CHAPTER ONE

1.0 INTRODUCTION

1.1 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. In Stickel (2009), a diagram of a double is shown in Fig. 1. 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.

Figure 1: Double Pendulum

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). The general equation for this method is:-

() – () = 0 …………………………………………………….. (1)

Runge-Kutta equation is generally to solve differential equation numerically and it’s very accurate also well behaved for wide range of problems. Generally, the general solution of Runge-Kutta for double pendulum is:-

w0 = α ……………………………………………………………….. .(2)

wi + 1 = wi +  (mi + m2 + 2m3 + m4) ……………………………….(3)

Which is wi ≈ y(ti) computes an approximate solution. Hamilton’s Equation is used when to solve the trajectories of double pendulum. The formula of Hamilton’s is:-

•  =  ……………………………………………………………(4)

Lastly, Euler-Lagrange equations for θ1 and θ2 are:

 () =  ……………………………………………………………(5)

1.2 STATEMENT OF RESEARCH PROBLEM

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.

1.3 AIM AND OBJECTIVES OF STUDY

The main aim of the research work is to examine double pendulum and its application. Other specific objectives of the study are:

1. to provide a simple quantitative description of the motion of a double pendulum
2. to determine the factors affecting the double  pendulum
3. to determine the moment of inertia of the double pendulum
4. to demonstrate each of the normal modes in a real double square pendulum
5. to demonstrate the appearance of chaos in the double pendulum

1.4 RESEARCH QUESTIONS

The study came up with research questions so as to ascertain the above stated objectives of the study. The research questions for the study are:

1.  How can a simple quantitative description of the motion of a double pendulum be achieved?
2. What are the factors affecting the double pendulum?
3. What is the moment of inertia of the double pendulum?
4. How can each normal mode in a real double square pendulum be demonstrated?
5. How can the appearance of chaos in the double pendulum be demonstrated?

1.5 ORGANISATION OF STUDY

This section deals with the organization of the research work in chapters; the chapter one of the research work will cover the background of the study, the statement of problem, the aims and objectives of study, significance and the scope of study, the chapter two will deal with the review of related literature on the effect of double pendulum and its application. The chapter three of the research work will cover the areas of materials and method. The chapter four will cover the experiment and discussion of result while the chapter five will cover the summary, conclusion and possible recommendation for the research work

1.6 METHODOLOGY

The motion of particles and rigid bodies is governed by Newton’s law. For the purpose of the research work, we will derive an alternate approach, placing Newton’s law into a form particularly convenient for multiple degree of freedom systems or systems in complex coordinate systems. This approach results in a set of equations called Lagrange’s equations and some part of Runge-kutta method. They are the beginning of a complex, more mathematical approach to mechanics called analytical dynamics. In this course we will only deal with this method at an elementary level. Even at this simplified level, it is clear that considerable simplification occurs in deriving the equations of motion for complex systems. These two approaches–Newton’s Law and Lagrange’s Equations–are totally compatible. No new physical laws result for one approach vs. the other. Many have argued that Lagrange’s Equations, based upon conservation of energy, are a more fundamental statement of the laws governing the motion of particles and rigid bodies. We shall not enter into this debate.

1.7 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. The study will serve as a repository of information to other researchers that desire to carry similar research on the above topic. Finally the study will contribute to the body of existing literature and knowledge in this field of study and provide a basis for further research

1.8 SCOPE OF STUDY

The study on the double pendulum and its application 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

1.9 DEFINITION OF TERMS

Pendulum: A pendulum is a weight suspended from a pivot so that it can swing freely.

Modes: The normal modes of a mechanical system are single frequency solutions to the equations of motion; the most general motion of the system is a superposition of its normal modes

Inertia: a tendency of the double pendulum to do nothing or to remain unchanged

REFERENCES

Stickel, L. (2009). The double pendulum. http://home2.fvcc.edu/ dhicketh/DiffEqns/spring09projects/LauraStickel/DoublePendulum.pdf .

S.Widnall, (2009). The double pendulum. http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamicsfall-2009/lecture/notes/MIT16/07F09/Lec20.pdf .

Nielsen, R., & B.T., N. (2013). The double pendulum. http://psi.nbi.dk/@psi/wiki/The-20-Double-20- Pendulum/files/projekt/201314/RON/EH/BTN.pdf .