Dynamic Characterization of Mechanical Components by Inverse Methods

Dynamic Characterization of Mechanical Components by Inverse Methods: Application to Bolted Joints Diogo André Azevedo Pontes Dissertação do Mestrado Integrado em Engenharia Mecânica Orientador: Professor José Dias Rodrigues Faculdade de Engenharia da Universidade do Porto Departamento de Engenharia Mecânica 2013 Dynamic Characterization of Mechanical Components by Inverse Methods: Application to Bolted Joints Diogo André Azevedo Pontes Dissertation submitted to Faculty of Engineering of University of Oporto to obtain the degree of Master in Mechanical Engineering Supervisor: Professor José Dias Rodrigues Laboratory of Vibrations of Mechanical Systems Department of Mechanical Engineering Faculty of Engineering of University of Oporto Oporto, July 2013 Acknowledgments First of all, I want to acknowledge Professor José Dias Rodrigues for his guidance and support during the development of this master thesis. Without his contribution, the execution of this work would never been possible. To my laboratory colleague Luis Cardoso for the help and opinions that he had given me during this work which were fundamental in some fields. To all my colleagues from Faculdade de Engenharia da Universidade do Porto (FEUP) for their friendship and attendance during these five years of course which is now ending. Lastly, thanks a lot to my dear family for all the support and conditions they have given and provided me along these years. v Resumo Na presente dissertação, componentes mecânicos denominados por juntas aparafusadas são analisados por métodos inversos, realizando-se assim uma caracterização dinâmica da referida junta de ligação. A caracterização dinâmica da junta de ligação realiza-se através de técnicas de desacoplamento que permitem obter as características de uma parte da viga sabendo-se o comportamento dinâmico de toda a viga e das restantes partes que serão desacopladas. De forma a realizar o desacoplamento, será considerada uma técnica de acoplamento de impedâncias, através da qual provêm as correspondentes técnicas de desacoplamento. As funções de resposta em frequência puras serão obtidas através do Método de Elementos Finitos, sendo posteriormente adicionado ruído de forma a simular dados experimentais. Para realizar a validação das técnicas implementadas, diferentes exemplos provenientes de obras científicas serão utilizados de forma a comprovar resultados. Com o objetivo de aumentar a eficácia destas técnicas de desacoplamento, introduzir-se-ão procedimentos experimentais tais como cancelamento de massas, análise de gamas de frequência e estimativa de funções de resposta em frequência não medidas. A performance e o impacto destes procedimentos serão avaliados e tidos em conta ao longo do trabalho. Posteriormente, serão adoptadas três vigas com configurações diferentes de forma a realizar uma análise experimental das mesmas. A aquisição experimental da informação da vigas será realizada de duas formas distintas, medindo todas as FRFs em todos os pontos e realizando uma análise modal experimental através de uma linha ou coluna, a partir da qual se obtém o modelo modal da estrutura global e as FRFs são obtidas por sintetização. As FRFs dos graus de liberdade de translação serão obtidas experimentalmente, enquanto que as FRFs correspondentes às rotações serão obtidas através de uma formulação que utiliza dados experimentais e teóricos provenientes do MEF. Uma vez realizada a caracterização dinâmica de toda a viga, as técnicas de desacoplamento abordadas anteriormente do ponto de vista teórico, serão implementadas com dados experimentais contaminados com ruído. vii Abstract In this dissertation, mechanical components called bolted joints are analysed through inverse methods, carrying out a dynamic characterization of the referred joint. This dynamic characterization makes use of uncoupling techniques which enable a dynamic identification of one part of the beam, knowing a priori the dynamic behaviour of the whole beam and the other substructures which will be uncoupled. In order to carry out the uncoupling of a bolted joint, an impedance coupling technique will be taken into account, whereby the corresponding uncoupling techniques are formulated. The Frequency-response functions will be generated through the Finite Element Method, and then some noise will be introduced to simulate experimental data. To perform the validation of these techniques which were implemented, different examples from scientific works will be done in order to compare the similarity of the results. Some experimental procedures were implemented with the purpose of enhance the effectiveness of these uncoupling techniques, such as a mass cancelation, an analysis of frequency ranges and an estimative of unmeasured FRFs. The performance and the impact of these procedures will be evaluated. Then, an experimental analysis of three beams with different configurations will be carried out. The experimental work will follow two different ways to obtain the dynamic information of the beam. Firstly, the FRFs will be measured in all translation degrees-of-freedom, in order to obtain the experimental matrix. Secondly, several modal analyses will be carried out using a row or a column of the main matrix, from which all the FRFs will be synthesised. The FRFs of the translation degrees-of-freedom are obtained experimentally, whereas the FRFs corresponding to the rotations are obtained through a formulation which uses experimental and theoretical data from the MEF. Once carried out the dynamic characterization of the whole beam, the uncoupling techniques, approached previously under the theoretical point of view, will be analysed with experimental data contaminated with noise. ix LIST OF FIGURES 3.12 Coupling of structure A with accelerometer B. . . . . . . . . . . . . . . . . . . . . 54 3.13 Finite element distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.14 Results obtained by the mass cancellation formulations. . . . . . . . . . . . . . . . 57 4.1 Experimental work diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Specimen1(Beam1).................................... 60 4.3 Specimen2(Beam2).................................... 60 4.4 Specimen 3 (Bolted joint). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Degrees-of-freedom considered per substructure. . . . . . . . . . . . . . . . . . . . 62 4.6 Measurement points considered per substructure. . . . . . . . . . . . . . . . . . . 62 4.7 Accelerance H97. ..................................... 64 4.8 Accelerance H97. ..................................... 65 4.9 Experimental results with different values of torque. . . . . . . . . . . . . . . . . . 66 4.10 Zoom taken from the sixth natural frequency. . . . . . . . . . . . . . . . . . . . . . 66 4.11 Accelerances H11 and H13................................. 67 4.12 Experimental LAC between transfer FRFs before and after of mass cancelation process,respectively. ................................... 68 4.13 Accelerance HB 11 obtained by uncoupling from experimental results. . . . . . . . . 69 4.14 Dynamic stiffness ZB 11 according to second and Wang formulations. . . . . . . . . 70 4.15 Accelerances HC 11 according to alternative coupling technique. . . . . . . . . . . . 70 4.16 Accelerances H1|1and H9|13................................ 71 4.17 Experimental LAC between transfer FRFs........................ 72 4.18 Force sensor used to measure the excitation (Force transducer 8203). . . . . . . . 72 4.19 Accelerances HC 1|1and HC 1|3. ............................... 73 4.20 Finite element mesh considered in this example. . . . . . . . . . . . . . . . . . . . 73 4.21 Accelerances H7|7and H2|7. ............................... 74 4.22 Accelerances H1|1and H1|2. ............................... 75 4.23 Experimental setup of the bolted joint. . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 Accelerances HC 1|1and HC 3|1. ............................... 80 5.2 Accelerance HB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3 Dynamic Stiffness ZB 11 of second and Wang formulations. . . . . . . . . . . . . . . 82 5.4 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 82 5.5 Experimentalsetup..................................... 83 5.6 Accelerances HC 1|15 and HC 15|15 experimental and canceled. . . . . . . . . . . . . . . 84 5.7 Accelerance HB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . . . . 85 5.8 Dynamic stiffness ZB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . 85 5.9 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 86 5.10 Accelerances HC 1|15 and HC 15|15............................... 87 5.11 Accelerance HB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . . . . 88 5.12 Accelerance HB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . . . . 88 5.13 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 89 5.14 Experimental setup with dummy masses. . . . . . . . . . . . . . . . . . . . . . . . 90 5.15 Accelerances HC 1|15 and HC 15|15............................... 91 5.16 Accelerance HB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . . . . 92 5.17 Dynamic stiffness ZB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . 92 5.18 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 93 5.19 Accelerances HC 1|15 and HC 15|15............................... 94 5.20 Accelerance HB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . . . . 95 xvi LIST OF FIGURES 5.21 Dynamic stiffness ZB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . 95 5.22 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 96 5.23 Accelerances HC 1|1and HC 3|1(row2)............................ 97 5.24 Accelerance HB 11 of the uncoupled joint (row 2). . . . . . . . . . . . . . . . . . . . . 98 5.25 Dynamic stiffness ZB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . 98 5.26 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 99 5.27 Accelerances HC 1|1and HC 3|1(row4)............................100 5.28 Accelerance HB 11 of the uncoupled joint (row 4). . . . . . . . . . . . . . . . . . . . . 101 5.29 Dynamic stiffness ZB 11 according to second and wang formulations. . . . . . . . . 101 5.30 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 102 5.31 Accelerances HC 1|1and HC 3|1(row5)............................103 5.32 Accelerance HB 11 of the uncoupled joint (row 5). . . . . . . . . . . . . . . . . . . . . 104 5.33 Dynamic Stiffness ZB 11 of the uncoupled joint. . . . . . . . . . . . . . . . . . . . . . 104 5.34 Accelerance HC 11 according to alternative coupling technique. . . . . . . . . . . . . 105 5.35 Comparison of estimated FRFs..............................106 5.36 Dynamic Stiffness ZB 11 obtained with all frequencies and with the selected frequencies, according to second formulation. . . . . . . . . . . . . . . . . . . . . . . 107 5.37 Dynamic Stiffness ZB 11 obtained with all frequencies and with the selected frequencies, according to Wang formulation. . . . . . . . . . . . . . . . . . . . . . . . 107 A.1 Euler-Bernoulli finite element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.1 Accelerances H91 and H93.................................124 B.2 Accelerances H95 and H97.................................124 B.3 Accelerances H99 and H9|11................................125 B.4 Accelerances H9|13 and H9|15. ..............................125 B.5 Accelerances H17 and H37.................................126 B.6 Accelerances H57 and H77.................................126 B.7 Accelerances H97 and H11|7................................127 B.8 Accelerances H13|7and H15|7. ..............................127 B.9 Accelerances of experimental analysis of beam 2 (1). . . . . . . . . . . . . . . . . . 128 B.10 Accelerances of experimental analysis of beam 2 (2). . . . . . . . . . . . . . . . . . 129 B.11 Accelerances of experimental analysis of beam 2 (3). . . . . . . . . . . . . . . . . . 130 B.12 Accelerances of experimental analysis of beam 2 (4). . . . . . . . . . . . . . . . . . 131 B.13 Accelerances of experimental analysis of beam 2 (5). . . . . . . . . . . . . . . . . . 132 B.14 Accelerances of experimental analysis of beam 2 (6). . . . . . . . . . . . . . . . . . 133 B.15 Accelerances of experimental analysis of beam 2 (1). . . . . . . . . . . . . . . . . . 134 B.16 Accelerances of experimental analysis of beam 2 (2). . . . . . . . . . . . . . . . . . 135 B.17 Accelerances of experimental analysis of beam 2 (3). . . . . . . . . . . . . . . . . . 136 B.18 Accelerances of experimental analysis of beam 2 (4). . . . . . . . . . . . . . . . . . 137 B.19 Accelerances of experimental analysis of beam 2 (5). . . . . . . . . . . . . . . . . . 138 B.20 Accelerances of experimental analysis of beam 2 (6). . . . . . . . . . . . . . . . . . 139 B.21 Accelerances H1|15 and H3|15. ..............................140 B.22 Accelerances H5|15 and H7|15. ..............................140 B.23 Accelerances H9|15 and H11|15...............................141 B.24 Accelerances H13|15 and H15|15. .............................141 B.25 Accelerances H1|15 and H3|15. ..............................142 B.26 Accelerances H5|15 and H7|15. ..............................142 B.27 Accelerances H9|15 and H11|15...............................142 xvii LIST OF FIGURES B.28 Accelerances H13|15 and H15|15. .............................143 xviii List of Tables 2.1 Beam characteristics of numerical example 1. . . . . . . . . . . . . . . . . . . . . . 27 2.2 Analysis of natural frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Beam characteristics of numerical example 2. . . . . . . . . . . . . . . . . . . . . . 29 2.4 Analysis of natural frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Geometry and material data of a beam finite element. . . . . . . . . . . . . . . . . 33 2.6 Analysis of natural frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 Geometry and material data of a beam finite element. . . . . . . . . . . . . . . . . 36 2.8 Comparison of uncoupling formulations . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Characteristics and properties of the beam. . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Characteristics and properties of the beam. . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Beam properties and characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1 Experimental measurements carried out in each beam. . . . . . . . . . . . . . . . 61 4.2 Characteristics and properties of the experimental beam. . . . . . . . . . . . . . . 62 4.3 NodesPositionperMode................................. 63 4.4 Natural frequencies obtained using a hammer. . . . . . . . . . . . . . . . . . . . . 64 4.5 Natural frequencies obtained using a shaker. . . . . . . . . . . . . . . . . . . . . . 65 4.6 Geometry and material data of a beam finite element. . . . . . . . . . . . . . . . . 73 5.1 Experimental modal analysis carried out in beam 2. . . . . . . . . . . . . . . . . . 78 5.2 Modal Parameters of the beam identification process. . . . . . . . . . . . . . . . . 79 5.3 MACCriterion. ...................................... 80 5.4 Different procedures for this experimental setup. . . . . . . . . . . . . . . . . . . . 83 5.5 Modal Parameters of the beam modal identification. . . . . . . . . . . . . . . . . . 84 5.6 MACCriterion. ...................................... 84 5.7 Modal Parameters of the beam modal identification. . . . . . . . . . . . . . . . . . 86 5.8 MACCriterion. ...................................... 87 5.9 Different procedures for this experimental setup with dummy masses. . . . . . . 90 5.10 Modal Parameters of the beam modal identification. . . . . . . . . . . . . . . . . . 90 5.11MACCriterion. ...................................... 91 5.12 Modal Parameters of the beam modal identification. . . . . . . . . . . . . . . . . . 93 5.13MACCriterion. ...................................... 94 5.14 Modal Parameters of the beam identification process (row 2). . . . . . . . . . . . . 97 5.15MACCriterion(row2)................................... 97 5.16 Modal Parameters of the beam identification process (row 4). . . . . . . . . . . . . 99 5.17MACCriterion(row4)...................................100 5.18 Modal Parameters of the beam identification process (row 5). . . . . . . . . . . . . 102 xix LIST OF TABLES 5.19MACCriterion(row5)...................................103 5.20 Input parameters of the algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 xx Nomenclature Abbreviations FRF Frequency-Response Function DOF Degree-of-freedom MAC Modal Assurance Criterium LAC Local Amplitude Criterium iInternal coordinate jJoint coordinate kInternal joint coordinate Matrix Variables BDeformation matrix LMatrix of the differential operators MMass matrix KStiffness matrix ¯ KComplex stiffness matrix NShape functions matrix (H)+Pseudo-Inverse Z(ω)Dynamic Stiffness ˜ Z(ω)Condensated Dynamic Stiffness H(ω)FRF matrix HX ij FRF matrix of structure Xwith coordinates iand j ΦModal matrix Scalar Variables EYoung’s modulus................................................................................................... IInertia Moment ρSpecific mass AArea of a surface lLength of a beam νPoisson coefficient ηrDamping coefficient associated to a natural frequency vWeight parameter of the algorithm of frequency analysis ωiFrequency of excitation riAmplification of matrix condition RiApproximation of ri xxi Vector Variables H(ω)Frequency-Response Function fc iLoad applied in substructure Cin coordinate i xc iDisplacement of coordinate iin substructure i Hpq(ω)FRF of a structure excited in point qand response measured in point p ˜ Hpq(ω)FRF with noise of a structure excited in point qand response measured in point p HA(ω)FRF obtained from a numerical model HX(ω)FRF obtained from a experimental analysis {ΨA}iModal vector of mode iof the numerical model {ΨX}jModal vector of mode jexperimentally measured Chapter 1 Introduction The conception and development of mechanical projects involves always joints to carry out the connection between different sub-structures. The two main joints types widely used in mechanical projects divide into bolted and adhesive joints. The bolted joints are the most used in mechanical construction, whereas adhesives have been improving their mechanical behavior along last decades although their application field is still a little limited. Normally, bolted joints provide a stronger mechanical connection, supporting high loads with good performances, being still the most used currently in main structures. On the other hand, adhesives joints have an application field more restrict than bolted joints, these have to be built with a determined size and thickness depending on the place where the connection joint will be applied. Therefore, the adhesive joints have their great applications in secondary substructures where there could be lower loads and the work conditions are not so demanding. A bolted joint could be regarded as a discontinuity in a structure which introduces high stress concentrations, changing the structure features and its behavior. These connection joints could have a big influence in the system natural frequencies and in the vibration modes, causing non-linear responses which are undesirable and too complicated to analyse and understand. The non-linearities in the joint interface are directly related to force-displacement relations. The added flexibility introduced by the joints heavily affects the structure behaviour and when subjected to dynamic loading, most of the energy is dissipated in the joints. The major source of damping in bolted joints comes from friction. When different substructures are assembled, we get a system which has higher damping values in comparison with individual members, and as this increase of damping is not caused by the members, some energy must be dissipated in the contact areas between the members when exposed to dynamic load. Due to these problems introduced by connection joints, there is the necessity of develop new methodologies to build structural models which could characterize physically these type of joints and make easier and predictable the comprehension of their behaviour. 1 CHAPTER 1. INTRODUCTION Nowadays, the analysis performed in the project of mechanical components are done using the Finite Element Method (MEF), which provides statics and dynamics analysis. This method makes estimations of natural frequencies and vibration modes, being possible the modeling of components with various shapes and sizes. The finite element method has an important role in the project of mechanical systems, serving as a prediction tool of dimensioning errors and high stress concentration zones in an anticipated working phase, performing a better knowledge about the prototype before its construction. Unfortunately, this method does not enable one approach about the dispersion or uncertainty of bolted joints which is important to understand once these joints have a great impact on the main structures behavior, namely, modifying the natural frequencies and the vibration modes, making impossible the acquisition of information of the physical properties of global structure. In this work, this method has a vast importance, once all the theoretical solutions are based on the assumptions of such method. Three innovative methodologies will be implemented in this work in order to improve the analysis of uncoupling techniques which will be approached throughout this work: • Mass Cancelation; • Estimative of Unmeasured FRFs. • Analysis of Frequency Ranges. These methodologies have an high impact and importance in the coherence and accuracy of results once without means such these, the final results would be even worse. The mass cancellation process allows to take into account the influence of the accelerometer’s masses in the response behaviour, such as in response amplitude and natural frequencies. The estimative of unmeasured FRFs provides the achievement of responses which can not be experimentally measured. Indeed, this process has an high importance once experimentally most of FRFs involving rotational degrees-of-freedom are difficult or even impossible to measure or analyse. The analysis of frequency ranges enables the knowledge of the frequencies where the noise level is lower and consequently a better linear regression could be performed to determine the corresponding dynamic stiffness. 1.1 Motivation This dissertation comes from another works done in the same area which promoted and raised problems and issues related to analysis of bolted joints once those researchers were not able to model these elements. For instance, a separated modal analysis of two substructures provides reliable results with modal shapes well defined and exact natural frequencies. When these two substructures are coupled using bolted joints to perform their connection, if another modal analyse is done, the results show an high discrepancy and lack of accuracy. Therefore, bolted joints are the cause of all this discrepancy and their behaviour needs to be studied and analysed to enable a better knowledge about their interference in the global system behaviour. 2 1.2. OBJECTIVES Therefore, this dissertation consists on a deep analysis of several uncoupling techniques and an attempt to introduce any modification or innovation which gives better results than those that have been achieved until the time. 1.2 Objectives The analysis of bolted joints has recognized problems which appear when these identification methods are implemented. The main objective of this work is to analyse, develop, apply and discuss identification techniques of bolted joints. Initially, in this work all the data necessary to test and validate the uncoupling techniques will be generated by the Finite Element Method, and then the global structure data will be replaced by experimental data which include noise and measurement errors. The introduction of experimental measured data requires the comprehension of problems related with experimental methods and devices used to process these data. These problems focus on questions related with numerical calculation, difficulties in the experimental data measurement and in the comparison between numerical and experimental results. Throughout this work some techniques will be approached in order to analyse the different data obtained: • Experimental Analysis Techniques; • Numerical methods, implementation of calculation programmes; • Coupling and Uncoupling Techniques; • New Experimental Methodologies; • Experimental Modal Analysis. The joint dynamic properties will be identified with uncoupling techniques. Four different uncoupling techniques will be considered with the purpose to evaluate and verify which provides better results when exposed to high noise levels and other experimental errors. In addiction, this work is carried out with the purpose of finding solutions to problems related with numerical uncertainties adjacent to matrix operations such inversions and differences. Furthermore, when noise is introduced, this problem is still more visible than without noise and numerical discrepancy increases a lot. This problem is intrinsic to experimental devices and to the process of experimental data acquisition. Then, an experimental work will be performed taking into account beams with bolted joints incorporated. One of most important objectives of this work is to apply all theoretical formulations to experimental data, replacing thus the origin of these data. Firstly, the global FRF matrix will be measured to achieve the beam dynamic characterization, and then we will measure a 3 CHAPTER 2. COUPLING / UNCOUPLING impedance coupling relates to an experimental way; having the information about the spatial model or the impedance model, it is possible to obtain the modal model and consequently the dynamic properties of a structure. Figure 2.1: Coupling techniques. In this work, several impedance coupling techniques will be approached owing to the facility associated with experimental and theoretical acquisition of FRFs and corresponding data treatment using a calculation software, MATLAB. The coupling process of subcomponents is normally a process more stable than the substructures uncoupling, principally due to matrix operations carried out in this process which interfere somewhat in the results obtained. Associated to these numerical problems is the noise introduced in experimental measurements by devices used to perform the measurements. Unfortunately, it is completely impossible to avoid the noise introduced by experimental measurements and eventually the only option to reduce noise levels comes by a technological improvement of such devices. 10 2.2. COUPLING TECHNIQUES 2.2 Coupling Techniques A coupling technique involves a connection between two structures in specific coordinates, assuming that the dynamic properties of both structures involved are known and well defined. The coupling of substructures A and B (Figure 2.2) can be performed with two different coupling techniques: • Classic Coupling Technique; • Alternative Coupling Technique. The theoretical assumptions of these two techniques will be analysed in the following sections of this chapter, as well as the numerical expressions which define both methods. 2.2.1 Classic Coupling Technique The classic coupling technique takes into account three different types of coordinates, as presented in figure 2.2. These coordinates are characterized as: • coordinates i, corresponding only to substructure A; • coordinates j, corresponding to the connection joint between substructures A and B; • coordinates k, corresponding only to substructure B. Figure 2.2: Coordinates to carry out the classic coupling. This method assumes that the coupling of two substructures, A and B, is rigidly connected through the coupling coordinates j, whereby the efforts are transmitted [2]. In this work, substructure B corresponds to the bolted joint whereas structure A represents the other two parts of the beam without physical connection. The assumptions of this technique focus on the following equations of force equilibrium and on the conditions of displacements compatibility [13]: (fA j+fB j=fC j xA j=xB j=xC j (2.1) 11 CHAPTER 2. COUPLING / UNCOUPLING The accelerance matrix Hwhich relate the structure accelerations with the dynamic loads could be defined in the following way: A(ω) = HF (ω)(2.2) The accelerance matrices of substructures A and B and structure C will be: HC=  HC ii HC ij HC ik HC ji HC jj HC jk HC ki HC kj HC kk  HA=HA ii HA ij HA ji HA jj HB=HB jj HB jk HB kj HB kk  Relating the forces equilibrium equations and the displacements compatibility conditions with the accelerance definition, the accelerance matrix HCis obtained. HC=    HA ii HA ji HA ij HA jj −10 0 0 0 0   +   0 0 0 0 0HB jj HB kj HB jk HB kk −1     −1 (2.3) The inversion process of the three matrices requires an high computational effort and furthermore, there is a high probability of having bad-conditioned matrices near the resonances and anti-resonance picks, which generate enormous numerical errors in their inverses. Therefore, the following matrix manipulation is performed: (HC)−1=  HA ii HA ji HA ij HA jj −10 0 0 0 I  +   I0 0 0 0HB jj HB kj HB jk HB kk −1  −  I0 0 0 0 0 0 0 I (2.4) Simplifying the second member of equation 2.4: (HC)−1= (HA #)−1+ (HB #)−1−I#(2.5) Multiplying in equation 2.5 at left by (HA #)−1HA #and at right by (HB #)−1HB #, comes: (HC)−1= (HA #)−1+ (HA #+HB #−HA #I#HB #)(HA #)−1(2.6) Thus, HCcould be calculated with one inversion only: HC=HB #(HA #+HB #−HA #I#HB #)−1HA #(2.7) 12 2.2. COUPLING TECHNIQUES Calculating the content inside the parenthesis: HC=  I0 0 0 0HB jj HB kj HB jk HB kk    I HA ij 0 0Hjj HB jk 0 0 I  −1 HA ii HA ji HA ij HA jj 0 0 0 0 I (2.8) Hjj =HA jj +HB jj (2.9) The inversion of the second matrix in the second member can be written in the form:   I HA ij 0 0Hjj HB jk 0 0 I  −1 =  I−HA ij (Hjj)−1HA ij (Hjj)−1HB jk 0 (Hjj)−1−(Hjj)−1HB jk 0 0 I (2.10) Replacing the equation 2.10 in equation 2.8 and carrying out the multiplication of the three matrices of the second member, gives:   HC ii HC ij HC ik HC ji HC jj HC jk HC ki HC kj HC kk  =  HA ii −HA ij (Hjj)−1HA ji HA ij −HA ij (Hjj)−1HA jj HA ij (Hjj)−1HB jk HB jj(Hjj)−1HA ji HB jj(Hjj)−1HA jj HB jk −HB jj(Hjj)−1HB jk HB kj(Hjj)−1HA ji HB kj(Hjj)−1HA jj HB kk −HB kj(Hjj)−1HB jk   (2.11) Simplifying the expression 2.11,   HA ii HA ij 0 HA ji HA jj 0 0 0 HB kk  −  HA ij (Hjj)−1HA ji(Hjj)−1HA ij (Hjj)−1HA jj −HA ij (Hjj)−1HB jk HA ji −HB jj(Hjj)−1HA ji HA jj −HB jj(Hjj)−1HA jj HB jj(Hjj)−1HB jk −HB jk −HB kj(Hjj)−1HA ji −HB kj(Hjj)−1HA jj HB kj(Hjj)−1HB jk   (2.12) The elements of the second line of the second matrix are constituted by differences of matrices, so it is possible to simplify these three elements: HA ji −HB jj(HA jj +HB jj)−1HA ji = (I−HB jj(HA jj +HB jj)−1)HA ji = = ((HA jj +HB jj)(HA jj +HB jj)−1−HB jj(HA jj +HB jj)−1)HA ji = = ((HA jj +HB jj −HB jj)(HA jj +HB jj)−1)HA ji =HA jj(HA jj +HB jj)−1HA ji With a similar process, we can also obtain the following results to the other elements of second line: 13 CHAPTER 2. COUPLING / UNCOUPLING HA ji −HB jj(Hjj)−1HA ji =HA jj(HA jj +HB jj)−1HA jj (2.13) HB jj(Hjj)−1HB jk −HB jk =−HA jj(HA jj +HB jj)−1HB jk (2.14) Replacing 2.13 and 2.14 in 2.11 results: HC=  HA ii HA ij 0 HA ji HA jj 0 0 0 HB kk  −  HA ij (Hjj)−1HA ji HA ij (Hjj)−1HA jj −HA ij (Hjj)−1HB jk HA jj(Hjj)−1HA ji HA jj(Hjj)−1HA jj −HA jj(Hjj)−1HB jk −HB kj(Hjj)−1HA ji −HB kj(Hjj)−1HA jj HB kj(Hjj)−1HB jk  (2.15) Simplifying the expression 2.15 in the following form:   HC ii HC ij HC ik HC ji HC jj HC jk HC ki HC kj HC kk  =  HA ii HA ij 0 HA ji HA jj 0 0 0 HB kk  −  HA ij HA jj −HB kj  (HA jj +HB jj)−1HA ji HA jj −HB jk(2.16) All these mathematical manipulations were performed with the objective of reduce the computational effort, making less matrices inversions and reducing the errors introduced by these operations. In expression 2.16 there is just one matrix inversion, restricting thus possible numerical problems. 14 2.2. COUPLING TECHNIQUES 2.2.2 Alternative Coupling Technique The alternative coupling technique considers only the intern coordinates iof substructure A and the joint coordinates jof substructure B. In many situations, the intern coordinates of substructure B does not have interest or sometimes these coordinates have a difficult access. Figure 2.3: Coordinates to the realisation of alternative coupling The dynamic stiffness of structure C could be written as the sum of the dynamic stiffness of structure A with the dynamic stiffness of substructure B: ZC=ZA+ZB(2.17) Collocating on evidence the matrix ZAin the second member of equation 2.17: ZC=ZA(I+HAZB)(2.18) Inverting both members, it gives: HC= (I+HAZB)−1HA(2.19) In the form of sub-matrices with the coordinates iand j: HC ii HC ij HC ji HC jj=Iii 0 0Ijj+HA ii HA ij HA ji HA jj0 0 0ZB jj−1HA ii HA ij HA ji HA jj(2.20) Simplifying the previous equation: HC ii HC ij HC ji HC jj=Iii HA ij ZB jj 0Ijj +HA jjZB jj−1HA ii HA ij HA ji HA jj(2.21) Developing equation 2.21 results: 15 CHAPTER 2. COUPLING / UNCOUPLING HC ii HC ij HC ji HC jj=HA ii −HA ij ZB jj(Ijj +HA jjZB jj)−1HA ji HA ij −HA ij ZB jj(Ijj +HA jjZB jj)−1HA jj (Ijj +HA jjZB jj)−1HA ji (Ijj +HA jjZB jj)−1HA jj (2.22) Putting on evidence the matrix HA: HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jjHA ij ZB jj(Ijj +HA jjZB jj)−1HA ji HA ij ZB jj(Ijj +HA jjZB jj)−1HA jj HA ji −(Ijj +HA jjZB jj)−1HA ji HA jj −(Ijj +HA jjZB jj)−1HA jj  HA ii HA ij HA ji HA jj−HC ii HC ij HC ji HC jj=HA ij ZB jj(Ijj +HA jjZB jj)−1HA ji HA ij ZB jj(Ijj +HA jjZB jj)−1HA jj HA ji −(Ijj +HA jjZB jj)−1HA ji HA jj −(Ijj +HA jjZB jj)−1HA jj (2.23) The elements of the second line of the second matrix of the second member are constituted by differences of matrices , so it is possible to simplify these two elements: HA ji −(Ijj +HA jjZB jj)−1HA ji = (Ijj −(Ijj +HA jjZB jj)−1)HA ji = = ((Ijj +HA jjZB jj)(Ijj +HA jjZB jj)−1−(Ijj +HA jjZB jj)−1)HA ji = = ((Ijj +HA jjZB jj −Ijj)(Ijj +HA jjZB jj)−1)HA ji = =HA jjZB jj(Ijj +HA jjZB jj)−1HA ji (2.24) From the second member, following a similar process, it gives: HA jj −(Ijj +HA jjZB jj)−1HA jj =HA jjZB jj(Ijj +HA jjZB jj)−1HA jj (2.25) Replacing the equations 2.24 and 2.25 in equation 2.23, it yields: HA ii HA ij HA ji HA jj−HC ii HC ij HC ji HC jj=HA ij ZB jj(Ijj +HA jjZB jj)−1HA ji HA ij ZB jj(Ijj +HA jjZB jj)−1HA jj HA jjZB jj(Ijj +HA jjZB jj)−1HA ji HA jjZB jj(Ijj +HA jjZB jj)−1HA jj (2.26) Reorganizing the previous expression 2.26: HA ii HA ij HA ji HA jj−HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jj0 0 0ZB jj(Ijj +HA jjZB jj)−1HA ii HA ij HA ji HA jj(2.27) 16 2.2. COUPLING TECHNIQUES Simplifying the element which contains the dynamic stiffness of B: ZB(Ijj +HA jjZB jj)−1= ((Ijj +HA jjZB jj)HB jj)−1= (HA jj +HB jj)−1(2.28) Replacing the equation 2.28 in equation 2.27, gives: HA ii HA ij HA ji HA jj−HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jj0 0 0 (HA jj +HB jj)−1HA ii HA ij HA ji HA jj(2.29) Lastly, simplifying the expression 2.29 it comes: HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jj−HA ij HA jj(HA jj +HB jj)−1HA ji HA jj(2.30) The equation 2.30 represents the final expression obtained to the coupling between two substructures A and B. This expression provides an advantage once it is only necessary one matrix inversion to achieve the dynamic characterization of structure C. 17 CHAPTER 2. COUPLING / UNCOUPLING 2.3 Uncoupling Techniques The uncoupling process of substructures possesses always an high complexity and difficulty when a joint analysis is required. The inverse process of coupling provides the achievement of the joint dynamic features through the coupling general equation (2.30). According to equation 2.30, it is possible to develop three different expressions to determine the joint dynamic stiffness using different coordinates. Another formulation will be exposed, Wang formulation in order to compare with the previous expressions. Along this work will be considered the following four formulations: • Formulation 1: without making use of the joint coordinates; • Formulation 2: making only use of joint coordinates; • Formulation 3: making only use of joint coordinates iand j; • Wang Formulation. 2.3.1 Formulation 1 For some reasons, sometimes it is not possible to use the joint coordinates j[2]. These coordinates could be inaccessible due to their location or other technical issue and their experimental measurement could not be performed. The corresponding measurement technique will not use the joint coordinates. Therefore, the term of equation 2.30 to take into account is HC ii : HC ii =HA ii −HA ij (HA jj +HB jj)−1HA ji (2.31) Organizing the expression 2.31: HA ii −HC ii =HA ij (HA jj +HB jj)−1HA jj (2.32) Multiplying at left by (HA ij )−1, and at right by (HA ji)−1: (HA ij )−1(HA ii −HC ii )(HA ji)−1= (HA jj +HB jj)−1(2.33) HA jj +HB jj = (HA ji)(HA ii −HC ii )−1(HA ji)(2.34) 18 2.3. UNCOUPLING TECHNIQUES HB jj =HA ji(HA ii −HC ii )−1HA ij −HA jj (2.35) This operation can only be applied if the number of coordinates iis equal to the number of coordinates j, once just with this condition the inversion operations are possible. In other words, the matrices must be square. On the other hand, if the number of coordinates are different, i.e. i>j,HB jj is obtained multiplying at left by Wji and at right by Wij: WjiHA ij (HA jj +HB jj)−1HA jjWij =Wji(HA ii −HC ii )Wij (2.36) Rearranging the equation 2.36: (HA jj +HB jj)−1= (WjiHA ij )−1Wji(HA ii −HC ii )Wij(HA jiWij)−1(2.37) Making Wji =HA ji and Wij =HA ij : (HA jj +HB jj)−1= (HA jiHA ij )−1HA ji(HA ii −HC ii )HA ij (HA jiHA ij )−1(2.38) The expression 2.38 simplifies itself considering (HA jiHA ij )−1HA ji as the pseudo-inverse of HA ij , and HA ij (HA jjHA ij )−1as the pseudo-inverse of HA ji. Replacing, respectively, by (HA ij )+and (HA ji)+, in the expression 2.38 results: (HA jj +HB jj)−1= (HA ij )+(HA ii −HC ii )(HA ji)+(2.39) Manipulating the expression 2.38: HA jj +HB jj =HA ij (HA ii −HC ii )−1HA ji (2.40) It results, HB jj =HA ij (HA ii −HC ii )−1HA ji −HA jj (2.41) Lastly, the operation could be more effective doing the inversion of both members: HB jj =HA jiHA ij (HA ji(HA ii −HC ii )HA ij )−1HA jiHA ij −HA jj (2.42) 19 CHAPTER 2. COUPLING / UNCOUPLING 2.5 Numerical Examples Throughout this work, several uncoupling techniques have been regarded that differ numerically and consequently the results obtained are different. In order to test and validate those techniques, some numerical examples will be considered with the aim of understand which is the best technique to use experimentally. For each example, the four techniques exposed above will be analysed, introducing noise in the global system’s FRF to simulate experimental data. Some formulations will have a better behaviour than other when exposed to the noise introduced by experimental measured data. This different behaviour depends on operations with matrix inversions which affect the matrix condition and consequently the results obtained. The matrix condition will be analysed along the frequency range to understand its dependence with resonance and anti-resonance picks. Furthermore, some examples from articles and Ph.D Thesis will be considered to validate the algorithms adopted and the finite element programmes used to extract the FRF of the structure and its substructures. 2.5.1 Numerical Example 1 In this example an uncoupling of a joint is performed, substructure B, from the global structure C [2]. The beam has the same section along its length, with different lengths for each substructure. In this example three substructures were considered, A1, A2 and B, forming the global structure C, while substructures A1 and A2 are considered as an only substructure A, for the sake of calculation. All the substructures are represented in figure 2.4. Figure 2.4: Coupling of substructures A and B, forming the structure C. The analysis of this example was carried out using Euler-Bernoulli finite elements, each element with four degrees of freedom. For each substructure were considered ten Euler-Bernoulli elements. The FRFs of substructures A and B, and structure C, are calculated separately, from the FE spatial model by using a direct frequency analysis. In this case it will be assumed that is not possible to perform measurements in the joint coordinates. Therefore, it should be used the formulation 1 to determine the joint dynamic characteristics: 26 2.5. NUMERICAL EXAMPLES HB jj =HA ji(HA ii −HC ii )−1HA ij −HA jj Table 2.1: Beam characteristics of numerical example 1. Beam Length Width Thickness νEρ A1270 mm 30 mm 5 mm 0.3 194 GPa 7562 Kg/m3 B 200 mm 30 mm 5 mm 0.3 194 GPa 7562 Kg/m3 A2370 mm 30 mm 5 mm 0.3 194 GPa 7562 Kg/m3 To simulate the errors which are common in the acquisition of experimental data, the global structure’s FRFs were contaminated with noise to evaluate the method stability. The introduction of noise is done according to the following expression, depending on the amplitude: Hpq(ωk) = Hpq(ωk) + Hpq(ωk)γ 50(rand(1) −0.5) (2.83) Where γis the noise level in percentage, whereas rand(1) corresponds to an uniform random distribution with values between 0 and 1. In this example 1 % is considered as noise level. Two different analysis were done, one without introduction of noise and other considering noise. Without noise in the FRF matrix HC ii , all the FRFs curves obtained by this uncoupling method fit very well the exact solutions, showing a good stability. If some noise is introduced, the method becomes quite unstable, showing a great discrepancy when compared with the exact solution. The inversion operation (HA ii −HC ii )could be considered as the major responsible for this instability. Furthermore, this matrix, which has to be inverted at each frequency, shows low values of matrix condition. The results obtained for this example could be seen in figure 2.5. As we can see, when noise is introduced, the level of discrepancy increases and the FRFs with noise do not match very well with the exact solutions obtained by the Finite Element Method. Another way to validate these results comes from an analysis over the natural frequencies, which are well visible in figure 2.5. Considering only the joint as beam, its natural frequencies could be obtained through the following theoretical expression [16]: ωi=(βil)2 2πsEI ρAl4/Hz (2.84) Taking into account boundary conditions as free-free, the first value to (βil)in expression 2.84 in which appears the first natural frequency is (βil)=4.7300. The comparison between the experimental and theoretical results is presented in table 2.2. 27 CHAPTER 2. COUPLING / UNCOUPLING 0 100 200 300 400 500 600 700 800 900 1000 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact Curve Formulation 1 Noise 1% 0 100 200 300 400 500 600 700 800 900 1000 10−2 100 102 104 106 ω [Hz] Accelerance HB 12 [m/s2N] Exact Curve Formulation 1 Noise 1% 0 100 200 300 400 500 600 700 800 900 1000 10−1 100 101 102 103 104 105 ω [Hz] Accelerance HB 13 [m/s2N] Exact Curve Formulation 1 Noise 1% 0 100 200 300 400 500 600 700 800 900 1000 10−2 100 102 104 106 ω [Hz] Accelerance HB 14 [m/s2N] Exact Curve Formulation 1 Noise 1% Figure 2.5: Uncoupling of substructure B from structure C. Table 2.2: Analysis of natural frequencies. FRF Theoretical Value Experimental Value Error / % HB 11 ω1= 677.10Hz ω1= 651.70Hz ε = 3.75 HB 12 ω1= 677.10Hz ω1= 651.70Hz ε = 3.75 Therefore, considering the results without the noise interference, it is possible notice that the natural frequencies are almost exact with a low error associated. This fact contributes to the validation of this numerical example. 2.5.2 Numerical Example 2 This numerical example was performed with the purpose of compare the following different uncoupling techniques: • Formulation 1, without using joint coordinates; • Formulation 2, using only joint coordinates; • Formulation 3, using intern and joint coordinates; • Wang Formulation. 28 2.5. NUMERICAL EXAMPLES As in previous example, here the same number of substructures is considered with an identic arrangement. Figure 2.6 shows the number of degrees of freedom considered and their position along the beam. The joint section dimensions are different from the other two substructures in order to test the programm used to analyse beams with different sections. All the dimensions and physical properties of substructures are referred in table 2.3. Figure 2.6: Coupling of substructures A and B, forming the structure C. Table 2.3: Beam characteristics of numerical example 2. Beam Length Width Thickness νEρ A1400 mm 25 mm 3 mm 0.3 210 GPa 7850 Kg/m3 B 250 mm 25 mm 6 mm 0.3 210 GPa 7850 Kg/m3 A2400 mm 25 mm 3 mm 0.3 210 GPa 7850 Kg/m3 The structure analysis based on a finite element approach, considering Euler-Bernoulli elements as in previous example. For each substructure were established eight Euler-Bernoulli elements, making a total of 24 elements for the global structure. Figures 2.7 and 2.8 present the results obtained with the four uncoupling techniques. Analysing both graphics, almost all the formulations provide FRFs which fit very well the exact FRF and just formulation 1 shows discrepancy and instability in some frequency ranges. The instability of formulation 1 can be related with numerical operations which could be affecting the results. 0 500 1000 1500 2000 2500 3000 10−2 10−1 100 101 102 103 104 105 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 1 Formulation 2 Formulation 3 Wang Formulation Figure 2.7: Accelerance HB 11 obtained with the uncoupling techniques. 29 CHAPTER 2. COUPLING / UNCOUPLING 0 500 1000 1500 2000 2500 3000 10−1 100 101 102 103 104 105 106 107 ω [Hz] Accelerance HB 12 [m/s2N] Exact FRF Formulation 1 Formulation 2 Formulation 3 Wang Formulation Figure 2.8: Accelerance HB 12 obtained with the uncoupling techniques. Once again, a verification of the natural frequencies could be done taking into account the equation 2.84. As we can see in figures 2.7 and 2.8, all the formulations have coincident natural frequencies. Table 2.4: Analysis of natural frequencies. Natural Frequency Theoretical Value / Hz Experimental Value / Hz Error / % ω1510.38 510.3 0.016 ω21406.92 1408.00 0.077 ω32758.12 2765.00 0.249 Table 2.4 shows a good correlation between the theoretical and experimental results, and the error calculated is lower than 1 %. Normally, experimental data owns always some errors caused by noise introduced during the corresponding measurements. To simulate these errors, a noise level of 1 % was introduced in previous FRFs obtained, considering noise which is dependant on the amplitude as in previous example. As we can see in figures 2.9 and 2.10, the noise introduced causes an high discrepancy and instability in results obtained, principally in resonance and anti-resonance zones. The first and third formulations show great instability along the frequency range and it is completely impossible to identify any natural frequencies as well as antiresonance picks. The second and Wang formulations provide much better results, despite the visible instability still present. 30 2.5. NUMERICAL EXAMPLES 0 500 1000 1500 2000 2500 3000 10−3 10−2 10−1 100 101 102 103 104 105 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 1 Noise 1% Formulation 2 Noise 1% Formulation 3 Noise 1% Wang Formulation Noise 1% Figure 2.9: Accelerance HB 11 obtained with 1% of noise. 0 500 1000 1500 2000 2500 3000 10−1 100 101 102 103 104 105 106 107 108 ω [Hz] Accelerance HB 12 [m/s2N] Exact FRF Formulation 1 Noise 1% Formulation 2 Noise 1% Formulation 3 Noise 1% Wang Formulation Noise 1% Figure 2.10: Accelerance HB 12 obtained with 1% of noise. The comparison between results obtained with noise and the clean data was performed with a correlation criterion, LAC. Figure 2.11 shows this criterion applied to the four formulations for the joint degrees-of-freedom. Observing figure 2.11, it is possible to conclude that formulation 2 possesses and provides better results once LAC values obtained are the highest, close to 1 which is the recommended. Relating to other formulations, only Wang formulation has sustainable LAC values whereas first and third formulations give lower values showing an high instability. Therefore, measurements in joint coordinates should be done whenever possible once formulation 2 provides 31 CHAPTER 2. COUPLING / UNCOUPLING better results than other formulations. Figure 2.11: Average LAC of joint coordinates’ FRFs. 32 2.5. NUMERICAL EXAMPLES 2.5.3 Numerical Example 3 This numerical example was carried out to evaluate the same four uncoupling techniques approached in previous example, considering a different beam with a round section along its length [8]. The number of substructures is the same and Euler-Bernoulli finite elements were used. The number of elements and degrees-of-freedom is available in figure 2.12. The FRFs of both substructures were calculated separately considering eight Euler-Bernoulli elements for the main structure. For the sake of agreement and equality, only one finite element was taken into account to represent the joint model. Figure 2.12: Coupling of substructures A and B, forming the structure C. Here in this example, the joint has a geometry and properties equal to the other parts of the structure. The properties and characteristics of each Euler-Bernoulli element are presented in table 2.5. Table 2.5: Geometry and material data of a beam finite element. Length L = 0.1 m Diameter d= 0.005 m Young’s modulus E = 2.07 ×1011 N/m2 Shear modulus G = 0.385.E Density ρ= 7547 Kg/m3 The results obtained for the joint uncoupling are presented in figures 2.13 and 2.14. A detailed analysis of the results indicates that all techniques converge with exact curve at blue, even though first formulation already shows a low discrepancy in some frequencies. Table 2.6 provides a study about the natural frequency value, which isn’t equal to the theoretical value calculated. Indeed, this fact is related to the number of elements considered to the joint, in this case just one. Thus, the natural frequencies depends also on the number of elements, decreasing the error with the raise of the number of elements taken into account. The accelerance curves obtained are similar to results published in [8], which validates and confirms the results of this example. Although a good visible coherence, it is possible to check that there is an high level of discrepancy, and this fact could be related with the low number of finite elements considered to calculate the FRFs. Therefore, another factor affects the stability of these methods which are really sensitive to external conditions. 33 CHAPTER 2. COUPLING / UNCOUPLING 0 500 1000 1500 2000 2500 3000 10−1 100 101 102 103 104 105 106 ω [Hz] Accelerance HB 11 [m/s2N] Exata Formulation 1 Formulation 2 Formulation 3 Wang Formulation Figure 2.13: Uncoupled accelerance HB 11. 0 500 1000 1500 2000 2500 3000 100 101 102 103 104 105 106 107 108 ω [Hz] Accelerance HB 12 [m/s2N] Exata Formulation 1 Formulation 2 Formulation 3 Wang Formulation Figure 2.14: Uncoupled accelerance HB 12. Table 2.6: Analysis of natural frequencies. Natural Frequency Theoretical Value / Hz Experimental Value /Hz Error / % ω12331.04 2795.00 19.89 A noise level of 1 % was introduced in the global system’s FRFs to obtain the accelerance HB 11. The noise introduced depends on the amplitude. Figure 2.15 presents the FRFs contaminated 34 2.5. NUMERICAL EXAMPLES with noise where once again second and Wang formulation provide better results than other formulations. Figure 2.15: Uncoupled FRFs HB 11 with 1 % of noise. The contaminated FRFs with noise were compared with the original ones using a correlation criterium, LAC. Figure 2.16 shows the mean LAC for the joint degrees-of-freedom; the second and Wang formulation give again better results than other formulations which reinforces that joint measurements should always be done whenever possible. Figure 2.16: Average LAC of joint coordinates’ FRFs. 35 CHAPTER 3. EXPERIMENTAL DATA ANALYSIS Formulation 1: HB jj =HA jiHA ij (HA ji(HA ii −HC ii )HA ij )−1HA jiHA ij −HA jj Formulation 2: HB jj = (HA jj(HA jj −HC jj)−1−Ijj)HA jj Formulation 3: HB jj =HA jj(HA ji(HA ij −HC ij ))−1HA jiHA ij −HA jj Wang Formulation: HB jj =HA ji HA jjHA ii HA ii +HA ii −HC ii HA ii HA ij +HA ij −HC ij HA jiHA ii +HA ji −HC ji HA jiHA ij +HA jj −HC jj−1HA ij HA jj−HA jj According to previous expressions, in each formulation the matrices which have to be inverted are the following: A1=HA ji(HA ii −HC ii )HA ij A2=HA jj −HC jj A3=HA ji(HA ij −HC ij ) A4=HA ii HA ii +HA ii −HC ii HA ii HA ij +HA ij −HC ij HA jiHA ii +HA ji −HC ji HA jiHA ij +HA jj −HC jj The condition of each matrix Aiwill be calculated for each frequency ωin order to analyse the FRF behaviour and understand in which ranges the uncertainty and discrepancy are lower to provide better results. Normally, high values of matrix condition are associated with an high level of noise in a determined frequency range. 3.2.1 Algorithm Once calculated the matrices condition along the frequency range, these data have to be processed and analysed. The function ri(ω)relates the condition logarithm, making a magnification of the matrix condition value: ri(ω) = τln(cond(Ai(ω))) (3.1) 42 3.2. ANALYSIS OF FREQUENCY RANGES The parameter τrepresents a magnification factor of the condition logarithm. To identify the ranges where this function is minimum was used a Matlab function that provides a regression along the frequency range: Ri(ω) = csaps(ω, ri(ω), v)(3.2) where vis a weight parameter that varies between 0 to 1. If νis equal to zero the regression is linear, whereas if vis equal to 1 we have an interpolation that passes in all points along the frequency range. To calculate the function minimums, the first and second derivative of Ri(ω)were analysed. It was established that the first derivative should be lower to a given parameter ξand the second derivative should be positive: dRi dω < ξ ^d2Ri dω2>0(3.3) 3.2.2 Numerical Example 5 This numerical example was carried out to implement the algorithm proposed in previous section. This algorithm provides a better comprehension and knowledge about the frequency ranges where the measurements should be done according to the assumptions of that algorithm. For substructures A1and A2were considered six Euler-Bernoulli finite elements, making a total of thirteen elements for the whole structure. Figure 3.1 shows the beam considered with the corresponding finite elements distribution and substructures. Figure 3.1: Coupling of substructures A and B, forming the structure C. The beam’s dimensions and characteristics are presented in table 3.1. Once again, in this example set up a noise level of 3 % dependent of the amplitude as considered in other examples. 43 CHAPTER 3. EXPERIMENTAL DATA ANALYSIS Table 3.1: Characteristics and properties of the beam. Beam Length [mm] Width [mm] Thickness [mm] E [GPa] ρ[kg/m3] A1 270 30 5 194 7562 B 200 30 10 194 7562 A2 370 30 5 194 7562 The algorithm parameters set up as ξ= 0.05 and v= 10−6in order to perform the different analysis and find the frequency ranges where Ri(ω)has local minimums. The following figures show the results obtained for each formulation when this algorithm is applied. The grey bars correspond to the frequencies where ri(ω)has local minimums. 0 500 1000 1500 2000 10−10 10−8 10−6 10−4 10−2 100 Receptance HB 11 [m/N] ω [Hz] Exact FRF F. 1 Noise 3% 0 500 1000 1500 2000 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 ω [Hz] ri(ω),Ri(ω) ri(ω) Ri(ω) Figure 3.2: Receptance HB 11,ri(ω)and Ri(ω)of formulation 1. 0 500 1000 1500 2000 10−10 10−8 10−6 10−4 10−2 100 ω [Hz] Receptance HB 11 [m/N] Exact FRF F. 2 Noise 3% 0 500 1000 1500 2000 0 100 200 300 400 500 600 700 800 ω [Hz] ri(ω),Ri(ω) ri(ω) Ri(ω) Figure 3.3: Receptance HB 11,ri(ω)and Ri(ω)of formulation 2. 44 3.2. ANALYSIS OF FREQUENCY RANGES 0 500 1000 1500 2000 10−10 10−8 10−6 10−4 10−2 100 ω [Hz] Receptance HB 11 [m/N] Exact FRF F. 3 Noise 3% 0 500 1000 1500 2000 200 300 400 500 600 700 800 900 1000 1100 ω [Hz] ri(ω),Ri(ω) ri(ω) Ri(ω) Figure 3.4: Receptance HB 11,ri(ω)and Ri(ω)of formulation 3. 0 500 1000 1500 2000 10−10 10−8 10−6 10−4 10−2 100 ω [Hz] Receptance HB 11 [m/N] Exact FRF Wang F. Noise 3% 0 500 1000 1500 2000 200 300 400 500 600 700 800 ω [Hz] ri(ω),Ri(ω) ri(ω) Ri(ω) Figure 3.5: Receptance HB 11,ri(ω)and Ri(ω)of Wang formulation . Through an analysis of previous figures, it is possible to get some conclusions. Wang and second formulations provide again better results than other formulations. The first formulation (figure 3.2) shows an high discrepancy along the frequency range, being completely impossible to understand and fit the real FRF. The curve rireflects this fact with few frequencies where Riis minimum. The second formulation (figure 3.3) provides much better results. The level of uncertainty and discrepancy is lower than in previous formulation and noise doesn’t affect so much the FRF. On the right side of figure 3.3 there are local minimums of Riwell visible and defined. The third formulation (figure 3.4) doesn’t provide good results and like in first formulation, it is possible to visualize an high discrepancy along the frequency range, although results are slightly better than in figure 3.2. Wang formulation (figure 3.5) gives better results than last formulation. The noise effect is not too visible and there isn’t a so huge discrepancy and uncertainty. On the right side of figure 3.5, it is possible to see some minimums of Ri. Despite the good results, formulation 2 provides 45 CHAPTER 3. EXPERIMENTAL DATA ANALYSIS better results than Wang formulation. Therefore, formulation 2 gives again better results and it reinforces that joints measurements should be carried out whenever possible. This algorithm serves as an interesting tool to choose the best frequency range to measure experimentally and its application gives information about matrix conditions for each frequency that is a complicated problem in this work. 3.2.2.1 Dynamic Stiffness Calculation Once obtained the frequency ranges where the noise level is lower and the function Rihas local minimums according to the algorithm presented, these frequencies were selected in order to obtain the joint dynamic stiffness. According to the results obtained in previous section, second and Wang formulations provide better results and consequently these two formulations were chosen to obtain the joint dynamic stiffness. Firstly, the joint dynamic stiffness was calculated at each frequency, ZB(ω) = (HB(ω))−1, and then with the selected frequencies a quadratic regression was carried out using those points. Figures 3.6 and 3.7 show the corresponding results. 0 500 1000 1500 2000 −4 −3 −2 −1 0 1 2x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] 0 500 1000 1500 2000 −4 −3 −2 −1 0 1 2x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Figure 3.6: ZB 11 obtained with all frequencies and with the selected frequencies for formulation 2. 46 3.3. UNMEASURED FRFS 0 500 1000 1500 2000 −4 −3 −2 −1 0 1 2x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] 0 500 1000 1500 2000 −4 −3 −2 −1 0 1 2x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Figure 3.7: ZB 11 obtained with all frequencies and with the selected frequencies for Wang formulation. The blue line corresponds to the exact dynamic stiffness whereas the black line is the dynamic stiffness obtained with the selected frequency ranges and the corresponding quadratic regression. Analysing the results obtained, formulation 2 provides again better results owing to the lower dispersion observed when noise is introduced in the samples. Therefore, selecting the frequency ranges where the noise level is lower has a great importance and impact once we can obtain results much more accurate and close to the exact solution. 3.3 Estimative of Unmeasured FRF An experimental analysis of a beam requires in most situations an estimative of FRFs, once some of those FRFs can not be measured due to experimental problems. Normally, in structures there are zones where is completely impossible to perform measurements owing to problems related to beam structure and geometry. Furthermore, measurements involving rotational degrees-of-freedom are difficult or even impossible to carry out and therefore, there is a need to find and develop formulations which allow the determination of these specific FRFs. In [9] and [13], authors have approached this problem and new formulations have been developed. These formulations assume that part of the structure is modelled numerically, for instance using the finite element method, and the unavailable experimental FRFs are determined by expressions which allow the evaluation of these FRFs. 47 CHAPTER 3. EXPERIMENTAL DATA ANALYSIS 3.3.1 Theoretical Formulation Maia and Batista [13] studied this problem and developed a new formulation to estimate unmeasured FRFs relating directly the dynamic stiffness of two substructures with the dynamic stiffness of the main structure, following a similar way as presented in section 2. The coordinates to take into account divide into two groups: experimentally measured coordinates, t, and coordinates rwhere the responses measurement is impossible to perform due to any reason, as shown in figure 3.8. Figure 3.8: Coupling of substructures A and B, forming the structure C. Using the alternative dynamic coupling expression to simplify, it comes: HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jj−HA ij HA jj(HA jj +HB jj)−1HA ji HA jj(3.4) From equation 3.4 it is possible to write four equations, which possess information from the three structures: HC ii =HA ii −HA ij (HA jj +HB jj)−1HA ji (3.5) HC ij =HA ij −HA ij (HA jj +HB jj)−1HA jj (3.6) HC ji =HA ji −HA jj(HA jj +HB jj)−1HA ji (3.7) HC jj =HA jj −HA jj(HA jj +HB jj)−1HA jj (3.8) Rearranging expression 3.5 and representing the pseudo-inverse (HA ij )+, (HA ij )+(HA ii −HC ii ) = (HA ii −HC ii )−1HA ji (3.9) 48 3.3. UNMEASURED FRFS Replacing in expression 3.7: HC ji =HA ji −HA jj(HA ij )+(HA jj +HB jj)(3.10) Reorganizing the expression 3.7: (HA jj)−1(HA ji −HC ji ) = (HA jj +HB jj)−1HA ji (3.11) Replacing in expression 3.24: HC ii =HA ii −HA ij (HA jj)−1(HA ji +HC ji )(3.12) Reorganizing the expression 3.6: (HA ij )+(HA ij −HC ij )=(HA jj +HB jj)−1HA jj (3.13) Iij −(HA ij )+HC ij = (HA jj +HB jj)−1HA jj (3.14) Replacing in expression 3.15: HC jj =HA jj(HA ij )+HC ij (3.15) Summarizing, it is possible to determine with the following expressions the matrix HC:      HC ji =HA ji −HA jj(HA ij )+(HA ii −HC ii ) HC ii =HA ii −HA ij (HA jj)−1(HA ji −HC ji ) HC jj =HA jj(HA ij )+HC ij The coordinates ifrom structure C will be composed by coordinates tand r. Therefore, the new matrix HCwill suffer the following transformation: HC=HC ii HC ij HC ji HC jj (3.16) i=t+r−→ HC=  HC tt HC tr HC tj HC rt HC rr HC rj HC jt HC jr HC jj  (3.17) 49 CHAPTER 3. EXPERIMENTAL DATA ANALYSIS In the new matrix HConly the sub matrix HC tt can be experimentally determined. Considering only coordinates ias translation coordinates, i=t: HC jt =HA jt −HA jj(HA tj )+(HA tt −HC tt )(3.18) Relating translations with rotations, i=rand i=t: HC rt =HA rt −HA rj(HA jj)−1(HA jt −HC jt )(3.19) i=t, i =r−→ HC jr =HA jr −HA jj(HA tj )+(HA tr −HC tr )(3.20) Considering only icoordinates as rotation coordinates, i=r: HC rr =HA rr −HA rj(HA jj)−1(HA jr −HC jr)(3.21) Lastly, considering coordinates ias translations, it gives: i=t−→ HC jj =HA jj(HA tj )+HC tj (3.22) The matrix HAis determined through the finite element method. Thus, starting from the responses known, all unknown FRFs can be estimated through a sequential process. 3.3.2 Alternative Formulation All the FRF of matrix HCcould be determined through a simpler way. Resuming the equations 3.23, 3.24 and 3.25: HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jj−HA ij HA jj(HA jj +HB jj)−1HA ji HA jj(3.23) HC ii =HA ii −HA ij (HA jj +HB jj)−1HA ji (3.24) i=t+r−→ HC=  HC tt HC tr HC tj HC rt HC rr HC rj HC jt HC jr HC jj  (3.25) 50 3.3. UNMEASURED FRFS From equation 3.24, it is possible to write: (HA jj +HB jj)−1= (HA ij )+(HA ii −HC ii )(HA ji)+(3.26) Replacing the equation 3.26 in equation 3.4: HC ii HC ij HC ji HC jj=HA ii HA ij HA ji HA jj−HA ij HA jj((HA ij )+(HA ii −HC ii )(HA ji)+)HA ji HA jj(3.27) If we use the coordinates idecomposition as presented in equation 3.27, the standard equation could be written as:   HC tt HC tr HC tj HC rt HC rr HC rj HC jt HC jr HC jj  =  HA tt HA tr HA tj HA rt HA rr HA rj HA jt HA jr HA jj  −  HA tj HA rj HA jj  ((HA tj )+(HA tt −HC tt )(HA jt)+)HA jt HA jr HA jj (3.28) 3.3.3 Numerical Example 6 In order to evaluate the feasibility of the alternative formulation exposed in previous section, the following example was carried out. Figure 3.9 shows the approached structure and each substructure was divided into six Euler-Bernoulli finite elements with two degrees-of-freedom per node. Figure 3.9: Coupling of substructures A and B, forming the structure C. The characteristics and properties of the beam are referred in table 3.2. The translational degrees-of-freedom are considered as coordinates where is possible to perform measurements, coordinates t, whereas coordinates rrepresent either coordinate where measurements are impossible for any reason, in this specific case the rotational degrees-of-freedom. 51 CHAPTER 3. EXPERIMENTAL DATA ANALYSIS 58 Chapter 4 Uncoupling with Experimental FRFs 4.1 Introduction Throughout this work, several uncoupling techniques have been presented with the corresponding numerical examples which were tested and validated through theoretical expressions and other sources, [2], [8], [9]. Therefore, at this point of the work it is important to apply those uncoupling techniques to experimental cases in order to evaluate their performance under experimental data. The analysis of these techniques was carried out with aluminum specimens which hold bolted joints incorporated. Figure 4.1 summarizes the treatment steps of experimental data acquired. Firstly, the experimental data acquired will be properly treated and organized. The following step is an estimative of unmeasured FRFs whereby the complete matrix HCis obtained using numerical data from substructure A and the experimental data of the main structure. Then, the four uncoupling techniques are applied in order to obtain the matrix HB jj which characterizes the joint. When the matrix HB jj is available, it is possible to achieve the joint dynamic stiffness carrying out the inverse of that matrix. Performing an alternative coupling between the joint information obtained through the uncoupling techniques and the matrix HA obtained numerically, we achieve the global matrix HCand consequently this matrix could be compared with the initial matrix HCto evaluate the process effectiveness. Three different specimens were considered in this experimental work. Firstly, the experimental work started with a simple beam without any joint (figure 4.2); the purpose of this arrangement is to analyse a beam for which there are theoretical and numerical solutions well defined in order to understand if the experimental setup is well tuned and totally prepared and capable of analyse the following beams with joints incorporated. 59 CHAPTER 4. UNCOUPLING WITH EXPERIMENTAL FRFS Figure 4.1: Experimental work diagram. Figure 4.2: Specimen 1 (Beam 1). Secondly, a beam with a bolted joint (figure 4.3) was analysed, using the setup previously prepared and tuned. Figure 4.3: Specimen 2 (Beam 2). Then, the bolted joint as considered in beam 2 (figure 4.4) was taken into account. Analysing this specimen, we want to measure directly the translation FRFs and then compare them with that obtained through the uncoupling techniques. The definition drawings of the three types of beams analysed are available in Appendix D. Figure 4.4: Specimen 3 (Bolted joint). 60 4.2. FINITE ELEMENT MESH In this working phase, lots of experimental measurements will be carried out in the three specimens, using both the shaker and the hammer as excitation means. Table 4.1 summarizes all the experimental measurements, as well as the theoretical methodologies applied to each experimental data set. Table 4.1: Experimental measurements carried out in each beam. Beam 1 Beam 2 Beam 3 Excitation Mean Excitation Mean Excitation Mean Methods Hammer Shaker Hammer Shaker Hammer Shaker Measurements X X X X X - Mass - - X X - - Cancelation Estimative of - - X - - - unmeasured FRFs Uncoupling - - X - - - techniques Analysis of - - X - - - frequencies ranges New coupling - - X - - - 4.2 Finite Element Mesh The experimental measurements need to be previously approached and studied in order to foresee the experimental results which we are looking for. Therefore, some theoretical studies were done at this step, such as a survey about the number of natural frequencies that could appear in a certain frequency range and the joint length once it is important to have at least one natural frequency in the frequency range experimentally taken into account. The experimental frequency range taken into account depends on the capacities of the experimental devices which are available to perform these experimental tests. Thus, a frequency range between 0 to 800 Hz was considered with a resolution of 801 lines. The finite element mesh considered is shown in figures 4.5 and 4.3, representing the degreesof-freedom and measurement points, respectively. The nomenclature of the experimental FRFs is set up according to DOF distribution, for the sake of coherence with the numerical formulations. The position of each measurement point coincides exactly with the corresponding node of the finite element mesh, and therefore experimentally we just measure the translation DOF in a node. The experimental FRFs obtained will define the matrix HCwhich corresponds to the global structure, whereas the theoretical FRFs which define the matrix HAwill be obtained through the finite element programme, BAPMEF. 61 CHAPTER 4. UNCOUPLING WITH EXPERIMENTAL FRFS Figure 4.5: Degrees-of-freedom considered per substructure. Figure 4.6: Measurement points considered per substructure. Each substructure Aidivides into eighteen Euler-Bernoulli finite elements. A study about the impact of the number of finite elements per substructure in the uncoupling techniques was carried out. In other words, a study about the mesh effect was done and it has been found out a number of finite elements from which there is no interference in the final results; the number of elements is close to the value considered. The properties and dimensions of each substructure are given on table 4.2. To the structure which corresponds to the joint, we just consider the length for the sake of coherence, once it has incorporated the bolt and the nut which change the beam properties in that area. Table 4.2: Characteristics and properties of the experimental beam. Beam Length [mm] Width [mm] Thickness [mm] E [GPa] ρ[kg/m3] A1 240 25 5 70 2700 B 220 - - - - A2 340 25 5 70 2700 The FRFs of both substructures were calculated through the finite element programme taking into account the assumptions presented previously. Nowadays, the experimental devices available just enable measurements of translation degrees-of-freedom and the corresponding FRFs of rotation have to be estimated considering the finite element results and the experimental results. The same methodology exposed in section 3.3 will be adopted in this case and then with all FRFs available, the joint uncoupling will be carried out using the four uncoupling techniques presented. A previous study about modal shapes was carried out in order to evaluate the position of the two suspension ropes. The two suspension ropes were placed on the position of the vibration nodes of a beam, considering the nodes of the first modal shape for a free-free beam. So, the wire ropes were placed at 179.2 mm of each extremity of the beam. Then, a beam with a concentrated mass, to take into account the influence of the bolted joint, was simulated in order to study the nodes position of each modal shape. Table 4.3 shows the results obtained for each mode. 62 4.3. EXPERIMENTAL ANALYSIS OF BEAM 1 Table 4.3: Nodes Position per Mode. Nodes Position per Mode / [mm] Mode 1 288 - 512 Mode 2 224 - 400 - 576 Mode 3 176 - 320 - 464 - 624 Mode 4 144 - 272 - 400 - 528 - 640 Analysing table 4.3, there is no node identified in this survey that coincides exactly with the position of suspension ropes. Thus, this fact avoids some experimental problems that could exist if one node of a mode coincides with the suspension ropes or would be in its neighborhood. 4.3 Experimental Analysis of Beam 1 As previously stated, Beam 1 will serve as a way of comparison between the results obtained with a shaker and a hammer, and also as a validation and adjustment of the experimental setup. Thus, some measurements were performed in this beam using a hammer and a shaker. Using the shaker in a fixed excitation node, the responses were measured in all nodes changing the corresponding position of the accelerometer; using the hammer to excite all the nodes, the response was measured in a fixed node varying the corresponding excitation node. All the results obtained in this experimental analysis are available in Appendix B. 4.3.1 Using the Hammer 5800SL Figure 4.8 shows the results obtained with a hammer as excitation mean. As we can see, the experimental FRF fits in a good way the theoretical FRF obtained by the finite element method. The behaviour of both curves is similar, although there is some discrepancy in certain frequency ranges. In the frequency range between 0 to 50 Hz, we can see some discrepancy due to the influence of extra factors, such as the suspension ropes. 63 CHAPTER 4. UNCOUPLING WITH EXPERIMENTAL FRFS 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H97 [m/s2N] H97 − Experimental H97 − BAPMEF Figure 4.7: Accelerance H97. Table 4.4 shows a study about the natural frequencies. The experimental values obtained are really near the theoretical values calculated, which indicates an excellent accuracy obtained with this excitation mean. Table 4.4: Natural frequencies obtained using a hammer. Natural Frequency Theoretical Value Experimental Value Error / % ω140.889 41.000 0.272 ω2112.715 113.000 0.253 ω3220.966 221.000 0.015 ω4365.270 367.000 0.474 ω5545.643 550.000 0.799 ω6762.102 764.000 0.249 4.3.2 Using the Shaker LDS210 (Force transducer 8200) The results obtained with shaker are presented in figure 4.8. Analysing this figure, we can see that the experimental curve fits very well the theoretical curve, despite some discrepancy present in the highest frequencies that could be related with the drive rod which introduces a natural frequency near the highest frequencies. Here again, it is possible to see some discrepancy in the first frequencies owing to external factors as stated in previous section. 64 4.4. EVALUATION OF THE TORQUE EFFECT 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H97 [m/s2N] H97 − Experimental H97 − BAPMEF Figure 4.8: Accelerance H97. Table 4.5 gives a study about natural frequencies. The results obtained experimentally are really similar to the theoretical values, despite an higher error in some natural frequencies. Table 4.5: Natural frequencies obtained using a shaker. Natural Frequency Theoretical Value Experimental Value Error / % ω140.889 42.000 2.717 ω2112.715 111.000 1.522 ω3220.966 221.000 0.015 ω4365.270 366.000 0.199 ω5545.643 543.000 0.484 ω6762.102 756.000 0.801 4.4 Evaluation of the torque effect The torque effect contributes somewhat to the behaviour of frequency-response functions. Thus, an experimental study was performed with four different torque values, 5 N.m, 10 N.m, 15 N.m and 20 N.m. A beam with a bolted joint (figure 4.3) was taken into account and a hammer was used to excite the beam, and the response was measured with an accelerometer. Using a dynamometric key, the measurements with four different torques were performed. The beam was suspended with two nylon ropes in order to simulate free-free boundary conditions. These ropes were collocated specifically in the vibrations nodes of the first vibration mode to get a better analysis without the interference of the other modal modes. Figure 4.9 shows the results obtained. 65 CHAPTER 4. UNCOUPLING WITH EXPERIMENTAL FRFS Analysing figure 4.9 it is possible to understand that the torque affects lightly the FRFs’ behaviour and just in the sixth natural frequency is visible some discrepancy between the curves. Thus, torque affects the FRFs’ behaviour and its interference could be considered null to certain values. Figure 4.13 provides a zoom over the sixth natural frequency zone. Figure 4.9: Experimental results with different values of torque. Indeed, the torque value has some interference and it becomes constant from values around 15 N.m. In the experimental measurements a torque of 20 N.m was taken into account in order to ensure that there is no interference of this factor in the final results. Figure 4.10: Zoom taken from the sixth natural frequency. 66 4.5. EXPERIMENTAL ANALYSIS OF BEAM 2 4.5 Experimental Analysis of Beam 2 This experimental analysis was carried out with the purpose of identify a joint using the beam 2 (figure 4.3) which holds a bolted joint in its structure. It took into account the same number of measurement points with an identic disposition as considered in previous experimental analysis. Once again, the measurements used as excitation means a hammer and a shaker. Furthermore, in this experimental analysis the four uncoupling techniques were implemented in order to obtain the joint dynamic behaviour. The effects of transducers’ mass were removed according to a mass cancelation process. Therefore, once the experimental FRFs obtained, the FRFs which involve rotational degrees-of-freedom were generated considering the algorithm proposed in section 3.3. Then, with the matrix HCcompletely available, the four uncoupling techniques were implemented. All the results obtained in this experimental analysis are available in Appendix B. 4.5.1 Using the Hammer 5800SL The frequency response functions were measured in all degrees-of-freedom, exciting all points and measuring the response in the corresponding points. The necessity of this process comes from the fact that we need the complete matrix HC, at each frequency ω, to perform the joint uncoupling with the four techniques. Figure 4.11 shows some results obtained in this experimental analysis. We can see a good coherence and accuracy both in direct and in transfer FRFs. 0 100 200 300 400 500 600 700 800 100 101 102 103 104 ω [Hz] Accelerance H11 [m/s2N] H11 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H13 and H31 [m/s2N] H13 H31 Figure 4.11: Accelerances H11 and H13. For the sake of coherence and accuracy, all the experimental FRFs were subjected to a mass 67 CHAPTER 4. UNCOUPLING WITH EXPERIMENTAL FRFS 7 to represent the force transducer effect and a mass of 1 g in the same point to represent the accelerometer effect. To the other FRF H2|7, it was considered the force sensor mass in the same point and the accelerometer mass was moved to the node 2. Indeed, this corresponds exactly to the experimental procedure taken into account. Figure 4.21 shows the results obtained in this simulation. This simulation represents exactly the experimental results and in fact, the accelerometer mass has some interference in the structure and can not be negligible in this particular case. The big point is that we may have always a system with the same mass distribution, either we want to cancel the accelerometer mass or the force sensor mass. Indeed, if we want to cancel the FRF Hij which has the accelerometer mass placed on node i, we need the direct FRF Hjj which now has the accelerometer mass placed on node j. If we want to cancel the force sensor mass, the same fact occurs oppositely. Indeed, this problem has a justified explanation and recently some authors have been studying this phenomenon, and for instance Si et al. [18] give some solutions to solve or avoid this kind of problems. 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω/Hz AcceleranceH77 [m/s2N] h77 − pure H77 C − canceled 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω/Hz AcceleranceH27 [m/s2N] h27 − pure H27 − canceled Figure 4.21: Accelerances H7|7and H2|7. 4.6 Experimental analysis of the bolted joint Lastly, the bolted joint was analysed in order to experimentally characterize the bolted joint included in previous specimen. The excitation of the joint was done with a hammer, whereas the response was measured with the accelerometer. For the sake of clarity, in this experimental analysis we just measure the translation degrees-of-freedom, once the rotational ones aren’t available due to reasons already stated in this work. Figures 4.22 shows a direct and a transfer FRF. 74 4.7. DISCUSSION OF RESULTS 0 100 200 300 400 500 600 700 800 100 101 102 103 104 ω [Hz] Accelerance H11 B H11 B 0 100 200 300 400 500 600 700 800 101 102 103 104 105 ω [Hz] Accelerance H12 B H12 B Figure 4.22: Accelerances H1|1and H1|2. Indeed, the behaviour of both FRFs is similar to the uncoupling techniques results, having a resonance placed on the same zone. The resonance found is lightly higher than that which was theoretically considered, once in fact the bolted joint and its overlap do not work entirely as a concentrated mass, but in a different way with a mass distribution more homogeneous. Even though, the FRFs are really similar and therefore this another point which validates the work carried out. Figure 4.23 shows the experimental setup of the bolted joint where it is possible to visualize the accelerometer 27A11 normally used to measure the response in this work. Figure 4.23: Experimental setup of the bolted joint. 4.7 Discussion of results The experimental analysis of beam 1 provided good results, although there are some differences between the two excitation means of the beam. The shaker provides results that fit very well the exact solution and the natural frequencies are near the exact values. On the other 75 CHAPTER 4. UNCOUPLING WITH EXPERIMENTAL FRFS hand, the hammer provides values for the natural frequencies much more close, but the curve doesn’t fit so very well the theoretical solution. Therefore, exciting the beam with a shaker provides better results than with an hammer, even though an higher error in natural frequencies which in most cases does not interfere so much in the results. The torque applied to the bolted joint affects slightly the FRFs’ behaviour along the frequency range. If the torque applied is low, any undesirable phenomenons such as slipping and trepidation between the two beam parts will occur, causing uncertainty and lack of accuracy in the dynamic responses acquired. Actually, this effect must be canceled in order to avoid even more uncertainty in this type of analysis. The value from what there is not influence of this extra factor was detected and considered to every bolted joints used in this work to carry out experimental measurements. The analysis of beam 2 provided a comparison between two excitation means as an application to perform the joint uncoupling. The hammer gave much better results than the shaker due to some technical reasons. The experimental analysis using the hammer demonstrated an excellent practical applicability, once the accelerometer mass almost hadn’t affected the beam behaviour but its cancelation was performed for the sake of coherence, and the natural frequencies showed a good accuracy. The joint uncoupling exceeded the results expected with two formulations providing good results, once it was expected an huge discrepancy and uncertainty caused by the noise which is automatically inserted in experimental data. The estimative of unmeasured FRFs gave an important contribute to these results, once they need to be estimated. The coupling between experimental and theoretical data revealed some uncertainty, although it was possible to identify some natural frequencies. On the other hand, some experimental problems appeared when the shaker was used as excitation mean. Firstly, the drive rod was introducing an extra natural frequency in the samples due to its interference. Different drive rod lengths were experimented without success and another drive road was considered. The high mass of the new force transducer that we had to use due to the new drive road changed the beam behaviour and consequently there were some differences in transfer FRFs. Then, a mass cancelation methodology was implemented without success and the reasons to this problem were properly explained with a theoretical formulation. This problem allows to understand that the accelerometer low mass could have some influence and thus its cancelation was carried out in the previous experimental analysis with the hammer. Because of the worst quality of the results, the uncoupling techniques and other weren’t applied to this data set. Lastly, the bolted joint was analysed and good results were achieved. Indeed, the FRFs obtained are really similar to that obtained with the uncoupling techniques. Furthermore, the natural frequency identified is close to what was expected, even though a bit higher than the theoretical one. This fact shows that the overlap and bolt/nut introduce a special mass distribution, once it is not exactly the same as if we considered a concentrated mass in the bolt position. 76 Chapter 5 Uncoupling with Experimental Modal Analysis 5.1 Introduction Another way to study a beam with a bolted joint involves the execution of an experimental modal analysis. To perform this analysis, the programme CADA-PC was used to process the experimental data. The main advantage of this technique is the direct determination of the coefficients related to the system modal properties which are usually the parameters sought [19]. Another advantage of this process is that we just need to measure a row or a line of the FRF matrix to perform the corresponding modal identification, instead of measure the FRFs in all points as was done in all experimental analysis in last chapter. In other words, a modal analysis gives the modal matrix [φ]with the corresponding vibration modes, as well as the natural frequencies and the damping coefficients for each vibration mode. Having all these data available, we can synthesise any FRF in either points of the structure which we desire. Basically, the modal analysis process consists on a theoretical curve fitting of the adopted model to the experimental curves, in order to obtain the parameters of theoretical curves which better fit the experimental curves obtained. These parameters are usually called modal parameters which describe the system under analysis. To obtain these modal parameters, the programme CADA-PC, which works with the experimental setup, was sought. The software CADA-PC implements a multi degree-of-freedom technique for parameter estimation. It uses a time domain algorithm to identify poles (frequency and damping values), called the Least Square Complex Exponential method. In a second phase, residues are identified with the Least Square Frequency Domain technique [20]. The synthesising of each FRF is performed with an expansion in simple fractions (Equation 5.1). This expression takes into account a viscous damping expressed by the coefficient ξand 77 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS the influence of the different modes, carrying out a sweep in frequency. Hjk(ω) = n X r=1 φjrφkr (ω2 r−ω2)+2jξrωrω(5.1) The implementation of this type of analysis has some advantageous points, principally a lower level of noise in the samples and an improvement of accuracy on the dynamic behaviour of corresponding FRFs which are calculated according to the expansion in simple fractions. Therefore, we are looking for better results than that we have got in previous experimental analysis. The experimental analysis made use of the same beam used in previous experimental analysis (beam 2) with the same finite element mesh in order to keep the correspondence of the final results for a later discussion. The acquisition of a row or a column of the FRF matrix was carried out with the same excitation means, hammer and shaker. In this chapter, the data treatment will follow a similar sequence as presented by diagram 4.1. The big difference is that the FRFs of the translation DOFs will be synthesised taking into account the modal analysis outputs, the matrix modal and the corresponding natural frequencies identified and damping coefficients. From this point, all the procedure sequence is the same as done in previous experimental works. Table 5.1 summarizes all the experimental modal analyses done and the formulations implemented in each specific case. Table 5.1: Experimental modal analysis carried out in beam 2. Estimative of Mass unmeasured Uncoupling Dynamic Analysis Cancelation FRFs Techniques stiffness Coupling Shaker Column 8 - X X X X Laser vibrometer (x2) X X X X X Dummy X X X X X Masses (x2) Hammer Row 2 - X X X X Row 4 - X X X X Row 5 - X X X X 78 5.2. MODAL ANALYSIS WITH SHAKER LDS210 5.2 Modal Analysis with Shaker LDS210 This experimental modal analysis used the shaker as excitation mean, using the force transducer 8200 to measure the excitation. Firstly, an experimental modal analysis will be carried out with pure data, using the eighth column. Due to problems related with transducer’s masses which had an huge interference in previous data treatment, here we will try to avoid these problems taking into account an experimental setup with some modifications. Thereunto, a beam with dummy masses will be analysed, as well as a laser vibrometer will be used to measure the response, once this transducer does not introduce extra mass in the system, which is a great advantage. Thus, it will be possible to compare different analyses which have the transducers influence and other where different mass cancelation processes are implemented. 5.2.1 Modal Analysis with column 8 The shaker was placed on an extremity of the beam, and the response was measured in all degrees-of-freedom. Therefore, the main structure has the influence of the force sensor mass, as well as the accelerometer mass which changes its positions according to the measurement point required. Using the identification software CADA-PC, the corresponding modal identification gives the modal parameters, the damping coefficients and natural frequencies (table 5.2). The frequency-response functions of the translation degrees-of-freedom were calculated according to expression 5.1 presented above. Table 5.2: Modal Parameters of the beam identification process. Frequency / [Hz] ξ[%] N.º of DOFs Mode 1 35.331 1.790 8 Mode 2 100.850 0.572 8 Mode 3 197.382 0.301 8 Mode 4 323.269 0.197 8 Mode 5 501.015 0.129 8 Mode 6 672.425 0.141 8 Table 5.3 shows the values of the correlation criterium MAC, which relates the identified mode shapes obtained in this modal analysis. 79 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS Table 5.3: MAC Criterion. Mode Frequency Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 [Hz] [%] [%] [%] [%] [%] [%] 1 35.331 100 1.3 8.9 13.9 6.5 4.4 2 100.850 1.3 100 19.3 10.1 6.6 2.9 3 197.382 8.9 19.3 100 9.9 2.0 2.2 4 323.269 13.9 10.1 9.9 100 5.8 0.8 5 501.015 6.5 6.6 2.0 5.8 100 10.4 6 672.425 4.4 2.9 2.2 0.8 10.4 100 Figure 5.1 shows two synthesised FRFs,HC 1|1and HC 3|1. The synthesized FRF matches in a good way the experimental FRF, despite a certain discrepancy in some frequency ranges. As we can see, in the synthesized FRF there is no interference in the first frequencies (0-100 Hz) which normally is related to boundary conditions, for instance the influence of suspension ropes, in the acquisition of an experimental FRF. Therefore, we can conclude that the modal analysis was carried out with success, although there are some differences in natural frequencies, as well as some lack of coherence in resonance and anti-resonance zones. Indeed, these differences are the consequence of the different position of the force transducer, because in the experimental curve it is placed on DOF 1, whereas in the synthesised FRF it is placed on DOF 15 (measurement point 8), causing different mass distributions that explain this difference in terms of natural frequencies. 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 ω/Hz AcceleranceH1|1[m/s2N] H1|1 experimental H1|1 synthesized 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 ω/Hz AcceleranceH3|1[m/s2N] H3|1 experimental H3|1 synthesized Figure 5.1: Accelerances HC 1|1and HC 3|1. •Joint Uncoupling The modal analysis process provided all the information about the beam dynamic behaviour that we need to obtain the necessary FRFs which are required to apply the uncoupling algorithms. Having this information, the FRFs were calculated in all degrees-of-freedom taking into account the extracted modal parameters. In this numerical approach we are looking for results much more accurate, because the influence of extra factors such as noise and boundary conditions is reduced or even annulled. 80 5.2. MODAL ANALYSIS WITH SHAKER LDS210 For the sake of coherence, it is important to notice that these modal parameters still have the influence of the transducer’s mass used to perform the experimental measurements. Having the translation FRFs available, the rotation ones were estimated according to the formulation used in previous numerical examples. Once obtained the matrix HC, the uncoupling techniques were implemented in order to obtain the joint dynamic characterization. Figure 5.2 shows the results obtained with the four uncoupling techniques. Analysing the results obtained, we can notice that there is an improvement on the level of accuracy in some formulations comparing to previous experimental examples. Once again, the second and Wang formulations provided better results than other formulations. The behaviour of first and third formulations was expected with an huge discrepancy and uncertainty. The second formulation gives good results, where is well visible an anti-resonance and a resonance pick, even though an anti-resonance better defined than the resonance. The Wang formulation gives an excellent result with a resonance very well defined, whereas the anti-resonance zone has some discrepancy and uncertainty. 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 1 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 2 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 3 0 200 400 600 800 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Wang Formulation Figure 5.2: Accelerance HB 11 of the uncoupled joint. •Dynamic Stiffness If we have the joint FRFs available, the corresponding dynamic stiffness can be calculated. The formulations with better results, second and Wang, were selected (figure 5.3). The dynamic stiffness obtained takes the excepted shape when there are extra factors (introduction of noise) which contaminate the experimental samples, although these results have a lower level of noise as said previously. 81 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS 0 200 400 600 800 −6 −4 −2 0 2 4 6x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Second Formulation 0 200 400 600 800 −1.5 −1 −0.5 0 0.5 1 1.5 2x 106 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Wang Formulation Figure 5.3: Dynamic Stiffness ZB 11 of second and Wang formulations. •Coupling with joint FRFs Once obtained the joint FRF matrix through the uncoupling process, the inverse process was executed in order to compare the results with the FRFs experimentally measured. The alternative coupling technique made use of matrix HB jj from second and Wang formulations because both provided good results. This process will introduce even more numerical uncertainty in the samples once new matrix inversions have to be performed. Figure 5.4 shows the results obtained. 0 100 200 300 400 500 600 700 800 10−4 10−2 100 102 104 106 108 ω/Hz Accelerance HC 11 [m/s2N] Experimental FRF Formulation 2 Wang Formulation Figure 5.4: Accelerance HC 11 according to alternative coupling technique. In terms of natural frequencies, there is a good correspondence between the FRFs obtained and the synthesized ones by coupling, even though the appearance of a spurious natural frequency close to 450 Hz which has appeared in other experimental examples without a concrete explanation. In terms of magnitude, there are differences in some frequency ranges, principally in the range where appears the new unknown natural frequency. In these examples, second formulation provides better results than Wang formulation, mainly in first frequencies, but this formulation introduces a spurious natural frequency at 80 Hz that doesn’t appear in Wang formulation. 82 5.2. MODAL ANALYSIS WITH SHAKER LDS210 5.2.2 Modal Analysis using the Laser Vibrometer and Shaker LDS201 The previous modal analysis, which used the eighth column, has the influence of the force sensor and accelerometer mass that cause a different mass distribution. In previous chapter, the numerical example that used the shaker as excitation mean gave bad results when a mass cancelation process was required to eliminate the transducers influence, once that process wasn’t coherent due to the accelerometer position. Therefore, the influence of this transducers must be canceled and in this section we will propose a different experimental setup capable of consider these effects. To measure the response, a laser vibrometer will be used once it doesn’t introduce extra mass in the system like the accelerometer normally introduces, and thus we just have to cancel the force sensor mass, becoming the process much more feasible. Two different ways will be followed, varying the order between the mass cancelation process and the modal analysis. Table 5.4 presents the two different procedures. Table 5.4: Different procedures for this experimental setup. Shaker LDS 210 + Laser Vibrometer First Procedure Second Procedure 1º Experimental modal analysis 1º Mass cancelation (force sensor) 2º Synthesising 2º Experimental modal analysis 3º Mass cancelation (force sensor) 3º Synthesising 4º Uncoupling 4º Uncoupling Figure 5.5 shows this experimental setup, where it is possible to see the shaker as well as the laser vibrometer. Figure 5.5: Experimental setup. 5.2.2.1 First Procedure: Synthesising + Mass Cancelation In the first procedure, we measured the experimental FRFs exciting the beam in an extremity (measurement point 8) and measuring the response in all points. Then, a modal analysis was carried out with these experimental data. Table 5.5 shows the results of this modal analysis. 83 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS Table 5.9: Different procedures for this experimental setup with dummy masses. Shaker LDS210 + Accelerometer 27A11 + Dummy masses First Procedure Second Procedure 1º Experimental modal analysis 1º Mass cancelation (force sensor) 2º Synthesising 2º Experimental modal analysis 3º Mass cancelation (force sensor) 3º Synthesising 4º Uncoupling 4º Uncoupling Figure 5.14 shows the experimental setup with the dummy masses in each measurement point. Figure 5.14: Experimental setup with dummy masses. 5.2.3.1 First Procedure: Synthesising + Mass Cancelation In this first procedure, the response was measured in all points with the shaker placed on an extremity (point 8), having all the points a dummy mass to simulate the accelerometer. Then, an experimental modal analysis was carried out and the final results are shown in table 5.10. Considering dummy masses, concentrated masses are introduced in the system and consequently the natural frequencies decrease and table 5.10 verifies this fact. Table 5.10: Modal Parameters of the beam modal identification. Frequency / [Hz] ξ[%] N.º of DOFs Mode 1 35.607 1.499 8 Mode 2 101.518 0.565 8 Mode 3 198.765 0.313 8 Mode 4 324.907 0.206 8 Mode 5 494.049 0.128 8 Mode 6 663.045 0.134 8 Table 5.11 gives a representation of the correlation criterion MAC that relates the identified 90 5.2. MODAL ANALYSIS WITH SHAKER LDS210 mode shapes, where it is possible to understand that there is good correspondence between the different modes. Table 5.11: MAC Criterion. Mode Frequency Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 [Hz] [%] [%] [%] [%] [%] [%] 1 35.607 100 1.9 16.5 15.5 4.9 0.8 2 101.518 1.9 100 9.3 15 4.9 1.8 3 198.765 16.5 9.3 100 1.3 7.0 7.8 4 324.907 15.5 15.0 1.3 100 4.7 8.0 5 494.049 4.9 4.9 7.0 4.7 100 5.2 6 663.045 0.8 1.6 7.8 8.0 5.2 100 Then, all the necessary FRFs were synthesised and implemented a mass cancelation process, considering only the force sensor mass to cancelation effects. Therefore, the main structure will have the influence of seven dummy masses and the accelerometer mass. Figure 5.15 gives the results of this process, comparing the experimental with synthesised and canceled FRF. As we can see, the mass cancelation process increases lightly the natural frequencies. The synthesised FRF shows good coherence in terms of natural frequencies, but the anti-resonances aren’t coherent with the experimental one. 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 ω/Hz AcceleranceHC 1|15 [m/s2N] H1|15 C experimental H1|15 C synthesized 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 ω/Hz AcceleranceHC 15|15 [m/s2N] H15|15 C experimental H15|15 C synthesized Figure 5.15: Accelerances HC 1|15 and HC 15|15. •Joint Uncoupling Then, the rotational FRFs were generated to have the necessary information available to carry out the joint uncoupling. Figure 5.16 gives the results of this uncoupling process, and once again second and Wang formulations provide the best results. In both formulations, it is possible to identify the resonance and anti-resonance zones, demonstrating however some discrepancy in certain frequency ranges. 91 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 1 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 2 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 3 0 200 400 600 800 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Wang Formulation Figure 5.16: Accelerance HB 11 of the uncoupled joint. •Dynamic Stiffness Inverting the matrix HB jj, the dynamic stiffness ZB jj can be immediately calculated. Figure 5.21 gives the results of the two formulations, where second formulation provides a curve more accurate than Wang formulation, reflecting the discrepancy present in the uncoupling results. The dynamic stiffness takes again the characteristic behaviour when the data are contaminated with noise. 0 200 400 600 800 −6 −4 −2 0 2 4 6x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Second Formulation 0 200 400 600 800 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Wang Formulation Figure 5.17: Dynamic stiffness ZB 11 of the uncoupled joint. •Coupling with joint FRFs Once obtained the joint dynamic identification, a coupling between this information and 92 5.2. MODAL ANALYSIS WITH SHAKER LDS210 the structure HAcan be performed. Figure 5.18 gives the results of this coupling for the second and wang formulations. Here again, there is an huge discrepancy and incoherence between the synthesised FRFs and the two other curves, but it is possible to identify four natural frequencies up to 400 Hz. Indeed, in this case is well visible the influence of the successive numerical manipulations conditioned by the noise introduced, making impossible the achievement of coherent results that theoretically would give identical FRFs that were taken into account in the beginning of the cycle. 0 100 200 300 400 500 600 700 800 10−4 10−2 100 102 104 106 108 ω/Hz AcceleranceHC 11 [m/s2N] Exact FRF Formulation 2 Wang Formulation Figure 5.18: Accelerance HC 11 according to alternative coupling technique. 5.2.3.2 Second Procedure: Mass Cancelation + Synthesising In this case, the cancelation of the force sensor mass was done first, and then performed the experimental modal analysis. For the sake of clarity, in this modal analysis the dummy masses are considered in the main structure, and no cancelation is done for this masses. The experimental setup is exactly the same, but the experimental data are treated before the modal analysis. Table 5.12 presents the results of this modal analysis. The natural frequencies are slightly higher than in previous modal analysis, once the mass cancelation had already been implemented. Table 5.12: Modal Parameters of the beam modal identification. Frequency / [Hz] ξ[%] N.º of DOFs Mode 1 37.779 1.16 8 Mode 2 107.477 0.238 8 Mode 3 203.601 0.153 8 Mode 4 335.535 0.154 8 Mode 5 506.300 0.076 8 Mode 6 674.711 0.130 8 Table 5.13 gives the correlation criterion MAC that relates the identified mode shapes. Analysing 93 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS the MAC table, we can except good results from this modal analysis once there is a good coherence in the final results. Table 5.13: MAC Criterion. Mode Frequency Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 [Hz] [%] [%] [%] [%] [%] [%] 1 37.779 100 0.2 27.0 0.7 10.2 1.7 2 107.477 0.2 100 2.0 20.9 0.0 6.4 3 203.601 27.0 2.0 100 0.1 17.5 1.0 4 335.535 0.7 20.9 0.1 100 0.3 19.5 5 506.300 10.2 0.0 17.5 0.3 100 0.5 6 674.711 1.7 6.4 1.0 19.5 0.5 100 Having the modal matrix and corresponding modal parameters available, the FRFs were synthesised in the required points. Figure 5.19 gives two samples, a transfer and a direct FRF, which compares the experimental curve already canceled and the synthesised FRF. Indeed, there is an excellent coherence between the two curves, once the synthesised FRF fits very well the experimental one both in resonance and anti-resonance zones. 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 ω/Hz AcceleranceHC 1|15 [m/s2N] H1|15 C experimental H1|15 C synthesized 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 ω/Hz AcceleranceHC 15|15 [m/s2N] H15|15 C experimental H15|15 C synthesized Figure 5.19: Accelerances HC 1|15 and HC 15|15. •Joint Uncoupling Then, the rotational FRFs were generated through the synthesised FRFs and the numerical matrix HA. Having the necessary information available, the uncoupling techniques were implemented in order to achieve the joint dynamic characterization. Here again, the formulations which gave the best results are the same, although there are an higher uncertainty if we compare with the previous modal analysis, once the mass cancelation process done before the modal analysis could have contaminated even more the experimental data. In second and Wang formulations, there are more than one resonance identified and the level of uncertainty really increases comparing to previous modal analysis. 94 5.2. MODAL ANALYSIS WITH SHAKER LDS210 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 1 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 2 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 3 0 200 400 600 800 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Wang Formulation Figure 5.20: Accelerance HB 11 of the uncoupled joint. •Dynamic Stiffness Once obtained the joint dynamic characterization, the dynamic stiffness is calculated inverting the matrix HB jj. Figure 5.12 gives the dynamic stiffness ZB 11 according to second and Wang formulations. Second formulation provides good results despite the uncertainty present above, whereas Wang formulation reflects the high uncertainty at the right of the joint natural frequency. 0 200 400 600 800 −6 −4 −2 0 2 4 6x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Second Formulation 0 200 400 600 800 −6 −5 −4 −3 −2 −1 0 1 2 3x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Wang Formulation Figure 5.21: Dynamic stiffness ZB 11 of the uncoupled joint. •Coupling with joint FRFs If the dynamic characterization of the bolted joint is available, a new coupling can be performed and then compare these results with the initial FRFs of the main structure. Figure 5.22 95 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS provides the results of this coupling, and the discrepancy verified during the uncoupling process affects these results. This discrepancy is a direct consequence of several mathematical operations and of the noise that is generally introduced during experimental measurements. Analysing the results, in both formulations is complicated to fit the exact FRF, although the second formulation gives better results. 0 100 200 300 400 500 600 700 800 10−2 100 102 104 106 108 ω/Hz AcceleranceHC 11 [m/s2N] Exact FRF Formulation 2 Wang Formulation Figure 5.22: Accelerance HC 11 according to alternative coupling technique. 5.3 Modal Analysis with hammer 5800SL This experimental modal analysis used the hammer as excitation mean. The data acquired during the previous experimental work with the hammer, in which we had measured the entire matrix HCwere used to perform these modal analysis. Once we have the entire matrix available, three different rows will be selected from that matrix and then a modal analysis will be performed for each matrix row. For each row, the accelerometer is always placed on the same position whereas the hammer varies its position. Thus, in each row we have always the same system with an identic mass distribution. 5.3.1 Modal Analysis using the second row In this modal analysis, the accelerometer was placed on the second measurement point, whereas the hammer excited the other measurement points to obtain the translation FRFs. The modal identification process provided the modal parameters, which are available in table 5.14. 96 5.3. MODAL ANALYSIS WITH HAMMER 5800SL Table 5.14: Modal Parameters of the beam identification process (row 2). Frequency / [Hz] ξ[%] N.º of DOFs Mode 1 38.484 1.593 8 Mode 2 111.060 0.593 8 Mode 3 210.193 0.449 8 Mode 4 350.983 0.195 8 Mode 5 534.058 0.128 8 Mode 6 725.716 0.124 8 Table 5.15 has a representation of the correlation criterion MAC that relates the identified mode shapes. Table 5.15: MAC Criterion (row 2). Mode Frequency Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 [Hz] [%] [%] [%] [%] [%] [%] 1 38.484 100 3.0 27.2 0.0 9.7 8.3 2 111.060 3.0 100 2.2 16.7 1.1 15.9 3 210.193 27.2 2.2 100 0.5 24.5 0.1 4 350.983 0 16.7 0.5 100 0.4 23.0 5 534.058 69.7 1.1 24.5 0.4 100 0.0 6 725.716 8.3 15.9 0.1 23.0 0.0 100 Once obtained the modal matrix and the corresponding parameters, either FRF could be synthesised in any point. Figure 5.23 compares the experimental FRFs with the synthesised ones. As we can see, there is a good correspondence in terms of natural frequencies and magnitude, although there are some differences in the anti-resonances zones. 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 ω/Hz Accelerance HC 11 [m/s2N] H11 C experimental H11 C synthesized 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 ω/Hz Accelerance HC 31 [m/s2N] H31 C experimental H31 C synthesized Figure 5.23: Accelerances HC 1|1and HC 3|1(row 2). 97 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS •Joint uncoupling Having the matrix HCcompletely available, the uncoupling techniques can be applied to obtain the joint dynamic characterization (figure 5.24). Here again, second and Wang formulations provided better results whereas the other formulations showed an huge discrepancy as expected. In terms of resonances, both formulations identify very well the natural frequency, and the anti-resonance just is distinguishable in second formulation. Therefore, in this case second formulation behaves better than Wang formulation. 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 1 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 2 0 200 400 600 800 10−4 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Formulation 3 0 200 400 600 800 10−2 100 102 104 106 ω [Hz] Accelerance HB 11 [m/s2N] Exact FRF Wang Formulation Figure 5.24: Accelerance HB 11 of the uncoupled joint (row 2). •Dynamic Stiffness Figure 5.25 gives the dynamic stiffness according to second and wang formulations once these formulations gave coherent results. Both curves are well defined, reflecting the impact of noise which normally influences the dynamic stiffness behaviour. 0 100 200 300 400 500 600 700 800 −6 −4 −2 0 2 4 6x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Second Formulation 0 100 200 300 400 500 600 700 800 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Wang Formulation Figure 5.25: Dynamic stiffness ZB 11 of the uncoupled joint. 98 5.3. MODAL ANALYSIS WITH HAMMER 5800SL •Coupling with joint FRFs Then, the results obtained through the second and Wang formulations were selected to carry out the coupling between these data and the theoretical matrix HA. Figure 5.26 gives one sample of this coupling where there is correspondence in some natural frequencies, although there is a new natural frequency that appears again close to 450 Hz. 0 100 200 300 400 500 600 700 800 10−4 10−2 100 102 104 106 108 ω/Hz Accelerance HC 11 [m/s2N] Experimental FRF Formulation 2 Wang Formulation Figure 5.26: Accelerance HC 11 according to alternative coupling technique. 5.3.2 Modal Analysis of forth row This modal analysis used the forth row of matrix HC, with the accelerometer placed on a joint coordinate, the node 4. Having the accelerometer placed on the forth node, the FRFs were measured in every nodes changing the hammer position. Table 5.16 gives the results of this modal analysis. Table 5.16: Modal Parameters of the beam identification process (row 4). Frequency / [Hz] ξ[%] N.º of DOFs Mode 1 38.649 1.597 8 Mode 2 110.751 0.595 8 Mode 3 209.611 0.312 8 Mode 4 351.136 0.185 8 Mode 5 532.682 0.120 8 Mode 6 725.282 0.115 8 Table 5.17 gives a representation of the correlation criterion MAC that relates the identified mode shapes. Analysing the MAC table, there are some high values out of the main diagonal, despite the good coherence of the synthesised FRFs in comparison with experimental ones. 99 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS 5.3.3.1 Evaluation of unmeasured FRFs The algorithm presented in section 3.3 allows an estimative of unmeasured FRFs which are required to carry out the joint uncoupling. Thereunto, the experimental modal analysis that uses the fifth row was taken into account, in order to compare the estimated FRFs with FRFs provided by the MEF programme, BAPMEF. Here, we are considering transfer and direct FRFs that involve rotations, once these correspond to the DOFs where is impossible to do experimental measurements. The degrees-of-freedom of the extremities were considered in this study to evaluate the performance of the algorithm in this specific experimental example. The theoretical example uses a beam with the same dimensions, with a concentrated mass of 40 g in the joint position to simulate its interference. Figure 5.35 gives the comparison of four FRFs that involve rotations and translation DOFs. 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 105 106 107 108 ω [Hz] Accelerance [m/s2N] H2|2 C BAPMEF H2|2 C Estimated 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 105 106 107 ω [Hz] Accelerance [m/s2N] H1|16 C BAPMEF H1|16 C Estimated 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 106 ω [Hz] Accelerance [m/s2N] H2|15 C BAPMEF H2|15 C Estimated 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 105 106 107 108 ω [Hz] Accelerance [m/s2N] H2|16 C BAPMEF H2|16 C Estimated Figure 5.35: Comparison of estimated FRFs. As we can see, there is indeed a good similarity between the theoretical and estimated FRFs in most cases, despite the FRF HC 1|16 where there isn’t correspondence in terms of amplitude and behaviour. However, despite some incoherence in any FRF, this algorithm provides excellent results, much better than expected when compared to the numerical example. Here again, we can point out that the natural frequencies of the theoretical FRFs are slightly lower than the natural frequencies of the estimated FRFs, meaning that the mass distribution of the bolted joint isn’t exactly the same as considering a bolted jointed, as previously discussed in this dissertation. 106 5.3. MODAL ANALYSIS WITH HAMMER 5800SL Therefore, the implementation of this algorithm was carried out with success, despite some differences in any FRFs which have a direct interference in the results of the joint uncoupling. This study validates experimentally this algorithm that had provided an indispensable tool along this dissertation. 5.3.3.2 Analysis of frequency ranges Up to now, in this work several experimental examples were carried out and different results were achieved. The algorithm that allows the selection of frequencies where the noise level is lower was applied in this example, once this provided a better approximation to the dynamic stiffness. Figures 5.36 and 5.37 show the results obtained with this algorithm. The blue curve represents the exact dynamic stiffness, whereas the black one represents the approximated function. 0 200 400 600 800 −6 −4 −2 0 2 4 6 8x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] 0 200 400 600 800 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2x 106 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Approximation Exact Figure 5.36: Dynamic Stiffness ZB 11 obtained with all frequencies and with the selected frequencies, according to second formulation. 0 200 400 600 800 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x 107 Dynamic Stiffness ZB 11 [N/m] ω [Hz] 0 200 400 600 800 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2x 106 Dynamic Stiffness ZB 11 [N/m] ω [Hz] Approximation Exact Figure 5.37: Dynamic Stiffness ZB 11 obtained with all frequencies and with the selected frequencies, according to Wang formulation. 107 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS The input parameters of the algorithm that selects the frequencies are available in table 5.20. The parameters were adjusted for each formulation in order to obtain a good approximation function. Table 5.20: Input parameters of the algorithm. τ v ξ Second formulation 30 40E−60.03 Wang formulation 30 60E−60.02 Analysing the results obtained, we have achieved a good approximation function for each formulation, which validates this algorithm. The exact curve corresponds to a beam with the same joint lenght, but it has no concentrated mass that would simulate the presence of the nut and respective bolt. Therefore, the curve is not completely equivalent in physical terms, but this represents indeed the behaviour of the joint dynamic stiffness. 5.4 Discussion of results The experimental modal analyses carried out and presented in this chapter revealed a good feasibility and effectiveness in terms of final results, as well as an alternative to the experimental procedure applied in previous chapter where is required the measurements of all FRFs. Firstly, the experimental modal analyses used the shaker as excitation mean and different ways of measuring the response. The force sensor introduced extra mass on the system and consequently mass cancelation procedures were required. The modal analysis which used the eighth column of the experimental matrix HC tt pointed out the important influence of the transducers’ mass. The synthesised FRFs, using the modal parameters of this column, showed a big difference in the natural frequencies values, principally when the FRFs of other columns were compared with the experimental ones. Indeed, this difference is directly related with the influence of the force sensor mass which can not be ignored. Despite the finding of this important fact, the results of the algorithm usually applied showed good coherence in the joint uncoupling. This fact helps to better understand why in previous chapter the measurements that used the shaker weren’t successful, due to the different force transducer position in each matrix column. In order to cancel the force sensor mass and become the structure as consistent as possible, two experimental alternatives were presented, using a laser vibrometer to measure the response and collocating dummy masses in all points to simulate the accelerometer influence. In these two alternatives, we were looking for eliminate the accelerometer influence and become the mass cancelation algorithm more effective, due to the problems related with the accelerometer influence properly explained earlier, and then cancel numerically the force sensor mass. The laser vibrometer provided a good experimental efficiency, once this transducer doesn’t introduce extra mass in the system. The cancelation of the force sensor mass was made in two 108 5.4. DISCUSSION OF RESULTS different ways and the moment of this mass cancelation process had a great impact in the results. Performing the mass cancelation process before the modal analysis, we are introducing even more numerical uncertainty in the samples and consequently the quality of the results decrease. In the modal analysis, the identification of natural frequencies became complicated and consequently the MAC table reflected the poor results of this modal analysis. The inverse process, doing first the synthesising of the FRFs and then the mass cancelation revealed a better effectiveness, once the mass cancelation is done with data much less contaminated with noise and there is correspondence between the natural frequencies. The second way showed therefore a better effectiveness, but these two processes would give theoretically the same results and here is well reflected the influence of one more mathematical operation. In terms of results, the second way provided better results, although we could visualize some discrepancy in the uncoupling results comparing to previous experimental results. The use of dummy masses showed a good availability and effectiveness once it was possible to simulate the accelerometer influence in all measurement points. Indeed, the mass distribution of the beam became different once basically eight concentrated masses were introduced in the structure and consequently it was verified that the natural frequencies decreased lightly. In this experimental setup, the mass cancelation process followed the same way, before and after of the modal analysis. The impact of the mass cancelation process wasn’t so perceptible, but it still had some influence. The results of the first way, doing first the modal analysis, gave better results once we were canceling data much less contaminated. Taking into account both ways, the use of dummy masses provided better results than the use of the laser vibrometer, but we were expecting better results with the laser vibrometer. The way to measure the response could have influenced the results, once there is no explanation to the bad results obtained with the laser vibrometer. Theoretically, the laser vibrometer would be the best procedure once just the force sensor mass is introduced in the system, and then the force sensor is canceled, supplying the information of the real beam. Another experimental modal analyses were carried out, this time using the hammer as excitation mean and the accelerometer to measure the response. Three different matrix rows were used in order to compare the results. In this specific case, the influence of accelerometer mass was ignored and evaluating the final results, this condition had’t almost any effect in the main structure. Therefore, in each row we are analysing always a system with the same mass distribution, whereas if another row is selected, just the accelerometer position is changed. Both modal analyses of the three rows provided good results with a good correspondence and coherence. The synthesised FRFs fitted very well the experimental ones, showing an exact coincidence in terms of natural frequencies whereas the anti-resonances zones were generally correspondent, although there were some discrepant cases. The forth row provided an excellent approximation to the uncoupled joint, using the Wang formulation, with a resonance and anti-resonance very well defined. On the other hand, the fifth row gave an excellent result to the uncoupled joint using the second formulation. The coupling between the uncoupled joint and the other beam substructure gave generally better results than the experimental setup using the shaker, being possible to achieve correspondence in some natural frequencies. However, the comparison between the results of this coupling and the initial FRFs revealed that the mathematical operations have an huge impact and influence in the experimental data quality. In many examples, the FRFs obtained with the new coupling process presented a new natural frequency close to 450 Hz, which indeed has an equivalent value to the joint natural frequency 109 CHAPTER 5. UNCOUPLING WITH EXPERIMENTAL MODAL ANALYSIS and can there can be some relationship between this two facts once this spurious natural frequency of the main structure was unexpected and absent. The comparison of estimated FRFs with the corresponding theoretical homonym, showed the high effectiveness of this algorithm that allowed the achievement of interesting results. In the four examples studied, only one had evidenced some discrepancy, despite a good similarity with the theoretical curve. The algorithm of selection of frequency ranges allowed a coherent quadratic approximation to the theoretical dynamic stiffness, which proves the correct implementation of this algorithm. 110 Chapter 6 Conclusion 6.1 Conclusions 6.1.1 Theoretical Work This dissertation presents a deep study about a dynamic characterization of mechanical components, in this case bolted joints, using uncoupling techniques to perform that dynamic identification. Indeed, these techniques are really sensitive to any kind of error or other external influence and experimental techniques have to be applied in order to enhance the performance of that uncoupling techniques. The uncoupling techniques showed a good performance and accuracy when exposed to clean data, but when contaminated with noise their behaviour changed a lot and consequently a formulation differentiation was made. The introduction of noise affects directly the matrix condition, and this is the main factor which affects their performance, as well as matrix inversions and differences which contributes even more to this fact. Associated to this important detail, the type of coordinates used in each formulation has an important role in the accuracy of these methods. An analysis was carried out taking into account these details, and therefore whenever possible we should measure in joint coordinates, as well as give more importance to that formulations which have a low mathematical manipulation effort. Between the formulations approached, the second and Wang formulations provided always better results once these techniques fulfill all the conditions which less affect their mathematical performance. The experimental techniques considered to enhance the performance of the uncoupling techniques revealed a great utility, once it was shown that they can help a lot in the analysis of experimental data which normally is polluted with noise. The estimative of unmeasured FRFs was in fact an important tool to perform the uncoupling, because without rotational FRFs these uncoupling techniques couldn’t be applied. The analysis of frequency ranges allowed a better approximation to the dynamic stiffness, which when exposed to contaminated data takes a strange shape. The mass cancelation process confirmed that the transducers used in experimental simulations have an important impact in the structure dynamic behaviour which can 111 CHAPTER 6. CONCLUSION not be ignored as we found out in the experimental work. The Euler-Bernoulli finite elements used showed a good applicability and feasibility. During the elaboration of the different numerical examples, it was stated that the number of elements of each substructure influences directly the accuracy of the uncoupling techniques. Increasing the number of elements, the accuracy of those techniques also improves to a certain number from what there is no need to use more elements. The correlation criterion LAC provided the comparison between theoretical and contaminated FRFs, being an important tool that made easier the evaluation on the uncoupling techniques effectiveness, once all of them have given different results. These theoretical formulations were validated with scientific work such as articles and other which verified the results and gave motivation to an experimental work based on this previous theoretical work carried out. 6.1.2 Experimental Work The experimental work focused basically on two different ways of achieve the global matrix HC, measuring all the FRFs and through a row or a column, perform a modal analysis that provide the information of the whole structure. The uncoupling techniques had a similar behaviour to what were obtained in the theoretical examples, proving that the numerical noise introduced has some similarity to the reality. Here again, the formulations that use joint coordinates provided better results, which reinforces again that joint measurements must be carried out. The measurements of all FRFs with the hammer as excitation mean gave coherent results. The mass cancelation process increased slightly the mean LAC between the transfer FRFs, once the accelerometer mass is really low. Unfortunately, the measurements with the shaker gave bad results principally due to the influence of the transducers’ mass which cause different mass distributions in each column and consequently large differences in the transfer FRFs as stated by indicator LAC. In this dissertation, the mass cancelation had an important role and this process was indeed much more sensible and complicated than initially expected. Actually, the influence of transducers’ masses has an extraordinary importance and a strong impact in the structures behavior. The knowledge and discovery of this important interference led us to adopt different experimental setups to avoid or decrease these type of errors. The use of dummy masses and a laser vibrometer were two solutions presented in this work that eliminated this problems and became the mass cancelation completely effective in numerical terms, but the results weren’t what we were really looking for. The experimental modal analyses were carried with success, providing good results when the hammer was used as excitation mean. In relation to the experimental work that measured all the FRFs, this process has important advantage once the noise level in this case is much 112 6.2. FUTURE WORKS lower than in other experimental way. Consequently, if we have a lower level of noise, the results will be much better because the noise has a strong impact in mathematical operations as seen along this work. Therefore, this experimental process is much more advantageous than the other one, and is in fact another working way in this area that gives better results. Furthermore, considering nas the number of measurement points, a modal analysis is much less time consuming than the first experimental procedure once in a modal analysis are required only n measurements whereas in first procedure are required n2measurements. The execution of an experimental modal analysis appears as an alternative with excellent results comparing to the usual experimental work that uses all the FRFs. In terms of excitation means, in this work the hammer provided often better results than the shaker. The use of the shaker introduced extra mass that caused the problems properly documented in this work, despite the good results obtained in terms of adjustment and coherence in the resonance and anti-resonance zones. Actually, the source of this problem is in the drive rod used with the shaker, once the force transducer had a mass that could not be ignored. On the other hand, the hammer gave always results with exact natural frequencies, despite the lack of accuracy in certain frequency ranges. The use of the shaker is more practical in terms of data acquisition, whereas using the hammer, the excitation is done manually. Both correlation criterions used in the experimental work, LAC and MAC, showed a great applicability and effectiveness when was required the comparison of results. The criterion LAC provided the comprehension of some problems, such as the differences between FRFs that were causing strange results, as well as the influence of concentrated introduced in the system by transducers. The correlation criterion MAC had evaluated all the experimental modal analysis, relating the different modal shapes identified in each analysis and giving an idea about the analysis effectiveness. The dissertation now ending contributed to deepen these subjects and understand the real boundaries of this knowledge field. These matters are principally influenced by noise that influences all the mathematical operations and consequently all procedures, as well as small experimental interferences that sometimes have an important impact in the results and must be avoided and eliminated whenever possible. 6.2 Future Works The uncoupling of mechanical components from their main structures is an extensive subject with lots of constraints that affect the performance of these methods. Indeed, these methodologies are really sensitive and vulnerable to extra factors as we have seen along this dissertation, and there is an urgent necessity of develop new techniques capable of reduce or even eliminate the source of some errors which have an high effect. Therefore, this scientific field requires much more research and a long way has still to be made to achieve better results. The technological progress could give an important help to these uncoupling techniques, once if the experimental devices improved their capacities, the high noise levels usually present in experimental data would decrease and consequently the level of uncertainty would not affect so much these techniques. This is an important point because noise causes bad conditioned 113 CHAPTER 6. CONCLUSION matrices which have a direct impact in mathematical operations such as inversions and differences of matrices. The influence of transducers’ masses appears as a subject which is still not very well studied, and in this work they caused some problems for what were presented some solutions. These details influence directly the structure’s natural frequencies once extra mass is added, and consequently the results hold with their interference. Indeed, an experimental measurement has these problems and therefore new methodologies need to be developed in order to remove the influence of that masses. Some work have already been done in this area ([18]), but new methodologies more feasible and practical are required. Another matter that need to be deeply approached is the joint mass distribution, because the beam overlap in conjunction with the bolt provides a different mass distribution, as well as dynamic behaviours which depend on mass distribution. The contact between the bolt and the two beam parts could introduce any non-linear behaviour even as other unknown phenomenons. In this work, it was point out that considering a beam with a concentrated mass, which would represent the extra mass introduced, is not exactly the same once the beam natural frequency differs a little from the value that the uncoupling techniques give. A matter that could be even more approached is the dynamic stiffness. In this work, we have calculated the dynamic stiffness in each experimental analysis, as well as from the theoretical point of view and a algorithm of identification of frequencies was applied. However, the real value of this physical quantity isn’t still well developed and approached. A depth study about this concept could provide extra information and knowledge about the real capacity and coverage of this matter. 114 References [1] M. H. Mayer and L. Gaul. Segment-to-segment contact elements for modelling joint interfaces in finite element analysis. Mechanical Systems and Signal Processing, 21:724–734, 2007. [2] F. C. Batista. Caracterização Dinâmica de Juntas Aparafusadas. PhD thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal, 2012. [3] J. M. A. N. C. Nóbrega. Modelação de estruturas por análise modal experimental e acoplamento dinâmico. Master’s thesis, Faculdade de Engenharia da Universidade do Porto, Portugal, 1996. [4] N. M. M. Maia, J. M. M Silva, A. M. R. Ribeiro, and P. L. C. G. C Silva. On the dynamic characterization of joints using uncoupling techniques. [5] H. Jalali, H. Ahmadian, and J. Mottershead. Identification of nonlinear bolted lap-joint parameters by force-state mapping. International Journal of Solids and Structures, 44:8087– 8105, 2007. [6] J. Kim, J. Yoon, and B. Kang. Finite element analysis and modelling of structure with bolted joints. Applied Mathematical Modelling, 31:895–911, 2007. [7] D. Celtic and M. Boltezar. Identification of the dynamic properties of joints using frequency-response functions. Journal of Sounds and Vibration, 317:158–174, 2008. [8] D. Celtic and M. Boltezar. The influence of the coordinate reduction on the identification of the joint dynamic properties. Mechanical Systems & Signal Processing, 23:1260–1271, 2009. [9] M. Wang, D. Wang, and G. Zheng. Joint dynamic properties identification with partially measured frequency response function. Mechanical Systems & Signal Processing, 27:499–512, 2012. [10] M. Mehrpouya, E. Graham, and S. S. Park. Frf based joint dynamics modeling and identification. Mechanical Systems and Signal Processing, 2013. [11] H. Grafe. Model Updating of Large Structural Dynamics Models Using Measured Response Functions. PhD thesis, Department of Mechanical, Imperial College of Science and Technology, University of London, U.K., 1998. [12] R. J. Allemang and D. L. Brown. A correlation coefficient for modal vector analysis. pages 110–116, 1982. [13] N. M. M. Maia and F. C. Batista. Estimation of unmeasured frequency response functions. ICSV19, 3:8–12, 2012. 115 APPENDIX A. EULER-BERNOULLI FINITE ELEMENTS presented above are formulated to a single element, whereas Kand Mare the final product of the assembly of all elementary matrices. H(ω)=(K−ω2M)−1(A.21) To overcome the necessity of invert the matrix at each frequency, the programme BAPMEF, calculates a FRF according to expression A.22. This way corresponds to a system of equations, where {H}jis the FRF that will be calculated, whereas {F}jis the force vector composed by a unitary load in excitation DOF and zeros in the other components. K−ω2M{H}j={F}j(A.22) Here, we obtain a column of the FRFs matrix corresponding to the excitation at degree-offreedom jand response for all the degrees-of-freedom, whereas according to expression A.21 we obtain the entire FRF matrix at each frequency ω. 122 Appendix B Experimental results 123 APPENDIX B. EXPERIMENTAL RESULTS B.1 Experimental FRFs B.1.1 Beam 1 B.1.1.1 Results obtained with Hammer 5800SL / Accelerometer 27A11 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H91 [m/s2N] H91 − Experimental H91 − BAPMEF 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H93 [m/s2N] H93 − Experimental H93 − BAPMEF Figure B.1: Accelerances H91 and H93. 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H95 [m/s2N] H95 − Experimental H95 − BAPMEF 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H97 [m/s2N] H97 − Experimental H97 − BAPMEF Figure B.2: Accelerances H95 and H97. 124 B.1. EXPERIMENTAL FRFS 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H99 [m/s2N] H99 − Experimental H99 − BAPMEF 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H11|9 [m/s2N] H9|11 − Experimental H9|11 − BAPMEF Figure B.3: Accelerances H99 and H9|11. 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H9|13 [m/s2N] H9|13 − Experimental H9|13 − BAPMEF 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H9|15 [m/s2N] H9|15 − Experimental H9|15 − BAPMEF Figure B.4: Accelerances H9|13 and H9|15. This experimental test was carried out with an impact hammer 5800SL. The response was measured on node five, corresponding to the degrees-of-freedom nine and ten, and all the measurement poitns were excited with the impact hammer. Only the translation degrees-offreedom have been measured. 125 APPENDIX B. EXPERIMENTAL RESULTS B.1.1.2 Results obtained with Shaker LDS201 / Accelerometer 27A11 This experimental test was carried out with the shaker LDS201. The excitation was taken into account in measurement point 4, having the shaker instrumented in this node. Then, the response was measured in all nodes along the beam. Once again, only the translation degreesof-freedom have been measured. 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H17 [m/s2N] H17 − Experimental H17 − BAPMEF 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H37 [m/s2N] H37 − Experimental H37 − BAPMEF Figure B.5: Accelerances H17 and H37. 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H57 [m/s2N] H57 − Experimental H57 − BAPMEF 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H77 [m/s2N] H77 − Experimental H77 − BAPMEF Figure B.6: Accelerances H57 and H77. 126 B.1. EXPERIMENTAL FRFS 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H97 [m/s2N] H97 − Experimental H97 − BAPMEF 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H11|7 [m/s2N] H11|7 − Experimental H11|7 − BAPMEF Figure B.7: Accelerances H97 and H11|7. 0 100 200 300 400 500 600 700 800 10−3 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H13|7 [m/s2N] H13|7 − Experimental H13|7 − BAPMEF 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 105 ω [Hz] Accelerance H15|7 [m/s2N] H15|7 − Experimental H15|7 − BAPMEF Figure B.8: Accelerances H13|7and H15|7. 127 APPENDIX B. EXPERIMENTAL RESULTS B.1.2 Beam 2 B.1.2.1 Results obtained with Hammer 5800SL / Accelerometer 27A11 0 100 200 300 400 500 600 700 800 100 101 102 103 104 ω [Hz] Accelerance H11 [m/s2N] H11 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H13 and H31 [m/s2N] H13 H31 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H15 and H51 [m/s2N] H15 H51 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H17 and H71 [m/s2N] H17 H71 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H19 and H91 [m/s2N] H19 H91 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H33 [m/s2N] Analysis of beam 2: excitation with an hammer H33 Figure B.9: Accelerances of experimental analysis of beam 2 (1). 128 B.1. EXPERIMENTAL FRFS This experimental analysis used the beam 2 as experimental specimen. The excitation was made with hammer 27A11 and the response measured with accelerometer 27A11. The FRFs were acquired in all measurement points, in order to obtain the whole experimental matrix HC tt . 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H35 and H53 [m/s2N] H35 H53 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H37 and H73 [m/s2N] H37 H73 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H39 and H93 [m/s2N] H39 H93 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H55 [m/s2N] H55 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H57 and H75 [m/s2N] H57 H75 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H59 and H95 [m/s2N] H59 H95 Figure B.10: Accelerances of experimental analysis of beam 2 (2). 129 APPENDIX B. EXPERIMENTAL RESULTS 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H77 [m/s2N] H77 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H79 and H97 [m/s2N] Analysis of beam 2: excitation with an hammer H79 H97 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H99 [m/s2N] H99 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H1|11 and H11|1 [m/s2N] H1|11 H11|1 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H1|13 and H13|1 [m/s2N] H1|13 H13|1 0 100 200 300 400 500 600 700 800 100 101 102 103 104 105 ω [Hz] Accelerance H1|15 and H15|1 [m/s2N] H1|15 H15|1 Figure B.11: Accelerances of experimental analysis of beam 2 (3). 130 B.1. EXPERIMENTAL FRFS 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H3|11 and H11|3 [m/s2N] H3|11 H11|3 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H3|13 and H13|3 [m/s2N] H3|13 H13|3 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H3|15 and H15|3 [m/s2N] H3|15 H15|3 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H5|11 and H11|5 [m/s2N] H5|11 H11|5 0 100 200 300 400 500 600 700 800 10−2 10−1 100 101 102 103 104 ω [Hz] Accelerance H5|13 and H13|5 [m/s2N] H5|13 H13|5 0 100 200 300 400 500 600 700 800 10−1 100 101 102 103 104 ω [Hz] Accelerance H38 and H83 [m/s2N] H5|15 H15|5 Figure B.12: Accelerances of experimental analysis of beam 2 (4). 131