rotor dynamics.pdf

A time-domain methodology for rotor dynamics: analysis

and force identification

Pedro Vaz Dias Lopes Paulo

Dissertação para obtenção do Grau de Mestre em

Engenharia Aeroespacial

Júri

Presidente: Professor Doutor João Manuel Lage de Miranda Lemos

Orientador: Professor Doutor Nuno Manuel Mendes Maia

Co-Orientador: Professor Doutor Fernando José Parracho Lau

Vogais: Professor Doutor Filipe Szolnoky Ramos Pinto Cunha

Professor Doutor Miguel António Lopes de Matos Neves

Outubro 2011

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ACKNOWLEDGEMENTS

This work was the result of intense work developed in the last 6 months. The effort was turned

much lighter with the cooperation carried out with my college Rafael Carvalho and the PhD student

Yoann Lage, who are developing projects in this subject, and have supplied me with valuable

information relevant to the content of this work.

I must also leave here a word of gratitude to Professor Miguel Neves for his support and

revision of the numeric methods employed during the course of this work. My thanks should also be

addressed to my advisers Professor Nuno Maia and Professor Fernando Lau. The first for is revision

and support in the area of the vibration theory and the second also for having reviewed this work and

for his advices concerning the Propfan engine case-study.

Finally I must thank my family to whom I owe it all.

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RESUMO

O estudo da dinâmica de rotors é essencial para a compreensão de componentes tão

determinantes em engenharia como bombas, compressores, turbinas e geradores. No âmbito da

engenharia aeronáutica, a dinâmica de rotors encontra ampla aplicação no contexto dos motores das

aeronaves.

Um dos presentes desafios nesta área prende-se com a identificação das forças que actuam

em sistemas rotativos, i.e. a capacidade de caracterizar a nível de localização, amplitude e forma as

forças que actuam em determinado ponto de uma estrutura rotativa, tendo como única fonte as

medições da sua resposta. Existem actualmente vários métodos disponíveis na literatura que

propõem soluções para o problema em rotors e que apresentam vários algoritmos de identificação de

forças que demonstraram bons resultados em estruturas não rotativas (e.g. vigas).

São exemplos disso três métodos de identificação de forças no domínio do tempo designados

por SWAT, ISF e DMISF. A identificação no domínio do tempo tem a vantagem de permitir a

obtenção quase em tempo real da amplitude e evolução das forças que actuam sobre uma dada

estrutura.

Este projecto de dissertação de Mestrado apresenta a proposta de uma aplicação númerica

destas novas metodologias de identificação no domínio do tempo em rotors simples e finalmente num

exemplo de aplicação aeronáutica: o motor Propfan desenvolvido no âmbito do projecto europeu

DUPRIN.

Palavras-chave:

Dinâmica de Rotors, Identificação de Forças, Domínio do Tempo, Propfan, SWAT, ISF,

DMISF

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ABSTRACT

The study of Rotordynamics is essential to understand the behavior of determinant

components in Engineering such as pumps, compressors, turbines and generators. In the field of

aeronautics, Rotordynamics finds wide application in the context of aircraft engines.

One of the actual challenges in this area is related with the identification of the forces acting in

rotary systems, i.e. the ability to characterize the forces acting in a specific point of the rotating

structure in what concerns its application point, amplitude and shape, using solely the measurements

taken from its response. Today there are various available methods in literature that propose solutions

to the problem in rotors and that present various force identification algorithms that have shown good

results in non-rotating structures (e.g. beams).

Examples are three time-domain force identifications methods dubbed SWAT, ISF and

DMISF. The time-domain identifications has the advantage of allowing the determination of the

amplitude and evolution of the forces acting in a given structure almost in real-time.

This Master thesis project presents the proposal of a numeric application of these new time-

domain identification methodologies to simple rotating systems and finally to a specific case-study of

aeronautics: the Propfan engine developed in the scope of the European project DUPRIN.

Keywords:

Rotordynamics, Force Identification, Time-domain, Propfan, SWAT, ISF, DMISF

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TABLE OF CONTENTS

LIST OF FIGURES .................................................................................................................................. vi

LIST OF TABLES .................................................................................................................................. viii

NOMENCLATURE .................................................................................................................................. ix

1 INTRODUCTION ............................................................................................................................. 1

1.1 Addressed Problem and Intended Goals ................................................................................ 1

1.2 Brief Scientific Publications Review ........................................................................................ 1

1.2.1 Rotordynamics Theory ........................................................................................................ 1

1.2.2 The Force Reconstruction and Identification Problems....................................................... 3

2 FUNDAMENTALS ........................................................................................................................... 6

2.1 Rotordynamics Fundamentals ................................................................................................. 6

2.1.1 Rotor Parts Fundamentals ................................................................................................... 7

2.1.2 Rayleigh-Ritz Analytical Solution – Application to a Multirotor Model ............................... 12

2.1.3 Rotordynamics Analysis – Application to the Multirotor Model ......................................... 16

2.1.4 Rotordynamics Modal Orbits and Sense of the Whirl........................................................ 20

2.2 Force Reconstruction Fundamentals .................................................................................... 22

2.2.1 FD – Frequency Domain Force Reconstruction ................................................................ 22

2.2.2 SWAT – Sum of Weighted Accelerations Technique ........................................................ 26

2.2.3 State-Space Fundamentals ............................................................................................... 27

2.2.4 ISF – Inverse Structural Filter ............................................................................................ 31

2.2.5 DMISF – Delayed Multi-step Structural Filter .................................................................... 34

3 NUMERICAL METHODS .............................................................................................................. 36

3.1 Numerical Aspects of the Rotordynamics Finite Elements ................................................... 36

3.1.1 The displacement vector ................................................................................................... 36

3.1.2 The finite elements of the rotor parts ................................................................................. 36

3.1.3 Solution of the Eigenvalue/Eigenvector Problem .............................................................. 40

3.2 Numerical Aspects of the Force Reconstruction Methods .................................................... 42

3.2.1 Structural Response Extraction ......................................................................................... 42

3.2.2 Frequency Domain Method ............................................................................................... 43

3.2.3 SWAT ................................................................................................................................ 43

3.2.4 ISF ..................................................................................................................................... 44

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3.2.5 DMISF ................................................................................................................................ 45

3.3 Modeling Forces in Rotordynamics using ANSYS® ............................................................. 46

4 APPLICATIONS AND DISCUSSION ............................................................................................ 48

4.1 Rotordynamics Analysis ........................................................................................................ 48

4.1.1 The Finite Element Model .................................................................................................. 49

4.1.2 Validation of the Finite Element Model .............................................................................. 49

4.1.3 Numerical Data .................................................................................................................. 50

4.1.4 Results and Results Discussion ........................................................................................ 51

4.2 Force Reconstruction ............................................................................................................ 63

4.2.1 Application to a Beam ........................................................................................................ 63

4.2.2 Application to a Symmetric Rotor ...................................................................................... 68

4.3 The Propfan Case Study ....................................................................................................... 73

4.3.1 Modal Analysis ................................................................................................................... 74

4.3.2 Reconstruction of a Force Caused by an Unbalance in the Fan ....................................... 79

5 CONCLUSIONS AND FURTHER DEVELOPMENTS................................................................... 82

5.1 Conclusions ........................................................................................................................... 82

5.1.1 Rotordynamics Analysis .................................................................................................... 82

5.1.2 Force Reconstruction Methods .......................................................................................... 82

5.1.3 The Propfan Case Study ................................................................................................... 82

5.2 Further Developments ........................................................................................................... 82

REFERENCES ...................................................................................................................................... 84

APPENDIXES ........................................................................................................................................ 86

Appendix 1 – Determination of the Whirl Sense of an Asymmetric Multirotor ...................................... 86

Appendix 2 – Energy Dissipation and Self-Excited Vibrations (Instability) ........................................... 88

Appendix 3 – Deviation Between Finite Elements and Rayleigh-Ritz Results ...................................... 95

Appendix 4 – Deviation Between the Natural Frequencies Obtained via Direct Method and the

Pseudo-modal Method .......................................................................................................................... 96

Appendix 5 – Used Beam Element’s Matrices ...................................................................................... 98

Appendix 6 – Brief Fundaments About Rotor Transient Motion ............................................................ 99

vi

LIST OF FIGURES

Figure 2.1 – Amplitude evolution with time on an unstable structural system subject to an initial

disturbance – source: (7) ......................................................................................................................... 7

Figure 2.2 – The disk and its reference frames – source: (4) ................................................................. 8

Figure 2.3 – The cross-section of the shaft – source: (4) ........................................................................ 9

Figure 2.4 – The bearing model – source: (4) ....................................................................................... 10

Figure 2.5 – Rotating mass with offset length – source: (4) .................................................................. 11

Figure 2.6 – The Multirotor model ......................................................................................................... 12

Figure 2.7 – Inertial reference frame coordinates – source: (4) ............................................................ 13

Figure 2.8 – Campbell Diagram example – source: (4) ........................................................................ 17

Figure 2.9 – Response diagram example – source: (4) ........................................................................ 19

Figure 2.10 – Possible orbits of a rotor’s shaft ...................................................................................... 20

Figure 2.11 – Generic elliptical orbit with displacement vector ..................................................... 21

Figure 2.12 – Position of a mass unbalance (in bold) in a disk performing Forward and Backward Whirl

............................................................................................................................................................... 22

Figure 2.13 – Illustration of a decomposition of a time-domain spectrum – source: (7) ....................... 25

Figure 3.1 – Shaft finite element – source: (4) ...................................................................................... 37

Figure 3.2 – Force components due to the presence of a mass unbalance in the disk of the rotor. .... 46

Figure 4.1 – The Multirotor model ......................................................................................................... 48

Figure 4.2 – Campbell Diagram convergence – the brackets above represent: (number of elements in

the inner shaft/number of elements in the outer shaft) .......................................................................... 50

Figure 4.3 – Campbell diagram – Symmetric rotor Example 1 ............................................................. 51

Figure 4.4 – Finite element curves make use of the direct method (chapter 3.1.3) .............................. 52


Figure 4.6 – Campbell diagram – Symmetric rotor Example 2 ............................................................. 54



Figure 4.9 – Campbell diagram – Symmetric Multirotor Example 3 ...................................................... 56

Figure 4.10 – Finite element curves make use of the direct method (chapter 3.1.3) ............................ 57


Figure 4.12 – Campbell diagram – Asymmetric Multirotor .................................................................... 59


Figure 4.14 – Campbell Diagram – Damped Multirotor Example 1....................................................... 61

Figure 4.15 – Results make use of the direct method (chapter 3.1.3) .................................................. 61

Figure 4.16 – Campbell Diagram – Damped Multirotor Example 2....................................................... 62

Figure 4.17 – Results make use of the direct method (chapter 3.1.3) .................................................. 63

Figure 4.18 – Beam Model with Force Impulse position represented by vector F ................................ 64

Figure 4.19 – Finite Element Model of the Beam .................................................................................. 64

Figure 4.20 – SWAT applied to the beam ............................................................................................. 66

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Figure 4.21 – ISF applied to the beam .................................................................................................. 67

Figure 4.22 – DMISF applied to the beam ............................................................................................ 68

Figure 4.23 – The Symmetric rotor model ............................................................................................. 68

Figure 4.24 – Finite element model with the element numbers in black and the nodal numbers in blue

............................................................................................................................................................... 69

Figure 4.25 – Error vs Force Combination Graphic .............................................................................. 71

Figure 4.26 – SWAT applied to the Monorotor ...................................................................................... 72

Figure 4.27 – ISF applied to the Monorotor ........................................................................................... 72

Figure 4.28 – DMISF applied to the Monorotor ..................................................................................... 73

Figure 4.29 – The DUPRIN Propfan in its test facility – source: (6) ...................................................... 74

Figure 4.30 – The Propfan revolution section ....................................................................................... 75

Figure 4.31 – The Propfan finite element model ................................................................................... 76

Figure 4.32 – Propfan Campbell diagram comparison .......................................................................... 77

Figure 4.33 – First two modal shapes of the Propfan; Author’s model above/Model in (6) below; FW

(Left) and BW(Right) .............................................................................................................................. 78

Figure 4.34 – ISF applied to an accelerating Propfan ........................................................................... 80

Figure A.1 – Whirl sense and orbit phase with .............................................................. 87

Figure A.2 – Whirl sense and orbit phase with .............................................................. 88

Figure A.3 – Generic periodic orbit of a rotor relative to a inertial reference frame– source: (7) - edited

............................................................................................................................................................... 90

Figure A.4 – Force components of the various matrices decomposed on its symmetric and skew-

symmetric terms – source: (7) ............................................................................................................... 92

Figure A.5 – First bending mode shape of the Multirotor ...................................................................... 96

Figure A.6 .............................................................................................................................................. 97

Figure A.7 .............................................................................................................................................. 97

Figure A.8 – Beam Element and its Degrees of Freedom .................................................................... 98

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LIST OF TABLES

Table 4.1 – Critical speeds of each mesh and the correspondent errors computed using the FE

solution as reference ............................................................................................................................. 50

Table 4.2 – Critical speeds associated with the previous Campbell diagram – RR/FE error ............... 51




Table 4.6 – Critical speeds associated with the previous Campbell diagram ....................................... 61

Table 4.7 – Critical speeds associated with the previous Campbell diagram ....................................... 62

Table 4.8 – Assembly between global and local dofs of the Beam mesh ............................................. 65

Table 4.9 – Numeric data of the beam .................................................................................................. 65

Table 4.10 – Transient analysis and force impulse data ....................................................................... 66

Table 4.11 – Assembly between global and local dofs of the Monorotor mesh .................................... 69

Table 4.12 – Numeric data of the Monorotor ......................................................................................... 70

Table 4.13 – Relevant data for the frequency domain force identification method ............................... 70

Table 4.14 – Relevant data for the time domain force identification methods ...................................... 70

Table 4.15 – List of the known and unknown nodes ............................................................................. 70

Table 4.16 – Numeric data of the Propfan ............................................................................................ 76

Table 4.17 – Error computed relative to the values obtained with the author’s finite element model ... 77

Table 4.18 – Transient analysis and mass unbalance data .................................................................. 80

Table A.1 – source: (7) - edited ............................................................................................................. 94

Table A.2 – Nodal displacements .......................................................................................................... 96

Table A.3 ............................................................................................................................................... 98

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NOMENCLATURE

Greek Letters

Modal damping

Generic constant

Eigenvectors of the finite element system

Nodal displacement vector

Virtual work

Virtual work performed by external forces

Virtual displacement along and directions

Error vector

Modal damping ratio

Modal displacements vector

Euler angles

Rate of nutation, rate of precession and rate of spin

Modal eigenvalue

Poisson ratio

Constant pi

Density

Mode shapes matrix

Mode shape vector

Modal vector

Constant rotating speed

Angular frequency

Angular speed components along and directions

Latin Letters

Cross-sectional area of the beam element

Residue matrix

Discrete time state-space system matrices

Continuous time state-space system matrices

ISF matrices

Coefficients of the damped system response equations

x

Gyroscopic term

Acceleration vector

Shear effect correction factor

Rigid body acceleration

Damping matrix of the system

Damping matrix of the elastic bearing

Damping matrix component at the ith line and j

th column

Distance between the mass unbalance position and the center of the shaft

Young modulus

Net energy-per-cycle exchange

Force vector

Axial force acting on the shaft

Force application point(s) selector

Force vector along the i

th generalized coordinates

Force component along the and directions

Shape function, first derivative of the shape function, second derivative of the

shape function

Shear modulus

Receptance matrix

Height of the beam cross-section

Routh-Hurwitz matrix

ith Markov parameter

ith Markov parameter of the inverse system

Area moment of inertial of the shaft cross-section about its neutral axis

Identity matrix

Inertia of the disk about its principal axis

Generic constant

Imaginary unit

Stiffness matrix of the system

Stiffness matrix of the elastic bearing

Stiffness matrix of the shaft

Disk stiffness correction matrix due to transient motion

Stiffness matrix due to the application of an axial force on the shaft

Shaft stiffness correction matrix due to transient motion

xi

Stiffness of the system

Sample number of the discrete state-space system

Stiffness matrix component at the ith line and j

th column

Length

Partial length of the shaft

– lead of the ISF

Mass matrix of the system

Mass of the disk

Mass matrix of the shaft

Mass of the system

Mass unbalance

Shape function

Number of force inputs

Number of measured degrees-of-freedom

Outer/inner shaft rotating speed ratio

Number of degrees-of-freedom

Number of inputs

Number of outputs

Amplitude of the vibrations along the generalized coordinate

Principal coordinate displacement amplitude

Principal coordinate displacement vector

DMISF Delay

ith generalized coordinate

ith generalized coordinate derivative

Radius

Roots of the characteristic equation

Cross-sectional area of the shaft

Generic matrix

Kinetic energy of the system

Transmissibility matrix

Kinetic energy of the disk

Kinetic energy of the shaft

Sampling time

Kinetic energy of the mass unbalance

Time

xii

Time at the kth sample

Strain energy of the system

Strain energy of the shaft

Input vector

Input vector derivative

Stacking input vector

Displacements along the and directions

State vector

State vector derivative

Response Vector

Output vector

Output vector second derivative

Stacking output vector

Weighting matrix

Wide of the beam cross-section

Dynamic stiffness matrix

Abbreviations

BW Backward whirl

DMISF Delayed multi-step inverse structural filter

Dof(s) Degree(s)-of-freedom

DUPRIN Ducted Propfan Investigation

FE Finite elements

FW Forward whirl

ISF Inverse structural filter

RB Rigid Body

RR Rayleigh-Ritz

Synch Synchronous

SWAT Sum of weighted accelerations technique

1

1 INTRODUCTION

1.1 Addressed Problem and Intended Goals

There are various applications on engineering, where rotors and rotating parts are of vital

importance. They play a determinant role on pumps, compressors, turbines, generators not to mention

other components. Due to its intrinsic dynamic state, vibration is especially relevant in this kind of

machines. Interaction between the rotating motion and the applied forces on the structure causes

some operation points to present intolerable vibration levels. Though the dynamic behavior of these

machines is rather complex, is nowadays considered well documented and understood.

As in any engineering applications, reliable and efficient operation is intended on the

machines with such components, and to achieve this, practical diagnosis methods are needed to

record and display accurately the performance that is actually being obtained, so that further

improvements can be implemented. A method of achieving a real-time depiction of the forces acting

on a rotor is a valuable tool for such a task, in what concerns the rotor structural performance

enhancement.

This work intends to provide firstly a revision of the literature regarding the subject of

Rotordynamics (Chapter 2.1) and of time-domain force identification methodology (Chapter 2.2).

Afterwards, a computational code developed by the author, make use of the implementation of several

force identification methods both in time and frequency domains, will be presented (Chapter 3.2). The

initial task of this code is to reproduce some of the examples presented in the literature, where only

non-rotating structures such as beams were found (Chapter 4.2.1). Finally the code will be applied to

the identification of forces in a simple rotor model (Chapter 4.2.2), and to a particular aeronautical

application of a Propfan engine (Chapter 4.3).

1.2 Brief Scientific Publications Review

1.2.1 Rotordynamics Theory

The first known rotor model used for a consistent study of the dynamic behavior and vibration

of this kind of structures was published in 1895 by the German engineer August Föppl [1]. This model

consisted of a single disk, centrally located on a shaft of constant circular cross-section and with

undamped rigid bearings placed at each end of the shaft. Föppl used this model to demonstrate that

such rotor operation was still stable even when its rotating speed exceeded the critical rotating speed,

i.e. he showed that in supercritical operation, tolerable vibration levels are observed.

Despite this fact, the first approach to the Rotordynamics study is often attributed to the

American Henry Homan Jeffcott, who has used a model such the one developed by Föppl. The

reason for this, was that Föppl published his work in a not so widely read German civil engineering

journal; in contrast, Jeffcott managed to publish his work in a well-known English journal, making it to

spread widely over the U.K. and the U.S.

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Since the Jeffcott publication was the first most spread work on this subject, a rotor model

consisting of one disk on a constant circular cross-section shaft, and with two rigid bearings at each

shaft end is called a Jeffcott rotor, although (as it will be seen later) in this work one has opted to

include such rotors under the designation Monorotor.

Soon, the scientists and engineers who have dedicated their work to the study of the dynamics

of this kind of structures, noted that the dynamics of a rotor is rather complex when we compare it for

instance to a spring-mass system. Since then Rotordynamics has been densely studied and

nowadays there are much broader applications in various rotor models of higher complexity than the

Föppl-Jeffcott rotor, and there is a more comprehensive knowledge of the dynamics involved.

The complexity of this dynamics can be confusing for those who venture for the first time in the

subject. Fortunately Dr. Frederick Nelson has turned it much easier with his “Rotor Dynamics without

Equations” [1]. This work by putting the focus more on the physics of the problem than in math,

becomes in my opinion an excellent introduction on the subject, and was important to clarify various

complex concepts, that were introduced in the course of this work.

One of the researchers who have devoted their work to the understanding of Rotordynamics

was D. J. Ewins. He has published recently an article where a very good review of the dynamic

problems associated with the turbomachinery of aircraft engines is included [2]. This article doesn’t

limit itself to the study of instability phenomena caused by the intrinsic structural nature of the rotor,

rather it includes a review on instability phenomena caused by the structure-aerodynamic interaction.

Another application example of Rotordynamics is presented in the work of J. J. Sinou et al. [3].

It includes a numeric prediction and an experimental validation of the non-linear dynamic behavior of a

Turbofan engine.

To gain physical intuition of this subject Nelson’s article is a good start, but to have a more

consistent work model to take proper engineering decisions, numeric methods are necessary. Maurice

Lalanne and Guy Ferraris have made this with their “Rotordynamics Prediction in Engineering” [4].

Their models and their numerical approaches were broadly used in the chapters of this work

dedicated to Rotordynamics analysis. Their work starts by defining the most common rotor models

with the help of the Rayleigh-Ritz method. Methods for the computation of the natural frequencies and

of the critical operation points are presented for each example. This work includes additionally numeric

predictions of the rotor response due to the action of various load configurations present in typical

rotor operation. Moreover the orbits described by the rotor in its vibration modes are also depicted for

each example, including as well a study about how these orbits change with changing bearing

properties. Everything is complemented with the application of the finite element method to the rotor

examples.

The book offers ultimately an application of these concepts to real case-studies where rotors

are used. In particular a Propfan engine developed in the scope of an European research project was

analyzed on the light of the concepts that were presented in the initial chapters of the work.

The studied Propfan engine will be reused on this work, not only to reproduce the associated

finite element model and resulting diagrams, but also to apply some new concepts of force

reconstruction in the diagnose of a common situation in jet engines: mass unbalance on the disks.

3

To achieve such a degree of understanding about the Propfan engine, a proper revision on

the matter was made. A short history about the Propfan research project can be found in [5].

Additional data was also found in the paper by Lalanne and Ferraris [6], where the followed models,

as well as the resultant solutions for the Propfan engine are condensed.

Also in Guy and Lalanne’s book, methods for unstable operation prediction are presented,

namely the Routh-Hurwitz method [4].

A more comprehensive and well documented work by Maurice Adams [7], “Rotating

Machinery Vibration”, has also been a source of information for this thesis. It was valuable mainly

because it offers additional and more complete information about rotor self-excitation behavior, than

the work of Lalanne and Ferraris. On the topic of dynamic analysis, this book has detailed many

specific parameters that have influence in the dynamics of rotors, so that more reality-accurate rotor

models could be programmed. They are also much more incisive in the modeling of the various

bearing characteristics, such as shaft/bearing misalignment, bearing type, and influence of lubrication.

It even contains correction factors used to predict fluid-structural interaction in turbo machines. It is

also much more detailed in several particularities of the numerical implementation of Rotordynamics,

namely in what concerns result tolerances and typical numeric errors. The book has also a dense

review on the data acquisition and usual experimental procedures in the context of rotor vibration

analysis and monitoring. The book ends with a series of interesting case studies where all the above

mentioned aspects are applied into real rotating machinery.

In what concerns Rotordynamics publication in Portuguese language, an academic book, with

many examples of application of Lalanne and Ferraris’ models was found [8]. This book was of great

importance due to the relevancy and quantity of its examples to understand the influence of the

bearing properties on the whirl motion and sense, and on the instability thresholds determination of the

rotors.

Despite they weren’t use on this work, two publications about the determination of the bearing

properties from experimental measures [9] and finite element method solutions [10] were referred.

Their connection to Rotordynamics and force reconstruction justifies the relevancy of this inclusion.

1.2.2 The Force Reconstruction and Identification Problems

In this work force identification is defined as being the capability to locate and determine the

time/frequency history of a force acting in a specific point of the structure, using as input the response

data of the considered structure. When the application point of the force is assumed to be known and

one is only interested in determining its time/frequency history from the measurement of the structure

response, then it is considered that the force is being reconstructed.

The concept of force identification based on the measurement of the structure response has

been one of the widest studied subjects in Mechanical Engineering, due to a myriad of reasons. Some

of them are related mainly with the fact that the use of force transducers sometimes is not practical in

complex system applications, and/or the dynamic loads of the considered system are too complex in

number, distribution and behavior in order to be accurately predictable. Moreover the determination of

the loads of a mechanic system is of unquestionable interest as an efficiency augmentation and

diagnostic tool.

4

Efforts in the last few decades are being made in the sense of achieving a practical method

capable of solving in a satisfactory way the inverse problem of dynamic systems [11]. The concept of

inverse problem is very simple, it is just based on the inversion of the conventional determination of

response from given loadings method (dubbed the forward or direct problem). The problem is, that

response is not always easy to characterize nor the accuracy of the system data is sufficient to obtain

a satisfactory result. In fact the inverse problem is more sensitive to inaccuracies in the model’s data

than the direct problem [12].

The Frequency Domain method (FD) is, at the present time, one of the most known

techniques in the force identification field. Hundhausen’s work contains a very good review in the

subject [13]. It makes use of the measured response in several points of the system and of the

frequency response function to compute an estimation of the applied forces.

A recent submitted article developed by the IDMEC Mechanic Engineering department in

Instituto Superior Técnico, extends this method to the localization of loads in a structure [14], [15]

making use of the transmissibility concept, based on the works of Maia et al [16] [17]. This will be used

in this work in a practical example applied to Rotordynamics.

Despite the fact of obtaining good results in many cases, the FD method is not suitable for

real-time force estimation on structural systems. In fact, when there is only relatively short duration

data available the FD method renders inaccurate results. Furthermore in several applications a

representation of the force evolution over time is pretended. The force identification in time domain

has been less studied as its frequency domain equivalent, but that is beginning to change, and there is

already a broad number of published works on the subject.

The inverse problem is posed differently in the time than in the frequency domain case. This

means that the faced challenges come from other sources previously unknown that require solution,

but it can also lead to more robust and accurate force identification if these difficulties are overcame.

The sum of the weighted accelerations technique (SWAT) is one of the time domain methods

present in the literature. It was first presented by Carne et al. [18], despite the fact it was first

developed previously by Priddy and Smallwood [19]. This method makes use of modal filtering to

estimate the applied forces on the structure. Firstly a set of multiple measures of the structure

response is obtained and then the resultant rigid body modal accelerations (the accelerations on the

center of gravity of the structure) are isolated through the application of a weighting matrix. If the

inertial properties of the structure are known the resultant forces or moments corresponding to the

translations or rotations defined by the isolated rigid body mode of the structure are determined.

This method pose several difficulties for force identification, although reconstruction is

possible, one can only determine the resultant inputs in the center of gravity of the structure, so in

many applications is impossible to deduce the application point of the forces. Additionally, there are

limitations related with the modal filtering process. This procedure requires that the used sensors are

well placed and in sufficient number in order to separate the response due to rigid body modes of that

related to flexible modes of the considered structure. Since a structure has an infinite number of

modes, one will always obtain an approximate solution, being the frequency band at which the rigid

body modes can be isolated wider, as the sensor set increases.

5

An extension to the SWAT method that allows the simultaneous determination of multiple

forces acting on a structure has been developed by Genaro and Rade [20]. This was not whatsoever

included in this work.

Other technique present in the literature is based on the inversion of the equations of motion

of the system. Kammer and Steltzner [21] have presented a method that uses this principle, named

the inverse structural filter (ISF). In this method the discrete time equations of motion are inverted, this

approach is more advantageous than the one that involves the inversion of the continuous time

equations of motion, since it avoids the integration and differentiation of the measured responses.

However, the signal processing should be the aim of extra careful handling. The authors begin by

representing the direct problem in the discrete state-space form, and propose a possible way of

representing the inverse system also in the discrete state-space form. The original formulation of the

ISF uses the measured responses to reconstruct the force input in the same time instant. This

procedure has therefore a causal relationship. In both the discrete and the continuous cases, the

inverse system is contaminated with unstable poles that yield to erroneous solutions of the estimated

forces. So, besides this algorithm, Steltzner and Kammer [21] presented an approach that creates a

stable ISF directly from the measured responses to reconstruct the force input. They also introduce

the concept of ‘non causal lead’, this means that the force input is reconstructed using responses from

future time instants. This approach has shown good results in the sense that better ISF stability is

achieved, but the force reconstruction become now only close to real-time determinable.

Recently Allen and Carne [12] presented an extension to the ISF that makes use of the non-

causal concept. The delayed multi-step inverse structural filter (DMISF) pretends to improve the state-

space representation of the original formulation of the ISF, using multiple time future response

samples to reconstruct the current force input and has shown a good performance in augmenting the

stability of the solution. In this work Allen and Carne used a state-space model built only from modal

parameters, which makes force reconstruction application to experimental cases much easier. Other

method to determine the state-space matrices from the typical mass, damping and stiffness matrices

is presented in the work of Unger and De Roeck [22].

At the time this was written no direct application of the ISF method to rotating systems was

found. The works found were either applied to simple spring-mass systems, beams, or other more

complex situations, such as the simulation of the docking process of the Space Shuttle in the MIR

space station [21].

Despite of this, there are already some publications on the subject of Rotordynamics force

identification. Zutavern and Childs have developed a force identification algorithm using magnetic

bearings [23], in its turn Verhoeven has published a work in this field making use of analytical

synthetized transfer functions [24]. Spirig and Staubli [25] have presented a method that determines

the forces acting in the rotor’s seals integrating the circumferential pressure distribution measured in

the seal coordinate.

In this work it is only performed the application of the FD, SWAT, ISF and DMISF methods to

Rotordynamics examples.

6

2 FUNDAMENTALS

2.1 Rotordynamics Fundamentals

On this chapter, the basic notions and topics related with Rotordynamics will be presented.

Before addressing some physical particularities of this kind of structures a study model will be

deduced. Based on this model, several typical Rotordynamics analysis options are explored, whose

obtained results are relevant for describing objectively the lateral dynamic behavior of rotors,

completing therefore, a comprehensive understanding to the reader who intends a basic introduction

on this topic. The effects caused by the action of a torsional moment applied on the shaft on its

bending behavior will not be considered.

Rotary systems such as rotors are constituted by several elementary parts that will be further

analyzed in the following section. These elementary parts are: the disk, the shaft and the bearing. A

common source of rotor excitation is resultant from an unbalance present on the rotor. A simple way of

including the effect of this excitation will be presented.

The elements that support the shaft of the rotor are the bearings. The bearings can be

classified into rigid or elastic. Rigid bearings are an engineering artifice that constrains any transversal

displacements of the shaft on the supporting point. In practice this is equivalent to a high stiffness

bearing. Outside this range of assumption, the bearing is considered elastic and is characterized by

finite stiffness properties and, if adequate, by damping properties.

Rotors have different designations depending on the characteristics of their bearings. If the

rotor’s bearings have symmetric stiffness properties it is called symmetric rotor, otherwise it would be

dubbed asymmetric rotor. If damping is added to the bearings of the rotor, it is designated as damped

rotor, whether its stiffness properties are symmetric or asymmetric.

Many machines in engineering, like jet engines, make use of two or more coaxial shafts use to

transmit power between a high or low pressure turbine and a compressor or fan. This systems

characterized by two or more coaxial shafts are known as Multirotors. Rotors with only one shaft are

dubbed Monorotors. Multirotors and Monorotors can have more than one disk in each shaft, what

really distinguishes one from the other is the number of shafts composing the rotary system.

As it will be explained later in depth on this chapter, while it rotates the rotor undergoes two

basic lateral vibratory phenomena that act simultaneously and determine the rotor’s vibration modes:

shaft bending and precessional whirling. The first, similarly to a beam, cause a lateral displacement of

the shaft sections relative to its reference state and results from the transversal loads applied to the

shaft, e.g. that caused by an unbalance on the rotor’s disk. The precessional whirling comes from the

fact that besides bending the shaft is also rotating, in fact, as the bended shaft rotates it describes

orbits around its initial reference state. The sense of this orbital motion depends on the sense of the

impact acting on the shaft. It is called Forward Whirl (FW) if the impact acts on the sense of the shaft

rotation or Backward Whirl (BW) if the impact acts on opposite sense to the rotation of the shaft. To

determine the parameters that influence the shape and frequency of the rotor’s vibration modes will

also be an aim of this chapter.

khagendrayadav

Highlight

7

The subject of self-excited vibrations will be treated in Appendix 2 to this work, since it is a

secondary aspect to this Thesis. Self-excited vibration is an unstable state of a structural system and

is responsible for severe machine damage due to the high vibratory levels that characterize this state.

Figure 2.1 shows the evolution of the amplitude of vibration of an unstable structural system

subjected to an initial disturbance.

Figure 2.1 – Amplitude evolution with time on an unstable structural system subject to an initial disturbance –

source: [7]

The ever growing amplitude of an unstable system raises the vibratory levels to intolerable

values. In the context of rotor lateral vibration, Crandall in 1983 [7], has shown that instability is

caused by dynamic forces which are perpendicular to the instantaneous radial displacement vector of

the rotor’s orbit. These aspects will be fully characterized and described on Appendix 2, and the non-

conservative nature of these destabilizing forces will also be referred. Finally, also in this appendix, a

method of predicting the stable boundaries in terms of the rotor rotating speed will be deduced.

Let us now proceed with the presentation of each of the elementary parts that constitute

rotors.

2.1.1 Rotor Parts Fundamentals

The rotor model that will be used in this work is constituted by some few basic parts. These

are: the disk, the shaft, the bearings and seals, and the mass unbalance, always present in machines

with rotors. To obtain the rotor model, the equation of Lagrange will be used in the form [4]:

(

)

(2.1.1.1)

In which denotes the number of the degree-of-freedom, the system’s generalized coordinates and

the generalized loads acting on it.

Thus the disk, shaft and mass unbalance kinetic energy must be computed, as well as the

strain energy of the shaft since it is the only rotor component considered as flexible.

In the subsequent pages is presented the detailed depiction of each rotor component and the

computation of the energies associated with each of them.

8

1. The Disk

The disk is considered rigid. This implies, since no strain energy is definable, that kinetic

energy is the only energy characterizing this component.

Figure 2.2 – The disk and its reference frames – source: [4]

Considering the disk represented in Figure 2.2, the angular speed vector [ ]

defined about its center of mass in the reference frame fixed to the disk, is as follows:

[

] [

] (2.1.1.2)

where and are the Euler angles and are its first time derivatives, dubbed rate of nutation,

rate of spin and rate of precession respectively.

Moreover, designating the displacements of the disk’s center of mass along the directions

and of the inertial reference frame respectively by and , the kinetic energy expression for the disk

results:

( )

(

) (2.1.1.3)

is the disk’s mass, and , , , represent the disk’s inertia about its principal axis .

The expression (2.1.1.3) can be simplified, noting that . Further simplification can be

achieved considering the angles and very small and constant rate of spin in (2.1.1.2).

Substituting these statements in expression (2.1.1.2) and then in (2.1.1.3) it results:

[

] [

] [

]

9

( )

( ⏟

( )

⏟

( )

⏟

)

( ( )

⏟

)

Giving the following expression for the kinetic energy of the disk:

( )

(

)

( )

(2.1.1.4)

where, the last term , is accounting for the Gyroscopic effect.

2. The Shaft

The shaft will be considered as a flexible beam with circular uniform cross-section, thus it is

characterized both by kinetic and strain energies. In the Figure 2.3, the shaft cross section is shown

along with two reference frames, the inertial ones with displacements and along the and axis,

and the rotating reference frame solidary with the shaft’s cross-section center, characterized by the

displacements and along the and axis.

Figure 2.3 – The cross-section of the shaft – source: [4]

The kinetic energy for such a shaft can be described exactly by the same course of reasoning

followed for the disk, with the slightly difference this time, that it should be applied to an element of

length of constant cross-section. The kinetic energy expression for the shaft is:

∫ ( )

∫ ( )

∫

(2.1.1.5)

Here denotes mass per unit volume, is the shaft’s cross-sectional area which is supposed

to be constant, and is the area moment of inertial of the shaft cross-section about its neutral axis,

10

which is once more considered as constant. As to the physical meaning of the terms of the equation

(2.1.1.5), it can be said that the second one is the secondary effect of rotary inertia, coherent with

Timoshenko beam theory, and the last term represents the gyroscopic effect.

Considering the strain caused by the rotation of the shaft, and the strain caused by a constant

axial force applied in the shaft’s ends, and if additionally the shaft is considered elastic, i.e. it is

inside the scope of validity of the Hook’s Law, it results for the strain energy of the symmetric shaft:

[4]

∫ [(

)

(

)

]

∫ [(

) (

)

]

(2.1.1.6)

The first term in the expression (2.1.1.6) accounts for the strain energy resultant from the

longitudinal deformation of the spinning shaft. This longitudinal strain due to rotation of the points

inside the constant cross-section of the shaft is considered approximately linear, since the

displacements are small and the cross-section has symmetry relative to the rotating reference frame.

The longitudinal strain caused by the presence of an axial force has only non-linear terms and the

associated strain energy is included in the second term of the equation (2.1.1.6). Constant, is the

Young’s Modulus of the shaft’s material.

3. Bearings and Seals

On this work the elastic stiffness and viscous damping properties of the bearings will be

assumed as known. Notwithstanding this fact, there are several methods to estimate the bearings

characteristics of rotors based both in its finite element model [10] as well as in direct measurements

in a test stand [9]. Besides that, the shaft bending influence will be neglected.

Figure 2.4 – The bearing model – source: [4]

The bearings will be modeled as represented in the Figure 2.4, the next step is to write the

virtual work of the forces acting in the springs and dampers constituting them, in order to relate the

components and of the force with the bearing’s characteristics.

11

The virtual work done by the forces acting on the bearings is given by:

(2.1.1.7)

The virtual work done by the forces acting on the springs and dampers of the bearing can be

written as:

(2.1.1.8)

Joining (2.1.1.7) and (2.1.1.8) together and writing the resultant expression in matricial form,

give us after elimination of the virtual displacements:

[

] [

] [

] [

] [

] (2.1.1.9)

4. The mass unbalance

The mass unbalance will be modeled as a punctual mass situated at a given distance

from the center of the shaft.

Figure 2.5 – Rotating mass with offset length – source: [4]

With the help of Figure 2.5 let us define first the displacement vector , in the inertial

reference frame :

|

| (2.1.1.10)

where is considered as constant.

Knowing the displacement vector the speed of the punctual mass can be computed:

|

| (2.1.1.11)

12

With (2.1.1.11), and observing that the punctual mass is much smaller than the disk’s

mass and thus can be neglected, the kinetic energy of the punctual mass yields:

( ) (2.1.1.12)

2.1.2 Rayleigh-Ritz Analytical Solution – Application to a Multirotor Model

Now that each rotor component has been presented along with its energy equations, the

Rayleigh-Ritz method will be applied to obtain the shape functions of the lateral vibration of rotors.

To achieve this, a birotor, i.e. a Multirotor configuration with two coaxial shafts, is employed in

order to obtain a general formulation of the dynamics behavior of Multirotors. In this model, only the

inner shaft is considered as flexible while the outer is considered as rigid.

The studied model is shown in the Figure 2.6.

Figure 2.6 – The Multirotor model

The Multirotor inner shaft is supported in two rigid supports, denominated A and D, while the

outer coaxial spool, is rigidly supported in B. The displacements in C are the same in the inner and

outer shafts. An external elastic support is also placed on this location.

Before applying the expressions obtained in the last section in the Lagrange equation, it is

necessary to write the displacements and of the rotor in terms of a shape function ( ), that

describes properly the lateral vibration behavior of the rotor. On this work the employed shape function

will be equal to the one of the first mode of vibration of a beam, and that is:

( )

(2.1.2.1)

In which is the rotor’s length.

For the first rotor, which is the inner one it results:

13

Figure 2.7 – Inertial reference frame coordinates – source: [4]

( ) ( ) (2.1.2.2)

( ) ( ) (2.1.2.3)

Where and denote the displacements in the and directions on the first rotor, and

are generalized independent coordinates, and ( ) denotes the shape equation for the first rotor,

and coincides with the definition in the equation (2.1.2.1).

If the angular displacements and of the first rotor, shown in Figure 2.7 are considered

small, the following relations are valid:

( )

( ) ( ) (2.1.2.4)

( )

( ) ( ) (2.1.2.5)

and: ( )

The needed second derivate of and to compute the strain energy, are given by:

( )

( ) ( ) (2.1.2.6)

( )

( ) ( ) (2.1.2.7)

and: ( ) (

)

The second rotor, the outer one, is assumed to be rigid, so the second rotor displacements (

and ) and angular displacements ( and ) are written in the following way:

( )

( ) ( ) (2.1.2.8)

14

( )

( ) ( ) (2.1.2.9)

( )

( ) ( ) (2.1.2.10)

( )

( ) ( ) (2.1.2.11)

It is observable from (2.1.2.9) and (2.1.2.10) that is a constant, since the second shaft is

rigid. For this reason there is no definable strain energy for the second rotor, in conformity with the

rigid rotor assumption:

(2.1.2.12)

Applying expressions (2.1.2.1-5) and (2.1.2.6-11) in the kinetic energy equations (2.1.1.4) and

(2.1.1.5), for each of the two shafts and two rotors, it results:

(

) (2.1.2.13)

with:

( )

( ) ( )

∫ ( )

∫ ( )

∫ ( )

∫ ( )

(2.1.2.14)

( ) ∫

( )

(2.1.2.15)

∫

( )

(2.1.2.16)

where is dubbed gyroscopic term, the index 1 and 2 concern rotor 1 and 2 respectively, and where

the constant terms of equations (2.1.1.4) and (2.1.1.5) were ignored since they will disappear anyway

after application of Lagrange equation.

The rotating speeds of the rotors are related by a constant ratio , which is integer, following

the relation:

(2.1.2.17)

15

In what concerns on this case the strain energy, the only contribution comes from the first

rotor. Recalling equation (2.1.1.6), substituting on it the expressions (2.1.2.5) and (2.1.2.6), and

considering the axial force null, it yields:

(

) (2.1.2.18)

where:

∫ ( )

(2.1.2.19)

Substituting expressions (2.1.2.1) and (2.1.2.2) in equation (2.1.1.9), and noting from the

Figure 2.6 that the bearing is located at a distance from the reference frame origin, the components

of the generalized force acting in the bearing become:

[

] ( ) [

]

⏟ [ ]

[

] ( ) [

]

⏟ [ ]

[

] (2.1.2.20)

Finally, the kinetic energy due to the presence of a punctual mass in a disk and at a given

distance from the axis of rotation of the Multirotor is written with the help of equation (2.1.1.12). If the

punctual mass is applied to the first disk in , its kinetic energy is given by:

( )( ) (2.1.2.21)

On the other hand, if it is applied to the second in , the equation becomes:

( )( ) (2.1.2.22)

If and are two generic transversal forces applied in a specific longitudinal coordinate of

the rotor, and acting along the and directions respectively, the virtual work done by these forces is

given by:

( ) ( ) (2.1.2.23)

Which after can be rewritten in the generalized coordinates in the following way;

( ) ( ) ( ) ( ) (2.1.2.24)

Eliminating the virtual displacements we obtain the generalized forces as follows:

( ) ( )

( ) ( ) (2.1.2.25)

16

So, for the most general case of two coaxial rotors with a viscously damped bearing, rigid disk,

symmetric shaft, with external transversal applied forces ( ) and ( ), Coriolis acceleration effect

and assuming no motion in the axial direction, the resulting motion equations, considering only the

degrees-of-freedom (Dofs) perpendicular to the axial axis are given by:

[

]⏟

[ ]

{

} ([

]

⏟ ( )

[ ( )

( ) ])

⏟ [ ]

{

} [

]

⏟ [ ]

{

}

{ ( )}

(2.1.2.26)

In the equation above, is the inertial matrix, and the gyroscopic effect on rotor 1 and 2

respectively, is the rotating speed, the inner-outer spool rotating speed ratio, [ ] include the

damping coefficients in the bearings, [ ] the total stiffness matrix (bearings plus strain stiffness), and

are the applied forces along the two defined generalized coordinates and .

Despite the previous deduction has been applied to a Multirotor configuration, it is easily

adapted to a Monorotor configuration if the outer rotor terms in (2.1.2.26) are considered null.

2.1.3 Rotordynamics Analysis – Application to the Multirotor Model

In this work, both undamped asymmetric and damped asymmetric bearings will be considered.

The computation of the natural frequencies of the rotors along with an expedite way of presenting their

variation with rotating speed will be first addressed. After that, the response of the considered rotors

due to mass unbalances is analyzed. The coupled terms in the stiffness and damping matrices will be

neglected on this section. The following explanation follows the procedure adopted in [4].

1. Campbell diagram

To obtain the Campbell diagram the vector in equation (2.1.2.26) is made equal to zero,

since the natural frequencies are obtained with the rotor free of loads (free vibration) and in harmonic

motion conditions, which can be expressed by:

(2.1.3.1)

(2.1.3.2)

with:

√

Where denotes the angular frequency of the free vibrations and denotes the modal

damping associated with each free vibration mode. Of course is null in undamped rotors. is the

imaginary unit defined as √ . denote the amplitude of the vibrations in the generalized

coordinates .

Introducing these equations in (2.1.2.26) it results the following eigenvalue/eigenvector modal

problem:

17

[ ] {

} {

} (2.1.3.3)

this gives the characteristic equation written just below:

( ) ( ( )

)

( ( ) ) ( )

(2.1.3.4)

The solution of the equation (2.1.3.4) leads to two pairs of complex conjugate roots,

√ and

√ , where and are the first and the second natural

frequencies of the rotor and are dependent of the rotating speed of the inner shaft ( ), and and

are the associated modal damping. This procedure is valid whether or not viscous damping is present.

These results yield to the equations that are used to obtain the Campbell Diagram. The Campbell

Diagram, exemplified on the Figure 2.8, is a graphic that show the variation of the natural

frequencies of the rotor with increasing rotating speed in turn of its axis, i.e. the frequency curves.

Alongside this representation, several lines can be traced from the graphic’s origin with an inclination

that establishes a relevant relation between the rotating speed and the natural frequencies. These

lines are generically defined by , where is the inner-outer spool rotating speed ratio and

the rotating speed in rotations per minute. The most common line traced on the Campbell Diagram

is the synchronous line, i.e. the line that includes the points at which the natural frequency equals the

rotating speed, and corresponds to the line defined by on Figure 2.8 The knowledge of the

intersection points of these lines with the frequency curves is very relevant for the prediction of critical

operation points as it will be shown later.

Figure 2.8 – Campbell Diagram example – source: [4]

18

2. Response to forces due to mass unbalances

Similarly to the description in the point 4 of the section 2.1.1, the effect of a punctual mass

‘ ’ at a given distance ‘ ’ from the center of the disk ‘ ’ in the force vector is:

{ ( )

( )} {

( ) ( )

( ) ( )

} (2.1.3.5)

in which ( ) is the value of the displacement function in the location of the disk.

Considering additionally that viscous damping effects are neglected ([ ] [ ]) the solutions of

the resulting equation are sought in the following form:

(2.1.3.6)

(2.1.3.7)

Substituting (2.1.3.4), (2.1.3.5) and (2.1.3.6) in (2.1.2.26), and considering that the unbalance

mass is situated in the disk 1 situated at , it results:

[ [

] ( [

( )

( ) ]) [

]] {

} {

( )

( ) }

(2.1.3.8)

Solving equation (2.1.3.8) for the variables and , it yields:

( ) ( ( ) )

( )( ) ( ) (2.1.3.9)

( ) ( ( ) )

( )( ) ( ) (2.1.3.10)

If the unbalance mass is situated in the disk 2 at , the speed in the solution’s equations

(2.1.3.6-7) must be the rotating speed of the respective shaft. Since disk 2 is located in the outer shaft

is the proper speed to use. The modal vector equations are then obtained by following the

procedure used before, and are shown just below:

( ) ( ( ) )

( )( ) ( ) (2.1.3.11)

( ) ( ( ) )

( )( ) ( ) (2.1.3.12)

If (asymmetric rotor), then one can conclude that the rotors describe elliptical

orbits; otherwise the orbits are circular. If the denominators are made equal to zero, we can obtain the

critical speeds by solving the resultant equation. There are two critical speeds when the rotor is

asymmetric and only one solution when the rotor is symmetric.

19

Since an unbalance on the inner disk is synchronous with the rotation of the inner shaft, these

critical speeds will correspond to the intersection points between the synchronous line and the

frequency curves on the Campbell Diagram if disk one is unbalanced. But if the unbalanced mass is

applied on disk 2, the imparted force can be asynchronous in relation to the inner shaft depending on

the relation between rotating speeds quantified by . So the critical speeds of such an unbalance will

correspond to the intersection points between the dependent asynchronous line and the frequency

curves on the Campbell Diagram.

If viscous damping is added to the model then the solutions can be sought in the following

form:

(2.1.3.13)

(2.1.3.14)

In its turn, using (2.1.3.13), (2.1.3.14) and (2.1.3.5) in the equation (2.1.2.26), the values of the

coefficients , , and are obtained by solving the resultant system.

The orbits will be elliptical in case of asymmetric stiffness in the bearings but the existence of

damping animates this ellipse with a precession motion. This means that the semi-major and semi-

minor axis of the ellipse will make an angle relative to the inertial reference frame axis.

Caution must be taken in the used value of the speed in the solution equations (2.1.3.13-14), if

the unbalance mass is placed on disk 2 then again is the rotating speed to use.

The Campbell Diagram can be used on the prediction of the critical speeds for this case

exactly in the same way as it was described previously, attending, of course, if the considered

excitation force is synchronous or asynchronous, i.e. affected by the factor .

Equations (2.1.3.9-14) can now be used to obtain the response diagram of the rotor with the

respective bearing conditions. The response diagram in Multirotor configuration is a plot of the

logarithm of the sum of the transversal vibration amplitude( ) against the rotating speed of the

inmost shaft, as exemplifies Figure 2.9.

Figure 2.9 – Response diagram example – source: [4]

20

2.1.4 Rotordynamics Modal Orbits and Sense of the Whirl

In the last section rotor vibration orbits were characterized for each Multirotor type.

Nevertheless, nothing was said about the sense of the orbit described by the rotor while it vibrates and

rotates simultaneously. In fact, as it will be addressed shortly, rotor orbits are not trivial phenomena. In

some rotors the transversal orbit described by the shaft vibration has an opposite sense to that of the

rotor’s rotating sense, it is then said that the rotor is in backward whirl. If the shaft vibration orbit has a

corotational sense with that of the rotor’s rotating speed, then the rotor motion is dubbed as forward

whirl.

On this section, it will be analytically shown, the existence of this phenomena as well as what

causes the rotor to be in backward whirl instead of forward whirl and vice-versa.

Let us first consider again the Multirotor represented in Figure 2.6. As previously discussed,

the displacements of the rotor in the and directions are given by and expressions such as

that displayed in the previous section. It was already mentioned in that section that the orbits could be

circular or elliptical depending if the rotor was symmetric or asymmetric.

Figure 2.10 (a-c) shows the various possible orbit shapes in Rotordynamics, summarizing also

the bearing characteristics behind each one of them.

Figure 2.10 – Possible orbits of a rotor’s shaft

Now let us show the way the sense of whirl is determined. For this let us consider the

displacement vector of the rotor rotating axis center relative to the inertial reference frame origin,

displayed in the Figure 2.11.

Shaft position

Shaft position

Shaft position

Shaft position

(c)

(b) (a)

21

Figure 2.11 – Generic elliptical orbit with displacement vector

It can therefore be written as:

[

] (2.1.4.1)

Consequently the displacement vector derivative yields, making use of the chain rule:

[

] (2.1.4.2)

The sense of whirl is deduced from the signal of the crossed product between the

displacement vector and its derivative:

[

] (2.1.4.3)

If the resultant vector in (2.1.4.3) is pointing to the positive semi axis of , then the rotor is

said to be in forward whirl (FW). On the contrary if it is pointing the other way around it is said to be in

backward whirl (BW).

To help the visualization of this phenomenon Figure 2.12 shows position of an unbalanced

mass relative to the inertial axis as the disk of the Monorotor there represented follows the path both

in forward and backward whirl. This depiction is only valid if the disk’s rotating speed is considered to

be equal to the orbit speed, i.e. synchronous. This assumption is consistent with what was predicted

on the last section. As we saw there, for a mass unbalance on the inner rotor’s disk of a Multirotor

configuration, whose vibration modes are given by the solutions (2.1.3.6-7), the orbit revolution speed

is synchronous with the disk’s rotating speed .

A Shaft position at 𝑦 𝑙

22

Figure 2.12 – Position of a mass unbalance (in bold) in a disk performing Forward and Backward Whirl

The expressions of and will of course depend of the bearing characteristics of the rotor.

Anyway, only for exemplificative purposes the determination of the whirl sense for a particular case of

the undamped Multirotor like the one shown in Figure 2.6 with asymmetric stiffness bearing properties,

and a mass unbalance on disk 1, is presented in the Appendix 1 to this work.

2.2 Force Reconstruction Fundamentals

In this chapter the theory behind various force reconstruction methods is presented, based on

the review explained on the chapter 1.2.2. Its limitations and special aspects are also subject to

analysis. The majority of the content of this chapter follows the paper of Allen and Carne [12].

As it was said, force reconstruction can be generically defined as the estimation of the

amplitude and frequency- or time-history of the loads acting on a structure using measurements of the

structure’s response resulting from the action of those loads.

Firstly an approach will be presented based on the frequency domain, i.e. the measurements

of the response are used to generate a load vector containing the amplitude of the estimated forces

and its frequency history. Additionally, a method of localizing the forces acting on a structure is also

presented.

Afterwards it is explained the theory background of the methods based on time-domain force-

reconstruction, i.e. the estimation of the force amplitude and time-history through real-time or almost

real-time structural response measurements. The SWAT method is perhaps the simplest time-domain

based force reconstruction method, so it is also the first time-domain method to be approached.

Before the ISF and DMISF methods are presented, a brief introduction is made to the state-space

fundamentals, since these two time-domain based methods are based on this mathematical

representation.

2.2.1 FD – Frequency Domain Force Reconstruction For a generic dynamic system the equations of motion are given by:

[ ]{ } [ ]{ } [ ]{ } { ( )} (2.2.1.1)

23

Where [ ] is the mass matrix of the system, [ ] is the damping matrix, [ ] the

stiffness matrix of the structure and the vector { } { } is the generalized coordinate

vector, being a positive integer representing the number of degrees-of-freedom (dofs) of the finite

element model. Vector { ( )} includes the loads exerted in the system’s dofs.

Equation (2.1.2.26) for instance, is a particular application of equation (2.2.1.1) to Multirotors.

The Mono- and Multirotor dynamic system’s matrices , and are computable through application

of the Rayleigh-Ritz method as shown in chapter 2.1.2, or by using the finite element model as it will

be shown later on the chapter 3.1.

For each frequency value, the response in the dof in steady-state conditions can be written

as:

(2.2.1.2)

Applying the first and second derivatives with respect to time, one obtains respectively:

(2.2.1.3)

(2.2.1.4)

After substituting (2.2.1.3) and (2.2.1.4) in equation (2.2.1.1) it follows that:

[ [ ] [ ] [ ] ]⏟ [ ( )]

{ } { ( )} (2.2.1.5)

where: { } { }

Observing equation (2.2.1.5) one should notice that the relation between the loads acting on

the system and the caused displacements is only dependent of the matrix [ ( )] . Due to its

determinant importance the [ ( )] matrix is known as the dynamic stiffness matrix.

If the dynamic stiffness matrix is computed the force reconstruction becomes possible in every

node simply by multiplying it by the dofs measured response as the following equation shows.

{ ( )} [ ( )]{ } (2.2.1.6)

Despite its simple aspect, equation (2.2.1.6) is hard to apply to complex systems. In the

majority of the applications there are a large number of dofs, or some measurement points are simply

inaccessible making the measurement of all the system’s dofs unpractical. Therefore a way of

determining the unknown system’s dofs response using the known ones is vital. The Transmissibility

concept in multi-dof systems is the tool that supplies that result. This means that measurements of the

response done in some few nodes are used to deduce the response in the dofs where the response is

unknown. A recent approach made by [15] allows not only the deduction of the responses in all the

24

structures dofs, but also the localization of the loads which are acting in the structure. This approach

will be briefly introduced in the end of this section.

Even knowing all the system’s dofs responses, it is not practical to reconstruct all the system’s

load vector. Instead it is more efficient to reconstruct the loads in the spots where we are sure they are

acting. So if the location of the applied loads is known, the problem can then be reduced to the

following equation:

{ ( )} [ ( )] { } (2.2.1.7)

where is the number of force inputs and the number of measured dofs. [ ( )] in this case is

a reduced dynamic stiffness matrix that relates only the involved dofs.

Equation (2.2.1.7) is solvable as long as the number of used responses is larger or equal to

the number of Forces acting on the structural system.

Force localization in the Frequency Domain

Some brief notes regarding the method of force localization explained in the recent work of

[15] will now be introduced.

The motivation for this procedure has been already mentioned before, and mainly it has to do

with the impossibility in many applications to determine the responses in all the system’s dofs and/or

because the point of application of the loads is unknown.

This methodology, begins by defining two response vectors { } and { } dividing among them

the responses extracted from a limited set of the structure’s dofs, each of them containing the

responses in the and coordinates respectively. A load vector { } containing the loads acting in

coordinates is also defined. The number of forces acting on the system will be dubbed as and the

number of extracted responses designated by . Considering additionally that the relation between

the responses and the forces is given by:

{ ( )} [ ( )]{ ( )} (2.2.1.8)

where: [ ] [ ] is the receptance matrix defined as the inverse of the dynamics stiffness

matrix.

Therefore, considering equation (2.2.1.8), { }, { } are related with { } by:

{ } [ ]{ }

{ } [ ]{ } (2.2.1.9)

Where the indices and refer to the established relation between the response

coordinates and , and the load’s coordinates respectively.

Solving equations (2.2.1.9) by relating each one of them through the load vector { }, it

results:

25

{ } [ ][ ] ⏟ [

]

{ } (2.2.1.10)

in which [ ] designate the pseudo-inverse matrix and where [ ] is the transmissibility matrix. This

matrix establishes a relation between the dofs responses under a specific load condition . The

proposed force localization method works by assuming random and different combinations for the

force point of application and to compute the resulting transmissibility matrix for each combination

since, as it was already mentioned, the matrix matrices are assumed to be known. After equation

(2.2.1.10) is applied in a user defined frequency range and a result is obtained for the response

vector { }. The symbol was added, to distinguish between the computed response vector from the

actually measured response vector { }.

For each combination of forces the accumulated error between vectors { } and { } is

computed component by component as the next expression states, and saved in an array :

∑ ( ( ( ( ))) ( ( ( ))))

{ } { }

(2.2.1.11)

The combination at which the error is minimized is expected to correspond to the real point of

application of the forces.

After this is accomplished the amplitude of the loads can be determined through equation

(2.2.1.7), keeping on mind that this is possible, as long as the number of used responses is larger

or equal to the number of forces acting on the structural system.

Transformation of the Load Vector from the Frequency Domain Method to the Time-Domain

If is intended a comparison between the force reconstruction obtained using the FD with the

results obtained by the time-domain methods presented below, an inverse FFT (Fast Fourier

Transform) can be applied to load vector estimated by the FD method in order to convert frequency-

domain results into time-domain results.

Figure 2.13 – Illustration of a decomposition of a time-domain spectrum – source: [7]

26

Figure 2.13 illustrates how a spectrum of a response signal can be obtained from the FRF

(Frequency Response Function) through the discrete sum of individual sinusoidal modal contributions.

This is the procedure followed by the inverse FFT. Likewise, if it is applied to the load vector in the

frequency domain obtained in (2.2.1.7), a time-domain representation of the loads can be achieved.

2.2.2 SWAT – Sum of Weighted Accelerations Technique

The SWAT method makes use of the concept of modal filter. Good revision of these topics can

be found in [18]. The whole method is based in the computation of a weighting matrix that after applied

to a structure’s response, isolate the rigid body accelerations of the structure from the influence of the

flexible modes response. The weighting matrix is the modal filter, since it is detaching the rigid body

accelerations from a response consisting of both rigid body and flexible modes. Following the

computation of the rigid body accelerations, the loads acting on the center of gravity of a structure can

be estimated from the multiplication between the rigid body accelerations and its mass properties.

The main limitation of this method is that it does not enable by itself the determination of the

spatial application point of the load; thus force identification is not possible without external

assumptions. Despite that fact, force reconstruction is possible provided that the applied loads are

less or equal to the number of rigid body modes. Another setback of this method is that, the number of

acceleration measurements must be at least as great as the number of rigid body plus the number of

flexible modes to be successful. Additionally in the experimental case, the sensors must be placed in a

way that enables a proper determination of the modes of the structure.

In this text, the approach followed by Carne et al. [26] to derive the SWAT method is

presented and implemented.

As previously mentioned the weighting matrix is applied to the measures to obtain the rigid

body accelerations. Following this statement it can established that:

{ } [ ] { } (2.2.2.1)

where the sought weighting matrix [ ] is used to extract the vector { } of rigid

body accelerations, being the number of rigid body modes, and the number of measurement

points.

The method is completed after multiplying the rigid body accelerations by the rigid body mass

properties to obtain the load vector { ( )} .

{ } { ( )} (2.2.2.2)

If this method is systematically applied to the measurements at each time instant, a time –

domain force reconstruction at the structure’s center of gravity is obtained.

Let us now introduce how the weighting matrix [ ] is computed.

Firstly, it begins by establishing a relationship between the measured accelerations and the

modal shapes using a sum of modal contributions representation.

27

{ } [ ]{ } (2.2.2.3)

where { } is a vector that contains the measured accelerations at the measurement points, { }

is the vector of modal displacements accelerations, [ ] is a matrix containing the mode

shapes, is the number of modes (both rigid body and flexible).It should be remarked that, according

to vibrations theory, any continuous system has an infinite number of modes, meaning that (2.2.2.3) is

just an approximation. Matrix [ ] can either be computed solving the Eigen problem if a finite element

model of the structure is available or through the algorithm of mode isolation (AMI) [27] if an

experimental mount is being analyzed.

Equation (2.2.2.3) can be rewritten in a more convenient way, if the rigid body mode vectors

are considered mass normalized, i.e. the modal vectors { } are such that make { } [ ]{ } [ ] and

{ } [ ]{ } [ ] , being [ ] the identity matrix and the eigenvalue, and if they are

assigned to the leading columns of [ ], it results:

{ } [[ ] [ ]] {{ }

{ }} (2.2.2.4)

where the and indices correspond to the rigid body and elastic modes, respectively and { }

represent the second derivative of the elastic modal displacements which will multiply by the [ ]

matrix columns. If (2.2.2.1) is introduced in (2.2.2.4), and taking into account once again that the

weighting matrix must isolate the rigid body modes, it yields:

[ ] [[ ] [ ]] [[ ] [ ]

]

[ ] ([[ ] [ ]] ) [

[ ]

[ ]

] (2.2.2.5)

It should be noted, that as long as the condition is verified, the columns of the

weighting matrix are able to extract the rigid body accelerations from the modal data.

After [ ] is determined equations (2.2.2.1-2) are ready to be computed, giving the load time-

history of the resultant in the structure’s center of gravity of the loads corresponding to the rigid body

motions introduced in (2.2.2.5) in the [ ] matrix.

To conclude this section it is worth mentioning that a SWAT variant was developed by Genaro

and Rade [20] that enhances this method allowing the possibility to determine multiple forces acting

simultaneously.

2.2.3 State-Space Fundamentals

Next, an introduction to the state-space fundamentals is done, as it is needed for the next two

identification methods.

State-Space system representation has applications in many fields of engineering. Particularly

in the analysis of dynamics systems, the state-space formulation is used to write in an alternative way

the multi-degree of freedom equations of motion.

In the most general case the continuous-time state-space system equations [12] are given by:

28

{ ( )} [ ]{ ( )} [ ]{ ( )}

{ ( )} [ ]{ ( )} [ ]{ ( )} (2.2.3.1)

Here, { } is the state vector of the system, { } contains the inputs acting on the system,

{ } is the vector that contains the outputs of the considered system, [ ] is the system’s

matrix, [ ] the input influence matrix, [ ] is the output influence matrix and [ ] is the

direct throughput matrix. The index denotes that the system is a continuous time state-space

system and { } is the first time derivative of vector . The first expression in (2.2.3.1) is called the

state-space equation and the second expression is dubbed the output equation.

In many applications, e.g. in experimental applications, a discrete time representation of the

state-space is preferred. The output of the discrete time system will exactly reproduce the one

obtained with the continuous time system, as long as the zero order hold (ZOH) condition is verified,

i.e. if the input is constant between samples [12]. The discrete time state-space equations are the

following ones [21]:

{ } [ ]{ } [ ]{ }

{ } [ ]{ } [ ]{ } (2.2.3.2)

The matrices and vectors of this system have an identical meaning to the ones on the

continuous time state-space system, and the index is the sample instant number. The conversion

from continuous to discrete time state-space imply a manipulation of the [ ] and [ ] matrices, while

[ ] and [ ] remain unaltered. The new matrices are displayed in the following equations. [12]

[ ] ( )

[ ] [ ] [ ( ) [ ]][ ]

[ ] [ ]

[ ] [ ] (2.2.3.3)

in equation (2.2.3.3) is the time between successive time samples. The time at which each sample

is obtained can be thus obtained simply by multiplying by the th time step at which time is intended

to be computed.

Let us now focus on the content of the vectors and matrices of the state-space systems in the

context of the dynamic of mechanical systems.

In the most general case of a degree of freedom dynamic system, with stiffness [ ], mass

[ ], damping [ ] and with a load { ( )} acting on it, the equations of motion are:

[ ] { ( ) }

[ ] { ( )} [ ] { ( )} [ ] { ( )}

(2.2.3.6)

29

where { } is the modal coordinates vector of the system and where { } and { } denote the second

and first time derivates of the modal coordinate vector. [ ] is simply a matrix of ones and zeros

that selects the subset at which forces are being applied, is the number of input loads.

One possible choice for the state-space vector is the following:

{ } {{ }

{ }} (2.2.3.7)

Let us now introduce the state vector { } on equation (2.2.3.6). It returns the following

representation:

[[ ] [ ]

[ ] [ ] ] { } [

[ ] [ ]

[ ] [ ] ] { } [

[ ]

[ ] ] [ ]{ }

(2.2.3.8)

If one works out this equation one obtains in the end the following expression:

{ } [[ ] [ ]

[ ][ ] [ ][ ] ]⏟

[ ]

{ } [[ ]

[ ] ] [ ]⏟

[ ]

{ }

(2.2.3.9)

Comparing equation (2.2.3.9) with the state-space equation in (2.2.3.1), matrices [ ] and [ ]

become determined by similarity.

The matrices of the output equation, i.e. the second expression in (2.2.3.2), depend on the

nature of the system as well as on the intended output variable. For structural systems such the ones

studied in this work, where the force locations are not always entirely known, measures are obtained

for degrees-of-freedom on which forces are not applied, the system is therefore called a non-

collocated system.

The state-space of non-collocated systems possesses non-minimum phase zeros, this

statement implies that the direct throughput matrix [ ] is dropped rank and thus the output equation

for this systems will only depend on the output influence matrix [21].

In control theory, a system is said to be minimum-phase if the system and its inverse are

causal and stable. This is not the case of the state-space system used in force identification methods,

since they may be associated to unstable inverse systems. This essentially happens because the

zeros of the non-collocated state-space systems have positive real parts, i.e. in the right-hand side of

the complex plane. This is a condition for a system to have an unstable inverse system according to

the control theory, since the eigenvalues of the inverse system are the zeros of its associated system.

Recall from vibration theory how positive real parts in the eigenvalues are closely related to unstable

phenomena (see Appendix 2).

The output influence matrix [ ], will work simply as an output selector, in which the selection

options are the state vector components. If the displacements on the degrees-of-freedom of structure

30

are considered as the output of the system, and recalling that the nodal displacements are included in

the first positions of the state vector in (2.2.3.7), it results for the output equation:

{ } [[ ] [ ] ]⏟ [ ]

{ } (2.2.3.10)

This is not whatsoever the only way to determine the matrices comprising the State-space

system. For instance, Allen and Carne [26] have derived the state-space system of the structures they

have tested from modal parameters. This has been made fitting to the experimentally measured

Frequency Response Function (FRF) [27] a state-space modal model that follows the standard

definition:

[ ( )] ∑ (

[ ]

[ ]

)

(2.2.3.11)

where ( ) denotes the complex conjugate, [ ( )] is the FRF matrix at frequency , is the modal

eigenvalue, i.e. √ , where is the modal damping ratio and the modal

undamped natural frequency for the th mode of vibration. [ ] is the residue matrix, defined as:

[ ] { } { }

(2.2.3.12)

where { } and { } are the modal vectors of the state-space of the response and drive (load)

locations. Considering that the loads are only applied to a subset of the response locations, it can be

expressed by:

{ } { }

{ } { }

[ ] (2.2.3.13)

The correspondent state-space system of this FRF [12] is:

[ ] [ ] , [ ] [ ] [ ] ,

[ ] [ ] (2.2.3.14)

where is a diagonal matrix containing the eigenvalues in ascending order and the columns of

contain the { } vectors in the same order as the eigenvalues. These matrices are then introduced in

the state-space equations (2.2.3.1). [ ] is zero since the system is a non-minimum phase one as was

justified before.

Now that the continuous state-space system has been completely characterized for a

structural system, and after presenting two ways of describing the system’s matrices, the topic of the

inverse state-space system dubbed the inverse structural filter will be addressed.

31

2.2.4 ISF – Inverse Structural Filter

The inverse structural filter (ISF) first deduced by Steltzner and Kammer [21], is a time domain

method which is based on the inversion of the state-space system depiction of the equations of

motion, in which the taken inputs are the structural response data and returned outputs are the

estimations of the input loads applied in a specific point of the structure. This is an advantage

compared to the SWAT method, which only computes the resultant forces and the center of gravity of

the structure.

Before inverting the system, an additional manipulation should be performed on the state-

space system. On experimental applications accelerations are more easily measured than

displacements. Therefore, the second derivative of the state and output equations will be computed in

order to obtain a state-space representation of the accelerations. They are expressed in the following

form:

{ } [ ]{ } [ ]{ },

{ } [ ]{ } [ ]([ ]{ } [ ]{ }), (2.2.4.1)

{ } [ ]{ } [ ]{ },

with:

[ ] [ ][ ] [ ] [ ][ ]

The discrete time representation is simply obtained for the acceleration representation after

the matrix transformations in (2.2.3.3) are applied. While [ ] and [ ] remain constant and equal to

[ ] and [ ] respectively.

{ } [ ]{ } [ ]{ },

{ } [ ]{ } [ ]{ } (2.2.4.2)

Steltzner and Kammer [21] have then manipulated the discrete time state-space equations

yielding to the interchange of the input with the output, obtaining the following inverse state-space

system:

{ } [ ]{ } [ ]{ }

{ } [ ]{ } [ ]{ } (2.2.4.3)

where,

(2.2.4.4)

are the inverse state-space system matrices.

32

It should be noted that what is obtained in equation (2.2.4.3) is the time derivative of the loads,

meaning that afterwards a numeric integration method should be applied in order to obtain the load

reconstruction.

As it has been told, non-collocated state-space systems are common practice in the field of

force reconstruction. It can be shown [21] that the non-minimum phase zeros that characterize such

state-space systems, have positive real parts, and become the eigenvalues of the inverse state-space

system. This generates an unstable inverse system. Due to this fact, the formulation displayed in

equation (2.2.4.3) has not been used, and alternative ways of representation of the structure were

implemented.

Steltzner and Kammer, [21], used expression (2.2.4.5) for the pulse response of the state-

space system assuming zero initial conditions for the structural acceleration, which is presented

below:

∑ (2.2.4.5)

where the Markov parameters proposed by Steltzner and Kammer are [21]:

Equation (2.2.4.5) is derived stepping forward the expressions in (2.2.4.2), i.e. substituting

successively the first expression in (2.2.4.2), into the second expression in (2.2.4.2).

Step 1: { } [ ]([ ]{ } [ ]{ }) [ ]{ }

Step 2: { } [ ][ ]( [ ]{ } [ ]{ } ) [ ][ ]{ } [ ]{ }

In the limit : { } [ ][ ] [ ]{ } [ ][ ][ ]{ } [ ][ ]{ } [ ]{ }

(2.2.4.5)

Though this expression involves a limitless number of Markov parameters, it is known [21] that

the system becomes fully described using only Markov parameters, being the number of

observable and controllable modes of the structural system.

Likewise, the same representation can be adopted for the inverse system displayed in

(2.2.4.3), again based on the assumption of zero initial conditions and following the same procedure

described for the demonstration of expression (2.2.4.5), but this time applied to the expressions in

(2.2.4.3). Following this procedure it results for the inverse system:

∑ (2.2.4.6)

The inverse system Markov parameters proposed by Steltzner and Kammer [21] are

presented below:

(2.2.4.7)

33

This representation does not always result in a converging sum, for the same reasons

explained previously involving non-collocated systems.

In the literature, various authors as in [21] as well as in [12] have opted to neglect the direct

throughput matrix [ ] of the discrete state-space system and to step forward the time of the system

times in order to improve the ISF performance. The direct throughput matrix result from the

multiplication between [ ] and [ ], which in case modal parameters are used to the construction of

the state-space, is highly dependent on the real part of [ ][ ][ ] . This term in its turn could be very

small when applied to lightly damped structures, also meaning lack of accuracy in its determination.

Applying these changes to equations (2.2.4.2) it results in the following equations:

Stepping 1 time instant:

{ } [ ]{ } [ ]{ }

{ } [ ]{ } [ ]{ }⏟ ( )

{ } [ ]([ ]{ } [ ]{ }) [ ][ ]⏟ [ ]

{ } [ ][ ]⏟ [ ]

{ }

Stepping 2 time instants:

{ } [ ]{ } [ ]{ }

{ } [ ][ ]{ } [ ][ ]{ }

{ } [ ][ ]([ ]{ } [ ]{ }) [ ][ ]⏟ ( )

{ }

{ } [ ][ ] ⏟ [ ]

{ } [ ][ ][ ]⏟ [ ]

{ }

Stepping the time forward time instants it yields:

{ } [ ][ ] ⏟ [ ]

{ } [ ][ ] [ ]⏟ [ ]

{ } (2.2.4.8)

The inversion of this state-space system follows exactly the same standard procedure used in

(2.2.4.3) and (2.2.4.4), with the slightly difference that instead of [ ]or [ ] the inverse system matrices

should be computed with [ ] and [ ] since the state-space system is now defined in a different way.

Using this inverse system (2.2.4.6) becomes:

∑ (2.2.4.9)

with:

This equation uses the responses measured at instants to estimate the loads at a

previous instant , i.e. the inverse system associated with (2.2.4.9) is non-causal. This solution has

made results obtained with ISF stable in structures where they otherwise were divergent.

34

2.2.5 DMISF – Delayed Multi-step Structural Filter

Based on the work of Steltzner and Kammer, Allen and Carne [12] developed the Delayed

Multi-step Inverse Structural Filter (DMISF). This method improves the performance of the common

ISF since it takes into account the stabilizing effect of the non-causal ISF, i.e. an ISF that uses future

values of the response to estimate the forces at a given time. This implies that a delay is involved

between the response data acquisition and the load reconstruction.

Recalling the stepping forward procedure displayed in the expression (2.2.4.8), Allen and

Carne reformulate the output equation of the state-space system by stacking this equations for a given

number of time steps and rewriting (2.2.4.8) for a step of 1 time instant as:

{ } [ ]{ } [ ]{ }

{ } [ ]([ ]{ } [ ]{ }) (2.2.5.1)

{ } [ ][ ]⏟ [ ]

{ } [ ][ ]⏟ [ ]

{ }

For the next time steps the output equation yields:

{ } [ ]{ } [ ]{ }

{ } [ ]{ } [ ]{ } (2.2.5.2)

{ } [ ][ ]{ } [ ][ ]{ } [ ]{ }

{ } [ ][ ] { } [ ][ ] [ ]{ } [ ][ ]{ } [ ]{ })

Stacking the time output equation for each time step, it results in the following modified output

equation.

{

{ }

{ }

{ }}

⏟ {

}

[

[ ]

[ ][ ]

[ ][ ]

]

⏟ [ ]

{ }

[

[ ]

[ ][ ]

[ ][ ] [ ]

[ ][ ]

[ ][ ] [ ]

[ ][ ]

[ ][ ]]

⏟ [ ]

{

{ }

{ }

{ }}

⏟ {

}

(2.2.5.3)

The state equation must be also slightly changed, in order to consider the new definition of the

input load stacking vector { }. It is called so because it literally stacks each time instant of the load

considered in the DMISF force reconstruction. The improved state-space system obtained using the

DMISF becomes:

{ } [ ]{ } [ ][ ]{ }

[ ] [[ ] [ ] [ ]] (2.2.5.4)

35

{ } [ ]{ } [ ]{

}

In equation (2.2.5.4) [ ] is a matrix that isolates the input sub vector { } out of the input

vector { }, in conformity with the definition of the state equation in (2.2.5.1) and (2.2.5.2).

The inverse structural filter procedure in (2.2.4.3) and (2.2.4.4) is then applied to the state-

space system in (2.2.5.4) by:

{ } [ ]{ } [ ]{ }

{ } [ ]{ } [ ]{

} .. (2.2.5.5)

where,

(2.2.5.6)

This inverse system estimates the time derivate of the applied loads at time instants between

to from the responses between to . If response data is recorded until a given time

instant , the above displayed inverse state-space system allows multiple load estimates.

DMISF has shown in [12] and [26] to be more stable and slightly more accurate than ISF,

SWAT and FD methods.

36

3 NUMERICAL METHODS

3.1 Numerical Aspects of the Rotordynamics Finite Elements

On this chapter, it is described how the finite element method implemented in the author’s

code were applied to each rotor part, becoming therefore possible to build and analyze rotor system

models such as Monorotors and Multirotors. This chapter follows the explanation made by Lalanne

and Ferraris [4].

3.1.1 The displacement vector

First of all the definition of the displacement vector must be done in order to apply the

Lagrange equation (2.1.1.1) to the rotor part energy equation defined in the chapter 2.1.1.

To define the displacement vector, four degrees of freedom will be considered per rotor node.

These dofs were already shown in Figure 2.7, and they are two translational displacements and

along the and axis of the inertial reference frame respectively, and two slopes about these two axis

designated by and respectively. The slopes are related to the displacements through the

equations:

(3.1.1.1)

Following these statements the displacement vector is:

[ ] (3.1.1.2)

The axial and the torsional dofs will not be considered.

3.1.2 The finite elements of the rotor parts

Let us now define each finite element composing the different parts of the rotor model: the

disk, the shaft and the bearings. A perturbation caused by the mass unbalance on the disk will also be

included in the finite element model.

The disk

The disk element will be considered as having only one node, representing therefore a rigid

disk. Thus the element matrices of the disk are obtained simply by applying the Lagrange equation

(2.1.1.1) to the equation (2.1.1.4) which includes the total energy of the disk of the rotor. If additionally

the displacement vector defined in (3.1.1.2) is introduced in (2.1.1.1) in spite of the generalized

coordinate vector it results:

(

)

[

] [

] [

] [

]

(3.1.2.1)

37

Therefore one can conclude that the disk element contributes with a mass matrix (the first in

equation (3.1.2.1)) and with a gyroscopic matrix which is the one that depend on the rotating speed .

This element corresponds to the MASS21 element in ANSYS®.

The shaft

The shaft element of length will be modeled as a two node beam element with constant

circular cross-section, and is illustrated in the Figure 3.1. The Euler-Bernoulli beam theory will be the

considered for the elastic shaft since the modeled shaft is long and slender. Other theories, like the

Timoshenko beam theory is proper for shorter beams and larger cross-sections, and includes the

influence cause by shear loads on the shaft. This affects its total stiffness. Since this effect could be

important in rotor shafts, a way to include a shear correction is presented.

Figure 3.1 – Shaft finite element – source: [4]

Both nodes are characterized by the displacement vector defined in equation (3.1.1.2). The

nodal displacement vector for the shaft element can be written as:

[ ] (3.1.2.2)

Now in order to build the shaft finite element the displacements must be related to shape

functions ( ), as the next equation states:

( )

( ) (3.1.2.3)

where and include the displacements contained in (3.1.2.2) in the and directions

respectively. Concretizing this last statement it results:

[ ]

[ ] (3.1.2.4)

38

The shape functions in (3.1.2.3) ( ) and ( ) will be those typically used in a beam in

bending, and they are:

( ) [

]

( ) [

]

(3.1.2.5)

Introducing the equations in (3.1.2.3) and its derivatives in the total kinetic energy expression

(2.1.1.5), computing the integrations and introducing the resulting expression on the Lagrange

equation, it results:

(

)

( ) (3.1.2.6)

where , and are:

[

]

(3.1.2.7)

[

]

(3.1.2.8)

39

[

]

(3.1.2.9)

Likewise, introducing again the equations in (3.1.2.3) in the strain energy expression in

(2.1.1.6), performing all the integral operations involved, and applying the Lagrange equation, it yields:

( ) (3.1.2.10)

with and equal to:

( )

[

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ]

(3.1.2.11)

[

]

(3.1.2.12)

where is the correction factor that takes into account the shear effect in the shafts stiffness, and it is

equal to:

40

(3.1.2.13)

with shear modulus defined as:

( ) (3.1.2.14)

where is the Poisson’s ratio of the shaft’s material, and is the reduced area of its cross-section

approximately equal to the cross-section area . The matrix represents the classical stiffness matrix

of the shaft while the matrix represents the stiffness due to application of an axial force on the

shaft.

Though based in the Timoshenko theory the ANSYS® element BEAM188 (2 node/6 dofs), is

the assigned element to model the shaft. The determinant reason to choose this element was the fact

that this is the only two node beam element available in ANSYS® that includes the Coriolis effect in its

computations. Besides that, it is recommended for slender to moderately slender beams, such the

ones considered to model the shafts of the rotors in this work.

The Bearings

The bearings in the finite element system are subjected to the same assumptions referred on

the chapter 2.1.1. Adapting equation (2.1.1.9) to the displacement vector considered in the finite

element model in (3.1.1.2), it results for the forces acting on the bearings:

[

] [

] [

] [

] [

] (3.1.2.15)

In ANSYS® this was modeled using the spring/damper element COMBIN14

Mass unbalance

The kinetic energy related to a mass placed with a specific offset in relation to the rotation axis

is shown in equation (2.1.1.12). Like has been done before if this kinetic energy expression is

introduced in the Lagrange equation the finite element model of this mass gives:

(

)

[

] (3.1.2.16)

As it is shown in equation 3.1.2.16 the mass unbalance effect acts along the two translational

directions stated on the displacement vector in equation 3.1.1.2. The way the mass unbalance is

implemented in ANSYS® will be subject of discussion on the chapter 3.3.

3.1.3 Solution of the Eigenvalue/Eigenvector Problem

In chapter 2.1.3 the most general case of the eigenvalue/eigenvector problem in

Rotordynamics was enunciated, and it was briefly mentioned how in general terms it should be solved

41

to give the natural frequencies and the modal vectors. Now in this section the exact procedure of the

solution is made clear, so that the numerical implementation become fully comprehended.

Two methods of solving this problem will be very shortly presented: The pseudo-modal

method, quicker and more efficient, and the direct method, more precise but more demanding of

computational effort.

The general equation used to compute the natural frequencies and the modal vectors is:

[ ] [ ] [ ] (3.1.3.1)

where is the modal displacement and [ ], [ ] and [ ] are the mass, damping and stiffness matrices

respectively.

The pseudo-modal method

The pseudo-modal method starts with the solution of the eigenvectors of the following reduced

system:

[ ] [ ] (3.1.3.2)

where [ ] is the stiffness matrix without the bearing stiffness terms.

The eigenvectors { } resulting from this system are then used to state the following equation:

{ } (3.1.3.3)

where contains the displacements in the principal coordinates.

Introducing (3.1.3.3) in the equation (3.1.3.1) it gives:

{ } [ ]{ } { } [ ]{ } { } [ ]{ }

(3.1.3.4)

In rotors, as it was seen, the damping matrix depends on the rotating speed , therefore

equation (3.1.3.4) should be solved for specific values of .

The solutions are sought as before:

(3.1.3.5)

Introducing equation (3.1.3.5) in (3.1.3.4) it yields:

[ ] (3.1.3.6)

where , and are given respectively by:

(3.1.3.7)

42

Manipulating equation (3.1.3.6) it results:

[

]

⏟

[

]

[

] (3.1.3.8)

Solving the Eigenproblem of matrix in (3.1.3.8), the modal vectors are extracted from the

semi-vector and the resulting eigenvalues are included in in the form already defined in the

beginning of the section 2.1.3.

The direct method

Direct method corresponds to the solution of the equation (3.1.3.1) without intermediate steps.

There are several numeric functions capable of that. In Matlab®, for instance, this is possible using

function polyeig that solves the polynomial eigenvalue/eigenvector problem of second degree stated

below:

[ ]{ } { } (3.1.3.9)

if

where correspond to the system’s eigenvectors and to the system’s eigenvalues.

3.2 Numerical Aspects of the Force Reconstruction Methods

On this chapter the particular aspects regarding the numerical implementation of the force

reconstruction methods described in the chapter 2.2 are presented. The methods were entirely

programmed in Matlab® by the author of this work, and the source data of force reconstruction in the

time domain – structural response measurements – was obtained through an ANSYS® finite element

model. Since this particular topic is transversal to SWAT, IDF and DMISF methods, this chapter will

begin with a description of the extraction and construction procedure of the response vector. In the

frequency domain the responses in permanent state were computed in the Matlab® code.

After this implementation, the force reconstruction methods itself follow exactly the procedure

explained for each one in chapter 2.2. Nevertheless, some important remarks about some numerical

operations are addressed, including some numerical integration methods used in methods like ISF

and DMISF in order to obtain the final solution. Before this chapter ends a brief description of the used

numerical integration methods is made.

3.2.1 Structural Response Extraction

Before the time domain force identification methods are applied, it is necessary to measure

the response data. The response data was extracted from a full transient analysis made on ANSYS®

to each of the studied finite element models.

The full transient analysis uses the Newmark method described on [28]. Of course a constant

time step should be chosen to drive the data acquisition of the variables from the analysis. This time

discretization is especially important in ISF and DMISF methods where discrete state-space systems

43

are employed as described in 2.2.4.-2.2.5. After the full transient analysis is performed, the

accelerations at the desire dofs are extracted using the ANSYS® time history postprocessor.

These measurements are then loaded in Matlab® where the response vector is built. The way

the measurements are organized inside the vector depend on the sequence of the dofs of the

numerical method used to build the inverse system of the structure.

After this is accomplished, the Matlab® code proceeds with the specific force identification

algorithms for each of the implemented time-domain force identification methods.

3.2.2 Frequency Domain Method

After the determination of the finite element model of the structure under study, the responses

in the permanent state resultant from generic harmonic force acting on it are calculated applying

equation (2.2.1.8) in the following way:

{ ( )} [ [ ] [ ] [ ] ] ⏟ [ ( )]

{ ( )} (3.2.2.1)

This is done in an arbitrary range of frequencies chosen by the user.

The next step is to choose what dofs responses should be included either in vector { } and

{ }, this can be also arbitrated by the user.

The code is programmed to scroll all the possible combinations of application points of the

force, and to successively apply equations (2.2.1.10-11). In each combination step the program gets

the proper values for the matrices [ ] and [ ] from the system’s [ ( )] matrix that was derived

using the finite element model in the chosen frequency interval.

By definition the code will get the nodes where the lowest error was registered and considers

that the loads are acting in the resulting nodes. Finally the amplitude of the loads is then computed

with equation (2.2.1.7).

3.2.3 SWAT

In this method the first step consists of determining the weighting matrix as described in

equation 2.2.2.4. For that the elastic modal vectors of the considered finite element model of the

structure are first determined through the solution of the respectively associated

eigenvalue/eigenvector problem using the direct method. This second finite element model must be as

close as possible to the ANSYS® finite element model where responses are extracted, i.e. with similar

element and equal discretization of the mesh, and it is built in a separate code. Then the rigid body

modes vectors are computed in conformity with the structural system under study. The user can

choose what rigid body vectors to include, since the SWAT method, as described in section 2.2.2

estimates the force in the direction of the correspondent rigid body motions.

To show this let us consider, for instance, a beam finite element. This beam element is

characterized by three dofs per node, so it has also three rigid body modes. Two of them are

translational (longitudinal translation is the first nodal dof and vertical translation the second), and a

third one is rotational (rotation about an axis perpendicular to the beam’s lateral section, which is the

also the third nodal dof). Figure 6.8 in Appendix 5 shows the mentioned beam finite element. In a

44

translational rigid body mode vector, the vertical for instance, every single dof in that direction moves

and the others have a null modal displacement. Generalizing for the other two rigid body modes, one

can conclude that for the beam element the rigid body vectors are:

[ ] [ ]

[ ] [ ] (3.2.3.1)

[ ] [ ]

Thus, one can conclude that in the developed code, the rigid body vectors are built by making

all the dofs in the direction in which the rigid body motion is being considered equal to one, while the

other vector positions are considered null.

As described, the elastic mode vectors are obtained solving the eigenvector problem, which in

a generic dynamic system is:

[ ]{ } { } (3.2.3.2)

with:

√

Here , and are the mass, damping and stiffness matrices of the studied system. And

the eigenvalues of the structure. In the case these { } are complex, only the real part is considered

as described in [12]. The number of flexible modes to include in the weighting matrix computation is

chosen by the user.

Now the weighting matrix is computable and the programmed Matlab® follows the sequence

already explained in chapter 2.2.2. As is visible in equation 2.2.2.1, the weighting matrix [ ] is applied

to the vector { } which includes the instant accelerations in every dof, in order to compute the

rigid body accelerations.

The extracted responses for a specific time instant will be placed in the vector { } attending

the sequence in the modal vectors of the structure.

Then, equation 2.2.2.5 is applied to determine the load vector in the considered instant, where

the inertial characteristics are immediately determined, since the geometry and the density of the

material is assumed to be known.

Finally a cycle is applied so that equations 2.2.2.2 and 2.2.2.5 are successively applied in the

considered time domain.

3.2.4 ISF

The ISF method begins with the definition of the state-space system matrices. As we saw

back on chapter 2.2.3 there’s actually two ways of doing so: through the system’s matrices, or through

modal parameters.

Only the sate space system based on the system matrices was numerically implemented. The

programmed code follows the explanation shown in chapter 2.2.3. The system matrices were built in a

45

separate finite element program, as was previously said for the SWAT method this finite element

model is similar to the finite element model from where the structure’s responses were extracted. The

user must also introduce matrix [ ], since it will indicate the location of the degree-of-freedom where

the forces are to be reconstructed, by introducing 1 in the array position(s) corresponding to the force

location and 0 in the remaining array positions.

The next step consists on defining the state-space system itself. This was done using the

Simulink® tools present in Matlab®. This tools are very important, since they include a function ‘s2d’

that automatically discretizes the state-space system giving its altered matrices. All the user needs to

introduce in the programmed is the constant time step used on the discretization. Note that the used

time step must be equal to the acquisition interval used in the transient analysis made on ANSYS®

from where the responses were extracted. So far the described procedure is shared between the ISF

and the DMISF techniques.

Before the inverse system is programmed, the non-causal lead must be introduced by the

user along with the number of Markov parameters he wants to use.

Finally the inverse system is ready for computation. The inverse system matrices must first be

built according to the equations (2.2.4.4). After this, the Markov parameters of the inverse can be

computed according to (2.2.4.7), at last equation (2.2.4.10) is cyclically applied to each time step

between and , being the total number of sub-steps of the extracted response.

Since only the first time derivative is obtained from equation (2.2.4.10), a numeric integration

method must be applied in order to obtain the force reconstruction. This aspect is common both to ISF

and DMISF methods.

The Newton-Cotes Simpson rule is the used numeric integration method, applied in the

context of the ISF method, and noticing that the time step is constant, it results:

[ ] (3.2.4.1)

where represents the time step number. The integration is applied between time step until time

step .

In the case a force impulse rather than a harmonic force was estimated, the Simpson rule has

proven to lead to wrong estimations. Therefore, for the examples where the force impulse is being

estimated, the trapezoidal integration method is used, following the equation (3.2.4.2) displayed just

below.

[ ] (

) (3.2.4.2)

3.2.5 DMISF

The first steps of the DMISF are common to the ISF method as it was mentioned above. After

the discrete space-state system matrices are defined, the user must introduce the delay he intends

to use to run the force reconstruction. For the chosen delay the code is now ready to build the

matrices [ ] and [ ] in equation (2.2.5.3), using the computed discrete space-state matrices.

46

The next step consists of obtaining the inverse space-state system following the procedure

and matricidal transformations described in equations (2.2.5.5-6). Vector { } is then obtained via

equation (2.2.5.5), which contains reconstructions of the load first time derivative. Only the last line

of this vector is considered for integration, i.e. the positions corresponding to { }, since they

correspond to the last and thus more perfected, force estimation.

Therefore the following integration necessary for the force reconstruction gives, according to

Simpson:

[ ]

(3.2.5.1)

where represent once more the time step. The integration is applied between time step until

time step .

Again for the estimation of force impulses the trapezoidal integration method was employed as

follows:

[ ] (

) (3.2.5.2)

3.3 Modeling Forces in Rotordynamics using ANSYS®

In this work only the vibrations caused by the presence of an unbalance mass in the disk of

the rotor are analyzed.

As demonstrated through equation 2.1.3.5 the force associated with a mass unbalance can be

described by two force components along the rotor’s generalized coordinates and dubbed

and respectively. This is illustrated in Figure 3.2.

Figure 3.2 – Force components due to the presence of a mass unbalance in the disk of the rotor.

In ANSYS® this was modeled by applying to punctual forces along the two directions

perpendicular to the rotating axis. The way these two components are defined depends on the

performed analysis.

47

Knowing that the forces related with mass unbalances of the disk are synchronous, one just

needs to specify the force amplitude and force component phase angle when performing a harmonic

analysis in ANSYS®.

When a transient analysis is performed there is no way to define a synchronous force

automatically. So it is necessary to compute the force components in each time instant and save them

in specific arrays in coherence with the force equations 2.1.3.5 associated with the mass unbalance.

The program will go through the saved arrays and apply the correspondent loads in the proper time

instants.

48

4 APPLICATIONS AND DISCUSSION

4.1 Rotordynamics Analysis

Typical Rotordynamics results presented in the chapter 2.1.3, namely the Campbell Diagram

and the Response Diagram for the case of an excitation caused by unbalances in the disks of the

rotor, are here presented to illustrate the reliability of the implemented codes.

The Problem

The problem, at which one will apply this kind of analysis, will be a Multirotor exactly such as

that represented in Figure 2.6, and it is again represented now in Figure 4.1.

Figure 4.1 – The Multirotor model

Making reference to Figure 4.1, note that the Multirotor consists of two coaxial shafts, the inner

shaft is simple supported in two rigid bearings in points A and D, the two shafts are connected in one

of the outer shaft’s ends corresponding to point C, and the other end of the outer shaft, which is

symbolized by point B, is considered to be simple supported. Each shaft drives one disk. The disks in

their turn are perfectly located and labeled in Figure 4.1 under the designations Disk 1 and Disk 2.

Additionally an elastic bearing is considered to be supporting the outer shaft in the coordinate

corresponding to point C.

Solving Methods

The used solution methods to obtain the Campbell and Response Diagrams are the Rayleigh-

Ritz and the finite element method, using for this last method a model made in ANSYS®. The

application of the Raleigh-Ritz method has been already described in the chapter 2.1.2. The finite

element model is presented in the next section along with the used elements. The Campbell Diagram

is obtained in ANSYS® through a pseudo-modal analysis (QRDAMP), such as that explained in

chapter 3.1.3, applied successively to an array of various rotating speeds. The Response Diagram is

obtained through a full harmonic analysis, based in the direct method (recall the meaning of the direct

method in chapter 3.1.3) modal solution of the finite element model.

49

4.1.1 The Finite Element Model

The modeling in ANSYS® of the considered problem is based on the examples of the

Rotordynamics chapter of the ANSYS® Manual. Here the shafts are modeled using a set of equally

spaced axial nodes, to which are attributed beam elements (BEAM 188 with quadratic polynomials)

with the section properties of each respective shaft. In this model the inner shaft is divided in 15 equal

parts (16 nodes) and the outer is divided in 4 equal parts (5 nodes).

The disks are modeled as punctual masses (MASS21), with the respective inertial properties,

in the nodes corresponding to their location and to their shaft.

The bearings are modeled using two spring-damper elements (COMBIN14), and are

connected to the node in the bearing location corresponding to the outer shaft.

The outer and the inner shafts are considered connected in the right end node of the outer

one. Here a constrain linking the displacements in the and directions between the inner and the

outer shafts nodes in this location is applied.

The inner shaft is simple supported in both ends so the displacements in the corresponding

nodes are constrained. The outer one is simple supported only in the left end node, meaning therefore

that the displacements in the corresponding node are subjected to the same constrains.

Additionally, the displacements and rotations about the axis are constrained in all nodes.

The only exception is in the case where an axial moment is applied in the inner shaft. Here the

rotations about are only constrained in the nodes of the outer shaft and in the right end node of the

inner shaft.

4.1.2 Validation of the Finite Element Model

To validate the previous presented finite element model were developed two additional finite

element models, one with a coarser mesh and another with a more refined mesh. After this, a

Campbell diagram drawing was performed and the critical speeds calculated for each of the three

meshes (these two and the one presented in the last section). The Campbell diagram construction

was again performed in ANSYS® making use of the pseudo-modal method QRDAMP. For each

analysis the used rotating speed ratio parameter was and no elastic bearing was considered,

i.e.:

The net change of the values of the critical speeds will work as convergence criteria in this

validation.

The elements that were used in these two additional meshes were the same used for each

rotor component as it was stated on chapter 4.1.1. In the coarser mesh was employed the half of the

elements used in the previous described mesh. In the most refined mesh was employed the double of

the elements per shaft used in the mesh described on chapter 4.1.1.

Figure 4.2 along with table 4.1 show the Campbell diagrams and the values of the inner and

outer shaft synchronous critical speeds for the three studied meshes.

50

Coarse (7/2) Used (15/4) Refined (30/8)

Figure 4.2 – Campbell Diagram convergence – the brackets above represent: (number of elements in the inner

shaft/number of elements in the outer shaft)

Table 4.1 – Critical speeds of each mesh and the correspondent errors computed using the FE solution as reference

Inner synch Critical Speeds (rpm) Coarse Used Refined

FW BW FW BW FW BW

Inner Synch 12339 14410 8710.453 11406.598 8710.451 11406.595

Outer Synch 2611 2693 1908.433 2011.715 1908.433 2011.714

Errors (%) Error coarse/used Error used/refined

FW BW FW BW

Inner Synch 41.7 26.3 2.296e-5 2.630e-5

Outer Synch 36.8 33.9 0.000 4.97e-5

From the mentioned Figure 4.2 and table 4.1 it is visible that the error between the used finite

element model and the most refined mesh is irrelevant (in the order of %). So the quality of the

used discretization is confirmed for the purpose of the natural frequencies calculation.

4.1.3 Numerical Data

The numerical data of the model is presented below referring to the terminology used on the

Figure 4.1.1:

The material constants are:

The mass unbalance data is:

The bearing properties and the shafts’ rotating speeds will differ from each particular solution.

51

4.1.4 Results and Results Discussion

It will be now presented the Campbell and Response Diagrams for various conditions of

elastic bearing properties, like stiffness asymmetry and damping. The influence of multiple inner and

outer rotating speed ratios, which relation is governed by the parameter that was first presented in

equation (2.1.2.17) will be also a focus of the following section. The effects caused by a torsional

moment placed in the inner shaft will also be considered late on this section.

The Campbell Diagrams will be drawn taking into consideration a step of between

each computation of the natural frequencies. In the Response Diagrams it is used a step of

between each response measurement. The term rotating speed in the diagrams refers to the rotating

speed of the inmost shaft, i.e. the value of . The Campbell Diagrams include the representation of

the inner shaft synchronous ( ) and the outer shaft synchronous ( ) lines.

The errors between the Rayleigh-Ritz and finite element methods presented in this section are

always computed relatively to the finite element numeric value.

Symmetric system

For the symmetric system the elastic bearing properties are:

Example 1

For :

Figure 4.3 – Campbell diagram – Symmetric rotor Example 1

Table 4.2 – Critical speeds associated with the previous Campbell diagram – RR/FE error

Critical Speeds Inner shaft synchronous

(rpm)

Outer shaft synchronous

(rpm)

FW BW FW BW

Rayleigh-Ritz 8743 12008 1940 2064

Finite Element 8710 11407 1908 2012

Error (%) 0.38 5.27 1.68 2.58

52

There are some remarkable aspects in Figure 4.3, namely:

1. The obtained curves using the finite element method have lower natural

frequencies comparing with the computed solutions using the Rayleigh-Ritz method. This result

has already been documented in the literature, particularly in [4]. Being the shape equation used in the

Rayleigh-Ritz method the first mode exact solution of a beam in bending, and the finite element

method an approximation method, it was expectable that the opposite was verified, since the exact

solution’s shape function would adapt better to the first mode shape. To explain this disparity let us

analyze the first mode shape. The rotor’s shaft shape in bending is deformed by the presence of the

disk, since it concentrates the majority of the rotor’s weight. The interaction with the bearings would

also play a role if the tested rotor had one. The modal shape obtained with the finite element method

therefore presents lower natural frequencies because the used mesh allows a better agreement

between the deformation computed numerically and the deformation that is actually being observed in

the shaft. To prove this way of reasoning, a detailed study of the influence of the disk’s weight and

position is present in the Appendix 3 to this work.

2. The BW natural frequencies grow and the FW natural frequencies decrease with

increasing rotating speed.. The gyroscopic effect is rather complex in a Multirotor configuration. In

this particular case , this means that the outer shaft is rotating 5 times faster and in the

opposite direction relative to the inner shaft. Consequently the skew-symmetric part of the damping

matrix governed by the term is negative. This means therefore that the gyroscopic effect will

act in the opposite sense it used to if the Multirotor was constituted solely by the inner shaft, reducing

the stiffness whenever it is in FW and increasing it when the Multirotor is in BW.

Response diagram – mass unbalance in disk 1:

Figure 4.4 – Finite element curves make use of the direct method (chapter 3.1.3)

The following can be said about Figure 4.4:

53

1. The Amplitude grows, and approaches asymptotically the inner shaft

synchronous FW critical rotating speed of the considered symmetric system, after which it

decreases and stabilizes. The reason why no resonance is verified at the BW critical rotating speed

is related with the fact that the rotor is symmetric. Consequently the rotor’s orbits will be circular.

Moreover since the unbalance is present in disk 1, the excitation force is synchronous and co-

rotational with the inner shaft who governs the overall rotor orbit. In this particular situation as it was

justified in chapter 2.1.3 there is only one resonance operating point, and it corresponds to the FW

mode, since it is being excited by a co-rotational excitation force.

2. The finite element results differ from those obtained with the Raleigh-Ritz

method. The asymptote of the finite element computed response is verified for slightly lower rotating

speeds since, as it was previously explained, the computed natural frequencies are lower using the

finite element method than using the Rayleigh-Ritz method. Moreover the absolute values of the

amplitude obtained with each of the employed methods differ slightly from each other. Again this has

to be with the fact that the finite element method results adapt better to the modal shape of the

Multirotor first bending mode (see Appendix 3).

Response diagram – mass unbalance in disk 2


It can be verified in Figure 4.5 that:

1. The critical speeds asymptotes occur only for the outer shaft synchronous BW

modes. This happens because the excitation force, caused by the mass unbalance in disk 2, is

counter-rotational to the inner shaft and outer shaft synchronous, and therefore excites only the outer

shaft synchronous BW orbits.

2. The finite element results of the Response Diagrams are not coherent with the

finite element results of the correspondent Campbell Diagram. Namely, the asymptote

corresponding to the BW critical speed for the finite element curve, does not agree with the computed

54

critical speed in the Campbell Diagram, i.e. . Additionally, close to the , another

asymptote is visible. This happens because, as it was said in the ‘Solving Methods’ in the beginning of

chapter 4.1, the harmonic solution of ANSYS® uses the direct method to compute the natural

frequencies and the pseudo-modal method was the employed method for the natural frequencies

computation to construct the Campbell diagrams on this work. As it is shown in the Appendix 4 to this

work, if the direct method was used to determine the natural frequencies and consequently to build the

Campbell Diagrams, the frequency curves would present a deviation from the solutions obtained using

the pseudo-modal method, especially at higher frequencies. This explains the deviation in the finite

element results. Additionally the direct method used in the harmonic analysis includes all the modes in

the frequency range under study, and not just the first mode as the Rayleigh-Ritz equations in chapter

2.1.3 contemplate. This explains the extra asymptote, which corresponds to the BW of the second

vibration mode of the symmetric Multirotor which is within the range of the harmonic analysis done for

the mass unbalance in disk 2.

3. The amplitude differences between the response curve obtained with the

Rayleigh-Ritz method and the curve obtained with the finite element method are larger for a

mass unbalance in disk 2 than for a mass unbalance in disk 1. This could be related with the fact

that when the Rayleigh.-Ritz method was deduced for the Multirotor on chapter 2.1.2, the outer shaft

was considered as rigid. This assumption was no longer taken into account when the finite element

model was built.

Example 2

For

Figure 4.6 – Campbell diagram – Symmetric rotor Example 2

55


Critical Speeds Inner/outer shaft synchronous

(rpm)

Rayleigh-Ritz BW FW

9658 10371

Finite Element 9436 10188

Error (%) 2.35 1.80

Comparing figures 4.3 and 4.6 it is visible that:

1. The BW and FW are now much closer. This is related to the gyroscopic effect.

Because the rotors are rotating at equal speeds but in opposite senses, making the gyroscopic term

present in the matrix very low and therefore diminishing the gap between the BW and FW

curves.

2. The FW mode is now increasing and the BW mode decreasing with increasing

rotating speed. This happens because , and thus the gyroscopic force vector is now

corotational with the inner shaft spin speed vector.



56



Figures 4.7-8 confirm what was previously said about the effect of the mass unbalance in the

disks of the rotor. The differences between the finite elements and the Rayleigh-Ritz theories were

already been justified.

Example 3

For

Figure 4.9 – Campbell diagram – Symmetric Multirotor Example 3

57



(rpm)

Outer shaft synchronous

(rpm)

BW FW BW FW

Rayleigh-Ritz 7809 16624 1883 2140

Finite Element 7774 15223 1853 2084

Error (%) 0.45 9.20 1.62 2.69

In Figure 4.9 the only new aspect worth discussion is the large gap between the FW and BW

curves. This is easily explained by the co-rotate operation of both shafts in this case. This will result in

the largest absolute value of in the set of results presented until now. Consequently the

gyroscopic effect is significant which will lead to a sharper increase in the distance between the

curves.



58



The deviation between the finite elements calculated critical speeds, using the direct and

pseudo-modal methods, was already subject to discussion. Anyway one should notice the increased

deviation between the finite element FW critical speed computed in the Campbell diagram in Figure

4.9 ( ), the same critical speed compute with the Rayleigh-Ritz method ( )and the

correspondent asymptote location in the response diagram 4.10 ( ). This could be related

either with numeric aspects of the finite element solution of the direct method, or with errors introduced

by the assumptions taken in the Rayleigh-Ritz and the finite element pseudo-modal approximate

methods. Recall that the Rayleigh-Ritz method, for instance, doesn’t include a correction for the disk’s

inertia influence on the mode’s shape.

Asymmetric system

For the asymmetric system the elastic bearing properties are:

59

For :

Figure 4.12 – Campbell diagram – Asymmetric Multirotor



(rpm)

Rayleigh-Ritz FW BW

9224 13039

Finite Element 9180 12177

Error (%) 0.48 7.08

Comparing the Campbell Figure 4.12 with Figure 4.3, built in the same rotating speed ratio

conditions in the symmetric system, there are some noticeable changes.

1. When the rotor is stopped there are two natural frequencies instead of one.

Since the bearing stiffness characteristics are now asymmetric, the bending natural frequencies will

now be sensible to the different stiffness properties in the and directions. The same is to say,

that the characteristic equation (2.1.3.4) will now have two solutions for .

2. The FW critical speeds are higher for the asymmetric Multirotor than for the

symmetric and the BW critical speeds are lower. We can conclude that the asymmetric stiffness

properties of the bearing causes the FW and BW curves to drift apart relative to the symmetric

Multirotor FW and BW curves.

60



In Figure 4.13 is interesting to note that when asymmetric stiffness properties in the bearings

are verified, a FW excitation force such as that caused by a mass unbalance in disk 1, can also cause

resonant behavior at BW critical speeds.

The response diagram differences caused by a mass unbalance in disk 2 relative to a mass

unbalance in disk 1 were already widely discussed in the previous results. For this reason we will skip

the response diagram for a mass unbalance in disk 2 from this point onwards.

Damped system

The previous results were sufficient to show the most relevant differences between the finite

element and the Raleigh-Ritz method. For this reason, in the damped system we will just consider the

solutions obtained via finite element method, again using the pseudo-modal method for the Campbell

diagrams and the direct method for the response diagrams.

For the damped system the elastic bearing properties are:

61

Example 1

For :

Figure 4.14 – Campbell Diagram – Damped Multirotor Example 1

Table 4.6 – Critical speeds associated with the previous Campbell diagram

Inner shaft synchronous

(rpm)

FW BW

9453 12153

The introduction of small amount of damping doesn’t change significantly the Campbell

diagram and the value of the critical speeds. This is visible if we compare the Figure 4.14 with Figure

4.12.


Figure 4.15 – Results make use of the direct method (chapter 3.1.3)

62

The Response diagram by the contrary shows substantial differences after the introduction of

damping. Asymptotes are now longer visible and the overall vibration amplitude has decreased.

Example 2

For

Figure 4.16 – Campbell Diagram – Damped Multirotor Example 2

Table 4.7 – Critical speeds associated with the previous Campbell diagram

Inner shaft synchronous

(rpm)

BW FW

9186 9837

The Campbell diagram shown in Figure 4.16, presents major alterations in its curves with

increasing damping. High damping cause the FW whirl curve to appear above the BW curve for

lower rotating speeds even when , and for higher rotating speeds the curves’ relative

position switches, resulting in a rather complex evolution

63


Figure 4.17 – Results make use of the direct method (chapter 3.1.3)

Figure 4.17 shows once more the general tendency of the response diagram of an

increasingly damped rotor: Lower amplitude and less sharp peaks.

4.2 Force Reconstruction

In this chapter one aims to apply the various force reconstruction methods presented in

chapter 2.2., to a finite element model of a symmetric rotor. But first of all an application to a simpler

system, consisting of a finite element model of a beam is presented, in order to provide a good

introduction to the application of each of the various time domain force reconstruction methods

included in this work.

The discussion of each example will include a comment about the quality of the solution when

the number and position of the studied system’s response is modified.

4.2.1 Application to a Beam

The Problem

Let us then start by reconstruction a force acting transversally to a beam in a previously known

location. The problem to be solved here is based on the beam that was subject of experimental

studies on the article of Allen and Carne about DMISF [12]. Figure 4.18 shows the beam in question.

In all examples the beam is considered free of any constrains.

64

Figure 4.18 – Beam Model with Force Impulse position represented by vector F

An impulse force is applied in the left-most end of the beam. In the preceding analysis it will be

tested the capacity of each time-domain method to reconstruct the force impulse.

Solving Methods

Every time-domain, namely the SWAT, ISF and DMISF methods will be applied to the

previous described beam following the numeric implementation explained on chapter 3.2, and the

fundamentals on chapter 2.2. The Frequency domain method was not implemented, because it was

only programmed for harmonic forces and not for impulse forces such as the one applied on this

example. An application of the FD method will be illustrated on the chapter 4.2.2.

The structure responses are extracted as mentioned on chapter 3.2.1. In the particular case of

the beam the translational and rotational accelerations in every node are extracted and used in the

reconstruction systems of each force reconstruction method.

The Finite Element Model

As was made clear on chapter 3, there are two finite element models. One for extract the

structure’s responses and another to build the systems matrices , and used in SWAT to

compute the flexible mode’s modal vectors and in the ISF and DMISF to build the state-space system

matrices. These finite element models have in common the discretization of the mesh. In the particular

case of this beam, 12 equally spaced beam elements were used as shown on Figure 4.19 along with

the finite element assembly table.

Figure 4.19 – Finite Element Model of the Beam

65

Table 4.8 – Assembly between global and local dofs of the Beam mesh

Element Number Element Type

1 Beam 1 2 3 4 5 6

2 Beam 4 5 6 7 8 9

3 Beam 7 8 9 10 11 12

4 Beam 10 11 12 13 14 15

5 Beam 13 14 15 16 17 18

6 Beam 16 17 18 19 20 21

7 Beam 19 20 21 22 23 24

8 Beam 22 23 24 25 26 27

9 Beam 25 26 27 28 29 30

10 Beam 28 29 30 31 32- 33-

11 Beam 31 32- 33- 34 35 36

12 Beam 34 35 36 37 38 39

In the ANSYS® finite element model, where the transient analysis of the beam subject to an

impulse was made, the beam model was built with BEAM188 (six degree-of-freedom per node)

elements, that makes use of quadratic polynomials. Though based in the Timoshenko beam theory it

is the recommended beam element for slender to moderately slender beams. The systems matrices

were built using a three degree-of-freedom per node beam element, whose element matrices and

characterization is stated on the Appendix 5 to this work.

To complete the beam model, it will be supported by two soft springs to simulate free-free

boundary conditions and the force will be applied on the leftmost node of the beam. In ANSYS® these

springs were introduced with COMBIN14 elements, while in the system’s matrices the springs’

stiffness was simply added in the proper positions of the system’s matrix.

Additionally a small value of internal damping is assigned to the beam to simulate the

experimental conditioning of the beam. This was done by considering a stiffness dependent damping

matrix, expressed by a positive integer factor , i.e. [ ] [ ] . Of course the spring stiffness is

not considered in the computation that’s why the designation is used to distinguish from the

general system’s stiffness matrix . In ANSYS® factor was introduced as a material property using

the ‘MP, DAMP’ function.

Numeric Data and Force Input Time-history

Table 4.2.1.1 includes all its relevant material and dimensional parameters.

Table 4.9 – Numeric data of the beam

Material Constants Section Dimensions Spring Stiffness

⁄

(Density)

⁄

(Young Modulus)

66

The Force input signal has the characteristics displayed in table 4.2.1.2.

Table 4.10 – Transient analysis and force impulse data

Recorded Time

Force Impulse Amplitude

Force Impulse Application Instant

Force Impulse Duration

Sampling Time Interval

Results and Results Discussion

1. SWAT

To apply the SWAT method to the beam a 15 elastic modes were included in order to apply

the force reconstruction method. The obtained solution is displayed in Figure 4.20 for the entire

analyzed domain on the left-hand side, and only for the first of the analysis on the right-hand

side. Both the measured and reconstructed forces time-history are displayed in the graphics. Recall

that SWAT estimates the resultant force impulse in the center of gravity of the structure.

The impulse was defined in the transient analysis performed in ANSYS® with the properties

described in Table 4.10.

Figure 4.20 – SWAT applied to the beam

Notice that the force impulse estimated by the SWAT method is about lower than the

force impulse actually acting in the beam. This is slightly higher than the error attributed to the

SWAT method in [12].

In this analysis there are several sources of error.

First of all the finite element model used to build the force reconstruction system have slightly

differences from the finite element model from where the responses were extracted. This means that

the force was estimated in a structural system with responses that not exactly match the response it

would have if the force impulse was applied on it.

67

Additionally in the mentioned work of Carne et al. the method was applied experimentally and

not numerically, which implies a different procedure in the way the elastic modes are obtained and

treated.

To conclude, it can be said that the SWAT method presents rather good results for the

estimation of the force impulse amplitude in the beam. After the force impulse is removed the SWAT

method stills estimating a non-zero force value, but even though the main estimated curve peak is

clearly located between the time intervals in which the force impulse was acting in the beam.

2. ISF

For the ISF method the lower delay parameter value at which stable results were obtained

was , the chosen number of Markov parameters to use in this analysis was . The force is being

estimated in the dof where the force is applied. It corresponds to dof number 2 as shown on Figure

4.2.1.1. So vector [ ], has a 1 in the second array position and 0 in the remaining. The results thus

obtained are displayed in Figure 4.21. Again with the entire time domain in the left-hand side and the

first time instants displayed on the right-hand side.

Figure 4.21 – ISF applied to the beam

As expected from the results in the literature [12], the ISF leads to very good estimations with

low errors. The estimated curve practically overlays the measured force impulse. Despite this fact, it is

noticeable the residual ringing conducted by the estimated force curve after the force impulse is

removed.

3. DMISF

The first stable solution for the beam using DMISF was obtained for . As in the ISF

method vector [ ] will make the force estimation to take place in the dof where we know the force is

applied. Figure 4.22 shows the obtained results for the entire time domain on the left-hand side and

the first time instants ‘zoom’ on the right-hand side.

68

Figure 4.22 – DMISF applied to the beam

The results obtained with DMISF are as good as that obtained with the ISF method. The

estimated force almost overlaps the measured force in the impulse application time instants. Despite

this, it is noticeable a slight reduction in the estimation curve ringing in the post-force impulse time

period.

This set of results globally suggests that the DMISF presents better results than the other

methods, since the estimated force follows in perfection the measured force and it has the less

significant residual ringing.

4.2.2 Application to a Symmetric Rotor

The Problem

Now let us consider a symmetric rotor, such as that shown in Figure 4.23 with a simple

supported shaft, no elastic bearings, and one disk.

Figure 4.23 – The Symmetric rotor model

The considered rotating rotor will be subject to a harmonic force excitation caused by a mass

unbalance in the disk. The rotor is considered to be rotating at constant speed.

69

Solving Methods

Firstly the Frequency domain method will be applied to the current problem in order to localize

the input load. Then the SWAT, ISF and DMISF methods will be successively applied to the rotor to

determine the time-history of the force. The method implementation follows the description made on

chapters 3.2 and 2.2.

Once more time the response is extracted from a finite element model as mentioned on

chapter 3.2.1. The translational and rotation accelerations about the two perpendicular axes to the

rotor rotating axis, in every model node, compose the response data from which forces are to be

estimated.


Right below in Figure 4.24, the used mesh in the finite element models is represented for this

rotor along with the assembly matrix used to build the matrices of the system, with every dof number

identified.

Figure 4.24 – Finite element model with the element numbers in black and the nodal numbers in blue

Table 4.11 – Assembly between global and local dofs of the Monorotor mesh

Element Number Element Type

1 Beam 1 2 3 4 5 6 7 8

2 Beam 5 6 7 8 9 10 11 12

3 Beam 9 10 11 12 13 14 15 16

4 Beam 13 14 15 16 17 18 19 20

5 Beam 17 18 19 20 21 22 23 24

6 Beam 21 22 23 24 25 26 27 28

7 Beam 25 26 27 28 29 30 31 32

8 Beam 29 30 31 32 33 34 35 36

9 Beam 33 34 35 36 37 38 39 40

10 Punctual Mass 13 14 15 16 - - - -

The element matrices for the rotor components are shown on chapter 3.1.2. In this example

the used element stiffness matrices are the ones which include a shear correction. Since no axial

force is applied in the rotor, the axial force stiffness influence is null.

In the ANSYS® finite element model the shaft is modeled with BEAM188 (six dof per node)

elements with quadratic polynomials, the disk is represented by a punctual mass of equivalent inertia

70

using the element MASS21, and the displacements in the transversal directions are constrained at the

shaft’s extremity nodes.

There’s no addition of damping factors.

Numeric Data and Force Input Time-history

Table 4.12 includes all its relevant material and dimensional parameters.

Table 4.12 – Numeric data of the Monorotor

Material Constants Dimensions Mass Unbalance

⁄

(Density)

(Young Modulus)

(mass unbalance position)

(Poisson Ratio)

Table 4.13 includes all relevant data used in the frequency domain analysis.

Table 4.13 – Relevant data for the frequency domain force identification method

Initial frequency

Final frequency

Number of solutions

The time domain analysis parameters as well as the value of the rotating speed that influences

the force input signal has the characteristics displayed in Table 4.14.

Table 4.14 – Relevant data for the time domain force identification methods

Recorded Time

Rotating speed



1. Frequency Domain

For the Frequency domain force localization problem it is necessary to define the nodes where

the response is known and where it is unknown as demonstrated on chapter 2.2.1. This information is

condensed in table 4.15.

Table 4.15 – List of the known and unknown nodes

Node number

Known

Unknown

71

Appling the Frequency domain method as stated in the parameters present in table 4.13, the

error graphic for each force combination, computed as demonstrated in equation (2.2.1.11), it results a

graphic such as Figure 4.25 displays below.

Figure 4.25 – Error vs Force Combination Graphic

As is perceptible in Figure 4.25, the error is minimized in the force combination. This

combination corresponds to the three forces acting respectively in the nodes number , and , i.e.

in the rigid supports and in the disk’s node.

The force position is therefore determined as being placed at the disk.

2. SWAT

The rotor is now rotating at constant speed and it is intended to determine the force time

history of the applied load at the disk. Recall that the estimated force determined with the SWAT

method estimates the resultant force at the structure’s gravity center. As the disk is rotating, so does

the force, since the mass unbalance that causes the unbalance rotates with it. Therefore the force will

be reconstructed in both rotor transversal directions. The following results were obtained using 20

elastic modes, and the two translation rigid body mode vectors (each for nodal translation dof) are built

as stated on chapter 3.2.3.

72

The results for each direction are displayed in Figure 4.26 below.

Figure 4.26 – SWAT applied to the Monorotor

Notice that the estimated force curve present higher values than the measured force when the

force amplitude approaches the higher peaks. As the force approaches its lower amplitude peaks, the

estimated force curve starts do adjust better the measured force curve results. In this particular case

the error is even less than the 20% error obtained when SWAT was applied to the beam.

Again the differences between the finite element models used to extract and to estimate the

force input induce some error into the solution.

These results are highly dependent of the number of elastic modes included in the analysis.

The best results for the rotor were obtained with the 20 elastic modes in the direction and 30 modes

in the direction.

As it was expectable the force time-history present minimum amplitude in one direction in the

time instant it is achieving the maximum amplitude in the other direction.

Despite the described handicaps SWAT allows a good understanding of the time-domain

evolution of the force acting on the structure.

3. ISF

Applying the ISF method with a delay parameter and Markov parameters, a stable

solution was obtained for the force applied at the disk’s node in both transversal directions. To obtain

a force estimation in each direction, it is necessary to generate two [ ] vectors with a in the array

position correspondent to the dof in the desired position and direction in each analysis. The obtained

graphics are represented in Figure 4.27.

Figure 4.27 – ISF applied to the Monorotor

73

In general the estimated force curve closely follows the measured force curve.

Despite of that it should be mentioned, the difficulty demonstrated by the estimated force

curve to follow the peaks at lower amplitudes, exceeding in some points the amplitude of the

measured force.

Also note the intense ringing of the estimated force in the first time instants of the

reconstructed force in the direction.

4. DMISF

Using the same [ ] vectors that were used in the ISF method in order to obtain an estimation

of the force in the and directions, and applying a delay parameter of , it results two

graphics. They are displayed in Figure 4.28.

Figure 4.28 – DMISF applied to the Monorotor

Notice that the ring is much less intense in the DMISF estimated curve when in comparison

with the ISF estimated curve. Nevertheless, the estimated force amplitude especially in the higher

force peaks fails to meet a perfect matching with the measured force curve.

Globally it can be said that time-domain force identification methods present good results

when applied to simple examples of Rotordynamics. SWAT presents good results qualitatively

speaking, but fails to present consisting force amplitude reconstruction. The ISF method despite

showing some ringing has proven to be highly reliable in force amplitude prediction, while the DMISF

method has shown difficulties in determining the force amplitude at the higher force amplitude peaks,

but diminishes the ringing of the force curve. This is consistent with what was obtained in the beam

application example.

4.3 The Propfan Case Study

In this chapter we will use the presented tools of Rotordynamics analysis and Force

identification on the Propfan jet engine. This engine has been part of a European research project

called Ducted Propfan Investigation or DUPRIN, and of course we will begin the chapter by a brief

presentation of the motivation and history of this project.

74

The DUPRIN project was launched with the support of the European Commission in 1990, in

which 14 European research partners were involved. They had the objective of developing a High

bypass, low fuel consumption and highly efficient jet engine to meet the growing needs of

sustainability in the aircraft propulsion market [5]. The achieved design included a compressed air fed

four stage turbine that supplied the shafts connected with two counter-rotating ducted propfans with

160kW allowing a maximum rotating speed of up to 16 000 rpm. [6]

Figure 4.29 – The DUPRIN Propfan in its test facility – source: [6]

The first task to perform will be the reproduction of the Campbell Diagram and the first two

modal, and to compare the achieved solutions with that present on the literature [6]. For that a self-

made finite element model in ANSYS® is used.

To finish the chapter, the ISF method will be applied to the finite model of a Propfan engine to

reconstruct a force whose location is known but whose amplitude and periodicity is ignored. This last

example intends to show that time- domain force identification can be of practical use in aircraft

propulsion applications.

4.3.1 Modal Analysis

In this section the modal analysis representation in Rotordynamics, the Campbell diagram, will

be presented for the DUPRIN Propfan. The BW and FW orbits of the first vibration mode are also

shown. This set of results intends to validate the coherence of the dynamic behavior of our Propfan

model.

The Problem

The structure to be analyzed is a Multirotor configuration based on the DUPRIN Propfan. The

exact dimensions (with exception of the its length and maximum radius indicated in [29] ) of the

Propfan engine, as well as its constituent materials were not provided. A trial to achieve an

approximate model becomes therefore the only available choice.

75

In what concerns the materials, a titanium alloy was assumed to the elements corresponding

to the fans, and a stainless steel was assumed in the remaining elements representing the Propfan’s

shaft.

Lalanne et al. show a picture of their Propfan finite element model when they performed

studies in the subject [6]. This picture was used to scale the Propfan engine finite element model used

in this work. To define the dimensions of the model of the engine a cross-reference between the scale

and the two known dimensions was made determining therefore the remaining dimensions of the

Propfan, resulting in the schematic shown in Figure 4.30.

Figure 4.30 – The Propfan revolution section

The Propfan model has three bearings whose longitudinal and radial coordinates are

displayed in Figure 4.30 as well, and they include only stiffness terms in the and components. A

material damping term was also considered, equivalent to a quality factor . This was performed

because the authors in [6] considered it appropriate to describe the real dynamic behavior of the

engine. The numeric values of the stiffness of the bearings were not provided as well, therefore its

values were adjusted until an acceptable agreement between the author’s results and that present in

[6] was met.

Solving Methods

In order to obtain the intended results, a finite element model was built in ANSYS® and a

modal analysis performed. After this the ANSYS® Rotordynamics Post-processing tools were used to

extract the Campbell diagram representation, to compute the correspondent critical speeds and to

display the orbit shapes associated with the first two vibration modes.


The developed finite element model was inspired in the one used by Lalanne and Ferraris [6].

The inner shaft was divided in 25 BEAM 188 elements and the outer one was divided in 15 BEAM 188

elements. Recalling the finite element model used in the Multirotor example on chapter 4.1, and

comparing it with this example, it is noticeable the degree of refinement used in this particular model.

76

Additionally, and as it was performed for the Multirotor example in chapter 4.1, the disks were

modeled by punctual masses of equivalent inertia, using element MASS 21 and the bearings’ stiffness

was modeled with COMBIN14 spring elements.

To simplify the finite element model the stiffness of the spring connecting the inner and the

outer shaft was considered infinite. This was also an assumption taken in the Multirotor example

present in this work, and it has given satisfactory results.

The quality factor was was accounted for making use of the function MP/ DMPR.

The resultant finite element model is shown in Figure 4.31 just below.

Figure 4.31 – The Propfan finite element model Numerical Data

The Propfan dimensions are already shown in the sketch shown in Figure 4.30. The remaining

data is described on Table 4.16 below.

Table 4.16 – Numeric data of the Propfan

Material Constants

Bearing

Properties Stainless Steel Titanium

⁄

(Density)

⁄

(Density)

(Young Modulus)

(Young Modulus)

(Poisson Ratio)

(Poisson Ratio)

77


Campbell Diagram

The obtained Campbell Diagram for the Propfan used in this work is shown in Figure 4.32

along with the Campbell Diagram obtained in [6] for the same structure. The associated critical speeds

are indicated in Table 4.17.

Figure 4.32 – Propfan Campbell diagram comparison

Table 4.17 – Error computed relative to the values obtained with the author’s finite element model

Critical Speeds Inner/Outer shaft synchronous

(rpm)

Finite Element FW BW

7146 8421

Lalanne et al. 7019 8357

Error (%) 1.78 0.76

Modal orbits

The modal orbits for the first two vibration modes are here depicted in Figure 4.33 and

compared with the results of Lalanne et al. This is done to confirm the coherence of the obtained

modal analysis solution.

78

Figure 4.33 – First two modal shapes of the Propfan; Author’s model above/Model in [6] below; FW (Left) and

BW(Right)

Several conclusions can be taken from the established comparisons evident in Figures 4.32-

33.

1. The dynamic behavior is coherent, i.e. in the frequency range contained in the

Campbell Diagrams in Figure 4.32 the whirl sequence at which the vibration modes appear is the

same in both finite element models. Adding to this the modal shapes associated with the first two

vibration modes computed with the developed finite element model coincide with the modal shapes

obtained by Lalanne et al. in their work about the DUPRIN project.

2. The evolution of the first two frequency curves as well as the critical speed

values is similar in both finite element models as Figure 4.32 and Table 4.17 state.

These two conclusions suggest the reliability of the developed finite element model and give

the necessary confidence to try the application of a time-domain force reconstruction method in this

Propfan model.

3. The verified differences for the third and fourth vibration modes should be related with

incorrect assumptions in the dimensioning and material constant choice for each Propfan part,

between the in-house developed Propfan finite element model and the one developed in the studies of

Lalanne et al. [6]

79

4.3.2 Reconstruction of a Force Caused by an Unbalance in the Fan

The Problem

The force excitation at which the Propfan shown in Figure 4.30 will be subjected is caused by

a mass unbalance in Fan 1. The Propfan will start from rest and will accelerate at constant rate to a

speed of 625 achieved 0.5 seconds afterwards. The two shafts constituting the Propfan are

running at the same speed but in opposite senses. The Fundaments Chapter 2.1 of this work didn’t

contain any mention to transient motion of rotors. Therefore a brief description of these motions is

provided in the Appendix 6 to this work.

Solving Methods

The finite element model presented in the last section (4.3.1) is slightly changed before being

applied to this problem. The couple between the two rotors used in the previous model to connect the

inner and the outer shafts was removed, and a small stiffness spring was introduced in both and

directions. This was performed because when the Transient analysis was carried out in the finite

element model of the Propfan, the force reported to be applied in the node of the FAN 1 differed from

the force actually being applied in that node, leading to erroneous values of the response. The author

could only overtake this issue, in the available time frame for this work, by removing the referred

coupling and by introducing the springs. This means that the modal analysis that was previously

undertaken not actually corresponds to the behavior of the finite element model employed in the

Transient analysis code.

A full transient analysis, in the previously described conditions, was performed in ANSYS® in

order to get the Propfan responses. The force was modeled as two punctual forces acting along the

and components as described on chapter 3.3, that follow the force functions present in equation

A6.8 for transient motion deduced in Appendix 6.

Since the Propfan model is much more complex than the Symmetric rotor considered in

chapter 4.2.2, the accelerations will be only registered in the translational dofs of the even nodes of

the inner rotor (13 nodes – 26 measurements), and in the translational dofs of each node of the outer

shaft (16 nodes – 32 measurements) limiting the response array in each time instant to 58 values.

Additionally in order to apply the ISF, the state-space system was built using the system’s

matrices. These matrices were built using the Substructure tool of ANSYS® using the same 58 dofs

from where the response was extracted as master dofs, i.e. ANSYS® built a simplified model from the

dofs the user wants to include in the force reconstruction process. ISF is now ready for application

according to chapters 2.2.4 and 3.2.4, with the nuance that the damping matrix [ ] that include the

gyroscopic terms is now varying from time instant to time instant, and that new stiffness matrix terms

appear.

This is achieved through cyclic computation.

One may think that the problem is simply solved by applying the ISF method with matrix

[ ] changing in each time instant. This however has led to unstable solutions. Another cyclic

methodology is proposed by the author.

80

Each cycle step will correspond to a time instant. Firstly the rotating speed is fixed with its

instantaneous value. Then the estimated force is computed for the time domain comprising the initial

time instant and the actual time instant with constant rotating speed equal to the instantaneous

rotating speed. When the force estimation is obtained, only the value corresponding to the step time

instant is saved to the force estimation array.

This methodology has presented stable solutions but increased computation time.

ISF will be the only force reconstruction method applied since it demonstrated to be the most

computer efficient method in the author’s experience.

Numerical Data

The data in Table 4.16 is again applicable to this case. It must be now added the Transient

analysis data, the mass unbalance data as well as the stiffness of the springs connecting the inner

shaft with the outer one, included in Table 4.18.

Table 4.18 – Transient analysis and mass unbalance data

Recorded Time


Mass unbalance

Offset

Spring stiffness data


The resultant estimated force curves are shown in Figure 4.34 along with the force actually

acting on the node corresponding to Fan 1. The elapsed analysis time is also shown.

Figure 4.34 – ISF applied to an accelerating Propfan

The estimated curves present a reasonable agreement with the measured force at lower force

amplitudes and a considerable deviation at the higher amplitudes. These deviations may be related

with the reduced set of measurement points and with imprecisions in the finite element model

developed in ANSYS®, from which the responses were obtained.

Additionally the elapsed analysis time starts to become an issue. In the average of the two

reconstructions the code needed 647 seconds to reconstruct 0.5 seconds of force time-history. The

81

ISF method was used at the first place and the number of Markov parameters was kept low, since this

configuration has shown the most computer efficient performance.

82

5 CONCLUSIONS AND FURTHER DEVELOPMENTS

5.1 Conclusions

5.1.1 Rotordynamics Analysis

The tests performed on chapter 4.1 confirm what was said and predicted in the Rotordynamics

fundamentals chapter 2.1.

In general the finite element results have led to better results, concerning the determination of

the natural frequencies and the structural response as it is justified in the Appendix 3 to this work.

The inclusion of a small amount of damping has shown very satisfactory results in the control

of the rotor’s vibration levels.

5.1.2 Force Reconstruction Methods

The beam example (chapter 4.2.1) shows a general trend in the applied force reconstruction

methods. SWAT underestimates the values of the measured force with higher errors comparing with

the ISF and DMISF methods. No evident advantage is taken from the usage of the DMISF method,

since ISF was enough to obtain a stable solution.

In the rotor application example (chapter 4.2.2) these trends were not observed, in fact SWAT

presented some values of the force estimation curve over the measured one and in the ISF and

DMISF results the opposite was verified. Once more there was no obvious advantage from the

application of the DMISF method in this specific example in comparison with the ISF example.

Stable results were easily obtained in every method applied to the rotor making time-domain

force reconstruction methods a valid option for constant spin speed rotors.

Despite all the reconstructed forces have origin in a mass unbalance, the author doesn’t see

any limitation on the application of the studied time-domain force reconstruction methods to other load

configurations such as punctual synchronous and asynchronous forces.

5.1.3 The Propfan Case Study

In this example present on chapter 4.3, the limitations of the developed Propfan finite element

are obvious, as stated in the performed modal analysis. Nevertheless coherent results were obtained,

allowing the generalization of the following force reconstruction results to other complex rotating

systems such as aircraft engines.

The force reconstruction results show not only the capability of ISF to reconstruct forces acting

in an accelerating Propfan but also that the reconstruction is possible when only a small subset of

structural measurements is employed.

This was however achieved at the expense of higher elapsed analysis time and considerable

force estimation deviations (see Figure 4.34).

5.2 Further Developments

It was not the purpose of this work to show how parametric changes of the time-domain force

reconstruction methods influence its solution, i.e. how a change on the delay would affect the quality

83

of the solution of ISF and DMISF and how the inclusion of more or less flexible modes would affect the

ability of SWAT to estimate better or worse the forces. It would be interesting though to develop this

kind of work in the future, for the sake of the optimization of the force identification process.

To improve the stability of ISF and DMISF methods, it would be interesting to employ non-

linear control optimization techniques as proposed in the Appendix of the paper by Allen and Carne

[26].

In what concerns the SWAT method, the generalization presented by Genaro and Rade [20] to

systems with multiple forces acting simultaneously, would be an important step in the improvement

reliability of this method.

It would be also interesting in the context of rotor transient operation to employ these methods

on the detection of changes in the rotor dynamic operation, i.e. to detect changes in the rotor’s spin

speed or even in the spin speed change rate (see Appendix 6).

One should recall finally that these methods were applied exclusively numerically.

Experimental application would imply necessarily adjustments in the followed procedure, namely in the

definition of the state-space system used in the ISF and DMISF methods. In the mentioned methods, it

would be more advantageous to use the state-space system based on modal properties (easily

accessed in experimental test mounts) rather than based on the system matrices (that implies the

usage of a numeric finite element model).

84

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[1] F. Nelson, Rotor Dynamics without Equations, vol. 10(3), 2007, pp. 2-10.

[2] D. J. Ewins, Control of vibration and resonance in aero engines and rotating machinery - An

overview, vol. 87, Imperial College London, United Kingdom: International Journal of Pressure

Vessels and Piping, 2010, pp. 504-510.

[3] J. J. Sinou, D. Demailly, C. Villa, M. Massenzio and F. Laurant, Rotordynamics Analysis:

Experimental and Numerical Investigations, Vols. Volume 5: 19th Biennial Conference on

Mechanical Vibration and Noise, Parts A, B, and C, ASME 2003 International Design Engineering

Technical Conferences and Computers and Information in Engineering Conference

(IDETC/CIE2003).

[4] M. Lalanne and G. Ferraris, Rotordynamics prediction in Engineering, Wiley; 2 edition, 1998.

[5] H. P. M. Ernst-Heinrich Hirschel, Aeronautical research in Germany: from Lilienthal until today,

vol. Volume 147, Springer, 2004.

[6] G. Ferraris, V. Maisonneuve and M. Lalanne, Prediction of the Dynamic Behavior of Non-

symmetric Coaxial Co- or Counter-Rotating Rotors, vol. 195(4), Journal of Sound and Vibration,

1996, pp. 649-666.

[7] M. Adams, Rotating Machinery Vibration, 2nd Edition ed., CRC Press, 2009.

[8] J. C. Pereira, Introdução à Dinâmica de Rotores, Florianópolis: Universidade Federal de Santa

Catarina, 2003.

[9] E. Assis and V. Steffen Jr., Inverse Problem Techniques For The Identification Of Rotor-Bearing

Systems, vol. 11, Inverse Problems in Engineering, 2003, pp. 39-53.

[10] Q. Han, H. Yao and B. Wen, Parameter identifications for a rotor system based on its finite

element model and with varying speeds, vol. 26, Acta Mech Sin, 2010, pp. 299-303.

[11] K. Stevens, Force Identification Problems - An Overview, Proceedings of the 1987 SEM Spring

Conference on Experimental Mechanics, 1987, pp. 838-844.

[12] M. S. Allen and T. G. Carne, Delayed, multi-step inverse structural filter for robust force

identification, vol. 22, Mechanical Systems and Signal Processing, 2008, pp. 1036-1054.

[13] R. J. Hundhausen, D. E. Adams, M. Derriso, P. Kukuchek and R. Alloway, Transient loads

identification for a standof metallic thermal protection system panel, Orlando, Florida, 2005.

[14] M. M. Neves and N. M. M. Maia, Estimation of Applied Forces Using The Transmissibility

Concept, vol. paper 377, IDMEC-IST, Technical University of Lisbon, Department of Mechanical

Engineering: ISMA 2010.

[15] Y. Lage, N. M. M. Maia, M. M. Neves and A. M. R. Ribeiro, A Force Identification Approach with

the Transmissibility Concept for Multi-Degree-Of-Freedom Systems, IDMEC, Department of

Mechanical Engineering, Instituto Superior Técnico.

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[16] N. Maia, M. Fontul and A. Ribeiro, Transmissibility of Forces in Multiple-Degree-of-Freedom

Systems, Leuven, Belgium: Proceedings of ISMA 2006, "Noise and Vibration Engineering", 2006.

[17] A. M. R. Ribeiro, N. M. M. Maia and J. Silva, On the Generalization of the Transmissibility

Concept, vol. 14, Mechanical Systems and Signal Processing, 2000, pp. 29-35.

[18] T. G. Carne, V. I. Bateman and R. L. Mayes, Force Reconstruction using the sum of weighted

accelerations technique, San Diego, CA: 10th International Modal Analysis Conference (IMAC X),

1992, pp. 291-298.

[19] D. L. Gregory, T. G. Priddy and D. O. Smallwood, Experimental Determination of the Dynamic

Forces Acting on Non-rigid Bodies, Long Beach, CA: Aerospace Technology Conference and

Exposition, 1986, p. SAE Paper 861791.

[20] G. Genaro and D. A. Rade, Input force identification in the time-domain, Santa Barbara, CA: 16th

International Modal Analysis Conference (IMAC XVI), 1998, pp. 124-129.

[21] A. D. Kammer and D. C. Steltzner, Input Force Estimation Using an Inverse Structural Filter, 17th

International Modal Analysis Conference (IMAC XVII), 1999, pp. 954-960.

[22] J. Unger and G. De Roeck, System Identification and Damage Detection in Civil Engineering,

unpublished, 2002.

[23] Z. S. Zutavern and D. W. Childs, Identification of Rotordynamic Forces in a Flexible Rotor System

Using Magnetic Bearings, vol. Vol. 130, Transactions of the ASME, 2008, pp. 022504/1-6.

[24] J. Verhoeven, Excitation force identification of rotating machines using operational rotor/stator

amplitude data and analytical synthesized tranfer functions, vol. 110, Boston: American society of

mechanical engineers, Design engineering technology conferences, 1987, pp. 307-314.

[25] M. Spirig and T. Staubli, Identification of Non-linear Rotor Dynamic Coefficients Using Multiple

Circunfrencial Pressure Measurements, Hochschule Luzern, 1997.

[26] M. S. Allen and T. G. Carne, Comparison of Inverse Structural Filter and Sum of Weighted

Accelerations Technique Time Domain Force Identification Methods, 47th AIAA-ASME-ASCE-

AHS-ASC Structures, Strustural Dynamics and Materials Conference, 2006.

[27] M. S. Allen and J. H. Ginsberg, A global, single-input–multi-output (SIMO) implementation of the

algorithm of mode isolation and application to analytical and experimental data, 2006, p. 1090–

1111.

[28] Release Documentation for ANSYS, Theory Reference, Chapter 17. Analysis Procedures, 17.2.

Transient Analysis.

[29] G. H. Hegen, W. Puffert-Meissner, L. Dieterle and H. Vollmers, Flow field investigations on a wing

installed Counter Rotating Ultra-high-bypass Fan engine simulator in the low speed wind tunnel

DNW-LST, Vols. NLR-TP-99104, National Aerospace Laborator y NLR, 1999.

86

APPENDIXES

Appendix 1 – Determination of the Whirl Sense of an Asymmetric Multirotor

The equations that depict the orbit motion of the first vibration of a undamped asymmetric rotor

are the expressions (2.1.3.9-10). If these equations are introduced on the expression (2.1.4.3), it

gives:

[

] (A.1.1)

From the inspection of the equations (2.1.3.9-10), it is visible that the sign of the product

is dependent of the sign of the following expression :

( ( ) )( ( )

) (A.1.2)

The points of sense of whirl change, are the point at which is null. Therefore the rotating

speeds at which this happens are easily obtained from (A.1.2), giving:

√

√

(A.1.3)

So we can conclude from a signal analysis of equation (A.1.2) that:

rotor in FW

rotor in BW

Another interesting aspect, is the location of the critical speeds relative to the whirl change

speeds and . As it was referred in the previous section, the critical speeds are determined

making the denominator of the expressions (2.1.3.9-10) equal to zero. Concretizing this statement, it

results:

( ) ( )( ) ( ) (A.1.4)

which after some symbolic operations become:

( ) ( ) ( ( ) ) (A.1.5)

87

Because function ( ) is a quadratic one we can conclude that it is positive if and

, being and the two critical speeds.

It can be shown that introducing the expressions in (A.1.3) in equation (A.1.4) it results:

( )

( )( )

( )

( )( )

(A.1.6)

So, if the asymmetry in the bearings is so that in both situations of signal (see

the definitions of in equations 2.1.2.15-16), the relations between critical speeds and whirl sense

change speeds are:

with

with

Additionally is curious to verify that the elliptical orbit become progressively flatter on the

axis and more elongated on the axis as the rotor passes the BW region. These situations are all

depicted on the Figures A.1-2.

Figure A.1 – Whirl sense and orbit phase with

𝛺 𝛺 𝜔 𝜔 Ω

88

Figure A.2 – Whirl sense and orbit phase with

Appendix 2 – Energy Dissipation and Self-Excited Vibrations (Instability)

Energy Dissipation and its Relation with Instability

Bearings are used to give rotors their structural support and to control its vibration levels. The

bearings achieve this effect because the energy associated with vibratory motion is dissipated through

them, reducing the overall energy demand of the rotor at a given operation point. The loss of energy

due to the damping effect of the bearings is intended to compensate the sharp increase on energy

demand caused by operation of the rotor close to resonance points (critical speeds), where the

vibration levels are more problematic. But if excessive damping is present on the bearings, the

increase in dissipated energy can lever up the overall energy to provide the motion of the rotor to a

point that exceeds the demanded energy in a poorly damped rotor.

Bearing characteristics are also capable of causing self-excited vibration phenomena, i.e.

unstable dynamics motion that result during normal rotor operation, with unchanged or non-present

external excitations. In an energy point of view, one can see self-excitation as caused by energy

imparted to the rotor per cycle of harmonic motion. This is known as an instability state.

In this section it is intended to reach an expression that quantifies and determines the sources

of dissipation and self-excitation in rotor’s bearings. That will be obtained following the course of

reasoning used in [7].

A compacted form for the equations of motion is present below:

[ ]{ } [ ]{ } [ ]{ } { ( )} (A.2.1)

whose meaning of the matrices and vectors are already known from previous sections.

To start with, the matrices of equation (A.2.1) are decomposed into their symmetric and skew-

symmetric parts. This is done so that the dynamical effects could be identified as energy conservative

and energy non-conservative.

From the matrix algebra theory, the decomposition of a generic matrix [ ], can be expressed

as follows:

𝛺 𝛺 𝜔

𝜔 Ω

89

[ ] [ ] [

] (A.2.2)

where:

[ ]

[ ]

[ ]

[ ] (A.2.3)

The and indexes designate the symmetric and skew-symmetric parts respectively.

In the same way the matrices of (2.1.5.1) are decomposed, resulting:

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] (A.2.4)

That [ ] and [ ] are conservative, and [ ] non-conservative, can be an accepted

statement [7]. The skew-symmetric parts are now further analyzed, regarding their influence in the

incremental work done on the rotor, so that they could be classified into causers of conservative or

non-conservative energetic effects.

Firstly [ ] will be analyzed. In equation (2.1.2.26), it is visible that the contributions to the

skew-symmetric part of the damping matrix come from two different sources. The first contribution is

always present in the considered model and that is the damping effect resultant from the gyroscopic

effect, the second contribution depends on the skew-symmetric characteristics of the bearings

( and ), coherent with the definition shown in Figure 2.4. The computation of the incremental

work done by these two contributions must be effectuated, so that the energetic effects can be

determined. The incremental work done on the rotor by the skew-symmetric terms of a bearing is

written in the following way:

[ ] {

} { } (A.2.5)

in which:

[ ] [

] (A.2.6)

Applying the chain rule ( ) and performing the multiplications in (2.1.5.5), it

results:

( ) (A.2.7)

The incremental work is null, since the force vector is always perpendicular to its associated

velocity vector. Let us now focus on the gyroscopic moment effects in what concerns the incremental

work done on the rotor. The expression that shows this relation is:

[

] {

} { }

⏟

[

] {

} { }

(A.2.8)

Again the resulting incremental work is null, because the gyroscopic moment vector is always

perpendicular to its associated angular velocity vector. The conclusion to be taken from (A.2.7) and

(A.2.8), is that the skew-symmetric part of the damping matrix includes only conservative force fields

and doesn’t contribute therefore to the addition or dissipation of energy of the rotor.

Considering again the course of reasoning used to obtain the expression (2.1.2.26), it is visible

that the strain energy is only responsible for the symmetric terms of the stiffness matrix [ ] . So, the

90

skew-symmetric terms [ ] can only be influenced by the skew-symmetric stiffness terms of the

rotor’s bearings.

Again, to determine the energy effect of these terms the calculation of the incremental work is

indispensable:

[

] { } { }

(A.2.9)

Where:

and

Hence, the incremental work done by the skew-symmetric part of the bearings is an exact

differential, meaning that the transferred energy between any two given points of the rotor’s trajectory

is path dependent and therefore, non-conservative.

FigureA.3 – Generic periodic orbit of a rotor relative to a inertial reference frame– source: [7] - edited

Now it must be determined if this non-conservative effect adds or dissipates energy from the

rotor. That can be achieved formulating the expression an expression of the net energy-per-cycle

exchange resultant from the [ ] terms:

∮ ∮( )

(A.2.10)

With the help of the Figure A.3, and considering two line integrals between the generic points

and , the previous integral becomes:

∫ ( )

(A.2.11)

Watching carefully the expression (A.2.11), it is perceivable that the integral represents the

orbit area, and that its value is positive when in co-rotational whirl, and negative otherwise. It should

be noted additionally, that when there’s dissipation of energy , and when there’s energy being

added to the rotor . Since , it can be concluded that the skew-symmetric terms of the

stiffness have a dissipative effect in backward whirl conditions and excites (energy addition) the rotor if

the whirl is forwards.

𝑧 (𝑥)

𝑧 (𝑥)

91

The effects of [ ] terms are now investigated. This terms can only result from the rotor’s

elements mass matrices, namely from those of the disk and the shaft. By similarity with the formulation

followed in equations (A.2.9), (A.2.10) and (A.2.11), it can be obtained for the energy-per-cycle

influence of the skew symmetric part of the mass energy the expression right below:

∮( )

(A.2.12)

If the rotor transversal vibration is considered as harmonic, such is the case of equations

(2.1.2.5) and (2.1.2.6). The equation (A.2.12) can be further simplified:

∮( )

(A.2.13)

The factor comes from the second time derivative of the trigonometric functions present in

(2.1.2.5) and (2.1.2.6). Similarly with what was done to obtain equation (A.2.11), the integral in (A.2.13

can be split in two line integrals giving:

∫ ( )

(A.2.14)

For , and recalling the considerations previously written regarding the nature and

signal of the line integral displayed in (A.2.14), one concludes that the skew-symmetric part of the

mass matrix dissipates energy ( ) when the rotor is in forward whirl, and an instability causer

( ) when the opposite is verified. Although this phenomena is theoretically possible, it has

never been documented [7](page 76), meaning that is consistent with physical reality. This

conclusion is significant, since it implies that the rotor’s mass matrix must be constrained to symmetry

in order to reproduce real results.

Neglecting therefore let us now write an expression that includes non-conservative

contributions of the skew-symmetric stiffness matrix and the symmetric damping matrix, in one single

expression, so that the total energy-per-cycle exchange of the rotor can be obtained. It is worth to

mention, that both non-conservative contributions depend only on the bearing stiffness and damping

characteristics.

For a better visualization of the mentioned effects of the symmetric and skew-symmetric parts

Figure A.3, shows two typical rotor orbits, both in FW (a) and (c) and BW (b) and (d).

92

Figure A.4 – Force components of the various matrices decomposed on its symmetric and skew-symmetric terms

– source: [7]

On the circular orbits in Figure A.4 (a) and (b), it is visible that the only force components that

change their sense when the rotor changes from FW to BW, are the vectors and . The first

acts always parallel, but in the opposite sense to the instantaneous velocity vector, thus contributing

for the stable behavior of the rotor. The second always acts radially to the rotor’s orbit but has a

centripetal sense if the rotor is in FW and centrifugal sense when the rotor is in BW. This happens

because is the gyroscopic effect term, and is this effect the responsible for the divergence of the

FW and BW branches of the Campbell Diagram as Figure 2.8 states. The frequency grows with the

rotating speed if the Monorotor is in FW because the gyroscopic effect force vector has the same

sense of the rotor stiffness force vector , causing virtual structure stiffening and therefore a natural

frequency growth. The opposite happens when the Monorotor is in BW.

It is also observable that the instability generator terms ( ) and ( ), are acting parallel

and in the sense of motion of the orbit, when it is in FW and BW respectively. This confirms what was

said concerning the non-conservative effects of these terms.

On the same Figure A.4 are represented in (c) and (d) an elliptical rotor orbit shape. The

general conclusions taken before to the circular shape orbits, regarding the sense of action and the

stability effects of each term, can be taken again into account for these orbits, excepting that this time

the forces acting on the rotor are harder to visualize, since the displacement vector is no longer

perpendicular to the instantaneous velocity vector , with the exception, of course, of the points where

vector overlap the semi-major and semi-minor axis of the ellipse. Consequently, force vector ( )

stops acting exclusively perpendicularly to the instantaneous velocity vector. This fact is particularly

relevant since it explains why, when the orbits are elliptical, and hence when the rotor is asymmetric, a

co-rotational excitation force such as that caused by a mass unbalance present in a rotor’s disk, can

induce both FW and BW vibrations, while it induces in a symmetric rotor (circular orbit) only FW

vibrations.

Let us recall the expression (2.1.2.19), which describes the radial interaction force vector

{ } on a bearing of the rotor. The incremental work done by the forces acting on the bearing is

simply calculated multiplying each component of this force vector by the respective incremental

displacements. The energy-per-cycle exchange on the bearings can then be written as:

93

∮( ) ⏟

∫ ( )

⁄

(A.2.15)

Let us consider a generic harmonic rotor orbit, so that an expression for the energy exchange

per harmonic cycle can be obtained. For this purpose, and will be defined as:

( )

( ) (A.2.16)

Substituting expressions, (A.2.16) and (2.1.2.19) in (A.2.15), solving the resultant integral, and

recalling that the incremental work done by the symmetric part of the stiffness matrix and the skew-

symmetric part of the damping matrix is null, it yields:

[ (

) ( )]

(A.2.17)

The parameters , , , , and are all positive, furthermore the phase angles

difference is positive for forward whirl ( ( ) ), and negative for backward whirl rotor motion

( ( ) ). Typically the dependent term of the equation (A.2.17), grows faster than the

dependent term, with increasing rotating speed. This means, that for corotational whirl (forward whirl),

the two effects can exactly balance at a given rotating speed . This speed is dubbed the instability

threshold speed, since an increment in the rotating speed would lead the rotor into an energy addition

state or self-excitation.

It becomes therefore clear, from the analysis of the previous equation that unstable

phenomena can only derive from forward whirl state. Moreover, it is worth noting, that instability will

occur first in the lowest-frequency mode, since it is characterized by the largest traversed orbit areas,

and consequently by higher products.

With these equations the fundamentals of energy are concluded. In the first sections of

chapter 2.1 the kinetic and strain energies resultant of the rotor operation were presented, those

equation together with the ones presented just now, that account for the dissipation self-excitation

energy with origin on the bearings, supply the necessary tools for a basic study on rotor’s energy

efficiency.

Instability Critical Speed Determination and Parameters that Affect its Performance

Despite the energy-per-cycle exchange in the bearing capability to detect the stability

threshold speed, it isn’t the usual method used to this end. In fact the eigenvalues of the dynamic

system are the most common tool for instability prediction. This is so because usually in computational

methods there’s an inherent error associated with the extraction of the eigenvectors [7]. This leads to

slightly different orbits and consequently the variables in equations (A.2.16) are affected by this error

contaminating the speed threshold computation through equation (A.2.17).

As it will be shown, the methods based on the determination of the eigenvalues in free motion

conditions are more simple and precise since they don’t involve so much intermediate calculations as

the energy based method of equation (A.2.17).

On a generic damped rotor in free motion, the solutions of the eigenvalue/eigenvector problem

are given as described by the equations (2.1.3.1-2) whose eigenvalues are in the form:

√ (A.2.18)

94

whose variables meanings were already explained.

Introducing equation (A.2.18) in the solution equations (2.1.3.1-2) it results:

(

√ )

( )

(

√ )

( ) (A.2.19)

where the second terms ( ) represent oscillatory motion and the first terms (

√ )

, represent

the amplitude decay or increase over time, depending if its exponent is negative or positive

respectively.

The Table A.1 summarizes how will the structural system behave with the change of the

values of and .

Table A.1 – source: [7] - edited

and Signal Modal Motion

Zero damped, steady state sinusoidal motion

Underdamped, sinusoidal, exponential decay

Negatively damped, sinusoidal, exponential growth

Rigid body mode

Overdamped, nonoscillatory, exponential decay

Statically unstable, nonoscillatory, exponential growth

As the table shows, a dynamic system will enter into instability whenever the value of is

negative, since it changes makes the real part of the eigenvalue positive.

So for a simple damped rotor, one can compute the eigenvalues for several rotating speeds

until the real part becomes real. This method besides giving the critical speed for instability would also

give the value of the natural frequency at which an unstable state is verified.

Another method present in the literature [4] is the Routh-Hurwitz criterion. In order to make this

method clear, let us consider again equation (2.1.3.4) for a Multirotor. This equation can be generically

described as the following polynomial equation:

(A.2.20)

where are the coefficients of the characteristic equation.

The Routh-Hurwitz method defines a matrix with the coefficients of the characteristic

equation, which is:

[

]

(A.2.21)

with the coefficients of equation (A.2.20), (A.2.21) becomes:

95

[

] (A.2.22)

from which are deductible the Routh-Hurwitz coefficients for Multirotor system:

[ ]

[ ] |

|

[ ]

[ ] (A.2.23)

Comparing (2.1.3.4) with (A.2.20), the coefficients give:

( )

( )

( )

( )

(( ) ( )( )

),

(A.2.24)

Substituting the expressions of (A.2.24) into the Routh-Hurwitz coefficients whose calculation

is expressed in (A.2.23), it yields:

( )

( )( )

( )

[

( )( )

]

( )

(A.2.25)

Routh and Hurwitz have demonstrated that the system is unstable whenever one of the Routh-

Hurwitz coefficients is negative [4].

Note that if the couple terms of the bearings’ stiffness are equal to zero, the Multirotor is stable

for any value of , which confirms the statements made after the deduction of equation (A.2.17).

Appendix 3 – Deviation Between Finite Elements and Rayleigh-Ritz Results

The objective of this appendix is to suggest an explanation for the fact of the natural

frequencies obtained with the finite element model in the example stated on chapter 4.1, are lower

than the ones obtained through the Rayleigh-Ritz analytic method,.

Let us begin by explaining the deviation in the natural frequencies. As it was shown in Chapter

2.1.2, a sine shape function was used to describe the rotor’s first bending mode and to build its

Rayleigh-Ritz model. This shape function is displayed in equation (2.1.2.1) and it is mainly used to

96

describe the first bending mode of a simple supported beam. For this function the deformed shape of

the rotor would have its maximum value at the middle point of the shaft.

In rotors with complex geometries, where, for instance, the disks and elastic bearings are

deviated from the shaft’s middle point, and an additional outer shaft is considered the first mode shape

can be slightly different from that described by the shape function in equation (2.1.2.1).

Let us watch closely the first bending vibration mode shape of a symmetric Multirotor such as

that used in chapter 4.1.4 at rest. For this purpose we will use a finite element model and the ANSYS®

toolset, and the refined mesh presented on chapter 4.1.2.

The model’s nodes along with the modal shape normalized to the mass matrix are presented

in the Figure A.5. while the total nodal displacement is presented in Table A.2 for each node number.

Figure A.5 – First bending mode shape of the Multirotor

Disk 1 is located on node 11 and disk 2 on node 36.

Table A.2 – Nodal displacements

Node nr. 1 2 3 4 5 6 7

Displ. (m) 0 0.025083 0.049917 0.074252 0.097839 0.12043 0.14178

Node nr. 8 9 10 11 12 13 14

Displ. (m) 0.16165 0.17978 0.19595 0.2099 0.22057 0.22825 0.23304

Node nr. 15 16 17 18 19 20 21

Displ. (m) 0.23507 0.23444 0.23127 0.22569 0.21781 0.20776 0.19567

Node nr. 22 23 24 25 26 27 28

Displ. (m) 0.1816 0.1658 0.14847 0.12979 0.10994 0.089121 0.067524

Node nr. 29 30 31 32 33 34 35

Displ. (m) 0.045342 0.022769 0 0 0.024573 0.049145 0.073714

Node nr. 36 37 38 39 40

Displ. (m) 0.098279 0.12267 0.14703 0.17136 0.19567

In Figure A.5 it is visible that the middle point of the inner shaft corresponds to node 16, but

the displacement Table A.2 shows that the maximum displacement is verified in node 15. This slight

difference in the mode’s shape can contribute decisively for the disparities in the computation of

natural frequencies of the rotor with each of the used methods.

Appendix 4 – Deviation Between the Natural Frequencies Obtained via Direct Method and the Pseudo-modal Method

Here it is intended to explain the reason way some additional asymptotes appear in the

Response diagrams where the finite element method was employed and to explain way some critical

Inner shaft

Outer shaft

Springs

97

resonance speeds in the response diagrams are slightly deviated from the predicted critical speeds in

the Campbell diagrams.

This is actually related with the methods used in ANSYS® to determine both the Campbell

and the Response diagrams. As it was said in chapter 4.1 the Campbell diagrams were computed

using a pseudo-modal method and the response diagram was obtained by performing a full harmonic

analysis which computes the structure’s modes through the direct method and considers all the

structure’s modes in its construction.

To show this, the response diagram shown in Figure 4.5 is again considered and reproduced

in Figure A.6.

Figure A.6

Now the resultant Campbell diagram for the correspondent situation calculated with the direct

method instead of the pseudo-modal method is displayed in Figure A.7. The relevant critical speeds

are in the annexed table A.3.

Figure A.7

98

Table A.3

Outer shaft synchronous critical speeds

(rpm)

Direct Method Pseudo-modal Error Direct/Pseudo-modal methods (%)

1-FW 2-BW 2-FW 1-FW 2-BW 1-FW 2-BW

1906 2009 5018 1908 2012 0.105 0.149

As it is visible in Figure A.7, the anti-resonance observed in the Response Diagram in Figure

A.6 is related with the proximity of the critical speed of the second bending vibration mode FW.

Additionally note that the natural frequencies obtained with the direct method have slightly

lower values than that obtained with the pseudo-modal method. This difference is accentuated with

growing rotating speed.

Appendix 5 – Used Beam Element’s Matrices

The Figure A.8 shows the three degree-of-freedom per node beam element implemented in

the Matlab® code used on the application chapter 4.2.1. As it is also visible in the mentioned figure,

the beam element has two nodes.

Figure A.8 – Beam Element and its Degrees of Freedom

The element’s dofs correspond to the following nomenclature:

[ ]

(A.5.1)

This element is characterized by the following mass and stiffness:

[ ]

[

]

(A.5.2)

99

[ ]

[

⁄

⁄

⁄

⁄

]

(A.5.3)

Appendix 6 – Brief Fundaments About Rotor Transient Motion

In transient motions of rotors the rotating speed becomes a function of time described, so the angular

velocity in equation (2.1.1.2) becomes:

( ) (A.6.1)

The disk’s kinetic energy becomes:

( )

(

)

( )

(A.6.2)

And the shaft’s kinetic energy is now given by:

∫ ( )

∫ ( )

∫

(A.6.3)

If the Lagrange equation is applied with the considered displacement vector (3.1.1.2) in

chapter 3.1.1, to the disk’s and the shaft’s kinetic energy expressions (A.6.2-3), it results.

(

)

(

)

( ) (A.6.4)

where is the mass matrix in equation (3.1.2.1) and the gyroscopic effect matrix also in

(3.1.2.1) but divided by . ( ) correspond to the matrices with the same nomenclature

presented on chapter 3.1.2 and is the matrix (3.1.2.9) divided by .

Matrices and are two additional terms present only in transient motion defined as:

[

] (A.6.5)

100

[ ]

(A.6.6)

In what concerns the mass unbalance, its kinetic energy expression for transient motions is

given by:

( ) (A.6.7)

whose resultant force components are thus obtained:

(

)

[

] (A.6.8)

To describe the evolution of the rotating speed with time, a linear law was chosen,

characterized by the following expression:

( ) (A.6.9)

in which and are given by:

where , , , are initial conditions of the transient problem and denote respectively the initial and

final rotating speeds and time instants of the transient motion.

Finally it is worth defining the following expressions:

(A.6.10)

( )

(A.6.11)

with:

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