VIBRATION ANALYSIS OF PIPELINE SUPPORTED BEAMON ELASTIC FOUNDATION USING FOURIER TRANSFORM METHOD

ABSTRACT

Elastic foundation can be refered to as soil surface that can expand and assume its original position on application of force. Numerous research works has been carried out on pipeline supported beam using different method of analysis such as Differential Transform Method, Finite difference Method Finite Quadrature Element Method and so on. 

This work examines the vibration analysis of pipeline supported beam on elastic foundation using Fourier Transform Method in form of non-prismatic Euler-Bernoulli beam of length L resting on Winkler foundation with uniformly partially distributed load on one hand and devoid of load on the other hand. The resulting vibrational behaviour of the system was described by two different partial differential equations of order four under different model of vibration that is forced vibration and free vibration models respectively.

Fourier Transform Method was used to transform the Partial Differential Equation (PDE) to ordinary equation in order to obtain the exact analytical solution for both models. An assumed solution was formed in form of series solution and exponential for models (I) and (II) respectively, Ms-excel computer package was used to obtain the graphical analysis of the solutions respectively. The graphs were used to obtain the dynamic response of the beam under different parameters and the result shows the following:

For model (I), it shows that increase in velocity leads to increase in both the displacement and bending moment of the beam under different parameters. Similarly for model (II), increase in mass of the beam also leads to upward decrease and downward increase in deflection of the beam for various values of damping coefficient (h) and Winkler coefficient (c1) under time varying and distance varying mode. The graphs also diminish to zero as a result of the damping effect in the system. 

Key words: Euler-Bernoulli, Partially distributed load, Forced vibration, Free vibration, Dynamic response, Winkler foundation.

Word Count: 291.

TABLE OF CONTENTS

CONTENTSPAGE

Title Pagei

Abstractii

Certificationiii

Dedicationiv

Acknowledgementsv

Table of contentvii

CHAPTER ONE

Introduction

1.1 Background to the Study1

1.2Statement of the Problem5

1.3 Aims and Objectives of the Study5

1.5 Scope of the Study5

1.6Definition of Terms6

CHAPTER TWO

Literature Review7

CHAPTER THREE

Materials and Method19

Mathematical Formulation19

3.1Method of Solution to Model121

3.2 Analytical Solution for Model one (I) Using Fourier Transform Method21

3.3 Modal Analysis of Model (I)28

3.4Assumptions Taken to Solve Model (I)28

3.5Analytical Solution to Model (I)29

3.6Solution to Model (II)35

3.7Computation of the Fourier Transform of 37

CHAPTER FOUR

Results and discussions42

4.1Model (I)42

4.2Model (II) 43

CHAPTER FIVE

5.0Summary Conclusion and Recommendation for Further Study80

5.1Summary80

5.2Conclusion80

5.3Recommendation for further study81

References82

Appendix87