Thesis manuel barrientos

Course of M.Sc. in Mechanical Engineering

EVALUATION OF OPERATIONAL MODAL ANALYSIS FOR ROTATING MACHINERY Thesis in collaboration with SKF INDUSTRIE S.p.A.

Tutor: Prof. Alessandro Fasana Co-Tutor: Eng. Angelico Approsio Prof. Sergio Diaz Candidate: Manuel A. Barrientos D.

October 2014

ACKNOWLEDGMENT

I would like to thank my Tutor, Prof. Alessandro Fasana for his expert advice, suggestions and guidance he has provided during this project. I have been lucky to have a friendly tutor who cared about my work and who responded to my questions so promptly.

I must express my gratitude to SKF Industries S.p.A, especially to Angelico Approsio from Industrial Drives Segment who has been also my guidance and support during the development of this thesis, my gratitude for his kindness and for trust in me in order to develop of this activity in SKF. Also, I want to include Fabrizio Mandrile from Engineering Consultancy Services, Franco Porzio from Field Maintenance Services and Tommaso Gribodo from Reliability Engineering, for all the technical assistance during the development of this activity. My gratitude to all the people in SKF Solution Factory in Turin for the friendly treatment during these months.

Also, I would like to thank Politecnico di Torino and Universidad Sim贸n Bolivar because they have selected me to participate in this Double Degree program, and especially to Prof. Sergio Diaz as local responsible for the revision of this work.

Finally, I would like thank God for my family, for all the support and sacrifice in order to obtain my degree in Italy, especially in these difficult times in Venezuela; without their help my participation in this project would not have been possible.

2

CONTENTS ABSTRACT ............................................................................................................................................ 5 INTRODUCTION .................................................................................................................................. 6 CHAPTER I: THEORETICAL BACKGROUNG ............................................................................. 8 1.1

MODAL ANALYSIS ............................................................................................................. 8

1.1.1

FREE UNDAMPED CONDITION ................................................................................. 8

1.1.2

FREE DAMPED CONDITION ....................................................................................... 9

1.1.3

HARMONIC EXCITATION ......................................................................................... 10

1.2

RANDOM PROCESSES ..................................................................................................... 11

1.3

MODAL PARAMETERS EXTRACTION........................................................................ 15

1.3.1

COMPLEX MODE INDICATOR FUNCTION ........................................................... 15

1.3.2

MULTIVARIATE MODE INDICATOR FUNCTION ................................................. 16

1.3.3

PEAK PICKING ............................................................................................................ 18

1.3.4

ALIAS-FREE POLYREFERENCE METHOD ............................................................. 19

1.3.5

COMPLEX EXPONENTIAL METHOD ...................................................................... 20

1.3.6

Z-POLYNOMIAL METHOD ........................................................................................ 22

1.3.7

STABILITY DIAGRAM ............................................................................................... 23

1.4

OPERATIONAL DEFLECTION SHAPES ...................................................................... 23

1.4.1

ODS DEFINITION ........................................................................................................ 24

1.4.2

MEASUREMENTS IMPLEMENTED FOR ODS ........................................................ 25

1.4.3

THE ODS LIMITATION ............................................................................................... 27

1.5

OPERATIONAL MODAL ANALYSIS ............................................................................. 28

1.5.1

OMA DEFINITION ....................................................................................................... 28

1.5.2

MEASUREMENTS IMPLEMENTED FOR OMA....................................................... 29

1.5.3

MODAL IDENTIFICATION TECHNIQUES .............................................................. 29

1.5.3

SEPARATION OF HARMONIC COMPONENTS ...................................................... 31

CHAPTER II: METHODOLOGY ..................................................................................................... 34 2.1

OBJECTIVES ...................................................................................................................... 34

2.2

DESCRIPTION OF THE PILOT CASE: A PAPER MILL GEARBOX ....................... 34

2.3

INSTRUMENTATION AND MEASUREMENT SETUP ............................................... 37

2.3.1

BUMP TEST ACQUISITION ....................................................................................... 40

2.3.2

RUN-UP TEST ACQUISITON ..................................................................................... 40

2.3.3

ODS ACQUISITION ..................................................................................................... 40 3

2.4

PROCESSING TECHNIQUES FOR CLASSICAL MODAL TESTS ........................... 41

2.5

OPERATIONAL MODAL ANALYSIS PROCEDURE .................................................. 42

2.5.1

SIGNAL PROCESSING AND ODS GENERATION................................................... 42

2.5.2 SELECTION OF MODAL INDICATOR FUNCTIONS AND CURVE FITTING ALGORITHMS.............................................................................................................................. 44 2.5.3

EVALUATION OF CURVE FITTING TECHNIQUES ............................................... 45

2.5.4

HARMONIC IDENTIFICATION ................................................................................. 45

CHAPTER III: OMA RESULTS AND ANALYSIS......................................................................... 46 3.1 CLASSICAL TESTS: BUMP AND RUN-UP TEST .............................................................. 46 3.2 OPERATIONAL MODAL ANALYSIS................................................................................... 50 3.2.1

SIGNAL PROCESSING AND ODS GENERATION................................................... 50

3.2.2 SELECTION OF MODAL INDICATOR FUNCTIONS AND CURVE FITTING ALGORITMS ................................................................................................................................ 55 3.2.3

EVALUATION OF CURVE FITTING TECHNIQUES ............................................... 66

3.2.4

HARMONIC IDENTIFICATION ................................................................................. 73

CHAPTER IV: FINITE ELEMENT MODAL ANALYSIS. ........................................................... 79 4.1

GEOMETRY ........................................................................................................................ 79

4.2

MESHING ............................................................................................................................ 81

CHAPTER V: FEM VS OMA COMPARISON. ............................................................................... 83 CONCLUSIONS .................................................................................................................................. 85 RECOMENDATIONS ......................................................................................................................... 86 APPENDIX A. Harmonics due to Mechanical Components in a Rotating Machinery ........................ 87 APPENDIX B. Pictures of the dismounted gearbox. ............................................................................ 88 APPENDIX C. Comparison between signal of Channel 1 and Channel 2. .......................................... 89 APPENDIX D. Matlab Codes. .............................................................................................................. 90 BIBLIOGRAPHY ................................................................................................................................ 94

4

ABSTRACT This thesis discusses the Evaluation of Operational Modal Analysis for the extraction of modal parameters in rotating machinery, analyzing a real industrial case: a paper mill parallel gearbox.

First, a theoretical revision of Classical Modal Analysis, Operational Deflection Shapes and Modal Analysis of Output-Only measurement were presented. Also, we discussed some Modal Indicator Functions (like CMIF and MMIF) and FRF-based curve fitting techniques (Alias Free Polynomial, Z Polynomial and Complex Exponential Method) for extraction of modal frequencies, modal shapes and modal damping ratios.

Then, a comparison between Classical Modal Tests and OMA results was done, and we found that OMA is able to identify a shorter number of modal frequencies (the excited modes at operative condition); nevertheless the harmonic components affect the reliability of the curve fitting methods. In consequence, it is important to implement post processing techniques for the separation of harmonics from real modes.

Next, we compared different output-only modal parameter extractors, and we reached an acceptable performance using single-reference techniques in comparison with the multireference methods, in special Alias Free and CE Method, that gave us larger number of real modes. Then, we have analyzed the effect of the minimum number of stable poles for the identification of real data using the stabilization diagram; at the end we identified a number that provides reliable data. A third analysis was the splitting of the frequency range in order to determine the relationship between performance and data length; our results suggest that larger frequency length provides a shorter number of real modes with a shorter number of false modes (harmonics), while the length splitting calculates a larger number of modes with lower accuracy (increase of spurious frequencies).

After the previous analysis, we have analyzed the performance of the most popular harmonic identification techniques: Short Time Fourier Transformation, Singular Value Decomposition and Probability Density Function. The results showed that PDF approach seems to be the best indicator of harmonic components, while STFT and SVD require a previous knowledge of the fundamental frequencies acting on the system.

Finally, a first attempt has been done in order to validate Operational Modal Analysis results with Finite Element Modelling. Unfortunately, the complexity of the pilot case and the absent of technical information affected the convergence between numerical modelling and testing.

5

INTRODUCTION Vibration in rotating machinery is very common in the industry. A machine is always vibrating, but without the correct control it can bring several problems. Some of the issues related to an uncontrolled vibrational level are:        

Decrease in productivity and generation of unsteady processes.

High levels of noise.

Lower efficiency.

Shorter machine life.

Reduction in safety conditions.

High maintenance and repair cost for the effect of the vibration.

Increase in energy consumption.

And many others.

The diagnosis, analysis and solution of any vibrational problem require the data acquisition of the actual behavior of the machine. Nowadays, the most popular and widely used method for the study of a vibrating system is the Experimental Modal Analysis (EMA), in which a controlled force is exciting a mechanical system through a hammer or a shaker, and the response of the system is obtained with transducers; then input-output methods are applied in order to extract the modal parameters from a Frequency Response Function (FRF). Also, it is becoming popular in the industry the application of Finite Element Modelling (FEM) as an alternative of numerical simulation for the vibration analysis, but nevertheless, it is important to validate the model with testing results. In the industry, the acquisition of the modal parameters is also useful for design verification, dynamic modification and optimization of a machine.

There are many problems arising from the application of EMA in the industry. First, this technique requires the stop of the analyzed rotating machinery, and in the real field this can generate an enormous loss of production and waste of time if the machine is part of a plant or a continuous process; for example: a hydraulic turbine in a hydroelectric power plant, a pump in a beverage plant or an electric motor supplying energy to a hospital. And also, sometimes the price of a critical equipment is so high, that a company does not have the availability of providing a twin machine for experimental purposes.

Within EMA, other techniques are implemented for the identification of natural frequencies like the Bump and Run Up–Coast down Test, but again these tools change the operative condition of a rotating machine: Bump Test implies the stop of the machine and the measurement of the response for identification of modal frequencies, and the Run Up-Coast down Test consist in the variation of the operating speed for extraction of modal frequencies.

Recently, an output-only approach has been developed and implemented with the objective of the acquisition of modal parameters in real operative conditions: Operational Modal Analysis (OMA).This technique seems very attractive because allows the acquisitions of modal parameters with the measurement of only output data with the machine running under

6

operational condition. The benefits derived from the application of this method instead of using EMA are evident: saving of time and saving of production losses.

However, it is a fact that the process of identification of modes in rotating machinery is not so easy because the presence of harmonics in the frequency domain of the response, and also because the forces from real condition do not guarantee the excitation of all the modes. So, it is important to evaluate the performance of OMA. This thesis has been developed thanks to the interest of the SKF Industrial Drives Service Segment in the evaluation of Operational Modal Analysis. Actually, SKF is leader in services related to vibration analysis, electrical analysis and condition monitoring of End User assets or Original Equipment Manufacturers; also the company has developed sophisticated data acquisition systems like SKF Microlog Series (a portable data acquisition system with 2/4 channels) and SKF Multilog On-line system IMx-P (a system with 16 channels), with their corresponding software for data processing and analysis. Inside the vibrational analysis, SKF implements the Experimental Modal Analysis, but as we discussed before, they have identified the unavailability and unwillingness of the customer for the stop of the rotating machinery, because of the impact on production.

Also, the thesis is done in order to obtain the degree of M.Sc. in Mechanical Engineering at Politecnico di Torino (Italy) and the B.Sc. in Mechanical Engineering at Universidad Sim贸n Bolivar (Venezuela), thanks to a Bilateral Agreement between both institutions.

The objectives of this thesis is the evaluation of Operational Modal Analysis in the extraction of modal parameters in a real industrial case: a rotating machinery; and then the comparison with different experimental techniques for modal extraction.

In the first chapter of this project, we will look over the theory of Classical Modal Analysis, Operational Deflection Shapes (ODS) and OMA. Then, we will provide a general description of the machine being analyzed: a gearbox operating in a Paper Mill company; including a presentation of the processes that happen inside this type of plant, and explaining the importance of the application of Operational Modal Analysis instead of the experimental approach.

Subsequently, in the second chapter, we will describe the methodology applied for the evaluation of Operational Modal Analysis; and in the third chapter we will present the analysis and results. Finally, the last two chapters describe the implementation of a Finite Element Model for the validation of testing results.

7

CHAPTER I: THEORETICAL BACKGROUNG It is impossible to understand Operational Modal Analysis without the base of Modal Analysis (deterministic and stochastic load), Experimental Modal Analysis and Modal Parameters Extraction; and for this reason it is necessary to recap the basic concepts and fundamentals in the two following sections. However, a deeper study of this subjects can be consulted in the bibliography. Then, the last part is focused in the theory of ODS and OMA. 1.1

MODAL ANALYSIS

Modal analysis is a technique implemented for the description of the vibration of a linear system, and it is widely used for design, control and analysis of a machines. According to modal theory, three parameters are required for describing a vibrating system: natural frequencies, modal shapes and damping ratios.

We can study a typical system with “nâ€? number masses (or n degree of freedoms), connected with springs and dampers, and with external forces applied in each mass, as we can see in next figure. Due to the fact that mass, stiffness and damping are constant, we can say that the system is time invariant. đ??š1 (đ?‘Ą)

đ??š2 (đ?‘Ą) đ?‘˜2

đ?‘˜1 đ?‘š1 đ??ś1

đ?‘˜đ?‘›

đ?‘˜3 ‌

đ?‘š2 đ??ś2

đ?‘Ľ1 , đ?‘ĽĚ‡ 1 , đ?‘ĽĚˆ 1

đ??šđ?‘› (đ?‘Ą)

đ??ś3 đ?‘Ľ2 , đ?‘ĽĚ‡ 2 , đ?‘ĽĚˆ 2

đ?‘˜đ?‘›+1 đ?‘šđ?‘› đ??śđ?‘›+1

đ??śđ?‘› đ?‘Ľđ?‘› , đ?‘ĽĚ‡ đ?‘› , đ?‘ĽĚˆ đ?‘›

Figure 1.1. Representation of a multi-degree of freedom system.

We know that, the behavior of this system is expressed as a second order differential system of equation, as follows: [đ?‘€]{đ?‘ĽĚˆ (đ?‘Ą)} + [đ??ś]{đ?‘ĽĚ‡ (đ?‘Ą)} + [đ??ž]{đ?‘Ľ(đ?‘Ą)} = {đ??š(đ?‘Ą)}

(1.1)

In which [đ?‘€] is the mass matrix of the system, [đ??ś] the damping matrix, and [đ??ž] the stiffness matrix, all of them symmetric and positive defined. Also {đ??š(đ?‘Ą)} is the vector of the external forces. It is important to assume a steady state condition.

1.1.1

FREE UNDAMPED CONDITION

8

First, we can start analyzing the free un-damped condition, so the equation 1.1 is simplified into the equation 1.1.2. [đ?‘€]{đ?‘ĽĚˆ (đ?‘Ą)} + [đ??ž]{đ?‘Ľ(đ?‘Ą)} = {0} (1.2) It has been proven that, the general solution of this problem is: {đ?‘Ľ(đ?‘Ą)} = {đ?œ“} ∙ cos(đ?œ” ∙ đ?‘Ą − đ?œ‘)

(1.3)

Here đ?œ” is the frequency of the response and đ?œ‘ is the phase. Substituting equation (1.3) in equation (1.2), and simplifying, we obtain: (−đ?œ”2 ∙ [đ?‘€]{đ?œ“} + [đ??ž]{đ?œ“}) ∙ cos(đ?œ” ∙ đ?‘Ą − đ?œ‘) = {0}

(1.4)

Due to the fact that it is expected to have a non-null solution, the cosine function can be removed from last expression, and we get: (−đ?œ”2 ∙ [đ?‘€] + [đ??ž]){đ?œ“} = {0}

(1.5)

Here, again we are looking for a non-null solution, so it is identified the eigenproblem equation, that is the equation 1.6. |−đ?œ”2 ∙ [đ?‘€] + [đ??ž]| = 0

(1.6)

Previous expression is the characteristic equation of the system, and for real situations the “nâ€? roots are positive, containing the natural frequencies đ?œ”1 , đ?œ”2 , ‌ , đ?œ”đ?‘› . Once that the natural frequencies are calculated we can substitute in the equation (1.5), so we obtain the modal vectors of the system:{đ?œ“}1 , {đ?œ“}2 , ‌ , {đ?œ“}đ?‘› . It is important to clarify that those values depends just in the mass matrix, the stiffness matrix and the boundary conditions of the system. Finding the natural frequencies and modal shapes for an undamped system is very important because allow us to determine the movement of all the system. We obtain for our case: {đ?‘Ľ(đ?‘Ą)} = ∑đ?‘›đ?‘&#x;=1 đ?‘Žđ?‘&#x; {đ?œ“}đ?‘&#x; ∙ cos(đ?œ”đ?‘&#x; ∙ đ?‘Ą − đ?œ‘đ?‘&#x; )

(1.7)

Where the magnitude đ?‘Žđ?‘&#x; and the phase đ?œ‘đ?‘&#x; are functions of the initial conditions (position and velocity). Also it is important to remember that the modal vectors are M-orthogonal and Korthogonal, and this allow us to define the modal mass and modal stiffness as follow: {đ?œ“}đ?‘&#x; đ?‘‡ [đ?‘€]{đ?œ“}đ?‘&#x; = đ?‘šđ?‘&#x; {đ?œ“}đ?‘&#x; đ?‘‡ [đ??ž]{đ?œ“}đ?‘&#x; = đ?‘˜đ?‘&#x;

đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1,2, ‌ , đ?‘›

đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1,2, ‌ , đ?‘›

(1.8)

(1.9)

Thanks to the modal masses, we can do the normalization of the modal vectors, so we obtain the new set of vectors: {đ?œ™}đ?‘&#x; =

1.1.2

1 √đ?‘šđ?‘&#x;

∙ {đ?œ“}đ?‘&#x;

đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1,2, ‌ , đ?‘›

(1.10)

FREE DAMPED CONDITION

9

The results in equation 1.8 and 1.9 allow us to go deeper and obtain the response of the system in free damping condition. Unfortunately the damping matrix is not always orthogonal with the modal vectors, so we can study the system assuming the condition 1.11, where the damping matrix is a linear combination of mass and stiffness matrix: [đ??ś] = đ?›ź ∙ [đ?‘€] + đ?›˝ ∙ [đ??ž]

(1.11)

Under this hypothesis, it is easy to identify the solution: {đ?‘Ľ(đ?‘Ą)} = ∑đ?‘›đ?‘&#x;=1 đ?‘’ −đ?œ‰đ?‘&#x; ∙đ?œ”đ?‘&#x; ∙đ?‘Ą ∙ (đ??´đ?‘&#x; {đ?œ“}đ?‘&#x; cos(đ?œ”đ?‘‘đ?‘&#x; ∙ đ?‘Ą) + đ??ľđ?‘&#x; {đ?œ“}đ?‘&#x; sin(đ?œ”đ?‘‘đ?‘&#x; ∙ đ?‘Ą))

(1.12)

In the last result, the constants đ??´đ?‘&#x; and đ??ľđ?‘&#x; depend on the initial conditions, đ?œ‰đ?‘&#x; is the modal damping ratio and đ?œ”đ?‘‘đ?‘&#x; is the damped frequency.

1.1.3

HARMONIC EXCITATION

Our original equation 1.1 has the contribution of forces; so, if we assume harmonic forces (as equation 1.13), then it will be possible to obtain the receptance between the response of d.o.f. “iâ€? and the forced d.o.f. “eâ€?. {đ?‘“(đ?‘Ą)} = {đ?‘“đ?‘œ } sin đ?œ” ∙ đ?‘Ą

(1.13)

The receptance is a function of the frequency, and represents the ratio between a response đ?‘Ľđ?‘œđ?‘– and a force input đ?‘“đ?‘œđ?‘’ (see equation 1.14), and it is also known as the dynamic compliance, frequency response function (FRF) or transfer function. đ?‘Ľ

đ?›źđ?‘–đ?‘’ (đ?œ”) = đ?‘“đ?‘œđ?‘– = ∑đ?‘›đ?‘&#x;=1 (đ?œ” đ?‘œđ?‘’

đ?‘&#x;

đ?œ™đ?‘’đ?‘&#x; đ?œ™đ?‘–đ?‘&#x; 2 −đ?œ”2 )+2đ?‘—∙đ?œ”∙đ?œ”

đ?‘&#x; ∙đ?œ‰đ?‘&#x;

(1.14)

Last definition allow us to define a receptance matrix (or FRF matrix), with dimension “n response d.o.f. and m forced d.o.f.â€? đ?›ź11 đ?›ź [đ?›ź(đ?œ”)] = [ 21 â‹Ž đ?›źđ?‘š1

�12 �22 ⋎ ��1

‌ �1� ‌ �2� ] ⋹ ⋎ ‌ ���

(1.15)

Equation 1.14 is true if we can obtain � and � for the condition 1.11. In another case, we can apply another methods for the solution of our system of equation, for example the Duncan Method [1], and in that case the expression of the equation of motion can be transformed using the next variable: {�} = {

{�} } (1.16) {�̇ }

So, the equation becomes: [[0]

[đ?‘š]]{đ?‘ŚĚ‡ } + [[đ?‘?] [0]]{đ?‘ŚĚ‡ } + [[đ?‘˜]

[0]]{đ?‘Ś} = {đ?‘“(đ?‘Ą)} (1.17)

10

And then, with the use of the auxiliary equation 1.18, we obtain the final system of equation 1.19. [[đ?‘š]

[

[0]]{đ?‘ŚĚ‡ } − [[0]

[đ?‘?] [đ?‘š] [đ?‘˜] ] {đ?‘ŚĚ‡ } + [ [đ?‘š] [0] [0]

[đ?‘š]]{đ?‘Ś} = {0} (1.18)

[0] đ?‘“(đ?‘Ą) ] {đ?‘Ś} = { } (1.19) −[đ?‘š] 0

As before we look for an eigenvalue problem (taking the homogenous approach), so it is possible to simplify the last expression as equation 1.20. [đ??´]{đ?‘ŚĚ‡ } + [đ??ľ]{đ?‘Ś} = {0} (1.20) And then, the eigenvalues and eigenvectors of the system are: đ?‘ = đ?‘ đ?‘&#x; đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1, ‌ ,2đ?‘› (1.21) {đ?œƒ} = {đ?œƒ}đ?‘&#x; đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1, ‌ ,2đ?‘› (1.22) Like the non-proportional case, we can prove that, thanks to the A-orthogonally and Borthogonally of eigenvectors we get: {đ?œƒ}đ?‘&#x; đ?‘‡ [đ??´]{đ?œƒ}đ?‘&#x; = đ?‘Žđ?‘&#x;

đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1,2, ‌ ,2đ?‘›

(1.23)

{đ?œƒ}đ?‘&#x; đ?‘‡ [đ??ľ]{đ?œƒ}đ?‘&#x; = đ?‘?đ?‘&#x;

đ?‘“đ?‘œđ?‘&#x; đ?‘&#x; = 1,2, ‌ ,2đ?‘›

(1.24)

Also that we can use both coefficients in order to obtain the eigenvalues, like the equation 1.25. đ?‘Ž

đ?‘ đ?‘&#x; = − đ?‘?đ?‘&#x; (1.25) đ?‘&#x;

The last equation allow us to obtain the expression for the receptance: đ?œ™ đ?œ™

đ?‘–đ?‘&#x; đ?‘’đ?‘&#x; đ?›źđ?‘–đ?‘’ (đ?œ”) = ∑đ?‘›đ?‘&#x;=1(đ?‘—∙đ?œ”−đ?‘† + đ?‘&#x;

đ?œ™đ?‘–đ?‘&#x; ∗ đ?œ™đ?‘’đ?‘&#x; ∗ đ?‘—∙đ?œ”−đ?‘†đ?‘&#x;

) (1.26)

Here, đ?œ™đ?‘&#x; ∗ is the conjugated of đ?œ™đ?‘&#x; , and đ?œ™đ?‘&#x; is the normalized expression of đ?œƒđ?‘&#x; , obtained with the division with the square root of đ?‘Žđ?‘&#x; .

1.2

RANDOM PROCESSES

The theory studied so far is related to deterministic processes; but in Operational Modal Analysis, most of the cases are associated with stochastic signals (see figure 2), especially because the excitation of the machine is unknown, so it is not correct to assume a harmonic force, for instance.

11

Figure 1.2. Example of a Random Signal.

It is important to clarify that next theory is assuming a Linear Invariant System. Then, assuming that the force acting in our system is like the previous plot; the signal đ?‘Ľ(đ?‘Ą) can be spitted in different samples đ?‘Ľđ?‘˜ (đ?‘Ą), each data with a length of time đ?‘‡. Then, the average of each sample is: 1

�

đ?œ‡đ?‘Ľ (đ?‘˜) = lim đ?‘‡ âˆŤ0 đ?‘Ľđ?‘˜ (đ?‘Ą) ∙ đ?‘‘đ?‘Ą (1.27) đ?‘‡â†’∞

Also, it is possible to define the autocorrelation function as in the equation 1.28.

1

�

đ?‘…đ?‘Ľđ?‘Ľ (đ?‘Ą, đ?œ?) = lim đ?‘‡ âˆŤ0 (đ?‘Ľđ?‘˜ (đ?‘Ą)đ?‘Ľđ?‘˜ (đ?‘Ą + đ?œ?)) ∙ đ?‘‘đ?‘Ą (1.28) đ?‘‡â†’∞

The correlation is the degree of similitude between two signals. For a specific sample đ?‘Ľđ?‘˜ (đ?‘Ą) , if the average and the autocorrelation is the same for each time t, then we will have a stationary process. And, if the similitude remains for each sample, then we will present an ergodic process.

In general, an ergodic process produce stationary process, but not all stationary process generate an ergodic process. We will continue the analysis of a random process assuming ergodic process, obtaining the response đ?‘Śđ?‘˜ (đ?‘Ą) for each input sample đ?‘Ľđ?‘˜ (đ?‘Ą); and with this we can define the auto-correlations (eq. 1.29 and 1.30) and cross-correlation (eq. 1.31) between force and response, but using the expected functions instead of the integral: đ?‘…đ?‘Ľđ?‘Ľ (đ?œ?) = đ??¸[đ?‘Ľ(đ?‘Ą)đ?‘Ľ(đ?‘Ą + đ?œ?)] (1.29) đ?‘…đ?‘Śđ?‘Ś (đ?‘Ą, đ?œ?) = đ??¸[đ?‘Ś(đ?‘Ą)đ?‘Ś(đ?‘Ą + đ?œ?)] (1.30) đ?‘…đ?‘Ľđ?‘Ś (đ?‘Ą, đ?œ?) = đ??¸[đ?‘Ľ(đ?‘Ą)đ?‘Ś(đ?‘Ą + đ?œ?)] (1.31) Without getting deeper into mathematical aspects, it has been proved that that the autocorrelation function is even but the cross-correlation is not.

Once that we have the expressions of the correlations, we are able to define the Auto and Cross Power Spectral Density (PSD) applying the Fourier Transform to the correlation function; and these functions are important because allow us to estimate the Frequency Response Function 12

(FRF) in a random vibrational problem. Auto-PSD of the input signal is defined in equation 1.32, while for the output signal we have formulation 1.33. Then, the cross-PSD between both signals are expressed in equation 1.34. +∞

đ?‘†đ?‘Ľđ?‘Ľ (đ?œ”) = âˆŤâˆ’âˆž đ?‘…đ?‘Ľđ?‘Ľ (đ?‘Ą, đ?œ?)đ?‘’ −đ?‘–đ?œ”đ?œ? ∙ đ?‘‘đ?œ” (1.32) +∞

đ?‘†đ?‘Śđ?‘Ś (đ?œ”) = âˆŤâˆ’âˆž đ?‘…đ?‘Śđ?‘Ś (đ?‘Ą, đ?œ?)đ?‘’ −đ?‘–đ?œ”đ?œ? ∙ đ?‘‘đ?œ” (1.33) +∞

đ?‘†đ?‘Ľđ?‘Ś (đ?œ”) = âˆŤâˆ’âˆž đ?‘…đ?‘Ľđ?‘Ś (đ?‘Ą, đ?œ?)đ?‘’ −đ?‘–đ?œ”đ?œ? ∙ đ?‘‘đ?œ” (1.34) It is important to remember two properties of the power spectral density that are useful in order to define the FRF: đ?‘†đ?‘Ľđ?‘Ľ (đ?œ”) = đ?‘†đ?‘Ľđ?‘Ľ (−đ?œ”) (1.35) đ?‘†đ?‘Ľđ?‘Ś (đ?œ”)∗ = đ?‘†đ?‘Ľđ?‘Ś (−đ?œ”) (1.36) Then, remembering the linear invariant system assumption, in a black box model representation like figure 1.3:

Figure 1.3. SISO Black Box Model Having defined the signal x(t) as the input, and y(t) as the response of the system, we can write the first approximation of the FRF function : đ?‘†đ?‘Ľđ?‘Ś (đ?œ”)

đ??ť1 (đ?œ”) = đ?‘†

đ?‘Ľđ?‘Ľ (đ?œ”)

(1.37)

And a second relation: đ?‘†đ?‘Śđ?‘Ś (đ?œ”)

đ??ť2 (đ?œ”) = đ?‘†

đ?‘Śđ?‘Ľ (đ?œ”)

(1.38)

In order to determine the degree of similitude between both approaches, the coherence function has been defined (equation 1.39). The result is a value between 0 and 1: if the coherence is equal or close to 1 (ideal system), then both relations will be identical. đ??ť (đ?œ”)

đ?›žđ?‘Ľđ?‘Ś 2 (đ?œ”) = đ??ť1 (đ?œ”) (1.39) 2

In the literature there are modifications of the FRF expressions taking into consideration the presence of noise in the input and/or output signal. Also there are analytic and empirical

13

expressions for the definition of an equivalent FRF in function of both approaches, but the most simples formulations are: đ??ťđ?‘Ł =

đ??ť1 +đ??ť2 2

(1.40)

Or

đ??ťđ?‘Ł = √đ??ť1 đ??ť2 (1.41) In the case of a multi-input multi-output system, as in figure 1.4, with M inputs and L outputs, we can rewrite the expression of the receptance function using matrices.

Figure 1.4. MIMO Black Box Model.

Here, it is possible to write the matrices of auto-PSD of the inputs (eq. 1.42) and cross power spectral density between each signal in the system (eq. 1.43). ��1�1 � [��� ] = [ �1�2 ⋎ ��1��

��2�1 ��2�2 ⋎ ��2��

‌ ����1 ⋯ ����2 ] (1.42) ⋎ ⋹ ⋯ �����

đ?‘†đ?‘Ľ1đ?‘Ś1 đ?‘† [đ?‘†đ?‘Ľđ?‘Ś ] = đ?‘Ľ1đ?‘Ś2 â‹Ž [đ?‘†đ?‘Ľ1đ?‘Śđ??ż

đ?‘†đ?‘Ľ2đ?‘Ś1 đ?‘†đ?‘Ľ2đ?‘Ś2 â‹Ž đ?‘†đ?‘Ľ2đ?‘Śđ??ż

‌ đ?‘†đ?‘Ľđ?‘€đ?‘Ś1 â‹Ż đ?‘†đ?‘Ľđ?‘€đ?‘Ś2 (1.43) â‹Ž â‹ą â‹Ż đ?‘†đ?‘Ľđ?‘€đ?‘Śđ??ż ]

Finally, the receptance matrix can be calculated with the expression: [đ??ť(đ?œ”)] = [đ?‘†đ?‘Ľđ?‘Ś ][đ?‘†đ?‘Ľđ?‘Ľ ]−1 (1.44)

14

1.3

MODAL PARAMETERS EXTRACTION

The practical implementation of modal analysis theory is limited because most of the calculation require the knowledge of the mass matrix, the stiffness matrix and the damping matrix.

Practically it is impossible to obtain a good estimation of those matrices from a real machine.

Modern engineering applies the use of Finite Element Modelling (FEM) for the calculation of those matrices; however, it is important to evaluate the similitude between the model and the real machine. For that reason, there is a branch of the Modal Analysis called Experimental Modal Analysis, which dedicates to the analysis of real systems.

Inside Experimental Modal Analysis, there are many types of techniques for the extraction of the modal parameters of mechanical systems: modal frequencies đ?œ”đ?‘&#x; , modal shapes {đ?œ“}đ?‘&#x; and modal damping đ?œ‰đ?‘&#x; . In the literature, we have methods that are applied in the time domain signal, and methods that only analyze the frequency domain. Also, some techniques work in presence of SDOF systems and others just evaluate MDOF systems.

In this section, we will introduce the functions for modal identification and parameter extractions that are applied in this work through the software ME´scopeVES (Vibrant Technology, Inc.), unfortunately it is impossible to cover and evaluate all the kind methods in this thesis, so we will limit the theory.

1.3.1

COMPLEX MODE INDICATOR FUNCTION

The Complex Mode Indicator Function (CMIF) is a method that indicates the position of peaks in a FRF, taking also into account the presence of repeated roots or closed modes [4]. An example is presented in the next figure, in which CMIF algorithm was applied to a FRF function using the software ME´scopeVES, in a range between 100Hz and 700 Hz.

Figure 1.5. Example of CMIF in a FRF. 15

In the article of A. Phillips related to CMIF method [5], we observe that the parameter calculation algorithm consists in two stages: first the identification of modal vectors and then the extraction of the modal frequencies and scalar factors; in consequence, CMIF is useful just in the first stage.

This method is defined applying the Singular Value Decomposition (SVD) in each frequency line of the FRF matrix (like in the equation 1.45); remembering that the elements of the FRF matrix can be obtained using equations 1.26 and 1.44. [đ??ť(đ?œ”)] = [đ?‘ˆ][đ?‘†][đ?‘‰]đ??ť (1.45) In the last expression the matrix [S] is the singular value matrix, and [ ]đ??ť is the Hermitian operator. If the number of effective modes is less than the number of inputs for the FRF, then the matrix [U] will approximate the mode shapes and the matrix [V] contains the participation factors. [5] In the next stage, CMIF generates a plot of the singular value in function of the frequency (with a discrete evaluation). The value obtained from each frequency is proportional to the 1 ratio (đ?‘—∙đ?œ”−đ?‘† , therefore the plot presents higher peaks if the discrete frequency is close to a đ?‘&#x;)

modal frequency. However, not all the peaks identified by CMIF method are indication of modes, because the presence of noise or leakage. For instance, in Operational Modal Analysis, there is an important presence of harmonics in the response signal, and CMIF is unable to differentiate mode peaks between harmonic peaks, but this problem will be analyzed later.

Another disadvantage of the CMIF method, presented in the bibliography, is that in the plotting process of the singular value magnitude, we can have a similarity between two eigenvalues (cross eigenvalue), generating a peak in the curve. This problem can be avoided comparing the vectors of each eigenvalue.

Finally, as an alternative procedure of the CMIF techniques, we can substitute the FRF matrix in the equation 1.45 with the Real part of the matrix if our measurement is displacement/force or acceleration/force; in another case (velocity/force) we can use the imaginary part. This improvement is done in order to generate real values. [6]

1.3.2 MULTIVARIATE MODE INDICATOR FUNCTION The Multivariate Mode Indicator Function (MMIF), like the CMIF, is a method used for the indication of modes in a FRF matrix. Also, it takes into account the presence of closely modes and repeated modes, but MMIF has the same disadvantages of the CMIF method. In figure 1.6 there is an example of the application of MMIF to a Frequency Response Function.

16

Figure 1.5. Example of MMIF in a FRF. The MMIF requires the decomposition of the response in real and imaginary part (frequency domain): {đ?‘‹} = đ?‘…đ?‘’({đ?‘‹}) + đ?‘– ∙ đ??źđ?‘š({đ?‘‹}) (1.46)

Then, substituting last expression into the equation of motion: đ?‘…đ?‘’({đ?‘‹}) + đ?‘– ∙ đ??źđ?‘š({đ?‘‹}) = (đ?‘…đ?‘’({đ??ť}) + đ?‘– ∙ đ??źđ?‘š({đ??ť})) ∙ đ??š (1.47)

Taking as reference the work of R. Williams, J. Crowley and H. Bold: “If a normal mode can be excited at a particular frequency, a force vector F must be found such that the real part Re(X) of the response vector is as small as possible compared to the total responseâ€? [7]. Consequently, we can express the norm of the real response (equation 1.48) and the total response (equation 1.49). ‖đ?‘…đ?‘’({đ?‘‹})‖2 = đ?‘…đ?‘’({đ?‘‹})đ?‘‡ ∙ [đ?‘€] ∙ đ?‘…đ?‘’({đ?‘‹}) (1.48) ‖đ?‘…đ?‘’({đ?‘‹}) + đ?‘– ∙ đ??źđ?‘š({đ?‘‹})‖2 = đ?‘…đ?‘’({đ?‘‹})đ?‘‡ ∙ [đ?‘€] ∙ đ?‘…đ?‘’({đ?‘‹}) + đ??źđ?‘š({đ?‘‹})đ?‘‡ ∙ [đ?‘€] ∙ đ??źđ?‘š({đ?‘‹}) (1.49) Finally, MMIF is defined as the eigenvalue solution of the equation resulting from the previous statement (minimization equation): đ?‘…đ?‘’([đ??ť]đ?‘‡ )đ?‘…đ?‘’([đ??ť])đ??š = đ?›ź(đ?‘…đ?‘’([đ??ť]đ?‘‡ )đ?‘…đ?‘’([đ??ť]) + đ??źđ?‘š([đ??ť]đ?‘‡ )đ??źđ?‘š([đ??ť]))đ??š (1.50) MMIF calculates each eigenvalue for each frequency line, and the frequency where the curves have a minimum is likely to be a natural frequency. [8]

17

1.3.3 PEAK PICKING This method is implemented for the extraction of the modal shapes from the frequency response matrix obtained in a hammer or a shaker test. It is used for single degree of freedom system and operates in the frequency domain.

Peak picking works with the equation 1.14, in which the known parameters are the modal frequencies and the modal damping that can be obtained with other parameter extraction methods; for example, the modal frequencies can be obtained with the MMIF and the CMIF method, while the damping can be obtained through the logarithmic decrement method or the half-power method [1].

Let´s assume that the element �11 of the receptance matrix is as follows:

Figure 1.6. Component �11 of the receptance matrix.

Now, in each peak of the curve we can assume that the bigger contribution of the function is due to the modal frequency located at that peak, so, we can do the approximation: đ?›źđ?‘–đ?‘’ (đ?œ”) ≈ (đ?œ”

đ?‘&#x;

đ?œ™đ?‘’đ?‘&#x; đ?œ™đ?‘–đ?‘&#x; 2 −đ?œ” 2 )+2đ?‘—∙đ?œ”∙đ?œ”

đ?‘&#x; ∙đ?œ‰đ?‘&#x;

(1.51)

Then, taking r=1 (it means, the first modal frequency), we obtain: đ?›ź11 (đ?œ”) ≈ (đ?œ”

1

đ?œ™11 đ?œ™11 2 −đ?œ”2 )+2đ?‘—∙đ?œ”∙đ?œ”

1 ∙đ?œ‰1

(1.52)

This approximation allows the evaluation at the first mode, so we can obtain the value of the first component of the first modal shape: đ?œ™11 . Then, we proceed again with the evaluation but now in the function đ?›ź21 , presented in the figure 1.7.

18

Figure 1.7. Component �21 of the receptance matrix.

The evaluation at the first mode: đ?›ź21 (đ?œ”) ≈ (đ?œ”

1

đ?œ™21 đ?œ™11 2 −đ?œ”2 )+2đ?‘—∙đ?œ”∙đ?œ”

2 ∙đ?œ‰1

(1.53)

Again, the expression in the denominator is known, and the only unknown variable is the component đ?œ™21 in the numerator, but this can be obtained because the component đ?œ™11 was calculated in the previous step. We can repeat the last procedure with each function đ?›źđ?‘–1 , and at the end we will be able to construct the first normalized modal shape{đ?œ™}1 .

After having found the first vector, we start with the equation 1.52, but now taking r=2: đ?›ź11 (đ?œ”) ≈ (đ?œ”

2

đ?œ™12 đ?œ™12 2 −đ?œ”2 )+2đ?‘—∙đ?œ”∙đ?œ”

2 ∙đ?œ‰2

(1.54)

In the previous equation we obtain the component đ?œ™12 ; and if we use the function đ?›ź21 we will obtain đ?œ™22 . So, we can see that this analysis can be done for each variable r, until the maximum number of mode of our system, and we will be able to obtain all the mode shapes.

In peak peaking method, we do not need to compute all the elements of the receptance matrix, we just need one row or one column; in consequence this method is suitable for the hammer with “m� inputs and one output; or a shaker test with one input and “m� outputs.

1.3.4 ALIAS-FREE POLYREFERENCE METHOD The Alias Free polyreference method (AFPoly), is a new technique for modal parameter extraction introduced in 2006 by ATA Engineering, Inc [9]. It is a frequency domain technique

19

that extracts the modal frequencies, modal shapes and the modal damping from a FRF column or row. The base of the method is the orthogonal polynomial rational fraction method.

Through rational fraction polynomial method, we can substitute the equation 1.26 in the next way: đ?›źđ?‘–đ?‘’ (đ?œ”) =

đ?‘&#x; ∑2đ?‘›âˆ’1 đ?‘&#x;=0 đ?›˝đ?‘&#x; (đ?‘—∙đ?œ”đ?‘&#x; ) đ?‘&#x; ∑2đ?‘› đ?‘&#x;=0 đ?›źđ?‘&#x; (đ?‘—∙đ?œ”đ?‘&#x; )

(1.55)

Then, the implementation of orthogonal polynomial in the classical method, provides an advantage because improves the numerical stability of the method in an entire frequency range that can contain a big amount of modes, so it is used with stability diagram (we will introduce this tool later). Also, a problem during curve fitting is that the residual effects of out-of-band modes (modes outside the analyzed frequency range) can generate aliasing, but the AFPoly method keeps the residual effects of out-of-band modes outside, avoiding aliasing (here the reason of the name).

Despite the recent creation of this method, AF Poly seems to provide reliable results in wide range of data [10]; and it has been quickly included in the most important softwares for EMA and OMA, like ME´scopeVES, LMS and Brßel and Kjaer softwares.

1.3.5 COMPLEX EXPONENTIAL METHOD The Complex Exponential Method (CE Method) is a popular time domain technique that works with MDOF systems. The outputs are the natural frequencies and damping of a non-proportional damping system. Equation 1.26 can be reformulated in: đ??´đ?‘&#x;

đ?‘–đ?‘’ đ?›źđ?‘–đ?‘’ (đ?œ”) = ∑2đ?‘› đ?‘&#x;=1 đ?‘—∙đ?œ”−đ?‘† (1.56) đ?‘&#x;

Also, đ?‘†đ?‘&#x; can be expressed as: đ?‘†đ?‘&#x; = −đ?œ‰đ?‘&#x; ∙ đ?œ”đ?‘&#x; + đ?‘— ∙ đ?œ”đ?‘&#x; √1 − đ?œ‰đ?‘&#x; 2 (1.57) Because the CE Method operates in time domain, it uses the Impulse Response Function, which can be obtained applying the inverse of Fourier Transform to the equation 1.56. This function is: đ?‘†đ?‘&#x;∙ đ?‘Ą â„Žđ?‘–đ?‘’ (đ?‘Ą) = ∑2đ?‘› (1.58) đ?‘&#x;=1 đ??´đ?‘&#x;đ?‘–đ?‘’ ∙ đ?‘’

Due to the discretization of the signal into samples, we can evaluate the response using a constant step, like in the equation 1.59. đ?‘Ą = đ?‘š ∙ ∆đ?‘Ą (1.59) So, substituting into the expression 1.58, we obtain: đ?‘†đ?‘&#x;∙ đ?‘šâˆ™âˆ†đ?‘Ą â„Žđ?‘š = â„Žđ?‘–đ?‘’ (đ?‘š ∙ ∆đ?‘Ą) = ∑2đ?‘› (1.60) đ?‘&#x;=1 đ??´đ?‘&#x; ∙ đ?‘’

20

For simplicity, it is better to rewrite the exponential part of previous equation into the form: đ?‘‰đ?‘&#x; đ?‘š = đ?‘’ đ?‘†đ?‘&#x;∙ đ?‘šâˆ™âˆ†đ?‘Ą (1.61) In consequence, the new expression of the impulse response function is: đ?‘š â„Žđ?‘š = ∑2đ?‘› (1.62) đ?‘&#x;=1 đ??´đ?‘&#x; ∙ đ?‘‰đ?‘&#x;

CE method applies some techniques in order to obtain each đ?‘‰đ?‘&#x; ; values that are essential for the calculation of the modal and damping frequencies.

Equation 1.62 allow us to evaluate each sample in the signal, with a maximum number of points equal to “pâ€?. Then, we will obtain the system of equation 1.63, in which the right part is completely unknown. â„Ž0 đ?‘‰1 0 1 â„Ž { 1 } = đ?‘‰1 â‹Ž â‹Ž â„Žđ?‘? [đ?‘‰ đ?‘?

1

đ?‘‰2 0 đ?‘‰21 â‹Ž đ?‘‰2 đ?‘?

â‹Ż đ?‘‰2đ?‘› 0 đ??´1 â‹Ż đ?‘‰2đ?‘›1 { đ??´2 } (1.63) â‹Ž â‹ą â‹Ž đ?‘? đ??´ â‹Ż đ?‘‰2đ?‘› ] đ?‘ƒ

Then, we multiply each row of the system by a constant coefficient β, obtaining: â„Ž0 ∙ đ?›˝0 đ?‘‰1 0 1 â„Ž ∙đ?›˝ { 1 1 } = đ?‘‰1 â‹Ž â‹Ž â„Žđ?‘? ∙ đ?›˝đ?‘? [đ?‘‰ đ?‘?

1

đ?‘‰2 0 đ?‘‰21 â‹Ž đ?‘‰2 đ?‘?

â‹Ż đ?‘‰2đ?‘› 0 đ??´1 ∙ đ?›˝0 â‹Ż đ?‘‰2đ?‘›1 { đ??´2 ∙ đ?›˝1 } (1.64) â‹Ž â‹ą â‹Ž đ?‘? đ??´ â‹Ż đ?‘‰2đ?‘› ] đ?‘ƒ ∙ đ?›˝đ?‘?

If we sum each component, then the resulting equation will appear: ∑đ?‘ƒđ?‘ž=0 â„Žđ?‘ž ∙ đ?›˝đ?‘ž = đ??´1 ∑đ?‘ƒđ?‘ž=0 đ?›˝đ?‘ž ∙ đ?‘‰1 đ?‘ž + đ??´2 ∑đ?‘ƒđ?‘ž=0 đ?›˝đ?‘ž ∙ đ?‘‰2 đ?‘ž + â‹Ż + đ??´2đ?‘› ∑đ?‘ƒđ?‘ž=0 đ?›˝đ?‘ž ∙ đ?‘‰2đ?‘› đ?‘ž (1.65) So, it is of interest the identification of the polynomial 1.66, which provides the annulation of each sum in the right side (p should be equal to 2n). đ?›˝0 + đ?›˝1 ∙ đ?‘‰ 1 + â‹Ż + đ?›˝đ?‘ƒ ∙ đ?‘‰ đ?‘ƒ = 0 (1.66) In consequence, substituting in the equation 1.65 we obtain: đ?›˝0 ∙ â„Ž0 + đ?›˝1 ∙ â„Ž1 + â‹Ż + đ?›˝đ?‘ƒ ∙ â„Žđ?‘? = 0 (1.67) Now, we start again with the procedure like in the equation 1.63, but from sampling 1 until p+1. Multiplying for the coefficients β it remains: đ?‘‰11 â„Ž1 ∙ đ?›˝0 2 â„Ž ∙đ?›˝ { 2 1 } = đ?‘‰1 â‹Ž â‹Ž đ?‘?+1 â„Žđ?‘?+1 ∙ đ?›˝đ?‘? [đ?‘‰ 1

đ?‘‰21 đ?‘‰2 2 â‹Ž đ?‘?+1 đ?‘‰2

đ??´1 ∙ đ?›˝0 â‹Ż đ?‘‰2đ?‘›1 2 â‹Ż đ?‘‰2đ?‘› { đ??´2 ∙ đ?›˝1 } (1.68) â‹Ž â‹ą â‹Ž đ?‘?+1 â‹Ż đ?‘‰2đ?‘› ] đ??´đ?‘ƒ ∙ đ?›˝đ?‘?

Equation 1.69 presents the sum of each equation in the previous system; but again, we can use the polynomial of expression 1.66 in order to annul the sums in right size. ∑đ?‘ƒđ?‘ž=0 â„Žđ?‘ž+1 ∙ đ?›˝đ?‘ž = đ??´1 ∙ đ?‘‰1 ∑đ?‘ƒđ?‘ž=0 đ?›˝đ?‘ž ∙ đ?‘‰1 đ?‘ž + â‹Ż + đ??´2đ?‘› ∙ đ?‘‰2đ?‘› ∑đ?‘ƒđ?‘ž=0 đ?›˝đ?‘ž ∙ đ?‘‰2đ?‘› đ?‘ž (1.69) 21

The result of the substitution with the equation 1.67 is: đ?›˝0 ∙ â„Ž1 + đ?›˝1 ∙ â„Ž2 + â‹Ż + đ?›˝đ?‘ƒ ∙ â„Žđ?‘?+1 = 0 (1.70) The idea of the Complex Exponential Method is that we can repeat each calculation and use the property in the equation 1.66, until arrive into a maximum sample M (with M≼2n). At the end we will be able to construct the system of equation 1.71; remembering that the matrix of impulse responses is known. â„Ž0 â„Ž [ 1 â‹Ž â„Žđ?‘€

â„Ž1 â„Ž2 â‹Ž â„Žđ?‘€+1

â‹Ż â„Ž2đ?‘› đ?›˝0 0 đ?›˝1 â‹Ż â„Ž2đ?‘›+1 ] { } = {0} (1.71) â‹Ž â‹Ž â‹ą â‹Ž 0 â‹Ż â„Žđ?‘€+2đ?‘› đ?›˝2đ?‘›

We can solve the last expression and obtain the β coefficients; for example, using the least square method (in that case the method is called the Least Squares Complex Exponential Method). Then, it will be easy to obtain the roots from equation 1.66: �1 , �2 , ‌ , �2� .

Finally, the modal frequencies are calculated with the expression: ln(đ?‘‰đ?‘&#x; )

đ?œ”đ?‘&#x; = |

∆đ?‘Ą

| (1.72)

And the damping ratios are obtained with: đ?œ‰đ?‘&#x; = −

ln(đ?‘‰đ?‘&#x; ) ) ∆đ?‘Ą

đ?‘…đ?‘’(

đ?œ”đ?‘&#x;

(1.73)

1.3.6 Z-POLYNOMIAL METHOD Like the Alias-Free Method, the Z-Polynomial method (Z-Poly) is a modification of the Rational Fraction Orthogonal Polynomial, and it is implemented for the extraction of modal frequencies and damping ratios, without the modal shapes. This method uses the Z transform to change the discrete time signal of the response into a complex frequency domain signal, providing more stable solutions. Also, Z-Polynomial is good for the evaluation of a big number of modes in a wide frequency range.

For the computational point of view, the curve fitting with AF-Polynomial and Z-Polynomial methods in a FRF follows the next algorithm [11]:  



First, a set of FRF expressions like the one in equation 1.55 is solved for the numerator and denominator coefficients.

As we know, the denominator have information of modal frequencies and damping ratios: so, the coefficients are inserted into a root solver, estimating the mentioned modal parameters.

Finally, for the curve fitting of the FRF, the numerator is used in a partial fraction expansion, containing the complex residue for each mode.

22

1.3.7 STABILITY DIAGRAM The stability diagram is a popular post-processing technique in modal parameter extraction and curve fitting of any kind of response signal: FRF, cross spectra density function, auto spectra density function, etc. It should be applied for the identification of the number of real physical modes in the frequency signal.

This technique will change the order of modes in the FRF expression 1.14, within a range of order one until a maximum desired order, and it will present all the poles of the equation at the same time. A simple representation of this plot is presented in figure 1.8.

Figure 1.8. Example of a Stability diagram.

In the last picture we see that some poles (green color) are converging to a vertical solution, then we say that those frequencies are stable and represent a real physical modes. Otherwise, the blue poles are computational results; it means that are solutions of the FRF equation, but do not represent a physical solution. So, the application of Stability diagram is important ot understand if the solutions of any modal parameter extraction method is real or not. Usually, in the most popular EMA and OMA softwares, it is possible to interact with the maximum and minimum allowable damping ratio, or frequency range.

1.4

OPERATIONAL DEFLECTION SHAPES

In this part we will discuss about Operational Deflection Shapes, because the acquisition procedure is the same for Operational Modal Analysis; also because ODS was used in the past for rough estimation of modal parameters in real condition.

23

1.4.1 ODS DEFINITION Operational Deflection Shapes, better known as ODS, is a computational technique implemented for the visualization of the deflections in a machine or mechanical structure under real operative condition. This visualization is very useful for understanding the machine dynamics and motion, the interaction between the different components in the machine, or the machine-environment interaction, also to identify conditions with high level of vibration.

Usually, ODS is implemented in constant or cyclostationary conditions: velocities, loads, pressure, power, time, etc.; also, this technique can be used in linear or non-linear systems. The main advantage of ODS is that does not require the stop of the machine or transportation/modification of the boundary condition.

It is possible to define a simple algorithm that is used in the ODS softwares: [12] 1. First, the 3D geometry of the machine is inserted.

2. Then, the user is able to define some specific points (nodes) over the geometry. Those nodes will be measured in real condition.

3. After the data acquisition, the user is able to import the data into the software.

4. ODS software generates an interpolation, using as reference the measured points. The interpolation estimates the movement (displacement, velocity or acceleration) of the unknown points.

5. Finally, the visualization is done.

A typical result of a ODS visualization is presented in figure 1.9: We can observe the deflections of a bench test for three different frequencies.

(a)

(b) 24

(c)

Figure 1.9. ODS shapes of a test bench at a) 30 Hz, b) 40 Hz and c) 50 Hz.

1.4.2 MEASUREMENTS IMPLEMENTED FOR ODS There are two different ways to acquire the data in the ODS process [13], but the main idea is always to obtain magnitude and phase of the response of a set of points in the machine: 

Simultaneous multi-channel acquisition, in which each desired point is measured for example with accelerometers. This technique is faster, but very demanding, because depending on the dimensions of our machine it is needed to use large amount of accelerometers. The advantage is that we obtain the correct magnitude and phase of each point related to the others.



Measurement set acquisition o Using one channel: obtaining the data point by point. This method is simpler but takes much time, and is just applicable if we have repeatable operations, otherwise the values of each DOF will not have a correct value relative to one another. o Using reference transducers: we require two or more acquisition channels. Here we are using one accelerometer (or more) fixed as a reference, and another one is moved to each desired position and direction. This type of measurement is time consuming and requires a steady state operation.

Now, independently of the type of methodology implemented for the acquisition, we can classify the type of data that can be measured/calculated for the ODS visualization: 

Time domain In this case the output of the measurement is the response of each point in the time domain, without any post processing technique. Here, the ODS will be also a time domain function.



FFT of the Response Fast Fourier Transform (FFT) of the time response that can be displacement, velocity or acceleration. It represents a function of the frequency, and is calculated through the Discrete Fourier Transform (DFT): 25

đ?‘›

1

−đ?‘–∙đ?‘˜âˆ™2đ?œ‹âˆ™ đ?‘ {đ?‘Œ}đ?‘˜ = ∙ ∑đ?‘ −1 đ?‘›=0 {đ?‘Ś}đ?‘› ∙ đ?‘’ đ?‘

k=0,¹1,¹2,‌ (1.74)

With N equal to the sampling value and đ?‘Śđ?‘› equal to the response at time "đ?‘› ∙ ∆đ?‘Ąâ€?, with constant time step. Then, FFT simplifies the calculation of DFT, using two properties: {đ?‘Œ}đ?‘ +đ?‘˜ = {đ?‘Œ}đ?‘˜

{đ?‘Œ}đ?‘ −đ??ž = {đ?‘Œ}đ?‘˜ ∗ (1.75)

And

In consequence, inserting the FFT of each point, the ODS will be a frequency domain function. 

Cross Power Spectra A two channel acquisition system allows us to preserve the exact phase between measurements. The Cross Power Spectra is computed with the expected value of the Fourier spectra of the roving response {đ?‘‹(đ?œ”)} multiplied by the conjugated of the Fourier spectra of the fixed reference response {đ?‘Œ(đ?œ”)}.

1

[đ?‘†đ?‘Ľđ?‘Ś (đ?œ”)] = lim đ?‘‡ đ??¸[{đ?‘‹(đ?œ”)} ∙ {đ?‘Œ ∗ (đ?œ”)}] (1.76) đ?‘‡â†’∞

The previous calculation is done for each node of the structure, and at the end we will have the information of one row (or one column) of the cross power spectra matrix. In case of a multi-reference measurement the number of the known rows (or columns) will match the number of reference channels.

In the last expression, the output vector {đ?‘‹(đ?œ”)} has a dimension equal to the number of measured response signals (đ?‘ đ?‘&#x;đ?‘’đ?‘ đ?‘? ), while the reference vector {đ?‘Œ(đ?œ”)} has a dimension equal to the number of reference channels (đ?‘ đ?‘&#x;đ?‘’đ?‘“ ). 

Transmissibility The transmissibility is the ratio between the cross power spectrum and the auto power spectrum of the response. đ?‘†đ?‘Ľđ?‘Ś (đ?œ”) đ??ťđ?‘Ľđ?‘Ś (đ?œ”) = (1.77) đ?‘†đ?‘Śđ?‘Ś (đ?œ”)

The advantage of using the Transmissibility for ODS is that ensures the phase matching between measurements. Nevertheless, resonances are not always presented in the transmissibility peaks, so they are present in flat regions. Last issue is a limitation for the use of transmissibility in Operational Modal Analysis. 

ODS FRF: ODSF FRF is calculated assuming steady state operations; generally is calculated taking a fixed reference, but it can be defined also for a multi-reference evaluation (like the cross spectra density matrix).

The definition of the ODS FRF is [14]: đ?‘‚đ??ˇđ?‘†đ??šđ?‘…đ??šđ?‘– = |đ?‘‹đ?‘– |đ?‘’ đ?‘—(âˆ?đ?‘Ľ đ?‘– −âˆ?đ?‘Ś đ?‘– ) (1.78)

26

Here |đ?‘‹| is the magnitude of the roving signal of the channel i, and đ?‘’ đ?‘—(âˆ?đ?‘Ľ −âˆ?đ?‘Ś ) is the phase difference between roving signal and the reference signal. Then, we can write the magnitude definition like in the equation 1.79. |đ?‘‹đ?‘– | = √đ?‘‹đ?‘– ∙ đ?‘‹ ∗ đ?‘–

(1.79)

And the phase difference can be written doing the next operations: đ?‘‹đ?‘– = đ??´ ∙ đ?‘’ đ?‘—(âˆ?đ?‘Ľ )đ?‘–

(1.80)

đ?‘Œđ?‘– = đ??ľ ∙ đ?‘’ đ?‘—(âˆ?đ?‘Ś )đ?‘–

(1.81)

Then, multiplying the response per the conjugated of the reference signal, we get: đ?‘‹đ?‘– ∙ đ?‘Œđ?‘– ∗ = đ??´ ∙ đ??ľ ∙ đ?‘’ đ?‘—(âˆ?đ?‘Ľ đ?‘– −âˆ?đ?‘Ś đ?‘–) (1.82) So, it is posible to obtain: đ?‘’ đ?‘—(âˆ?đ?‘Ľ đ?‘– −âˆ?đ?‘Ś đ?‘– ) =

đ?‘‹đ?‘– ∙đ?‘Œđ?‘– ∗ đ??´âˆ™đ??ľ

đ?‘‹ ∙đ?‘Œ ∗

= |đ?‘‹đ?‘– ∙đ?‘Œđ?‘– ∗| (1.83) đ?‘–

đ?‘–

Finally, substituting expressions 1.79 and 1.83 into the equation 1.78, we obtain: đ?‘‹ ∙đ?‘Œ ∗

đ?‘‚đ??ˇđ?‘†đ??šđ?‘…đ??š = √đ?‘‹đ?‘– ∙ đ?‘‹đ?‘– ∗ |đ?‘‹đ?‘– ∙đ?‘Œ ∗đ?‘– | đ?‘–

đ?‘–

(1.84)

Now, dividing by the period and applying limit of the period to infinite (ideally), the resulting expression is: ���

đ?‘‚đ??ˇđ?‘†đ??šđ?‘…đ??š(đ?œ”) = |đ?‘‹đ?‘– | ∙ |đ?‘†

đ?‘Ľđ?‘Ś |

(1.85)

At the end, ODS FRF has the correct magnitude of the roving point and the phase is the same as the cross power spectral density, so also it has the correct relative phase of each point. If we do the last calculation for each channel, then we will obtain a vector of deflections. In case of multi-reference signals, the ODSFRF should be calculated for each reference, so we will have a number of vectors equal to the number of references.

1.4.3 THE ODS LIMITATION The implementation of ODS technique has a very strong limitation: in the practice it is impossible to assume that all the peaks are evidence of modal frequencies due to the presence of harmonics . In figure 1.10 there is a comparison between the FRF and the ODSFRF; as we know the machine is stopped during the hammer test, then as a consequence the FRF peaks are just related to the excited modes. Indeed, in ODS the machine is operating, so we have many harmonic contributions in the response signal. In conclusion, practically it is not straightforward the identification of the modal frequencies and extraction of modal parameter from a ODS. 27

Figure 1.10. Comparison between FRF and ODS FRF So, an ODS shape is a superposition of excited modal shapes, random vibration and harmonics.

Most of the harmonics are associated with internal rotating parts of the machine, and a short summary of the effect of those components is presented is Appendix A. L. Hermans and H. Van der Auweraer [15] said that in the past the first approach used for the identification of modal frequencies in operative conditions was linked to the ODS technique.

The first step was the extraction of some assumed modal shapes, and then the comparison with some laboratory modal shapes results. Of course, this is not practical because we will not have always the availability of a rotating machinery for technical evaluation in a laboratory.

However, we can compare the deflection shapes with an analytical model (finite element model, for instance), but it is important that the model is as similar as possible to the real system.

1.5

OPERATIONAL MODAL ANALYSIS 1.5.1 OMA DEFINITION

Operational Modal Analysis (OMA) is a modern technique implemented for the extraction of modal parameters through the data acquisition of the response of a mechanical system in operational condition (output only measurement). The main requirement is the presence of broadband excitation in order to excite all the natural frequencies [15].

The implementation of OMA instead of EMA gives the following advantages:

28

   

The acquisition is simpler, because OMA does not need the measurement of input forces.

OMA implies a faster measurement.

EMA´s results differ significantly from the real conditions because the laboratory conditions do not have real loads or real boundary condition. Moreover OMA does need the modification of the operative condition.

Time, cost and production saving.

Like in ODS, in case that we have a white noise input, the identification of modal parameters will be easier because all the peaks are associated with a specific modal frequency. But again, the aforementioned situation is not common in rotating machinery due to the presence of harmonics; these harmonics will generate some errors during the modal parameter extraction, because the curve fitting techniques will not differentiate harmonics form true structural modes, but the description of some suggested techniques applied for the extraction of harmonics in OMA signal will be described later.

A simple workflow diagram of Operational Modal Analysis is illustrated in the following figure:

Data Acquisition

Signal Processing

Curve Fitting

Separation of Harmonics

Figure 1.11. Simplified workflow of OMA 1.5.2

MEASUREMENTS IMPLEMENTED FOR OMA

As we mentioned before, the steps applied for the acquisition of data for Operational Modal Analysis are the same of the ODS technique. However, in OMA, special considerations should be applied for the selection of the roving points and reference points in the structure: The roving locations have to be chosen such that they contain enough modal information. For this reason we cannot locate our transducers close to the base or a fixed locations. Also, we should select those points taking into account the presence of nodal points (like in the experimental modal analysis test). After the data acquisition, the next step is the implementation of the modal identification techniques.

1.5.3 MODAL IDENTIFICATION TECHNIQUES In Operational Modal Analysis there is a big number of modal identification techniques that have been developed for the extraction of modes in only-output problems. Companies like Brüel & Kjær and LMS have done a significant progress over the last decades in the development of 29

some OMA techniques. These companies, together with Vibrant Technology, Inc (USA) provide the most popular softwares for the application of OMA in the industry.

In general, we can divide OMA techniques depending in the domain of calculation: frequency and time domain. Also, most of these methods are modifications of typical EMA techniques like Peak Picking, CE Method, Z-Poly and AF-Poly (techniques presented before), in order to do curve fitting in only-output data [16]. Unfortunately, it is not possible to explain and apply each method in this work, so we will just mention some popular techniques in the following table: ([15],[17] and [18]).

Domain Frequency

Time

Technique Frequency Domain Decomposition (FDD) Enhanced Frequency Domain Decomposition (EFDD) Curve Fitting Frequency Domain Decomposition(CFDD) PolyMax Stochastic Subspace Identification (SSI) Natural Excitation Technique (NExT) Least Square Complex Exponential Method Eigensystem Realization Algorithm (ERA) Extended Ibrahim (EI)

Popular Software Provider B&K, SVIBS B&K, SVIBS B&K, SVIBS LMS B&K, SVIBS -

Table 1.1. Other popular OMA techniques in the market.

In the case of FRF-based (or IRF) curve fitting techniques in OMA, extraction should be applied using the cross-power spectra density function or ODSFRF, considering all the roving channels and the reference(s) channel(s). However, we might remember that the presence of harmonics will strongly affect the performance of those methods, so we will define some useful pre and post processing techniques later.

In equation 1.76 we have defined the matrix of the cross power spectral density between the roving response and the reference response. In case of OMA, it is possible to modally decompose the matrix in the form [15]: {đ?œ“đ?‘&#x; }{đ??żđ?‘&#x; }đ?‘‡

[đ?‘†đ?‘Ľđ?‘Ś (đ?œ”)]=∑đ?‘ đ?‘&#x;=1 (

đ?‘—∙đ?œ”−đ?‘†đ?‘&#x;

+

{đ?œ“đ?‘&#x; }∗ {đ??żđ?‘&#x; }đ??ť đ?‘—∙đ?œ”−đ?‘†đ?‘&#x;

∗

+

{đ??żđ?‘&#x; }{đ?œ“đ?‘&#x; }đ?‘‡ −đ?‘—∙đ?œ”−đ?‘†đ?‘&#x;

+

{đ??żđ?‘&#x; }∗ {đ?œ“đ?‘&#x; }đ??ť −đ?‘—∙đ?œ”−đ?‘†đ?‘&#x; ∗

) (1.86)

In the last expression, {đ??żđ?‘&#x; } is a reference factor vector for the rth mode with dimension đ?‘ đ?‘&#x;đ?‘’đ?‘“ .

Also, the previous modal decomposition can be applied to the ODSFRF matrix and auto-power spectra density matrix. Using this procedure, EMA-based modal parameter extraction techniques will operate under the definition 1.86 instead of the equation 1.26. Obviously, after the application of curve fitting, special attention must be dedicated to the identification of harmonics in the result.

30

1.5.3

SEPARATION OF HARMONIC COMPONENTS

As we discussed before, the main issue of Operational Modal Analysis is the identification of the harmonics in the spectra response. Several investigations and approaches have been developed in order to deal with this problem. The most popular techniques presented in the bibliography are: 

The implementation of the Stabilization Diagram, like in the EMA case, is used to distinguish spurious modes from physical modes [15]. Moreover, in modern OMA softwares, it is possible to set up tolerances for the identification of nonrealistic damping or low damping from modal frequencies [19]; this is because forced excitation can be considered as low damped vibration [20]. On the other hand, it is expected a better performance of this technique in the case of heavily damped structures.



The Singular Value Decomposition (SVD) plot is the base of another well know output-only modal identification techniques: FDD, CFDD and CFDD. This technique is used as a harmonic indicator, implementing the decomposition on the auto-spectra density function matrix for each frequency [20], as follows: [đ?‘†đ?‘Ľđ?‘Ľ ] = [đ?‘‰][đ?‘†][đ?‘‰]đ??ť (1.87) In which [V] are the singular vectors and [S] is the singular matrix. Then, the curve is done plotting the singular values for each frequency. In this case we will have a number of curves equal to the number of channels. In presence of harmonic components, narrow peaks will be found in multiple curves, and abrupt changes happen at those frequencies, but in the modal peaks the curves are smoother [19].



The Short Time Fourier Transformation (STFT) was presented by BrĂźel and Kjaer as a way to identify modes and harmonics, and “also disclosure information of the structural behavior of the test object and the natural of the excitationâ€? [20]. This technique consists in the division of the time history signal into smaller length, computing the FFT of each block, and then displaying the FFT autospectra in a 3D plot as a function of the rotational speed. The plot is an equivalent to the Run Up Test Plot but at constant speed, so the user will be able to see the behavior of the modes and harmonic peaks.



The Probability Density Function (PDF) seems to be an easy and attractive way for the identification of modes and harmonics from a frequency response plot (auto or crosspower spectra and ODSFRF). This method is based in the statistic theory that takes into account the random behavior of any peak in the frequency signal ([20] and [21]). The definition of the normalized probability density function is:

31

1

2 /2đ?œŽ 2

đ?‘“(đ?‘Ľ) = đ?œŽâˆš2đ?œ‹ đ?‘’ −(đ?‘Ľâˆ’đ?œ‡)

(1.88)

Where x is the stochastic variable, đ?œ‡ is the mean value and đ?œŽ is the standard deviation.

First step is the isolation of the peak, procedure that can be done with a band-pass filter, and subsequently, the calculation of the PDF must be done with the result. Then, the shape of the PDF will be different for two cases: in case of a modal peak, the PDF will present a Gaussian distribution like in figure 1.12a, but the harmonic component distribution will present two peaks, like in figure 1.12b.

(a)

(b) Figure 1.12. Probability Density Function of a) Structural Mode b) Harmonic Component



The Visual Mode Shape Comparison can be applied for the validation of the shapes obtained from a modal extraction technique or also in an ODS visualization. This methodology reveals whether a computed modal shape is from a true mode or not. As we know from the theory of classical modal analysis, in case of harmonic frequencies, the deflection patter will be a combination of the excited modes and the forces acting on the machine; however, in case of harmonics close to a modal peak, the deflection patter will be similar to the mode, so in this case the harmonic will be not identified.

32

It is not simple to have a previous knowledge of the real mode shapes in case of complex machines or complex boundary conditions, so in those cases a visual comparison is difficult. The main suggestion is to have an accurate Finite Element (FE) model and apply the Modal Assurance Criterion (MAC) between the analytical modal shapes and the calculated shapes. MAC calculation between two vectors is defined as: ‖{đ?‘‹}∙{đ?‘Œ}‖2

đ?‘€đ??´đ??ś({đ?‘‹}, {đ?‘Œ}) = ‖{đ?‘‹}‖∙‖{đ?‘Œ}‖ (1.89) MAC returns a number between 0 and 1 for each pair that is related to the degree of similitude between both vectors. A value close to 1 means a good correlation, while a value close to 0 means null similitude.

In the next graphic we present a 3D MAC comparison between the modal shapes obtained from two different curve fitting methods. The big values at the diagonal columns show the high similitude between both models.

Figure 1.13. Example of a 3D MAC.

33

CHAPTER II: METHODOLOGY

2.1

OBJECTIVES

After the discussion of the theory required for the application of Operational Modal Analysis, we proceed with the second chapter, relative to the description of the methodology applied to achieve the following objectives: 1. Evaluation of the applicability of the Operational Modal Analysis Technique in a real industrial case: a rotating machinery.

2. Comparison of the results between classical modal tests and OMA. 3. Evaluation of the performance and comparison of different methods applied for Modal Parameter Extraction in Output-Only measurement.

4. Analysis of the harmonic identification capacity of different techniques.

5. Validation of OMA results implementing a Finite Element Analysis.

In order to satisfy these objectives, we will apply Classical Modal Tests and Operational Modal Analysis in a pilot case. Next lines describe the Tested Machine, the applied acquisition and post processing, and the methodology useful for analysis.

2.2

DESCRIPTION OF THE PILOT CASE: A PAPER MILL GEARBOX

For the application of OMA in a real industrial case, the SKF Industrial Drives Services had the availability of a SKF customer asset in Lucca (Italy): The gearbox which drives the Yankee Roll of a paper mill plant.

The paper mill machine is a continuously operating plant with typical dimensions of 10 meters wide, 20 meters high and 200 meters long [22]. The processes inside this plant are characterized by high speeds (1000-1500 m/min) and temperatures up to 200 ÂşC; operating 24 hours, and it has a reduced number of stops per year for maintenance purposes. For this reason, the application of Operational Modal Analysis instead of classical test is very important for this kind of processes, in which a reduced time of stop represents a tremendous loss of production (up to 5000-8000 â‚Ź/hr). Figure 2.1 shows a real paper mill machine.

Generally, this machine is divided into the following sections: a forming section, a press section, a drying section, a coating section, a calendar and a reeler section. In the drying section, an important number of drying cylinders are required in linear configuration for the reduction of the paper humidity applying pressure, changing the humidity of the paper from 50-65% to 510%; but in the case of tissue paper the drying level is low, therefore the drying cylinders are substituted by a big cylinder: the Yankee roll. A simplified scheme of the tissue machine with the Yankee roll is presented in figure 2.2.

34

Figure 2.1. A Paper Mill Machine

Figure 2.2. Tissue Machine scheme Moreover, the Yankee cylinder operates at low speed and it can have a diameter of 4-9 m.

Generally, the design steam pressure is 11 bar and the maximum steam temperature can reach 200ยบC. In our case, the analyzed gearbox transforms the power received by two synchronized electrical motors into a higher level of torque in order to rotate the Yankee Roll. Both motors are located on a granite base to isolate the vibration, while the motor shafts are coupled through two flexible couplings. Moreover, the shafts of both motors are protected by a special cover for security reasons but they do not have any kind of contact with the gearbox, and do not represent a boundary condition. The gearbox is connected to the ground by a cylindrical joint, allowing for rotation like a pendulous. In another side, the output hollow shaft of the gearbox is coupled with 35

the Yankee cylinder; this shaft passes through the machine and at the end is covered by a steam press fixed to the gearbox. Finally, an auxiliary drive is even connected the other side of one input shaft, aimed to allow slow rotations during maintenance operations. In the following picture we present the scheme of the mechanical system, for a better understanding:

Figure 2.3. Scheme of the Mechanical System Figure 2.4 shows the real gearbox operating in the tissue plant, and in figure 2.5 we present the connection of the gearbox and the ground through a cylindrical joint.

YANKEE CILYNDER

GEARBOX

AUXILIARY DRIVE

STEAM PRESS

INPUT OF THE ELETRIC MOTOR

Figure 2.4. Location of the Gearbox in the tissue plant 36

Figure 2.5. Cylindrical joint of the gearbox Finally, table 2.1 contains the information relative to the properties of the gearbox. Appendix B instead, presents some pictures of the dismounted gearbox.

Property Manufacturer Mass Material Dimensions

Description Valmet 2200 Kg Steel 420x940x1805 mm続 Two electrical Motors; flexible couplings 1033.25 rpm (17.2 Hz)

Drive Input Velocity Transmission i1=1:2.48 i2=1:1.71 Ratios Table 2.1. Properties of the paper mill gearbox

2.3

INSTRUMENTATION AND MEASUREMENT SETUP

We will describe in this part all the instrumentation and measurement applied for the fulfillment of the objectives. We applied two classical modal tests for the identification of modal frequencies at non-operative condition: Bump Test and Run-Up Test. It is important to clarify that the plant stop is done only twice per year for maintenance purposes. The acquisition for this work was developed during the completely stop of the plant on June 18th, 2014, with a 37

duration of 30 minutes; after that, the plant was maintained in a very low speed for a couple of hours, and then it reached the normal condition. During the half an hour stop it was just possible to perform Bump Test, and then the preparation of the Data Acquisition System for the Run-up Test; so unfortunately it was not possible to apply the Hammer Test for acquisition of Frequency Response Functions and this will be a limitation for this thesis. Finally, an ODS acquisition at operative condition was taken for the development of Operational Modal Analysis.

For the measurement, two data acquisition systems were used: A 4 channel SKF Microlog Analyzer GX for the realization of Bump Test and Run-Up Test, and a 16 channel SKF Multilog On-line System IMx-P for the output-only measurement at operative condition.

(a)

(b)

Figure 2.6. Data Acquisition systems: a) SKF Microlog Analyzer GX b) SKF IMx-P Unit Also, 20 industrial monoaxial accelerometers SKF CMSS200 were employed for the acquisition. These accelerometers are designed for installation robustness, with magnetic bases and a sensitivity of 100mV/g.

Figure 2.7. Accelerometer SKF CMS200.

Finally, for the Run-Up test, it was used the SKF CMS 6155XK-U-CE Optical phase reference kit, for the measurement of the velocity. This kit consists in a remote optical sensor like in figure 2.8, an interface module and a magnetic holder.

38

Figure 2.8. Laser Tachometer.

In order to define the maximum frequency of interest, we take into account the gear mesh frequencies of the input shaft, because it is logical to expect harmonics due to the presence of gears in the operative condition. In appendix A, we can see that gear harmonics can include up to a 3rd GMF. In our case, for the input shaft we have: 3đ?‘Ľđ??şđ?‘€đ??š = 3 ∙ (đ?œ”đ?‘–đ?‘›đ?‘?đ?‘˘đ?‘Ą ∙ đ?‘ đ?‘Ąđ?‘’đ?‘’đ?‘Ąâ„Ž đ?‘ â„Žđ?‘Žđ?‘“ ) = 3 ∙ (17.2đ??ťđ?‘§ ∙ 25) = 1290 đ??ťđ?‘§ Then, the next allowable maximum frequency in the data acquisition system was 2000 Hz. The following step was the definition of 16 points on the Gearbox. The position of transducers were selected in order to avoid nodes in the modal shapes or acquisition of undesired local modes of components (shafts, cover plates, bearings, bolts, auxiliary drive, etc.). In the next figure we observe the position of the transducers in a fixed reference system, in which each transducer has a measurement direction orthogonal to the surface.

13Z

14Z 15Z

3X 6X 5X 2X 8X

1X Z X

11 X

4X 7X

10 X

16Y 12Z

Y 9X

Figure 2.9. Labeling of points for data acquisition.

39

2.3.1 BUMP TEST ACQUISITION For the Bump Test we used SKF Microlog, for simultaneous acquisition using 4 channels, implementing the following channel set up:

Channel CH1 CH2 CH3 CH4 Point 8X 11X 15Z 16Y Table 2.2. Channel configuration for the Bump Test.

After placing the accelerometers, forces were applied with a hammer close to point 15; also it has been done a different acquisition changing the direction of the applied force (directions X, Y and Z) in order to ensure the correct excitation of all the modes. Location 15 was selected because that point is far from the cylindrical joint of the gearbox, so we expect a higher excitation. Again, the maximum frequency was 2000 Hz and the number of lines was 6400.

2.3.2 RUN-UP TEST ACQUISITON Following acquisition was during the Run-Up Test for the gearbox, taking as reference the velocity of the second electric motor shaft within a range of 408-1213 rpm. Like in previous case, we used SKF Microlog but the Ru-Cd module only allowed us to acquire two channels and the tachometer connected. The channel configuration is presented in the next table.

Channel Point

CH1 15Z

CH1 11X

Table 2.3. Channel configuration for the Run-Up Test.

2.3.3

ODS ACQUISITION

As it was mentioned before, ODS Acquisition was done with SKF Multilog IMx-P for the simultaneous acquirement of 16 channels during the operation of the gearbox at 1033 rpm (input shaft). We selected a maximum frequency range of 2000 Hz and a number of lines of 6900, providing a maximum allowable length of 3.2 seconds, a sampling frequency of 5120 Hz (2.56 per the maximum frequency) and 16384 as sampling number. Following table includes the implemented configuration of the channels: Channel CH1 CH2 CH3 CH4 CH5 CH6 CH7 CH8 CH9 CH10 CH11 CH12 CH13 CH14 CH15 CH16 Point 1X 2X 3X 4X 5X 6X 7X 8X 9X 10X 11X 12Z 13Z 14Z 15Z 16Y

Table 2.4. Channel configuration for the ODS Acquisition.

40

Unfortunately, after the acquisition process, we discovered an electrical issue in the cable of Channel 1 that affected the signal and as a consequence the data was completely neglected for analysis. In appendix C, we present the difference observed between Channel 1 and a correct signal (Channel 2).

Once that the acquisition step has been completed, the techniques and processes described in following section are applied. The softwares useful for this part are: SKF @ptitude Analyst Monitor for data process of the SKF Microlog system, SKF @ptitude Observer for processing the ODS data, ME´scopeVES for modal parameter extraction of output-only data using curvefitting techniques, MATLAB for the harmonic identification, Ansys Workbench for the modelling of our analytical system and Solidworks 2013 for the generation of the model for FEM analysis and the creation of simplified 3D model for ODS simulation.

2.4

PROCESSING TECHNIQUES FOR CLASSICAL MODAL TESTS

After the data acquisition, we process the signal obtained in the bump test, in order to identify the modal frequencies of the gearbox. The acquisition system exports the .csv files for each excitation direction. Then, using Matlab, a code is written for the plotting of the bump curves of the nodes in function of frequency. In the plot, we overlap for each point the curves of each excitation direction, this can help us to identify and compare modal frequencies excited due to a different hammer direction and to differentiate peaks of modal frequencies and noise.

Appendix D contains the Matlab code useful for the generation of bump curves from the csv files, while the corresponding plots and results will be exposed in next chapter.

For the Run-Up test, we use SKF @ptitude Observer for the exportation of data from the acquisition system and the corresponding generation of the waterfall diagrams for both channels. The waterfall is done applying hanning window for the FFT calculation, taking 8192 samples of the time domain data with a 10% of overlap and considering a RMS detection of the amplitude.

Figure 2.10. Generation of waterfall diagrams in SKF @ptitude Analyst.

41

2.5

OPERATIONAL MODAL ANALYSIS PROCEDURE

Once that the data acquisition for ODS simulation and the Classical Modal Tests of the Gearbox have been done, we proceed with the following methodology for achieving the objectives:  





First, we apply Signal processing and ODS generation in order prepare the data for the curve fitting part.

Then, we will do an analysis for the selection of the available Mode Indicator Functions and Curve Fitting Methods with higher performance in the extraction of modal parameters; during this process we compare with the result of Bump and Run-Up test.

A second analysis includes the evaluation and comparison of each selected algorithm, analyzing the influence of the minimum number of stables poles in the stability diagram and the frequency length during curve fitting.

Finally, we proceed with the evaluation of the popular harmonic-identification method: Short Time Fourier Transformation, Singular Value Decomposition, Probability Density Function and the visual comparison of the operational deflection shapes.

2.5.1

SIGNAL PROCESSING AND ODS GENERATION

In order to advance with OMA, we should transform the data acquired in the testing step. SKF @ptitude observer allow us to transfer all the time domain signal in an unique UFF file, and then, it is possible to import it into the software ME´scopeVES for data processing. A workflow diagram of the signal processing step is presented in the following figure:

Selection of a reference signal

Resolution and sampling number

DeConvolution Window

ODS generation

Figure 2.11. Work flow diagram of the Signal Processing Step.

Inside ME´scopeVES, the first step is the transformation of the time domain signals into ODSFRF data for the generation of ODS visualization and curve fitting, but we need to remember that the calculation of ODS-FRF spectra requires the selection of a reference signal (see equation 1.61), satisfying the following conditions:  

It should be chosen at a DOF where the machine has lots of motion.

The reference must be located at a position that dos not generate a null node. [13]

Following step should be the selection of a good resolution and sampling number to obtain a cleaner frequency domain signal. As we can see in appendix C, it is expected a big presence 42

of harmonics in our gearbox, and as a consequence the performance of the curve-fitting methods can be poor, due to an increase in the time of computation or the calculation of spurious modes.

However, the selection of a correct resolution and an optimal number of samples can remove many lightly-damped harmonics and optimize the curve-fitting step.

Then, it is important the application of a De-Convolution window to the set of ODS-FRF signals as it is suggested by Schwarz and Richardson [23]. The main idea is that each element of a column of the Cross-Spectra or ODS-FRF is a summation of roving frequency response by the complex conjugate of the reference response signal. Because a multiplication in the frequency domain signal represents a convolution in the time domain signal, then the equivalent Inverse of ODS-FRF looks like the next curve:

Figure 2.12. Inverse of an ODS-FRF signal.

The left side of the previous signal is due to the contribution of the roving channel, while the right side is an effect of the reference channel. So, if we desire to apply a FRF-based curve fitting technique, then we will apply a De-Convolution window in order to remove the right side. The final signal will looks like a typical IRF function corrected for IRF or FRF curve fitting (figure 2.13).

Figure 2.13. Inverse of an ODS-FRF signal with the De-Convolution window.

Final step correspond to the creation of a 3D model for the ODS visualization. In this thesis we use Solidworks for the creation of a simplified gearbox model, and after that the 3D geometry is exported into ME´scopeVES for Operational Deflection Shapes. ME´scopeVES allows the creation of points on the geometry for the insertion of ODS data and the generation of points interpolation (calculation of the acceleration of non-measured points).

43

2.5.2

SELECTION OF MODAL INDICATOR FUNCTIONS AND CURVE FITTING ALGORITHMS.

After the signal processing, we can continue with the curve fitting of the output-only data for the extraction of the modal parameters using ME´scopeVES. Following workflow diagram is represented as an expansion of the Curve Fitting step (Figure 1.11).

Application of a Mode Indicator Function

Curve Fitting

Stabilization Diagram

Figure 2.14. Work flow diagram of the Curve Fitting Step.

In ME´scopeVES we are able to use the following Mode Indicator Functions (those were described in the theoretical chapter):    

Complex Mode Indicator Function (CMIF) Multivariate Mode Indicator Function (MMIF) Multi-Reference CMIF Multi-Reference MMIF

Also, we can apply different Curve Fitting Methods that are FRF-based techniques for the acquisition of modal parameters:     

Alias Free Polynomial (AF Polynomial) Complex Exponential Method (CEM) Z Polynomial Multi-Reference Complex Exponential Method Multi-Reference Z Polynomial

Now, our first step is the selection of those algorithms that have a faster and better performance for the extraction of modal parameters, including both single and multi-reference methods. In order to do this we will use the work flow line like in Figure 2.14 with each possible combination of Mode Indicator Functions and Curve Fitting Methods in the complete frequency range of acquisition: This can help us to identify the methods that work better with a high frequency range (higher amount of data and peaks). During the process we will compare the calculated modal frequencies with the Classical Modal Test and in the last part, the results from the curvefitting methods will be filtered using the stabilization diagram; the last one, as already known, removes false modes and eliminate lightly-damped harmonics present in the result.

44

Moving on the last two methods, Complex Exponential and Z Polynomial, it is important to underline that they cannot extract modal shapes by themselves, which instead can be obtained combining pick picking 2.5.3

EVALUATION OF CURVE FITTING TECHNIQUES

After the selection of the most efficient methods, we proceed with the second part of the analysis.

First, we would like to understand in the stability diagram, what is the minimum number of stable poles which give us reliable and meaningful results. Then, it is important to evaluate the performance of the curve fitting methods. As we know, the minimum number of stable poles works as a reference in order to save stable poles and neglect non-stable poles; however, this value is introduced by the user and an incorrect estimation of this value can affect the performance of the modal extraction method. We intend to vary this parameter to figure out how much it can affect the accuracy and quality of the results.

Once identified the minimum number of stable poles, we will apply curve fitting splitting the frequency range (0-2000 Hz) in two different sets: 0-1000 Hz and 1000-2000 Hz. This operation can help us to understand the influence in the reduction of the frequency signal for curve fitting and the accuracy in the identification of modes. Finally, we will compare the modal parameters from each curve-fitting method applying MAC between each of them.

2.5.4

HARMONIC IDENTIFICATION

As we can remember from the theoretical part, the presence of harmonic components in the computed modes of curve fitting techniques is expected. The methodology applied in this part will be used in order to evaluate the identification capacity of different techniques in the first chapter. The following methods, already described in the theoretical part, will be applied:   



Short Time Fourier Transformation.

Singular Value Decomposition.

Probability Density Function.

Visual comparison of Operation Deflection Shapes.

For the application of these techniques, we develop the corresponding application in Matlab, importing the .wav files of the ODS data and doing the corresponding signal processing. In appendix D we attach the codes.

45

CHAPTER III: OMA RESULTS AND ANALYSIS In the following chapter, we present all the results and analysis in the same order as it was explained in the methodology part for the Evaluation of Operational Modal Analysis for Rotating Machinery.

3.1 CLASSICAL TESTS: BUMP AND RUN-UP TEST The following pictures contain the Bump Curves for each measured point at the bump test. We can see that curves of excitation at direction X present a large noise, and of course the identification of modes through this curve is difficult. Also, the peaks of modal frequencies are located at high frequency range (higher than 400 Hz), this results is logic because it represents a good design of the gearbox, in which the modal frequencies should be located far from the operative condition.

Figure 3.1. Bump curves of Point 8X.

Figure 3.2. Bump curves of Point 11X. 46

Figure 3.3. Bump curves of Point 15Z.

Figure 3.4. Bump curves of Point 16Y. Also, we show the waterfall diagrams from our Run-Up test.

47

Figure 3.5. Waterfall at point 11X.

Figure 3.6. Waterfall at point 15Z. In figure 3.5 we observe the presence of harmonics in the waterfall diagram, that are the inclined straight lines in which the frequency position of peaks change according with the velocity. Also, it is not so easy to locate the exact position of modal frequencies, because the picture have some not-well defined vertical regions that suggest the presence of modes. It is important to remember that theoretically, the modes should be represented by vertical straight lines in the waterfall diagram, but in our case we have vertical regions like in the range 600-800 Hz or 1800-1900 48

Hz. In Figure 3.6 we can observe some harmonics and vertical regions, but unlike the previous case we get some well-defined vertical lines inside the range 0-300Hz. Obviously the lines at 200 and 300 Hz were generated by the electric motor connected at the auxiliary drive that was running during the Run-Up test; the lines at lower frequency can be some real modal frequencies, however we must remember that we did not obtain any mode at low frequency for the bump test.

The problem of not-well defined vertical regions can be the result of a weak excitation of the modes, due to the presence of closed modes at those regions or due to the selection of an incorrect sensor position for the Run-Up case. Next table present all the modal frequencies identified at Bump and Run-up test.

Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

8X

BUMP TEST 11X 15Z

423

693 828

948

660 689 766 828 876 909 948

1035 1067 1193

660 766

895 948 1035 1067

1193 1262 1311 1319

1193 1262 1311 1319

1333

1798

423 548 660 699 766 828 876 909 948 1010 1035 1067 1124 1192 1265

Run Up 15Z 11X 19 23 52 61

669 765

733

909

1315 1333

1341 1417 1469

1469 1483 1513

1677

16Y

1487 1513 1525 1583 1635 1649 1681

1583 1634

1514

1583

1518 1589 1634

1649 1674 1703

1730 1795

1672

1688

1703 1806 1813

1857 1890 1952

1857 1890 1915 1952

1830 1857 1890 1923 1975

1830 1857 1917 1952 1975

1919 1953

1986

Table 3.1. Modal Frequencies obtained with Bump and Run-Up Test (Units in Hz).

49

In the table 3.1 the Bump test highlights some modes present in all acquisitions (1857 Hz), instead there are others which are identified in just some channels; also some modes were identified with only one method, and others with both techniques. Besides, the Run-Up test gives us lower number of modes with respect to the Bump Test, and also, estimation of modal frequencies through this method present some errors because we have vertical regions instead of well-defined vertical lines in the spectrogram.

3.2 OPERATIONAL MODAL ANALYSIS Now, we will focus on the results and analysis for Operational Modal Analysis and during this discussion we will compare the results obtained within the Classical Modal Tests.

3.2.1

SIGNAL PROCESSING AND ODS GENERATION

In the operative condition, we have a large amount of motion at channels 11X and 15Z, it can be justified because both points are far from the boundary conditions, represented by the cylindrical joint and the input shafts (connected to the motors). So, we will take both channels as references. From the ODS-FRF calculated for each channel, we observe that there is a similitude between both generated shapes; some results are presented in next figures:

(a)

(b)

50

(c)

(d)

(e)

(f)

Figure 3.7. ODS-FRF of point 2X with reference: a) 11X b) 15Z. ODS-FRF of point 3X with reference: C) 11X d) 15Z. ODS-FRF of point 3x with reference e) 11X f) 15Z. Due to the fact that the ODS-FRF does not have a significant variation with respect to the selection of the reference channel, it is not easy to choose between the points, in order to proceed with our work we take the reference 11X for single-reference curve-fitting and both channels for multi-reference curve fitting.

Also, we can see in the ODS response function that the input velocity 17.2 Hz has a peak of low amplitude, so we can guarantee that this operative frequency is a free-resonance condition.

51

17.2Hz

Figure 3.6. 17.2Hz at the ODS-FRF of point 3X. Then, knowing the number of gear teeth on the shafts, we calculate that the operative speeds of the middle and output rotor are 7 Hz and 4.3 Hz respectively. Again, we observe that the peaks of both conditions present a low amplitude level (Figure 3.7).

4.3Hz 7Hz

Figure 3.7. Frequencies 4.7 Hz and 7 Hz at the ODS-FRF of point 3X. As it was discussed in the description of the methodology, the actual resolution might be not appropriate for OMA-curve fitting, and a correct resolution must be defined in order to obtain a smoother signal and good number of samples. In our case, it was applied a hanning window for the creation of ODS data with 10 averages.

52

Now, we will do a variation in the number of samples, in order to analyze the signal behavior at 1024, 2048 and 4096 samples.

(a)

(b)

(c) Figure 3.7 ODS FRF of point 2X with a) 1024 b) 2048 and c) 4096 samples.

The third case (4096 samples) presents a large number of peaks, and this can be a problem for the operational modal analysis. In the first case with 1024 samples we obtain a frequency resolution of 2.5 Hz/ line, while with 2048 samples the value is 1.25 Hz/line. So, we decide to take the first case with two risks: first, there will be an error associated with the resolution and second it will be not possible to distinguish closed modes separated by 2.5 Hz; however, this resolution should be better for the performance of curve fitting techniques.

53

After the selection of an appropriate of samples, we apply the De-Convolution window, in order to eliminate the contribution of the reference signal. The inverse of the ODS-FRF before the De-Convolution windows looks like the following plot:

Figure 3.8. Inverse ODS FRF of point 2X without the De-Convolution window.

But in the next image we can see the effect of the De-Convolution window in order to obtain an Inverse ODS-FRF with a similar behavior of an IRF.

Figure 3.9. Inverse ODS FRF of point 2X with the De-Convolution window.

54

3.2.2

SELECTION OF MODAL INDICATOR FUNCTIONS AND CURVE FITTING ALGORITMS

For the curve fitting process we follow the methodology described in the previous chapter, in order to estimate the modal parameters from the output-only data. So, once that each Mode Indicator Function and Curve Fitting algorithm has been applied at the range 0-2000 Hz, we set up the stabilization diagram with a desired limit of iteration of 100 modes, ensuring that each possible mode will be included for each model. As a first approximation it is required a maximum damping of 10 Hz as a frequency tolerance and damping tolerance of 2.5 Hz (same as the frequency resolution). Finally, we will ask to ME´scopeVES to identify those modes which were repeated at least 20 times for each iteration, as previous setup for the satiability diagram. In case of the multi-reference methods, we use two reference channels: point 11X and 15Z. In the following pages, we present the stabilization diagram and result table for each case. The table contents information of the calculated modal frequencies and in the column of observation we present the result of the comparison with Classical Modal Test

55



Stabilization applying CMIF method: o AF Polynomial

Figure 3.10. Stabilization diagram using CMIF and AF Polynomial.

Frequency (Hz) Observation 22.12 Mode 1 26.37 Mode 1 51.16 Mode 2 70.95 198.2 Harmonic of E. Motor 304 Harmonic of E. Motor 695.8 Mode 7 912.1 Mode 11 1064 Mode 15 1217 1348 Mode 22 1725 Mode 33 1799 Mode 34 1861 Mode 37 1940 1977 Mode 41 1979 Mode 41 Total Identified Modes: 10

Table 3.2 Results of the application of CMIF and AF Polynomial

56

o CE Polynomial

Figure 3.11. Stabilization diagram using CMIF and CE Polynomial

Frequency (Hz) Observation 303.9 Harmonic of E. Motor 449.4 695.8 Mode 7 761.2 Mode 8 912.9 Mode 11 1065 Mode 15 1217 1348 Mode 22 1490 Mode 25 1652 Mode 30 1725 Mode 33 1798 Mode 34 1939 Total Identified Modes: 9

Table 3.3 Results of the application of CMIF and CE Polynomial

57

o Z Polynomial

Figure 3.12. Stabilization diagram using CMIF and Z Polynomial

Frequency (Hz) Observation 695.6 Mode 7 760.8 Mode 8 912.9 Mode 11 1065 Mode 15 1216 1347 Mode 22 1725 Mode 33 1798 Mode 34 1939 Total Identified Modes: 7

Table 3.4 Results of the application of CMIF and Z Polynomial

58



Stabilization applying MMIF method: o AF Polynomial

Figure 3.13. Stabilization diagram using MMIF and AF Polynomial

Frequency (Hz) 22.12 26.37 51.16 70.95

Observation Mode 1 Mode 1 Mode 2 Harmonic of E. 198.2 Motor Harmonic of E. 304 Motor 695.8 Mode 7 912.1 Mode 11 1064 Mode 15 1217 1348 Mode 22 1725 Mode 33 1799 Mode 34 1861 Mode 37 1940 1977 Mode 41 1979 Mode 41 Total Identified Modes: 10

Table 3.5 Results of the application of MMIF and AF Polynomial

59

o CE Polynomial

Figure 3.14. Stabilization diagram using MMIF and CE Polynomial

Frequency (Hz)

Observation Harmonic of E. 303.8 Motor 450 695.8 Mode 7 761.1 Mode 8 851.7 913 Mode 11 1065 Mode 15 1217 1348 Mode 22 1367 1391 1490 Mode 25 1608 1652 Mode 30 1725 Mode 33 1798 Mode 34 1939 Total Identified Modes: 9

Table 3.6 Results of the application of MMIF and CE Polynomial

60

o Z Polynomial

Figure 3.15. Stabilization diagram using MMIF and Z Polynomial

Frequency (Hz) Observation 695.6 Mode 7 760.8 Mode 8 912.9 Mode 11 1065 Mode 15 1216 1347 1725 Mode 33 1798 Mode 34 1939 Total Identified Modes: 6

Table 3.7 Results of the application of MMIF and Z Polynomial

61



Stabilization applying Multi reference CMIF method: o CE Polynomial

Figure 3.16. Stabilization diagram using Multi-CMIF and CE Polynomial

Frequency (Hz) Observation 198.8 Harmonic of E. Motor 304 Harmonic of E. Motor 609.2 Harmonic of E. Motor 695.8 Mode 7 695.9 Mode 7 761.3 912.6 Mode 11 1065 Mode 15 1217 1725 Mode 33 1797 Mode 34 1939 Total Identified Modes: 5

Table 3.8 Results of the application of Multi-CMIF and CE Polynomial

62

o Z Polynomial

Figure 3.17. Stabilization diagram using Multi-CMIF and Z Polynomial

Frequency (Hz) Observation 695.7 Mode 7 760.5 Mode 8 912.5 Mode 11 1065 Mode 15 1797 Mode 34 Total Identified Modes: 5

Table 3.9 Results of the application of Multi-CMIF and CE Polynomial

63



Stabilization applying Multi reference MMIF method: o CE Polynomial

Figure 3.18. Stabilization diagram using Multi-MMIF and CE Polynomial

Frequency (Hz) Observation 695.7 Mode 7 760.6 Mode 6 912.6 Mode 11 912.6 Mode 11 1065 Mode 15 1065 Mode 15 Total Identified Modes: 4

Table 3.10 Results of the application of Multi-MMIF and CE Polynomial

64

o Z Polynomial

Figure 3.19. Stabilization diagram using Multi-MMIF and Z Polynomial Frequency (Hz) Observation 695.7 Mode7 761 Mode 8 912.3 Mode 11 1725 Mode 33 Total Identified Modes: 4

Table 3.11 Results of the application of Multi-MMIF and Z Polynomial After the generation of all the stabilization diagrams, we can observe the following aspects: 





We obtain in the single-reference methods a larger number of real modes in comparison with the multi-reference methods. For example, applying CMIF with AF Poly and CE poly we get 10 and 9 real modes respectively, but using Multi-CMIF with CE and Z Poly we obtain 5 modes for each case. So, it seems that these polyreference methods are not so good working with high amount of data and we decided not to continue our analysis with this kind of method because is time consuming.

AF Poly and CE method give us larger number of modes than the Z poly. However, we can see that not all the same modes are obtained with each method; for instance, inside MMIF results, we get the modal frequency 1977 Hz using the CE (10 modes) but we did not find it using AF Poly (11 modes). This result suggests us to simultaneously apply the three curve-fitting methods in order to obtain the higher possible amount of real modes, complementing each techniques.

Unfortunately, even after the application of the stabilization diagram we get some calculated frequencies that do not correspond to results from the Bump or Run-Up test.

This might suggest that the curve-fitting methods are not so effective for excluding harmonic peaks. For example, in the tables we can see that some methods included some harmonics due to the electric motor: 2x and 3x.

65



Moreover, there is not a visible difference between the application of CMIF or MMIF in the modal parameter acquisition. The similitude can be visualized for example using MAC between the results of the AF Poly.

Table 3.12. MAC between CMIF-AF Poly and MMIF-AF Poly. (Frequencies in Hz and Damping Rations in Hz) As we can see, the previous MAC evidences a high correlation between both results, and there is only one difference for the mode at 1979 Hz that seems to be the repetition of mode at 1977 Hz. So, we will decide to continue our analysis with the CMIF method only.

3.2.3

EVALUATION OF CURVE FITTING TECHNIQUES

In the following tables we present the modal frequencies and modal damping ratios obtained using Complex Exponential Method (CE), Z Polynomial (Z Poly) and Alias Free Polynomial (AF Poly) methods changing the minimum number of stable poles in the stability diagram, i.e. taking values 10, 20, 30 and 40. Values highlighted in blue are those modes that were also identified using Bump and Run Up test, and it is important to clarify that some values are repeated, so the total number of modes calculated does not include the repetition of a mode; the value in parenthesis is the number of equivalent modal frequencies presented in the table 3.1.

Besides, the red highlighted values are frequencies closed to the harmonics generated by the electric motor on the auxiliary drive (200 Hz and 300 Hz).

66

n= 40 Freq (Hz) Damp (Hz) 51.16 (2) 2.038 1064 (15) 0.476 1725 (33) -0.396 1799 (34) 1.044 Obtained Modes: 4

n= 30 Freq (Hz) Damp (Hz) 22.12 (1) 2.288 51.16 (2) 2.038 304 -0.852 695.8 (7) 0.223 1064 (15) 0.476 1217 1.154 1725 (33) -0.396 1799 (34) 1.044 1940 2.168 Obtained Modes: 6

n= 20 Freq (Hz) Damp (Hz) 22.12 (1) 2.288 26.37 1.935 51.16 (2) 2.038 70.95 2.154 198.2 4.028 304 -0.852 695.8 (7) 0.223 912.1 (1) 1.268 1064 (15) 0.476 1217 1.154 1348 (22) 2.923 1725 (33) -0.396 1799 (34) 1.044 1861 (3)7 0.865 1940 2.168 1977 (41) 2.371 1979 (41) 1.962 Obtained Modes:10

n= 10 Freq (Hz) Damp (Hz) 22.12 (1) 2.288 26.37 1.935 34.03 2.702 47.56 3.193 51.16 (2) 2.038 70.95 2.154 105.8 0.394 151.4 0.95 198.2 4.028 304 -0.852 447.7 2.98 452.3 1.789 695.8 (7) 0.223 761 (8) 2.747 761.3 (8) 0.056 912.1 (11) 1.268 914.5 (11) 1.739 1064 (15) 0.476 1217 1.154 1348 (22) 2.923 1652 (30) 0.464 1687 (31) -1.169 1725 (33) -0.396 1773 1.494 1799 (34) 1.044 1802 (34) 1.2 1822 4.457 1826 (36) 3.353 1861 (37) 0.865 1886 (38) 0.969 1908 1.095 1913 (39) 1.646 1933 3.459 1936 3.054 1940 2.168 1943 1.134 1952 (40) 0.115 1956 0.397 1963 2.467 1967 0.871 1972 1.626 1977 (41) 2.371 1979 (41) 1.962 Obtained Modes: 18

Table 3.13. Variation of the minimum number of stable poles using AF Poly.

67

n= 40

n= 30

n= 20

n= 10

Damp Freq (Hz) (Hz) 303.9 0.758 695.8 (7) 1.066 761.2 (8) 1.668 912.9 (11) 2.807 1065 (15) 1.702 1217 1.454 1725 (33) 0.919 1798 (34) 1.53 1939 1.215 Obtained Modes: 6

Damp Freq (Hz) (Hz) 303.9 0.758 449.4 4.182 695.8 (7) 1.066 761.2 (8) 1.668 912.9 (11) 2.807 1065 (15) 1.702 1217 1.454 1348 (22) 2.223 1725 (33) 0.919 1798 (34) 1.53 1939 1.215 Obtained Modes: 7

Damp Freq (Hz) (Hz) 196.5 5.613 303.9 0.758 449.4 4.182 695.8 (7) 1.066 761.2 (8) 1.668 912.9 (11) 2.807 1065 (15) 1.702 1217 1.454 1348 (22) 2.223 1490 (25) 1.095 1652 (30) 1.934 1725 (33) 0.919 1798 (34) 1.53 1939 1.215 Obtained Modes: 9

Damp Freq (Hz) (Hz) 23.57 3.804 51.27 3.005 151.2 2.722 196.5 5.613 199.3 5.57 303.9 0.758 449.4 4.182 452.3 4.493 607.9 5.313 611.4 5.254 670.9 4.394 695.8 (7) 1.066 761.2 (8) 1.668 851.2 3.766 912.9 (11) 2.807 943.2 (12) 3.128 1065 (15) 1.702 1149 4.718 1152 3.598 1183 4.746 1217 1.454 1348 (22) 2.223 1349 5.107 1367 2.034 1387 5.103 1391 2.486 1392 5.064 1490 (24) 1.095 1534 4.813 1561 3.644 1608 4.539 1652 (30) 1.934 1686 (31) 3.67 1725 (33) 0.919 1798 (34) 1.53 1823 4.822 1891 (38) 2.646 1895 4.462 1939 1.215 Obtained Modes: 14

Table 3.14. Variation of the minimum number of stable poles using CE Poly.

68

n= 40

n= 30

n= 20

n= 10

Freq (Hz)

Damp (Hz)

Freq (Hz)

Damp (Hz)

Freq (Hz)

Damp (Hz)

Freq (Hz)

Damp (Hz)

695.6 (7)

0.782

695.6 (7)

0.782

695.6 (7)

0.782

451.5

6.506

760.8 (8)

2.006

760.8 (8)

2.006

760.8 (8)

2.006

695.6 (7)

0.782

912.9 (11)

2.866

912.9 (11)

2.866

912.9 (11)

2.866

760.8 (8)

2.006

1725 (33)

0.97

1065 (15)

2.355

1065 (15)

2.355

912.9 (11)

2.866

1798 (34)

1.079

1725 (33)

0.97

1216

3.117

1065 (15)

2.355

1798 (34)

1.079

1347 (22)

5.114

1216

3.117

1725 ( 33)

0.97

1346 (22)

1.582

1798 (34)

1.079

1347

5.114

1939

1.505

1537

5.332

1725 (33)

0.97

1727

3.465

1798 (34)

1.079

1829

4.786

1939

1.505

1940

4.742

Obtained Modes: 5

Obtained Modes: 6

Obtained Modes: 7

Obtained Modes: 7

Table 3.15. Variation of the minimum number of stable poles using Z Poly.

After the generation of the tables we can reach the following results: 







The accuracy of the methods increases if the minimum number of stable poles is higher.

This is evident looking the results of “n” equal to 40 for instance: in the case of AF and Z Poly we obtain 100% of modes associated with results from the Bump and Run-Up test, and in the CE Poly we obtain 6 real modes from 9 calculated modal frequencies; however, if we use a lower number of stable poles (10 for instance), we get higher number of modes but also many non-realistic modes that can be due to the presence of strong harmonics.

Also, the results suggest that the Alias Free Method and the Complex Exponential Method are more susceptible to the presence of harmonics. This is because some calculated modes are harmonics of the electric motor or non-associated with Classical Modal Test results (200 Hz and 300 Hz).

The AF Poly method has some inconsistencies in the estimation of modal damping ratios because some damping ratios of expected real modal frequencies are negative. This was not the case for the other methods and for that reason we cannot neglect those results.

We consider that the selection of a minimum number of stable poles equal to 20 generates reliable data, with a relative large number of modal frequencies and short number of harmonics.

Next, selecting a minimum number of 20 for the stability condition, we apply the Modal Assurance Criterion (MAC) between the computed operational shapes of each mode (we prefer to use this term because not all the vectors are real modal shapes due to the presence of harmonics) 69

Table 3.16. MAC between AF Poly and CE Method. (Frequencies in Hz and Damping Rations in Hz)

Table 3.17. MAC between AF Poly and Z Poly. (Frequencies in Hz and Damping Rations in Hz)

70

Table 3.18. MAC between CE Method and Z Poly. (Frequencies in Hz and Damping Rations in Hz) Previous MAC tables highlight a high correlation degree between the calculated modal shapes of different methods, but they show us a non-consistence result for the harmonics: deflection shapes of electric motor harmonics (table 3.16) are different, while the deflection for harmonics at 1217 Hz and at 1939 Hz are similar; however, we consider that this is not a problem because the objective of OMA is the extraction of modal parameters and we are not interest in the deflection of harmonics (that can be correctly evaluated using ODS for instance). Also, in the last analysis we will deal with the identification of harmonics in order to exclude those results.

Now, we apply curve fitting dividing the analyzed regions in two equal frequency lengths: range 0-1000 Hz and 1000-2000 HZ. As it was mentioned in the methodology part, this is done in order to understand the influence in the length and amount of data during the estimation of modal parameters; in next table we present the results (again the blue marked calculated modal frequencies are equal or close to modal frequencies from Classical Modal Tests and in parenthesis we refer to the equivalent mode).

AF Poly

CE

Z Poly

Freq.(Hz)

Damp.(Hz)

Freq. (Hz)

Damp. (Hz)

Freq. (Hz)

Damp. (Hz)

51.65 (2)

2.072

51.22 (2)

2.791

303.4

3.452

71.44

1.802

71.02

3.296

450.2

4.689

151.9

1.102

151.7

2.285

519

6.365

301.9

-0.127

201.3

2.645

538.1

4.964

304

0.281

302.9

1.445

670.2

4.412

609.2

1.998

391.9

2.137

695.8 (7)

1.823

695.5 (7)

1.336

449.9

3.683

719.4

5.035

71

738.7

2.129

457.2

1.935

738.5

4.754

761 (8)

1.49

546.2 (5)

1.457

761.8 (8)

1.534

827.4 (9)

0.476

609.2

2.79

828.4 (9)

3.432

899.7 (11)

1.874

668 (6)

3.469

913.3 (11)

1.988

913.3 (11)

1.6

695.7 (7)

1.554

1723

1.536

944.3 (12)

2.273

712.7

2.272

1727 (33)

2.43

958

1.187

739.3

2.532

1746

3.379

966

1.026

761.1 (8)

1.716

1797 (34)

1.14

969.6

1.634

778.3

3.081

1803 (34)

3.553

973.4

2.219

826.6 (9)

1.222

1829 (36)

4.317

976.4

2.761

898.5 (11)

2.757

979.7

3.326

913.2 (11)

2.112

984.1

2.874

944 (12)

2.315

1014 (13)

2.706

1014 (13)

2.94

1020

2.416

1020

2.776

1025

2.276

1025

2.463

1029 (14)

2.262

1030

2.392

1050

0.577

1035 (14)

1.723

1060

0.017

1061

-0.081

1066 (15)

2.033

1065 (15)

1.948

1084

1.053

1084

1.057

1150

0.018

1152

0.213

1217

0.949

1217

0.944

1348 (22)

1.631

1348 (22)

1.651

1368

1.175

1368

1.127

1391

0.463

1391

0.492

1415 (23)

-0.341

1414 (23)

-0.332

1532 (27)

0.726

1532 (27)

0.903

1561

0.12

1560

0.05

1651 (30)

1.609

1652 (30)

1.581

1725 (33)

1.149

1725 (33)

1.185

1797 (34)

1.431

1797 (34)

1.507

1827 (36)

0.84

1827 (36)

1.192

1860 (37)

1.302

1860 (37)

1.166

1886 (38)

0.437

1886 (38)

0.799

1941

2.96

1918 (39)

0.753

1954 (40)

0.938

1939

3.731

1961

0.79

1942

2.72

1967

1.519

1954 (40)

1.367

1971

2.312

1961

0.899

1977

3.246

1966

1.813

1981

3.013

1972

2.549

1986 (42)

2.756

1977 (41)

3.379

1989 (42)

2.293

1982

3.109

1985 (42)

2.894

1987 (42)

2.764

Obtained Modes: 20

Obtained Modes: 7

Obtained Modes: 24

Table 3.19. Modal Parameters using a shorter frequency length during curve fitting.

72

As we can see from last table, the application of a shorter frequency length at the curve fitting step brings a larger number of recognized real modal frequencies (24 in total), but also a larger number of harmonics. So in this case it is important the identification of harmonics and modes (process that will be analyzed in the next part of this chapter). On the other hand, AF and CE methods are more efficient for the computation of larger number of modes instead of the Z Poly method.

3.2.4

HARMONIC IDENTIFICATION

Now, we will deal with the recognition of harmonics during Operational Modal Analysis applying different techniques. 

Short Time Fourier Transformation.

First, we start applying the STFT method, remembering that this technique is recommended by Bruel and Kjaer as a practical way for the identification of modes and harmonics. The following image present the results of the STFT using Matlab.

Figure 3.20. Short Time Fourier Transformation for each signal.

After the generation of previous plots we try to identify the frequencies with higher amplitudes, i.e. values frequencies with a well-defined horizontal line at the spectrogram. Then we compare with the modal frequencies obtained in the bump and Run-Up test and with the harmonics of the fundamental frequency 17.2 Hz (speed of the input shaft). In next table we present the results:

73

MODE 7 17.2x43 8 11 15/17.2x62 17.2x71 22 17.2x81 23 24 33 34 36 37 28 40/17.2x113

2X 696

3X 696

4X 696

764

1221

5X 696 765 915

1221 1372 1396

6X

915 1066 1221 1349

7X 696 764 915

8X

9X

10X

11X 696

12Z 696

13Z

765 915 1066 1221

915

1395 1396 1395 1394

915 1066 1221 1349 1372 1395

14Z 696

15Z

16Y

738

738

765 915

1221

1428 1473 1500 1730 1730 1730 1731 1730 1802 1802 1802 1802

1538 1730 1730 1730 1730 1730 1730 1730 1802 1802 1802 1802 1802 1830 1828

1866 1904 1904 1945

1945 1945

1945 1945 1945

1946 1945 1945 1946

Table 3.20. STFT Results.

As we can see in the previous table, it is possible to identify some modal frequencies and harmonics of the fundamental frequency 17.2 Hz; however the use of this method present following disadvantages: some frequencies can be modal frequencies but also harmonics (like 1066 Hz or 1945 Hz), so in that case it is not possible the identification. Also, there are some frequencies that are not associated with Classical Modal Tests or harmonics of 17.2 HZ, and in that case it is important to compare with other fundamental frequencies present in the system in order to determine if the frequency is a mode or not. In the case of our gearbox, the estimated number of fundamental frequencies is unknown due to lack of information; for example, we do not know the fundamental frequencies of the auxiliary drive connected to the input shaft.

However, we consider that STFT is useful used as a first step in order to estimate modal frequencies and harmonics, but it is important to identify all the fundamental frequencies that can affect the response for a good identification.



Singular Value Decomposition.

After the application of STFT procedure, we plot the Singular Value Decomposition curves of the Auto-Spectra matrix using Matlab. As we can remember from the theory (see section 1.5.3) the behavior of the plot can be analyzed in order to identify modes from harmonics. In next picture we present the plot:

74

Figure 3.20. SVD Plot.

According to B&K in presence of harmonic components narrow peaks will be found in multiple curves while for modal peaks the curves are smoother. For the generation of the plot we reduced the number of samples in order to obtain a smoother signal. So, using the rule of previous statement we found the results in table 3.21:

Figure 3.21. SVD Plot with Possible peaks of modal frequencies.

75

MODE 1 7 12 18 24 31 34

Frequency (Hz) 25 697 940 1260 1465 1675 1790

Table 3.21. SVD Results.

Modal frequencies identified with SVD are consistent with classical modal tests results, however the big presence of harmonic peaks makes difficult the recognition of more modes, especially because the procedure implies a visual identification. For example, we can see in the figure 3.21 that harmonics at 200 Hz and 300 Hz can be easily assumed as modal frequencies without a previous knowledge of the system; so we consider that a good use of the Singular Value Decomposition requires the knowledge of possible fundamental frequencies affecting the system. Also, we can conclude that this method is not practical for our case because the total number of modes recognized was low and the large presence of harmonics makes difficult the recognition of modes.



Probability Density Function.

In appendix D we present the code used for the application of PDF. We applied PDF to the frequencies obtained using CE, Z Poly and AF Poly with a minimum number of stables poles equal to 30. In Matlab we isolated each peak in the auto-spectra of channel 2 and then we applied the probability function. At the end we compared the results with those obtained from classical tests. In next table we present the results and then the plots of each PDF:

76

Frequency (Hz)

Method applied for the identification of the mode

PDF Result

Bump Test Comparison

22.12 51.16 70.95 449.4 695.8 761.2 912.9 1065 1217 1348 1490 1652 1725 1799 1861 1940 1977

AF/Bump AF/Bump AF CE AF/CE/Z/Bump CE/Z AF/CE/Z/Bump AF/CE/Z/Bump AF/CE/Z AF/CE/Z/Bump CE CE/Bump AF/CE/Z/Bump AF/CE/Z/Bump AF/Bump AF/CE/Z/Bump AF

Mode Mode Harmonic Harmonic Mode Mode Mode 11 Harmonic Harmonic Mode Mode Mode Harmonic Mode Mode Mode Harmonic

Mode 1 Mode 2 Harmonic Harmonic Mode 7 Mode 8 Mode 11 Harmonic Harmonic Mode 22 Mode 24 Mode 30 Mode 33 Mode 30 Mode 37 Mode 40 Harmonic

Table 3.22. PDF Results.

(a)

(b)

(c)

(d)

(e)

(f)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(p)

(r)

77

(s)

(t)

Figure 3.22. PDF at the frequencies (in Hz): a)22.12 b)51.16 c)70.95 d)449.5 e)680 f)775 g)912 i)1065 j)1217 k)1348 l)1490 m)1652 n)1725 o)1799 p)1861 q)1950 r)1977 With this method we obtain a good result for the identification of modes and harmonics, with the exception of the frequency at 1725 Hz, but maybe that case can be the overlapping of a harmonic in a mode, affecting the result. One advantage of the PDF method is that provides an analytical solution of the identification problem, instead of previous techniques that requires a visual comparison and a knowledge of the system.

After the correct identification of the modal frequencies we present in next images the estimated modal shapes using curve fitting:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 3.23. Modal Shapes at (frequencies unit in Hz): a)22.12 b)51.16 c)695.8 d)761.2 e)912.9 f)1348 g)1490 h)1652 i)1725 j)1799 k)1861 l)1950 78

CHAPTER IV: FINITE ELEMENT MODAL ANALYSIS. As final goal, we would like to validate OMA results with the modal parameters obtained through a Finite Element Model. Therefore, we will build up a simplified 3D model of the paper mill gearbox using the software SolidWorks 2013, and then we will import this geometry in Ansys Workbench v15. A pre-stress analysis is necessary in order to calculate the stiffness matrix related to the mounting process, afterward to implement the modal analysis.

Unfortunately, the End User customer, as usual, does not have the asset drawings; there was just the availability of the assembly drawing (in Figure 4.1) without information of dimensions, materials or masses. However, a twin machine (without some connected equipment under operative conditions like the Steam Press and the Auxiliary Drive) was in SKF Solution Factory (Italy) for maintenance operations and it was possible to collect the principal dimensions.

Figure 4.1. Assembly Drawing

4.1

GEOMETRY

Due to this lack of information, several assumptions have been made during the modelling:    

Assumption of asset material: Structural Steel.

Simplification of the gears, modelling cylindrical volumes without teeth, connected with the shaft by bonded connections.

Substitution of the cover plates by equivalent point masses.

Dimensions and components of the auxiliary drive and steam press were unknown, so we inserted equivalent concentrated masses (estimated) on the corresponding contact surface (see next figure).

79

Figure 4.2. Equivalent Point Mass of the Steam Press. 

In order to simplify the bolt effect for the joint of the casing, we have followed the recommendation of SKF Engineering Consultancy Services described in the following figure. For the bottom part we introduced a bolt contact in Ansys to simulate the bolt rigidity and at the upper part we introduced a frictional contact, allowing the displacement of the upper half shell. In that region, we assume an asymmetrical contact behavior with a friction coefficient of 0.15

Figure 4.3. Bolt modelling.  

Frictional coefficient of 0.15 between both sides of the shell.

In Ansys is not possible to simulate bearings, so we simplified these parts with equivalent point masses and we inserted springs with equivalent stiffness. The stiffness of the bearings were obtained using the software SKF Beast, knowing the type of bearing (618/500 and 22318), and assuming for each one the 30% of maximum radial load and 10% of axial load. This is a very important assumption driven by SKF experience, due to the lack of additional information. As we know, the stiffness of the bearing is not linear and change in function of the frequency, and this will be a limitation for our model.

80

Figure 4.4. Modelling of the radial bearing stiffness. 





4.2

Assumption of bolt pretensions of 40 kN during the assembly, in order to calculate the stiffness matrix related to the mounting process, afterward to implement the modal analysis.

The lubrication system inside the gearbox was neglected for the modelling because we do not consider that it has a real impact in the modal shapes. Instead, the End User was able to provide the liters of oil inside the mechanical system, modelled as concentrated mass inside the shell.

The Boundary Conditions of the machine were simulated in the following way: o Overhang of the gearbox with the Yankee Roll shaft. o Fixed support of the cylindrical joint with the structure working as foundation (See Figure 2.5). o Cylindrical support of both electric motor drives.

MESHING

Once that geometry and the boundary conditions are defined, we insert the mesh in order to proceed with Finite Element Analysis. For the overall structure we define an automatic triangle mesh in Ansys with a relevance of 80% (See Next figure).

Figure 4.5. FEM Meshing.

81

In another way, in order to correctly evaluate the change of stiffness due to the assembly load, a fine mesh in the contact bolt-casing is required. So, we have defined a mesh refinement of 3 mm on the casing contact surface and a refinement mesh of 2 mm on the bolt bottom surface, as we can see in the next figure:

Figure 4.6. Casing Surface and Bolt Bottom Surface Meshing.

After the meshing generation, we compute in Ansys the modal parameters inside the frequency range 0-2000 Hz, and then we compare with the OMA results of table 3.22. In the next chapter we present the results.

82

CHAPTER V: FEM VS OMA COMPARISON. In the Finite Element Analysis, we obtained 100 modes inside the frequency range 0-2000 Hz. Ansys computed many modes related to the overall behavior of the casing, but also local modes related to the shafts, bolts and local regions of the shell. Next table presents the modal frequencies obtained with the numerical simulation.

52,079 89,649 110,95 135,11 177,56 218,09 248,37 290,93 344,64 381,61 407,01 418,6 425,94 454,28 479,19 484,03 494,06 507,07 516,69 521,78

Modal Frequencies 540,63 981,8 1342,9 599,27 992,17 1354,1 618,71 999,24 1374,7 636,8 1008,5 1395,5 666,75 1023,1 1403,7 684,42 1029,1 1428,6 690,29 1055,3 1440, 701,51 1074,1 1467, 739,03 1078,2 1471,7 749,68 1097,7 1501,2 756,65 1125,4 1531, 810,92 1141,6 1537,3 820,27 1166,8 1556,9 858,29 1190,7 1560,1 881,73 1200,9 1605,5 898,66 1213,3 1617,4 930,09 1261,3 1633,7 943,07 1285, 1647,8 960,81 1304,6 1686,6 965,41 1333,5 1689,7

1703,1 1727, 1743,4 1777,8 1797,5 1824,2 1826,1 1827,8 1828,3 1829,4 1830, 1830,3 1832,2 1834,6 1858,1 1861,2 1874,1 1883,6 1943,2 1965,8

Table 5.1. Modal Frequencies Obtained with FEM Analysis.

In the comparison of modal parameters, we did not obtain a convergence between the FE Modes and the OMA results, which suggest that the finite element model is not reliable. First, the modal frequencies of table 3.22 do not coincide with modal frequencies in table 5.1, and even if the modal frequencies were close, the modal shapes present a completely different behavior.

Next figure contents some examples of modal shapes computed in Ansys.

83

(a)

(c)

(b)

(d)

Figure 5.1. FEM Modal shapes at a) 507.07 Hz, b) 1166.8 Hz, c) 494.06 Hz and d) 1428.6 Hz. The previous images suggest the presence of local deformations in the machine, and due to the limitation of the number of channels during the acquisition, the identification and evaluation of those local deflections in ODS visualization are not possible.

After the comparison between OMA and FEM, we can discuss some limitations presented during the construction of the 3D model: 







The absence of the asset drawings can induce several errors during the construction of the 3D model, because the measurement was done with the machine assembled and some unknown dimensions were assumed, generating mistakes.

As we can remember, the dimensions and weight of the auxiliary drive and steam press were unknown. We consider that the simplification with estimated concentrated masses generate a big error during the computation of analytical modes; especially because the dimensions of those components are considerable with respect to the gearbox dimensions.

Also, the assumption of bearing stiffness may produce errors during the analytical simulation for the construction of the stiffness matrix. In real operation the bearing stiffness is not linear, and classical FEM softwares have limitations for a correct modelling.

As a last point, the supposition of cylindrical supports for the input and output shafts can be wrong, because in real life those shafts are flexible.

84

CONCLUSIONS This thesis evaluates the application of Operational Modal Analysis for rotating machinery.

Experimental test was performed in order to implement OMA in a real gearbox operating in a paper mill plant. After the development of this project we have reached to the following conclusions: The number of modes identified at Classical Tests was higher than OMA results and this is because OMA performance depends in the number of excited modes at operative condition.

Also, we conclude that single-reference methods work better than multi-reference methods for a large amount of data, and this is evident if we compare the number of modes calculated using both approaches. As we know from the theory, multi-reference methods are useful for the identification of repeated or closed modes, but unfortunately they have a limitation for parameter extraction in a large frequency band. Moreover, we observe that that the implementation of the Complex Mode Indicator Function and Multivariate Mode Indicator Function produce similar results.

During curve fitting, we have found larger number of real modes using the AF Poly and CE method, instead of the Z Poly; however, all the results presented non-real modes (harmonics) at the modal parameter extraction, and in the case of AF Poly there were errors in the estimation of modal damping ratios. Nevertheless, modal shapes of calculated modes were similar for all the methods. Besides, the application of Stability Diagram is useful for the reduction of the number of harmonics present at the curve fitting results, this can be done selecting a correct minimum number of stable poles or increasing the frequency bandwidth of ODS data. However, the splitting of frequency length during curve fitting brings larger number of real modes but also a larger number of harmonics.

In case of the analyzed methods for harmonic identification, the results showed that the Probability Density Function is the most reliable and accurate method. However the use of Short Time Fourier Transformation and Singular Value Decomposition can be correctly applied if the enough number of fundamental frequencies in the system is provided, in another case some wrong estimations can be generated.

Unfortunately, we were not able to validate OMA results using a Finite Element Analysis, because the information provided by the End User was not enough for the generation of a reliable model.

85

RECOMENDATIONS We consider that the knowledge acquired during the development of this work is enough for the presentation of some recommendations useful for the application of Operational Modal Analysis or future researches. 

First, as we saw during the acquisition of our test, technical errors were presented at the testing step, so it is important to verify the correct performance of Data Acquisition Systems, wiring and transducers before the ODS measurement.  Unfortunately the time of stop of the machine was not large enough for the application of Hammer Test, and as a consequence a comparison of modal damping ratios and modal shapes was not possible. We suggest for future evaluations the comparison with FRF results.  A technical issue was the maximum acquisition time available within SKF IMx-P: 3.2 second, but we recommend the acquisition of larger recording time histories in order to ensure that modal frequencies are sufficient excited.  The information supplied by the customer was not enough for a proper evaluation of OMA, especially during the harmonic identification. So, for future cases we will suggest the analysis in a more controllable asset, with enough information of the system and the disposition of the 3D drawings for the generation of a reliable FEM analysis  Also, we will suggest for future works the comparison of LMS and B&K Methods.

86

APPENDIX A. Harmonics due to Mechanical Components in a Rotating Machinery (The following table is a simplification of the Vibration Diagnostic Wall Chart of the Technical Associates of Charlotte, p.c. The table does not contain information related to bearings, lubrication or electric motors because the location of harmonic frequencies depend in a complicated analysis.)

87

APPENDIX B. Pictures of the dismounted gearbox.

Figure B.1. Input side of the gearbox

Figure B.1. Output side of the gearbox

88

APPENDIX C. Comparison between signal of Channel 1 and Channel 2.

(a)

(c)

(b)

(d)

Figure C. a) Time signal of Channel 1 b) Time signal of Channel 2 c) FFT of channel 1 d) FFT of Channel 2

In the last figure, we can see the difference between the incorrect signal of channel 1 and a correct signal (channel 2). First, in the time domain, we observe that channel 1 presents an unsteady condition (a), while the correct channel gives a steady response (b); also, we can see that Channel 1 presents a very low acceleration in comparison with the correct signal and this is not correct. In another way, at the frequency domain, it is evident the difference between both channels: Channel 1 has a large noise, with an unusual distribution of harmonics in respect with the correct signal.

89

APPENDIX D. Matlab Codes.

D.1 BUMP TEST %BUMP TEST clear all clc %Importating the .csv files of a local folder. Each file corresponde to a %different direction of the excitation at point 15 DATAx=csvread('C:\...\1X.csv',61,0); %Direction X DATAy=csvread('C:\...\1Y.csv',61,0); %Direction Y DATAz=csvread('C:\...\1Z.csv',61,0); %Direction Z %Point 8X frq=DATAx(:,1); %[Hz] frequency X8=DATAx(:,2); %[g] measured magnitude Y8=DATAy(:,2); %[g] measured magnitude Z8=DATAz(:,2); %[g] measured magnitude %Point 11X X11=DATAx(:,3); %[g] measured magnitude Y11=DATAy(:,3); %[g] measured magnitude Z11=DATAz(:,3); %[g] measured magnitude %Point 15Z X15=DATAx(:,4); %[g] measured magnitude Y15=DATAy(:,4); %[g] measured magnitude Z15=DATAz(:,4); %[g] measured magnitude %Point 16Y X16=DATAx(:,5); %[g] measured magnitude Y16=DATAy(:,5); %[g] measured magnitude Z16=DATAz(:,5); %[g] measured magnitude %Following code is for the plot of each point plot(frq,X8,frq,Y8,frq,Z8); title('BUMP POINT 8X'); xlabel('Frequency (Hz)'); ylabel('Acceleration (g)'); legend('X excitation','Y excitation','Z excitation'); figure; plot(frq,X11,frq,Y11,frq,Z11); title('BUMP POINT 11X'); xlabel('Frequency (Hz)'); ylabel('Acceleration (g)'); legend('X excitation','Y excitation','Z excitation'); figure; plot(frq,X15,frq,Y15,frq,Z15); title('BUMP POINT 15Z'); xlabel('Frequency (Hz)'); ylabel('Acceleration (g)'); legend('X excitation','Y excitation','Z excitation'); figure; plot(frq,X16,frq,Y16,frq,Z16); title('BUMP POINT 16Y'); xlabel('Frequency (Hz)'); ylabel('Acceleration (g)'); legend('X excitation','Y excitation','Z excitation');

90

D.2 SHORT TIME FOURIER TRANSFORMATON (STFT). %Short Time Fast Fourier Transform %This procedure is done for each channel measurement, in order to check the stationary condiiton of our system and also to identify %natural frequencies and harmonic component clear all clc %reading the wav files from a local folder %Fs is te samplig frequency [AcCh(:,1),Fs,Nbit]=wavread('C:\...\2X.wav'); AcCh(:,2)=wavread('C:\...\3X.wav'); AcCh(:,3)=wavread('C:\...\4X.wav'); AcCh(:,4)=wavread('C:\...\5X.wav'); AcCh(:,5)=wavread('C:\...\6X.wav'); AcCh(:,6)=wavread('C:\...\7X.wav'); AcCh(:,7)=wavread('C:\...\8X.wav'); AcCh(:,8)=wavread('C:\...\9X.wav'); AcCh(:,9)=wavread('C:\...\10X.wav'); AcCh(:,10)=wavread('C:\...\11X.wav'); AcCh(:,11)=wavread('C:\...\12Z.wav'); AcCh(:,12)=wavread('C:\...\13Z.wav'); AcCh(:,13)=wavread('C:\...\14Z.wav'); AcCh(:,14)=wavread('C:\...\15Z.wav'); AcCh(:,15)=wavread('C:\...\16Y.wav'); Nsamples=size(AcCh,1); %number of samples Nchannels=size(AcCh,2); %number of channels %Applying Short Time-FFT for each channel F=0:2000; %[Hz]Frequency range for i=1:15; %Now we call the spectrogram function, applying a hanning window and 50% %of overlap [S,F,T]=spectrogram(AcCh(:,i),[],[],F,Fs); %lines for plot subplot(3,5,i); surf(T,F,((abs(S))),'EdgeColor','none'); axis xy; axis tight; colormap(jet); view(0,90); xlabel('Time'); ylabel('Frequency (Hz)'); title(sprintf('STFT FOR SIGNAL %i', i+1)); end

91

D.3 Singular Value Decomposition (SVD). %SVD (Singular Value Decomposition) %SVD method for evaluating the harmonic frequencies clear all clc %Reading all the WAV files [AcCh(:,1),Fs,Nbit]=wavread('C:\...\2X.wav'); AcCh(:,2)=wavread('C:\...\3X.wav'); AcCh(:,3)=wavread('C:\...\4X.wav'); AcCh(:,4)=wavread('C:\...\5X.wav'); AcCh(:,5)=wavread('C:\...\6X.wav'); AcCh(:,6)=wavread('C:\...\7X.wav'); AcCh(:,7)=wavread('C:\...\8X.wav'); AcCh(:,8)=wavread('C:\...\9X.wav'); AcCh(:,9)=wavread('C:\...\10X.wav'); AcCh(:,10)=wavread('C:\...\11X.wav'); AcCh(:,11)=wavread('C:\...\12Z.wav'); AcCh(:,12)=wavread('C:\...\13Z.wav'); AcCh(:,13)=wavread('C:\...\14Z.wav'); AcCh(:,14)=wavread('C:\...\15Z.wav'); AcCh(:,15)=wavread('C:\...\16Y.wav'); %calling the welch periodogram method h=spectrum.welch; Nsamples=size(AcCh,1); %number of samples Nchannels=size(AcCh,2); %number of channels %Calculating the Auto Power Spectral Density for each signal for i=1:Nchannels; [Apsd(:,i),freq] =cpsd(AcCh(:,i),AcCh(:,i),[],686,1024,Fs); end %Applying SVD for i=1:size(Apsd(:,1),1); %Building the matrix for each frequency for j=1:Nchannels; vector(j)=Apsd(i,j); end Sxx=diag(vector); [V,SVD_S,V]=svd(Sxx); %Applying SVD for each frequency SV(:,i)=diag(SVD_S); end %superposing all the curves plot(freq,20*log(SV(10,:)),freq,20*log(SV(11,:)),freq,20*log(SV(12,:)),freq,20*log(SV(13,:)),f req,20*log(SV(14,:)),freq,20*log(SV(15,:)) ); axis([0 2000 -400 0]); xlabel('Frequency [Hz]'); ylabel('Amplitude [dB]');

92

D.4 Probability Density Function (PDF). clear all clc %In next lines we import the time domain signal of the acc. measurement in %one channel. Then we calculate the cross-power spectral density taking %N=1024, and then we plot the function. [x,Fs,Nbit]=wavread('C:\Users\Manuel\Desktop\Acquisition\ODS\CARCASE\With taco\2X.wav'); [X,f]=cpsd(x,x,[],686,1024,Fs); %We plot the the auto-spectra plot(f,20*log(X)); figure; %Maximum frequency fmax=f(size(f,1));

%Cutting the spectra to the desired peak f_l=2025; %start f_h=2040; %end %saving the frequency signal in the range for i=1:size(f); if f(i)>=f_l; i_start=i; break end end for i=1:size(f); if f(i)>=f_h; i_end=i; break end end n=1; for i=i_start:i_end; new_f(n)=f(i); new_X(n)=X(i); n=n+1; end %Applying PDF Function ksdensity(new_X)

93

BIBLIOGRAPHY [1] A. Fasana and S. Marchesiello. “Meccanica delle vibrazioni”. CLUT, Torino, 2006.

[2] N. Maia and J. Silva. “Theoretical and Experimental Modal Analysis”. Research Studies Press LTD. England, 1997 [3] C. W. De Silva. “Vibration: Fundamentals and Practice”. 2nd Edition, CRC Press, Boca Raton, FL, 2007.

[4] M. Richardson and B. Schwarz, “Modal Parameter Estimation from Operating Data”.

Jamestown, California, 2003.

[5] A. W. Phillips, R.J. Allemang and W. A. Fladung. “The Complex Mode Indicator Function (CMIF) as a Parameter Estimation Method”. University of Cincinnati, USA. Presented at IMAC XVI, 1998.

[6] R.J. Alemang and D. L. Brown. “A Complete Review of the Complex Mode Indicator Function (CMIF) with Applications”. Proc. Of the International Conference on Noise and Vibration Engineering. Leuven, 2006.

[7] R. Williams, J. Crowley and H. Vold. “The Multivariate Mode Indicator Function in Modal Analysis”. Presented at IMAC III, 1985.

[8] M. Rades. “Performance of various mode indicator functions”. Universitatea Politehnica Bucuresti. Rumania, 2010.

[9] H.Vord, K. Napolitano, D. Hensley and M. Richardson. “Aliasing in Modal Parameter Estimation: An Historical Look and New Innovations”. Presented at IMA XXV, 2007.

[10] R. Brillhard, K. Napolitano, D. Osterholt. “Utilization of Alias Free Polyreference for Mixed Mode Structures”. Presented at IMAC XXVI, Orlando, Florida, 2008.

[11] “ME´scopeVES Application Note #19: Using the Z-Polynomial Method with the Stability Diagram”. Vibrant Technology, Inc. 2004 [12] B. Schwarz, and M. Richardson, “Introduction to Operating Deflection Shapes”. CSI Reliability Week, Orlando, FL, 1999.

[13] B. Schwarz, and M. Richardson, “Measurements Required for Displaying Operating Deflection Shapes”. Presented at IMAC XXII, Jamestown, California, 2004.

[14] B. Wonki, K. Yongsoo, J. Dayou, K. Park and S. Wang. “Scaling the Operating Deflection Shapes Obtained from Scanning Laser Doppler Vibrometer”. Journal of Nondestructive Evaluation. Volume 30, Issue 2, pp 91-98. 2011 [15] L. Hermans, and H. Van Der Auweraer. “Modal testing and analysis of structures under operational conditions: Industrial Applications”. LMS International, Belgium, 1999.

[16] M. Richardson, B. Shwarz. “Modal Parameter Estimation from Operating Data”.

Proceedings of the 29th IMAC Conference. California, 2003.

[17] H. Herlufsen, S. Gade, N. Møller. “Identification Techniques for Operational Modal Analysis-An Overview and Practical Experiences”. Proceedings of 1st IOMAC Conference, Copenhagen, Denmark. 2005 [18] L. Zhang, R. Brincker, and P. Andersen. “An overview of Operational Modal Analysis: Major Development and Issues”. Aalborg Universitet, Denmark, 2005.

[19] S. Gade, R. Schlombs, C. Hundeck and C. Fenselau. “Operational Modal Analysis on a Wind Turbine Gearbox”. Proceedings of IMAC XXVII. Orlando, Florida, 2009 [20] N. Jacobsen. “Separating Structural Modes and Harmonics Components in Operational Modal Analysis”. Proceedings of IMAC XXIV Conference, 2006.

94

[21] B. Emiliano, J. Ferreira and T. C. de Freitas. “Effect of Harmonic Components in an Only Output Based Modal Analysis” Proceedings of COBEM 2009. Brazil, 2009 [22] “Rolling bearings in paper machines”. SKF, 2002.

[23] B. Schwarz & M. Richardson. “Using the De-Convolution Window for Operational Modal Analysis”. Proceedings of the IMAC 2007. 2007

95