Developing the Second Order Differential Equation that Models Simple Pendular Motion
A Note About This Lesson
This optional appendix lesson is meant to give the reader an overview of the theoretical development of the second order non linear differential equation that models a simple pendulum.
It is meant to be read after completing the interactive project on simple pendulums earlier in the book. That said, if you already are familiar with pendulums, this article is readable on its own.
Introduction
A simple pendulum is an object of mass kilograms attached to the end of a rod of length meters, when the other end of the rod is attached to a fixed point. When the object is displaced radians from equilibrium (i.e. a "plumb line) and released, the function which describes the angular displacement of the object from plumb in radians is called pendular motion. The units of are seconds.
We will develop a second order differential equation governing pendular motion by studying the forces that act on the object of mass in a simple pendulum.
Move to see the forces on . You can also adjust using the slider. We will discuss below.
Developing the Second Order Differential Equation Model
To develop the second order nonlinear differential equation that governs , as is standard practice in Newtonian physics, we'll express the force acting on the object of mass in two ways and then equate the two expressions.
First, we'll express the force acting on the object of mass with Newton's second law, (force equals mass times acceleration).
The acceleration, , of the object of mass is the second derivative of , the arc length of circular path travelled by (shown above). From geometry, we know that the circular arc is the length of the rod, (the radius of the circular arc), times the angle of displacement from the plumb line, . Therefore:
Substituting this into yields the following expression of the force acting on the object of mass :
On the other hand, we can also calculate the force of gravity acting on the object of mass . Based on inspection of the applet above, the force of gravity acting on the object of mass is the component of gravity that is tangent to (The component perpendicular to is in balance with support provided by the rod of length ).
As can be seen in the applet above, the component of the force of gravity tangent to is . Therefore we can express the force of gravity acting on the object of as:
The force is negative because positive (and ) is in the opposite direction.
Finally, we can equate these two expressions of the forces acting on the object of mass to obtain this second order differential equation:
Dividing both sides by yields the standard form of the second order (nonlinear) differential equation governing undamped pendular motion:
This is nonlinear because of the sine that encloses on the right hand side.
We can add damping by adding an additional term on the right, proportional to the rate of change of . Just as in damped spring motion, damping is proportional to velocity, but now damping is also inversely proportional to mass . Thus the second order equation governing damped pendular motion is
where , a damping coefficient. Note that is the viscosity of the fluid the system is immersed in, and is the mass of the object.
You may also see this second order differential equation written as:
or if all the terms are collected on the left:
Any function which satisfies the above equation is said to be a model of pendular motion. For small initial displacements, , and so the above can be estimated using traditional second order algebraic methods for solving homogeneous second order differential equations with constant coefficients. However if initial displacement is larger than about a half a radian, this approximation is very poor, and numerical methods should be used instead such as Runge Kutta 4 (RK4). There are algebraic methods to solve this non linear equation, but it's vastly easier to estimate solutions with a numerical method such as RK4.
Estimating Solutions with Numerical Methods
As always, first we need to convert the second order differential equation to a system of two first order equations by way of systemification. We'll use the substitution: and , and cut out
x_1 and x_2.
Differentiating and substituting yields .
Differentiating and substituting yields .
As usual, putting it all together yields the first order system form of the second order equation governing pendular motion:
The following applet illustrates 100 iterations of the RK4 numerical methods applied to this first order system of differential equations. In the left pane, the x coordinates of the dots are the numerical estimates of ; the y coordinates of the dots are estimates of . In the right pane, these coordinates are transformed into a visualization of a pendulum.
On the left you can adjust:
- , the length of the rod (in meters)
- the damping coefficient equal to
- stepsize, the amount of time that advances between estimates. This is sometimes also called .
- The initial displacement of (in radians; drag the point Initial
- to adjust)