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Scientist: Examples: Differential Equations

Driven Pendulum

Suppose we suspend a ball of mass m at the end of a rod of length L and set it in motion to swing back and forth subject to a driving force f. The equation to describe the system is:

u" + f*u' + sin(u) = 0

If we set u = up, then this equation can be reduced to:

u' = up

up' = -sin(u) - f*up

We model this system in Scientist as follows:

Micromath Scientist Model Driven Pendulum

To solve the equations and see the results, we run a simulation from time, t = 0 to time, t = 50 by setting the the independant variable range and number of intervals in the varable and parameter interface.

Micromath Scientist Variable and Parameter Set Driven Pendulum Simulation

When the simulation is run Scientist calculates the result at each interval and outputs the result in a new spreadsheet. We use the quick chart function look at the results.

Micromath Scientist Driven Pendulum Chart Graph

Notice that the force applied to the pendulum drives it from one oscillatory mode to another in the first 10 seconds, after which it settles down to a normal sinusoidal oscillation.