The Inverted Pendulum

1 Introduction

The purpose of this lab is to explore the dynamics of the simple harmonic oscillator (SHO). To make things a bit more interesting, we will model and study the motion of an inverted pendulum (IP), which is a special type of tunable mechanical oscillator. As we will see below, the inverted pendulum contains two restoring forces, one positive and one negative. By adjusting the relative strengths of these two forces, we can change the oscillation frequency of the pendulum over a wide range.

As usual (see Chapter 1), we will first make a mathematical model of the inverted pendulum. Then you will characterize the system by measuring various parameters in the model. And finally you will observe the motion of the pendulum and see that it agrees with the model (to within experimental uncertainties).

The inverted pendulum is a fairly simple mechanical device, so you should be able to analyze and characterize the system almost completely. At the same time, the inverted pendulum exhibits some interesting dynamics, and it demonstrates several important principles in physics. Waves and oscillators are everywhere in physics and engineering, and one of the best ways to understand oscillatory phenomenon is to carefully analyze a relatively simple system like the inverted pendulum.

2 Modeling the Inverted Pendulum (IP)

2.1 The Simple Harmonic Oscillator

We begin our discussion with the most basic harmonic oscillator – a mass on a spring. We can write the restoring force F = −kx in this case, where k is the spring constant. Combining this with Newton’s law, F = ma = mẍ, gives ẍ = −(k/m)x, or

ẍ + ω_{0}^{2}x = 0  (1)
with ω_{0}^{2} = k/m

The general solution to this equation is x(t) = A_{1} cos(ω_{0}t) + A_{2} sin(ω_{0}t), where A_{1} and A_{2} are constants. (You can plug x(t) in yourself to see that it solves the equation.) Once we specify the initial conditions x(0) and ẋ(0), we can then calculate the constants A_{1} and A_{2}. Alternatively, we can write the general solution as x(t) = A cos(ω_{0}t + φ), where A and φ are constants.

The math is simpler if we use a complex function x̃(t) in the equation, in which case the solution becomes x̃(t) = Ãe^{iω_{0}t}, where now Ã is a complex constant. (Again, see that this solves the equation.) To get the actual motion of the oscillator, we then take the real part, so x(t) = Re[x̃(t)]. (If you have not