3. Deriving the EOM
Setup from FBD:
Defining generalized moment of inertia for rod:
Initial Conditions:
Solve for the EOM:
ω
mg
l
θ
4. Deriving the Mathieu Equation
Inverted Pendulum EOM:
Dimensionless driving frequency,
time, amplitude:
Mathieu Equation:
HangingInverted
-ω +A
Stability
Region
Constant ω
ConstantA
5. Deriving the Effective Potential
The EOM is split into rapid oscillation (ε) and slow oscillation (θ0) components:
Solving for the slow component EOM:
Defining the Effective Potential:
Hanging
WellInverted
Well