Coupled Double Pendulum Simulation and Testing

Created a theoretical model of a double pendulum and compared to measured results

Animation of double pendulum model

As part of one of my mechanical engineering courses I created a theoretical model of a coupled double pendulum. The predicted trajectory of this model was then compared to experimental data.

Diagram of system
Free body diagrams used to obtain acceleration expressions

I chose to use a mechanics (sum force & moments) approach to solve for the angular acceleration of each linkage. This results in 6 linearly independent expressions with 6 unknowns that can be solved for. While an energy based approach may have resulted in simpler expressions, I was using MATLAB to symbolically solve the resultant expressions. In the end, the 6x6 matrix of accelerations and reaction forces spit out the following monster expression for the angular acceleration of each link.

Expressions for the angular acceleration of each link that were spit out by MATLAB. It's possible that some of this trig could be simplified in ways that MATLAB doesn't recognize.

Despite how complicated this looks, it did look about how you would expect when simulated. I used the ode45 numerical integration routine to calculate linkages’ positions over time given the expressions for acceleration. Additionally, the energy of the system is conserved over time, which is a good sign. The simulated linkages were then compared to experimental results.

A tracking software called Kinovea was used to get the position of each link over time from a high speed video.
A comparison between experimental and simulated data

Overall the model had pretty good correlation with the experimental data. Error bars were generated for the theoretical data based on a Monte Carlo simulation, where the uncertainty of each parameter (mass, lengths, starting angle, etc.) was taken into account. Experimental uncertainty was measured based on the standard deviation over multiple trials. The location of the second linkage especially deviates from the theoretical model over time. This is probably because of the frictionless bearing assumption in the model.