Chapter 6.5, Problem 18E

Interpretation Introduction

Interpretation:

To define an appropriate dimension-less version of $\text{b}$, and to plot all the qualitatively different phase portraits that occur as $\text{b}$ and $\text{γ}$ vary for the $\text{mr}\ddot{\varphi }\text{ = - b}\dot{\varphi }\text{ - mg sin }\varphi {\text{ + mrω}}^2\text{sin }\varphi \text{ cos }\varphi$, where $\text{b}$ is arbitrary.

Concept Introduction:

The bead’s motion is governed by $\text{mr}\ddot{\varphi }\text{ = - b}\dot{\varphi }\text{ - mg sin }\varphi {\text{ + mrω}}^2\text{sin }\varphi \text{ cos }\varphi$.

Fixed points of the system are the points where $\dot{\varphi }=0$.

To find the dimensionless system, consider a dimension-less time $\text{τ}$ such that $\text{τ = }\frac{\text{t}}{\text{T}}$.