# Double pendulum

## Project

The goal of this project is to demonstrate the main characteristics of chaos, and the study methods of it on a real double pendulum.
The rotation of the upper pendulum is measured by a 3-axis accelerometer, and of the lower pendulum by a digital rotation encoder. A bluetooth module sending the data to a PC, and the software displays the chaotic parameters and sections along with the spectacular movements.
The system demonstrates different measurement-methods, using numerical analysis and the hallmarks of the chaos like sensitive dependence to the initial conditions, so its suitabilty in high school education. Analysing the difference between this system with friction, and the theoretical, conservative double pendulum is an intstructive and exciting excersise for undergraduate students.

See pictures and parameters of our double pendulum

## Software

The software was written in LabVIEW. It displays the following main elements:
• Signals of the accelerometer, and the calculated upper angular displacements,
• The filtered upper-angle data (using Bessel-Lowpass Filter) and the calculated angular velocity,
• The lower angular displacements and the filtered data (using Bessel-Lowpass Filter) and the calculated angular velocity,
• The spectrums of the upper and lower-angle data,
• The comparison of different movements (started from near-to same initial conditions),
• The phase-portraits of the two pendula, and a phase-space section.

## Measurement results

We measured 8 different movements (started from near-to same initial conditions), and after the offline synchronization, we compared them by calculating the Ljapunov-exponents and a phase-space section.
Analysing the difference between these results and the theoretical ones, (calculated for a conservative double pendulum, chosen the appropiate energy-levels) is a complex, but instructive problem for researchers and undergraduate students.

## Our studies

[1] G. Vadai, Z. Gingl and J. Mellár, Real-time demonstration of the main characteristics of chaos in the motion of a real double pendulum, European Journal of Physics 33 907-920, 2012

[2] Kettős inga kaotikus mozgásának vizsgálata virtuális méréstechnikával - Report of scientific student's circle - Gergely Vadai, 2010

[3] Kettős inga kaotikus mozgásának vizsgálata virtuális méréstechnikával - BSc Thesis - Gergely Vadai, 2010