Damped Pendulum

Opening credits

• Can Thymio become a pendulum?

General concept presentation

• Simple pendulum
• A pendulum consists of a mass m fixed by a rigid shaft to a pivot, so that it can swing freely. The mass moves along a circle, and can therefore be described by the angle THETA, which uses the vertical as reference.
• To understand how a pendulum works, let's write its equation of motion.
• To do so, let's start by observing which forces apply on the mass: its weight P and the force T that the shaft applies on the mass.
• Let's apply Newton's 2nd law: net force = m a in the setup that we have drawn.
• The acceleration in circular motion is a = R d2THETA/dt2. In our case, R=L, so a = L d2THETA/dt2.
• We define the x and y axes and decompose the forces along these axes so as to apply Newton's second law. Along the x axis, we find: -mgsinTHETA = mLd2THETA/dt2 and along the y axis, we find: mgcosTHETA - T = 0.
• After simplifying, we get the equation of motion of a simple pendulum : d2THETA/dt2 = - g/L sinTHETA
• Damped pendulum
• In reality, we must take into account the environment's friction (here, the air). We then observe that the pendulum's oscillations get smaller and smaller, until the pendulum actually stops.
• To take friction into account in the equations, we must add an element that has an effect on the angular speed: LAMBDA dTHETA/dt.
• With this information, the equation of motion of a damped pendulum is: d2THETA/dt2 = - g/L sinTHETA - LAMBDA dTHETA/dt

Concept presentation with Thymio

• We have created a lego structure to illustrate the damped pendulum with Thymio. The rotation axis goes through Thymio's pencil hole so as to be as central as possible. To stabilise the oscillation, we have added an extra weight at the bottom.
• We can detect Thymio's position relative to the vertical using its accelerometer. This enables us to represent the oscillations described by the pendulum.

Closing credits