Pinion-rack dynamical model: Euler-Lagrange equations with kinematic constraints

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 17:13

Summary:

This video discusses the Euler-Lagrange modeling of a 1 degree of freedom rack-and-pinion mechanism.

The mechanism is identical to the one seen in the video [pinionmEN], where Newton’s equations were used, with force and torque balances where reaction forces intervened in the contact.

The Euler-Lagrange equations do not involve reaction forces, but rather kinetic and potential energies (the latter is zero in this specific simple example).

The video uses the Euler-Lagrange formalism in two ways:

• Expressing the Lagrangian $L=T-V$ as a function of a single coordinate (1 degree of freedom model), obtaining a model with (inertia/equivalent mass) times acceleration equal to a certain balance of torques or generalized forces.

• Expressing the Lagrangian in terms of two generalized coordinates (2 degree of freedom model) and introducing the mechanical link between both, $f(q)=q_1\rho -q_2=0$, through a Lagrange multiplier. Due to the way it is written, the multiplier ends up being equal to the reaction force, but that need not happen in a more complex problem setup, in general.

As a result of both developments, exactly the same models are obtained as these in the video referred to above (which used Newton’s mechanics), obviously, given that they are equivalent.

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