Theory

where:

• τ is the restoring torque (N·m),

• κPLA​ is the torsion constant of the PLA line (N·m/rad),

• θ is the angular displacement (radians).

The negative sign indicates that the torque acts in the opposite direction to the displacement.

 Determination of PLA Torsion Constant: 

The encoder records the angular displacement over time, allowing precise measurement of θ(t). The torsion constant κPLA can be determined by analyzing the system's oscillatory motion. Since the system undergoes simple harmonic motion (SHM) when no external damping is present, the period T of oscillation is given by:

where:

• I is the moment of inertia of the rotating plate.

By measuring the period T and knowing the moment of inertia III, the torsion constant κPLA​ can be calculated.

Damped Oscillations with Magnets: 

In the second part of the experiment, magnets are added to the rotating plate to introduce magnetic damping. As the plate rotates, the magnets interact with nearby metal or magnetic materials (or the ambient magnetic field), creating eddy currents that oppose the motion, thus introducing a damping force.

The equation of motion for the system with damping is:

where:

• b is the damping coefficient due to the magnets, representing the strength of the damping force.

This leads to a damped harmonic motion, where the amplitude of oscillations decreases over time. The behavior of the system depends on the damping coefficient:

• Underdamped: The system oscillates with gradually decreasing amplitude.

• Critically damped: The system returns to equilibrium as quickly as possible without oscillating.

• Overdamped: The system slowly returns to equilibrium without oscillating.

Measuring Damping Effects: 

The encoder will record the time evolution of θ(t), and from this data, the damping effects caused by the magnets can be analyzed. The logarithmic decrement method can determine the damping ratio ζ, which is related to the damping coefficient b. The logarithmic decrement δ is given by:

where n is the number of oscillations between successive peaks. The damping ratio ζ is related to the damping coefficient as:

By measuring the decay of the angular displacement over time, the strength of the damping force and the effectiveness of the magnets in creating eddy currents can be evaluated. This provides insight into the damping behavior and how it affects the system's oscillations.