An Investigation On The Factors Affecting Beat Frequency Of A Coupled Pendula System
An investigation on the factors affecting beat frequency of a coupled pendula system.
Abstract
In the course of this experiment, I investigate how the mass of the counter-weight affects the beat frequency, when the system is put into the perfect superposition of the symmetric and anti-symmetric modes. I then go onto to explore different factors and other ideas involved with coupled pendula systems.
Essential Theory
In order to gain understanding on the workings of these phenomena, I will be starting with the mathematical expressions for both a single pendulum and the coupled pendulum system, in particular the equation of motion:
Where J is the moment of inertia (=), is the length of the pendula, and is the length down the pendulum of the coupling device. If we then use the identity, and assume that, for small angles, we can simplify this equation to:
As the coupled pendula system exhibit two modes, this equation can be written in two ways:
We can then write and as follows, by using trigonometric rules:
Where is the angular frequency, and A, B and are constants chosen to satisfy the initial conditions. When we substitute in these identities to the above equations of motion, we get the two homogenous equations:
These two equation yield two equations for , and . When we use in the above equations, we find that . This is the symmetric mode of the system, where A and B have the same amplitude and phase. When using , we get the result . This is the anti-symmetric mode, where A and B have the same amplitude, but are in opposite phases. From this knowledge, we can devise equations of motion for both the modes separately, which turn out as follows:
For my experiment, I will be setting up the system so that the motion of the two pendula will be a perfect superposition of the two modes. In this situation, , and , which simplifies the equations of motion to:
using simple trig. identities, these can be written as:
from these equations we have
Abstract
In the course of this experiment, I investigate how the mass of the counter-weight affects the beat frequency, when the system is put into the perfect superposition of the symmetric and anti-symmetric modes. I then go onto to explore different factors and other ideas involved with coupled pendula systems.
Essential Theory
In order to gain understanding on the workings of these phenomena, I will be starting with the mathematical expressions for both a single pendulum and the coupled pendulum system, in particular the equation of motion:
Where J is the moment of inertia (=), is the length of the pendula, and is the length down the pendulum of the coupling device. If we then use the identity, and assume that, for small angles, we can simplify this equation to:
As the coupled pendula system exhibit two modes, this equation can be written in two ways:
We can then write and as follows, by using trigonometric rules:
Where is the angular frequency, and A, B and are constants chosen to satisfy the initial conditions. When we substitute in these identities to the above equations of motion, we get the two homogenous equations:
These two equation yield two equations for , and . When we use in the above equations, we find that . This is the symmetric mode of the system, where A and B have the same amplitude and phase. When using , we get the result . This is the anti-symmetric mode, where A and B have the same amplitude, but are in opposite phases. From this knowledge, we can devise equations of motion for both the modes separately, which turn out as follows:
For my experiment, I will be setting up the system so that the motion of the two pendula will be a perfect superposition of the two modes. In this situation, , and , which simplifies the equations of motion to:
using simple trig. identities, these can be written as:
from these equations we have