Dynamics of planar movement in intrinsic (Tangent, Normal) coordinates (2 DoF point mass)

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 16:40

*Enlace a Spanish version

Summary:

This video discusses how to obtain the equations in Frenet intrinsic coordinates (tangent, normal) of the movement of a point mass subjected to forces. If the direction of the velocity vector is $\overrightarrow{T}:=(\text{cos}<!--FUNCTION\; APPLICATION-->𝜃,\text{sin}<!--FUNCTION\; APPLICATION-->𝜃)$, then $\overrightarrow{v}=\nu \overrightarrow{T}$, where $\nu$ is the velocity of the “speedometer ” in, say, a car (i.e., $\nu$ is a scalar). Then $\frac{d\nu }{dt}=\frac{F_T}{m}$, and $\frac{d𝜃}{dt}=\frac{F_{N_L}}{m\nu }$ where $F_T$ is the tangential force, and $F_{N_L}$ is the normal force (positive if it points to the left, counterclockwise, of the trajectory). The proof of these expressions and the discussion of their meaning are the objectives of the video.

The final part recalls that a rigid body has an extra degree of freedom (angular) and, therefore, does not have to be oriented with the center-of-mass path angle $𝜃$, using airplane or sports car maneuvers as an example. The equations with that additional degree of freedom are not covered, for brevity, in this introductory video, only devoted to “point mass” dynamics: we assume “path frame” and “body frame” are identical.

These equations will be used, for instance, in video [fugoid1EN] to model an aircraft/glider phugoid flight mode.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.