examples involve linear friction and stiffness elements, it is a simple matter to replace the appropriate terms by the nonlinear element laws when we encounter a nonlinear system. For example, a linear frictional torque $\backslash ($ B $\backslash$omega $\backslash )$ would become $\backslash (\backslash$tau$\_\{$B$\}(\backslash$omega$)\backslash ),$ which is a single$-$valued algebraic function that describes the nonlinear relationship.

EXAMPLE $5\mathrm{.}1$

Derive the state$-$variable model for the rotational system shown in Figure $5\mathrm{.}13($a$)$ when the outputs are the angular acceleration of the disk and the counterclockwise torque exerted on the disk by the flexible shaft. Also find the input$-$output equation relating $\backslash (\backslash$theta $\backslash )$ and $\backslash (\backslash$tau$\_\{$a$\}($t$)\backslash ).$