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 All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, 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 of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Dragana Manestar Technical Editor Teodora Smiljanic Cover Designer Jan Hyrat Image Copyright Eskemar, 2011. Used under license from Shutterstock.com First published August, 2011 Printed in Croatia A free online edition of this book is available at www.intechopen.com 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 free online editions of InTech Books and Journals can be found at www.intechopen.com 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, [...]...  wr r l (2 .12 ) where  represents the kinetic properties of the constraint as  (s)  kc  cc s  mc s 2 (2 .13 ) Note that kc, cc, and mc in Eq (2 .13 ) are the non-dimensional spring constant, damping coefficient, and the attached mass, respectively, defined by 2 kc  Kc L T cc  Cc c0 T mc  Mc c0 TL (2 .14 )  (r  t  1)  i t (2 .15 ) Equation (2 .12 ) leads to Combining Eqs (2 .11 ) and (2 .15 ), the wave... 2 Define Rir as the global wave reflection coefficient which relates the amplitudes of negative- and positive-traveling waves on the right side of discontinuity i such that   C ir  RirC ir  C(i 1) r C il f( i 1)  C1 R1r 1 (2 .19 )  C ir  Cn R( i 1) l  C1 R( i 1) r Ril Rir   C(i 1) r Cil Cir i 1 i  Cn Rnl n Fig 2 Waves traveling along a multi-span string   Since C ir  f i R( i  1) lC... span Based on the definition of the global wave reflection coefficient, at the boundary   C1  R1rC1 (2.27)   However, recalling C1  r1C1 from Eq (2.24), it can be found that  (r1R1r  1) C1  0 (2.28) F(s )  r1 R1r  1  0 (2.29) For nontrivial solutions, which is the characteristic equation in terms of the Laplace variable s for the multi-span string with arbitrary discontinuities and boundaries... R1r recursively expands to include all the effects of constraints in the remaining side of the string until its expansion terminates at the rightmost boundary which yields Rnlrn This equation simply states that when the string system vibrates at one of its natural frequencies, r1R1r 1 As a simple example, for a single span uniformly damped string fixed at both ends, r1r2 1 from Eq (2 .18 ) and R1re2i... Ril  ri  ti2 ( Rir1  ri ) 1 (2.23) Rir and Ril progressively expand to include all the global wave reflection coefficients of discontinuities along the string before terminating its expansion at the boundaries where     C 1  r1C 1 C n  rnC n (2.24) 8 Recent Advances in Vibrations Analysis While the global wave reflection coefficient relates the amplitudes of waves traveling in the opposite direction... determined in terms of wave reflection and transmission coefficients For example, consider an infinitely long string constrained at a local coordinate   0 as shown in Fig 1, where the constraint is a point support consisting of an attached mass (Mc), a transverse spring (Kc), and a viscous damper (Cc) tC  C rC   0 Fig 1 Wave reflection and transmission at a discontinuity When a positive-traveling... 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., 19 73; Graff, 19 75; Fahy, 19 87) While the subject of wave propagation has been extensively studied in the fields of acoustics in fluids and solids rather than vibrations of... transfer function Denoting this interspan wave transfer coefficient as the global wave transmission coefficient Ti, define  C ir  TiC(i  1) r (2.25)    Rewriting Eq (2. 21) by applying C il  f( i  1) C(i  1) r and Cir  RirCir , and then comparing it with Eq (2.25), the global wave transmission coefficient at discontinuity i can be found as Ti  (1  ri Rir ) 1 ti f( i  1) (2.26) The global wave... discontinuities along its traveling path, it is more computationally efficient to employ the concepts of global wave reflection and transmission coefficients, in particular when the free or forced vibration analysis of a multi-span string is sought These coefficients relate the amplitudes of incoming and outgoing waves at a discontinuity Consider wave motion in a multi-span string as illustrated in Fig... & Mace, 2005) Applying the concept of wave reflection and transmission, Mace (19 84) 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., 19 73), Mead (19 94) determined natural frequencies of Euler-Bernoulli beams This principle states that . 2 011 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Recent Advances in Vibrations. (1) ir C   1 C  (1) ir R  (1) il R  (1) ir C   (1) i f  n C  n i 1i  1 C  1  ir C 1r R Recent Advances in Vibrations Analysis 8 While the global wave reflection coefficient relates the amplitudes of waves traveling in the. recalling 11 1 CrC    from Eq. (2.24), it can be found that 11 1 (1) 0 r rR C    (2.28) For nontrivial solutions, 11 () 1 0 r Fs rR   (2.29) which is the characteristic equation in