Equilibrium System Behavior: Mass-Spring-Damper Model

[From Bruce Abbott (2015.02.11.0955 EST)]

I’ve augmented the mass-spring-damper simulation I posted earlier in order to provide an additional way of viewing the behavior of this equilibrium system. There is now the option to display a phase-plane plot. It is this kind of plot that reveals system dynamics in terms of so-called “attractor basins” and related features. In the screen-capture below, I’ve kept all the default settings of the program but selected the phase plot for display during the simulation.

The simulation begins by applying a steady force to the mass, which moves the mass to the left (in the model display) and compresses the spring. The phase-space diagram begins with the graph at center-screen, the resting position of the mass-spring system. The applied force accelerates the mass, increasing its velocity and moving its position to the left. The system then begins to oscillate around the new equilibrium position, with position and velocity changing out of phase. In the absence of damping the plot would follow a circle, but with damping, energy is lost and the graph spirals toward its new equilibrium position. Because the system will eventually converge onto a single point, in system dynamics terms this is called a “point attractor.”

After a short while, the applied force is removed and the system then begins to converge on a new point attractor, this one centered on the original equilibrium position that held sway before the force was applied.

The mass-spring-damper system will follow a circle around the point attractor in the absence of damping, indicating stable oscillation. With damping and the attendant loss of energy, the circle becomes a spiral toward the point attractor, demonstrating that this is a stable system that eventually comes to rest at its equilibrium point.

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In chaotic systems the plot may circle a given point, but the path followed around the point varies “chaotically.” In such as case, the point around which the plot orbits is called a “strange attractor.” The smoothly converging plot shown above demonstrates that the mass-spring-damper system is not chaotic.