Uneven pendulums

Last crawled date: 1 year, 11 months ago

Made with Autodesk Inventor 2014 as a Dynamic simulation. The action is due to the gravitation force only.

There are two kinds of pendulums:
• Mathematical pendulum - a punctate mass suspended by an in-extensible thread, that has no weight
• Physical pendulum - any other (real) pendulum

The mathematical pendulum is characterized by:
1 / The length ratio is equal to the ratio pendulum oscillation periods at the power two (for instance, the period of an oscillation is tripled if the length is multiplied by 3 at the power 2)
2 / The oscillation period is independent of the hanging body mass and the body nature (isochronous)
3 / When the amplitude oscillation is small, the period does not depend on amplitude

Here we have a set of ten physical pendulums with different lengths of the wire, which oscillate for 140 seconds without any friction. The wires are just like the bicycle spokes.

The moment of start (the zero seconds) is when all ten pendulums are at 20° to the left from the vertical. Even if the lengths of wires are here increased by 2 mm (from 142 to 160 mm), their oscillation periods do not grow strictly in arithmetical progression, because they are physical and not mathematical pendulums.

By working with Inventor you can use Output Grapher to export the velocity values corresponding to each pendulum on the same Excel file. The file is included here and it was created on a step basis of 0.01 seconds, so the file has 140 x 100 = 14000 lines.

If you study this file you can find two moments (marked in colors):
• Yellow: The moment when the velocities for all the ten pendulums are near zero at around the 68.10 seconds, but this is when they are directly opposed.
• Red: At the double time (136.20 seconds) you will find indeed the first moment when all the pendulums are in a position approximately similar with the start moment.

These „moments” are actually ranges, because they are extended on more than one single step. If you want to reduce the ranges you have to increase the similarity of the physical pendulums with mathematical pendulums. To do this you have to increase the length and the mass of the weight, as well as to reduce the start angle from 20° to at most 3°. Another way is to calculate or find the position of the center of gravity for every of the ten pendulums and to assign them values in an arithmetic progression.

Note that the start moment of the oscillation is at the 10 seconds in the movie. In other words, if you want to identify a precise step in the Excel file, you have to add 10 seconds when you look to the movie, so that 68.10 seconds is 78.10 seconds in the movie.

As usual, if you work with Inventor you will find all the .iam and .ipt files so you can download them to enter Environments / Dynamic simulation and see for yourself what's happening. You will have all the settings at your will, so you can change any settings you want. You can also modify the dimensions of parts - like the lengths of wires, for example.

Note: The idea for this stand comes from https://youtu.be/V87VXA6gPuE. If you manufacture the experimental stand using this data, then you will start the oscillation by using a bar to keep the pendulums aligned at the launching moment.

Enjoy!