1.1 Introduction
Pendulum refers to the weight suspended from a pivot so that it can swing freely. The double pendulum is the simplest example of a time-independent Hamiltonian system that exhibits chaotic behaviour. It consists of two point masses attached by massless rods and free to rotate in a plane. Its motion is governed by a set of coupled ordinary differential equation and its phase-space is four dimensional. The planar double pendulum serves as a paradigm for chaotic dynamics in classical mechanics. The character of its motion changes dramatically as the energy is increased from zero to infinity. At low energies, the system represents a typical case of coupled harmonic oscillators, and therefore, can be treated as an integrable system. At intermediate energies, however, it exhibits more or less chaotic features (Poschel, 2009).
As a prelude to other parts of this study, this chapter will discuss the background upon which this study was initiated, the statement of problems that led to this study, the Aim and Objectives of the study. Others are Significance of the study, Scope of work, Research questions, Limitation of the study and Definition of technical terms.
1.2 Background of Study
Pendulum that attach with another pendulum is called double pendulum. The area of dynamical system in physic and mathematics, a rich dynamic behavior of a strong sensitivity is exhibits from the double pendulum of simple physic system with initial conditions. Double pendulum has a difference types whether same mass or different mass that declare as m1 and m2 and same length or different length that declare as L1 or L2. It’s also have different angles.
The conservative system happens when double pendulum is friction-less that allows a conservation of energy, that is Energy in = Energyout. Furthermore, Stickel (2009) mentioned that double pendulum is two masses attached to rigid, mass less, rod with the base at a stationary location. In other words, the double pendulum become a linear system when angle is small and become non linear when angle is big.
To predict the behavior of double pendulum is very limited in certain regimes that is initial condition because the extreme sensitivity towards even small perturbations. In addition, Nielsen & B.T. (2013) said that the double pendulum is considered as a model system exhibiting deterministic chaotic behavior and the motion is governed by a set of coupled differential equations. This project we will use four types of methods to solve the double pendulum and its application which are Lagrangian Equation, Range-Kutta Equation, Hamilton’s Equation and lastly Euler Equation. In Stickel (2009), the Lagrangian is representation system of motion and can be used when system is conservative. Determine expressions for the kinetic energy and the potential when apply the Lagrange’s equation (S. Widnall, 2009).
Therefore, in Nigeria where the research was carried out, the activities that was conducted is to know the Double Pendulum and Its Application.
1.3 Statement of Problems
Investigation revealed that there have been series of studies on double pendulum that means it is a system that although the equations are known and you know of an instant in time the position and velocity of the pendulum exact, it is still not possible to predict how it will behave in the future. Secondly other studies have focused on double pendulum but not even a single study has been carried out on double pendulum and its application in Nigeria.
1.4 Aim and Objectives of Study
The aim of the study is to examine the Double Pendulum and Its Application. In achieving this aim, the following specific objectives were laid out as follows:
- To demonstrate each of the normal modes in a real double square pendulum;
- To assess the moment of inertia of the double pendulum;
- To evaluate the appearance of chaos in the double pendulum;
- To determine the factors affecting the double pendulum; and
- To provide a simple quantitative description of the motion of a double pendulum.
1.5 Research Questions
The study came up with research questions so as to be able to ascertain the above stated objectives. The specific research questions for the study are stated below as follows:
- How can a simple quantitative description of the motion of a double pendulum be achieved?
- How can each normal mode in a real double square pendulum be demonstrated?
- How can the appearance of chaos in the double pendulum be demonstrated?
- What are the factors affecting the double pendulum?
- What is the moment of inertia of the double pendulum?
1.6 Significance of Study
The study on the double pendulum and its application will be of immense benefit to the physics and mathematics department in Universities and other tertiary institutions in Nigeria as the findings of the study will educate the entire population under the umbrella of the study on the double pendulum, the factors affecting the performance of the double pendulum, the demonstration of normal modes in the double pendulum and also the demonstration on the appearance of chaos in the double pendulum.
This study will be of immense benefit to other researchers who intend to know more on this study and can also be used by non-researchers to build more on their research work. This study contributes to knowledge and could serve as a guide for other study.
1.7 Scope of Study
The scope of the research is focused on the Double Pendulum and Its Application. This research will cover on the factors affecting the performance of the double pendulum, the demonstration of normal modes in the double pendulum and also the demonstration on the appearance of chaos in the double pendulum.
1.8 Limitations of the Study
During the course of this study, many things militated against its completion, some of which are:
- Time Constraint: The time frame given to accomplish this project was very short due to school academic calendar and it was carried out under pressure which made the researcher not to implement some necessary features.
- Research material: availability of research material is a major setback to the scope of the study.
- Frequent power failure: This made the researcher append more money on fuel to ensure sustainable power.
- Financial Constraint: Insufficient fund tends to impede the efficiency of the researcher in sourcing for the relevant materials, literature or information and in the process of data collection (internet, questionnaire and interview).
1.9 Definition of Terms
Pendulum: A pendulum is a weight suspended from a pivot so that it can swing freely.
Inertia: a tendency of the double pendulum to do nothing or to remain unchanged.
SDP: Is an acronym for Spatial Double Pendulum
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