3 mass, 4 spring mechanical system: modal analysis of free response oscillations

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 15:10

*Enlace a Spanish version

Summary:

This video presents an analysis of the free response modes of a 3-mass, 4-spring system whose modeling was detailed in the videos [moll3modEN] and [moll3mod2EN].

The Matlab code to visualize an animation of the movement of said masses was detailed in the video [moll3aniEN]; this code will be used (with minor variations) also in this video to present the results, but its detail will not be explained since it was already done in the aforementioned video.

A simpler code for a single mass-spring-damper animation apperas in [masmuAnimEN].

Basically, the modes of the free response are given by eigenvalues and eigenvectors of the matrix $A$ in a representation $\frac{dx}{dt}=Ax+Bu$, where, obviously, the term $Bu$ has no relevance in the free response studied here.

In this case, $A$ has complex eigenvalues and eigenvectors. The real part of the igenvalues (decay rate) allows approximating the duration of the transient (time until the oscillations disappear); it is the same in all modes, although it need not be in a general case. The imaginary part is the frequency of the oscillations in radians per second.

In this system, the free response has three oscillatory modes that last 65 seconds to disappear, of different frequencies and with the masses either in phase or in phase opposition.

Eigenvalues and eigenvectors are analyzed, and the real part of the eigenvectors is loaded as initial condition to simulate each mode, and to display an animation of it, so that the meaning of the mode is better understood. These modes are similar but not the same as ”resonance” modes (forced response to sinusoidal input).

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