Mass-Spring-Damper System

This example demonstrates how to model a classical Mass-Spring-Damper system in Simit. This system is a fundamental building block in mechanical engineering and control theory.

Mathematical Model

The system is governed by Newton’s Second Law:

Rearranging for the acceleration $\ddot{x}$:

Where:

• $m$: Mass (kg)
• $c$: Damping coefficient (N·s/m)
• $k$: Spring stiffness (N/m)
• $x$: Position (m)
• $F(t)$: External force input (N)

Simit Implementation

To implement this in the Simit Canvas:

1. Integrators: Use two Integrator blocks to convert acceleration $\ddot{x}$ to velocity $\dot{x}$ and position $x$.
2. Sum: Use a Sum block to compute the net force.
3. Gain: Use Gain blocks for $1/m$, , and .

Simulation Results

For a critically damped system ($c^2 = 4mk$), the response should settle without oscillation. For an underdamped system ($c^2 < 4mk$), you will observe decaying oscillations.

Parameter Sweep

You can run a parameter sweep in the console to see the effect of varying damping $c$:

// Console script
const results = [];
for (let c = 0.1; c <= 2.0; c += 0.5) {
  sys.parameters.c = c;
  const res = sys.run(10.0); // Run for 10 seconds
  results.push(res);
}
plot(results);