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

95 PAGES (16787 WORDS) Dissertation
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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.

CONTENTS PAGE

Title Page i

Abstract ii

Certification iii

Dedication iv

Acknowledgements v

Table of content vii

CHAPTER ONE

Introduction

1.1 Background to the Study 1

1.2 Statement of the Problem 5

1.3 Aims and Objectives of the Study 5

1.5 Scope of the Study 5

1.6 Definition of Terms 6

CHAPTER TWO

Literature Review 7

CHAPTER THREE

Materials and Method 19

Mathematical Formulation 19

3.1 Method of Solution to Model1 21

3.2 Analytical Solution for Model one (I) Using Fourier Transform Method 21

3.3 Modal Analysis of Model (I) 28

3.4 Assumptions Taken to Solve Model (I) 28

3.5 Analytical Solution to Model (I) 29

3.6 Solution to Model (II) 35

3.7 Computation of the Fourier Transform of 37

CHAPTER FOUR

Results and discussions 42

4.1 Model (I) 42

4.2 Model (II) 43

CHAPTER FIVE

5.0 Summary Conclusion and Recommendation for Further Study 80

5.1 Summary 80

5.2 Conclusion 80

5.3 Recommendation for further study 81

References 82

Appendix 87

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APA

Makinde, T (2019). VIBRATION ANALYSIS OF PIPELINE SUPPORTED BEAMON ELASTIC FOUNDATION USING FOURIER TRANSFORM METHOD. Afribary.com: Retrieved November 25, 2020, from https://afribary.com/works/my-msc-project-complete

MLA 8th

Taiwo, Makinde. "VIBRATION ANALYSIS OF PIPELINE SUPPORTED BEAMON ELASTIC FOUNDATION USING FOURIER TRANSFORM METHOD" Afribary.com. Afribary.com, 24 Aug. 2019, https://afribary.com/works/my-msc-project-complete . Accessed 25 Nov. 2020.

MLA7

Taiwo, Makinde. "VIBRATION ANALYSIS OF PIPELINE SUPPORTED BEAMON ELASTIC FOUNDATION USING FOURIER TRANSFORM METHOD". Afribary.com, Afribary.com, 24 Aug. 2019. Web. 25 Nov. 2020. < https://afribary.com/works/my-msc-project-complete >.

Chicago

Taiwo, Makinde. "VIBRATION ANALYSIS OF PIPELINE SUPPORTED BEAMON ELASTIC FOUNDATION USING FOURIER TRANSFORM METHOD" Afribary.com (2019). Accessed November 25, 2020. https://afribary.com/works/my-msc-project-complete