Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Recitations: 1 session / week, 1 hour / session
Laboratories: 1 session / week, 1 hour / session
Description
We will start by applying Newton’s Laws and Work-Energy principles to the motion of single particles, systems of particles and rigid bodies in planar motion. Then we use virtual displacements and virtual work to introduce Lagrange’s formulation of the equations motion for systems of particles and rigid bodies in planar motion. We will uncover a system’s equilibrium points and perform linear stability analyses. Lastly, we consider free and forced vibrations of linear multi-degree of freedom models of mechanical systems. Throughout the course, we will use MATLAB® to practice numerical methods for solving dynamics and vibrations problems.
Goals
After this course you will be able to:
- Apply knowledge of 8.01 and 18.03 to new problems
- Define a coordinate system for the system under consideration
- Derive equations of motion using either Newton’s momentum principles or Lagrange’s equations
- Analyze the equations of motion for the existence of equilibrium points and characterize the stability of those points
- Solve the equations of motion
Prerequisites
The prerequisites are Physics I (8.01) and Differential Equations (18.03).
Related Courses
After this course, related courses include:
2.004 - Dynamics and Control II
2.050J/12.006J/18.353J - Nonlinear Dynamics I: Chaos
2.032 - Dynamics
Required Texts
Williams, J. H., Jr. Fundamentals of Applied Dynamics. New York, NY: John Wiley and Sons, Inc., 2006. ISBN: 9780470133859.
Recommended Texts
Bedford, A., and Wallace L. Fowler. Engineering Mechanics: Dynamics. 2nd ed. Menlo Park, CA: Addison-Wesley Publishing, Inc., 1998. ISBN: 9780201180718.
Examinations
There will be two mid-term exams (1.5 hours each) scheduled 2 days after Ses #L12 and 5 days after Ses #L19. The final exam is 3 hours long.
All the exams (including the final) will be closed book. One sheet of handwritten notes will be allowed at the first mid-term exam, two sheets at the second mid-term exam, and three sheets at the final exam.
There will be two optional review sessions, one before each mid-term exam.
Recitations
The purpose of the recitations is to give students experience in the subject by working out examples and expanding on the material presented in the lectures. Attendance and participation in the recitations is obligatory.
MATLAB®
There will be MATLAB® sessions on Friday, covering material relevant to the course and problem sets. In addition there will be MATLAB® office hours to help with the homework materials. Typically, the MATLAB® problem to be worked through on Friday will be distributed through the course website earlier in the week.
Homework
Homework problems will typically be assigned every Monday and will be due at the beginning of lecture on Monday of the following week. The problem sets will be provided in the assignments section. No late homework will be accepted. You may discuss the problems with others in class, but you must (a) write up your eventual solution independently, and (b) list the names of students with whom you discussed the problem set. Problem sets may contain a MATLAB® component.
Grading
| ACTIVITIES | PERCENTAGES |
|---|---|
| Homework (including MATLAB® problems) | 30% |
| Mid-term exams (2) | 40% |
| Final exam | 30% |
List of Topics
| SES # | Topics |
|---|---|
| I. Motion of a Single Particle | |
| L1-L3 |
Kinematics: Trajectory, Velocity, Acceleration, Inertial Frame, Moving Frames Forces and Torques Linear Momentum Principle Angular Momentum Principle Work-Energy Principle Equations of Motion |
| II. Motion of Systems of Particles | |
| L4-L6 |
Internal and External Forces Linear Momentum Principle Angular Momentum Principle Work-Energy Principle Conservative Systems Impulsive Forces Collisions |
| III. 2D Motion of Rigid Bodies | |
| L7-L12 |
Kinematics: Angular Velocity, Instantaneous Center of Rotation Linear Momentum Principle Angular Momentum Principle Work-Energy Principle Parallel Axis Theorem |
| IV. Introduction to Lagrangian Dynamics | |
| L13-L19 |
Constraints and Forces Ideal Constraints Virtual Displacements Virtual Work D’Alembert’s Principle Generalized Coordinates and Forces Lagrangian Equations of Motion For 2D Holonomic Systems of Particles Equilibria Linearization of Equations Stability |
| V. Vibrations | |
| L20-L24 |
1-Degree-of-Freedom Oscillations: Natural Frequencies, Free-, Damped-, and Forced Response Multi-Degree-of-Freedom Oscillations: Natural Frequencies, Normal Modes, Free-, Damped-, and Forced Response Time Response Frequency Response Bode Plots |
Calendar
The calendar below provides information on the course’s lecture (L), recitations (R) and MATLAB® laboratory (M) sessions.
| ses # | TOPICS | KEY DATES |
|---|---|---|
| I. Motion of a Single Particle | ||
| L1 | Newton’s Laws, Cartesian and Polar Coordinates, Dynamics of a Single Particle | |
| M1 | Introduction and Overview of MATLAB® | |
| L2 | Work-Energy Principle | Problem set 1 out |
| L3 | Dynamics of a Single Particle: Angular Momentum | |
| R1 | Kinematics | |
| M2 | Lab 1: MATLAB® Interface and Matrix Multiplication | |
| II. Motion of Systems of Particles | ||
| L4 | Systems of Particles: Angular Momentum and Work Energy Principle |
Problem set 1 due Problem set 2 out |
| L5 | Systems of Particles: Example 1: Linear Momentum and Conservation of Energy, Example 2: Angular Momentum | |
| R2 | Systems of Particles: Linear and Angular Momentum, Solutions in MATLAB® | |
| M3 | Lab 2: Conditionals I | |
| L6 | Collisions |
Problem set 2 due Problem set 3 out |
| III. 2D-Motion of Rigid Bodies | ||
| L7 | 2D-Motion of Rigid Bodies: Kinematics | |
| R3 | Collisions and Problem Set 3 Hints | |
| M4 | Lab 3: Conditionals II | |
| L8 | 2D-Motion of Rigid Bodies: Kinematics - Instant Centers; Kinetics |
Problem set 3 due Problem set 4 out |
| L9 | 2D-Motion of Rigid Bodies: Kinetics, Parallel Axis Theorem | |
| R4 | Instant Centers and Problem Set 4 Hints | |
| M5 | Lab 4: Functions I | |
| L10 | 2D-Motion of Rigid Bodies: Falling Stick Example, Work-Energy Principle |
Problem set 4 due Problem set 5 out |
| L11 | 2D-Motion of Rigid Bodies: Finding Moments of Inertia, Rolling Cylinder with Hole Example | |
| R5 | Moments of Inertia and Problem Set 5 Hints | |
| M6 | Lab 5: Functions II | |
| L12 | 2D-Motion of Rigid Bodies: Rolling Cylinder and Rocker Examples |
Problem set 5 due Exam 1 two days after Ses #L12 |
| R6 | Exam 1: Problems 1 and 2 | |
| M7 | Lab 6: Algorithms | |
| M8 | Lab 7: ODE I | |
| IV. Introduction to Lagrangian Dynamics | ||
| L13 | Lagrangian Dynamics: Generalized Coordinates and Forces | |
| L14 | Lagrangian Dynamics: Virtual Work and Generalized Forces | |
| R7 | Virtual Work, Generalized Forces, Problem Set 6 Hints | |
| M9 | Lab 8: ODE II | |
| L15 | Lagrangian Dynamics: Derivations of Lagrange’s Equations and Examples |
Problem set 6 due Problem set 7 out |
| L16 | Lagrangian Dynamics: Examples | |
| R8 | Problem Set 7 Hints | |
| M10 | Lab 9: Eigenvalue Problems | |
| L17 | Lagrangian Dynamics: Examples and Equilibrium Analysis |
Problem set 7 due Problem set 8 out |
| R9 | Problem Set 8 Hints: Problem 1 | |
| M11 | Lab 10: Project | |
| L18 | Lagrangian Dynamics: Examples and Equilibrium Analysis | |
| L19 | Lagrangian Dynamics: Examples and Equilibrium Analysis |
Problem set 8 due Exam 2 five days after Ses #L19 |
| R10 | Rolling Disk: Sample Exam Question | |
| M12 | Lab 11: Project | |
| V. Vibrations | ||
| L20 | Vibrations: Second Order Systems with One Degree of Freedom - Free Response | Problem set 9 out |
| R11 | Review of Lecture 20: Second Order Systems with One Degree of Freedom | |
| M13 | Lab 12: Project | |
| L21 | Vibrations: Second Order Systems with One Degree of Freedom - Forced Response | |
| L22 | Vibrations: Free Response of Multi-Degree-of-Freedom Systems |
Problem set 9 due Problem set 10 out (optional) |
| R12 | Problem Set 10 Hints: Problem 3 | |
| M14 | Lab 13: Project | |
| L23 | Vibrations: Two Degrees of Freedom Systems - Wilberforce Pendulum | |
| L24 | Vibrations: Forced Response of Multi-Degree-of-Freedom Systems | Final exam 8 days after Ses #L24 |
