Introduction MEAM 535. What is MEAM 535? Audience. Advanced topics in dynamics

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1 What is MEAM 535? Advanced topics in dynamics Audience Review of Newtonian mechanics MEAM 535 Introduction Analytical mechanics: Lagrangian and Hamiltonian Special topics: Stability of dynamical systems, simulation Applications to linkages (planar, spatial), robot arms, mobile robots, rotor crafts Required course for all MEAM Ph.D. students Basics for Ph.D. qualifying exams for students in mechanical systems, robotics, mems, manufacturing, controls Elective for undergraduates, Core course for master s students Pre-requisites Mathematical maturity: differential equations, intro to linear algebra, vector calculus Basic computational skills Undergraduate dynamics course (particle dynamics, Newton s laws, work, energy, momentum,.) 1

2 Topics MEAM 535 Course Outline 3-D vectors, rotation matrices Vector calculus Three dimensional kinematics Generalized coordinates, generalized speeds, constrained systems Kinetics of particles: review Inertia dyadic Principle of virtual work, D Alembert s principle Newton-Euler equations for rigid bodies Lagrange s equations of motion Integrals of motion Hamiltonian mechanics Variational calculus May be Examples Tentative syllabus and other details 2

3 Logistics Instructors Vijay Kumar Michael A. Carchidi Teaching assistant Caitlin Powers Outside the class Office hours (what is a good time?) Appointments through Google groups Course material is on-line Text and reference material Excerpts from text books and papers (on-line) [meam535/dynamics] Slides (on-line) Homeworks (on-line) 3

4 Meetings MEAM 535 Logistics (continued) Monday, Wednesday 12:00-1:30 pm Review sessions (as needed) Work required 2 midterms final exam Problem solving Weekly homework (midterms will reuse homework problems) Grading Homework 10% Midterm I 20% Midterm II 25% Projects 10% Final 35% 4

5 Project 1 Kinematics and dynamics of a robot manipulator 7 degrees of freedom Modeling (geometry) Kinematics (joint motions and hand motions) Dynamics (joint torques and accelerations) 5

6 Project 2 Hummingbird Quadrotor (Ascending Technologies) 55 cm diameter 8 cm height Carbon fiber, Mg frame 500 gm (3 LiPo cells) 1. Dynamic Model 2. Matlab Simulation 3. Controllers for the quadrotor 4. Get it to fly interesting trajectories 6

7 Questions? 7

8 Introduction: Historical Perspective Three chapters in mechanics Chapter I Newtonian mechanics or classical mechanics Kepler - planetary motion Galileo - importance of acceleration Newton - Principia Mathematica (1687) 8

9 The Basis of Classical Mechanics Newtonian = Classical = non relativistic Assumptions (DG, 1988) Reasonable length scale ignore interactions at the atomic, molecular level not consider large scale systems - astronomy Reasonable time scale speeds are much smaller than speed of light Summary of Newtonian mechanics Aristotle, Kepler, Galileo, Newton 3 laws of motion inertial or Newtonian frame (Galileo) frame fixed to the earth frame fixed to the center of sun frame fixed to the center of the universe space is three-dimensional, Euclidean (homogeneous and isotropic) 9

10 The Basis of Classical Mechanics Three fundamental assumptions (Arnold, 1989) Space is 3-D, Euclidean (Newton) Principle of relativity (Galileo) There exist coordinate systems, called inertial reference frames such that: All the laws of nature are the same at all instants All coordinate systems that are in uniform, rectilinear motion with respect to the inertial one are also inertial Principle of determinacy (Newton) The initial state (all positions and velocities at some time) and a knowledge of external forces acting on the system uniquely determines the motion of the system. 10

11 Chapter II MEAM 535 Historical Perspective (Continued) Lagrangian mechanics Bernoulli - principle of virtual work, statics Euler - dynamics of rotating rigid bodies D Alembert - extension of virtual work to dynamics Lagrange - Mechanique Analytique (1788) Basis of Analytical mechanics Analytical because it is based on a few fundamental principles Lagrange describes it as an approach which does not require drawing any diagrams (e.g. free body diagrams) + simpler (once you know how to use it) - requires interpretation, maturity etc. 11

12 Why analytical mechanics (and mathematical approaches)? Analytical mechanics Deal with scalar quantities (e.g. energy), as opposed to vectorial quantities (e.g. acceleration) Lends itself to symbolic manipulation (automated via the computer) Useful in simulation, control Newtonian mechanics Arguably less powerful Can solve any problem by Newtonian mechanics Easier to understand Freshmen, sophomores learn it Lends itself to experimental validation Key: Important to be able to derive the equations of motion for systems! 12

13 Chapter III MEAM 535 Historical Perspective (Continued) Hamiltonian mechanics Poisson - canonical descriptions, impulse models Hamilton - first order equations, variational principles Jacobi - integrals of motion Gauss - principle of least curvature Analytical mechanics consists of Lagrangian mechanics Hamiltonian mechanics 13

14 History Focus of MEAM

15 Inverse Dynamics or Analysis MEAM 535 Engineering Problems Given an observed motion, what forces must have been applied in order to produce the motion Forward Dynamics Given a set of forces acting on the system, what is the ensuing motion of the system Control Given a desired motion, what kind of forces need to be applied to the system Fundamentals: Focus of MEAM 535 Equations of motion for all three problems. 15

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