First-principle modelling of a 3 mass, 4 spring mechanical system: recap and normalised matrix form

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 14:00

*Enlace a Spanish version

Summary:

This video is a continuation/complement of the video [moll3modEN], which details how to obtain each equation of motion of a system with 3 masses and 4 springs, and the signs of the force balances.

This additional video reviews the concept of a model in ’absolute’ units (with natural lengths of springs) and its equilibrium point (zero derivatives). Then the model is presented in incremental units and its expression in normalized form $ẋ=Ax+Bu$, $y=Cx+Du$ calculating the matrices $A$ and $B$; regarding output equation, do note that $C$ and $D$ matrices depend on each application, here $C=I$, $D=0$ are proposed as an example (full information output).

The simulation of the above mass-spring system (with a generic ode45 solver and also a specialised linear system simulator lsim) is discussed in video [moll3sim1EN], which continues this case study.

Eigenvalues of the $A$ matrix are key in analysing the properties of the free response (zero input), as discussed in the video [moll3freeEN].

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