If there is, we'll let you know.
Of the options that require some work, they are using Flash, or creating an animated image of some kind.
Typically though, the best animated cog logos come from people who know something about animation. Most animations are one-off, and become the property of the client who commissions them. You might find some of those people who are willing to dedicate their time and resources for free to create a one-off piece for you, but then the end result may not be to your liking.
What you could do if you want it done for free, is to do the work yourself. Go to an animation forum and get their help so that you can do the work to create the animation that you require.