A philosophy of (bio)mechanics

A final speech delivered to the students of BIOMEDE 231, December 11, 2019, 10:30 a.m.

Some people, at the beginning of these things, cannot believe I am going to stand up here for the next hour reading off a piece of paper, generally without pause. Those that know me, know that indeed I am the sort to stand here, prattle on, wax poetic, try to bring a tear to an eye or two (if but my own), and spend the next hour reading off this piece of paper, generally without pause. It is a comment on what I feel it means to comment on things I think are important. And I think lining up a few words to express how I feel about a topic I care about to an audience I have come to know and respect is a good use of my time (as much as I have remaining), my talents (such as they are), and my professional capacities (may I always meet and exceed them!). Those who had not really known me prior to this day, will, by the end, know at least that much.
The purpose of this speech is multifold. First, it serves as a draping in which to refresh in the minds of those participating, the contents of the class. Second, it reassess the original questions posed at the beginning of the course, including who we all are, what this class was about, where we can plunge into the deepest darkest depths of this material, when have been able to, and why the content of this course is some you should ponder and appreciate pondering from time to time. Finally, it allows me to share a few otherwise wise words. Though the advice present throughout this speech is unsolicited, having gone through as much of life as I have and been approximately where you were not too long ago, it is my hope such advice may help you on your course.
Having spent my fair share of minutes in excruciating boredom, if this does not sound like something you are in engaging with, I would like to point out, that those there are the doors and they are not locked. I point this out for three distinct reasons. The first, you are free to leave. And to stay. Always. That is freedom, and I am in no position to curb yours. Please use yours freely. The second, it means that anyone could come through those doors. How we welcome a stranger is always a sign of who we are. How we respect those we share the room with, another aspect. How we come and go, more or less, is how we are judged in society. The third, allows me a segue to describe the precisely two times working for this class in which I was a chickenshit coward.
I wished to begin this semester with both a syllabus and a constitution. That is, I wished to have all planned out, all the things I intended for us to learn this semester and all the thing I hoped we would stand for. I put together the syllabus, pretty much as I wanted, though, those keen-eyed among you need not have it pointed out that things did in fact change. I never got the “constitution” much past a draft phrase. I had resolved to map the jist with our resident country’s, first, free speech, second, well, in this country, it’s guns. And I was going to have a part in there saying that should anyone come through those doors with bullets for us, I’d try my darnedest to make sure none of them got to you. I’d say this, but of course those of you who have witnessed me teach the first part of this class, it was always you watching my back. I’d say this, but I was too skittish to say it at the beginning of this class. Broaching the subject of mass shootings is generally, maybe, not the best way to start a first day in a (bio)mechanics class, but when is it ever?
What’s more, the harm from this sword of Damocles is not that it falls, but that it hangs. Why remind ourselves that, out there, there exists darkness when we may was easily warm our hands by the light of our sciences? A sensible approach, which I elected for.
Chickenshit coward moment number two: I wanted to talk about September 11th and I didn’t. All I did was write the date on the board, and somewhere in the notes include “a remembrance” (syllabus too!). I had every intention of explaining how it changed things. It’s importance to our nation’s history and how it’s being forgotten, left unmentioned. History is erased when we stop talking about it. The words “I have a dream” invoke a living breathing history of civil rights that while expressed in a moment describes the movements of time. It is repeated, relived, relearned, remembered. In an age of irony, we said we would never forget. In an age of post-irony, we can’t recall. A dream forgotten, lost. I had a dream, but I lost the spirit. Again, some things are not for me to comment upon. Thinking on what I should say regarding the importance of doing well what we learn here, I opted for not so fine a point, and I again safely took the coward’s way out. Thinking on what I was to say that day to you, given the relevant scientific and engineering parallels to be shown, I was drawn. Ultimately, I opted for silent reflection. When the words don’t come, let the moment.
Most every time we generally meet, I am expected to have a few remarks prepared. That’s a stranger place to be in than you might expect. Most minutes of your day, you will will spend without consideration. There is no quote unquote “plan” of the variety one might expect of the script I have here before me. Though planned, your life is rarely rehearsed. Though each time we meet, I have generally given a good deal of forethought. I’ve gone through it all time and again. This here, this cadence, these facts, this order, this possible question, this follow up, supplementals, variations. I have read the texts, cajoled the facts, synthesized disparate materials. Tried to at least. And but so that means I had to have prepared remarks, every day on the following ABET-accredited subject matter: “an introduction to topics in biomechanics, including statics, dynamics, and deformable body mechanics with applications to biological tissues and systems.” The topics to be included from the outset included, force, moment, and torque vectors, systems in equilibrium and applications to biomechanics, linear kinematics and kinetics, angular kinematics and kinetics, deformable body mechanics, stress and strain, stress and strain analysis, and mechanical properties of biological tissues. Our objectives, 
  • to review methods of vector mathematics and mechanics, 
  • to learn methods of statics with applications to biomechanical systems,
  • to learn methods of dynamics with applications to biomechanics,
  • to introduce ourselves to methods of solid mechanics, including stress and strain, and
  • to become partly acquainted basic mechanical properties of tissues

Working towards those objectives, I intended the students of my teaching to 

  • review mathematical concepts and methods relating to force, moment, and torque vectors,
  • learn methods of statics for systems in equilibrium, including Newton’s laws, free-body diagrams, constraints, and reactions,
  • apply methods of statics to biomechanical systems,
  • learn methods of dynamics, including linear and angular kinematics and kinetics,
  • learn basic principles and methods of solid mechanics, including stress, strain, stress-strain diagrams, elastic and plastic deformations, models of material behavior, multi-axial deformation, Mohr’s circle, torsion, and bending, and
  • learn about mechanical properties of biological tissues, including viscoelastic ones.
Now, as we face our inevitable ends, we may say we have learned some portion well, some sliver less, some not at all.
It’s okay to be confused. It’s okay to get frustrated. I say this aoud as much for you as I do for me. It’s okay to not understand something the first, second, third time you make a run at it. Fourth, fifth, and sixth time either, so long as you’re willing try for a seventh. All this to say, that it is perfectly fine and reasonable at this point in your academic career not to fully command this material. It’s expected even. You are learning and that means finding the brain space you’re going to carve out for this material, carving out that brain space with the help of others (present professor included), and taking the time to do it right. With time and effort all things – most things – can be done. And it’s okay not to know how just yet.
I am thee and we are ourselves equals on this and many other things
And so long as you’re putting your best effort in, there isn’t much you shouldn’t be proud of. You have achieved and that achievement should feel something like greatness. To do your best is certainly to be doing something in the direction of good, and I’m not sure any one has got any right to ask anything more of you in this life. So do your best and be proud.
You hadn’t yet done this before.
At some point in life, you will be asked to something you have never done before then – CPR, design a device, implant a splint, document a procedure, streamline a machining process, speak in front of congress – and you will want to be prepared. Preparation comes from forethought and retained knowledge. What you know helps you to figure out what you don’t. At some point we couldn’t count, we can now do calculus. With calculus you know how fast something will go, know how it will change. You can, with the right formulations, see an object’s past, present, and future. At some point in history this was mystical. Now routine.
Indeed, many of us like when the problems have a routine to them, a simple algorithmic approach. Here is but a small smattering that we attempted to master in this class:
  • Solving vector equations – (1) Understand the rules of vector algebra, (2) tally up the knowns and the unknowns, (3) have some quick shortcuts memorized, and (4) apply what you know;
  • Drawing free body diagrams – (1) Define the system, (2) sketch the system, (3) stare at each cut, (4) fool the body, and (5) replace gravity with a force;
  • Analyzing via a method of joints – (1) Draw FBD of total system and write independent equations, (2) draw FBD of all joints, (3) for each joint FBD, write force balance equations, and (4) solve the joint equations.
  • Analyzing via a method of sections – (1) Draw FBD diagrams, (2) apply laws of mechanics to each FBD, and (3) solve the mechanics equations for unknowns of interest; and
  • Determining motion of a particle – (1) Draw a FBD, (2) find the forces on the particle in terms of its position, velocity, and time, (3) write the linear momentum balance equation for the particle, and (4) break the vector equation into components.
I, myself, prefer the murkier more ambiguous type of problems. The ones without a simple algorithm. The ones that show where one’s focus is drawn. Do they seek reaction forces? Is it a balance of energies? About which point might they consider a moment’s thought.
I stop here for a brief aside about the consideration of a point in space and its ramifications for our world or at least a part of the world we inherited. In explaining the title of his novel, Umberto Eco had two characters exchange a brief aside in the presence of a narrator (hiding in a museum):
“It’s Foucault’s Pendulum,” [–the title of the book–] he was saying. “First tried out in a cellar in 1851, then shown at the Observatoire, and later under the dome of the Panthéon with a wire sixty-seven meter long and a sphere weighing twenty-eight kilos. Since 1855 it’s been here,in a small version, hanging from that hole in the middle of the rib.”
“What does it do? Just hang there?”
“It proves the rotation of the earth. Since the point of suspension doesn’t move…”
“Why doesn’t it move?”
“Well because a point…the central point, I mean, the one right in the middle of all the points you see…it’s a geometric point; you can’t see it because it has no dimension, and if something has no dimension, it can’t move, not right or left, not up or down. So it doesn’t rotate with the earth. You understand? It can’t even rotate around itself. There is not ‘itself.’ ”
“But the earth turns.”
“The earth turns, but the point doesn’t. That’s how it is. Just take my word for it.”
“I guess it’s the Pendulum’s business.”
The narrator comments:
Above her head was the only stable place in the cosmos, the only refuge from the damnation of the panta rei, and she guessed it was the Pendulum’s business, not hers. A moment later the couple went off – he trained on some textbook that had blunted his capacity for wonder, she, inert and insensitive to the thrill of the infinite, both oblivious of the awesomeness of their encounter – their first and last encounter – with the One, the Ein-Sof, the Ineffeable. How could you fail to kneel down before this alter of certitude?
Foucault’s Pendulum as a concept and as a book are well worth your time, if you have it to spare. Because that is a neat, murkier – at one point mystical – situation to consider, whose consequences are profound and whose set up, you are plenty capable of recapitulating, given what you have learned in this class (And others, if you were studious in your early physics classes). To the extent we could not consider the arcania of the field, I place blame squarely on those dozen of hours I have at my disposal. It is not a lot of time to learn, digest, retain, and use this stuff. It just isn’t. And that time wouldn’t be most efficiently spent if I was simply trying to wile and bamboozle you with all the astounding facts to which this subject matter now makes you privy. 
However, at your leisure, ponder.
Given the title of this speech, some are possibly curious as to the my strictly philosophic notions regarding the class’s content. To that end, my conceptions of reality in this class have been of the pragmatic sort residing often in a convenient representation of Euclidian space in a Cartesian coordinate system, modeling mechanics classically in a way that is “good enough for government work”. Which is to say it is the sort of reality designed to be replicated by students taking notes. The constraintment of our reality to the four corners of the page (such as the ones before me here) necessitate a specific grammar to the mechanical arts. First the mathematical notations. Then the drawings of systems, their forces, their velocities, their rotations, their deformations. Arrows, letter’s numbers, hexes, the spells written out to conjure the mechanical world we can see all around us. All this to say, through this class we have taken a simple perspective of a complex world and I am proud to say, learned much thereby. A philosophy conducive to further knowledge acquisition is the sort necessary for learning, and here I have attempted to erect just such a structure.
To the extent any of it has been successful will reside in the quiet of every learner’s heart, but at the very least, I hope you recognize that your professor tried, at the very least tried, to get us to learn some of the important realities of the world. Consider all that we learned during lectures, which included but which was not limited to the following:
  1. Who the instructors were, what sort of extra help they offer outside of the class room, went over how our grades would be determined (homework, exams, some readings, effort), agreed to a democratic process of revision, and collectively assessed what we knew of the topics covered, the objectives of, and the expected outcomes for those participating in the course;
  2. Three pillars of general mechanics (mechanical behavior, geometry of motion, and the relations of force to motion), Newton’s laws (even reading them in the “original” from the Principia itself), models in engineering, scalars, vectors, notation, relative positions, addition, subtraction, multiplication, division; 
  3. The dot product (with special emphasis on parallel and perpendicular vectors), its applicable features (including commutative, scalar, and distributive laws), the cross product, a few special cases, its algebraic rules, its geometric interpretation, its myriad uses, with it finding normal planes and optimal distances, moments about axes, the laws of sines and cosines, graphical, trigonometric, cartesian, and arbitrarily referenced methods of using such formulations;
  4. Free body diagrams, how to draw them, what they show and what they do not (including the system, the body, each force’s source and target, motion caused or prevented by forces, rotation caused or prevented by torques, velocity and acceleration left out), types of support systems (roller, pinned, fixed, simple) and related them to a few joints in the body (diathroid – ball and socket of the hip, hinge of the knee, pivot of the radius-ulna; amphithrioidal – ribs at the sternum; synarthoses – holding our heads together), to equivalent supports we added equivalent force sets, made couples, centered mass and gravity;
  5. Static equilibrium of a participle in one dimension using free-body diagrams to tell us what all adds up to zero, static equilibrium of a particle in two dimensions, in three, how the number of equations must match the number of unknowns (two forces and a moment, two moments and a force, three moments), and we saw a few special cases where equilibrium is achieved trivially: concurrent forces, one-force bodies, two force bodies, something hanging from a point; saw how linearity and superposition here, as elsewhere, aid in solution;
  6. Expanding statics of participles to a method of joints that enabled static equilibrium of an object using a force balance of tensions, shears, bending moments, and torsion; from that a method of sections could be further developed to consider more complicated structures in static equilibrium; and we briefly considered the hierarchical form of many biological materials including collagen’s multiple different forms at different levels of composition and how such methods many facilitate their biomechanical understanding;
  7. The nuanced technical differences between structures, trusses, and frames; redundancy; static determinacy;
  8. The effects of distributed loads along a rigid body such as cantilever beams; seeing those effects through shear and bending moment diagrams; 
  9. Internal forces within frames; the integral relationship between shear, bending moment, slope and displacement of a rigid beam; saw the relationships included those of singularity functions like the delta function, like the step function, like the ramp function;
  10. From the relationship between shear and bending moments we could predict the consequences of a particular loading condition, whether it be a concentrated force, a constant distribution, and a linearly increasing/decreasing distributed load;
  11. Formal definitions of a particle, mass, acceleration, force, work, energy (both kinetic and potential), and power, demonstrating how force might change as a function of a gravitation field (such as here on earth), a distance (such as with springs), a velocity (such as with dampers), oscillations (such as a mass-spring-damper system), and/or a combination of each;
  12. The force-distance relationship of springs, a simple energy carrier in a field, was used to investigate how its behavior would be modified if used in series or parallel with an equivalent spring (or dashpot); used such a force-field example to demonstrate linear momentum balance, angular momentum balance, the relationship between power and work, power and the rate-of-change of kinetic energy, the conservation of energy
  13. Harmonic oscillation, its physical manifestation in a mass-spring-damper system, its modeling as an ordinary differential equation whose general solution can reveal ways of characterizing the intensity of that oscillation (angular frequency, period, frequency, amplitude); multiple-degree-of-freedom-systems whose linear momentum we could balance; dashpots and the concept of dampedness, which includes undampedness (a spring and a mass), underdampedness (a spring and a mass and a weak damper), critically dampedness (a spring and a mass and a just-right damper), and overdampedness (a spring and a mass and a strong damper);
  14. Vectors describing position, velocity, and acceleration with respect to time and their derivatives; 
  15. Linear momentum balances; angular momentum balances; energy and work and power balances; each of those balances examined as rates of change to find maxima and minima of maximal and minimal interest; planar motion (including rectilinear translation, curvilinear translation, fixed axis rotation, and general planar motion); representations of relative velocity for a point in motion via a direct vector representation, a component representation, and a changing base representation; 
  16. Vectors in rotation including via the handy polar coordinates and their corresponding unit vectors; angular velocity of a particle; angular acceleration of a particle; dynamics of a particle in circular motion; a pendulum’s (and its inverted variant’s) swing;
  17. Rotation of a rigid object in rigid circular motion; rotated coordinate axes; the rotation matrix; angular velocity of a rigid object; angular acceleration of a rigid object; relative motion for rigid objects; the fundamental angular velocity equation (where the rate of change of some quantity is equivalent to crossing it with the angular velocity); the relation of angular velocity and acceleration to the rotation matrix;
  18. How to work together in small teams to agree upon what we as a peer group consider our level of competence to in part design an assessment of our skillset;
  19. Moments of inertia in two dimensions; the radius of gyration; the parallel axis theorem; the inclusion of moments of inertia into linear momentum, angular momentum, and energy balances;
  20. Representations of differential mass and its inclusion into general motion equations; relative and absolute velocity and acceleration for a rigid object in a plane;
  21. A summarization of dynamics enabling complete description of rigid objects per the laws of classical mechanics and a canonical approach to its use;
  22. Normal stress under axial loading; shear stress under the same conditions; sign conventions for each; units for each; stresses and strains on inclined sections as seen through various “cuts” in a body; displacement, deformation and the concept of strain; normal and shear strain; the stress-strain diagram and various points on it including portions of the curve representing elastic behavior, yield, strain hardening, ultimate strength, necking, and fracture; Hooke’s law; Poisson’s ratio; the relations between the elastic modulus, the shear modulus, and Poisson’s ratio;
  23. Normal and shear stress along Cartesian coordinates; notation and sign conventions thereof; static equilibrium on a differential volume subject to normal and shear stresses; planar stresses; transformation of stress along any arbitrary set of axes; stress invariance; principal stresses and maximum shear stresses; principal planes of maximum stress and maximum shear; their interrelation; double angles; and
  24. Mohr’s circle representation of plane stress including the general derivation of the circle equation and how to plot it out graphically; using it to find points of maximum stress and maximum shear; axial strains; planar strain, transformation of planar normal and shear strain along arbitrary axes; appreciating the similarity of the transformation equations of stress and strain; principal strains and maximum shearing strain; the measurement of strain by strain gages; and the use of strain rosettes to find the planar strain of an object.
I hope you, as a list of topics like that rolls past you, reflect on your own knowledge on the matters. Recognizing that as with all things knowable there exists a range within the general human population from a complete and absolute ignorance of a subject to a stupefying brilliance and expertise, if you can accept where you are on that spectrum with regard to the material in this class, where you came from, and ultimately, quo vadis, where you are going, then that should be enough to keep you resting easy at night. As with most journeys we take, every new day will not be our final step. Let us keep up our resolve to put one intellectual foot in front of the other, happy with what we learned, and capable of learning more.
Consider to that list of topics we did cover in this class, I originally intended to also include more biomechanical forays (such as the kinetics of the hand, an interesting study found by our own M. B.), models of viscoelasticity in time and frequency, torsion and tensors, arbitrarily distributed loads over arbitrary volumes, an overview of why the physics of our planet and the inheritance of traits for species over time constrains the growth and form of all living creatures here on earth (as expounded upon by D’Arcy Thompson), how given our biological constraints we can design biomedical equipment, and finally I even meant to show how the physics of the body (as manifest through evolution) has led the the ever more child-like incarnation of Mickey Mouse we find ourselves inundated with, but time, ever dwindling in its finitude, got the better of us. At last we find ourselves here at the end. 
And if you’ll allow me, would like to share a word or two regarding each one of you and your contribution(s) to the class. I have you listed here (and numbered) in the order in which you first registered for a class of mine.
  1. 14. T. T. As my “oldest” student here, it could be said that you are thereby wisest. Some might just as easily see its converse. All the same, I hope now having taken two biomedical engineering classes from me, you are developing a sense of what the discipline involves. May you use it effectively to become a neurosurgeon.
  2. 216. E. C. Another veteran of my first-year engineering course, though from a different year, you have not doubly entrusted me to provide an opportunity for learning excellence and in both instances you have exceled. May you continue on your academic journey, doubly sure of yourself.
  3. 218. H. K. As yet another veteran of the same class, you knew generally how I wanted to teach this class. You also generally had a notion of what you wanted this class to be. At the times that I asked you for feedback and advice, I appreciate that it was given candidly, and I believe the class has been the better for it. May you continue to candidly make better the situations you are in. 
  4. 256. Z. S. As sharp a mind as any that’s ever cut across my material. You have demonstrated smarts and kept your wits about you. May you continue to grow that mind of yours. 
  5. 327. R. S. You are a multitalented, multidimensional human being and I hope you continue to expand along every dimension that compels you. Along that path, may you know multitudes.
  6. 356. J. B. One of the keenest eyes from one of the farthest seats, with this material you demonstrate proficiency in both global and detailed problems. May you continue to take the detailed wide view.  
  7. 357. M. C. I wish we saw more of you. As an LSA student, your perspectives into the material would be interesting to include and helpful for making the material more broadly engaging. May you in the future to engage broadly with that which interests you.
  8. 358. B. L. I have seen your furrowed brow, I have seen your raised hand, I have seen your chipper face at office hours. I have seen you time and again put in effort and time and everything you got. May others continue to see such efforts.
  9. 359. M. D. You consistently asked keen questions, giving voice to the curiosities, confusions, and contemplations of others in the room. May you continue to speak with the acuity for all.
  10. 360. C. M. Every class has an individual who quietly dominates the assessments given. Though they do not say much in class, their knowledge shown on homework and exams says plenty. This class, this semester, that individual is you. May your skills continue to say plenty for you.
  11. 361. A. R. Personability and preparedness will get one far in life. From what I have learned of you through this course, you have both in spades. Such skills have places on teams, in projects, for betterment. May you continue do well with others so that you might do good in this world.
  12. 362. E. D. I apologize for the recent confusion regarding your homework submission. However, it allows me the opportunity to commend your ability to efficiently and effectively address an unknown problem. In figuring out what had happened you should level-headedness and a humble certainty. May you always solve problems so effectively.
  13. 363. B. M. I hope you got something out of that field trip to the University of Michigan Orthotics and Prosthetics Center. Moreover, I hope you get many more such experiences on your way to researching greatness. May you continue in the field with greatness.
  14. 364. R. A. You have regularly shown a genuine desire to understand the material and a natural inclination for reflection on your learning process. If you have known much, it comes as no surprise. That you will know more, there are even fewer doubts. May you reflect what you learn and learn from your reflections.  
  15. 365. C. H. It may be an odd thing to compliment, but I want to call out your second reading summary as of particularly exceptional quality. From the clean layout incorporating mathematics and multiple figures to including the essence of an example of circular motion kinematics by considering the acceleration of the earth, it was all top notch. May you ever summarize complexity efficiently.
  16. 366. B. R. A feat of human performance you highlighted as impressive in this class were backflips, indicating that your “developing sense for vector math allows [you] to understand how the motion changes throughout flight and the forces/direction of forces applied”. With such competent description, I am liable to start doing backflips here. May you continue to ably describe the impressive world around you.
  17. 367. E. B. Diligent, capable, participatory. Ask me to triangulate you in three words and that’s the best approximation I can give. In this class you have learned much and helped others learned more. Such is the best of what we “leaders and best” do: sharing what we can with others it might help. May you be ever diligent in doling out your capable participation.
  18. 368. L. W. There is, and will always be, a special place in the heart of instructors for those students who consistently sit in the first row. Those bold enough to venture forth first are the intrepid souls who mark our progress. May you continue to be first and foremost in most everything you do.
  19. 369. L. B. When you found yourself “struggling at the moment” with the material from the class, you reached out and wanted to shore up your strategies to approaching the content. For every impediment, seeking another way through (and asking those who might know!) is a righteous approach. For every moment, a moment just beyond. May find an approach that works best for your and resume the struggle.
  20. 370. G. G. For some, showing up bright-eyed and bushy-tailed day after day can be difficult. You have made it in this class look effortless. Such good energies when boundless, make good solutions inevitable. May you always show the energy of your talents.
  21. 371. J. O. Though we as instructors maybe shouldn’t admit it, there are those students we use as barometers to get a sense of the room. Throughout the semester, when mathematical notation needed to be cleared up, whether my modeling was right, or one of the other variations on error I was able to demonstrate in this class. In many such times I have looked to you in the back to see if I am finding my way. Your quiet guidance has helped this class. May your counsel, verbal or nonverbal, continue to help those around you.
  22. 372. A. H. From the sounds of it, you have had a rougher semester than most extracurricularly. At least while here, you have participated well in this class and put forth effort for each and every assignment. May you face every adversity with perseverance. 
  23. 373. M. C. In relaying a feat of human motion you found impressive, you chose our ability to lift weights, relating the movement to the tension in the muscles, the growth and repair of muscle fibers through exercise, and vector mechanics of the system. All true, all that which Sisyphus used to push the boulder. May you always lift the impressions and impressiveness of those around you.
That was it. That was us. That was the class.
Perhaps you noticed a distinct lack of biological problems. This was, after all, “bio” mechanics class, after all, and there were only cursory inclusions of biology periodically. I agree. I recognize this. It is the nature of the class as inherited throughout the development of our biomedical engineering discipline and department. One of the difficulties in shepherding a foremost biomedical engineering programs is that to make our students the kind of capable we want them to be – to make sure they have that Michigan Difference – we have to teach them a lot. In this case, we learned the subjects of statics, dynamics, and mechanics of materials, subjects that might otherwise span semesters each, we span in a semester, capably: with capable instructors, capable aides, and capable students, this all aligns for a maximizing of content for future biomechanical development. 
So why the minimal inclusion of biological examples in this particular class? In reality, it’s because the reality of biological examples gets complicated quickly. It’s hard enough to consider moments about a lever, let alone trying to model a single human person doing a single human chin up. Each take effort and to simplify matters will not do. To do it right, one must get all the way up to the bar, and for the content of this class (the aforementioned statics, dynamics, and mechanics of materials), that is really hard to make really simple. And simple is what is needed for fundamentals, the very thing we wanted to shore up through the course.
Such an approach was to make possible, I hope, a broad base of knowledge in each one of you to create your personal biomedical engineering expertise. We are not all going on to be orthotists, but some of us might and they are now prepared for that training. Some of us might never go on to conduct finite element simulations, but those of that do will know how to interpret principal stresses. We can intuit how composite materials (such as most biological materials) can transform those stresses and strains along preferred axes such as through muscle fibers or along tendons. We likely won’t all go on to take atomic force microscopy measurements, but those that do will be ready to analyze the results. So while our specific bio inclusions have been limited to a few nominal examples in class, on homework, and through exams, the mechanics undergirding these and further biomedical situations are now comprehended, for the most part, with competence. 
Why is such an understanding of mechanics important? Because all good things have moved. All bad things have moved. Into their place and where they are going. Movement arises from action, action from intent, intent the first mover of a cause, causes by control, purpose from control, meaning from purpose, something to know, something worth knowing. With mechanics you can understand how energy flows through a physical system. Some used here. Some there. You may measure the movements of cells, the contractions of muscles, the dilation of blood vessels. The whole world in and around you can be described more or less usefully by careful extension of the second law of mechanics as posited by Newton. And that is quite a hell of a thing to know.
With that, I thank you for the opportunity to dismiss this class one final time. I hope you all do well on your fast-approaching exams and on your ever-to-the-horizon journeys. I have done what I can for you here and I trust you can do more.
Good luck.