Explore the intricacies of friction and phase portraits in this comprehensive lecture on analytical dynamics. Delve into Coulomb's model of friction, examining its effects on dynamic systems through phase portraits. Investigate the cone of friction concept and participate in live experiments to determine friction coefficients for various materials. Analyze the impact of Coulomb friction on spring-mass systems and whirling beads, observing how it transforms isolated equilibrium points into continuous ranges. Examine the fascinating case of a falling broom or ladder, where friction enables leaning against a wall without falling, and degrees of freedom suddenly change. Conclude with an in-depth analysis of a spinning symmetrical top, exploring its spin, precession, and nutation motions while applying the Routh procedure to reduce a complex 3-degree freedom system to a single degree.
Friction and Phase Portraits - Coulomb Friction - Cone of Friction - Falling Broom - Spinning Top
Overview
Syllabus
We start out with some descriptions of Coulomb's experiments and observations, including the important parameter the coefficient of friction .
The cone of friction is described .
Live experiments to determine the coefficient of friction for various materials in contact, using a ramp that can tilt.
We summarize Coulomb's law of friction in one formula and show how the idealized law differs from realistic friction, which has some dependence on the relative velocity of the two materials.
The effect of Coulomb friction on phase portraits, illustrated first with a spring-mass system, sliding along a surface with friction. Generally, friction makes isolate equilibrium points into continuous ranges of equilibrium points.
Another example, with a bead in a whirling tube with friction present.
Example of falling broom, falling ladder or the idealized falling baton. Once the baton releases from the wall, the degrees of freedom jumps from 1 to 2, and the dynamics changes discontinuously (non-smooth mechanics). Friction in this case allows the baton or ladder to lean against the wall without falling..
Spinning symmetrical top is analyzed. Inspired by experiments, we consider the different types of motion: spin, precession, and nutation. Using conserved angular momenta and the Routh procedure, this 3 degree of freedom system reduces to just 1 degree of freedom, the angle the top axis makes with the vertical.
Taught by
Ross Dynamics Lab
