1. ME 407 Advanced Dynamics We will learn to model systems that can be viewed as collections of rigid bodies Common mechanical systems Robots Various wheeled vehicles The focus will be on engineering applications Divers and gymnasts 1 I’m open to applications you all care about

2. 2 I expect you to be comfortable with mathematics and abstract thinking in general even though our applications will be concrete I expect you to be familiar with geometry trigonometry linear algebra systems of ordinary differential equations vectors Prerequisites and some basic physics

3. 3 YOU NEEDTO INTERRUPT ME IF YOU DON’T KNOW WHAT IS GOING ON THIS IS IMPORTANT

4. Boilerplate There’s a web site: www.me.rochester.edu/courses/ME407www.me.rochester.edu/courses/ME407 (NOT UP TO DATE — STAY TUNED) My email, which I read regularly: gans@me.rochester.edu Text: Engineering Dynamics: From the Lagrangian to Simulation available in preprint form from Jill in the department office. Weekly problems sets Probably two midterms Meirovitch and/or Goldstein will be useful at the beginning both on two hour reserve in Carlson Office hours Tuesday-Thursday 2 – 4 or by appointment. 4

5. 5 We will go from very fundamental to very applied conservation of momentum and angular momentum What is a rigid body? Moments of inertia internal and external forces and torques work and energy geometry of three dimensional motion angular velocity and angular momentum coordinate systems

6. 6 We will go from very fundamental to very applied Hamilton’s principle The Euler-Lagrange equations Hamilton’s equations Kane’s method The null-space method Computational tricks: the method of Zs

7. 7 We will go from very fundamental to very applied engineering mechanisms: linkages, gears, etc. robots and their relatives wheeled vehicles of different sorts I’m open to applications you all care about

8. 8 Let me show you a couple of hard problems so you can see where we are going

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11. 11 We will also need mathematical and computational tools We need notation to understand ourselves better Most of the interesting problems are wildly nonlinear and we’ll need to integrate differential equations numerically I’m perfectly happy to use commercial code to do this but you do need to have an idea of what to expect so you can figure out if it’s right.

12. You will find Mathematica very useful. It’s available on many UR computers. We can take part of a class to deal with this if necessary. The following link will get you to more information than you need. http://www.me.rochester.edu/courses/ME201/websoft/softw.html Mathematica 12

13. 13 A little bit about notation vectors will be lower case bold face matrices will be upper case bold face “Vector notation” Matrix/linear algebra notation vectors will be column vectors, their transposes row vectors Indicial notation vectors have one superscript, their transposes have one subscript “real matrices” have one superscript and one subscript denoting row and column respectively

14. 14 Matrices do not have to be square. Examples of the notations

15. 15 Vector-matrix multiplication

16. 16 Summation convention “Metric tensor”

17. 17 ??

18. 18 The inertial coordinate system: coordinates x, y, z; unit vectors i, j, k i j k We will also have body coordinates, but not today We have to do physics in the inertial coordinate system

19. Start from the very basic: “f = ma” and consider a single particle/ point mass — moments of inertia all zero Conservation of momentum 19

20. 20 Angular momentum This doesn’t mean much for a particle, but we might as well start here This angular momentum is defined wrt the inertial origin, but any reference will do — different reference, different angular momentum

21. 21 Its rate of change which we call the torque. The torque depends on the point of reference — remember this

22. 22 Example: a particle falling under gravity i j k

23. 23 ??

24. 24 WORK AND ENERGY work = force times distance, so The kinetic energy of a particle

25. 25 and we can go back to the beginning and note that

26. 26 i j k 1 2 In general the integral will be different for the red path and the blue path If the integral is the same for all paths, we’ll have and the force is conservative

27. 27 Conservative forces come from potentials A force is conservative iff Potentials can be time-dependent; we will not deal with time-dependent potentials There’s a discussion of potentials in the text, and I’ll do a little on the board Bottom line The total energy, T + V, is conserved for a single particle under conservative forces

28. 28 An aside regarding potentials M m

29. 29 For celestial mechanics we do not include the m in the potential We associate the potential with the gravitating body There are several simple orbital examples in the text.

30. 30 SYSTEMS OF PARTICLES

31. 31 The particles can interact — including action at a distance Split each force into an external part and an interaction part, within the system momentum of the system the rate of change is equal to the force, so we have

32. 32 cancel All such pairs cancel by Newton’s third law of action and reaction This is called The weak law of action and reaction

33. 33 from which we deduce or, more generally Only the external forces change the momentum of a system under the weak law of action and reaction

34. 34 What is the momentum of a system? writethen

35. 35 If the sum of the external forces acting on a system is zero, the momentum of the system is conserved For example: the contents of a shotgun shell fired in a vacuum

36. 36 We can do the same thing for torque and angular momentum, and we’ll find we need a new law Look at a pair for simplicity’s sake

37. 37 The internal torques will cancel if the forces are parallel to a line connecting the two particles

38. 38 reference point r2r2 r1r1 r 1 – r 2 if f 12 is parallel to r 1 – r 2 Gravity works this way, as does electrostatics Not all internal forces work this way, but all the ones we care about do

39. 39 That is the strong law of action and reaction I will assume that throughout. We have the following for systems

40. 40 The angular momentum of a system can be written where You can establish this for homework. It’s not hard and it’s a good exercise.

41. 41 angular momentum of the system wrt the reference angular momentum of the system wrt the CM

42. 42 ??

43. 43 Kinetic Energy

44. 44 these are equal this is zero so the kinetic energy is as on the previous slide

45. 45 kinetic energy of the center of mass internal kinetic energy

46. 46 Let’s try to summarize today’s beginning

47. 47 ??