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,

$\dot{x}_{1}$ = $x_{2}$

$\dot{x}_{2}$ = $x_{1}$ (β² $x²_{1}$ – 1)

Question

Calculate the equilibrium position for the nonlinear system defined by [latex]\ddot{x}[/latex] + x - β²x³ = 0, or in state equation form, letting [latex]x_{1}[/latex] = x as before,

[latex]\dot{x}_{1}[/latex] = [latex]x_{2}[/latex]

[latex]\dot{x}_{2}[/latex] = [latex]x_{1}[/latex](β²[latex]x²_{1}[/latex] - 1)

Accepted Answer

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

[latex]x_{2}[/latex] = 0

[latex]x_{1}(β²x²_{1} - 1)[/latex] = 0

There are three solutions to this set of algebraic equations corresponding to the three equilibrium positions of the soft spring. They are

[latex]x_{e}[/latex] = [latex]\left[\begin{matrix} 0 \ 0 \end{matrix} ight] , \left[\begin{matrix} \frac{1}{β} \ 0 \end{matrix} ight] , \left[\begin{matrix} -\frac{1}{β} \ 0 \end{matrix} ight] [/latex]

Chapter 1

Q. 1.10.2

Q. 1.10.2

Step-by-Step

Verified Solution