Q35E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Swinging Door. The motion of a swinging door with an adjustment screw that controls the amount of friction on the hinges is governed by the initial value problem

$\mathbf{I\theta }\text{'}\text{'}\mathbf{+}\mathbf{b\theta }\text{'}\mathbf{+}\mathbf{k\theta }\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{\theta }\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\theta }}_0\mathbf{,}\mathbf{\theta }\text{'}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{v}}_0$ ,

where $\mathbf{\theta }$ is the angle that the door is open, $\mathbf{I}$ is the moment of inertia of the door about its hinges, $\mathbf{b}\mathbf{>}\mathbf{0}$ is a damping constant that varies with the amount of friction on the door, $\mathbf{k}\mathbf{>}\mathbf{0}$ is the spring constant associated with the swinging door,${\mathbf{\theta }}_{\mathbf{0}}$  is the initial angle that the door is opened, and ${\mathbf{v}}_{\mathbf{0}}$ is the initial angular velocity imparted to the door (see figure). If  $\mathrm{I}$ and $\mathbf{k}$  are fixed, determine for which values of $\mathbf{b}$  the door will not continually swing back and forth when closing.

Step-by-Step Solution

Answer

The door will not oscillate at the condition of  ${\mathrm{b}}^2\ge 4\mathrm{IK}$ .

Step 1: Differentiating the values of θ

Given differential equation is  $\mathrm{I\theta }\text{'}\text{'}+\mathrm{b\theta }\text{'}+\mathrm{k\theta }=0$ 

 

Let  $\mathrm{\theta }={\mathrm{e}}^{\mathrm{rt}}$ ,

 

Then  $\mathrm{\theta }\text{'}={\mathrm{re}}^{\mathrm{rt}}$

 

 $\mathrm{\theta }\text{'}\text{'}={\mathrm{r}}^2{\mathrm{e}}^{\mathrm{rt}}$

Step 2: Finding the condition for not oscillating

Then the auxiliary equation is:

 

 $\begin{array}{c}{\mathrm{Ir}}^2+\mathrm{br}+\mathrm{k}=0\\ \mathrm{r}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\times \mathrm{I}\times \mathrm{k}}}{2\times \mathrm{I}}\end{array}$

 

Then the condition for not oscillating is:

$\begin{array}{c}{\mathrm{b}}^2-4\mathrm{IK}\ge 0\\ {\mathrm{b}}^2\ge 4\mathrm{IK}\end{array}$