2.003J | Spring 2007 | Undergraduate

# Dynamics and Control I

## Syllabus

### 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).

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.

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.

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

Spring 2007
Problem Sets
Exams
Lecture Notes