First-principle modelling of a 3 mass, 4 spring mechanical system (state-space internal representation)

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 10:00

*Enlace a Spanish version

Summary:

This video models the state-space differential equations of a mechanical system, consisting on three masses connected to each other by means of springs, and also connected both to a fixed wall and to a mobile end, whose position will be the input to the system. A simpler mass-spring-damper system is modelled in video [masmusym2EN ] and animated in [masmuAnimEN]. If you are unfamiliar with the topic, please watch these videos prior to this one.

Although initially the equations are proposed in “absolute” coordinates, including the natural length of the springs, we quickly switch to “incremental” coordinates (with respect to the equilibrium position).

Six first-order differential equations are obtained that constitute the normalized internal representation (state variables are three positions and three velocities). As it is linear, it could be expressed as $ẋ=Ax+Bu$, although, for brevity and convenience, that step will be carried out in following videos, where the model will be simulated and animations of it will be made.

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