When did you first encounter the subject of your book, engineering dynamics? From whom/what book did you learn the subject? What was your own first reaction to the subject?
[NJK] I never actually took engineering dynamics until graduate school. As an undergraduate engineering physics major, I took classical mechanics in the physics department. Frankly, I was lost and confused. It was when I took both general relativity and orbital mechanics that I truly fell in love with the subject, both the historical connection and the beauty of the solar system. It was when I was a teaching assistant for the advanced dynamics in graduate school that I first saw the struggle students had transitioning from traditional engineering dynamics to Lagrangian methods. It was taking Spacecraft Dynamics from Thomas Kane that brought me to the connection between rigorous notation and the physics of the problems. I truly fell in love with rigid body dynamics. When I became a professor at Princeton, I actually taught our graduate course first before taking on the sophomore introductory course that led to the text. It was there that I began to see the need to introduce the ideas of coordinates and reference frames early in a student’s education so that Lagrangian and Hamiltonian methods don’t seem so mysterious.
[DAP] Although my initial encounters with engineering dynamics were not positive, it has grown to become one of my favorite subjects to teach—and a field that I rely on in my academic research. I first encountered the subject as an undergraduate student in a sophomore physics course. The course instructor handed out copies of his lecture notes—which I appreciated—but the barely visible date on the notes was nearly thirty years prior. Discouraged by this observation, and other factors, I subsequently dropped the class and changed my major to applied physics. My next encounter with engineering dynamics was in a first-year graduate class that focused on the Langrangian and Hamiltonian formulations. Although I did well in the class, a deficiency in my understanding of the Newtonian formulation of dynamics was exposed in my preparation for my PhD comprehensive exam. The professor who exposed this deficiency was none other than my eventual co-author. Jeremy assigned to me the remedial task of serving as his teaching assistant in his undergraduate dynamics course. Although I was initially frustrated by this outcome, it was through my TA service that I began to master the topic. Jeremy and I together formalized our version of the notation that we now consider so essential. The course evaluations were positive. After a second round of TA service, Jeremy casually proposed the idea of writing a textbook. Five years later, it was done.
Beyond the fact that this is a vital subject for engineering students to master, what do you think is inherently most interesting or fascinating about this subject?
[NJK] There are two things I find inherently interesting about dynamics. First is the historical importance. The need to describe and predict the functions of the heavens goes back 14 centuries. It wasn’t until Galileo and Newton that we truly had the tools necessary to explain the behavior of the planets. This overturned centuries of misconceptions and ushered in the era of modern science. The impact of this new science was immeasurable, yet the elegance and simplicity astounds.
Second, I never cease to wonder at the beauty of the mathematics and its connection to the physical world. With only Newton’s simple statements of mechanics, so much of what we observe and question can be understood. This is particularly true in the area of rigid bodies. That such complex and interesting behavior can arise from a small number of basic principals is truly astounding. Related to this is the importance of modeling. We try to emphasize in the book that the predictions we are making are for models of the actual system, not the systems themselves. Understanding when to make appropriate approximations and how to describe complex systems with models we know how to analyze is an important skill for young engineers to learn. The fact that these relatively simple models are so successful at describing behavior is truly astounding!
When did you first begin teaching this course to students? What was your experience the first time you taught the course? How has your method of teaching this subject changed over the years?
[DAP] I first taught engineering dynamics at the University of Maryland in the fall of 2008. My experience was overall positive, and the teaching experience dovetailed nicely with our ongoing work on the project at that time. I have continued to teach the course every fall, typically covering chapters 1-11 during a 15-week semester. This fall I will cover less material (probably only chapters 1-10), as I have made a conscious effort to apply practices from recent results in engineering education research. I now focus less on theoretical derivations and give the class more time to work out problems during lecture.
[NJK] I first taught dynamics to students in 2003. It was not a positive experience. I found that we were trying to cover too much in the time we had and that there was a tremendous disconnect between the rigor I was trying to achieve and the approaches taken by the traditional textbook. My approach to teaching it has evolved a good deal in the intervening years. I cut back on the scope of material, eliminating Lagrangian mechanics in the first course, and focusing on the most important concepts in order to include three-dimensional rigid bodies. I developed an active teaching style, where I call on students in class to solve problems at the board. I developed the idea that certain key concepts should be introduced at the introductory level rather than postponing to advanced classes. These include the importance of reference frames, the distinction between coordinates and frame directions (as defined by unit vectors), the idea that multiple combinations of coordinates can describe the configuration of a system (hence generalized coordinates later), and the importance of defining the number of degrees of freedom before choosing coordinates. Combining this with our explicit notation allowed me to employ “spaced repetition” in approaching the material. Though students sometimes find it frustrating, particularly for “easier” problems, I insist that they always follow the same process when solving problems of drawing frames, determining the number of degrees of freedom, choosing coordinates, writing the kinematics, drawing free-body diagrams, and then writing the equations of motion. This becomes second nature by the time we reach the more complicated topics. Over a period of five years, I found students became much more engaged and the course moved from one of the lowest rated to the highest, including a student teaching award in 2009. It was most gratifying to have seniors come to me as they graduated and tell me that dynamics was their favorite course. It is also wonderful to sit in on senior thesis presentation and see how students are able to apply the basic principals they learned as sophomores to their projects.
Do you notice that students ask the same kinds of questions you did when you first took the course -- or do students surprise or challenge you with the questions they ask?
[NJK] Without a doubt I am continually challenged and surprised by student questions. That is what makes teaching the course interesting year after year! I still find myself learning dynamics and every year I find myself challenged by a student question.
What inspired you to write your own textbook on this subject?
[NJK] There was really no one particular reason I decided to write a textbook. As I developed my notes over several years and experimented with different books, I found myself relying on the texts less and less and focusing more on making my notes stand alone. As I began to type up my notes and organize my thinking, it seemed natural to turn them into a new textbook. I found that I didn’t really understand the subject and the various connections between geometry, physics, and different methodologies until I was in more advanced classes in graduate school. My goal in teaching was to bring some of these more sophisticated concepts to introductory students, using the mathematical background they had, without sacrificing rigor and while making the presentation approachable and understandable. As I developed that approach I realized that no other books were doing that, so I decided to write my own.
As you say in your Preface, “dynamics is difficult.” How do you clarify some of the most difficult concepts to your students?
[NJK] Dynamics is difficult for two reasons. First, from the student’s perspective, there is just an enormous amount of content. There just seems to be so much to know to successfully navigate the topic. This includes kinematics in three dimensions, rotating frames, Newton’s laws, energy, angular momentum, Coriolis and centrifugal forces, rigid bodies, Euler’s laws, Euler’s equations, and a myriad of other small and large concepts. As students move to their graduate courses, the list grows to include Lagrangian and Hamiltonian mechanics. Students can become overwhelmed by the long list of ideas and how they are related. Second, all of these ideas need to be called upon to solve new problems. Sorting through the concepts to find the best tools for a given problem can be a struggle for the novice, particularly when there are usually multiple approaches that will work for any given problem.
Our book was written with both of these challenges in mind. Our goal was to make clear how each idea is really just an extension of a few basic and important concepts. We start simple and begin solving problems from the beginning, when only a small number of ideas have been introduced, to show how they can already be used to solve problems. We repeatedly solve problems throughout, both in class and in the book. And we introduce a very precise and rigorous methodology for students to follow so they can spend their energy on understanding how the concepts relate to the problem rather than rebuilding a solution approach.
I also use a very active learning approach in the classroom. My view is that students learn by doing. I try to avoid just deriving formulas in class; those are in the book. I use physical examples or videos accompanied by frequent problem solving. I have the students work together to solve problems and then present to the class. We also introduce all of the important concepts early in the course and have students repeatedly use them on progressively more challenging problems. This spaced repetition allows them to frequently revisit the most challenging ideas.
Your approach to the subject is very problem-driven. You have students solving dynamics problems from the very start of the text. Why do you feel it is critical for students to begin solving problems immediately upon beginning the study of dynamics?
[NJK] Solving problems from the beginning accomplishes a number of things. First and foremost, dynamics is ultimately about solving problems. We aim to provide students the skills and tools they need to solve real engineering problems. Starting them early gives us the entire course to practice and build problem-solving skills.
Second, solving problems provides context for the theoretical concepts and actively engages the student in having to learn them. Just introducing kinematics, for instance, can be dry and difficult to remember. Gradually introducing each concept in the context of real-world problems gives students a tangible reason for learning each idea and real problem to reinforce their memory.
Third, solving real problems early motivates students to engage with the course and material as they see that even the most basic concepts have application and relevance. As the material becomes more complicated and problems more difficult, they have the simpler problems from early in the course to fall back on and to motivate them for the more challenging ones.
Your style of writing is very conversational and accessible, and you take a slow, “bottom-up” approach, to make sure that students encountering the subject for the first time never get lost. Simultaneously, you innovate by using notation that is generally taught to and used by more advanced students and scientists. Why did you make this choice? How have your students responded to your unique combination of rigor and accessibility?
[NJK] In exploring other texts early in my teaching of this course, I found that students responded well to an informal, second person presentation but were very confused by the simplifications of the material. They had trouble not only following the development but more importantly were unable to apply what they learned to more difficult problems. My goal was to find a way to integrate the approachable presentation with a rigorous development of the physics and mathematics. The students have responded extremely well. They no longer complain that examples don’t prepare them for problems and they highly rate the book in their evaluations. Many students upon graduation have told me that dynamics was their favorite course in the department.
For what level of student is your textbook ideally suited? How do you feel your approach helps students to really master this subject?
[DAP] Our textbook is ideally suited for undergraduates (typically sophomores and juniors) though it may be of use for advanced undergraduates and first-year graduate students. The approach is to describe the fundamental concepts of Newtonian dynamics, using a rigorous notation throughout to aid clarity and assist in solving problems. The book provides examples, both short and long (the longer examples are called Tutorials), notes for further reading, a summary of key ideas, and practice problems to help the student. Our intent was to make the book as self-contained as possible, which is why we included several lengthy appendices on vector calculus and differential equations. Although our focus is on analytical problem solving, we included numerous examples of numerical problem solving with MATLAB. As a result, an engineering student with little or no physics or mathematical background can use our textbook to master the Newtonian formulation of dynamics as used in theory and in practice.
[NJK] I believe that our emphasis on rigor and explicit notation provides enormous pedagogical benefit to students. By tying the key underlying physical and geometric concepts to explicit notational markings, students are continually reminded of the fundamental basis for solving problems. Many students complain early on but we have received much feedback that the notation is extremely valuable. I recently read a PhD thesis from a graduate student who helped with the course and it was lovely to see him do all of the dynamics with the notation and approach from our book.
Some faculty might feel that we ask much of the student when we introduce concepts such as degrees of freedom, constraints, and multiple coordinates this early in their dynamics education. However, we find that our sophomores quickly master it through repeated problem solving and are much better equipped to understand advanced methods because they see how these concepts are not unique to analytical mechanics but are part of the foundation for understanding dynamics.