff you might enjoy the artiNote that if the entries of the linear cle from our Spring 2019 issue of the Engi- fractional transformation matrix are complex neering Quarterly entitled Ethereal Delights numbers, rather than integers, the transformawhich explores repeated exponential complex tion is called a Möbius transformation. These maps and shows the cool pictures that appear. are used in modern image processing to create Here is an assignment problem from our EEL 360◦ spherical effects. For example Google 461 Advanced Programming course that you Street View and virtual tours of buildings are
The BBC produced an excellent documentary about Dr. Wiles struggles to prove Fermat’s Last Theorem which can be found here: https://www.dailymotion. com/video/x621ukl
Engineering Quarterly WINTER 2020
VOL.II. . . No.2
created by taking pictures from various positions and then patching them together using transformations like these. For some cool videos and pictures of Möbius transformations applied to images take a look here: ima.umn.edu/˜arnold/moebius/
Complex Numbers Problem: The solution is a program producing a Möbius transformation for user supplied integers a,b,c, and d so that the output from the command ./moebius 0 1 1 0 is the following picture
S OLUTIONS TO P ROBLEMS Electronics II problem: First we find R2 VT H = Vcc = 3.66V R1 + R2 R1 R2 RT H = = 9.56kΩ R1 + R2 VT H − VBE = 2.63mA IE = RE − RT H /βDC and using these, the solutions are (a) Rin(base) = βac (25mV /IE ) = 665Ω (b) Rin = R1 ||R2 ||Rin(base) = 622Ω (c) Av =
RC ||RL re0
= 261
(d) Ai = βac = 70
The following is a program written in C using OpenGL and glut which produces the plot moebius.c
(e) Ap = Aµ Ai = (261)(70) = 18270 Circuits II Problem:
/ / g c c −o m o e b i u s m o e b i u s . c −lm // −lGL −lGLU − l g l u t # i n c l u d e <t i m e . h> # i n c l u d e <math . h> # i n c l u d e < s t d i o . h> # i n c l u d e <GL / g l . h> # i n c l u d e <GL / g l u . h> # i n c l u d e <GL / g l u t . h> # define # define # define # define # define
RLC damped oscillator in NGSpice ∗ RLC c i r c u i t L1 b a 100mH IC =1mA R1 a d 1000 C1 d c 200 p V0L b 0 0V V0R c 0 0V . control t r a n 0 . 5 u 500 u u i c g n u p l o t v o l t a g e V( a ) g n u p l o t c u r r e n t s i ( V0L ) i ( V0R ) . endc . end
Lx 5 Ly 5 lx 1 ly 1 N 20000
int a , b , c , d ; void I n i t i a l i z e ( void ) ; void d i s p l a y ( void ) ; void draw axes ( void ) ; void d r a w p o i n t s ( void ) ; v o i d key ( u n s i g n e d char , i n t , i n t ) ; f l o a t U( f l o a t , f l o a t ) ; f l o a t V( f l o a t , f l o a t ) ; v o i d Timer ( i n t ) ; f l o a t U( f l o a t x , f l o a t y ) { f l o a t n u m e r a t o r = ( a ∗ x+b ) ∗ ( c ∗ x+d ) + a ∗ c ∗y ∗ y ; f l o a t d e n o m i n a t o r = ( c ∗ x+d ) ∗ ( c ∗ x+d ) +c ∗ c ∗ y ∗y ; f l o a t transformed = numerator / denominator ; return ( transformed ) ; } f l o a t V( f l o a t x , f l o a t y ) { f l o a t n u m e r a t o r = a ∗ y ∗ ( c ∗ x+d ) − c ∗y ∗ ( a ∗ x+b ) ; f l o a t d e n o m i n a t o r = ( c ∗ x+d ) ∗ ( c ∗x+d ) +c ∗ c ∗ y ∗y ; f l o a t transformed = numerator / denominator ; return ( transformed ) ; } void I n i t i a l i z e ( void ) { glClearColor (0.8 , 0.8 , 0.9 , 1.0) ; g l u O r t h o 2 D (−Lx , Lx , −Ly , Ly ) ; } void d i s p l a y ( void ) { g l C l e a r ( GL COLOR BUFFER BIT ) ; draw points () ; draw axes ( ) ; glutSwapBuffers () ;
8
} void draw axes ( void ) { glColor3f (0.0 , 0.0 , 0.0) ; g l E n a b l e ( GL LINE SMOOTH ) ; glLineWidth ( 1 . 0 ) ; g l B e g i n ( GL LINES ) ; g l V e r t e x 2 f (−Lx +1 , 0 ) ; g l V e r t e x 2 f ( Lx−1 , 0 ) ; g l V e r t e x 2 f ( 0 , −Ly + 1 ) ; g l V e r t e x 2 f ( 0 , Ly−1) ; glEnd ( ) ; g l B e g i n ( GL LINES ) ; f o r ( i n t k = 0 ; k <20; k ++) { g l V e r t e x 2 f ( c o s ( 2 ∗ M PI ∗ k / 2 0 ) , s i n ( 2 ∗ M PI ∗ k / 2 0 ) ); g l V e r t e x 2 f ( c o s ( 2 ∗ M PI ∗ ( k + 1 ) / 2 0 ) , s i n ( 2 ∗ M PI ∗ ( k +1) / 2 0 ) ) ; } glEnd ( ) ; } void d r a w p o i n t s ( void ) { glPointSize (2.0) ; g l B e g i n ( GL POINTS ) ; f o r ( i n t i = 0 ; i <N ; i ++) { i n t s i g n 1 = r a n d ( ) %2; i f ( ! s i g n 1 ) s i g n 1 =−1; i n t s i g n 2 = r a n d ( ) %2; i f ( ! s i g n 2 ) s i g n 2 =−1; float x = sign1 ∗(( float ) ( r a n d ( ) %10000) ) / 5 0 0 0 ; float y = sign2 ∗(( float ) ( r a n d ( ) %10000) ) / 5 0 0 0 ; f l o a t u = U( x , y ) ; f l o a t v = V( x , y ) ; glColor3f (0.4 , 0.8 , 0.4) ; glVertex2f (x , y) ; glColor3f (0.4 , 0.4 , 0.4) ; glVertex2f (u , v) ; } glEnd ( ) ; } v o i d key ( u n s i g n e d char key , i n t x , int y) { s w i t c h ( key ) { case ’q ’ : exit (0) ; default : break ; } } i n t main ( i n t a r g c , char ∗ a r g v []) { i f ( a r g c != 5) { f p r i n t f ( s t d e r r , ” Usage : %s a b c d\n” , argv [ 0 ] ) ; f p r i n t f ( s t d e r r , ” where a b c d a r e i n t e g e r s \n” ) ; return 1; } a = a t o i ( argv [ 1 ] ) ; b = a t o i ( argv [ 2 ] ) ; c = a t o i ( argv [ 3 ] ) ; d = a t o i ( argv [ 4 ] ) ; s r a n d ( t i m e (NULL) ) ; g l u t I n i t (& a r g c , a r g v ) ; glutInitDisplayMode ( GLUT DOUBLE | GLUT RGB ) ; glutInitWindowPosition (200 , 100) ; glutInitWindowSize (1000 , 800) ; g l u t C r e a t e W i n d o w ( ” Complex Map z −> z ˆ 2 ” ) ; glutDisplayFunc ( display ) ; g l u t K e y b o a r d F u n c ( key ) ; Initialize () ; glutMainLoop ( ) ; return 0; }
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