Question 1.10.2: Calculate the equilibrium position for the nonlinear system ......
Calculate the equilibrium position for the nonlinear system defined by \ddot{x} + x – β²x³ = 0, or in state equation form, letting x_{1} = x as before,
\dot{x}_{1} = x_{2}
\dot{x}_{2} = x_{1}(β²x²_{1} – 1)
These equations represent the vibration of a “soft spring” and correspond to an approximation of the pendulum problem of Example 1.4.2, where sin x ≈ x – x³/6. The equations for the equilibrium position are
x_{2} = 0
x_{1}(β²x²_{1} – 1) = 0
There are three solutions to this set of algebraic equations corresponding to the three equilibrium positions of the soft spring. They are
x_{e} = \left[\begin{matrix} 0 \\ 0 \end{matrix}\right] , \left[\begin{matrix} \frac{1}{β} \\ 0 \end{matrix}\right] , \left[\begin{matrix} -\frac{1}{β} \\ 0 \end{matrix}\right]