Analytical Solution

What quantities do you think characterize the motions of oscillating systems such as a pendulum? More specifically, if you wanted to compare the behavior of an actual pendulum and a model pendulum, what quantities would you compare? This section will help you find out.

The mathematical model for the simple pendulum system can be written as

with
and

We want to solve this initial value problem. To see what method to use, it helps to classify the differential equation.

What is the general solution of this differential equation?

where

The parameter is the natural frequency of vibration. Using the initial conditions to determine A and B, we find the solution

A formula that is easier to understand and to visualize is

where and . Click here to see a derivation of this type of formula.

According to this formula, the mass undergoes simple harmonic motion of amplitude R and frequency , with phase angle . Both the displacement and the velocity are quantities of interest.

What is an expression for the velocity in terms of t?

A graph of the angular displacement for typical initial conditions can be seen by Clicking here

theta vs. t

The corresponding graph of the angular velocity can be seen by Clicking here

velocity vs. t

What do you think was the value of used for these plots?

Another way of presenting this information is by a parametric plot with t as a parameter, using the displacement as the horizontal coordinate and the velocity as the vertical coordinate. Click here to see this parametric plot.

phase plane

Does this graph remind you of anything?

How was the equation for the graph derived?

The versus representation is a plot in the Phase Plane or State Space.

What do the arrows represent?

Why are the arrows pointing in the clockwise direction?(Hint: if the velocity is positive, is the displacement increasing or decreasing?)


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