Kane's Method, Kane's Equations, Avoiding Lagrange Multipliers - Quasivelocities & Dynamic Equations

Explore an advanced lecture on analytical dynamics focusing on Kane's method and quasivelocities. Delve into efficient approaches for modeling systems with constraints, avoiding Lagrange multipliers. Learn about the general approach to defining quasivelocities and formulating dynamics of unconstrained variables. Examine Kane's method for deriving equations of motion based on d'Alembert's principle. Apply these concepts to practical examples, including a 2-particle baton system, vehicle stability in a skid (Chaplygin sleigh), and a semi-tractor-trailer truck model. Analyze equilibria and stability, including the jackknife instability. Gain insights into advanced dynamics concepts and their applications in mechanical systems and vehicle dynamics.

Introduction of topics.
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..
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..
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!)..
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..
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..
Example: vehicle stability in a skid; Chaplygin sleigh. The resulting equations can be analyzed in a phase plane which shows lines of equilibria..
Example: model of semi-tractor-trailer truck or roller racer. Analysis of equilibria reveals the jackknife instability..