Module 5: Applications of Second-Order ODEs
Model vibrations, resonance, RLC circuits, and explore the powerful analogy between mechanical and electrical systems.
Your Progress
Lessons 1-4
Practice
Module Quiz
Review
Learning Objectives
By the end of this module, you will be able to:
- Model undamped, damped, and forced spring-mass systems with second-order ODEs
- Classify damping as overdamped, critically damped, or underdamped
- Analyze resonance and beating in forced vibration systems
- Set up the ODE for an RLC series circuit and solve for charge and current
- Map mechanical quantities to electrical quantities using the analogy table
Module Lessons
1
Spring-Mass: Free Vibrations
Undamped and damped free vibrations: mx''+cx'+kx=0. Overdamped, critically damped, underdamped cases.
40-50 minutes
2
Forced Vibrations and Resonance
External forcing, resonance when forcing matches natural frequency, beating phenomenon.
40-50 minutes
3
RLC Circuits
Circuit equation Lq''+Rq'+q/C=E(t). Transient and steady-state solutions, impedance.
35-45 minutes
4
Mechanical vs Electrical Analogies
Side-by-side comparison: same math, different physics. Force-voltage analogy table.
30-40 minutes
After the Lessons
Practice Problems
10 problems covering vibrations, resonance, circuits, and analogies.
Practice Problems