The nonlinear spring in the mechanical system shown in Figure 8.36 is identified by the spring force \(f_{\text {spr }}=x|x|\). The system is subjected to a unit-step externally applied force \(u(t)\), and initial conditions \(x(0)=0, \dot{x}(0)=1\).

a. Derive the state-variable equations in vector form.

b. Using the RK4 method, plot the displacement \(x(t)\) versus \(0 \leq t \leq 10 \mathrm{sec}\).

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FIGURE 8.36 Problem 14. fspr=x|x| m x c 1 N's/m f (t) = u(t) m = 1 kg