FEM updating clamped beam

8/7/2019 FEM updating clamped beam

1/163

Model updating of a

clamped-free beam system

using FEMTOOLS

C.S. Kraaij

DCT 2006.128

Traineeship report

Coach(es): dr.ir. R.H.B. Feyir. L. Kodde

Technische Universiteit EindhovenDepartment Mechanical EngineeringDynamics and Control Technology Group

Eindhoven, January, 2007


2/163

2


3/163

Contents

Abstract 8

1 Introduction 9

2 Dynamic analysis of initial numerical beam models 11

2.1 Euler beam elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Timoshenko beam elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Geometrical and Physical properties of beam a and beam b . . . . . . . . . . . . . . . 13

2.4 Selection sensor/excitation location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Eigenfrequencies and mode shapes of beam a and beam b . . . . . . . . . . . . . . . 16

2.6 Comparison of frequency response functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Experimental modal analysis 21

3.1 Experimental test set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Experimental test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Modal-parameter fit procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Model updating 29

4.1 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Comparing numerical and experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Comparing initial FE model and experimental data of beam a . . . . . . . 31

3


4/163

4.2.2 Comparing initial FE model and experimental data of beam b . . . . . . . 32

4.3 FE Model updating process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Update process based on the experimental data of beam a . . . . . . . . . . 34

4.3.2 Update process based on the experimental data of beam b . . . . . . . . . . 38

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Conclusions and recommendations 41

Bibliography 43

A Sensor placement to identify 8 modes 45

B Frequency response functions FEM models 49

C Flowchart of data 55

D Frequency response function and coherency plots of Siglab 57

E Parameter Estimation from Frequency Response Measurements Using Ratio-nal Fraction Polynomials 67

F Global Curve Fitting of Frequency Response Measurements Using RationalFraction Polynomials 83

G Reformulation of transfer function 93

H Frequency Response Function general viscous damping estimation 95

I Integration of Residues 105

J Derivation of the UMM mode shapes 107

K Frequency Response Function proportional damping estimation 109

L Experimental mode shapes 119

M Mode shapes of FEMTOOLS models 125

N Influence of the stiffness on the number of iterations 131

O Bayesian Parameter Estimation method in FEMTOOLS 135

P MAC matrixes after the update strategies based on beam a 137

Q FRF of the FEMTOOLS updates with regards to beam a 141

4


5/163


6/163

6


7/163

Abstract

Model updating is used to improve the match between the dynamic properties (eigenvalues andeigenmodes) of a Finite Element (FE) model and test data. In this report some parameters ofinitial FE models of two designs of a clamped-free beam system, are updated based on thefirst five eigenfrequencies of the first five bending modes of an experimental model by usingFEMTOOLS 3.1.1. The goal of this traineeship is to learn about the possibilities of the FE modelupdating process inside the commercial software package FEMTOOLS. The beam models that areused for this purpose are a simple straight beam (beam a) and a slightly different beam (beamb). The difference between the beams is the height of the beams right half.

In order to create an interesting updating situation a mismatch of approximately 10 15%between the eigenfrequencies of two beam models is chosen. In MATLAB for both beams a FEmodel is built of50 Euler beam elements. The difference between the eigenfrequencies can beinfluenced by changing the height of beam b. This is done until the requirement is satisfied.

Further initially it was required that the influence of the shear stress on the eigenfrequencies isless than 1%. Therefore, both beams model are also built using 50 Timoshenko beam elements.The influence of the shear stress can be investigated by comparing the eigenfrequencies of theFE models built using Timoshenko beam elements with the eigenfrequencies of the FE modelsbuilt using Euler elements. By changing the relationship between the height and the length of thebeam the influence of the shear stress on the eigenfrequencies can be reduced. Both beams aremanufactured for carrying out experimental modal analysis. Eight frequency response functions(FRFs) of both beam systems are determined with a Signal Analyzer and the software packageSIGLAB. Therefore the beams are excited on eight optimal locations and the acceleration responseis measured on a fixed location. These FRFs are exported to the software package MESCOPEwhich is used to carry out the Orthogonal Polynomial modal-parameter fit procedure. The real

parts of the eigenvalues are very small in comparison with the imaginary parts. Therefore theundamped angular eigenfrequencies are used in the updating process. In FEMTOOLS the initialFE model is built (based on beam a) of fifty LINE2 elements. This element type takes the shearstress into account. The initial FE element model is updated, with different update strategies,based on the first five experimental eigenfrequencies corresponding to the five bending modesof beam a and beam b. Typical parameters which are updated are the clamping stiffnesses, thecross-section and the second moment of area of the right half of the beam. The results show aclear decrease in the discrepancies between the numerical and experimental eigenfrequencies.The correctness of the updated parameters depends on the chosen strategy.

It can be concluded that FEMTOOLS is capable to improve the match between the dynamic

7


8/163

properties of a FE model and experimental data. In order to carry out a physically meaningfulmodel update it is important that the user has knowledge about the updating process and about

the system that is updated.

8


9/163

Chapter 1Introduction

Model updating is used to improve the match between the dynamic properties (eigenvalues andeigenmodes) of a Finite Element (FE) model and test data. This can be realized by making aphysically meaningful change in the values of the model parameters. The updated model canbe used for analyzing the effect of changes in the configuration or various loads, without newexperimental testing.

FEMTOOLS is a package which contains a model update utility. The goal of this traineeshipis to learn about the possibilities of the FE model updating process inside FEMTOOLS. A simple

straight beam (beam a) and a similar beam model, which has a somewhat different geometry(beam b), will be used for this purpose. Before the actual model updating will be carried out,the eigenfrequencies of both beams shall be determined experimentally and numerically. Theexperimental eigenfrequencies will be compared with numerical eigenfrequencies, based on aninitial finite element model. Updating of the FE model will take place, based on differencesbetween the experimental and numerical eigenfrequencies.

In chapter 2 the dimensions of the beams will be determined using FE and theoretical analy-sis. Beams a and b are selected so that a mismatch between 10% and 15% exists in the eigen-frequencies. Further the influence of shear on the eigenfrequencies has to be negligibly small( 1%). The experimental tests and the modal parameter fit procedure will be described in chap-

ter 3. Both beams will be investigated experimentally. Hereby S IGLAB will be used to determinethe FRFs and ME SCOPE for carrying out the modal parameter fit procedure. In chapter 4 severalupdate strategies for both beams will be discussed. These strategies are implemented in FEM-TOOLS to update the FE models. The results are compared and discussed. In the last chapterconclusions will be drawn and recommendations for future research will be given.

9


10/163

1. introduction

10


11/163

Chapter 2Dynamic analysis of initial numerical

beam models

In Chapter 4 two simple beam models will be updated based on experimental data. For realize aninteresting update situation a mismatch of approximately 10 15% between the eigenfrequen-cies of two beam models is aimed at deliberately. Further it is required that the influence of theshear on the eigenfrequencies is less than 1%. In this chapter the dimensions of the two initialnumerical beam models will be determined. In a later stage, these beams will be manufacturedand analyzed experimentally. In this stage it is assumed that the clamp is infinitely stiff. Figure

2.1 shows the side views of the two numerical beam models. The difference between beam a andbeam b lies in the height of the second half, i.e. free end of the beam. To determine appropri-ate dimensions of the beams they are modeled using the finite element method to calculate theeigenfrequencies. The height of the second part of model b will be decreased so that the mis-match requirement is satisfied. The 2D situation will be analyzed. Axial and torsional vibrationswill not be considered in the FE model.

(a) Beam a (b) Beam b

Figure 2.1: Numerical beam models

2.1 Euler beam elements

To determine the eigenfrequencies and modes shapes of the two beam FE models, mass andstiffness matrices need to be build, using a finite number of beam elements. A possible beamelement is the Euler beam element.

11


12/163

2. dynamic analysis of initial numerical beam models

The mass and stiffness matrices of an Euler beam element are:

Me =Ale420

156 22le 54 13le22le 4l

2e 13le 3l

2e

54 13le 156 22le13le 3l

2e 22le 4l

2e

, Ke = EIl3e

12 6le 12 6le6le 4l

2e 6le 2l

2e

12 6le 12 6le6le 2l

2e 6le 4l

2e

where:

: Density [kg/m3]A : Cross-section [m2]le : Element length [m]

E : Youngs modulus [N/m2]I : Second moment of area [m4]

The mass and stiffness matrix of the system can be build by assembling the Euler beam ele-ments. Because of the initial assumption of infinite stiff clamping, the displacement and rotationat the clamping are zero. Hence the first two rows and columns of the mass en stiffness matrixof the system can be deleted. Using the assembled mass matrix and the assembled stiffness ma-trix of the total system, the eigenfrequencies and mode shapes of both beams can be determinedby solving the eigenvalue problem. In this stage we assume that there is no damping, so thedynamics behaviour of the system can be described by:

Mq(t) + Kq(t) = 0 (2.1)

This leads to the following eigenvalue problem:

[K 2i M]ui = 0 [K (2fi)2M]u = 0 (2.2)

where,

fi =i2 : Eigenfrequency of mode i [Hz]

ui : Mode shape of mode i []For the straight beam (beam a) based on Euler theory also an analytical solution of the eigen-

value problem is available. The following formulae describe the eigenfrequencies and modeshapes of a clamped-free beam system [Blevins, 1979]:

fi =2i

2L2

EI

A

1/2(2.3)

ui(x) = coshixL cosix

L isinhix

L sinix

L , i = 1, 2, 3...(2.4)

12


13/163


where [Blevins, 1979]:

i i i1 1.87510407 0.7340955142 4.69409113 1.0184673193 7.85475744 0.9992244974 10.99554073 1.0000335535 14.13716839 0.999998550

2.2 Timoshenko beam elements

Influence of shear stress is neglected in the Euler formulation. To determine the effect of the

shear stress on the eigenfrequencies and mode shapes, Timoshenko beam elements can be used.The mass matrix of the Timoshenko beam element is identical to the mass matrix of the Eulerbeam element. The element stiffness matrices differ. In the Timoshenko beam element stiffnessmatrix the dimensionless coefficient is added, to correct for the shear influence.

Ke =EI

le(1 + )

12/l2e 6/le 12/l2e 6/le

6/le 4 + 6/le 2 12/l2e 6/le 12/l

2e 6/le

6/le 2 6/le 4 +

, = 12EIGAsl2e

where:

G : Shear modulus [N/m2]As : Shear-resisting area [m

2]

2.3 Geometrical and Physical properties of beam a and beam b

The material and geometric properties are selected based on the eigenfrequencies. The dimen-sions of beam a and beam b will be chosen so that the difference in eigenfrequencies is between

10% and 15%. Furthermore it is required that influence of the shear on the eigenfrequencies isless than 1%. The analyses of the eigenfrequencies will follow in section 2.5. The selected materialproperties (the beams are made ofS235) and geometrical properties are collected in table 2.1.

For the calculation of the Timoshenko element stiffness matrix the following parameter val-ues are used:

Shear modulus (G) : G = 12(1+)E [N/m2]

Effective shear area (As) : As = A [m2]

In table 2.1 also the mass of one accelerometer is given. The number and selection of the axiallocation of the accelerometers will be discussed in section 2.4. The masses of the accelerometers

are taken into account by the calculations of the eigenfrequencies and mode shapes.

13


14/163


Table 2.1: Beam properties

Material properties

Youngs modulus (E) 2.1 1011 [N/m2]Poisson number () 310 []

Density () 7850 [kg/m3]

Geometrical properties

Timoshenko shear coefficient () 5/6[]Height (h) 10 [mm]

Height 2nd half beam b (h2) 7.5 [mm]Width (w) 60 [mm]Length (l) 500 [mm]

Accelerometer

Mass (sensor+wire) M 40 [kg]

2.4 Selection sensor/excitation location

The experiments to experimentally determine the frequency response functions are carried outusing hammer excitation and three accelerometers. These three accelerometers will be positionedat the same axial position of the beam. They will be distributed over the width of the beam in or-der to detect possible torsional modes (which are absent in the FE models) in the experiment.The added mass and the stiffness of the wire of the accelerometers have some influence on theeigenfrequencies. The influence of the added mass can be taken in to account in the models.

Therefore the sensor location needs to be determined in this fase. The sensor locations are se-lected based on the linear independency of the mode shapes Ue of interest. These mode shapesare calculated using the parameter values oftable 2.1. Obviously, the masses of the accelerometerswere not taken into account in this calculation. The selection method is known as the EffectiveIndependence Method (EIM). The first step is to delete from Ue all the rotations and inaccessibledofs. After this there remain m candidate dofs. Inaccessible dofs for beam b from figure2.1(b)are on the end of the beam and in the middle where the height is reduced. On these locations it isnot possible to place the sensors. For beam a only the dof on the end of the beam is inaccessible.The next step is to select the number of mode shapes (e) which are of interest and need to bedetected. Based on this information the Fisher Information Matrix Fee [de Kraker, 2004] can bedetermined:

Fee = UtmeUme (2.5)

The Fisher Information Matrix can be used to calculate the idempotent matrix Gmm [de Kraker, 2004]:

Gmm = UmeF1ee U

tme (2.6)

The idempotent matrix has the property that G2mm = Gmm and its trace equals its rank. The

diagonal terms represent the partial contribution to the rank of Gmm of each measurement dof.

14


15/163


The selection method is an iterative process of subsequently deleting the dof with the smallestterm on the diagonal, followed by updating Ume, Fee and recomputing Gmm. This will be done

until the number of sensors (s = 8) is reached (m e). The selected mode shapes are the first 5bending modes(e = 5).

Actually, it was decided that in the experiments only the transversal response was measuredat the last node but one at the free end of the beams. As stated before, at this axial position threeaccelerometers are placed over the width of the beam in order to detect possible torsional modes.Hammer excitation was carried out at the 8 selected sensor locations. The reason for this simplywas that the dynamics of the system are not changed because the three sensors stay fixed. Figure2.2 shows the result of the selection procedure.

0 500

0.2

0.4

0.6

0.8

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr

0 501

0.5

0

0.5

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr

Mode Shape

Excitation location

(a) Model a

0 500

0.2

0.4

0.6

0.8

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr

0 501

0.5

0

0.5

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr

Mode Shape

Excitation location

(b) Model b

Figure 2.2: Excitation location

It may seem a bit striking that some excitation places are located close to each other. This,however, may be due to the fact that 8 positions are used to identify only 5 modes. In case 8positions would have been used to identify 8 modes, a more uniform axial distribution of thelocations would have been obtained (Appendix A). This is a result of the selected method, whichonly looks at the partial contribution of each measurement dof to the linear independency. Theexcitation locations, which are measured relative to the clamping, are given in table 2.2.

Locations Beam a [mm] Beam b [mm]

1 100 1102 110 1203 200 2204 210 2305 300 3206 400 4107 410 4208 490 490

Table 2.2: Excitation locations

15


16/163


The location numbers in table 2.2 will be used in section 2.6to indicate the response locationi and the excitation location j of Hij . The accelerometers are located 490 mm of the clamping,

this location is chosen because the response on this point is large in general. Figure 2.3 shows thepositions of the three accelerometers (si).

Figure 2.3: Location of accelerometers on the beam (top view)

2.5 Eigenfrequencies and mode shapes of beam a and beam b

In this section the eigenfrequencies and mode shapes are calculated using the material and geo-metric properties oftable 2.1. First it is verified if the mean difference in the eigenfrequencies ofbeam a and beam b, is between 10% and 15%. The calculations are carried out with a FE modelof50 elements, built in MATLAB. The FE models contain the masses of the 3 accelerometers (40grams for each accelerometer+wire). The results are shown in table 2.3.

Table 2.3: Eigenfrequencies Euler beam

fi Analytic (Hertz) Beam a (Hertz) Beam b (Hertz) difference a,b (%)

1 33.4 33.1 36.3 9.72 209.4 207.7 178.2 14.23 586.4 582.4 506.3 13.14 1149.2 1142.5 962.0 15.85 1899.7 1890.4 1632.2 13.7

Herein the last column is derived as follow:

fbfafa

100%

The small difference (order 1%) between the analytic eigenfrequencies and the eigenfrequen-cies of beam a can be explained by the addition of the accelerometer masses in beam a, whichresults in a small decrease in the eigenfrequencies. The mean difference between the eigenfre-quencies between both models is 13.3%. This satisfies the previously mentioned requirement.

The influence of the shear on the eigenfrequencies can be analyzing by comparing the eigen-frequencies of the Euler FE models with those of the Timoshenko FE models. Hence a FE modelbased on the Timoshenko beam elements is build, of 50 elements, in MATLAB. The eigenfre-quencies of beam a and beam b calculated with the Timoshenko FE model are compared withthe results of the Euler FE model in table 2.4.

16


17/163


Table 2.4: Eigenfrequentie Euler and Timoshenko models

fi Beam a Beam bTimoshenko Euler diff. (%) Timoshenko Euler diff. (%)

1 33.1 33.1 0 36.3 36.3 02 207.4 207.7 0.1 177.9 178.2 0.23 580.1 582.4 0.4 504.2 506.3 0.44 1134.1 1142.5 0.7 955.5 962.0 0.75 1868.5 1890.4 1.2 1612.6 1632.2 1.2

Herein:

diff.= fEfTfT 100%

The mean difference between the eigenfrequencies of beam a and beam b is respectively0.48 and 0.50%, which satisfy the requirement ( 1%). This means that the influence of theshear stress, for the indicated beam dimensions, is insignificant. Hence the analyses of the modeshapes will carried out with the Euler FE model.

The lowest 5 bending mode shapes of beam a and beam b are shown in figure 2.4. Thesemode shapes are normalized by dividing the mode elements by the element with the largestmodulus. The difference between beam a and beam b lies in the height of the second half ofthe beam with the free end. So for beam b the first half is stiffer and heavier in relation to thesecond part. This explains the difference in the mode shapes.

0 500

0.2

0.4

0.6

0.8

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr

0 501

0.5

0

0.5

1

DOFnr0 50

1

0.5

0

0.5

1

DOFnr

Mode Shape Beam a

Mode Shape Beam b

Figure 2.4: Mode shapes (Euler beam models)

17


18/163


To compare the mode shapes several techniques are available. The Modal-Assurance-Criterion(MAC) is a widely use method. The MAC-number is a representation of the correlation between

two mode shapes. The MAC-number is defined as follows [de Kraker, 2004]:

M ACij =|ue

H

i unj |

2

(ueH

i uei )(u

nHj u

nj )

(2.7)

where,

uei : Numerical mode shape beam b

un

i : Numerical mode shape beam aueH

i : Hermitian transpose of the numerical mode shape beam b

unH

i : Hermitian transpose of the numerical mode shape beam a

Although the MAC criterion originally was developed for comparison of experimental and nu-merical mode shapes, here modes resulting from two different numerical models are compared.Because the mode shapes are real in this case, the Hermitian transposed of the mode shapes canbe replaced by the normal transposed of the mode shapes. Figure 2.5 shows a graphical represen-tation of the MAC-matrix.

0

2

4

6

0

2

4

6

0

0.2

0.4

0.6

0.8

Beam bBeam a

Figure 2.5: Graphical representation MAC-matrix Euler models

The graphical representation (figure 2.5) of the MAC-matrix shows that the correlation be-tween the corresponding mode shapes is high (diagonal elements close to 1). This means thatthe applied change in the height of the free end of the beam, in beam b, doesnt radically change

the mode shapes compared to beam a.

18


19/163


2.6 Comparison of frequency response functions

In this section the FRFs of beam a and beam b are compared. All models are undamped.Therefore both the Euler and Timoshenko beam models are used. All FRFs are shown in appen-dix B. To give an example H6,8, based on Euler beam elements, is shown in figure 2.6. In all FRFsin this section and in appendix B, 0 dB = 1 m/N.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H6,8

|

frequency [Hz]

(

displacement/force)[dB]

Beam a Euler beam model

Beam b Euler beam model

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H6,8

)

frequency [Hz]

angle[o]

Figure 2.6: H6,8, Euler beam models

The FRFs clearly show the difference in dynamic behaviour between both beams. From the2nd mode onwards the eigenfrequencies of beam b have a lower value then the eigenfrequenciesof beam a (see also table 2.3). The first eigenfrequentie, however, is clearly smaller for beam acompared to beam b. The influence of inclusion of shear in the beam model is investigated in

figure 2.7 for H6,8. Here, FRFs based on an Euler beam model are compared with FRFs based ona Timoshenko beam model.Its clear that the discrepancies between the FRFs based on the Euler

and Timoshenko models are very small in the low frequency range but grow for larger excitationfrequencies. This can be explained because shear modeling becomes more important for shorterwavelengths, i.e. higher modes. However, for the frequency range of intrest (first five modes) thedifferences can be neglected.

19


20/163


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H6,8

|

frequency [Hz]

(displacement/force)[dB]



Beam a Timoshenko beam model

Beam b Timoshenko beam model

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H6,8

)

frequency [Hz]

angle[o]

Figure 2.7: H6,8, Euler and Timoshenko beam models

2.7 Conclusion

The models of beam a and beam b satisfy the requirements. Assuming that the clamping isinfinitely stiff, the mean difference between the eigenfrequencies between both models is 13.3%.In reality the clamping of the experiments will have a finite stiffness, so in reality the eigenfre-quencies probably will decrease somewhat. Further the influence of the shear stress is negligiblysmall in the frequency range of intrest.

20


21/163

Chapter 3Experimental modal analysis

This chapter describes the experimental test set-up, the experimental test procedure and themodal parameter fit procedure to determine the frequency response functions, the eigenvaluesand corresponding mode shapes of beam a and beam b in an experimental way. The tests aredone in the DCT-lab of the Department of Mechanical Engineering at Eindhoven University ofTechnology. There an experimental set-up is realized. Data acquisition will be done with S IGLAB.The frequency response functions, determined in SIGLAB, will be fitted with the Orthogonal Poly-nomial modal-parameter fit procedure of the experimental modal analysis package ME SCOPE.Appendix Cdescribes the data flow between the used packages.

3.1 Experimental test set-up

The experimental set-up is shown in figure 3.1. The excitation hammer that is used has a synthetictip and a mass of 154 gram. The hammer can generate an impact signal up to approximately2500 Hz.

(a) Set-up

(b) Clamp

(c) Sensors

Figure 3.1: Experimental set-up

21


22/163

3. experimental modal analysis

The beam is oriented in vertical direction (figure 3.1(c)). The upper end of the beam is clampedbetween two metal blocks. The blocks are equipped with holes, perpendicular measurement to

the direction, for fixation to the wall. Three accelerometers are located 490 mm from the clamping(figure 3.1(b)). The accelerometer in the middle will be used to determine the required FRFs. Theother two FRFs, measured on the other accelerometers, will be used for recognizing possibletorsional modes. The beam on the picture is the so called beam a. Obviously, the same set-up is used for beam b. Although it is tried to make the clamping as stiff as possible, it willnever be infinitely stiff. Of course the finite clamping stiffness will have some influence on theeigenfrequencies. The experimental situation is schematically shown in figure 3.2.

(a) Beam a (b) Beam b

Figure 3.2: Experimental Beams

3.2 Experimental test procedure

The excitation and response sensors which are used during the experiments are shown in figure3.3. The hammer is used to excite the beam. This will be done on eight locations, see table 2.2. The

accelerometers are connected to the beam using wax. The hammer and accelerometers are usedto determine FRFs of the system. The digitized time signals are collected in a Signal Analyzerwhich is connected to a computer.

(a) Hammer with force transducer (b) Accelerometers

Figure 3.3: Excitation and response sensors

22


23/163


The software that is used to process the measured time signals which are collected by theSignal Analyzer, is SIGLAB. With the Dynamics Signal Analyzer (vna) utility, the FRFs can be

determined. The first step is setting SIGLAB data acquisition parameters. This can be done in theSIGLAB measurement setup window (see figure 3.4, [Sig, 1999]).

Figure 3.4: Siglab Measurement Setup Window

SIGLAB measures volts, hence a scaling unit has to be set for each measurement channel.The scaling units for the used sensors can be found in table 3.1. The accelerometer in the middleis of type 303A3, the accelerometers at the outside are of type 303A2.

Table 3.1: Scaling units

Sensors Scaling unit

Force sensor 435 [N/V]Accelerometer 303A2 954 [(m/s2)/V]Accelerometer 303A3 885 [(m/s2)/V]

The sample frequency to be used in SIGLAB depends on the chosen bandwidth (frequency

range of intrest). The first 5 eigenfrequencies are lower than 2.0KHz, which is chosen as thebandwidth. The sample frequency and the frequency resolution can be determined by the follow-ing formulae [Sig, 1999]:

fs = 2.56 BW (3.1)

f =2.56 BW

N(3.2)

The record length (N) is set on the maximum value, 8192 points. The beams are lightly

damped, hence the transient signal is not damped out in the measurement time (N/fs = 1.6 [s]).

23


24/163


Therefore numerical damping is introduced, via application of a so-called exponential window(figure 3.5(a)). The measured hammer impulse signal will be followed by a small noise signal.

To eliminate this noise a rectangular force window (figure 3.5(b)) is used. This is a rectangularwindow which is multiplied with the measured force signal. By using these windows signalleakage is avoided as much as possible. Figure 3.5 shows the principle of the force and exponentialwindow. To elucidate up the principle a simulation signal is added.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.62

1.5

1

0.5

0

0.5

1

1.5

2

2.5

time [s]

x(t)

Transient signal

Simulated transient signal

Exponential window

(a) Exponential window

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time [s]

x(t)

Impact signal

Simulated impact signal

Force window

(b) Force window

Figure 3.5: Windows

So, in SIGLAB a combination of the exponential and force windows is applied. The force win-dow is used on channel 1 (hammer) and the exponential window is used on the other (response)

channels. There are two exponential windows available, Exponential 0.1 and Exponential 0.01.The first one has decayed to 10% of the maximum value at the end of the frame. The second onehas decayed to 1% of the maximum value. The experiments are carried out with the exponential0.1 window. The force window has value 1 for the first 20% of the measured time. Hereafter theimpact signal is set to zero. The trigger delay (1%) ensures that the complete excitation signal andresponse signal are captured. By transforming the time signals to the frequency domain, usingFFT, the experimental FRFs and the corresponding coherence can be determined. To increasethe reliability, the estimates will be done based on the average of4 measurements. Estimates ofthese functions are calculated using the auto power spectrum (Pxx) of the excitations and thecross spectrum (Pxy) of the excitation and response ([Sig, 1999]):

H() =Pxy()

Pxx()(3.3)

The accuracy of the measured FRFs can be evaluated partly using the coherence function:

C() =|Pxy()|

2

Pxx()Pyy()(3.4)

The plots of the measured FRFs and the corresponding coherences are shown in Appendix

D. H6,8 of beam a is shown in figure 3.6as an example. The plots make clear which resonance

24


25/163


peaks belong to torsional modes. The influence of the torsional modes on the FRFs of intrest isnegligible. Furthermore the coherences show that the measurement results are reliable.

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50

(acceleration/force)dB

Hz

|H6,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H6,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

Figure 3.6: H6,8 beam a and corresponding coherence

3.3 Modal-parameter fit procedure

The experimental frequency response functions are used to determine the eigenvalues and modeshapes of the system. The Orthogonal Polynomial modal-parameter fit procedure, is carried out,in the package MESCOPE. The FRFs are imported in MESCOPE using an interface with SIGLAB.After importing the FRFs the excitation and response points have to be indicated for each FRF.

The fit procedure starts with an indication of the number of modes (5) in the frequency range ofinterest. The program selects all modes whose magnitude in the FRF is above a adaptable thresh-old. After the modes of interest are selected the eigenvalue Pk = k + jk (k is the dampedangular eigenfrequency, k represents the amount of damping) of each mode can be determined.Therefore the Orthogonal Polynomial method is used. The Orthogonal Polynomial method is aglobal fit procedure that fits a set of FRFs. This methode makes use of the orthogonality prop-erty. Appendices E [Richardson and Formenti, 1982] and F [Richardson and Formenti, 1985] containmore information about the Orthogonal Polynomial method. If the eigenvalues of the modesare known, the corresponding residues can be calculated. Therefore also the Orthogonal Polyno-mial method is used. The Orthogonal Polynomial fit procedure in MESCOPE assumes generalviscous damping. Moreover, light damping is assumed so that the eigenvalues and eigenmodes

occur in complex conjugate pairs. The matrix of FRFs for a general viscously damped system

25


26/163


with undercritically damped modes is formulated as follows [MEs, 2005]:

H() = 12j

nk=1

Rk

j Pk R

k

j Pk

(3.5)

Where,

Rk : Residue matrix (complex)Pk : Complex eigenvalueRk : Complex conjugate of the residue matrixPk : Complex conjugate of the eigenvalue

Equation 3.5 can also be written in the form:

H() =n

k=1

Ak +j( k)

+n

k=1

Ak +j( + k)

(3.6)

Which is used in [de Kraker, 2004], see Appendix G

The plots of these fit results are shown in Appendix H. MESCOPE makes a distinction be-tween residues in displacement/force unit, residues in velocity/force unit and residues in accel-eration/force unit. In this case the unit of the residues is acceleration/force. ME SCOPE containsa function that integrates the residues to change the dimension of an FRF. The results are dis-cussed in Appendix I. An expression for the FRF of a proportionally damped system is:

H() =n

k=1

uOkuTOK

mk(2Ok + 2jkOk

2)(3.7)

Where,

Ok =

2k + 2k, k =

Ok

Herein k is the real part of the eigenvalue Pk, k is the imaginary part of the eigenvalue

and Ok is the real undamped angular eigenfrequency of the undamped system, k is the dimen-sionless damping coefficient and mk = u

TOkM uOk is the modal mass. MESCOPE calculates the

residues and the Unit Modal Mass (UMM) mode shape, i.e. mk = 1. The following formula (forthe derivation, see appendix J) is used to calculate the UMM mode shapes based on the residuein MESCOPE [Richardson, 2000]:

u(k) =

(k)

rij(k) r(k) (3.8)

Herein rij(k) is the driving point residue. The modal parameter fit is done on the basis of

the assumption of viscous damping. Hence the UMM mode shape are complex. To use the FRF

26


27/163


estimation, for an proportionally damped system, the imaginary part of the mode shapes will bedeleted. The plots of these FRFs are shown in Appendix K. The real and imaginary part of the

UMM mode shape are plotted in Appendix L. Notice that the UMM mode shapes in MEscope arenormalized on the mass matrix (uTOkM uOk = 1) but in the graphical representing ofAppendix Lthe mode shapes are normalized by dividing the mode elements by the element with the largestmodulus . The frequencies and damping of the modes are shown in table3.2.

Table 3.2: Frequencies and Damping

Beam a Beam b

Mode Frequencyk2

[Hz] Damping (k) [-] Frequency

k2

[Hz] Damping (k) [-]

1 31.9 0.0107 35.2 0.00122 197.9 0.0025 167.4 0.0035

3 553.0 0.0013 473.2 0.00134 1082.2 0.0020 898.3 0.00145 1781.5 0.0028 1535.6 0.0044

3.4 Conclusions

In this chapter the dynamics characteristics of the beam systems are experimentally determined.The FRFs which are measured in the centre of the beam are reliable. The influence of the torsionmodes can be neglected. The modal-parameter fit procedure is carried out with the OrthogonalPolynomial method. The FRFs of the modal-parameter fit are compared with the experimentaldata, assuming viscous and proportional damping. Comparison of the FRFs shows that propor-tional damping is a legitimate assumption for this system. Because the damping values k arevery small, the update process can be carried out based on the undamped eigenfrequencies.

27


28/163


28


29/163

Chapter 4Model updating

Model updating is used to improve the match between the dynamic properties (eigenfrequenciesand mode shapes) of a FE model and test data. In this traineeship the update is carried out withthe commercial software package FEMTOOLS 3.1.1, and will be based on the eigenfrequencies.In this chapter the FE models will be build in FEM TOOLS. First the dynamic properties (eigen-frequencies and mode shapes) of the infinitely stiff clamped FE model, build in FEM TOOLS, willbe compared with the dynamic properties of the infinitely stiff clamped FE models build in M AT-LAB. Subsequently the initial FE model used in FEM TOOLS is introduced. Difference updatestrategies will be discussed. The results of the update strategies are analyzed and compared with

dynamics properties of the experimental beams.

4.1 Numerical models

Before the update process is started the infinitely stiff clamped FE models from Chapter 2, beama and beam b build in MATLAB of Timoshenko elements, will be compared with the infinitelystiff clamped FE models build in FEMTOOLS. The FE models, in FEMTOOLS, are build up from50 one-dimensional LINE2 elements. This element type is taken the shear stress into account.More complex element types are available in FEMTOOLS. Nevertheless this simple element type

is chosen. FEMTOOLS used the Lanczos Subspace Method (LSM) to obtain the dynamics proper-ties of the FE models. LSM solves the following eigenvalue problem [FEM, 2005]:

[K 2i M]ui = 0 (4.1)

The same eigenvalue problem is applied in Chapter 2 for the infinitely stiff clamped FE modelsin MATLAB. In FEMTOOLS the first 5 eigenfrequencies and corresponding mode shapes aredetermined. The eigenfrequencies are shown in (table 4.1) and the mode shapes are shown inAppendix M.

29


30/163

4. model updating

Table 4.1: Eigenfrequencies Matlab and FEMtools FE models

Matlab FEMtools

fi

Model a (Hz) Model b (Hz) Model a (Hz) Model b (Hz)

1 33.1 36.3 33.1 36.52 207.4 177.9 207.6 179.53 580.1 504.5 581.8 507.54 1134.1 955.5 1140.9 966.05 1868.5 1612.6 1887.1 1641.6

The eigenfrequencies of the FE models, built in FEMTOOLS, are slightly different from theeigenfrequencies determined with the FE model in MATLAB. This may be caused by applicationof different eigenvalue solvers or by application of a slightly different beam element.

4.2 Comparing numerical and experimental data

A schematic drawing of the initial FE model built in FE MTOOLS is shown in figure 4.1

Figure 4.1: Initial FE model

The unknown parameters in the initial FE model are the radial and transversal stiffness. Thenumber of update iterations strongly depends on the initial guess of the clamping stiffnesses(Ky and Kr), see appendix N. Therefore it is essential to choose a reasonable initial radial andtransversal clamping stiffness. The updates are carried out for the experimental beam a andbeam b, using the initial FE model. The chosen parameters (Ax,Iz,M) in the initial FE modelare based on the measured properties of beam a and beam b. The measurement results areshown in table 4.2. In the initial FE model Ax and Iz are set equal to the measured geometricproperties of the first half of the beam. In other words the initial FE model is based on beam a.

30


31/163

4. model updating

Table 4.2: Properties experimental beams

Geometric properties Beam a Beam b

h [mm] (1st half) 9.9 10h2 [mm] (2

th half) 7.3b [mm] 59.9 60

Ax [m2] (1st half) 5.93 1004 6 1004

Iz [m4] (1st half) 4.8434 1009 5 1009

Ax,2 [m2] (2th half) 4.83 1004

Iz,2 [m4] (2th half) 1.95 1009

Mass 3 accelerometers Beam a Beam b

M [kg] 0.012 0.012

In this traineeship only the first 5 undamped angular eigenfrequencies corresponding to thefirst 5 bending mode shapes are updated. The damping of the modes is eliminated in the up-dating process. So, only the undamped angular eigenfrequencies (OK =

2 + 2) of the

damped eigenvalues (Pk = k + jk) of the imported experimental data is used in the updating

process. The damping coefficient

k = /

2k 2k

of the experimental data will used later

on, in order to determine of the FRFs of the updated models.

4.2.1 Comparing initial FE model and experimental data of beam a

First the initial FE model (in FEMTOOLS) and the experimental data of beam a will be compared.Therefore an initial guess for the radial and transversal stiffness is necessary.

Ky = 107 : Initial transversal stiffness [N/m]

Kr = 105 : Initial radial stiffness [N/rad]

The eigenfrequencies of the initial FE model and the experimental data of beam a are com-pared in table 4.3. Herein:

Table 4.3: Eigenfrequencies of the experimental and initial FE model

fi Experiments [Hz] FEMtools FE model [Hz] diff. [%]

1 31.9 33.1 3.82 197.9 207.7 5.03 553.0 582.4 5.34 1082.2 1142.5 5.65 1781.5 1890.4 6.1

fFEMfExpfExp.

100%

The results show a mismatch of approximately 5%. The difference between the corresponding

mode shapes can be analysed with the MAC matrix, see figure 4.2 and table 4.4.

31


32/163

4. model updating

Figure 4.2: Graphical representation MAC-matrix

Table 4.4: MAC-matrix

FE mode nr.Experimental mode nr.

1 2 3 4 51 99.7 2.5 3.3 0.2 0.42 3.2 99.7 1.7 2.7 5.03 2.9 0.0 96.2 0.2 10.44 0.6 0.0 14.0 86.4 0.15 0.2 6.0 0.4 65.3 42.5

The MAC-matrix shows the correlation between the mode shapes. Obviously, the diagonalof the MAC-matrix is representing the correlation between the corresponding mode shapes. Thecorrelation on the diagonal is clearly decreasing after the 3rd mode shape. Apparently, the 5th

numerical modes has a higher correlation with the 4th experimental mode than with the 5th

experimental mode. A model updating procedure has been carried out in FEMTOOLS to improvethe clamping stiffnesses leading to Ky = 1.25 10

8N/m and Kr = 8.86 104N/rad. More

information on the model updating process will be given in section 4.3.

4.2.2 Comparing initial FE model and experimental data of beam b

In the initial FE model the transversal and radial clamping stiffnesses are chosen based on theupdating results with regard to beam a as discussed in the previous subsection:

Ky = 1.25 108 : Initial transversal stiffness [N/m]

Kr = 8.86 104 : Initial radial stiffness [N/rad]

The eigenfrequencies of the initial FE model and the experimental data of beam b are com-pared in table 4.5. The results show a mismatch of approximately 16.5%. This mismatch is dueto the (deliberate) bad initial guesses ofA2 and I2 for the right half of beam b. For A2 and I2 thesame values are taken as for the left part of beam b. So the (bad) initial FE model for beam b isin fact a good model for beam a.

32


33/163

4. model updating

Table 4.5: Eigenfrequencies of the experimental and FE model

fi Experiments [Hz] FEMtools FE model [Hz] diff. [%]

1 35.1 31.6 -102 167.1 198.5 18.83 473.2 555.1 17.34 898.2 1082.9 20.65 1535.5 1775.5 15.7

Herein:

fFEMfExpfExp.

100%

The correlation between the experimental mode shapes and the mode shapes of the initial FEmodel can be analyzed with the MAC matrix, see figure 4.3 and table 4.6

Figure 4.3: Graphical representation MAC-matrix

Table 4.6: MAC-matrix

FE mode nr.Experimental mode nr.

1 2 3 4 5

1 99.1 7.2 1.5 0.1 0.42 4.4 96.7 0.0 0.6 1.53 3.9 2.1 97.2 0.7 9.04 0.6 0.8 5.3 95.8 0.05 0.3 2.7 12.1 12.6 97.2

The MAC-matrix shows a surprisingly good correlation between the experimental and nu-merical corresponding mode shapes.

4.3 FE Model updating process

In this section the FE model updating process, carried out with FEM TOOLS, is discussed. Up-

dating is carried out using a Bayesian Parameter Estimation (BPE). The BPE error which is to be

33


34/163

4. model updating

minimized is defined as follows:

E = RTCRR + PTCPP (4.2)

where,

E : ErrorCR : Weighting matrix experimental data (eigenfrequencies, eigenmodes, MAC values, etc)CP : Weighting matrix updated model parameters

The minimization is realized by the iteratief method which is discussed in appendix O. Thestop criterion in FEMTOOLS is based on the Correlation Coefficients (CC). Several CC areavailable in FEMTOOLS. Because the updates in this traineeship are only based on the eigenfre-

quencies the chosen stop criterion is based on CC_ABS which is defined as follows:

CC_ABS =1

N

Ni=1

Cri|fi|

fi(4.3)

where,

N : Number of eigenfrequencies (N=5)Cri : Expected relative errorfi = fFEi fei : Difference between the (updated) FE frequency and the experimental frequency

fi = fei : Experimental frequency

For the expected relative error (Cri = 100cri) the standard value (1%) is used for all eigenfre-quencies. An absolute or relative stop criterion may be used in F EMTOOLS: CCt < 1 and|CCt+1 CCt| < 2. In our case CCt = CC_ABSt. Here, a relative stop criterion is used (so,1 = 0 and 2 = 0.001%) because this gives the best impression of the degree of convergence.

4.3.1 Update process based on the experimental data of beam a

In this subsection the update strategies of beam a are discussed. The parameters of the initial

FE model of beam a are given in the 3rd column oftable 4.8. The two parameter values of theclamping stiffness can be considered to be inaccurate whereas the other geometrical propertiescan be considered to be accurate. Also the physical properties can be considered to be accurate,see table 2.1. The experimental target eigenfrequencies of beam a and the eigenfrequencies ofthe initial FE model are given in table 4.3.

34


35/163

4. model updating

To update the initial FE model the following update strategies are selected:

Strategy 1a : Update the initial values of parameters Ky and Kr.

Strategy 1b : Update the initial values of parameters Ky, Kr and M (accelerometer mass).

Strategy 1c : Update the initial values of parameters Ax and Iz (Ax = Ax2 and Iz = Iz2).

Strategy 1d : Update in two steps, first the initial values of parameters Ky, Kr and M areupdated. Subsequently, the initial values of parameters Ax and Iz (Ax = Ax2 and Iz = Iz2)are updated.

Strategy 1e : Update in two steps, first the initial values of parameters Ax and Iz (Ax = Ax2and Iz = Iz2) are updated. Subsequently, the initial values of parameters Ky, Kr and M

are updated.

Strategy 1f : Update the initial values of parameters Ax, Iz , Ky, Kr and M (Ax = Ax2 andIz = Iz2).

In the updates of strategie 1d, Strategie 1e and Strategie 1f the values of Ax and Iz arebounded (5%). The geometrical parameters b and h of beam a are also measured (2nd columnoftable 4.8). The results of the applied strategies can be evaluated and compared by consideringthe final values ofCC_ABS, the final differences in the eigenfrequencies and the resulting up-dated parameter values, when the stop criterion is satisfied. The results are shown in table 4.7and table 4.8.

Table 4.7: Updating results of the applied strategies for beam a

Strategy 1a Strategy 1b Strategy 1c Strategy 1d Strategy 1e Strategy 1f

N1 17 17 12 17 2 44N2 2 44

f1 [%] 0.68 0.68 1.33 0.85 0.83 0.78f2 [%] 0.45 0.45 0.69 0.28 0.31 0.36f3 [%] 0.50 0.50 1.07 0.33 0.36 0.41f4 [%] 0.14 0.14 0.49 0.05 0.01 0.04f5 [%] 0.29 0.29 0.93 0.52 0.46 0.41

2 [%] 0.001 0.001 0.001 0.001 0.001 0.01

CC_ABS [%] 0.41 0.41 0.9 0.41 0.39 0.40* N is the number of iterations

35


36/163

4. model updating

The stop criterion of Strategy 1f (2 = 0.01) is chosen higher compared with the otherstrategies (2 = 0.001). The reason is that the stop criterion 2 = 0.001 can not be satisfied.

This is illustrated in figure 4.4. After reaching a minimum the correlation continuous increasing.

Figure 4.4: Correlation CC_ABS Strategie f

Table 4.7shows no perceptible difference between the results of strategy 1a and strategy

1b. This is because the influence of changing the accelerometer mass on the eigenfrequenciesis small. The correlation between the experimental and numerical mode shapes can be analysedwith the MAC matrices (appendix P). All the MAC matrices of the updated models give a sat-isfying result. The goal of the update is to improve the match between the dynamic properties(eigenvalues and eigenmodes) of the FE model and the experimental setup. However, a physi-cally meaningful change in the values of the model parameters is required for a useful updateresult. Hence the updated parameters are compared with the average measuring results of theexperimental beam a. The updated beam height and width can be calculated by formulae 4.4.The measurements results and the updated parameter values are shown in table 4.8.

Iz = 112bh3

Ax = bh

h =

12IzAx

; b =Axh

(4.4)

36


37/163

4. model updating

Table 4.8: Measured parameters of beam a compared with the results of the update strategies

Parameters Measuring Initial Updated values

result model a b c

Transversal stiffness (Kt) [N/m] 1 107 1.23 108 1.23 108

Radial stiffness (Kr) [N/rad] 1 105 9.24 104 8.97 104

Width beam a (b) [mm] 59.9 60 7.9Height beam a (h) [mm] 9.9 10 10.1

Mass accelerometer (M) [g] 12 12 10.3


result model d e f

Transversal stiffness (Kt) [N/m] 1 107 1.23 108 1.25 108 1.26 108

Radial stiffness (Kr) [N/rad] 1 105 8.97 104 9.17 104 9.30 104

Width beam a (b) [mm] 59.9 60 63 63 63

Height beam a (h) [mm] 9.9 10 10 10 10Mass accelerometer (M) [g] 12 12 10.3 11.4 11.5

* means that this parameter is not updated

The different update strategies, with the exception of strategy c, give approximately the sameresults. This could be expected because the only incorrect initial parameters are the clampingstiffnesses. The FRFs of the updated FE models are shown in Appendix Q (herein 0 dB =1 m/N). Based on the FRFs there can be concluded that all strategies, with the exception ofstrategy c, gives a good approximation of the experimental determined FRFs. The results of

strategy c attract the attention because the divergent value of the width. This can be explained bylooking at the tracking of the parameters (Ax and Iz). The track of the relative parameter changeof the cross area and the 2nd moment of area are approximately equal. So, only the width of thebeam is seriously changing.

0 2 4 6 8 10 12100

90

80

70

60

50

40

30

20

10

0

Iteration

Parameterchange(%

)

Crosssection

Second moment of area

2 4 6 8 10 12

98

96

94

92

90

88

86

Figure 4.5: Parameter tracking

37


38/163

4. model updating

The width b, however, does not influence the eigenfrequencies of a clamped-free beam, asfollows from the analytical expression[Blevins, 1979]:

fi =2i

2L2

EI

A=

2i2L2

Ebh3

12bh= fi =

2i2L2

Eh2

12(4.5)

So, the modal updating procedure may given an arbitrary value for b as a result!

4.3.2 Update process based on the experimental data of beam b

In this subsection the update strategies of beam b are considered. The experimental eigenfre-

quencies of beam b and the eigenfrequencies of the initial FE model are given in table 4.5. Theparameters of the initial FE model of beam b are given in the 3rd column of table 4.10. For,the initial values of the clamping stiffnesses the initial values obtained in the previous subsection4.3.1 are taken (for decrease the necessary iterations the Kr is chosen somewhat lower). Thesevalues can be considered to be reasonably accurate. However, nuts may be tightened by differentmoments for beam a and beam b. Please note that the initial height of the right half of beamb has been chosen inaccurate on purpose in order to see if the model updating procedure workswell. To update the initial model the following update strategies are selected:

Strategy 2a : Update the initial values of parameters Ky and Kr.

Strategy 2b : Update the initial values of parameters Ax2 and Iz2 .

Strategy 2c : Update the initial values of parameters Ax2 , Iz2 , Ky and Kr.

Strategy 2d : Update in two steps, first the initial values of parameters Ky, Kr and M areupdated. Subsequently, the initial values of parameters Ax2 and Iz2 are updated.

Strategy 2e : Update in two steps, first the initial values of parameters Ax2 and Iz2 are updated.Subsequently, the initial values of parameters Ky, Kr and M are updated.

Strategy 2f : Update the initial values of parameters Ax2 , Iz2 , Ky, Kr and M.

Notice that the parameters Ax2 and Iz2 for the right half of the beam, in these update strate-gies, are uncoupled of respectively Ax and Iz for the left half of the beam. Also the boundary onthe parameters Ax2 and Iz2 are deleted. The results are shown in table 4.9 and table 4.10.

38


39/163

4. model updating

Table 4.9: Updating results of the applied strategies for approaching beam b

Strategy 2a Strategy 2b Strategy 2c Strategy 2d Strategy 2e Strategy 2f

N1

14 6 47 14 6 44N2 12 6

f1 [%] 10.57 0.01 0.01 0.85 0.56 0.02f2 [%] 14.49 0.44 0.06 0.28 0.00 0.09f3 [%] 5.84 0.44 0.33 0.33 0.13 0.37f4 [%] 1.06 0.04 0.00 0.05 0.17 0.00f5 [%] 8.41 0.91 0.23 0.52 0.79 0.24

2 [%] 0.001 0.001 0.001 0.001 0.001 0.001CC_ABS [%] 8.1 0.37 0.13 0.40 0.33 0.14

* N is the number of iterations

The results of updating strategies 2a and 2d are unsatisfactorily. The reason is clear, thedifference in the eigenfrequencies depended on the height of the right half of the beam. Inthe update strategy 2a only the radial and transversal clamping stiffnesses are updated. Thisalso explains why strategy 2d is not a suitable update strategy. The correlation between thenumerical and experimental mode shapes can be analysed with the MAC matrices (appendixR). The MAC matrices confirm the comments about the update strategie 2a and strategie 2d.To verify physical meaning of the updated parameters these are compared with the measuredparameters of the experimental beam b in table 4.10.

Table 4.10: Measured parameters of beam b compared with the results of the update strategies


result model 2a 2b 2c

Radial stiffness (Kr) [N/rad] 1.25 108 9.92 106 2.70 108

Transversal stiffness (Kr) [N/m] 8.86 104 8.13 104 8.49 104

Width beam b (b) [mm] 60 60 61.4 61.5Height beam b (h) [mm] 10 10

Height right half beam b (h2) [mm] 7.3 10 7.2 7.1Mass accelerometers (M) [g] 12 12


result model 2d 2e 2f

Radial stiffness (Kr) [N/rad] 1.25 108

9.65 106

1.54 108

2.47 108

Transversal stiffness (Kr) [N/m] 8.86 104 7.68 104 7.86 104 9.59 104

Width beam b (b) [mm] 60 60 53.1 61.4 61.5Height beam b (h) [mm] 10 10

Height right part beam b (h2) [mm] 7.3 10 7.8 7.2 7.1Mass accelerometers (M) [g] 12 12 5.3 12.6 16.4

* means that this parameter is not updated

Good results are obtained for updating strategies 2b,2c,2e and 2f. The FRFs of the updated

FE models are shown in Appendix S (herein 0 dB = 1 m/N).

39


40/163

4. model updating

4.4 Conclusions

In this chapter FE model updates have been carried out in FEM TOOLS for two beam systemsbeam a and beam b. Therefore first initial FE models were built. These initial FE models areupdated using different strategies in order to approximate the experimental eigenfrequencies ofbeam a and beam b. The number of iterations depends on the quality of the initial guessesof the parameters. The quality of the update depends on the chosen strategy. Obviously, themost important point of the strategy is that the right parameters are selected. If two-step modelupdating is carried out, it is wise to first update the parameters whose expected mismatch ishighes.

40


41/163

Chapter 5Conclusions and recommendations

The software package FEMTOOLS is capable to improve the match between an initial FE modeland experimental data by updating the values of selected parameters of the FE model. The analy-ses in this report are based on FE models of a clamped-free beam system and the first 5 un-damped angular eigenfrequencies of two slightly different experimental beams. The updates arecarried out by using different updating strategies. The iterative updating process is based on thedifference between the eigenfrequencies of the FE model and the experimental data. Obviously,it is important that the user has knowledge about the updated system and the updating process.By selecting the right parameters and a well considered updating strategy, FEMTOOLS is capable

to approach the parameters of the experimental model. The initial FE element model is built inFEMTOOLS using fifty LINE2 beam elements. This element takes the shear stress into account.The results show that for the used beam dimensions the influence of the shear stress on theeigenfrequencies is negligibly small. The modes of the experimental models are weakly damped.Therefore undamped angular eigenfrequencies are used in the updating process.

In this report we only discuss updates based on the experimental eigenfrequencies. In thefuture updates based on experimental eigenfrequencies, experimental mode shapes and MAC-values could be considered. Also the influence of using different confidence values for the eigen-frequencies and model parameters would be investigated.

41


42/163

5. conclusions and recommendations

42


43/163

Bibliography

[Sig, 1999] (1999). Siglab User Guide, Section 5. Spectral Dynamics, San Jose, CA, USA.

[FEM, 2005] (2005). FEMtools Theoretical Manual Version 3.1. Dynamic Design Solutions, Leu-ven, Belgium.

[MEs, 2005] (2005). MEscopeVES Application Note #28. Vibrant Technology, Inc., Scotts Valley,CA, USA.

[Blevins, 1979] Blevins, R. D. (1979). Formulas for natural frequency and modeshape. Van NostrandReinhold Company.

[de Kraker, 2004] de Kraker, B. (2004). A Numerical - Experimental Approach in Structural Dy-namics. Shaker Publishing B.V.

[Richardson, 2000] Richardson, M. H. (2000). Modal mass, stiffness and damping. Technicalreport, Vibrant Technology, Inc.

[Richardson and Formenti, 1982] Richardson, M. H. and Formenti, D. L. (1982). Parameter esti-mation from frequency response measurements using rational fraction polynomials. 1st IMACConference, Orlande, FL, USA. (Appendix E).

[Richardson and Formenti, 1985] Richardson, M. H. and Formenti, D. L. (1985). Global curvefitting of frequency response measurements using rational fraction polynomials. 3rd IMACConference, Orlande, FL, USA. (Appendix F).

43


44/163

BIBLIOGRAPHY

44


45/163

Appendix ASensor placement to identify 8 modes

45


46/163

a. sensor placement to identify 8 modes

0 10 20 30 40 50

0

0.5

1

DOFnr0 10 20 30 40 50

1

0.5

0

0.5

1

DOFnr

0 10 20 30 40 501

0.5

0

0.5

1

DOFnr0 10 20 30 40 50

1

0.5

0

0.5

1

DOFnr

0 10 20 30 40 501

0.5

0

0.5

1

DOFnr0 10 20 30 40 50

1

0

1

2

DOFnr

0 10 20 30 40 50

2

1

0

1

DOFnr0 10 20 30 40 50

1

0

1

2

DOFnr

Mode Shape

Excitation location

Figure A.1: Sensor placement beam a

46


47/163


0 10 20 30 40 50

0

0.5

1

DOFnr0 10 20 30 40 50

1

0.5

0

0.5

1

DOFnr

0 10 20 30 40 501

0.5

0

0.5

1

DOFnr0 10 20 30 40 50

1

0.5

0

0.5

1

DOFnr

0 10 20 30 40 501

0.5

0

0.5

1

DOFnr0 10 20 30 40 50

2

1

0

1

2

DOFnr

0 10 20 30 40 50

2

1

0

1

2

DOFnr0 10 20 30 40 50

2

1

0

1

2

DOFnr

Mode Shape

Excitation location

Figure A.2: Sensor placement beam b

47


48/163


48


49/163

Appendix BFrequency response functions FEM

models

This Appendix shows an overview of the frequency response function of four FEM models. TheTimoshenko beam models and Euler beam models are built of 50 elements. The resolution ( f)of the plots is 5Hz and 0dB is equal to 1m/N.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H1,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H1,8

)

frequency [Hz]

angle[o]

49


50/163

b. frequency response functions fem models

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H2,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H2,8

)

frequency [Hz]

angle[o]

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H3,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H3,8

)

frequency [Hz]

angle[o]

50


51/163


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H4,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H4,8

)

frequency [Hz]

angle[o]

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H5,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H5,8

)

frequency [Hz]

angle[o]

51


52/163


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H6,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H6,8

)

frequency [Hz]

angle[o]

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H7,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H7,8

)

frequency [Hz]

angle[o]

52


53/163


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000250

200

150

100

50

|H8,8

|

frequency [Hz]






0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

50

100

150

200

(H8,8

)

frequency [Hz]

angle[o]

53


54/163


54


55/163

Appendix CFlowchart of data

The Flowchart is started in SIGLAB,where the experimental FRFs are determined. From

SIGLAB these data can be exported to MATLAB. The data that will be exported can be selected

55


56/163

c. flowchart of data

on the File Storage drop-down menu in the plotting window. To extract the acquired data toa MATLAB workspace, type the following command in the workspace: name=vna(get,meas).

The interface between SIGLAB and MESCOPE is realized by using the dynamically linked libraryWrtBlk32.dll which is used to write MESCOPE compatible Data Blocks disk files. For the in-terface between MESCOPE and FEMTOOLS some adaptation will be carried out. At first, theUMM mode shapes and the DOF can be saved in two separated .uff files which can be joinedtogether in one .uff file. Subsequently, the imaginary part of the eigenvalue is replaced by theundamped natural frequency. The data of the updated FE model can exported in a .uff file.To convert .uff to .mat some MATLAB script can be found on the internet site of Mathworks(http://www.mathworks.com/matlabcentral/fileexchange/loadFileList.do).

56


57/163

Appendix DFrequency response function and co-

herency plots of Siglab

This Appendix shows an overview of the frequency response function of the experimental models.Herein 0dB is equal to 1(m/s2)/N.

57


58/163

d. frequency response function and coherency plots of siglab

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H1,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H1,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H1,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H1,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.1: H1,8

58


59/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H2,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H2,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H2,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H2,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.2: H2,8

59


60/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H3,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H3,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H3,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H3,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.3: H3,8

60


61/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H4,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H4,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H4,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H4,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.4: H4,8

61


62/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H5,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H5,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H5,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H5,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.5: H5,8

62


63/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H6,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H6,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H6,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H6,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.6: H6,8

63


64/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H7,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H7,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H7,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H7,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.7: H7,8

64


65/163


0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H8,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H8,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

Center

Left

Right

(a) Beam a

0 200 400 600 800 1000 1200 1400 1600 1800 200050

0

50


Hz

|H8,8

|

0 200 400 600 800 1000 1200 1400 1600 1800 2000200

100

0

100

200

[]angle

Hz

(H8,8

)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Coherence

Hz

CenterLeft

Right

(b) Beam b

Figure D.8: H8,8

65


66/163


66


67/163

Appendix EParameter Estimation from Frequency

Response Measurements Using Ra-tional Fraction Polynomials

67


68/163


69/163

e. parameter estimation from frequency response measurements using rational

fraction polynomials

69


70/163



70


71/163



71


72/163



72


73/163



73


74/163



74


75/163



75


76/163



76


77/163



77


78/163



78


79/163



79


80/163



80


81/163



81


82/163



82


83/163

Appendix FGlobal Curve Fitting of Frequency

Response Measurements Using Ra-tional Fraction Polynomials

83


84/163

f. global curve fitting of frequency response measurements using rational fraction

polynomials

84


85/163


polynomials

85


86/163


polynomials

86


87/163


polynomials

87


88/163


polynomials

88


89/163


polynomials

89


90/163


polynomials

90


91/163


polynomials

91


92/163


polynomials

92


93/163

Appendix GReformulation of transfer function

In this Appendix the reformulation of the transfer function estimation for generally viscouslydamped systems with undercritically damped modes is discussed. The follow formulation isused in [MEs, 2005]:

H() =1

2j

nk=1

Rkj Pk

Rk

j Pk

(G.1)

Where,

Rk : Residue matrix (complex)Pk : Complex eigenvalueRk : Complex conjugate of the residue matrixPk : Complex conjugate of the eigenvalue

The residue matrix and complex eigenvalue can be split up in a real and imaginary part:

Rk = RkR +jRkIPk = k +jk

By using this in equation G.1 the follow formula can be obtained:

H() =n

k=1

1/2(RkI jRkR)j (k +jk)

+1/2(RkI +jRkR)

j (k jk)

By rewriting the denominator and changing the order, this expression can be written in the followform:

H() =

nk=1

1/2(RkI +jRkR)k +j( k) +

1/2(RkI jRkR)

k +j( + k)

93


94/163

g. reformulation of transfer function

By replacing numerators by Ak and A

k the transfer function estimation for generally viscouslydamped systems with undercritically damped modes is:

H() =n

k=1

Ak +j( k)

+n

k=1

Ak +j( + k)

(G.2)

This formula is used in [de Kraker, 2004].

94


95/163

Appendix HFrequency Response Function gen-

eral viscous damping estimation

This Appendix shows an overview of the frequency response function assuming general viscousdamping compared with the experimental data. Herein 0dB is equal to 1(m/s2)/N.

95


96/163

h. frequency response function general viscous damping estimation

0 200 400 600 800 1000 1200 1400 1600 1800 2000

40

20

0

20

40

60

80


Hz

|H1,8

FEM updating clamped beam

Documents

Transcript of FEM updating clamped beam