Lecture on Dynamics of Moving Pendulum
Introduction
- Focus on dynamics of a moving pendulum.
- Comparison between static and moving pendulum systems.
- Importance of understanding pendulum dynamics in various applications, including robotics and walking.
Concepts Covered
Moving Pendulum Dynamics
- Moving pendulum as a model for walking or dancing.
- Hip moves in motion, leg rotates simultaneously.
- Conceptualization as a robot with a moving base and rotating arms.
Equation of Motion
- Derived equations for pendulum dynamics:
- Fixed Pendulum: Moment equation around a fixed point.
- Moving Pendulum: Moment equation not applicable at the moving point O; valid around Mass Center (MC).
- Explained why traditional moment equations are not applicable when the base is moving.
Forces and Acceleration
- Reaction forces (Fx, Fy) and their effects on the pendulum.
- Use of derivative principles to understand acceleration impacts.
- Derivation of velocity and acceleration vectors with respect to time.
Application of Newton's Laws
- Utilization of Newton's Second Law for deriving force equations in X and Y directions.
- How to factor in the unknown forces Fx and Fy using these laws.
Linear and Nonlinear Dynamics
- Linearization of non-linear dynamics using Taylor series.
- Comparison of linearized and actual dynamics of the pendulum.
- Discussion of homogeneous vs. non-homogeneous differential equations.
Practical Example and Simulation
- Simulation of pendulum motion using MATLAB.
- Creation of state-space equations and simulation of moving pendulum dynamics.
- Demonstration of how variations in base motion affect pendulum dynamics.
Advanced Topics
Chaotic and Intermittent Dynamics
- Introduction to chaotic systems and their relevance in higher-order dynamics.
- Visualization of dynamics through phase portraits.
Control Systems
- Brief introduction to control applications and disturbance handling in dynamic systems.
- Future applications in control classes (M530).
Real-World Understanding
- Analogy to human motion and dance.
- Historical context and personal anecdotes to aid understanding.
- Encouragement to engage with code for better practical understanding.
Conclusion
- Emphasis on understanding dynamic systems and preparing for advanced studies in dynamics and control.
- Encouragement to explore additional applications and simulations for deeper learning.
Note: These notes are based on a lecture discussing the dynamics of moving pendulums and their applications in both theoretical and practical frameworks. The lecture included equations, simulations, and physical analogies to real-world systems.