Influence of bolted items on the results and consistency of Modal Analysis Master’s Thesis in the Master’s programme in Sound and Vibration MIGUEL COLOMO GONZÁLEZ Department of Civil and Environmental Engineering Division of Applied Acoustics Vibroacoustics Group CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2007 Master’s Thesis 2007:148 MASTER’S THESIS 2007:148 Influence of bolted items on the results and consistency of modal analysis MIGUEL COLOMO GONZÁLEZ Department of Civil and Environmental Engineering Division of Applied Acoustics Vibroacoustics Group CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2007 Influence of bolted items on the results and consistency of modal analysis © MIGUEL COLOMO GONZÁLEZ, 2007 Master’s Thesis 2007:148 Department of Civil and Environmental Engineering Division of Applied Acoustics Vibroacoustics Group Chalmers University of Technology SE-41296 Göteborg Sweden Tel. +46-(0)31 772 1000 Reproservice / Department of Civil and Environmental Engineering Göteborg, Sweden 2007 INFLUENCE OF BOLTED ITEMS ON THE RESULTS AND CONSISTENCY OF MODAL ANALYSIS Master’s Thesis in the MIGUEL COLOMO GONZÁLEZ Department of Civil and Environmental Engineering Division of Applied Acoustics Vibroacoustics Group Chalmers University of Technology Abstract Modal analysis testing is commonly performed on Body in Greys (BIGs), which are automobile structures in the designing/manufacturing stage where the body is formed by assembled metal sheets, and the main components as chassis, powertrain, doors, seats, etc. are not still mounted. Some bolted items are included in the BIG to better take into account their influence on body stiffness. However, their contribution to the stiffness is not impor- tant in the frequency range accessible for modal analysis (usually up to 70 Hz on a BIG). Moreover, they increase the dispersion in modal parame- ters obtained for nominally identical test objects. The questions that arises are whether the items should be included in the BIG definition when per- forming modal analysis or not, and, in this case, how the items in detail influence the results? Multi-input-multi-output (MIMO) measurements of inertances were carried out on three Volvo S80 BIGs. Several configurations were measured for each BIG, starting from the complete body, the bolted items were progressively removed. A modal analysis Matlab programme, MACOL, was developed following the poly-reference Least-Squares Fre- quency Domain (p-LSCF) method, well-known as PolyMAX. Modal anal- ysis results have proved bolted items influence. The biggest bolted item, the grill-overhanging-reinforcement (GOR) has standed out as the major source of the results inconsistency. It introduces modes highly affected by coupling. High coupling yields unreliability of the estimated modal pa- rameters. The GOR removal has been suggested to improve the accuracy of modal analysis results. Comparison of MACOL results with PolyMAX ones has validated the developed programme. Keywords: Modal analysis, Body-in-grey, Body-in-white, PolyMAX, p- LSCF, Least-squares, MIMO, FRF, Mode shape, MAC i Contents 1 Introduction 1 1.1 Thesis background . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Modal Analysis Theory 5 2.1 Experimental Modal Analysis phases . . . . . . . . . . . . . 6 2.2 System identification phase . . . . . . . . . . . . . . . . . . . 7 3 Modal Analysis Programme 10 3.1 The p-LSFD method . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1 Right matrix-fraction model . . . . . . . . . . . . . . . 11 3.1.2 System poles calculation . . . . . . . . . . . . . . . . . 12 3.1.3 Physical poles selection. Stabilization diagrams . . . 14 3.1.4 Modal Parameters Extraction. LSFD method . . . . . 15 3.1.5 Modal Validation . . . . . . . . . . . . . . . . . . . . . 16 3.2 Matlab p-LSCF version: MACOL . . . . . . . . . . . . . . . . 18 3.2.1 MACOL use . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.2 MACOL validation by LMS PolyMAX . . . . . . . . . 27 4 Measurements 33 4.1 Measurements planning . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 Test subjects . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 Support configurations . . . . . . . . . . . . . . . . . . 35 4.1.3 Excitation system . . . . . . . . . . . . . . . . . . . . . 38 4.1.4 Response points . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Measurements set-up . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Measurements procedure . . . . . . . . . . . . . . . . . . . . 47 5 Results and Analysis 48 5.1 Frequency limit of Modal Analysis on BIGs . . . . . . . . . . 48 5.2 MACOL study . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Brackets effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Dispersion introduced by bolted items . . . . . . . . . . . . 54 ii 5.4.1 Tunnel brace . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4.2 Radiator beam . . . . . . . . . . . . . . . . . . . . . . 56 5.4.3 Grill over-hanging reinforcement . . . . . . . . . . . . 57 5.5 Modes evolution. General results . . . . . . . . . . . . . . . . 58 6 Conclusions 62 7 Future work 63 A MACOL code structure 66 B Mode shapes validation 71 C Response points 74 D Excitation points 86 E Equipment pictures 87 F Trigger Happy code updated for MIMO testing 89 G Data structure 94 H Stabilization diagrams for frequency limit study 99 I MAC values used to build up modes evolution 104 iii Acknowledgements First of all, I would like to express my deepest gratitude to my supervisor Andrzej Pietrzyk from Volvo Car Corporation (VCC), for his unconditional support and his efforts to be there whenever it was needed. I am very grateful to my supervisor at Chalmers University, Wolfgang Kropp, for his contribution to important aspects as thesis organization and programme development. Many thanks to all members of Chalmers Applied Acoustics division for creating the best working (and non-working) enviroment I have ever experienced. Special thanks to Patrik Andersson for the interest shown in my work, it has encouraged me to keep on track with programming. My special thanks also goes to Börje Wijk, I cannot find enough words to ex- press how helpful and kind he has been. Muchas gracias, thanks a lot, to all my friends. To those who have been essential in my life, always making me feel appreciated and necessary. To those who have made Sweden my home for this year and a half. To the one beyond friendship, for understanding my departure and heartening me overseas. Lastly, the most important acknowledgement, to my parents. They have led me to become everything I am. They have sacrificed many things for their sons and daughter. They have given me all, support, love and life. Gracias. iv Acronyms BIG Body In Grey BIGn Body In Grey number n BIW Body In White EMA Experimental Modal Analysis FBM Flexible Body Mode FEM Finite Element Method FRF Frequency Response Function GOR Grill Overhanging Reinforcement MAC Modal Assurance Criterion MACOL Modal Analysis Colomo MIF Mode Indicator Function MIMO Multi Input Multi Output MISO Multi Input Single Output MMIF Multivariate Mode Indicator Function MPC Modal Phase Collinearity MPE Modal Parameters Extraction p-LSCF Poly reference Least-Squares Frequency Domain RB Radiator Beam RBM Rigid Body Mode SDOF Single Degree Of Freedom SIMO Single Input Multi Output v SISO Single Input Single Output TB Tunnel Brace VCC Volvo Car Corporation VTF Vibrational Transfer Function vi Symbols and Notation Matrix symbols [ ]: Matrix { }: Column vector < >: Row vector a×b: Dimensions (a rows, b columns) −1: Inverse T: Transpose tr(A): Trace ⊗: Kronecker product Complex operators Abs(x): x absolute value Re(x): x real value Im(x): x imaginary value ‖x‖: x norm x∗: x conjugate xH: x hermitian General notation [0]: Zeroes matrix [A]: Numerator matrix vii [Ac]: Companion matrix [B]: Denominator matrix Ar: Modal constant of rth mode C: Correlation C: Complex number E: Normalized error f : Frequency fp: Frequency of pth pole {F( f )}: Force frequency spectrum [GAA( f )]: Autopower spectrum of A signal [GAB( f )]: Crosspower spectrum between A and B signals H1: FRF estimation for uncorrelated input forces and noise Ĥ: Measured FRFs Hsum: FRFs sum Hsynt( f ): Synthesized FRFs [I]: Identity matrix j: Complex number unit Lp: pth modal participation factor LRoi: Lower residual for oth output and ith input [M]: Reduced normal equations matrix n: Number of resonance modes Na: Number of averages Ni: Number of inputs No: Number of outputs p: Numerator/denominator matrix polynomial order R: Real number URoi: Upper residual for oth output and ith input viii w: Weighting function X( f ): Displacement frequency spectrum Ẍ: Displacement second derivative (acceleration) frequency spectrum z: z-value [αr]( f ): Denominator polynomial coefficient for rth power < βor > ( f ): Numerator polynomial coefficient for rth power and oth output εo: Error between measured and modelled FRF for oth output [mγ2 o ]( f ): Multi-coherence of oth output ηp: Damping ratio of pth pole λr: rth system pole (in Laplace-domain) ∆t: Samping time θ: Matrix containing all polynomial coefficients ψor: Mode shape for oth output and rth mode σ: Standard deviation [Ψ]: Mode shapes matrix ix x Chapter 1 Introduction Modal analysis is a powerful tool to characterize the dynamic properties of a vibrating structure. Data quality and limitations of the used method have been generally established as the main factors determining modal analysis accuracy. However, in some cases the inherent differences between sup- posedly identical structures could be an important factor as well. In such cases modal models have to be built up from testing a set of nominal objects in order to diminish models inaccuracy. In addition, it is of convenience to determine the dispersion of the modal parameters due to the differences between structures. A Body In White (BIW) is the automobile designing/manufacturing stage where the body is formed by assembled metal sheets, and the main components as chassis, powertrain, doors... are not still mounted. A Body In Grey (BIG) is plainly a BIW containing front and rear window. BIGs will be the test objects along the thesis. The designers aim of performing modal analysis on a BIG is avoiding resonances in the frequency range excited by the engine when idling, which is 15-30 Hz. By combining Modal Analysis and Finite Element Method (FEM) techniques one can redesign the car in order to shift in frequency a disturbing mode or modify its damping. This is an advantageous solution as it spares money in prototypes. 1.1 Thesis background During GPDS project at Volvo Car Corporation (VCC) three Volvo S80 BIGs were measured without fixing the Grill Overhanging Reinforcement (GOR) by mistake. When system response was studied, a significant peak around 20 Hz was discovered. This resonance had not appeared in previous tests (done with fixed GOR). Moreover some resonance frequencies were shifted, 1 2 Chapter 1. Introduction their damping factors were changed or/and some modes were split. In- terest about the influence of bolted items on modal analysis results was aroused. Besides GOR, two other bolted items had potential to be decisive in the results, the radiator beam (RB) and the tunnel brace (TB). The three bolted items can be observed in figure 1.1: Figure 1.1: Bolted items: Grill Over-hanging Reinforcement (polygon), Radia- tor Beam (dash-line) and Tunnel Brace (elipse) The three items are included in modal testing to better take into account their influence on body stiffness. However, their contribution to the stiff- ness might not be important in the frequency range accessible for modal analysis (usually up to 70 Hz on a BIG). Every structure has its own resonance modes (self-modes). Joining a sub-structure to an original one (base structure) yields the appearance of new modes in the base object and the modification of its self-modes. Cou- pling between each structure self-modes is the cause for modification of the original ones. Same phenomenon is observed when bolted items are at- tached to the BIG. Furthermore, these bolted items increase the dispersion CHALMERS, Master’s Thesis 2007:148 Chapter 1. Introduction 3 between results obtained for nominally identical BIGs. 1.2 Objectives Section 1.1 introduces the consequences of using bolted items when modal analysis is performed. The objectives of the master’s thesis were set accord- ing the analysis of their influence: - Investigate the dispersion produced by the bolted items. - Recommend the best option for further measurements: Including or excluding the bolted items from the BIG concept. - Determine how high in frequency body modes can be consistently identified. As a beginner in modal analysis one has secondary goals. On the one hand, getting acquainted on modal analysis testing, on the other hand gain- ing a deep insight into the newest modal analysis algorithm, the p-LSCF method. With this last aim on mind, a Matlab programme is developed fol- lowing p-LSCF algorithm, it is called MACOL (Modal Analysis Colomo). One more goal is set, MACOL’s validation by cross-checking its results with a “professional“software, the well-known LMS PolyMAX. 1.3 Overview The thesis has been structured following the chronological order of the work done: - Chapter 2 gives the overall theoretical foundation of modal analysis. Its phases are presented. Special attention is paid to the system iden- tification stage as it is key to better understand theory behind Modal Parameters Extraction (MPE). - The third chapter covers all aspects referring modal analysis pro- gramme development. Firstly, p-LSCF method steps are described. Secondly, MACOL use is explained. Lastly, tests are run in order to validate MACOL by PolyMAX software. - Chapter 4 deals with all facets related to measurements. Measure- ments planning reveals the tests performed to decide the final mea- surements set-up. Additionally, measurements procedure can be found. CHALMERS, Master’s Thesis 2007:148 4 Chapter 1. Introduction - The fifth chapter shows the results obtained during the thesis. They are divided in five sections: the frequency limit of modal analysis on BIGs, results from studying MACOL ins and outs, the brackets effect on modal analysis, the dispersion introduced by the bolted items, and their influence on body modes. - Chapter 6 discusses results. Conclusions are summarized. - The seventh chapter proposes future work. CHALMERS, Master’s Thesis 2007:148 Chapter 2 Modal Analysis Theory Modal Analysis is an experimetal technique to characterize the dynamic properties of a vibrating system. The properties, known as modal param- eters, are estimated from measured Frequency Response Functions (FRFs). The most important modal parameters are presented below: - Resonance frequency: Frequencies at which the system tends to vi- brate at maximum amplitude. - Mode shapes: Form adopted by the system when it is excited at the resonance frequency. - Damping ratio: Actual damping over the amount of damping re- quired to reach critical damping. The critical damping can be inter- preted as the minimum damping that results in non-periodic motion, i.e. simple decay. 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 9 10 r=f/fo FR F Am pl itu de 0 0.5 1 1.5 2 2.5 3 0 20 40 60 80 100 120 140 160 180 r=f/fo FR F Ph as e η = 0 η = 0.2 η = 0.5 η = 1 Figure 2.1: FRF for a SDOF: Amplitude and Phase 5 6 Chapter 2. Modal Analysis Theory Figure 2.1 shows the damping effect on the amplitude and the phase of a resonance mode ( f0). The figure corresponds to the FRF of a vibrat- ing SDOF (Single Degree Of Freedom) structure. The extreme cases are damping ratio equal to zero and equal to one (critical damping). Damping ratio equal to zero yields an infinite amplitude at the resonance frequency whereas damping greater than or equal to critical damping yields perma- nent decay. 2.1 Experimental Modal Analysis phases Modal Analysis is known as Experimental Modal Analysis (EMA) in more precise terms when input forces can be measured. EMA comprises multiple fields of engineering as reflected in the characteristics of its phases: 1. Set-up: Preliminary studies are done to decide the excitation source, the test object suspension, the output points location...etc. A wrong set-up could derive in missing some modes, nonlinearities appear- ance... 2. Data acquisition: Optimal signal processing is essential to assure the suitability and quality of the data. Measurements data allows esti- mating FRFs and coherence functions. 3. System identification: Procedure whereby measured FRFs are ana- lyzed to find the theoretical model which mostly resembles the be- havior of the actual test object [1]. The modal model leads to modal parameter estimation. 4. Validation: Results reliability is evaluated. Parameters, as Modal As- surance Criterion (MAC) or Mode Phase Collinearity (MPC), quantify the rightness of the selected modes. 5. Application: Modal analysis can be used for prediction purposes. The aim is improving the structural dynamic behavior. Nowadays the main tool to apply EMA results are Finite Element Method (FEM) techniques. System identification phase is actually the one covered by the devel- oped modal analysis programme MACOL. Hence, this phase is of high in- terest to the thesis. Section 2.2 provides more information about system identification in order to introduce the basics of modal analysis methods. Further explanations about the other phases are given in [2]. CHALMERS, Master’s Thesis 2007:148 Chapter 2. Modal Analysis Theory 7 2.2 System identification phase The Vibrational Transfer Functions (VTFs) are FRFs which describe the sys- tem response to vibrations. The use of Fourier transform creates a VTF easy to handle. The well-known VTFs are: inertance (output acceleration over input force), mobility (output velocity over input force) and recep- tance (output displacement over input force). The results of the thesis are based on inertance VTFs. From now onwards, FRFs are used to referred to inertance VTFs. A FRF is a function over frequency which has different dimensions de- pending on the number of inputs and outputs. A test can be classified ac- cording to its inputs and outputs: SISO, SIMO, MISO, and MIMO (where: I = input, O = output, S = single, M = multi). The extreme cases are SISO where the FRF is unidimensional and MIMO where the FRF at a certain frequency is a matrix [H( f )] which has as many rows as outputs (No) and as many columns as inputs (Ni):    Ẍ1( f ) ... ẌNo( f )    =   H1,1( f ) . . . H1,Ni( f ) ... . . . ... HNo ,1( f ) . . . HNo ,Ni( f )      F1( f ) ... FNi( f )    {Ẍ( f )} = [H( f )]{F( f )} (2.1) The equation 2.1 shows the transfer matrix of a MIMO system at a cer- tain frequency. F( f ) and Ẍ( f ) vectors are the frequency spectrum of the force and the acceleration, respectively. There are different approaches to estimate the FRF matrix. The approach followed along the thesis assumes no noise on the input forces and uncorrelated noise on the outputs. The notation commonly used for this FRF estimation is H1. [H( f )]No×Ni = [GXF( f )]No×Ni [GFF( f )]−1 Ni×Ni (2.2) In equation 2.2 the transfer matrix is calculated by the “H1 estimation“. Crosspower spectrum between force and acceleration signals ([GXF( f )]) and autopower spectrum of force signal ([GFF( f )]) are involved in the esti- mation. When performing measurements is it of interest to have an indicator of data quality. The coherence function indicates the degree of consistent linear relationship between output and input signals during the averaging process for each frequency [3]. The coherence varies between 0 and 1, in other words no consistent linear relation and perfect consistent linear rela- tion, respectively. When there is more than one input, the coherence of an CHALMERS, Master’s Thesis 2007:148 8 Chapter 2. Modal Analysis Theory output signal is related to all the outputs and it is called multiple coherence ([mγ2 o ]( f )) [4]: [mγ2 o ]( f ) = Ni ∑ s=1 Ni ∑ t=1 Hos( f )GFsFt( f )H∗ ot( f ) Goo( f ) (2.3) Where GFsFt is the crosspower spectrum between forces sth and tth and Goo is the autopower spectrum of the oth output. Once the transfer matrix has been estimated, it is studied to better select the appropriate model, which is finally curve-fitted to the measured FRFs. There are several models but they share the same principle, a vibrating system response can be determined by the summation of the contribution of each mode. The principle comes along with the main limitation of modal analysis, modal models cannot accurately reflect the effect of modes out of the frequency range where modal analysis is performed. Each mode is dominating around its own resonance frequency, therefore inaccuracy is mainly noticeable between resonances. Figure 2.2: FRF breakdown into single modes contribution Figure 2.2 illustrates the base of modal models [5]. The top curve shows the FRF of a system which has three resonance modes in the considered fre- quency range. The lower curves show the contribution of each mode to the system response. The modes dominance in the frequency range nearby their resonance frequencies is significant. In addition, the antiresonances CHALMERS, Master’s Thesis 2007:148 Chapter 2. Modal Analysis Theory 9 (FRFs minima) can be observed. They appear at the modes contribution intersection. The general form of the FRFs model is a consequence of the principle. As one can observe in equation (2.4), modeled transfer function (for oth output,ith input) is formed by adding each mode contribution: Hoi( f ) = Ẍo Fi = n ∑ r=1 Ar,oi λ2 r − (2π f )2 (2.4) Where λr is the Laplace pole corresponding to the rth mode and Ar,oi is its modal constant for oth output and ith input, whereas n represents the number of modes. Modal models can be more complex as shown in equa- tion 3.16. CHALMERS, Master’s Thesis 2007:148 Chapter 3 Modal Analysis Programme The poly-reference Least Squares Complex Frequency-Domain Method (p- LSCF) is the latest algorithm proposed for performing modal analysis. It has successfully settled among the industry world as the most effective modal analysis technique thanks to the programme LMS Test.Lab, devel- oped by LMS International. LMS Test.Lab contains a p-LSCF module called PolyMAX. Research work done at thesis beginning revealed p-LSCF as most pow- erful modal analysis method. Hence, it was decided to develop and imple- ment a code based on this theory in Matlab. The resulting modal analysis programme is called MACOL (Modal Analysis COLomo) with reference to the author’s family name. To understand the advantages of p-LSCF method is key to be aware of the difficulties involved in Modal Analysis, according to [6]: - High order systems are problematic due to their high modal overlap- ping. - The damping of fully trimmed structures is not properly estimated. - Difficulty to avoid mathematical poles. - Uncertainty about the optimal poles selection. - Inconsistencies of the measured data. - Inconsistency of the modal participation factors obtained from differ- ent analysis. The main strength of p-LSCF method is being able to handle with the “rough“cases, i.e. high order or trimmed systems. Its very stable poles identification is the reason why it can overcome these cases. In addition, modal parameters are precisely extracted via short operational time. 10 Chapter 3. Modal Analysis Programme 11 3.1 The p-LSFD method The p-LSFD method is explained following the description provided by [7]. Nevertheless research from other sources was needed as well. 3.1.1 Right matrix-fraction model The right matrix-fraction model expresses the FRFs matrix as the division of two matrices. In order to simplify method presentation, the oth row of FRFs matrix is taken. It represents the oth output over all inputs: [H( f )]No×Ni = [B( f )]No×Ni [A( f )]−1 Ni×Ni < Ho( f ) >1×Ni = < Bo( f ) >1×Ni [A( f )]−1 Ni×Ni (3.1) Both, A( f ) and B( f ), are matrices built by polynomials. The polynomi- als variable is z (from z-domain): z( f ) = e−j2π f ∆t (3.2) In equation 3.2 ∆t is the sampling time. Polynomials are presented as follows: < Bo( f ) > = p ∑ r=0 < βor > zr( f ) ∈ C1×Ni [A( f )] = p ∑ r=0 [αr]zr( f ) ∈ CNi×Ni (3.3) The denominator matrix (A( f )) roots are actually the structure poles. The polynomial order p is a key parameter in curve fitting. Its importance is highlighted in sections 3.1.3 and 5.2. The polynomial coefficients shown in equation 3.3 are not the standard ones but a vector (in the case of the numerator) and a matrix (in the case of the denominator). Polynomial co- efficients can be assembled in matrices (equation 3.4): βo =   βo0 βo1 ... βop   ∈ R(p+1)×Ni α =   α0 α1 ... αp   ∈ RNi(p+1)×Ni θ =   β1 ... βNo α   ∈ R(No+Ni)(p+1)×Ni (3.4) CHALMERS, Master’s Thesis 2007:148 12 Chapter 3. Modal Analysis Programme Where matrix θ includes all the polynomial coefficients which must be calculated to curve-fit the measured FRFs matrix. 3.1.2 System poles calculation The curve-fitting is performed by least-squares means. The error between the measured FRFs (Ĥ) and the right matrix-fraction modeled FRFs (H) is taken in a way which linearizes the problem: εo( f , θ) = wo( f ) ( Bo( f , βo)− Ĥo( f )A( f , α) ) (3.5) The error (εo) in equation 3.5 can be weighted by a function w for each frequency and output. The weighting function for each FRF matrix row can be determined by the FRF variance [8]. Statistical error serves for estimat- ing variance [2]: wo( f ) = 1√ var(Ĥo( f )) = 1√ 1−mγ2 o ( f ) 2Namγ2 o ( f ) (3.6) The equation 3.6 illustrates the weighting function dependency on the coherence (mγ2 o) and the number of averages over each measurement (Na). The cost function all over the frequency points and FRFs can be derived from equation 3.7, where tr refers to matrix trace. ł(θ) = No ∑ o=0 Nf ∑ k=1 tr((εo( fk, θ))Hεo( fk, θ)) (3.7) The values participating in the cost function are gathered in a special system, the so-called reduced normal equations. Matrices used to build the system are defined in equation 3.8. Symbol ⊗ denotes Kronecker product. Xo =   wo( f1)(1 z( f1) . . . zp( f1)) ... wo( fNf )(1 z( fNf ) . . . zp( fNf ))   ∈ CNf×(p+1) Yo =   −wo( f1)(1 z( f1) . . . zp( f1))⊗ Ĥo( f1) ... −wo( fNf )(1 z( fNf ) . . . zp( fNf ))⊗ Ĥo( fNf )   ∈ CNf×Ni(p+1) Ro = Re(XH o Xo) ∈ R(p+1)×(p+1) So = Re(XH o Yo) ∈ R(p+1)×Ni(p+1) To = Re(YH o Yo) ∈ RNi(p+1)×Ni(p+1) (3.8) Least-squares solution is found by equaling the cost function derivative to zero. Matrices defined in equation 3.8 simplify the expression: CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 13 ∂l(θ) ∂βo = 2(Roβo + Soα) = 0, ∀o = 1, . . . , No ∂l(θ) ∂α = 2 Nf ∑ o=1 (ST o βo + Toα) = 0 (3.9) First equation from 3.9 allows expressing the numerator coefficients as a function of the denominator coefficients: βo = −R−1 o Soα (3.10) The reduced normal equations are gathered by a matrix, M, by applying the equation 3.10 on the second equation from 3.9: ( 2 Nf ∑ o=1 ( To − ST o R−1 o So )) α = 0 ⇔ Mα = 0 (3.11) Where M ∈ RNi(p+1)×Ni(p+1) is the reduced normal equations matrix. Last α coefficient is imposed to avoid the trivial solution and parameters redundancy in the right matrix-fraction model: αp = INi×Ni . The rest of the α coefficients are derived from equation 3.12:   α0 α1 ... αp−1   = −M(1 : Ni p, 1 : Ni p)−1M(1 : Ni p, Ni p + 1 : Ni(p + 1)) (3.12) Once the denominator coefficients are obtained, system poles can be found. The suggested method to calculate them is solving the eigenvalue problem of the denominator polynomial companion matrix (Ac) [9]: Ac =   0NixNi INixNI 0NixNi . . . 0NixNi 0NixNi 0NixNi 0NixNi INixNI . . . 0NixNi 0NixNi ... ... ... . . . ... ... 0NixNi 0NixNi 0NixNI . . . 0NixNi INixNi −[αT 0 ] −[αT 1 ] −[αT 2 ] . . . [−αT p−2] [−αT p−1]   (3.13) The eigenvalue problem solution also provides the modal participation factors L. Each modal participation factor is a vector which corresponds to a system pole. Taking for instance the rth pole, the modal participation factor is the rth column and the last Ni rows of the eigenvectors matrix. CHALMERS, Master’s Thesis 2007:148 14 Chapter 3. Modal Analysis Programme Modal participation factors relate to system inputs and they are propor- tional to mode shapes. In section 3.1.4 more information is given about modal participation factors. In addition, equation 3.16 presents the role of modal participation factors in a modal model. The poles are obtained in z-domain (zp). Then, they are converted to Laplace domain (λp) in order to calculate the resonance frequencies ( fp) and the damping ratios (ηp): z = e−λ∆t ⇒ λp = − ln zp ∆t (3.14) Resonance frequency and the damping ratio formulas (3.2.1) are straight forward when deriving them from poles in Laplace domain [11]: fp = Abs(Im(λp)) 2π ηp = − Re(λp) Abs(λp) (3.15) 3.1.3 Physical poles selection. Stabilization diagrams Years of research in Modal Analysis have not found yet a modal analy- sis algorithm capable to calculate only physical poles. Non-physical poles appear in calculations as a result of the over-estimation of the polynomial order necessary to find all the physical poles. They are called spurious or mathematical poles as well. The tool used to discriminate physical poles from the spurious ones is the stabilization diagram. It is obtained by repeating the poles calculation for decreasing model order. The Laplace poles which are not stable, ac- cording to Laplace stability criterion (Re(λp) < 0), are not shown in the stabilization diagram. Each set of poles calculated for a certain model or- der is compared with the set of poles calculated for the lower modal order. If rth pole at pth modal order is not found for the order below (p-1) is labeled as new or unstable pole in the stabilization diagram. A pole is considered not found when there is no pole in a 1% frequency margin for the lower or- der. In MACOL the symbol used is a red circle (o). When the pole is found there are different stability cases: CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 15 Table 3.1: Stability borders of damping ratios (η) and modal partipation factors (< L >) for Stabilization Diagrams ηr,p−ηr,p−1 ηr,p−1 < 5% ‖−‖ ‖‖ < 2% Symbol no no blue square yes no blue diamond no yes blue triangle yes yes black cross ∆ηr ∆ ‖< Lr >‖ The default background of MACOL stabilization diagram is the sum- mation of FRFs (equation 3.21). The summation is scaled in order to have similar dimensions to the stabilization diagram ones. The last case in table 3.1 will be from now onwards referred as a stable pole. After plotting the stabilization diagram the next step is selecting the physical poles. Further comments about poles selection are given in section 3.2.1. In addition a stabilization diagram example is shown in figure 3.4. 3.1.4 Modal Parameters Extraction. LSFD method The poles and the modal participation factors are the modal parameters already obtained at this point. The rest of the modal parameters can be estimated by identifying the FRFs with a modal model. In equation 2.4 a simple modal model was shown. When the poles and the modal participa- tion factors of a MIMO system are known the following modal model is of convenience: Hsynt oi ( f ) = n ∑ r=1 ( ψorLT ir j2π f − λr + ψ∗orLH ir j2π f − λ∗r ) − LRoi (2π f )2 + URoi (3.16) The FRF built from the estimated modal parameters is called synthe- sized FRF (Hsynt). Equation 3.16 shows the synthesized FRFs dependence on modal parameters. The stable poles appear as conjugate couples. Setting the modes in conjugate couples reduces to the half the quantity of modal parameters to calculate. The modal parameters included in the modal model are: - System poles(λ): The poles of the system in Laplace domain. Damp- ing ratios and frequencies can be obtained from poles by using equa- tion 3.2.1. - Modal participation factors (L): Modal participation factors relate to system inputs and they are proportional to mode shapes. There is a modal participation factor for each mode (r) and input (i). CHALMERS, Master’s Thesis 2007:148 16 Chapter 3. Modal Analysis Programme - Mode shapes (ψ): A mode shape is the specific pattern of vibration adopted by the system at a resonance frequency. There is a mode shape for each output (o) at each resonance mode (r). - Upper residual (UR): The upper residual is a term introduced to compensate the modes above the analyzed frequency range. It is a constant term defined for each transfer function (one per input- output combination). In MACOL the UR is implemented as a real value but could be taken as complex value as well. - Lower residual (LR): The lower residual is a term introduced to com- pensate the modes below the analyzed frequency range. Assuming proportional damping one can estimate the lower residual by a con- stant term over the squared frequency (in rad s ) of the considered point. There is a lower residual term for each transfer function (one per input-output combination). In MACOL the LR is implemented as a real value but could be taken as complex value as well. The residual terms (UR and LR) formulation is done for receptance FRFs, i.e. for output displacements over input forces. The equality of the measured FRFs and the modal model yields a lin- ear system of equations where the unknowns are the mode shapes and the residuals. The number of equations is higher than the number of un- knowns, therefore least-squares technique is once more needed. This step of modal analysis is commonly called LSFD method. 3.1.5 Modal Validation The modal validation phase intends to verify the results obtained in the Modal Parameter Extraction (MPE) stage. There have been developed sev- eral parameters to perform modal validation. A complete description of them can be found in [2]. In this section the parameters calculated by MA- COL are presented. Normalized Error and Correlation between measured and synthesized FRFs The synthesized FRFs comparison with the measured FRFs is the first veri- fication to be done in modal analysis. The selection of a spurious pole as a stable one can be easily observed at first sight. The accuracy of the synthe- CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 17 sis is quantized by the normalized error (3.17) and the correlation (3.18): EHoi = Nf ∑ n f =1 ∣∣∣Ĥoi( fn f )− Hsynt oi ( fn f ) ∣∣∣ 2 Nf ∑ n f =1 ∣∣∣Hsynt oi ( fn f ) ∣∣∣ 2 (3.17) CHoi = ∣∣∣∣∣ Nf ∑ n f =1 Ĥoi( fn f )H∗synt oi ( fn f ) ∣∣∣∣∣ 2 ( Nf ∑ n f =1 Ĥoi( fn f )Ĥ∗ oi( fn f ))( Nf ∑ n f =1 Hsynt oi ( fn f )H∗synt oi ( fn f )) (3.18) A proper modal analysis should not expect less than a 0.85 of correla- tion and more than a 0.1 of error. Modal Assurance Criterion (MAC) The Modal Assurance Criterion (MAC) compares different sets of estimated mode shapes. MAC is plainly the correlation between mode shape vectors. It can be used to check the relation between modes within the same set [2] as well, then it is called auto-MAC. It is based on the theoretical orthogo- nality between different physical modes. Ideally two equal modes would have a MAC of 1 (or 100%) and two different physical modes would have a MAC of 0. MAC([Ψ1], [Ψ2]) = ∣∣[ΨH 1 ][Ψ2] ∣∣2 ([ΨH 1 ][Ψ1])([ΨH 2 ][Ψ2]) (3.19) Equation 3.19 allows the calculation of the MAC-matrix. The MAC ma- trix contains the MAC values between all possible pairs between the modes of the two sets. Ψ1 ∈ CNo×n1 and Ψ2 ∈ CNo×n2 are matrices formed by all the mode shapes from first and second set respectively. The MAC value between two estimates of the same physical mode should be above 90% and the MAC value of two different modes should be below 10%. Other phenomena are explained below [2]: - MAC<90% between estimates of the same physical pole: At least one of the estimates is poor. This could be due to a poorly excited mode or a low amplitude mode. - MAC>35% between estimates of different modes at close frequencies: The modes are similar. It might be that they are the same modes but CHALMERS, Master’s Thesis 2007:148 18 Chapter 3. Modal Analysis Programme one part of the system is vibrating in phase. Small frequency shifts during measurements might have happened as well. - MAC>35% between estimates of different modes at distant frequen- cies: Measurements set-up error. An insufficient number of outputs or an erroneus setting of them could cause this phenomenon. Modal Phase Collinearity (MPC) The Modal Phase Collinearity (MPC) is a measure of the complexity degree of a mode. It quantizes the linear relation between real and imaginary parts of the mode shape coefficients. Its derivation is shown in equation 3.20. ψ̃or = ψor − No ∑ s=1 ψsr No ε = ∥∥Im{ψ̃}r ∥∥2 − ∥∥Re{ψ̃}r ∥∥2 2 ( Re{ψ̃}T r · Im{ψ̃}r ) θ = arctan ( |ε| + sign(ε) √ 1 + ε2 ) MPCr = ∥∥Re{ψ̃}r ∥∥2 + (Re{ψ̃}T r ·Im{ψ̃}r)(2(ε2+1) sin2 θ−1) ε(∥∥Im{ψ̃}r ∥∥2 + ∥∥Re{ψ̃}r ∥∥2 ) (3.20) The physical modes have a MPC close to 1. If the mode has a low MPC is either a mathematical or a noisy pole. The MPC is a valid tool when the modes are normal, i.e. when the damping is proportional. 3.2 Matlab p-LSCF version: MACOL The literature study done at the start of this master thesis revealed p-LSFD method as the most advanced and capable for cases of high modal over- lapping. This is the case of the thesis test object, the Volvo S80 BIG. The programming of a Matlab version became a challenge although initially it was not one of the thesis objectives. Due to its secondary role it has not been possible to complete the programme and, what is more important, to test it for other test objects than the 3 BIGs. Fortunately the modal analysis performed during the thesis has allowed testing MACOL capabilities. In addition, some cross-checking has been performed using PolyMAX from LMS Test.Lab, the software which intro- duced p-LSCF method in industry. CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 19 In section 3.2.1 MACOL use is explained and some details are given about the programme development. Section 3.2.2 presents the tests done to validate MACOL results by PolyMAX ones. 3.2.1 MACOL use The structure of the programme is found in appendix A and can serve as a reference to follow this chapter. The programme needs a data file to be started. The file contains the following inputs: - Frequency vector (fa): It is a vector containing all measured frequency points. It is highly recommended to have enough frequency resolu- tion otherwise modal analysis algorithms fail. - Transfer matrix (H): Matrix containing the inertance FRFs of a sys- tem. It is a three-dimensional matrix, where the first dimension re- lates to the frequency components, the second one to the input signals and the third one to the output signals. - Coherence: Matrix containing the coherence between the ouput sig- nals and the contribution over all system inputs. It is a two-dimensional matrix, where the first dimension relates to the frequency compo- nents and the second one to the output, as it is defined in equation 2.3. Section 3.1.2 explains the use of the coherence in FRFs weighting. It is not a necessary input because programme default settings estab- lish no weighting, i.e. all the components of the coherence matrix are one. MACOL steps are described in following points: Frequency range selection Modal analysis is seldom performed over all the measured frequency range. The standard action is dividing it into smaller frequency ranges which are independently analyzed. MACOL must be run for each range. The divi- sion leads to improvement on the results accuracy. Two suggestions are given [2]: - Each partition of the measured frequency range should have less than ten resonance modes. - Frequency limits are recommended to have low levels. Adittionally, resonance slopes should be avoided in order to diminish out-of-band modes influence. CHALMERS, Master’s Thesis 2007:148 20 Chapter 3. Modal Analysis Programme The influence of frequency range width on poles estimation has been observed. Checks performed discovered that using very narrow frequency ranges implies overlooking stable poles. Figures 3.1 and 3.2 illustrate this phenomenon. Narrowing the frequency range entitles the lost of frequency points, i.e. number of equations is reduced. This could be improved in- creasing the polynomial order (MACOL intends to find more poles) al- though increasing frequency range has proved to be more effective. 44 46 48 50 52 54 56 20 25 30 35 40 45 50 55 60 f (Hz) Po lyn om ia l o rd er a nd T Fs (s hi fte d) Figure 3.1: Stabilization diagram for the standard configuration of BIG1 wider range 44 45 46 47 48 49 50 20 25 30 35 40 45 50 55 60 f (Hz) Po lyn om ia l o rd er a nd T Fs (s hi fte d) Figure 3.2: Stabilization diagram for the standard configuration of BIG1 nar- rower range CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 21 In order to select a proper frequency range, measured data can be stud- ied in three different forms: - SDOF: The most of the modes can be clearly found by checking one of the system FRFs although a few modes could be overlooked. - FRFs Sum: According to one’s experience it is the best tool when choosing the frequency range although local modes or highly coupled modes could be hard to identify. The summation of FRFs is done separately in real and complex terms (3.21): Hsum = No ∑ o=1 Ni ∑ i=1 |Re(Hoi)| + j No ∑ o=1 Ni ∑ i=1 |Im(Hoi)| (3.21) - Multivariate Mode Indicator Function (MMIF): It is a derivation of the common Mode Indicator Function (MIF). The difference is that it takes into account all inputs. The MIF expresses the ratio of the kinetic energies of in-phase response to total response. It is a value from 0 to 1. The minimum of this function indicates the presence of modes that can be excited as real normal modes [10]. The MMIF is the minimum of the eigenvalues in equation 3.22. λ ( Re(H)TRe(H) + Im(H)T Im(H) ) v = Im(H)T Im(H)v (3.22) 0 20 40 60 80 100 120 140 −100 −50 0 50 f(Hz)FR F of a S DO F ov er 1 st s h 0 20 40 60 80 100 120 140 0 0.5 1 f(Hz) M M IF 0 20 40 60 80 100 120 140 −100 −50 0 50 f(Hz) FR Fs S um Figure 3.3: Measured data representations: Inertance SDOF FRF, Inertance FRFs Sum, MMIF CHALMERS, Master’s Thesis 2007:148 22 Chapter 3. Modal Analysis Programme The use of any of this indicators is acceptable. The three possibilities are shown in figure 3.3. System poles calculation. Polynomial orders selection The system poles are calculated in z-domain and then converted into Laplace- domain. Poles are found for different polynomial orders as explained in section 3.1.3. MACOL allows choosing the minimum and maximum poly- nomial order (p) to create stabilization diagrams. The poles are found by solving the companion matrix eigenvalue prob- lem (3.13), where the matrix dimension is Ni × p. Therefore the operational time is approximately proportional to p2, which means that it is advisable avoiding very high polynomial orders. A reasonable maximum polynomial order is 50 but this is dependent on the frequency range width, the number of FRFs and the measurements quality. Hence, it is worth to do some testing when performing modal anal- ysis on a data set for the first time. The lower polynomial orders do not con- tend all the poles as many of the ones found are unstable in laplace terms. Therefore, the minimum can be usually set from 10 to 20. In short, a proper selection of the frequency range and polynomial order limits is based on experience, engineer judgment and trial and error. System poles identification. Stabilization diagrams The system poles identification is the key step in modal analysis. Indeed, this is the only stage which needs of human intervention. Programs are able to find system poles for different polynomial orders but only mankind, through experience, is capable of selecting the physical poles among the ones found as stable. Table 3.1 contains the different poles one can find in a stabilization dia- gram according to MACOL notation. One physical pole is easy to identify when it appears for the most of the polynomial orders. In this case, a verti- cal line of black crosses can be seen. When two poles are close in frequency or the modal overlapping is too high, the vertical line is not seen. Instead, few stable poles appear in be- tween blue marks (either squares, diamonds or triangles). The coupling between close-in-frequency modes hinders the stabilization of modal pa- rameters in calculations. From now onwards, the modes which are not CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 23 clearly identifiable will be called “unclear modes“. The selection of a stable pole for an unclear mode can still be done but its reliability is doubtful. A discontinuity of pure stable poles (black crosses) appears for these poles. From one stable pole (black cross) to the next one, poles unstable in damping factor and/or modal participation fac- tor are found. The discontinuity entitles that the extracted modal parameters are sig- nificantly different depending on the polynomial order selected. Section 5.2 compares the results consistency of the unclear modes and the “easy- to-identify“ones. 38 40 42 44 46 48 50 52 54 56 15 20 25 30 35 40 45 50 55 f (Hz) Po lyn om ia l o rd er a nd T Fs (s hi fte d) Figure 3.4: Stabilization diagram. BIG1 standard configuration (f:38-55.5) In Figure 3.4 the two kind of modes can be found: easy-to-identify modes (at 40.26, 43.15, 45.79, 50.74 and 53.46 Hz) and unclear ones (at 40.83 and 48 Hz). Figure 3.5, on next page, shows how poles are selected. MACOL poles selection works by introducing, in the Matlab Command Window, the poles one by one. Firstly, the pole frequency, and then the polynomial order used to calculate it (figure 3.5). CHALMERS, Master’s Thesis 2007:148 24 Chapter 3. Modal Analysis Programme Figure 3.5: MACOL selection of stable poles MACOL’s modal parameters extraction Modal participation factors and poles are extracted from the stabilization diagram. The frequency and damping factor of each pole can be estimated as explained in equations and . The residuals and the mode shapes are the modal parameters left to esti- mate. Equation 3.16 can be used to create a linear equations system where these modal parameters are the unknowns. The measured transfer func- tions are the system constant terms. As the modal model is formulated for receptance FRFs, MACOL converts the inertance FRFs to receptance FRFs in order to solve the system. MACOL operates the system to speed the solving process up. The dia- gram in figure 3.6 explains the procedure followed. It starts from the initial situation, where there is one equation for each frequency point, input and output. The number of unknowns is related to the mode shapes (one per output for each of the resonance modes) and the residuals (one per output for each of the inputs). CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 25 Figure 3.6: Equations system operation diagram As Matlab is not able to handle matrices over a certain size is wise to shorten the number of equations by separately solving the equations for each output. It can be done because the equations generated by each out- put can be decoupled from the equations generated by the others. More- over, Matlab is proved to be faster when solving a real system instead of a complex one. Hence MACOL splits the equation system. Each equation becomes two, one from the real components and another one from the complex compo- nents. Modal Validation A complete description of the modal validation is given in section 3.1.5. When the MPE is done, MACOL offers checking the synthesized FRFs in comparison to the measured ones (figure 3.7). The modal parameters cal- culation is highly dependent on fitting accuracy at resonance frequencies, and surroundings. Therefore, it is the most important feature to analyze when comparing measured and synthesized FRFs. This modal validation step provides the operator a first view of the results quality and, specially, which modes are not well identified. In addition, MACOL calculates the FRFs normalized error and their correlation according to equations 3.17 and 3.18. CHALMERS, Master’s Thesis 2007:148 26 Chapter 3. Modal Analysis Programme 38 40 42 44 46 48 50 52 54 56 −40 −35 −30 −25 −20 −15 −10 −5 0 f(Hz) In er ta nc e (d B re 1 m s2 /N ) FRF of dof number 20 over the reference number 1 measured TF synthesised TF Figure 3.7: Synthesized and Measured FRF for a single DOF The second step of MACOL’s modal validation is the auto-MAC matrix study. All the elements contained in the matrix diagonal equal 100% (they are the auto-correlation of each mode shape). Coupling phenomena can be studied from auto-MAC values. The coupling between modes close in frequency yields a MACs over 10% although it should not be higher than 40-50%. A MAC over 50% between two close-in-frequency modes is a sign of having selected twice the same mode. Hence, one of them should be re- moved from analysis. Another issue to be checked is the high MAC values between far in frequency modes. In this case a MAC over 35% indicates an inappropriate measurement set-up, as commented in section 3.1.5. Figure 3.8 shows an example of MACOL’s modal validation. The modes under validation are the ones selected from figure 3.4. The study of the MAC values and the synthesized FRFs could suggest the modification of certain poles. MACOL allows the user to remove or modify poles, for in- stance by taking another polynomial order. When none of the options leads to a satisfactory accuracy level for a certain pole, a zooming of the pole is advised. However, the frequency range width cannot be extremely reduced as explained in section 3.2.1. The correlation and the normalized error can be observed at the bot- tom of figure 3.8. The fitting accuracy of synthesized FRFs is high (correla- tion: 92.6%, normalized error: 7.6%) as expected. At Command Window’s top, the modes frequencies and damping ratios are listed. The mode index corresponds to the number of row/column in the auto-MAC matrix. The MAC values study reveals a strong coupling between the 1st and 2nd modes (MAC=28.52%), and between the 5th and 6th modes (MAC=38.92%). MAC CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 27 values (around 30%) between the 5th and 9th modes, and between the 6th and 9th modes are of relevance as well. The 9th mode shape is similar to two other mode shapes. The cause might be the lack of output signals to totally capture body movement for the 9th mode. Figure 3.8: Modal Validation by MACOL 3.2.2 MACOL validation by LMS PolyMAX MACOL could not be completely finished by the end of the thesis although the most important modal analysis features can be performed except for modes animation. In programmes development is mandatory the verifica- tion of the results. PolyMAX from LMS Test.Lab is well-known in industry as a reliable modal analysis software. As PolyMAX and MACOL come from same method (p-LSCF), PolyMAX was chosen as benchmark in order to assure MACOL’s correct operation. CHALMERS, Master’s Thesis 2007:148 28 Chapter 3. Modal Analysis Programme Stabilization diagrams comparison Figures 3.9 and 3.10 show an example of stabilization diagrams from Poly- MAX and MACOL, respectively. The example was taken for same body and configuration. The notation in PolyMAX diagram is different. The red s means stable pole and is the equivalent to the black cross in MACOL. Same modes are found from both diagrams. Therefore MACOL search of poles works correctly. However, other differences between MACOL and PolyMAX operation have been observed. For instance, MACOL needs a higher polynomial order to find all the poles PolyMAX does. The consequence is reflected in operational time, which is higher for MACOL. Additionally, PolyMAX shows the damping ratio and the scattering to facilitate poles selection. More diagrams were investigated to reassure the reliability of MACOL diagrams. Figure 3.9: BIG3 without GOR&RB Stabilization diagram by PolyMAX CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 29 35 40 45 50 55 60 65 15 20 25 30 35 40 45 50 55 60 65 f (Hz) Po lyn om ia l o rd er a nd T Fs (s hi fte d) Figure 3.10: BIG3 without GOR&RB Stabilization diagram by MACOL Estimated poles validation The stabilization diagrams checking was somehow visual. The comparison between resonance frequencies and damping factors allows making a more solid judgment about the calculations validity up to this point. Table 3.2 compares the modes selected from the stabilization diagrams in figures 3.9 and 3.10. MACOL shows slightly lower values than PolyMAX for both resonance frequencies and damping ratios. The low differences between results certify MACOL’s capability for finding the correct physical poles. Table 3.2: fr and ηr for PolyMAX and MACOL fr(LMS) fr(MACOL) ∆ fr ηr(LMS) ηr(MACOL) ∆ηr 41.48 41.34 0.13 1.49 1.37 0.12 46.21 46.08 0.13 1.33 1.11 0.22 47.79 47.69 0.11 0.87 0.92 0.05 47.92 47.84 0.08 0.54 0.55 0.01 51.62 51.55 0.07 1.04 0.95 0.09 53.62 53.53 0.09 0.94 0.90 0.04 55.38 55.29 0.09 0.34 0.35 0.01 60.28 60.19 0.09 0.32 0.31 0.01 Modal Parameters Extraction (MPE) checkings The MPE leads to the estimation of the residuals and the mode shapes. MA- COL calculates real residuals and PolyMAX complex residuals. Therefore CHALMERS, Master’s Thesis 2007:148 30 Chapter 3. Modal Analysis Programme residuals cannot be numerically compared. Moreover, residuals are terms meant to compensate out of band modes, therefore they do not have an ab- solute value. The residuals aim is improving the fitting in the lower frequencies and the ranges between resonances. Synthesized FRFs provide information about residuals contribution. Reader can use figure 3.7 to check residu- als effect on synthesize FRFs. MACOL shows a reasonable fitting although it could be improved by using complex residuals. The mode shapes are vectors which have different magnitudes depend- ing on the weighting system used. The weighting (scaling) is done depend- ing on the system information one has. For instance, the mass associated to each output. PolyMAX can perform weighting whereas MACOL does not. However, weighting is not necessary to animate system mode shapes. The comparison between animations would be the best tool to validate MACOL mode shapes. Unfortunately, MACOL modes animation is not implemented yet. Therefore, the comparison of mode shapes magnitudes is the only possibility. Mode shapes amplitude have been compared for several cases. One of them is shown in appendix B. A reasonable approximation is observed for most of FRFs, although few of them (16 DOFs out of 279) present significant errors (over 40%). More- over, two FRFs cannot be taken into account because their error is over 1000%. If they are excluded, the averaged normalized error is less than 20%. The DOFs which have high errors might have a very low level of move- ment, or they might be poorly excited. As a result mode shapes are innacu- ratedly estimated for these DOFs. In spite of them, MACOL overall re- sults are considered acceptable although further studies are left to be done whenever animations are available. Modal validation comparison Section 3.2.1 highlighted the importance of an accurate fitting at the reso- nance peaks when considering synthesized FRFs. There is no parameter to quantify the fitting quality around the resonance peaks. Therefore the study has to be reduced to general results, i.e. the correlation and the nor- malized error. Table 3.3 gives a comparison between programmes for a single case, BIG3 without GOR and RB. Results are slightly in favor of MACOL, spe- CHALMERS, Master’s Thesis 2007:148 Chapter 3. Modal Analysis Programme 31 cially if one looks at the normalized error. Once again MACOL results are validated. Table 3.3: Coherence and Normalized Error by PolyMAX and MACOL LMS MACOL Correlation (%) 91.66 89.07 Normalized Error (%) 17.91 11.13 The auto-MAC values are compared as well. Values from figure 3.12 are cross-checked with the ones from LMS PolyMAX (figure 3.11). It cannot be expected to have same MACs as mode shapes correspond to different poles selections done by different programmes. Still values should be close, specially the ones having a high magnitude. For instance the MAC val- ues between the 6th mode and 7th one (MACMACOL = 64.99%,MACLMS = 61.04%) or the 7th mode and the 8th one (MACMACOL = 16.52%,MACLMS = 20.78%). These values are a coupling sign. The resemblance is acceptable. Hence, PolyMAX results support MACOL ones. Figure 3.11: Auto-MAC values extracted from LMS Test.Lab Figure 3.12: Auto-MAC values extracted from MACOL CHALMERS, Master’s Thesis 2007:148 32 Chapter 3. Modal Analysis Programme The poly-reference least squares frequency-domain method, p-LSFD, and its Matlab implementation, MACOL, have been explained along this chapter. Further comments and suggestions about MACOL will be given in section 5.2 in order to provide modal analysis operators a deeper under- standing of the method and help them to improve their use of programmes. CHALMERS, Master’s Thesis 2007:148 Chapter 4 Measurements 4.1 Measurements planning Any modal analysis test entails an extensive planning phase in which im- portant decisions have to be taken. The concerns related to this thesis were: - Test subjects: Sufficient quantity to ensure the characterisation of the bolted items influence. - Supports configuration: Minimisation of external influences. - Excitation system: Effective excitation of modes. - Response points: Positioned to capture all body modes and clarify their identification. 4.1.1 Test subjects The main goal of the thesis is studying the influence of three bolted items on the results and consistency of modal analysis performed on a Volvo S80 BIG. In order to observe their influence, modal analysis must be run over different body configurations. These configurations shall reveal the items contribution to body modes. Ideally, all possible configurations (sixteen) should be tested although it is not time-feasible. Therefore, it was decided to measure the configurations resulting from a progressive removal of the bolted items: - Standard BIG (all bolted items on). - Standard BIG without grill overhanging reinforcement (radiator beam and tunnel brace on). - Standard BIG without grill overhanging reinforcement and radiator beam (tunnel brace on). 33 34 Chapter 4. Measurements - Standard BIG without grill overhanging reinforcement, radiator beam and tunnel brace (all bolted items off). An additional configuration was measured to study the effect of fixing the GOR with brackets. The bolted items on the body are shown in figure 1.1. Figures 4.1, 4.2, 4.3 and 4.4 offer an image of them when they are not attached to the body: Figure 4.1: Grill Overhanging Reinforcement: GOR Figure 4.2: Brackets fixing GOR CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 35 Figure 4.3: Radiator Beam: RB Figure 4.4: Tunnel Brace: TB Modal Analysis tests on BIGs have dispersion added by the differences between nominally identical car bodies. Such differences come from man- ufacturing process of the structure components and their assembling. Dur- ing Volvo GPDS project bolted items were identified as a source of disper- sion as well. In order to discern the bolted items dispersion from the one coming from the structure, it seems advisable to measure more than one body. In addition, the thesis aims for a general conclusion which could be extrapolated to any Volvo S80 BIG. The bigger the number of measured bodies is, the better the results generalization is. Once again the ideal case, measuring a huge number of bodies, it is not feasible. Time restrictions derived in the measurement of three bodies: BIG1, BIG2 and BIG3. 4.1.2 Support configurations An inappropiate support configuration could lead to high noise levels, in other words to a poor estimation of the modal parameters, specially for the first flexible mode. The ideal support condition is free-free, i.e. structure CHALMERS, Master’s Thesis 2007:148 36 Chapter 4. Measurements freely suspended in space. In this case the structure would not be affected by any external forces or noise. The free-free support is not viable. In prac- tice very soft springs are used to approach the free-free supporting. The standard way to check the supports configuration quality is finding the Rigid Body Modes (RBMs) frequency. Every system has 6 RBMs. Each of them corresponds to the movement in one of the natural space DOFs (3 translations and 3 rotations). The main characteristic of a RBM is that the hole body moves as a rigid structure. The RBMs of a structure are de- termined solely by its mass and inertia properties. When having free-free support conditions the resonance frequencies of the RBMs are coincident at 0 Hz. The use of soft springs yields RBMs over 0 Hz. If the highest RBM in frequency is less than 10 to 20% the frequency of the lowest Flexi- ble Body Mode (FBM), one can still derive the mass and inertia properties of the body [1]. Modal analysis performed on automotive industry uses air-mounts to support car bodies due to their heavy weight. The air-mounts are air-filled balloons. They have a very low stiffness and are capable of supporting heavy structures. Figure 4.5 shows one of the air-mounts used during the thesis. The air-mount appears attached to a wood structure to stabilize its contact with the floor and the body. Figure 4.5: Air-mount Preliminary tests were done to assure a correct supporting system. The standard configuration was measured supported by three air-mounts (triangle- positioning) and four air-mounts (rectangle-positioning). The goal was testing whether three air-mounts would be sufficient to approach free-free supporting conditions or an extra air-mount would be needed. CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 37 Figure 4.6 shows the 3 air-mounts configuration, which is 2 air-mounts at the front and one at the rear. The 4 air-mounts configuration was built replacing the centered rear air-mount by two air-mounts placed where the arrows mark. Figure 4.6: Supports configuration The test was performed using two electrodynamic shakers exciting in vertical direction. They were placed on the right side, one at the front and the other one at the rear. Attention must be paid to figure 4.7. The shak- ers are on two height-adjustable supports. When modal analysis was per- formed using these supports unexpected resonances were found from 15- 20 Hz. The body is actually designed not to have resonances below 35 Hz, except for the RBMs. The resonance peaks were significantly stronger for the FRFs over the front shaker. Therefore, unexpected resonances investigation was directed towards the shakers set-up. Firstly, repeteability was tested confirming re- sults. In the next step, shaker supports were exchanged observing that res- onances were then appearing at the rear shaker. Hence, it was concluded that an internal resonance from the front shaker support was causing the unexpected resonances. Measuring experts were consulted about this be- haviour. They advised using rigid supports for shakers, therefore height- adjustable supports must be avoided. Figure 4.9 shows the shakers on the rigid supports used for the “production“measurements. The unexpected resonances disappeared when these supports were used. CHALMERS, Master’s Thesis 2007:148 38 Chapter 4. Measurements Figure 4.7: Preliminary test shakers set-up (left: rear shaker, right: front shaker). Shakers on height-adjustable supports Unexpected resonances were not considered as flexible modes in the RBMs study. Table 4.1 presents the results from modal analysis performed on this preliminary test. The table contains the six RBMs and the first Flex- ible Body Mode (FBM). Table 4.1: RBMs for 3 and 4 air-mounts & 1st FBM frequencies (Hz) Air−mounts RBM1 RBM2 RBM3 RBM4 RBM5 RBM6 FBM1 3 2.10 2.63 3.86 4.16 4.35 6.03 39.03 4 2.04 2.53 3.17 4.03 4.50 6.12 39.03 A closer look to the table leads to the following conclusions: a) 3 air-mounts are enough to guarantee free-free conditions. Last RBM represents 15.4% (<20%) of the first FBM. b) An extra air-mounts does not improve the system behaviour. Last RBM is the one setting the quality of the supporting system. Actually, it can be observed that is slightly higher when 4 air-mounts are used. 4.1.3 Excitation system Once the number of air-mounts was decided, the aim was finding the best excitation system among the possible ones. Impact hammer or electrody- namic shakers are the two methods to excite systems for EMA. The use CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 39 of shakers was determined by the need of having enough energy density over a wide frequency range. Two different points suggested the use of more than one shaker: a) All modes of interest cannot be excited from a single reference or shaker. b) The structure has highly coupled modes. Hence, more references help for identifying them [12]. The laboratory where the thesis was done counted on two electrody- namic shakers. Therefore, excitation systems were limited to a maximum of two shakers. The criterions to follow when selecting the best excitation system are: - All modes sufficiently excited ⇒ No mode is missed. - Low correlation between input signals ⇒ Higher linearity of the sys- tem response. - High coherence between output and input signals ⇒ Less noise. According to the criterions and the number of shakers one counted on, the following excitation systems were tested (pictures are found in figure 4.8): 1º One shaker excitation: One shaker at right rear with vertical excita- tion. 2º Right side excitation: One shaker at right rear and another one at right front with vertical excitation. 3º Diagonal excitation: One shaker at right rear and another one at left front with vertical excitation. 4º 45-degrees excitation: One shaker at right rear and another at left front with 45º excitation (direction ∈ Πyz and 45∠ to y and z). The use of a single shaker was meant to study the goodness of having a single input in terms of coherence. Moreover, the 45º excitation was of high educational interest to observe the structural behaviour in comparison to the vertical excitation. One could think that having excitation in two differ- ent directions (y and z) would derive in having modes better excited. CHALMERS, Master’s Thesis 2007:148 40 Chapter 4. Measurements Figure 4.8: Excitation systems tested: 1º One shaker 2º Right side 3º Diagonal 4º 45-degrees Figure 4.9 shows the shakers for vertical excitation. They are set on rigid supports which are formed by a heavy iron block and some wooden boards on the top. Figure 4.9: Preliminary test shakers set-up for vertical excitation (left: rear shaker, right: front shaker) CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 41 The first tests performed on the 45-degree excitation system were not succesful. Unexpected resonances were found in the frequency range be- tween RBMs and first FBM again. The experience obtained during the sup- ports configuration study draw the attention to shakers supports. Modifications done to improve the stability of the shaker supports are reflected in the FRF of a single DOF (figure 4.10). Figure 4.10: Single DOF inertance for different 45º-excitation set-ups Initially, the shaker supports used were the ones shown in figure 4.9 for vertical excitation (first configuration in figure 4.10). First hypothesis was that resonances were generated by boards sliding, therefore boards were glueded and clamped in vertical direction (z-direction) (second configu- ration). This modification actually made results worse, unexpected reso- nances were amplified. Second hypothesis was that shaker supports were not capable to firmly hold shakers in y-direction. Hence, two heavy iron blocks were clamped in y-direction to the initial support (final configuration). The unexpected resonances dissapeared althought higher levels were found around 30 Hz. The improvement was considered sufficient. Therefore, the final configura- tion (figure 4.11) was the one used to be compared with the other excitation systems. CHALMERS, Master’s Thesis 2007:148 42 Chapter 4. Measurements Figure 4.11: Preliminary test shakers set-up for 45-degrees excitation (left: rear shaker, right: front shaker) The four excitation systems were measured on the standard configura- tion of the BIG3. The study was based on the criterions presented in this section: Modes proper excitation Modal analysis was performed by MACOL yielding the results shown in table 4.2. The aim was checking the first requirement for the excitation system, missing no mode. The modes in bold could hardly be identified, i.e. modal analysis had to be done for a narrow frequency range and higher polynomial orders to find them. All excitation systems were able to excite modes, although it was ob- served that 45-degrees excitation (number 4) did not excite properly three modes. Surprisingly having excitation in two directions is not a guarantee to better excite modes. No clear indication could be extracted from these results to decide the best excitation system. Table 4.2: fr and ηr for excitation systems Index f1 η1 f2 η2 f3 η3 f4 η4 1 38.97 0.204 39.03 0.253 39.03 0.205 39.01 0.190 2 41.19 1.471 41.24 1.452 41.26 1.506 41.24 1.452 3 43.32 0.512 43.34 0.433 43.38 0.446 43.33 0.425 4 46.07 1.288 46.07 1.182 46.11 1.361 46.10 1.187 5 47.54 0.392 47.63 0.598 47.68 0.685 47.89 0.270 6 47.93 0.179 48.04 0.158 48.02 0.204 48.04 0.149 7 48.43 0.226 48.53 0.256 48.53 0.246 48.35 0.237 CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 43 8 51.15 0.959 51.20 0.944 51.23 0.976 51.28 1.012 9 53.20 0.912 53.25 0.855 53.29 0.883 53.33 0.935 10 54.37 0.260 54.41 0.233 54.44 0.247 54.38 0.220 11 57.96 0.396 58.01 0.388 58.03 0.377 57.98 0.373 12 61.93 0.503 61.92 0.485 61.93 0.450 61.95 0.428 13 63.59 0.346 63.59 0.344 63.57 0.343 63.58 0.319 14 65.00 0.150 65.00 0.212 64.84 0.188 64.98 0.149 15 68.32 0.359 68.31 0.344 68.33 0.341 68.39 0.366 16 70.60 1.272 70.60 1.196 70.59 1.236 70.63 1.234 17 73.92 1.064 74.01 0.855 73.96 1.015 73.99 1.077 18 74.60 0.908 74.61 0.762 74.62 0.896 74.70 0.863 19 75.92 0.706 75.99 0.637 75.95 0.712 75.80 0.769 Input signals correlation In Figure 4.12 the coherence (correlation) between the measured forces of the two shakers, for each system, is shown. Obviously the excitation sys- tem with a single shaker is missing. The high coherence up to 10 Hz is explained by the shakers signal offset and the fact that the shakers pro- duce a strong response at each other when RBMs are excited. The shakers correlation is very low for all systems except for the 45-degrees excitation one which has strong peaks at certain frequencies (some of them coincident with resonance modes). Figure 4.12: Shaker signals correlation for each excitation system The high correlation between shakers worsens the FRFs estimation which loses linearity. Therefore, the 45-degrees excitation system should be avoided. The shakers correlation do not point out any of the other three systems as CHALMERS, Master’s Thesis 2007:148 44 Chapter 4. Measurements better than the others. Output-inputs signals coherence The output-input coherence is the last element of comparison left to an- alyze. The average over all outputs and both inputs appears in figure 4.13. The coherence of all configurations resembles except for the frequency range between RBMs and FBMs. One could think beforehand that having just a single input would induce better coherence. It turns out that the sin- gle shaker system is the worst in terms of coherence. This might be due to insufficient input power applied to the structure when using just one shaker. The 45-degrees excitation is the most coherent but it was discarded by the forces correlation criterion. The second one is the diagonal system. No excitation system was highlighted by the other criterions. Hence, the di- agonal excitation system was considered as the best among the available options and it has been the one used for the “production“measurements. Figure 4.13: Averaged acceleration-force coherence for each excitation system 4.1.4 Response points Response points must be set in order to cover all structure and register all body modes. During GPDS Volvo project, 101 points were measured on BIGs. The animations obtained from measurements did not perfectly defined the body movement at certain frequencies. Before the thesis “pro- duction“measurements, a study was done to add extra points. It was based on the observation of resonance modes animations. Points were added at CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 45 the following sub-structures: grill-overhanging-reinforcement (GOR), radi- atior beam (RB), roof beams, beams between doors and trunk. In total 126 points were measured. Pictures showing all measured points are shown in appendix D and coordinates are given in table C.1. 4.2 Measurements set-up Previous section (4.1) described the process followed to take the main deci- sions related to measurements set-up. This section presents the equipment used and its arrengement in order to perform measurements. In [13], reader can found a complete description of the equipment nec- essary for carrying out vibroacoustic measurements and the explanation about how to mount it. Figure 4.14, on next page, is a sketch of the measurements set-up used for the “production“measurements. The numbers appearing in the figure relates to table 4.3, a list of the set-up equipment. Equipment pictures are shown in appendix E. Table 4.3: Equipment Index Manufact. Item Num. Serial num. 1 FireStone Air-mount 3 W01-M58-6008 2 B&K Triaxial accelerometer Type 4524B 4 30231 30234 30284 30285 3 B&K Mounting clips UA 1564 126 n/o 4 Agilent E8408 Acquisition system VXI 1 n/o 5 Acer Computer VXI2 station 1 n/o 6 LDS Power amplifier 2 PA100E 7 LDS V406 Electrodynamic shaker 2 57835/2 57835/3 8 B&K Force transducer Type 8200 2 1948760 2071279 9 B&K Charge amplifier Type 2635 2 872470 986722 CHALMERS, Master’s Thesis 2007:148 46 Chapter 4. Measurements Figure 4.14: Measurements set-up CHALMERS, Master’s Thesis 2007:148 Chapter 4. Measurements 47 4.3 Measurements procedure Measurements were performed by the author of the thesis, although one counted on Chalmers Acoustics Department members, specially when mount- ing the set-up. The signal analysis software used is a programme developed in Mat- lab, the Trigger Happy software which was programmed at the Applied Acoustics department of Chalmers University. During this Thesis an up- date of Trigger Happy was done to implement MIMO transfer function and coherence using equations 2.2 and 2.3. The Trigger Happy code up- date is presented in appendix F. Further details about this software can be found in [13]. The signal analysis settings used are listed in table 4.4. Two different sets of measurements were done. The one at lower frequencies for modal analysis (up to 625 Hz) and the other one at higher frequencies for Volvo GPDS project (up to 1250 Hz). The study of the different excitation signals proposed in [2] derived in the use of pure random signal to excite BIGs. The pure random signal is a non-periodic stochastic signal with a Gaussian probability distribution. The stochastic property implies the necessity of data averaging. The ad- vantage of using this signal is that it has a low peak to RMS ratio. There- fore, it yields the best linear approximation of non-linear systems, as BIGs. Its main drawback is leakage. Hanning window is performed by Trigger Happy software to diminish it. Sygnal analysis settings are presented in table 4.4. Table 4.4: Signal analysis settings Blocksize f range Samping f f resolution Modal Analysis 16384 625 1600 0.0977 Volvo GPDS 4096 1250 3200 0.7813 32 measurement sets were performed to cover the 126 points of the standard configuration and the one without brackets. The channels corre- spondance is provided in appendix G. Configurations not having all bolted items have less points: 114 points for configuration without GOR, 109 points for configurations without GOR and RB and without both and the TB. When less points were measured, the accelerometers and sets order in fig- ure G were kept. In addition, the points belonging to the same set were not separated into different sets when one of the points was removed. CHALMERS, Master’s Thesis 2007:148 Chapter 5 Results and Analysis The results obtained during the thesis are presented in this chapter. Firstly, frequency limit is set. It defines the boundary up to which results can be considered as reliable. Secondly, a study of MACOL’s poles identification is done to find the most consistent way to perform it. Afterwards, brackets results are analyzed to better understand their function. Sections 5.4 and 5.5 contain the key results in order to achieve the thesis objectives. The dispersion introduced by each bolted item is studied. Then, the modes evolution is derived, from the configuration without bolted items to the standard one. It illustrates the effect of having each bolted item on and their contribution to unclear modes. 5.1 Frequency limit of Modal Analysis on BIGs Modal analysis essence is characterizing the system dynamic response by the contribution of each resonance mode. Modal analysis methods intend to isolate each mode contribution and determined its characteristics (modal parameters). Nevertheless algorithms do not always succeed. The difficul- ties they have to overcome were presented in chapter 3. Among difficul- ties it stands out the results inconsistency when having high modal over- lapping. A high modal over-lapping, i.e. a high modal density, entitles modes coupling. The frequency limit when performing modal analysis is actually re- lated to modal over-lapping. More modes are found when frequency is increased. Hence, modal density is higher as one moves up in frequency. At certain point, modes coupling is too strong and modes contribution can- not be clearly identified. Stable poles can still be found although modal parameters extracted are not reliable. 48 Chapter 5. Results and Analysis 49 No analytical calculations can be done to set a modal analysis frequency limit. It is a common-sense matter which relies on the operator experience. When a clear trend of unclear modes is observed, one can considered that the limit has been exceeded. A study has been done to define the limit of the modal analysis tests performed in this thesis. The study was based on BIG3. Stabilization dia- grams of its five possible configurations were analyzed. The stabilization diagrams of the standard configuration are shown in figures 5.1 and 5.2. The rest of the stabilization diagrams can be found in appendix H. 35 40 45 50 55 60 65 0 10 20 30 40 50 60 70 80 f (Hz) Po lyn om ia l o rd er a nd T Fs (s hi fte d) Figure 5.1: Stabilization diagram for BIG3 at “middle“frequencies 60 65 70 75 80 85 90 10 20 30 40 50 60 70 80 90 f (Hz) Po lyn om ia l o rd er a nd T Fs (s hi fte d) Figure 5.2: Stabilization diagram for BIG3 at high frequencies CHALMERS, Master’s Thesis 2007:148 50 Chapter 5. Results and Analysis Firstly, “middle“frequencies are observed (figure 5.1). Starting from the first flexible mode, all modes are easy to identify (they are stable for all polynomial orders) up to 48 Hz. Three unclear modes are found around this frequency. One could have set limit at 47 Hz, however, the following modes are easily identifiable and too much information would have been lost. Analysis continues using figure 5.2. Unclear modes (around 64 and 71 Hz) are found in between clear modes, but they do not constitute a trend. A strong trend is observed over 75 Hz. From that point onwards, modes are mostly unstable either in damping ratio and/or in modal participation factor. Same trend has been observed for all configurations. Hence, the thesis results will be considered valid up to 75-77 Hz. The exact value de- pends on the configuration and the body. For instance, the mode at 74.73 Hz in BIG3 for the standard configuration corresponds to the mode at 76.11 Hz in BIG3 for the configuration without GOR and RB (correspondence is illustrated in figure 5.8). 5.2 MACOL study Polynomial order (p) plays an important role in physical poles identifica- tion, especially for closely spaced modes. The method understanding ob- tained from programming allows studying the ins and outs of poles se- lection. The modal parameter extraction dependence on polynomial order arises the following questions: - How sensible are the modal parameters to the polynomial order se- lection? - Which modal parameters are more affected? - Is it advisable choosing poles found for high or for low polynomial orders? The questions will be answered using MACOL software. Results ob- tained cannot be directly extrapolated to LMS PolyMAX although same phenomena are likely to be found, as both programmes are based on p- LSCF algorithm. The objective now is quantifying the influence of the selected polyno- mial order on the magnitudes of the modal parameters. Modes consistently found as stable for all polynomial orders are likely to show better results than those considered as unclear modes. Therefore, two extreme cases are investigated, an “easy-to-identify“mode and an unclear one. Poles selec- tion was done using figure 3.4. Results are shown in tables 5.1 and 5.2. The CHALMERS, Master’s Thesis 2007:148 Chapter 5. Results and Analysis 51 last line in tables provides the maximum variance obtained for each modal parameter. The limits set for ∆ fr, ∆ηr and ∆ ‖< Lr >‖ when polynomial orders are consecutive can be checked in table 3.1. Table 5.1: Poles selection dependence on polynomial order for an “easy-to- identify“mode p fr ηr(%) < Lr > 22 43.15 0.443 <-0.2826,-0.0731 + 0.0015i> 24 43.15 0.431 <-0.2761,-0.0686 - 0.0013i> 39 43.15 0.440 <0.2577,0.0669 + 0.0034i> 58 43.15 0.432 <0.2488,0.0651 + 0.0030i> ∆ fr(max) = 0.00% ∆ηr(max) = 2.71% ∆ ‖< Lr >‖ (max) = 12.00% Table 5.2: Poles selection dependence on polynomial order for an unclear mode p fr ηr(%) < Lr > 24 47.99 0.249 <0.0863 + 0.0068i,0.2406> 29 47.99 0.242 <0.0833 + 0.0215i,0.2256> 38 47.99 0.210 <0.0716 + 0.0277i,0.2041> 51 47.97 0.218 <0.0604 - 0.0072i,0.1989> ∆ fr(max) = 0.04% ∆ηr(max) = 12.45% ∆ ‖< Lr >‖ (max) = 19.97% Results are revealing. High accuracy in modes frequency is observed for both cases. Damping factors show different behavior in each case. In the “easy-to-identify“mode case, damping factors estimation is very con- sistent. Whereas for the unclear mode the maximum variation is over the double of the limit set for consecutive polynomial orders, 5%. The differ- ence is not extreme but one should be aware of it. Modal participation factors are not reliable in both cases. Its stability criterion for consecutive poles is 2%. A relative difference five times over the limit proves the lack of consistency. Modal participation factors relate to mode shapes (equation 3.16). Modal parameters sensibility to polynomial orders can be resumed as follows: - Poles location in frequency is practically perfect. - Damping ratio is not totally reliable for unclear modes. - Mode shapes and modal participation factors are the most affected modal parameters. Their values should be considered as an orienta- tion rather than an absolute number. CHALMERS, Master’s Thesis 2007:148 52 Chapter 5. Results and Analysis The difference between selecting poles estimated for high or low poly- nomial orders is analyzed. The extreme cases in figure 3.4 are taken, maxi- mum and minimum polynomial orders available for each mode. In order to carry out the investigation, the error and the correlation between measured and synthesized FRFs are used. Table 5.3: Coherence and Normalized Error for low and high polynomial or- der (p) Low p High p Correlation % 90.78 91.64 Normalized Error % 9.45 8.47 Table 5.3 shows insignificant differences between calculations, although results are slightly favorable to low polynomial order use. The correlation and the normalized error have been averaged over all frequency range. However, it should not be forgotten that fitting accuracy is mostly impor- tant at frequencies around resonance modes. Therefore, it is wise check- ing the synthesized and measured FRFs plot (figure 5.3). Figure observa- tion does not reveal any trend. Indeed, some poles appear better fitted for high polynomial orders, and other poles for low polynomial orders. Hence, there is no advantage in selecting poles of high or low polynomial orders. 38 40 42 44 46 48 50 52 54 56 −10 −5 0 5 10 15 20 25 30 35 f(Hz) FR Fs S um (d B re to 1 /k g) Low p High p Measured Figure 5.3: Synthesized FRFs Summation Comparison: low polynomial order, high polynomial order and measured FRFs CHALMERS, Master’s Thesis 2007:148 Chapter 5. Results and Analysis 53 5.3 Brackets effect The brackets implementation provides the best example to explain how to apply modal analysis on a BIG combined with FEM. In Volvo S80 develop- ment phase, measurements were done on a standard BIG, in which brackets had not been included yet. Results discovered a resonance mode around 20 Hz. The mode was par- ticularly disturbing because it was situated in the frequency range excited by the idling engine. The mode was related to GOR vibration. FEM was used to design some fixation able to damp the mode. Brackets were found as a good solution and were successfully implemented. Figure 5.4 show the mentioned peak when brackets are ON and when they are OFF, from the data measured on BIG3 during the thesis. 15 20 25 30 −65 −64.5 −64 −63.5 −63 −62.5 −62 f(Hz) In er ta nc e FR Fs S um Brackets ON Brackets OFF Figure 5.4: FRFs Summation with brackets ON and OFF In Volvo GPDS project modal analysis was performed on BIG1, BIG2 and BIG3. Brackets were forgotten by mistake. GPDS results constitute the perfect benchmark to cross-check the thesis results. The aim is guarantee- ing the quality of both, measuring process and modal analysis. In figures 5.5 and 5.6, resonance frequencies and damping ratios are compared for the three bodies. V subindex relates to Volvo and T subindex relates to Thesis. CHALMERS, Master’s Thesis 2007:148 54 Chapter 5. Results and Analysis Figure 5.5: Resonance frequencies for BIG1,BIG2,BIG3. Volvo and Thesis re- sults Same resonance modes are found for both analysis. Differences in fre- quency are lower than 2%. However, damping ratios resemblance is poor for certain modes. The averaged difference is 20%, which could be consid- ered as reasonable since damping ratio estimation is not highly accurate. The use of different set-ups might be the discrepances cause. Figure 5.6: Damping ratios for BIG1,BIG2,BIG3. Volvo and Thesis results 5.4 Dispersion introduced by bolted items Standard deviation (σ) is the variable selected to quantify the dispersion. Calculations could be done for all modal parameters although it has been considered sufficient to analyze the frequency deviation. CHALMERS, Master’s Thesis 2007:148 Chapter 5. Results and Analysis 55 The standard deviation is defined as the square root of the variance. Therefore, it measures the data spread around the mean and it has same units as the data. In most cases, the standard deviation is estimated by examining a random sample from a entire population. Equation 5.1 was used to estimate the standard deviation. N is the number of samples (three in this case) and fmean the averaged frequency of the mode. Figure 5.7: Standard deviation definition fmean − σ < f < fmean + σ σ = √√√√ 1 N − 1 N ∑ i=1 ( fi − fmean) (5.1) One of the main objectives of the thesis is studying the inconsistency of the modal analysis results due to the bolted items inclusion. The incon- sistency has been related to the results dispersion. This section intends to describe the dispersion added by each bolted item in order to possibly sug- gest the removal of any of them. 5.4.1 Tunnel brace Table 5.4 presents the modes found for the configuration without GOR and RB of BIG1, BIG2 and BIG3. Frequencies are averaged and the standard deviation is estimated for each mode. Last line shows the mean deviation value. For instance, one can focus on the most deviated mode, number seven. Figure H.5 is used to check how easily can the mode be identified. CHALMERS, Master’s Thesis 2007:148 56 Chapter 5. Results and Analysis It is unexpected having an “easy-to-identify“mode with the highest dis- persion. Opposite case is taken, one of the lowest deviation is found for mode number 2. However, this is one of the hardest modes to identify in figure H.5. Therefore, initial expectations are contradicted, dispersion is not proportional to the difficulties to identify a mode. Table 5.4: Modes for configuration without GOR and RB. Dispersion Index fr(1) fr(2) fr(3) fmean σf 1 40.99 42.34 41.34 41.56 0.696 2 45.90 46.41 46.08 46.13 0.262 3 47.53 47.81 47.84 47.73 0.172 4 47.76 47.94 47.69 47.79 0.130 5 50.91 52.28 51.55 51.58 0.687 6 53.98 53.74 53.53 53.75 0.228 7 58.60 58.53 55.29 57.47 1.892 8 61.66 61.51 60.19 61.12 0.807 9 64.85 64.76 64.51 64.71 0.179 10 68.82 68.85 68.92 68.87 0.053 11 70.96 72.45 70.96 71.45 0.861 12 74.03 74.73 74.26 74.34 0.353 13 76.13 77.06 76.11 76.43 0.542 σf (avg) 0.528 5.4.2 Radiator beam The data shown in table 5.5 is taken from the configuration which contains both the radiator beam and the tunnel brace. The averaged deviation is decreased after adding the radiator beam on although it was expected the opposite trend. Joining this result to the ones for the tunnel brace, it can be stated that dispersion is not an indicator of the inconsistencies of modal analysis results. Table 5.5: Modes for configuration without GOR. Dispersion Index fr(1) fr(2) fr(3) fmean σf 1 40.48 39.35 38.86 39.56 0.831 2 40.98 42.28 41.33 41.53 0.673 3 45.18 45.49 45.47 45.38 0.090 4 45.95 46.38 46.13 46.15 0.172 5 47.70 47.85 47.69 47.75 0.217 6 50.58 51.94 51.19 51.23 0.681 CHALMERS, Master’s Thesis 2007:148 Chapter 5. Results and Analysis 57 7 53.90 53.64 53.49 53.68 0.206 8 58.10 58.04 59.68 58.61 0.926 9 62.12 62.31 62.19 62.21 1.017 10 64.50 64.38 64.12 64.34 0.096 11 68.64 68.67 68.75 68.68 0.194 12 70.90 72.50 70.82 71.41 0.057 13 74.05 74.64 74.21 74.30 0.951 14 75.85 76.64 75.81 76.10 0.303 σf (avg) 0.458 5.4.3 Grill over-hanging reinforcement Table 5.6 corresponds to the standard configuration of the three BIG. Ac- cording to the average deviation, the difference of having the GOR on and off is practically non-existent in terms of dispersion. Unclear modes (5,6,7,14,15) are of high interest. Figures 5.1 and 5.2 show the discontinu- ity of stable poles at these frequencies. In section 5.2 the results inconsis- tency for unclear modes were revealed. Nevertheless, the unclear modes deviation is very low in comparison with other modes. This reaffirms that dispersion is not proportional to inconsistency. Table 5.6: Modes for standard configuration. Dispersion Index fr(1) fr(2) fr(3) fmean σf 1 40.29 39.34 38.94 39.52 0.693 2 40.82 42.20 41.35 41.46 0.694 3 43.15 43.45 43.45 43.35 0.135 4 45.78 46.22 46.05 46.02 0.174 5 47.53 47.76 47.67 47.65 0.225 6 47.98 48.09 47.91 47.99 0.114 7 48.50 48.74 48.52 48.59 0.089 8 50.74 51.96 51.36 51.35 0.610 9 53.46 53.34 53.33 53.37 0.074 10 56.23 56.12 54.39 55.58 1.033 11 61.11 60.82 58.02 59.98 1.709 12 61.91 62.07 62.03 62.01 0.083 13 63.97 63.85 63.58 63.80 0.200 14 64.90 65.00 64.61 64.84 0.203 15 65.15 65.14 64.94 65.08 0.121 16 68.21 68.30 68.38 68.29 0.085 17 70.59 72.36 70.69 71.21 0.993 18 73.82 74.40 74.06 74.09 0.290 19 74.63 75.49 74.74 74.95 0.471 σf (avg) 0.430 CHALMERS, Master’s Thesis 2007:148 58 Chapter 5. Results and Analysis 5.5 Modes evolution. General results The modes evolution study has been run over all configurations for BIG3. The aim is defining the effect of each bolted item on the body modes. To perform the modes evolution study MAC values were used. As explained in section 3.2.1, MAC values indicate the correlation between mode shapes. MAC values were calculated between consecutive configurations, i.e. con- figurations differing on a single item. The starting point was the plain body (body where all bolted items are off), then one-by-one bolted items were added. Phenomena involved when two structures are joined are described in chapter 1. A modes evolution diagram is shown in figure 5.8. MAC values used to develop the diagram can be found in appendix I. Reader must keep on mind that different structures are compared, therefore points which do not belong to both configurations are not taken into account to calculate MACs. However “usual“MAC values cannot be expected. Having attached an extra-item does influence a mode shape because body mass and damping are modified. Therefore, it is hard to find MAC values over 90% even if a couple of mode shapes resemble. Different relations has been defined in the modes evolution diagram: - High correlation (MAC > 60%): Modes highly correlated are joined by arrows. - Weak correlation (40% < MAC < 60%): Modes slightly correlated are joined by dotted arrows. The original mode, the one of the struc- ture before the bolted item is added, suffers a strong modification. - New mode (MAC < 40%): Modes not correlated to the ones from previous configuration. They are circled. - Swapped mode: The modes order is modified due to a bolted item inclusion. They are surrounded by a polygon. - Unclear mode: Hard-to-identify modes. They are marked by a slop- ing arrow. In following subsections the influence of each bolted item is studied, to later summarize in the general results. CHALMERS, Master’s Thesis 2007:148 Chapter 5. Results and Analysis 59 Figure 5.8: Modes evolution diagram CHALMERS, Master’s Thesis 2007:148 60 Chapter 5. Results and Analysis Tunnel brace influence Tunnel brace (TB) is the bolted item less important. The modes evolution diagram highlights a higher contribution of the other items. Only three body modes, from the “all items off“configuration, are modified when TB is set on. Two body modes (56.92 and 62.56 Hz) experience coupling with a TB self-mode, yielding three body modes in the “GOR&RB off“configuration. Radiator beam influence Two modes (38.86 and 62.19 Hz) appear when the Radiator Beam (RB) is added. They are not correlated to any of the modes for the “GOR&RB off“configuration. In this case bolted item self-modes are not close (in fre- quency) to body modes. Therefore coupling phenomenon does not occur. Four modes in the “GOR&RB off“configuration are strongly modified after adding the RB. It is of significance the case of the second one (47.84 Hz). One could think this mode is one of the unclear ones appearing in the standard configuration. However, modes evolution diagram shows how the mode is moved down in frequency due to RB effect. Indeed, it is swapped with the two previous modes (46.08 and 47.69 Hz) after setting the RB on. Grill over-hanging reinforcement influence The GOR is definitely the most influencing bolted item. Its heavier weight and its position might be the cause. The addition of the GOR introduces four new body modes, probably GOR self-modes. It is also a matter of size, the bigger the bolted item is, the earlier its self-modes are found in frequency. Therefore, when a bigger bolted item is joined to the body, more new modes are found in the re- sulting structure. Additionally, four modes from the configuration without GOR are strongly modified. General results Modes evolution diagram is an optimal tool to track unclear modes. The sloping arrows in diagram mark them. They mainly appear for the stan- dard configuration. As it was commented in section 5.4 they are situated in two different frequency areas, one around 48 Hz and the other one at 64 Hz. Stabilization diagrams observation, together with the studies in section 5.2, discovered the key role of unclear modes in the inconsistency of modal CHALMERS, Master’s Thesis 2007:148 Chapter 5. Results and Analysis 61 analysis results. Avoiding them would probably improve the results ac- curacy, especially for the mode shapes estimation. The modes evolution diagram provides a straight answer to the problem. Four of the five unclear modes appear when GOR is set on. Hence, modal analysis should be performed on BIGs excluding GOR. The other unclear mode, at 47.67 Hz, cannot be avoided as it comes from one of the modes from the plain body. Nevertheless this mode is not unclear when GOR is not on, as there is not coupling between itself and the modes intro- duced by the GOR. The bolted items influence on body modes can be studied from another point of view. Chapter 1 explained the reason to include bolted items in the BIG concept. Their inclusion is meant to better account their influence on body stiffness. However, it was not clear whether the contribution to the stiffness is noticeable in the frequency range accessible for modal analysis. The modes evolution diagram can clarify this issue by the study of modes “movement“when bolted items are included. In other words, by checking if modes are shifted up or down in frequency when bolted items are in- cluded. Damping factor variations have been studied as well. Body modes are slightly shifted up in frequency when tunnel brace is attached. Damping ratio is also slightly increased. Therefore, it can be concluded that TB is adding stiffness to the body. Radiator beam effect is also moderate. In its case, modes are slimly shifted down in frequency and damping factors become lower. Hence, RB contributes mass rather than stiffness. Grill overhanging reinforcement is once again the most impor- tant bolted item. From modes diagram can be clearly seen that GOR “pushes down“body modes. Nevertheless, its influence on stiffness is small because damping factors are slightly decreased. Both characteristics determine a mass be- havior. In short, bolted items contribution to stiffness cannot be considered important for t