Abstract

Citation

Researchers should cite this work as follows:

• Lukasz Grabarek (2022). 3-064-GearTrain-ModelingScenario. SIMIODE, QUBES Educational Resources. doi:10.25334/06KW-W970

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Article Context

Description

In this scenario you will model a series of interfacing gears, often called a gear train (see Figure 3). Your model will involve a second order linear differential equation with constant coefficients. The coefficients will encode the physical properties of the system such as moments of inertia of the gears and rotational friction.

Students should have some experience working with torque.

First approximation: gears with no teeth as an example of approach.

We are at this point only interested in the rough geometry of the problem, so let's smooth out the scenario and remove a complication: teeth. To this end, consider two disks of different radii, r₁ and r₂. Let us suppose that the smaller disk of radius r₁ is driving the larger disk with radius r₂. If you are able to build a model, fix your disks to a base so that they contact at a point on their circumferences and are able to rotate about their centers. You will rotate the smaller (driving) disk and observe a rotation in the larger (driven) disk. Let theta₁' and theta₂' be the rates, i.e. angular velocities (revolutions per minute), at which the disks revolve, respectively.

If you have access to a physical model, experiment with it. Otherwise, you may simply imagine such a set up or use an online tool like Gear Generator to aid your investigation.