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In his 1927 paper On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers, Udny Yule considers a thought experiment where there is a simple pendulum hanging from a fixed point and whose natural path is disturbed by pea shots from children; the example motivates the first use of the autoregressive model for time series analysis.
Is there a data set (with real data) available that will illustrate this pendulum example?
Since you are looking for only a pendulum, and not a generic time series with random noise (like @blairchristian posted), then I would consider simulating the data.
Here is an example of a pendulum code, although you don't need the visualization part. It's a pretty common coding assignment (not as common as double pendulum it seems), so you can probably find your favorite code.
To add the random perturbations, you just need to determine which dimensions are allowed (i.e. if the pendulum is like the Foucault Pendulum, the perhaps the peas can only add random force in the horizontal plane of motion.)
just for fun, consider using quasi-random perturbations that have burst or breakout behavior, like
stock market volume deviation from the mean of that day of the week (example, Apple stock over last 10 years)
live data, i.e. from the twitter public stream, that perturbs the pendulum based on the occurrence of tweets or aggregate data from tweets
This paper, "Sunspot cycle simulation using random noise", might be a good place to start:
http://tf.boulder.nist.gov/general/pdf/230.pdf
Actual sunspot data here:
