Introduction to Free Vibration

Problem Solving Procedure

1. Model: Create a simplified schematic of the system (noting assumptions) and find equivalent values, may use Lagrange equations

2. EOM: Create EOM(s) with desired DOF(s) in standard form: dependent variables on LHS, independent variable on RHS

3. Identify: Based on EOM, identify the type of system (e.g. forced/free, damped/undamped) to find the form of solution

4. Parameters: Calculate natural frequency, etc. for system

5. ICs: Find initial conditions based on DOF(s) of EOM(s)

6. Assemble Solution: Perform arithmetic to find the closed-form solution

Undamped Free Vibration

The general solution to these systems is

where

and

See https://www.intmath.com/analytic-trigonometry/6-express-sin-sum-angles.php for information for converting from sin/cos solution to amptude and phase solution

Damped Free Vibration

The general solution to these systems is

Underdamped

where

Critically Damped

Overdamped

Examples

Code