Lecture Notes on Kinematics and Dynamics

Key Concepts:

• Velocity and Displacement:

  • Velocity is the change in displacement over time (ds/dt).
  • Derivative of a displacement vs. time graph gives velocity.
  • Velocity vs. time graph derivative gives acceleration (dv/dt).
  • Integration of velocity over time (∫vdt) gives displacement.

• Derivatives and Integrals in Control:

  • Derivative in control indicates the future.
  • Integral represents the past.
  • PID Controllers:
    • P for Proportional (Present)
    • I for Integrator (Past)
    • D for Derivative (Future)

Types of Motion:

1. Constant Velocity (v = constant):

   • Acceleration (a) is 0.
   • Equation: s2 = vt + s1

2. Constant Acceleration (a = constant):

   • Equations:
     • v = at + v0
     • s2 = 1/2 at² + v0t + s1
     • v² - v0² = 2a(s2 - s1)

3. Acceleration as a Function of Time (a = f(t)):

   • Velocity: v = v0 + ∫f(t)dt
   • Example: If a = t³, then v = v0 + ∫t³dt

4. Acceleration as a Function of Velocity (a = f(v)):

   • Time: t = ∫dv/f(v)

5. Acceleration as a Function of Displacement (a = f(s)):

   • Velocity squared relation: v² = v0² + 2∫f(s)ds
   • Time for displacement: t = ∫ds/v(s)

Practical Examples:

• Example of a Ball on an Incline:

  • Acceleration is constant and directed down the incline.
  • Use kinematic equations to determine maximum distance and time to return.

• Homework Example Problem (226):

  • Spring compressed, releasing a block:
    • Plot acceleration versus displacement.
    • Find displacement and velocity over a given distance.

Additional Notes:

• Importance of setting up coordinate systems in dynamics.
• Ensure units are consistent (homogenization of units).
• Real-world applications such as NASA and US Navy projects.

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Remember, understanding the problem is crucial in dynamics. Always read and set up your coordinate system correctly, and be mindful of unit conversions. Use these fundamentals to approach more complex dynamics problems.