2014 inverted pendulum_presentation

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The document derives the equations of motion (EOM) for an inverted pendulum. It first sets up the free body diagram and defines terms like the generalized moment of inertia. It then solves the EOM and derives the Mathieu equation, which describes the inverted pendulum with dimensionless terms for driving frequency, time, and amplitude. Finally, it defines an effective potential well for the system and shows how the potential well strength varies with the pendulum angle.

Deriving the EOM
Setup from FBD:
Defining generalized moment of inertia for rod:
Initial Conditions:
Solve for the EOM:
ω
mg
l
θ

Deriving the Mathieu Equation
Inverted Pendulum EOM:
Dimensionless driving frequency,
time, amplitude:
Mathieu Equation:
HangingInverted
-ω +A
Stability
Region
Constant ω
ConstantA

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

~70° ~30° ~5°
Viewing Potential Well Strength

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