CHAPTER ONE
1.0 INTRODUCTION
1.1 BACKGROUND OF STUDY
The differential equations governing the transverse free vibration of non-prismatic Timoshenko beams resting on a two-parameter elastic foundation are two coupled differential equations in terms of transverse displacement and bending rotation angle with variable coefficients. Except for some special cases, there exists no closed-form solution in the literature; hence, approximate methods have played a notable role in the solution of this problem. Employing Chebyshev polynomials, Ruta, (2006) solved the free vibration and stability problems of non-prismatic Timoshenko beams resting on a two-parameter elastic foundation, providing that the moment of inertia and cross-sectional area can vary as general functions along the beam length. He introduced the effects of distributed normal, tangential, axial, and moment loads in the governing differential equations. De Rosa, (2005) used a geometrical approach to obtain the differential equations of motion for Timoshenko beams. He proposed two models for the free vibration analysis of Timoshenko beams resting on a Pasternak foundation, which is characterized by two parameters, namely Winkler spring and shear layer constants. In one of the models, the shear layer constant is interpreted as the proportion between bending moment and bending rotation, while in the other one, it is considered as the proportion between bending moment and total rotation.
Deriving approximate shape functions, Klasztorny, (2005) studied the vibration problem of non-prismatic Euler–Bernoulli and Timoshenko beams. Considering the presence of subtangential follower force, Auciello and De Rosa, (2005) carried out the dynamic analysis of beams on two–parameter elastic soil via application of the differential quadrature method (DQM) and the Rayleigh-Ritz method. Chen, (2002) studied the vibration of prismatic beams on one-parameter elastic foundation using the differential quadrature element method (DQEM). Later, Chen, (2002) employed DQEM for free vibration analysis of non-prismatic shear deformable beams resting on elastic foundations
The Classical Beam Theory (CBT), i.e. Euler Bernoulli Beam Theory, is the simplest theory that can be applied to slender FGBs. The first order shear deformation theory (FSDT), i.e. Timoshenko Beam Theory, is used for the case of either short beams or high frequency applications to overcome the limitations of the CBT by accounting for the tranverse shear deformation effect. Bhimaraddi and Chandrashekhara (1991) derived laminated composite beam equations of motion using the first-order shear deformation plate theory (FSDPT). Dadfarnia (1997) developed a new beam theory for laminated composite beams using the assumption that the lateral stresses and all derivatives with respect to the lateral coordinate in the plate equations of motion are ignored.
1.2 STATEMENT OF PROBLEM
The displacements functions for bending and shear of Timoshenko beam are assumed to be polynomials of third degree. However, cross-sectional tapering functions are not taken into account in the derivation of the finite element displacement or shape functions used in the aforementioned models, while they are considered as geometrical properties in energy integrals. Cleghorn and Tabarrok, (2005) developed a two-noded finite element formulation for a tapered Timoshenko beam for free lateral vibration analysis. In their model, the shape functions are obtained from the homogeneous solution of the governing equations for static deflection. They judged that the inclusion of the shear strain in nodal variables as in model used by To, (2004) is superfluous, and then selected the lateral displacement and rotation of cross-section as the nodal variables. Most of the researchers centered their work on cross section of the Timoshenko beam but not even a single study has been carried out on the vibration analysis of rotating Timoshenko beam using differential transform method.
1.3 AIM AND OBJECTIVES OF THE STUDY
The main aim of the research work is to examine the vibration analysis of rotating Timoshenko beam using differential transform method. Other specific objectives of the study are:
- to derive the differential transform equation governing the rotating Timoshenko beam
- to determine the potential and the kinetic energy expression of rotating Timoshenko beam
- to investigate on the factors affecting the application of differential transform method on rotating Timoshenko beam
- to proffer solution to the above stated problem
1.4 RESEARCH QUESTIONS
The study came up with research questions so as to answer the above objectives of the study. The research questions for the study are:
- What is the differential transform equation governing the rotating Timoshenko beam?
- What are the potential and the kinetic energy expression of rotating Timoshenko beam?
- What are the factors affecting the application of differential transform method on rotating Timoshenko beam?
1.5 RESEARCH METHODOLOGY
For the purpose of the research work, we will derive an alternate approach, placing Euler beam and Timoshenko beam 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 differential transform 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.
1.6 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 vibration analysis of rotating Timoshenko beam using differential transform method. The chapter three of the research work will cover the areas of materials and method. The chapter four will cover the area of experiment; while the chapter five will cover the summary, conclusion and possible recommendation for the research work.
1.7 SIGNIFICANCE OF STUDY
The study on the vibration analysis of rotating Timoshenko beam using differential transform method will be of immense benefit to the mechanical engineering department in the sense that the study will educate the students and other researchers on how to derive the differential transform equation for rotating Timoshenko beam, to also determine the potential and the kinetic energy expression of Timoshenko beam and also to teach them on the factors that is affecting the differential transform method. The study will serve as a repository of information for other researchers that desire to carry out a 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.
1.7 SCOPE OF STUDY
The study will focus on the vibration analysis of rotating Timoshenko beam using differential transform method. The study will cover on how to derive the differential transform equation for rotating Timoshenko beam, to also determine the potential and the kinetic energy expression of Timoshenko beam and also to teach them on the factors that is affecting the differential transform method.
1.8 LIMITATION OF STUDY
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).
Time constraint- The researcher will simultaneously engage in this study with other academic work. This consequently will cut down on the time devoted for the research work
1.9 DEFINITION OF TERMS
Vibration: Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point.