HW1, MAE 200A. Fall 2005. UCI
Nasser Abbasi
Given this simple pendulum, compute the equilibrium points and determine the linearized dynamics at each equilibrium point
Answer
part(1)
The system equation is given by
Since this is a second order ODE, there are 2 state variables. Convert this to state space formulation:
Let and , hence
At the equilibrium points we must have,
Hence we obtain 2 equations
Solving these equations, we obtain or for
Since the period is , then or , but is state variable that represents the angle hence
note: equilibrium at is stable, while at is unstable.
part(2)
starting with the nonlinear system equation
Near the equilibrium points, we express the nonlinear term in taylor series.
Suppose the penulium is at angle near the angle so it is a distance
Hence now
For the first equilibrium point, so the above becomes
For the first equilibrium point, so equation (1) becomes
Hence near , the linearized system equation is
and near the system equation is