Dr. Shane Ross, Virginia Tech. Lecture 28 of a course on analytical dynamics (Newton-Euler, Lagrangian dynamics, and 3D rigid body dynamics). We introduce quasivelocities and their use in an efficient approach to model systems with constraints in what is called Kane's method or Kane's equations, which skips the use of Lagrange multipliers.
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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Ross Dynamics Lab: [ Ссылка ]
► Chapters
0:00 Introduction of topics
0:43 Usual method of handling constraints using Lagrange multipliers in Lagrange's equations. If we have n generalized coordinates and S constraints, we end up with n+S equations and n+S unknowns.
5:17 Quasivelocities are introduced, and some examples mentioned. (1) Body-axis components of the angular velocity for Euler's rigid body dynamics; (2) Body-axis components of the inertial velocity in aircraft dynamics.
19:47 General approach: defining the last S quasivelocities as the constraints, and formulating the dynamics of the remaining unconstrained n-S quasivelocities. The main thing is we get to skip the use of Lagrange multipliers, and simulate the dynamics using a smaller number of dynamic ODEs (n-S instead of n+S, so a savings of twice the number of constraints!).
26:10 Kane's method of getting the equations of motion for the n-S unconstrained quasivelocities, based on d'Alembert's principle. See also the Jourdain Principle.
36:27 Example using this method. The 2-particle baton with a wheel or skate under one mass. For the 2 unconstrained quasivelocities, we get fairly simple 1st order ODEs. A Matlab simulations shows that we get the same results as before.
55:54 Example: vehicle stability in a skid; Chaplygin sleigh. The resulting equations can be analyzed in a phase plane which shows lines of equilibria.
1:10:53 Example: model of semi-tractor-trailer truck or roller racer. Analysis of equilibria reveals the jackknife instability.
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► Courses and Playlists by Dr. Ross
📚Attitude Dynamics and Control
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📚Nonlinear Dynamics and Chaos
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📚Hamiltonian Dynamics
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📚Three-Body Problem Orbital Mechanics
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📚Lagrangian and 3D Rigid Body Dynamics
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📚Center Manifolds, Normal Forms, and Bifurcations
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► Textbook:
Engineering Dynamics: A Comprehensive Introduction
by N. Jeremy Kasdin and Derek A. Paley
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Lecture 2020-12-03, Fall 2020
#dynamics #KanesMethod #nonholonomic #JourdainPrinciple #mechanics #quasivelocities #Chaplygin #ChaplyginSleigh #baton #knifeEdge #quasivelocity #BoltzmannHamel #Equilibrium #Equilibria #RelativeEquilibrium #DAlembert #Stability #Jackknife #semiTruck #jackknifeInstability #EulersEquation
