Recent Advances in Vibrations Analysis Part 1 potx

Contents Preface IX Part 1 Analytical Methods 1 Chapter 1 Exact Transfer Function Analysis of Distributed Parameter Systems by Wave Propagation Techniques 3 Bongsu Kang Chapter 2 Phas

RECENT ADVANCES IN VIBRATIONS ANALYSIS

Edited by Natalie Baddour

Recent Advances in Vibrations Analysis

Edited by Natalie Baddour

Published by InTech

Janeza Trdine 9, 51000 Rijeka, Croatia

Copyright © 2011 InTech

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referencing or personal use of the work must explicitly identify the original source Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher No responsibility is accepted for the accuracy of information contained in the published articles The publisher assumes no responsibility for any damage or injury to persons or property arising out

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Technical Editor Teodora Smiljanic

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Image Copyright Eskemar, 2011 Used under license from Shutterstock.com

First published August, 2011

Printed in Croatia

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Additional hard copies can be obtained from orders@intechweb.org

Recent Advances in Vibrations Analysis, Edited by Natalie Baddour

p cm

ISBN 978-953-307-696-6

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Contents

Preface IX Part 1 Analytical Methods 1

Chapter 1 Exact Transfer Function Analysis of

Distributed Parameter Systems by Wave Propagation Techniques 3

Bongsu Kang Chapter 2 Phase Diagram Analysis for Predicting

Nonlinearities and Transient Responses 27

Juan Carlos Jáuregui Chapter 3 A Levy Type Solution for Free Vibration

Analysis of a Nano-Plate Considering the Small Scale Effect 47

E Jomehzadeh and A R Saidi Chapter 4 Second Order Shear Deformation Theory

(SSDT) for Free Vibration Analysis on a Functionally Graded Quadrangle Plate 59

A Shahrjerdi and F Mustapha

Part 2 Vibrations Analysis for Machine Maintenance 79

Chapter 5 Maintenance of Reducers with an

Unbalanced Load Through Vibration and Oil Analysis Predictive Techniques 81

Aparecido Carlos Gonçalves, Daniel Fabiano Lago and Maria da Consolação Fonseca de Albuquerque Chapter 6 Probabilistic Vibration Models in

the Diagnosis of Power Transformers 103

Pablo H Ibargüengoytia, Roberto Liñan,

Alberth Pascacio and Enrique Betancourt

VI Contents

Chapter 7 Measurement of Satellite Solar Array

Panel Vibrations Caused by Thermal Snap and Gas Jet Thruster Firing 123

Mitsushige Oda, Yusuke Hagiwara, Satoshi Suzuki, Toshiyuki Nakamura, Noriyasu Inaba, Hirotaka Sawada, Masahiro Yoshii and Naoki Goto

Part 3 Modelling and Analysis of Complex Systems 141

Chapter 8 Modelling and Vibration Analysis

of Some Complex Mechanical Systems 143

Tadeusz Markowski, Stanisław Noga and Stanisław Rudy Chapter 9 Torsional Vibration of Eccentric Building Systems 169

Ramin Tabatabaei Chapter 10 Beam Structural Modelling in Hydroelastic

Analysis of Ultra Large Container Ships 193

Ivo Senjanović, Nikola Vladimir, Neven Hadžić and Marko Tomić Chapter 11 Stochastic Finite Element Method

in Mechanical Vibration 223

Mo Wenhui

Preface

This book covers recent advances in modern vibrations analysis, from analytical methods to applications of vibrations analysis to condition monitoring

The book opens with a section on recent advances in analytical methods Dr Kang Bongsu contributed a chapter that presents an alternative technique for the free and forced vibration analysis of one-dimensional distributed parameter systems This approach is based on the idea of superimposing the amplitudes of the constituent travelling waves, rather than the traditional approach of normal mode expansion that relies on the apriori calculation of eigensolutions or assumed normal modes

In the second chapter, Juan Carlos Jáuregui presents an application of phase space to the identification of nonlinearities and transients In this interesting approach, a phase diagram is represented as a three-dimensional plot which can then be used for frequency and dynamic identification of a system The application of this approach to nonlinear mechanical systems such as gears, bearings and friction is also included in the chapter

The next chapter presents an analytical solution for a nano-plate with Levy boundary conditions The free vibration analysis is based on a first order shear deformation theory which includes the small scale effect The governing equations of motion, reformulated as two new equations called the edge-zone and interior equations, are based on the nonlocal constitutive equations of Eringen

A Shahrjerdi and F Mustapha co-authored the fourth chapter, which discusses second-order shear deformation theory applied to a plate with simply supported boundary conditions The material properties of the plate are graded in the thickness direction by a power law distribution and the equations of motion are derived via the energy method and then solved by applying Navier's method It is interesting to note that the authors demonstrate that the results of the second-order theory are very close

to those reported in the literature using a third-order theory

The next section of the book deals with the application of vibrations analysis to the condition monitoring and maintenance of various machines The first chapter in this section deals with the maintenance of reducers that have unbalanced loads The most

X Preface

commonly used maintenance approaches for reducers are oil analysis (via laboratory chemical analysis) and separately, vibrations analysis In this chapter, a novel way of combining the two approaches for more accurate results is presented

The second chapter in this section presents an alternative method for detecting failures

in transformers via the analysis of the vibrations produced inside the transformer under operation Normally, the transformer produces vibrations in the windings and the core, and these vibrations vary according to operating conditions However, in the presence of mechanical failure, the vibration patterns are different from those produced by normal conditions This idea is used as the basis for a failure detection mechanism, with the promise that this approach makes it possible to design an on-line real-time diagnosis system

The final chapter in this section describes an interesting method for monitoring the thermal snap of satellites, an effect which has been known to cause attitude disturbance in Low Earth Orbit satellites The difficulty with these types of thermally induced vibrations is that they are very slow and cannot be monitored via a traditional sensor-driven approach This chapter thus describes a novel approach to this problem via an onboard monitoring camera Images taken in space and the image processing

of these images are explained

The third and final section of the book deals with the modelling and analysis of various complex mechanical systems In particular, the first chapter of this section deals with the vibrations analysis of several mechanical systems possessing complex design and geometry Specific systems considered include a fatigue test rig for aviation gear boxes, a gas turbine blade and finally an annular membrane resting on

an elastic foundation of a Winkler type

The next chapter in this section considers the free vibration of eccentric building systems In particular, the coupled torsional-translational vibrations of both symmetric and eccentric one-storey building systems subjected to ground excitation are modelled and then analysed

In the subsequent chapter, the structural modelling of beams as part of the hydroelastic analysis of large container ships is presented In developing these models, it is important to appropriately account for the contribution of transverse bulkheads to hull stiffness and the behaviour of the relatively short engine room structure The application of this approach to the hydroelastic analysis of a very large container ship is then illustrated

The final chapter deals with the use of the stochastic finite element method for vibrations analysis Although the finite element method analysis of complicated structures has become generally accepted, regarding the given factors as known constants does not always correspond to the reality that material properties, geometry parameters and applied loads of the structure are often modelled as stochastic Thus,

how to incorporate the stochastic nature of these parameters into a finite element model is shown

I would like to express warm thanks to all the contributors, in particular for their efforts to ensure that difficult material is made accessible to wider audience

Dr Natalie Baddour

Department of Mechanical Engineering

University of Ottawa

Canada

Part 1

Analytical Methods

1

Exact Transfer Function Analysis of Distributed Parameter Systems by Wave Propagation Techniques

Bongsu Kang

Indiana University – Purdue University Fort Wayne

USA

1 Introduction

The vibrations of elastic structures such as strings, beams, and plates can be described in terms of waves traveling in waveguides (Cremer et al., 1973; Graff, 1975; Fahy, 1987) While the subject of wave propagation has been extensively studied in the fields of acoustics in fluids and solids rather than vibrations of elastic structures, wave analysis techniques have been employed to reveal physical characteristics associated with structural vibrations of elastic media (Argento & Scott, 1995; Kang & Tan, 1998) One of the advantages of the wave analysis technique, when applied to the structural vibration analysis, is its compact and systematic approach to analyze complex structures with discontinuities (Mace, 1984; Yong & Lin, 1989; Kang et al., 2003; Mei & Mace, 2005) Applying the concept of wave reflection and transmission, Mace (1984) obtained the frequency equations of Euler-Bernoulli beams including waves of both propagating and near-field types By the phase-closure principle, also referred to as the wave-train closure principle (Cremer et al., 1973), Mead (1994) determined natural frequencies of Euler-Bernoulli beams This principle states that if the phase difference between incident and reflected waves is an integer multiple of 2, then the waves propagate at a natural frequency and their motions constitute a vibration mode Based on the same principle, Kang (2007) presented a systematic approach to the free and forced vibration analysis of multi-span beams

The classical method, known as the normal mode or eigenfunction expansion, of solving the forced vibration problem of a distributed parameter system involves expansion of the forcing function into the eigenfunctions of the associated free vibration problem While this method is theoretically sound and powerful, the method is difficult to implement when the problem to be solved is a non-self-adjoint system typically due to complicating effects such

as damping, discontinuities, or non-classical boundary conditions, for which case obtaining the exact eigensolutions is not often feasible Although approximate eigensolutions may be used instead of exact ones, the problem still persists in the form of poorly convergent solution and/or significant error in the solution As an alternative approach to solve forced vibration problems, Yang and Tan (1992) presented a method for evaluating exact

closed-Recent Advances in Vibrations Analysis

4

form transfer functions for a class of one-dimensional distributed parameter systems Applying the energy functionals of constrained and combined damped systems, Yang (1996a, 1996b) presented a method to obtain a closed-form transient response solution in eigenfunction series for a distributed damped system

The dynamic displacement of any point in an elastic waveguide can be determined by superimposing the amplitudes of the constituent waves traveling along the waveguide, which is a basis of wave propagation Based on this simple fact, an alternative technique for the free and forced vibration analysis of one-dimensional distributed parameter systems is presented The method of normal mode expansion is often difficult to implement for nonself-adjoint systems with complicating effects such as mode couplings, non-proportional damping, discontinuities, or arbitrary boundary conditions, since the method requires eigensolutions or assumed normal modes as a priori However, this alternative analysis technique based on the elastic wave propagation does not pose such a requirement and leads to the exact, closed-form, distributed transfer function of a distributed parameter system The general wave solution of the equation of motion governing the dynamics of a waveguide is cast into a matrix form in terms of the constituent waves defined in the Laplace domain The spatial amplitude variation of the traveling wave is represented by the field transfer matrix and the amplitude distortion of the traveling wave incident upon a discontinuity due to geometric or kinetic constraints is described by the local wave reflection and transmission matrices Combining these matrices in a progressive manner along the waveguide by applying the concepts of global wave reflection and transmission matrices leads to the exact characteristic equation and corresponding mode shapes for the free response analysis and the transfer function of the system for the forced response analysis The transient response solution for a complex system can be obtained through the Laplace inversion of the transfer function using numerical inversion algorithms The exact frequency response solution, which includes infinite normal modes of the system, can be obtained in terms of the complex frequency response function from the transfer function One of the main advantages of this analysis technique is its systematic formulation resulting

in a recursive computational algorithm which can be implemented into highly efficient computer codes This systematic approach also allows modular formulation which can be readily expandable to include additional discontinuities with little alteration to the existing formulation In addition, it is also computationally advantageous that the technique always results in operating matrices of a fixed size regardless of the number of discontinuities in a waveguide This analysis technique is applicable to any one-dimensional waveguides (strings, axial rods, torsional bars, beams, and frame structures), in particular systems with multiple point discontinuities such as viscoelastic supports, attached inertias, and geometric/material property changes The analysis technique is demonstrated using the second order wave equation, fourth order beam equation, and sixth order curved beam equation

2 Second order systems

The free transverse vibration of a taut string, longitudinal vibration of a thin bar, and the torsional vibration of a shaft are governed by the equation of motion in the same form, the wave equation, and thus they are mathematically analogous Therefore, with no loss of generality, the transverse vibration of the string is taken as a representative problem for the description of the present analysis technique based on wave propagation The equation

governing the transverse motion of a uniformly damped string of span length L is