# Flow Induced Vibration

Topics: Finite element method, Finite element method in structural mechanics, Direct stiffness method Pages: 59 (16180 words) Published: November 8, 2012
FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH

IVAN GRANT

Bachelor of Science in Mechanical Engineering Nagpur University Nagpur, India June, 2006

submitted in partial fulﬁllment of requirements for the degree MASTERS OF SCIENCE IN MECHANICAL ENGINEERING at the CLEVELAND STATE UNIVERSITY May, 2010

This thesis has been approved for the department of MECHANICAL ENGINEERING and the College of Graduate Studies by:

Thesis Chairperson, Majid Rashidi, Ph.D.

Department & Date

Asuquo B.Ebiana, Ph.D.

Department & Date

Rama S.Gorla, Ph.D.

Department & Date

ACKNOWLEDGMENTS

I would like to thank my advisor Dr.Majid Rashidi and Dr.Paul Bellini, who provided essential support and assistance throughout my graduate career, and also for their guidance which immensely contributed towards the completion of this thesis. This thesis would not have been realized without their support. I would also like to thank Dr.Asuquo.B.Ebiana and Dr.Rama.S.Gorla for being in my thesis committee. Thanks are also due to my parents,my brother and friends who have encouraged, supported and inspired me.

FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH

IVAN GRANT ABSTRACT Flow induced vibrations of pipes with internal ﬂuid ﬂow is studied in this work. Finite Element Analysis methodology is used to determine the critical ﬂuid velocity that induces the threshold of pipe instability. The partial diﬀerential equation of motion governing the lateral vibrations of the pipe is employed to develop the stiﬀness and inertia matrices corresponding to two of the terms of the equations of motion. The Equation of motion further includes a mixed-derivative term that was treated as a source for a dissipative function. The corresponding matrix with this dissipative function was developed and recognized as the potentially destabilizing factor for the lateral vibrations of the ﬂuid carrying pipe. Two types of boundary conditions, namely simply-supported and cantilevered were considered for the pipe. The appropriate mass, stiﬀness, and dissipative matrices were developed at an elemental level for the ﬂuid carrying pipe. These matrices were then assembled to form the overall mass, stiﬀness, and dissipative matrices of the entire system. Employing the ﬁnite element model developed in this work two series of parametric studies were conducted. First, a pipe with a constant wall thickness of 1 mm was analyzed. Then, the parametric studies were extended to a pipe with variable wall thickness. In this case, the wall thickness of the pipe was modeled to taper down from 2.54 mm to 0.01 mm. This study shows that the critical velocity of a pipe carrying ﬂuid can be increased by a factor of six as the result of tapering the wall thickness.

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ABSTRACT LIST OF FIGURES LIST OF TABLES I INTRODUCTION 1.1 1.2 1.3 1.4 II Overview of Internal Flow Induced Vibrations in Pipes . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition of Thesis . . . . . . . . . . . . . . . . . . . . . . . .

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FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 2.1 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . 4 4 4 12 12

Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 2.2.2 2.2.3 Shape Functions . . . . . . . . . . . . . . . . . . . . .

Formulating the Stiﬀness Matrix for a Pipe Carrying Fluid 14 Forming the Matrix for the Force that conforms the Fluid to the Pipe . . . . . . . . . . . . . . . . . . . . . 21

2.2.4 2.2.5

Dissipation Matrix Formulation for a Pipe carrying Fluid 26 Inertia Matrix Formulation for a Pipe carrying Fluid . 28

III FLOW INDUCED VIBRATIONS IN PIPES, A FINITE ELEMENT APPROACH 31

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3.1

Forming Global Stiﬀness Matrix from...