Dynamics Lecture Notes

Introduction

• Support MIT OpenCourseWare for free educational resources.
• Overview of mechanical engineering courses (Subjects 21-29).

Foundational Engineering Science (Subjects 21-25)

• Focus on model building and understanding the world through observations.
• Inquiry-based learning:
  • Develop models to explain problems.
  • Make observations to validate models.
  • Iterate based on results.

Modeling Process in Dynamics

1. Describe the Motion
   • Assign a coordinate system.
2. Choose Physical Laws
   • Examples: F = ma, conservation of energy, conservation of momentum.
3. Apply Mathematics
   • Solve equations of motion.

Historical Context in Dynamics

• Key Figures in Dynamics:
  • Cernus (1500s): Proposed heliocentric model.
  • Brae (circa 1600): Mathematician who collected astronomical data.
  • Kepler (1600): Developed laws of planetary motion using Brae's data.
  • Galileo (1609): Observed moons of Jupiter using a telescope.
  • Descartes (1630s): Developed analytical geometry.
  • Newton (1666): Formulated the three laws of motion.
  • Euler (1707-1783): Advanced the understanding of angular momentum and torque.
  • Lagrange (1788): Utilized energy methods to derive equations of motion.

Course Outline

• Key Topics:
  • Kinematics
  • Newton's laws and direct methods
  • Angular momentum and torque
  • Energy methods (Lagrange's contributions)
  • Applications to vibration problems.

Example Problem: Vibration

• Vibration System: Mass-spring system.
• Key Steps in Modeling:
  1. Describe the motion with a coordinate system.
  2. Apply Newton's Second Law (sum of forces = mass x acceleration).
  3. Construct Free Body Diagrams (FBDs).
     • Consider forces such as gravity and spring forces.

Free Body Diagram Methodology

• Draw forces in the direction they act.
• Define positive direction for deflections and velocities.
• Apply constitutive relationships:
  • Spring Force (FS) = kx
  • Damping Force (FD) = bẋ.

Equation of Motion Derivation

• Use FBD to set up the equation:
  FS + FD - mg = ma
  Rearranged to MẌ + Bẋ + Kx = mg.

Energy Method in Dynamics

• Total energy of the system: Kinetic + Potential Energy.
  • Kinetic Energy (T) = 1/2 * m * ẋ².
  • Potential Energy (U) = 1/2 * k * x² - mgx.
• Deriving equations of motion via energy considerations when no external forces act.

Kinematics and Reference Frames

• Introduction to kinematics:
  • Fixed inertial frames vs. moving frames.
  • Vectors and their derivatives:
    • Velocity and acceleration calculations.

Rigid Body Motion

• Rigid body motion involves both translation and rotation.
• Translation: All points move parallel.
• Rotation: All points rotate at the same rate.
• General Motion: Combination of translation and rotation, with the center of mass possibly moving.

Next Steps

• Further exploration of kinematics.
• Review readings up to Chapter 16 for vector derivatives and dynamics.