(Solved): Problem 2 A pet door (Fig. a) is modeled as a uniform pendulum (Fig. b) with length =0.4m ...

Problem 2 A pet door (Fig. a) is modeled as a uniform pendulum (Fig. b) with length $\mathbf{\ell }=0.4\mathbf{m}$ and mass $\mathbf{m}$. The friction at the pivot point $\mathrm{O}$ is modeled as rotational viscous damping with rotational damping coefficient ${\mathbf{b}}_{\mathbf{r}}$. The opening force is modeled as a horizontal force $\mathbf{f}(\mathbf{t})$ at the middle of the pendulum as shown. We assume small displacements about the vertical equilibrium. a) Find the transfer function from the input force $\mathbf{f}(\mathbf{t})$ to the angular displacement $\theta (t)$ of the pendulum. b) We consider the angular response to a unit step input force $\mathbf{f}(\mathbf{t})$. We observe that the unit step response $\theta (t)$ is oscillatory with the following properties: i) The angular displacement $\theta (t)$ reaches a maximum value ${\theta }_{\max }=6\mathbf{deg}$ ii) The angular displacement $\theta (\mathbf{t})$ converges to a steady-state value ${\theta }_{\mathrm{ss}}=4$ deg when $\mathbf{t}\to \infty$. Using this information, determine the values of the mass $m$ and the rotational damping coefficient $b_{\mathrm{r}}$. c) Roughly sketch the unit step response $\theta (\mathbf{t})$ that corresponds to the values of $\mathbf{m}$ and ${\mathbf{b}}_{\mathbf{r}}$ calculated in b). We are interested to adjust the rotational viscous damping at the pivot point $\mathrm{O}$ such that the angular response becomes non-oscillatory. The system mass $\mathbf{m}$ would remain the same. d) What is the smallest value of the new rotational damping coefficient ${\left({\mathbf{b}}_{\mathbf{r}}\right)}_{\text{new }}$ at the pivot point $\mathrm{O}$ that would result in a non-oscillatory response $\theta (t)$ ? e) What is the steady-state angular displacement ${\theta }_{\text{ss }}$ for this new value of the rotational damping coefficient? f) Roughly sketch the unit step response $\theta (t)$ that corresponds to the values of $m$ and ${\left({\mathbf{b}}_{\mathbf{r}}\right)}_{\text{new. }}$. Fig. a: Pet door Fig. b: Pet door model