Combining Ordinary Differential Equations with Rigid Body Dynamics: Teaching a Second-year Engineering Dynamics Course to Two-year College Graduates

Students graduating with a two-year technical diploma from vocational colleges are entering a new Energy Engineering Program in Spring 2015 Term at Schulich School of Engineering, University of Calgary. The program provides a path for students with hands-on skills to pursue an engineering bachelor degree. The need to reduce the program completion duration and to provide mathematics courses with sufficient practical aspects has led the School to design a second-year course that combines ordinary differential equations with rigid body dynamics. This course, named ENER 340, has a prerequsite of another course, ENER 240, which combines differential and integral calculus with particle dynamics. We the ENER 340 instructor team find that even with such prerequisite course that exposes the students to more elementary concepts of particle dynamics and calculus, students struggled with developing problem solving skills further to solve rigid body dynamics. We find that teaching ordinary differential equations is much easier than teaching rigid body dynamics due to clear logical procedures for solving the former. Students also find it easier to learn the former. We consider several topics that can help blend the two course subjects better based on our teaching experience. We also find that such course blending initiative requires slower teaching and learning speed to allow students to digest the course material better. In addition, a more dedicated textbook that combines both topics is required. 1. Cultures of Learning in Polytechnics and Universities A new Energy Engineering Program was launched by Schulich School of Engineering, University of Calgary in 2015 in order to provide a pathway toward a Bachelor of Science (BSc) degree for applicants with a Diploma in Engineering Technology. The applicants normally come from nearby polytechnic colleges (polytechnics), such as Southern Alberta Institute of Technology (SAIT) Polytechnic and Northern Alberta Institute of Technology (NAIT), and have two-year engineering technology diplomas ranging from Mechanical Engineering Technology to Power Engineering Technology. SAIT Polytechnic states on its Mechanical Engineering Technology program's website [1] that its admission requirements are at least 60% grades in high school mathematics, physics, and chemistry courses. NAIT states on its Mechanical Engineering Technology program's website [2] that the admission requirements average at about 74% from these high school courses except chemistry. These admission requirements suggest that some applicants to the Energy Engineering program have lower high school averages than their peers in other engineering programs at the University of Calgary, which typically require at least 85% average from the same high school courses. It is worth noting that majority of students in these programs, including those at SAIT and NAIT, are either from Calgary or province of Alberta, in which Calgary is located, so that these averages are largely taken from the same pool of students and thus can be compared directly. The high school average gap between students in the other engineering programs and applicants of Energy Engineering program reflects a lower academic readiness of the latter group. We believe the academic readiness gap is widened after completing a twoyear diploma program at a polytechnic. Anecdotally, we heard from colleagues on campus that mathematics and physics courses taught at polytechnics focus on using ready-made formulas for a fixed number of problems and do not teach therefore how to analyze a problem using a set of governing equations from scratch. Polytechnic students are thus conditioned in their two-year programs to regard and use mathematics as a formulaic tool that has only one-step process to obtain a solution to a problem. They are not well trained to use mathematics as an analytical tool to translate physical insights, visualize problem geometrically, and make sense of the solution obtained. The limited use of mathematics to solve problems in polytechnics is widespread [3]; the limited use is actually sensible since the focus of polytechnic education is to deliver "experiential and hands-on" education [4]. Four-year university engineering programs, in contrast, build each of the programs' foundations with first-year mathematics and physics courses that are later used to construct mathematical theories of physical processes relevant for each program. Polytechnic students entering Energy Engineering program therefore have to learn a new culture of learning that relies heavily on mathematical and physical concepts and analyses instead of practical (experiential) and hands-on learning. This culture of learning difference can be seen from style and depth of textbooks used. The textbook titled Applied Mechanics for Engineering Technology [5] is used for firstyear Engineering Statics in Mechanical Engineering Technology programs in SAIT, while the textbook titled Vector Mechanics for Engineers [6] is used for first-year Engineering Statics in Mechanical Engineering program at the University of Calgary. The Applied Mechanics textbook has little theoretical discussion on applied mechanics, such as the nature of potential energy, and focuses on practical problem solving using free body diagrams. It is concerned with forces (torques) and acceleration (angular accelerations) and doesn't delve into the differential equation structure of the equations of motion. The non-calculus approach of the Applied Mechanics textbook allows the problem solving approach to be presented clearly and logically since all equations involved are algebraic. It is expected that polytechnic students using the Applied Mechanics textbook will gain clarity and learn good problem solving skills – albeit limited – from the non-calculus approach. In fact, the Applied Mechanics textbook is able to combine engineering statics and dynamics in one book due to the non-calculus and focused approach on problem solving. The Vector Mechanics textbook used in the Mechanical Engineering program, in contrast, uses vector calculus in presenting applied mechanics so that the presentation may seem convoluted and thus do not offer clarity at first glance. Unit vectors in different coordinate systems have to be discussed. It becomes necessary to cover the differential equation structure of the governing equations, and more importantly obtaining solutions become much more complicated than that in the non-calculus approach. One can argue that this complication is required to generalize the formulation so that all mechanics problems can be at least expressed mathematically as a set of differential equations. Engineering students taking applied mechanics course may feel dismayed, however, at the difficulty level they face if they ever compare their course content with their polytechnic friends who take practically the same course but learn using the non-calculus approach. Qualitative comparison of the two textbooks suggests that the level of difficulty of the Applied Mechanics textbook's problems is about 2/3 of the problems' difficulty level given to Mechanical Engineering students in their exams. This decreased difficulty level seems proportional to the high school average gap between students enrolled in polytechnics and those in engineering. The class presentation of applied mechanics using vector calculus and differential equations, however, will increase the difficulty level by at least a factor of two. While exam problems presented to engineering students are 1/3 more difficult, the theory presented to them in class is at least twice more difficult. It is our opinion that engineering students have a much tougher (and possibly more confusing) applied mechanics course. Our teaching experience in engineering mechanics has taught us that the increased course complexity doesn't often translate to much more difficult exam problems because of either exam time constraint, or a lack of solvable problem databank, or the understanding from increased complexity–not just practical, testable skills–is what university should give to students. Two first-year mathematics courses in SAIT Mechanical Engineering Technology program cover basic algebra and trigonometry, plane analytical geometry, single-variable differential and integral calculus using a textbook titled Basic Technical Mathematics with Calculus [7]. Two first-year mathematics courses in Mechanical Engineering program at University of Calgary start with single-variable differential and integral calculus and end with vector algebra and multivariable calculus using a textbook titled Calculus: A Complete Course [8]. Their comparison reveals that polytechnic students do not learn vector calculus during the first year, which is consistent with the non-calculus approach to applied mechanics. Engineering Technology programs in NAIT and SAIT don't offer a differential equation course which is a mandatory course for any four-year engineering program. Mathematically, culture of learning in polytechnics differs from that in engineering programs by the absence of courses in vector calculus and differential equation in the former. Vector calculus helps integrate geometry with calculus, hence making formulation of a problem more visual, while differential equation provides a platform to formulate a problem and to test whether its solution under some assumptions is satisfactory when compared with experiments. Polytechnic students are not taught to use mathematics as an analytical tool to translate physical insights, visualize problem geometrically, and make sense of the solution obtained. This topic omission is logical since the students focus on acquiring hands-on skills and therefore use mathematics as a calculation tool. Geometrical and analytical skills normally derived from vector calculus and differential equations will have to be acquired from intuition developed by practice and experience.