ir. L. Kodde Department Mechanical Engineering Dynamics ...mate.tue.nl/mate/pdfs/8195.pdf · Bachelor final project. Maarten J. Beelen. April 2007.Page 3 of 33 e e+1 ue, F e φe,

Bachelor final project

Coordinators: dr. ir. R.H.B. Fey

ir. L. Kodde

Technische Universiteit Eindhoven

Department Mechanical Engineering

Dynamics and Control

Eindhoven, April, 2007

i

Summary Target

The target in this investigation is to validate the method of errormatrices. This method

locates and identifies errors in Finite Element (FE) models using the differences in the

dynamic properties (eigenvalues and eigenmodes) between the FE-model and test

data. The method is applied to a simple straight beam (beam A) and a similar beam

with a weld in the center (beam B). Using these simple structures information can be

obtained about the usefulness of this method for more complex structures.

Procedure

First some FE-models for beam A and beam B are constructed in Matlab. Second,

their number of degrees of freedom is reduced so that they can be compared with the

test data. This data is obtained from Impact Hammer modal testing: A known

excitation is applied on eight different points and the responses are measured. From

these responses eight frequency response functions (FRF’s) are determined with a

Signal Analyzer and the software package Siglab. The experimental eigenvalues and

eigenmodes are determined from the FRF’s using the software package ME’scope.

Results

When the FE-model for beam A is compared with the test data from beam B, the

largest difference is not caused by the weld, but by the clamp that is modeled as rigid.

This is because the weld does have fewer implications on the dynamic behavior of the

beam than was expected on beforehand. Therefore it is difficult to distinguish the

differences caused by the weld from other irregularities. A better approach is to make

sure in advance that the numerical models are less similar. Because when they are

similar, it is likely that the experimental models will also barely differ.

But, in spite of the near similarity of the beams, the method of errormatrices succeeds

in recognizing the errors caused by the weld. These errors almost disappear when a

FE-model for beam B is compared with test data from beam B (the weld is taken into

account in the model).

Conclusion

Obviously, finding modeling errors in this simple model is relatively easy. Therefore

it is a suitable way to validate the procedure. The error matrices prove to work for this

simple structure, so they should also be able to give an indication of where to search

for modeling errors in more complex structures.

A recommendation for a continuation of this investigation is to use more advanced

model updating strategies, like FEMtools.

ii

TABLE OF CONTENTS

SUMMARY i

1. INTRODUCTION 1

2. NUMERICAL ANALYSIS 2

2.1. MODELING BEAM A 3 2.1.1. STRATEGY 1 (MODEL A1) 3 2.1.2. STRATEGY 2 (MODEL A2) 5 2.2. MODELING BEAM B 6 2.2.1. STRATEGY 1 (MODEL B1) 6 2.2.2. STRATEGY 2 (MODEL B2) 8 2.2.3. STRATEGY 3 (MODEL B3) 8 2.3. COMPARISON OF NUMERICAL MODELS, REDUCTION AND ERRORMATRICES 9 2.3.1. EIGENFREQUENCIES AND EIGENMODES 9 2.3.2. FREQUENCY RESPONSE FUNCTIONS 11 2.3.3. REDUCING THE NUMBER OF DEGREES OF FREEDOM 12 2.3.4. SYSTEM ERROR MATRICES 15

3. EXPERIMENTAL ANALYSIS 18

3.1. EXPERIMENTAL SET-UP 18 3.2. EXPERIMENTAL MODAL ANALYSIS 19 3.2.1. SIGLAB SETTINGS 21 3.2.2. APPLIED WINDOWS 22 3.2.3. EXPERIMENTAL MODAL ANALYSIS 23 3.3. COMPARISON OF THE EXPERIMENTS 25 3.3.1. EIGENFREQUENCIES AND MODE-SHAPES 25 3.3.2. FREQUENCY RESPONSE FUNCTIONS 27

4. COMPARISON OF NUMERICAL AND EXPERIMENTAL DATA 28

4.1. EIGENFREQUENCIES AND MODE-SHAPES 28 4.2. FREQUENCY RESPONSE FUNCTIONS 29 4.3. SYSTEM ERROR MATRICES 30 4.3.1. SYSTEM ERROR MATRICES MODEL A 30 4.3.2. SYSTEM ERROR MATRICES MODEL B 31

5. CONCLUSIONS AND RECOMMENDATIONS 33

BIBLIOGRAPHY 34

iii

APPENDIX

A. NUMERICAL ANALYSIS A-1

A.1. MAC MATRICES A-1

A.2. ERRORMATRICES A-2

A.3. EIGENMODES A-4

A.4. NUMERICAL FREQUENCY RESPONSE FUNCTIONS A-5

B. ERRORMATRIX VALIDATION B-1

B.1. SIMPLE TEST B-1

B.2. INFLUENCE OF THE DYNAMIC REDUCTION AND THE NUMBER

OF MODES USED B-2

B.3. INFLUENCE OF THE NUMBER OF MODES ON X B-3

B.4. DYNAMIC REDUCTION B-4

B.5. STATIC REDUCTION B-5

C. COMPARISON NUMERICAL AND EXPERIMENTAL DATA C-1

C.1. MAC MATRICES C-1

C.2. ERRORMATRICES C-2

C.3. EIGENMODES C-3

C.4. FREQUENCY RESPONSE FUNCTIONS C-6

C.5. TORSIONAL MODES C-10

D. FLOWCHARTS OF DATA D-1

E. INDICATION OF THE MATLAB FILE E-1

Bachelor final project. Maarten J. Beelen. April 2007. Page 1 of 33

1. Introduction

A numerical model for the dynamic behavior, characterized by the eigenfrequencies

and eigenmodes, of a structure can be made using the finite element method (FEM).

Obviously, this FE-model should reflect the real dynamic behavior when it is used to

predict the dynamic behavior for different loading cases. Therefore it is necessary to

verify the model using experimental data obtained from modal test studies. Using the

differences between the numerical and experimental data, the parts of the structure

which are poorly modeled can be located. A method to find these parts in the

numerical model is the method of System Error Matrices. These matrices indicate the

difference in the stiffness- or mass- matrices between the numerical and experimental

data. In this report this verification method is introduced and evaluated.

Also, two methods are used for indicating differences between the models. These

methods are not able to locate errors:

Comparison of eigenfrequencies and eigenmodes, and the Modal Assurance Criterion

(MAC). This method shows the correlation between different mode-shapes.

The methods are applied to a simple structure: A cantilever beam (‘clamped-free’).

Two different steel cantilever beams are used: A simple straight beam (beam A) and a

similar beam with a weld halfway the beam length (beam B).

When comparing a numerical model for beam A with experimental results from beam

B, differences occur. It is obvious that the center of the beam (where the weld is

located) and the clamping are the parts that are responsible for most of these

differences, because in the numerical model the weld in the center is ignored, and the

clamping is assumed to be infinitely stiff. In this study it is investigated whether the

verification methods can locate the weld in the center as a poorly modeled part. On

the basis of these results an improved model is made, which also takes the weld into

account.

Chapter 2 will provide several numerical models of the beams made with the finite

element method. These models are compared with one another.

Chapter 3 will discuss the experiments that are made using Impact Hammer modal

testing: The excitation force and its acceleration response will be measured.

The experiments will be carried out for a beam without and a beam with a weld.

The experimental results are also compared.

In chapter 4 the numerical and experimental data will be compared

The problems that arise in these comparisons will be discussed too, for instance the

difference in the number of degrees of freedom in the numerical and experimental

data, or the modeling of damping.

Validating the methods applied to a simple system gives information about the

usefulness of the methods when they are applied to more complex systems.

The main problem definition can be formulated as follows:

Finite element models can be verified with error matrices and experimental modal

analyses.

Is the method of error matrices able to locate those parts of the structure which are

poorly modeled?


2. Numerical analysis

In this chapter, two different beams are modeled and compared. A ‘clamped’-free

straight beam (Beam A) and a similar beam, but with a weld halfway the length of the

beam (Beam B).

The numerical models of the beams are made using the Finite Element Method

(FEM). The 2D situation is analyzed for bending vibrations. Axial and torsional

vibrations are not being considered in the FE model. Figure 2.1 shows the two

numerical beam models.

(A) The straight beam (A). (B) The welded beam (B)

Figure 2.1: Numerical beam models

The model of the beam that is used for the experiments in chapter 3 is displayed in

figure 2.2. The 8 points that are indicated are the measured points from the

experiments. Therefore the relevant Degrees Of Freedom (dof’s) in the numerical

model are the ones that correspond to these eight points. Point number 4 corresponds

with the position of the weld in beam B.

Figure 2.2: Measurement points for the experimental model

Figure 2.3 shows the element distribution for the numerical model. The beam is

divided in x-direction into 100 elements of 5 mm. This size is small enough to

determine the eigenfrequencies and modes accurately. So there are 101 equally

distributed nodes, each containing 2 dof’s: a transversal displacement in y-direction

(u) and a rotation around the z-axis (φ). The dof numbers of the FE-model that

represent the deflections of the 8 experimental points are also indicated in the figure.

Figure 2.3: Element distribution for the numerical model

Dof-nr: 23 47 71 95 119 143 167 191

l=5 mm Element 2

dof-nr. 3&4 (u2 & φ2)

dof-nr. 1&2 (u1 & φ1) x

y

z

1 2 3 4 5 6 7 8

60 mm

L=500 mm

H=10 mm W=60 mm Weld position


e e+1

ue, Fe ue+1, Fe+1 φe, Me φe+1, Me+1

2.1. Modeling beam A

Mass and stiffness system matrices are required for the FE-models. They are derived

from the assembly of mass and stiffness element matrices. Two different strategies for

modeling beam A are introduced.

2.1.1. Strategy 1 (model A1)

For model A1 Euler beam element matrices are used. These element matrices only

take pure bending of the beam into account. Consider these equations:

eee fuK = , eee fuM =&& (2.1)

where,

=

+

+

1

1

e

e

e

e

e

u

u

u

ϕ

ϕ,

=

+

+

1

1

e

e

e

e

e

M

F

M

F

f (2.2)

Herein ue is the deflection and φe the rotation of node e, Fe is the force and Me the

moment acting on node e. Using the sign conventions as presented in figure 2.4 the

element matrices are derived.

Figure 2.4: Sign conventions

The Euler beam element stiffness matrix and the Euler beam element mass matrix are

given by:

−

−

−

=

2

22

3

4

612

264

612612

lsym

l

lll

ll

l

EIK

e ,

−

−

−

=

2

22

4

22156

3134

135422156

420

lsym

l

lll

ll

AIM

e ρ (2.3)

where,

E = Young’s modulus [N/m2]

3

12

1WHI = = Second moment of area [m

4]

en

Ll = = Element length [m]

L = Total length [m]

ne = Number of equidistant elements

ρ = Density [kg/m3]

A=WH = Cross-section [m2]


A summary of the parameters used for the beams is given in table 2.1.

Table 2.1: Beam properties

Geometric properties

L Length beam 500 [mm]

H Height beam 10 [mm]

W Width beam 60 [mm]

ne Number of

equidistant elements

100

Material properties

E Young’s modulus 11101.2 ⋅ [n/m2]

ν Poisson’s ratio 0.3 [-]

ρ Density 7850 [kg/m3]

The Euler beam element matrices are implemented into the system matrices. Because

there are 101 nodes with 2 degrees of freedom, the mass and stiffness matrices are of

size (202 x 202). The beam is clamped at the left end; node 1 is fixed, so u1=0 and

φ1=0. Therefore the first two rows and the first two columns of the mass and stiffness

matrices are deleted. After this partitioning the size of the system matrices becomes

(200 x 200).

In the experimental set-up three acceleration-sensors (with a total mass of 5.4 gram)

are attached at 20 mm from the free end of the beam. This extra mass is included in

the numerical models.


2.1.2. Strategy 2 (model A2)

For model A2 Timoshenko beam element matrices are used. In addition to bending

stress, these element matrices also take the influence of shear stress into account. This

influence becomes more important with higher frequencies (shorter wavelengths).

The Timoshenko beam element stiffness matrix is given by:

( )

Φ+

−

Φ−−Φ+

−

Φ+=

4

/6/12

2/64

/6/12/6/12

1 2

22

sym

ll

l

llll

l

EIK

w

e , 2

12

lGA

EI

s

=Φ (2.4)

where,

l = Element length [m]

G = Shear modulus [N/ m2]

As = Shear-resisting area [m2]

Ф = Correction factor for the shear influence [-]

EG)1(2

1

ν+= [N/ m

2] , AAs κ= [m

2] (2.5)

where,

ν = Poisson number [-]

κ = Timoshenko shear coefficient [-]

ν

νκ

56

55

+

+= [-] (2.6)

For rectangular beams equation (2.6) gives a good approximation of the shear

coefficient. [3]

The mass matrix of the Timoshenko beam element is identical to the mass matrix of

the Euler beam element.


3

3

2/

2/

3

2/

2/ 0

2

2

23

2

12

1

3

12

−−=

==

=

−−∫ ∫

∫

w

H

hH

H

hH

W

w

hH

WWH

Wydxdyy

dAyI

ww

hw x

y W

Figure 2.7: Cross section weld

H

hw

W

2.2. Modeling beam B

Modeling beam B is a bit more complicated due to the weld halfway the beam. Three

different strategies for modeling beam B are compared.

2.2.1. Strategy 1 (model B1)

Model B1 describes the geometric properties of the weld accurately using physical

modeling. Therefore a more detailed description of the weld is needed. Figure 2.5

shows the pictures of the weld in the beam that is used for the experiments in chapter

3. This weld is very unusual, and will not often be used in practice. The parts are only

welded together at the top and the bottom, leaving a pocket in the middle. A little

spacing between the two parts prevents them from rubbing against each other,

avoiding non-linear effects such as dry friction and contact. How this is modeled is

shown in figure 2.6.

Figure 2.5: Picture of the weld front view

Figure 2.6: Model of the weld front view cross-section A-A’

For the simulation of the weld a Timoshenko beam element is inserted, because the

influence of shear stress is important in this element. This Timoshenko beam element

has different parameters than the ones used for beam A strategy 2. The length of this

element is the length of the weld lw, and the cross section ww WhA 2= [m2]

The second moment of inertia is derived as follows:

(2.7)

hw

e

lw

A’

A


The weld is produced using Tungsten Inert Gas welding (TIG-welding). The additive

material used is steel P15. The material properties are practically the same as the steel

used for the beams.

The mass matrix of the Timoshenko beam element is identical to the mass matrix of

the Euler beam element, except with a different cross section (Aw) and element-

length (lw).

A summary of the weld properties is given in table 2.2.

Table 2.2: Weld properties

Geometric properties

hw Height weld ± 0.5 [mm]

lw Length weld ± 1.5 [mm]

Material properties

E Young’s modulus 11101.2 ⋅ [n/m2]

ν Poisson’s ratio 0.3 [-]

ρ Density 7850 [kg/m3]


E2, ρ2

lw


In model B2, the beam is divided into two parts; an extra node is inserted at the

position of the weld. A transversal spring and a rotational spring connect the two

parts. These springs simulate the weld.

Figure 2.8: numerical model for beam B, strategy 2.

The stiffness matrix of the inserted element is:

=

−

−

−

−

⇒=

+

+

+

+

1

1

1

1

00

00

00

00

e

e

e

e

e

e

e

e

RR

TT

RR

TT

eee

M

F

M

F

u

u

kk

kk

kk

kk

fuK

ϕ

ϕ (2.8)

where ue is the deflection and φe the rotation of node e, Fe is the force and Me the

moment acting on node e. The transversal spring stiffness kT and the rotational spring

stiffness kR are identified experimentally in section 4.3.2.

The mass matrix is modeled as a null matrix of size (4 x 4) because the extra element

is assumed to be mass less.

Because of the extra element, model B2 has 2 extra dof’s. The size of the system

matrices after partitioning is (202 x 202). The elongation of the beam due to the weld

(lw) is not taken into account in model B2.


In model B3 the element at the position of the weld has a different young’s modulus

E2 and/or a different density ρ2. The exact values of these different parameters depend

on the material and geometric properties of the weld. They are identified

experimentally in section 4.3.2.

The length of this element is lw, the length of the weld.

Figure 2.9: numerical model for beam B, strategy 3.

kT kR

Node e Node e+1


2.3. Comparison of numerical models, reduction and errormatrices

In this section the numerical models will be compared. Most attention will be given to

the comparison of models A1 and B1. First the eigenfrequencies and eigenmodes of

the clamped-free beam will be compared in section 2.3.1. Next, the frequency

response functions will be compared in section 2.3.2. The reduction of the number of

degrees of freedom is explained in section 2.3.3, and the differences between the

numerical models are investigated using System Error Matrices in section 2.3.4.

2.3.1. Eigenfrequencies and eigenmodes

The system is weakly damped (as will be seen in section 3.3.1.), therefore it may be

approximated with the undamped differential equations of motion (see also [1]):

0)()( =+ tKqtqM && (2.9)

Next, the eigenfrequencies and eigenmodes of the FE-models are determined by

solving the corresponding eigenvalue problem:

[ ] 02 =− rr MK ϕω (2.10)

where,

rr fπω 2= = Angular eigenfrequency of mode r [rad/s]

fr = Eigenfrequency of mode r [Hz]

φr = Eigenmode of mode r [-]

The eigenfrequencies and mode-shapes of a clamped-free beam can be found

analytically for verification purposes. Note that the analytical solutions are based on

bending only (no shear). [6]

M

EI

L

rr 2

2λω = ,

−−−=

L

x

L

x

L

x

L

x rrr

rrr

λλσ

λλϕ sinsinhcoscosh (2.11)

where, Table 2.3: Parameters λr and σr

mode r λr σr

1 1.87510407 0.734095514

2 4.69409113 1.018467319

3 7.85475744 0.999224497

4 10.99554073 1.000033553

5 14.13716839 0.999998550

Table 2.4 shows the first 5 eigenfrequencies ( πω 2/r ) of the models, and the

analytical solution.


Table 2.4: Numerical eigenfrequencies

Mode Analytic [Hz] Beam A

Strategy 1 [Hz]

Beam B,

Strategy 1 [Hz]

Difference A1, B1

(B1-A1)/A1

1 33.421 33.285 33.039 -0.73 %

2 209.44 208.82 206.51 -1.11 %

3 586.45 585.20 581.61 -0.61 %

4 1149.2 1147.6 1135.0 -1.09 %

5 1899.7 1898.1 1885.4 -0.67 %

So the differences in the eigenfrequencies between the models are very small.

As expected, the eigenfrequencies of model B1 are smaller than the eigenfrequencies

of model A1 due to incorporation of shear in model B1.

Note that the eigenfrequencies of model A1 are smaller than the analytical values.

This is opposing to the Rayleigh quotient iteration which converges downwards to

determine the numerical eigenfrequencies. This is due to the extra masses of the

sensors that are included in the numerical models

The differences in the mode-shapes are also very small: in appendix A.3 the first 5

mode-shapes are plotted. The graphs of the different models almost coincide.

The modal assurance criterion (MAC) is a method to compare the mode-shapes in a

quantitative manner.

It gives the degree of correlation between φie (experimental mode i) and φj

n (numerical

mode i). Here, the method is used to compare two numerical models.

( )( )n

j

nH

j

e

i

eH

i

n

j

eH

ijiMAC

ϕϕϕϕ

ϕϕ2

],[ = (2.12)

where eH

iϕ is the Hermitian transpose of e

iϕ . The MAC matrices are normalized so

that all the values are between 0 and 1. The MAC matrix for the correlation between

model A1 and B1 is shown in Figure 2.10.

Figure 2.10: Graphical representation MAC matrix

Table 2.5: MAC matrix

Model B1 mode nr. Model A1

mode nr. 1 2 3 4 5

1

2

3

4

5

1.0000 0.0032 0.0032 0.0025 0.0018

0.0036 1.0000 0.0021 0.0021 0.0017

0.0028 0.0022 0.9999 0.0013 0.0012

0.0024 0.0015 0.0014 0.9998 0.0003

0.0021 0.0014 0.0007 0.0003 0.9997

The diagonal of the MAC matrix is representing the correlation between the

corresponding mode-shapes. These diagonal values are all nearly 1 and the off

diagonal values are almost zero. So the correlation between the numerical modes is

very good.

The MAC matrices for some other models are displayed in appendix A.1.


2.3.2. Frequency response functions

The eigencolumns are mass-matrix normalized, by selecting all mass parameters mr

equal to 1:

1== r

T

rr Mm ϕϕ , r=1,2, … , n (2.13)

Implying that:

2

rr

T

rr Kk ωϕϕ == , r=1,2, … , n (2.14)

Next, the frequency response functions (frf’s) of the beams are calculated.

The transfer function matrix for a proportionally and under critically damped multi

degree of freedom linear system is:

(see also [1])

( )[ ] [ ]

( )∑= Ω+Ω−

=Ωn

r rrrr

rrlk

jm

lkH

122,

2 ωξω

ϕϕ (2.15)

where,

[ ]kr

ϕ = the kth

element of the eigencolumn φr.

This is the degree of freedom that is measured.

[ ]lr

ϕ = the lth

element of the eigencolumn φr.

This is the degree of freedom which is excited.

rr fπω 2= = the rth

angular eigenfrequency.

fπ2=Ω = the evaluated angular frequency

ξr = the rth

dimensionless damping-factor

Equation (2.15) is evaluated on a frequency range ]2000,10[∈f Hz, with a resolution

of 0.5 Hz. The damping factors ξr are low, therefore the system is weakly damped.

They are determined experimentally (see table 3.2).

( )ΩlkH , gives the frf’s in displacement per force [m/N]. But to compare them with the

experimental frf’s from chapter 3, ( )ΩlkH , has to be multiplied with –Ω2, to obtain

the frf’s in acceleration per force [(m/s2)/N].

The 8 transfer functions, which will be measured in chapter 3, are evaluated for the

numerical models A1 and B1 in appendix A.4. They are displayed in decibel:

( )ForceonAcceleratidB /10log20 ⋅= (2.16)

So the value 0 dB corresponds with 1 ( )

N

sm2/

.


2.3.3. Reducing the number of degrees of freedom

When two models are compared it is often necessary for them to have the same

number of degrees of freedom. But this number is not always the same between two

models.

For example the models A1, A2 and B3 have 200 dof’s, but B1 and B2 have 202 dof’s

because of the extra inserted node. The experimental models in chapter 3 only have 8

degrees of freedom. In these cases, the model with the most dof’s has to be reduced.

This is done in two different ways: with the static reduction or with the dynamic

reduction. The reduction of the stiffness matrix K is demonstrated. The reduction for

M is done analogously.

The unreduced stiffness matrix K has n degrees of freedom. So n is 200 or 202 in the

numerical models determined in the section 2.1 and 2.2. The target is to reduce this

number to m degrees of freedom. In chapter 3, the experimental models have 8

degrees of freedom (so m is 8). These m degrees of freedom have to correspond with

the measured displacements in the experimental model. The relevant dof-numbers in

the numerical models A1, A2 and B3 are 23, 47, 71, 95, 119, 143, 167 and 191. (See

figure 2.) And the relevant dof-numbers are 23, 47, 71, 95, 121, 145, 169 and 193 in

the models B1 and B2 because there an extra node is inserted in the center.

),(),( mmKnnK red⇒ nm < (2.17)

The unreduced displacement field )1,(nu , is partitioned:

=

o

g

u

uu ,

− )1,(

)1,(

mn

m (2.18)

where ug contains the relevant displacements and uo contains the rest.

K and M are partitioned analogously:

=

ooog

gogg

KK

KKK ,

−−−

−

),(),(

),(),(

mnmnmmn

mnmmm (2.19)

Static reduction

This method is also referred to as the Guyan reduction. Assuming that dof’s u0 are not

loaded, the following static relation can be written:

=

O

FT

KK

KKa

s

ooog

gogg (2.20)

where Fa (m,m) consists of m linearly independent load cases, Ts (n,m) is the static

reduction transformation matrix that has to be determined and O is an (n-m,m) null

matrix. Ts is constructed as follows:

Φ=

s

gg

s

IT (2.21)


where Igg is an (m,m) identitiy matrix, and Φs is yet to be determined.

Substituting equation (2.21) in equation (2.20) yields:

ogoos

sooggog

KK

OKIK

1−−=Φ⇒

=Φ+ (2.22)

The equation transforms into the reduced equation as follows:

gg

red

S fuKfuK =⇒= (2.23)

The transformation matrix TS is used to reduce u to ug, f to fg , K to red

SK and M to red

SM .

gS uTu = (2.24) S

T

S

red

S KTTK = (2.26)

fTfT

Sg= (2.25) S

T

S

red

S MTTM = (2.27)

A disadvantage of this method reveals itself when comparing the solutions of the

unreduced eigenvalue problem:

[ ] 02 =− rr MK ϕω (2.28)

And the reduced eigenvalue problem:

[ ] 02 =− r

red

Sr

red

S MK ϕω (2.29)

The solutions of the reduced model become inaccurate for higher modes. The

resulting eigenfrequencies of the unreduced and the reduced model (model A1) are

compared in table 2.6.

Table 2.6: Eigenfrequencies of the unreduced and static reduced model

Mode Unreduced

model A1 [Hz]

Static reduced

model A1 [Hz]

Error

1 33.285 33.285 0.00027 %

2 208.82 208.84 0.0070 %

3 585.20 585.57 0.063 %

4 1147.6 1151.0 0.30 %

5 1898.1 1918.1 1.05 %

6 2836.6 2923.3 3.06 %

7 3962.9 4240.2 7.00 %

8 5276.8 5769.1 9.33 %

As stated before, the error in the eigenfrequencies grows fast for higher

eigenfrequencies. The 8th

mode has an error of over 9%. So applying a static reduction

changes the eigenfrequencies and eigenmodes unacceptably in the high frequency

range.


Dynamic reduction

An alternative to the static reduction is the dynamic reduction method. Proceeding

with the sorted K and M from equation (2.19) and solving the eigenvalue problem

yields the eigencolumns φr:

[ ] 02 =− rr MK ϕω (2.30)

DT~

contains the first m of these eigencolumns:

],...,[~

1 mDT ϕϕ= ),(~

mnTD (2.31)

The displacement field u(n,1) is approximated by a linear combination of these m

eigencolumns:

pTu D

~= (2.32)

where p (m,1) contains the m generalized dof corresponding to these eigencolumns.

u and DT~

are partitioned as follows:

pT

T

u

u

Do

Dg

o

g

=

),(

),(

mmnT

mmT

Do

Dg

− (2.33)

The reduced model could be described in terms of p, but to enable direct comparison

with measured responses (dof ug) the following regular transformation is carried out:

gDg uTp1−= (2.34)

Substituting (2.34) in equation (2.33) gives:

g

DgDo

gg

o

gu

TT

I

u

u

=

−1 (2.35)

gD

o

guT

u

u=

(2.36)

This TD is the transformation matrix for the dynamic reduction.

The dynamically reduced equations of motion are given by:

gg

red

Dg

red

D fuKuM =+&& (2.37)

where,

D

T

D

red

D KTTK = , D

T

D

red

D MTTM = , fTfT

Dg= (2.38)

The dynamic reduction is preferred because in contrary to the static reduction the

lowest m eigenvalues and eigenmodes of the system are exactly preserved.


2.3.4. System Error Matrices

System Error Matrices [2] can be used to find those parts in the finite element model

which are responsible for the differences between the numerical and experimental

data. This will be applied later in section 4.3.

They can also be used to investigate the differences between two numerical models,

which will be discussed in this section. First, the theory behind system error matrices

will be given.

Theory

An error matrix is the difference between two system matrices. The case for the

stiffness is demonstrated here.

NE KKK −=∆ (2.39)

where KE is the stiffness matrix from the experimental ‘model’, and K

N is the stiffness

matrix from the numerical model.

Usually KE is unknown. But the eigenmodes and eigenfrequencies of the experimental

model can be determined, as will be explained in section 3.2.4. The (pseudo-) inverse

of KE may be constructed using the measured modes and eigenfrequencies. In practice

we have e measured modes Une for n measured dof with the corresponding e measured

eigenfrequencies Ωee.

( ) ( ) TE

ne

E

ee

E

ne

EUUK

21 −−Ω≈ (2.40)

( ) ( ) TN

ne

N

ee

N

ne

NUUK

21 −−Ω≈ (2.41)

These inverses are used in the 1st order Taylor approximation for the error matrix:

( ) ( ) NENN KKKKK11 −−

−≈∆ (2.42)

Whether this approach is valid is checked with the eigenvalue magnitudes of

( ) KK N ∆−1

which matrix (raised to higher powers) appears in the higher order terms

of the Taylor approximation of K∆ . The maximum of these eigenvalue magnitudes is

called X.

( )( )KKeigX N ∆=−1

max (2.43)

When X is smaller than 1, (then all of the eigenvalues are smaller than 1), the higher

order terms of the Taylor approximation may be neglected.

1<X (2.44)


Determining the error matrix for the mass matrix is done analogously to the approach

given above.

Only the mass matrix (pseudo-) inverses are determined differently:

( ) TE

ne

E

ne

EUUM ≈

−1 (2.45)

( ) TN

ne

N

ne

NUUM ≈

−1 (2.46)

The approximation for the mass error matrix becomes:

( ) ( ) NENN MMMMM11 −−

−≈∆ (2.47)

Method validation

First a simple test is done to check the method: Model A1 is compared with model B3.

The only difference between these models is the element halfway the beam. This test

is found in appendix B.1. The error matrices succeed in pointing out the element in the

center.

In general, not all the modes have to be used to calculate the (pseudo-) inverses

(equation (2.40)). This is an advantage of the method, because in practice only a

limited amount of experimental data is present. The influence on the number of modes

used is shown in appendix B.2. Also the influence of the dynamic reduction on the

error matrices is investigated there.

What is remarkable is that when more modes are used, X increases. This phenomenon

is examined in appendix B.3. A first observation of this fact could be that the first

order approximation of the error matrix doesn’t improve when more modes are used.

In appendices B.4 and B.5 the stiffness error matrices and the mass error matrices are

compared for two cases:

• Only a difference in the stiffness matrices. The center element has a different

E-modulus in the models. The mass error matrix should be zero.

• Only a difference in the mass matrices. The center element has a different

density.

The stiffness error matrix should be zero.

The reduction of the stiffness matrix also causes errors in the mass matrix. Therefore,

errors in the stiffness matrix are also distinguished in the mass error matrix. Also the

limited number of modes used is causing errors.

Modeling the mass is generally easier than modeling the stiffness. So it’s better to use

the stiffness error matrix for K, and neglect the differences in M.

This method of errormatrices gives an indication of where to search for modeling

errors in the numerical model. After the modeling errors are located, more advanced

model updating strategies could be used, like FEMtools. (see also [7]). These updating

strategies were not taken into account in this report.


Obviously, finding the error in this model is relatively easy. Therefore it was a

convenient way for checking the procedure. The approach of error matrices works for

this simple structure, so the approach should also work for more complex structures.

But for complex structures it becomes more important to have a lot of degrees of

freedom. So many measurements have to be made.

Comparison of numerical models

The error matrices for the comparison between the numerical models A1 and B1 are

displayed in figure 2.11. They are all normalized; the maximum value (corresponding

to 1) is displayed in the captions.

(A) ∆K, using dynamic reduction, 1~4.59·10

6 (B) ∆M, using dynamic reduction, 1~5.57·10

-3

(C) ∆K, using static reduction, 1~3.14·106 (D) ∆M, using static reduction, 1~4.52·10

-3

Figure 2.11: Error matrices numerical models

Both dynamic and static reduction is investigated; the results are very similar. The

highest peak is at the 4th

dof-nr which corresponds with the weld position. Both the

stiffness and the mass are different in models A1 and B1. So the error matrices succeed

in pointing out the position that is modeled differently. In appendix A.2 more error

matrices for the numerical models are shown.


Figure 3.1: Beam in the clamp with sensor

1 2 3 4 5 6 7

60 mm

L=500 mm

H=10 mm

W=60 mm 20 mm

8

3. Experimental analysis

This chapter discusses the experiments that are made using Impact Hammer modal

testing: The hammer excitation force and the resulting acceleration response are

measured.

3.1. Experimental set-up

One side of the beam is clamped between two metal blocks

with 8 bolts and attached to the wall, as can be seen in

figure 3.1.

The beam is oriented in vertical direction so the gravity has

no influence.

The acceleration-sensor that measures the response is

attached (using wax) at 20 mm from the free end of the

beam;

This position corresponds with the measurement point 8 of

the experimental model (see figure 3.2).

The sensor is positioned in the center of the width of the

beam, to minimize the torsional influence. For investigating

torsional modes, 2 extra sensors can be attached closer to

the sides.

The masses of the sensors are included in the FE-models.

Figure 3.2: Measurement points for the experimental model


Figure 3.4: Impact hammer and accelerometer

Figure 3.5: The signal analyzer Siglab

and the accompanying software

on the laptop

3.2. Experimental modal analysis

The beam is excited with the hammer at the 8 points that are

shown in figure 3.2. Each time the response in point 8 is

measured (figure 3.3). So this results in 8 transfer functions.

With these functions the eigenvalues and mode-shapes of the

system are determined.

The devices used are the impact hammer (PCB 86B03) and the

acceleration sensor (KISTLER 8732A500), see figure 3.4. They are

both connected to the signal analyzer Siglab (see also [5]) shown

in figure 3.5.

The hammer is connected to channel 1, and the accelerometer is

connected to channel 2.

Siglab acquires the time signals and sends them to the laptop

where the time signals are translated into the frequency response

functions, using Fast Fourier transformation.

Figure 3.3: Excitation of the beam


Every measurement has errors, caused by measurement noise and not hitting the beam

at exactly the same point and in exactly the same direction every time. Therefore, for

each frf 5 measurements are averaged. In this way the errors are cancelled out as

much as possible.

The quality of the averaged frequency response functions is expressed using the

coherence function. This function shows what part of the output is coming from the

real input, and what part is coming from the measurement noise. When there is not

much influence of the noise the value of the coherence function is close to one. When

the signal is dominated by noise, the value is close to zero.

The input signal for Siglab is in voltages. Scaling factors have to be used to convert

the voltages into the desired accelerations and forces. In fact, these scaling factors are

the sensitivities of the sensors.

The force sensor in the hammer has a sensitivity of 3103.2 −⋅

N

V,

and the accelerometer has a sensitivity of 920

V

sm2/

.

Using these sensitivities, the scaling factor becomes 2.12 [-]


3.2.1. Siglab settings

There are some settings for Siglab that are worth mentioning. They are listed in table

3.1. A brief explanation of these settings is given.

Table 3.1: Siglab settings

Channel 1 and channel 2 setup

Input voltage 2.5 V

Voltage type Bias

Frequency range

Bandwidth 0-2000 Hz

Record length 4096

Anti-aliasing filters On

Trigger

Channel 1 9 %

Delay -5 %

Processing

Stop at count 5

Modal parameters

Double hit amplitude 50.0%

Double hit delay 20.0 %

Force window size 10.0 %

Exponential window decay 5.0 %

The input range for the channels is set to 2.5 V; this is the lowest range where there

generally won’t be an overload. The type of voltage is set to Bias because the

accelerometer and the hammer have to receive their power supply from Siglab.

The bandwidth defines the measured frequency range. The interesting frequency

range is chosen to be from 10 to 2000 Hz, because the interesting eigenmodes (the

first 5) are in this domain. Also, the hammer is suitable to excite frequencies up to

20.000 Hz.

The record length is the number of points to be measured in this frequency range.

The bandwidth along with the record length determines the frequency resolution ∆f:

∆f = 2.56 x Bandwidth / Record length = 1.25 [Hz]

This means that the measuring time is 1 / 1.25 = 0.8 [s].

The anti-aliasing filters minimize the distortion artifacts caused by aliasing. This

aliasing comes into effect because a high-frequency signal is represented at a lower

frequency. The filters remove the signal components above the folding frequency,

before the signal is represented at a lower resolution.

The trigger is set on channel 1, on which the hammer is connected. When 9% of the

input range is exceeded the signal is recorded.

The trigger-delay starts this recording 5% of the measuring time earlier. This ensures

that the signal is captured at the beginning of the impulse.

The stop at count 5 ensures that each time 5 measurements are averaged in order to

get more accurate results.

Hzf 25.1=∆


3.2.2. Applied windows

A combination of windows has to be used in Siglab.

The system is weakly damped, so the response signal is not sufficiently damped to

avoid signal leakage at the end of the measuring time that is only 0.8 seconds. This is

easily noticed because when the beam is excited, it resounds for a couple of seconds.

Therefore an exponential window has to be used. This exponential window multiplies

the acceleration-signal with an exponential function, that is one at t=0 [s] and 0.05 at

t=0.08 [s]. The resulting signal is sufficiently damped within the measuring time.

Now the signal can be seen as periodic with the period time equal to the measuring

time.

In figure 3.6 a demonstration of the effect of the exponential window is shown.

It is applied to a weakly damped response signal which represents the response signal

from the accelerometer. The amplitude of this signal is still large at the end of the

measuring time. Multiplying this signal with the exponential function results in a

sufficiently damped signal.

The force window multiplies the hammer-signal with a function that is one for the first

10% of the measured time, and zero for the rest of the time. This is because the time

of the hammer impulse is small compared with the measuring time. The signal that

follows the hammer impulse is a noise signal that has to be filtered.

Figure 3.7 shows a demonstration of this window on an arbitrary function.

Figure 3.6: Exponential window Figure 3.7: Force window

The settings for the double hit remove a second hit when the hammer accidentally hits

the beam twice. The percentage is specified as a percentage of the first hit.

These settings for the windows where chosen because they yielded the best results,

with as little noise as possible.

0.8 t

05 . 0 x(t) =

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Exponential w indow

Measuring time t

x(t

)

Exponential window

Response signal

Damped signal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Force window

Measuring time t

x(t

)

Force window

Response signal

Adapted signal


3.2.3. Experimental modal analysis

Determining the experimental eigenvalues and mode-shapes of the system is done

using the package ME’scope (see also [4]). The measured frf’s are imported from

Siglab and then a threshold has to be set. This threshold is the value of the smallest

detectable mode. So the modes whose magnitudes in the frf are above this threshold

are selected.

Figure 3.8: Selecting the modes

Figure 3.8 shows the procedure of selecting the modes in ME’scope. Hence 5 modes

are selected in the frequency domain of 10-2000 Hz. Now a modal parameter fit

procedure is applied (see also [4]).

Since the beam system is lightly damped the frf’s of the proportionally damped

system may be used:

( ) ( )∑= Ω+Ω−

=Ωn

r OrrOrr

T

OrOr

jmH

122 2 ωξω

ϕϕ (3.1)

Here, Orω is the real undamped angular eigenfrequency and rξ is the dimensionless

damping coefficient. The eigenvalues λr which are estimated in the fit procedure are

represented by:

rrr jωµλ += (3.2)

Here, rµ is the real part of the eigenvalue and rω is the imaginary part representing

the damped angular frequency.

In our case of light damping equation (3.1) forms a good approximation for the real

frf (so proportional damping is assumed). In this case, Orω and rξ can be calculated

from equation (3.2) as follows:

22

rrOr ωµω += , Or

rr ω

µξ

−= (3.3)

Threshold


Normalizing the eigenmodes

The experimental (complex) eigenmodes are listed in appendix C.3, for both beam A

and beam B.

Each calculated eigenmode rϕ consists of i elements. Each element has magnitudes

( )iMϕ and accompanying phase angles ( )

iPϕ .

This is represented by a complex number:

( ) ( )iPj

iMi eϕϕϕ −

= (3.4)

The phase angle is practically the same at every point, except for some 180 degree

shifts. This was expected because this system is weakly damped.

Each eigenmode is divided by its kth

element with the highest absolute magnitude, to

normalize the eigenmodes. Due to weak damping, the remaining imaginary parts are

very small compared to the real parts, so the imaginary parts are neglected:

( ) ( )

( ) ( )

=

−

−

kP

iP

j

kM

j

iM

ie

eϕ

ϕ

ϕ

ϕϕ Re~ (3.5)

Obviously the kth

element of the normalized mode will have the largest magnitude

which is equal to 1.

The normalized eigenmodes are also listed in appendix C.3, and are plotted along with

the analytical solution.


0 10 20 30 40 50 60 70 80 90 100-1

0

1

dof-nr

Experimental model A Experimental model B Analytical

3.3. Comparison of the experiments

The experimental results of beam A and beam B are compared in this section.

3.3.1. Eigenfrequencies and mode-shapes

The eigenfrequencies and the dampingfactors that were determined using ME’scope

are listed in table 3.2.

Table 3.2: Experimental eigenfrequencies and damping factors

Beam A Beam B

Mode

Frequency

π

ω

2

kkf = [Hz]

Damping

kξ [%]

Frequency

π

ω

2

kkf = [Hz]

Damping

kξ [%]

Frequency

difference

A

AB − [%]

1 31.6 1.979 31.0 1.977 -1.65

2 197.7 0.340 193.5 0.345 -2.11

3 551.6 0.160 542.9 0.151 -1.58

4 1078.8 0.370 1057.4 0.153 -1.98

5 1775.0 0.208 1748.0 0.189 -1.52

The eigenfrequencies from the beams only differ about 2%. As expected the

eigenfrequencies of beam B are lower (increased flexibility due to the weld and

increased length). The damping is very low, mostly less than 1%. These experimental

dampingfactors are used in the FE-models, so that there won’t be a difference in the

damping levels.

The comparison of the 5th

normalized mode-shape is displayed as an example in

figure 3.9, the plots of the other mode-shapes are displayed in appendix C.3.

Figure 3.9: Comparison eigenmode no. 5

(The normalized experimental mode is multiplied by -1

in order to match with the analytical mode shape.)

Both graphs coincide, so the mode-shapes match very well.


The correlation between the experimental mode-shapes is analyzed using a MAC

matrix:

Figure 3.10: Graphical representation MAC matrix

Table 3.3: Mac matrix

Model B mode nr. Model A

mode nr. 1 2 3 4 5

1

2

3

4

5

0.9987 0.0011 0.0022 0.0028 0.0025

0.0026 0.9994 0.0009 0.0022 0.0018

0.0007 0.0012 0.9981 0.0013 0.0014

0.0012 0.0009 0.0009 0.9968 0.0013

0.0014 0.0001 0.0003 0.0031 0.9932

The diagonal of the MAC matrix is representing the correlation between the

corresponding mode-shapes. These diagonal values are all nearly 1 and the off

diagonal values are almost zero. So the correlation between the experimental modes is

very good.

From these results can be concluded that the dynamic behavior of both beams is very

similar despite the presence of the weld in beam B. This was already noticed because

the beams sounded alike when they were excited.


0 500 1000 1500 2000-40

-20

0

20

40

|H1,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 500 1000 1500 2000-200

0

200

∠∠∠∠ |H1,8

|

Frequency [Hz]

An

gle

[0]

0 500 1000 1500 20000

0.5

1

Frequency [Hz]

Co

he

ren

ce

Experimental model A

Experimental model B

3.3.2. Frequency response functions

The plots of the experimental FRF’s are shown in appendix C.4. H1,8 is shown in

figure 3.11 as an example.

Figure 3.11:H1,8 for beams A and B with corresponding coherence

The magnitude shows the 5 resonance peaks at the 5 eigenfrequencies. The phase

shows a 180 degree shift at these eigenfrequencies.

Also 2 torsional modes are present in this frequency domain; the first torsional mode

at 505 Hz, and the second at 1529 Hz. They are hard to distinguish in the plot because

the averaging canceled them out. The torsional modes are further analyzed in

appendix C.5.

The coherence is near one for frequencies above 200 Hz. Therefore the measurement

results are reliable for these high frequencies. But for frequencies below 200 Hz, the

coherence shows that the signal is distorted. Therefore the resonance peak for the 1st

mode (31 Hz) is hard to recognize from these plots. The modal parameter fit

procedure from ME’scope still was able to locate this first mode.


0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

dof-nr

FE-model B1 Experimental model B Analytical

4. Comparison of numerical and experimental data

This chapter treats the differences in the numerical and experimental data.

4.1. Eigenfrequencies and mode-shapes

A comparison of the numerical and experimental eigenfrequencies is shown in table

4.1.

Table 4.1: Comparison of the eigenfrequencies

Numerical Experimental Differences

Mode Beam A1

[Hz]

Beam B1

[Hz]

Beam A

[Hz]

Beam B

[Hz]

Beam A

[%]

Beam B

[%]

1 33.285 33.039 31.6 31.0 -5.20 -6.07

2 208.82 206.51 197.7 193.5 -5.35 -6.31

3 585.82 581.61 551.6 542.9 -5.74 -6.67

4 1147.6 1135 1078.8 1057.4 -6.00 -6.84

5 1898.1 1885.4 1775.0 1748.0 -6.48 -7.29

The differences of about 6% are quite large; this is probably impute to wrong

assumption that the clamp is infinitely stiff in the numerical models. From the results

presented in table 4.1 it can also be suspected that the finite stiffness of the clamping

has a larger influence on the dynamic behavior than the weld. In fact this will also

become clear from section 4.3 where error matrices will be discussed.

The experimental mode-shapes are shown in appendix C.3. The 5th

experimental

mode-shape of beam B and the 5th

numerical mode-shape of model B1 are displayed

in figure 4.1 as an example. The experimental mode-shape is only described in 9

points (1 at the clamp, and 8 measured points) while the numerical mode-shape has

100 displacement dofs. But the shapes match quite well.

Figure 4.1: Comparison eigenmode no. 5


4.2. Frequency response functions

The experimental and numerical frequency response functions for beam A and beam

B are displayed in appendix C.4. H1,8 is shown in figure 4.2 as an example.

Figure 4.2:H1,8 for beams A and B

The resonance peaks of the numerical models are shifted a little in comparison with

the resonance peaks of the experimental models. This is caused by the 6% difference

in the eigenfrequencies.

Also, the peaks of the numerical models are less sharp, which indicates higher

damping. However the damping coefficients used in the FE-models where

experimentally determined (see table 3.2).

0 500 1000 1500 2000-40

-20

0

20

40

|H1,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 500 1000 1500 2000-200

-100

0

100

200

∠∠∠∠ |H1,8

|

Frequency [Hz]

An

gle

[0]

Experimental model AExperimental model BFE-model A1

FE-model B1


4.3. System error matrices

It this section the numerical and experimental models are compared using error

matrices.

4.3.1. System error matrices model A

First model A1 is compared with the experimental model B (see figure 4.3).

The result is expected to show a peak at the 4th

dof-nr because that is the position of

the weld that is not modeled.

(A) Using dynamic reduction, 1~7.90·10

7 (B) Using static reduction, 1~6.10·10

7

Figure 4.3: Error matrix FE-model A1 – Experimental model B.

However the largest error is at the 1st dof. This is caused by the clamp that is modeled

as rigid. Apparently, this assumption is not entirely valid.

However, the second largest peak is indeed at the 4th

dof. This peak is a little higher

when using dynamic reduction compared to using the static reduction.

These results confirm the conjecture mentioned in subsection 4.1.


4.3.2. System error matrices model B

The peak at the 4th

dof can be reduced by taking the weld into account in the model.

So now the models B1, B2, and B3 that were developed in section 2.2 are compared

with the experimental model B.

In figure 4.4 this comparison is made for model B1.

(A) Using dynamic reduction, 1~7.81·10

7 (B) Using static reduction, 1~6.11·10

7

Figure 4.4: Error matrix FE-model B1 – Experimental model B.

The highest peak is still at the 1st dof because the model didn’t change at that position.

But indeed the peaks near dof 4 are reduced, so model B1 that takes the weld into

account represents the experimental model better.

The models B2 and B3 also reduce the peaks in the center. These error matrices are

displayed in appendix C.2. The following parameters were used:

Model B2: Transversal spring stiffness kT=1019

[N/m]

Rotational spring stiffness kR=104

[Nm/rad]

Model B3: Different young’s modulus E2=0.1E.

Different density ρ2=0.1ρ.

These parameters were chosen because they best reduced the peaks in the center.

In subsection 2.3.1 the differences between the numerical models A1 and B1 were

discussed. Their eigenfrequencies only differed about 1% and also the MAC matrices

showed a very good correlation between the eigenmodes.

So the dynamic behavior of the two beams A and B is very similar, which was

confirmed experimentally in table 3.2. The errors between models A and B introduced

by the weld are small compared with the errors induced by the clamp.


The magnitude of the modeling errors can be investigated by comparing the values of

the error matrix with a stiffness matrix.

In figure 4.5 the values of the error matrix between A1 and B1 are compared with the

corresponding values of the stiffness matrix A1, and expressed in percentages.

(A) Applied on stiffness matrix (B) Applied on mass matrix

Figure 4.5: Magnitude of the error matrix

These results show that the error due to not modeling the weld is only about 10

percent of the stiffness.


5. Conclusions and recommendations

The error matrices are able to locate the differences between a numerical and an

experimental ‘model’, two numerical models and two experimental ‘models’.

A number of facts may hamper the interpretation of the error matrices, like the limited

number of modes that is used, the first order approximation that is used, the number of

degrees of freedom that has to be reduced in case of an experimental ‘model’,

measurement noise, not modeled physics such as torsional modes, etc.

The weld does not have big implications on the dynamic behavior of the beam that is

used for the experiments in this investigation. This can already be seen from the

similarity of the numerical models; the eigenfrequencies hardly differ, the MAC

matrices show a very good correlation between the corresponding eigenmodes and the

frequency response functions almost coincide. This is confirmed by the good

correspondence between the dynamic characteristics (eigenfrequencies and

eigenmodes) of the beam with and the beam without weld resulting from the

experimental modal analyses. In addition the beams sound very alike when they are

excited. In conclusion: their dynamic behavior is very similar.

The largest difference between the numerical model for beam A and the test data from

beam B appeared to be not caused by the weld, but by the clamp that is modeled as

rigid in numerical model A.

In spite of the dominating errors due to the clamp model, the errors caused by the

weld are recognized by the error matrices. The latter errors almost disappear when the

weld is taken into account in the numerical model. Obviously, finding errors in this

simple beam model is relatively easy. Therefore it is a suitable way to validate the

procedure. The error matrices work for this simple structure. But for more complex

structures, much more measurement points are probably needed. The structure has to

contain enough degrees of freedom to point out the exact error locations.

The method of error matrices gives an indication where to search for modeling errors

in the numerical model. Once the locations of the modeling errors in the FE model are

known, more advanced model updating strategies, like FEMtools, can be applied to

further improve the numerical model.

The method of error matrices for localizing model errors in a FE model using

experimental data could be further verified using the same beam models (with and

without weld) as used in the current study, but with different boundary conditions.

Instead of investigating the clamped-free beam, it is probably better to study the free-

free beam (hanging in soft springs), so that the modelling errors of the weld become

dominant.

1

Bibliography

[1] Bram de Kraker, Dick H. van Campen, Mechanical Vibrations, Shaker

Publishing BV, April 2001.

[2] J. Sidhu and D.J. Ewins, Correlation of finite element and modal test studies

of a practical structure, 2nd

International Modal Analysis Conference, Florida,

1984.

[3] T. Kaneko, An experimental study of the Timoshenko's shear coefficient for

flexurally vibrating beams. D: Appl. Phys., Vol. 11, 1978.

[4] ME’scopeVES Application Note #28. Vibrant Technology, Inc., Scotts Valley,

CA, USA, 2005.

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

[6] R.D. Blevins, Formulas for natural frequency and modeshape. Van Nostrand

Reinhold Company, 1979.

[7] C.S. Kraaij, Model updating of a ‘clamped’-free beam system using FEMtools.

Eindhoven University of Technology, DCT 2006.128, the Netherlands,

January 2007.

A-1

APPENDIX

A. Numerical analysis

A.1. MAC matrices

The MAC matrices of all the combinations of numerical models are very similar.

Their diagonal values are near one, and the off diagonal values are near zero.

Therefore only 3 combinations are displayed in this section.


mode nr. 1 2 3 4 5

1

2

3

4

5

1.0000 0.0032 0.0032 0.0025 0.0018

0.0036 1.0000 0.0021 0.0021 0.0017

0.0028 0.0022 0.9999 0.0013 0.0012

0.0024 0.0015 0.0014 0.9998 0.0003

0.0021 0.0014 0.0007 0.0003 0.9997


mode nr. 1 2 3 4 5

1

2

3

4

5

0.9989 0.0005 0.0031 0.0034 0.0023

0.0081 0.9947 0.0029 0.0056 0.0017

0.0030 0.0014 0.9998 0.0008 0.0010

0.0021 0.0002 0.0023 0.9766 0.0024

0.0024 0.0018 0.0010 0.0003 0.9965


mode nr. 1 2 3 4 5

1

2

3

4

5

0.9998 0.0011 0.0030 0.0039 0.0022

0.0052 0.9984 0.0022 0.0017 0.0016

0.0030 0.0020 1.0000 0.0015 0.0010

0.0024 0.0002 0.0014 0.9937 0.0007

0.0023 0.0015 0.0010 0.0000 0.9998

A-2

A.2. Error matrices

The visualization of the stiffness errormatrices for all combinations of numerical

models is given in this section. The following table gives an overview of the figures:

The dynamic reduction is used for all the errormatrices. The static reduction results in

very similar bar plots.

(A)

(B)

(C)

(D)

A2 B1 B2 B3

A1 A B C D

A2 E F G

B1 H I

B2 J

A-3

(E)

(F)

(G)

(H)

(I)

(J)

A-4

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

dof-nr

Comparison eigenmode no. 1

FE-model A

1FE-model B

1 Analytical

0 10 20 30 40 50 60 70 80 90 100-1

0

1Comparison eigenmode no. 2

dof-nr

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

A.3. Eigenmodes

The eigenmodes are normalized for visualizing the mode-shapes.

A-5

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H2,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-400

-200

0

200

∠∠∠∠ |H2,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H1,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-400

-200

0

200

∠∠∠∠ |H1,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

A.4. Numerical frequency response functions

A-6

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H3,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-400

-200

0

200

∠∠∠∠ |H3,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H4,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-400

-200

0

200

∠∠∠∠ |H4,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

A-7

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H5,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-400

-200

0

200

∠∠∠∠ |H5,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H6,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-400

-200

0

200

∠∠∠∠ |H6,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

A-8

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H7,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H7,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H8,8

|

Frequency [Hz]

Ac

ce

lera

tio

n/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

∠∠∠∠ |H8,8

|

Frequency [Hz]

An

gle

[0]

FE-model A1

FE-model B1

B-1

B. Error matrix validation

In this section, the error matrix method is validated. The influence of the number of

modes used is investigated, along with the reduction of the number of degrees of

freedom. First a simple test is made.

B.1. Simple test

An error matrix for the stiffness is made. It is a comparison between two models

where only the E-modulus differs in the center element. These models have 200

degrees of freedom which makes it hard to visualize the error matrix.

( )( ) =∆=−

KKeigX N 1max 1.9266e-006

This result was expected because one element stiffness matrix is different. The 4 x 4

element stiffness matrix can be distinguished in the figure. 4 values are especially

high; they represent the displacements.

In this comparison only 5 eigenmodes are used for determining the (pseudo-) inverses.

(e=5 in equation (2.40))

X=0.00012526

This shows that errors occur when a limited number of eigenmodes is used.

B-2

B.2. Influence of the dynamic reduction and the number of modes used

In this section also the dynamic reduction of the models is taken into consideration.

This is an extra cause for errors. All the errormatrices are normalized.

The model is reduced to 50 degrees of freedom:

All modes used (50) 5 modes used

X=1.8279e-006 X=1.9391e-009

The model is reduced to 8 degrees of freedom:

All modes used (8) 5 modes used

X=0.026392 X=0.00023829

So both using a limited amount of modes, and reducing the number of dof are causes

for errors. But the method is still able to locate the modeling errors.

B-3

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

-6

Number of modes used

X

50 dofs

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

0.03


X

8 dofs

0 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25


X

200 dofs

B.3. Influence of the number of modes used on X

In all the former comparisons X decreases when fewer modes are used. This

phenomenon is examined further in this section. Three cases are distinguished, the

error matrix for an unreduced model with 200 dofs and for two reduced models, with

respectively, 50 dofs and 8 dofs.

These figures all show an increasing X with an increasing number of modes used.

This was not expected because intuitively the errormatrix should be better constructed

when more modes are used for determining the stiffness inverses. Apparently the first

order approximation of the error matrix doesn’t improve when more modes are used.

B-4

B.4. Dynamic reduction

Different stiffness matrices

Only a different E-modulus (factor 2) is taken, so there is only a difference in the

stiffness matrix. ∆M should be zero because there is not difference in the mass matrix.

But the dynamic reduction causes errors in the mass matrix, therefore ∆M is nonzero.

X=0.025656 X=0.0020401

Different mass matrices

Now only a different density (factor 2) is taken, so there is only a difference in the

mass matrix.

∆K should be zero, but it has large values. (about half of the ∆K above)

X= 0.00023829 X= 0.044773

All these plots have their highest values in the center. Even when there is only a

difference in the mass matrix, the stiffness error matrix points the part in the center as

the poorly modeled part.

x 105

x 105

x 10-3

x 10-3

x 105

x 105

x 10-3

x 10-3

B-5

x 105

x 105 x 10

-4

x 10-4

B.5. Static reduction

Now the same comparison is made using the static reduction in stead of the dynamic

reduction. This yields almost the same results.

Different stiffness matrices

X=0.025226 X=0.0020798

Different mass matrices

X=0 X=0.044864

x 105

x 105 x 10

-3

x 10-3

C-1

C. Comparison numerical and experimental data

C.1. MAC matrices


mode nr. 1 2 3 4 5

1

2

3

4

5

0.9978 0.0035 0.0012 0.0018 0.0018

0.0015 0.9965 0.0065 0.0002 0.0006

0.0044 0.0001 0.9864 0.0104 0.0006

0.0021 0.0040 0.0026 0.9776 0.0188

0.0020 0.0006 0.0053 0.0068 0.9535


mode nr. 1 2 3 4 5

1

2

3

4

5

0.9978 0.0034 0.0019 0.0044 0.0031

0.0023 0.9961 0.0062 0.0010 0.0032

0.0037 0.0001 0.9878 0.0115 0.0000

0.0017 0.0033 0.0021 0.9767 0.0154

0.0021 0.0005 0.0053 0.0065 0.9653


mode nr. 1 2 3 4 5

1

2

3

4

5

0.9978 0.0036 0.0018 0.0044 0.0031

0.0021 0.9960 0.0061 0.0008 0.0032

0.0040 0.0001 0.9869 0.0121 0.0000

0.0017 0.0037 0.0021 0.9759 0.0151

0.0019 0.0007 0.0061 0.0070 0.9617

C-2

C.2. Error matrices

Overview of figures:

Exp.

model A

Exp.

model B

A1 A B

A2 C D

B1 E

B2 F

B3 G (A)

(B)

(C)

(D)

(E)

(F)

(G)

C-3

C.3. Eigenmodes

Experimental eigenmodes beam A

Mode 1 Mode 2 Mode 3

Mag Phase (o) Mag Phase (

o) Mag Phase (

o)

1

2

3

4

5

6

7

8

5.0014

14.705

28.775

44.158

62.362

81.943

103.29

122.38

-177.65

179.83

179.82

178.67

-179.79

179.88

179.17

177.6

122.28

305.13

453.66

494.26

390.3

159.4

172.95

537.07

-0.50189

-1.4215

-1.1212

-0.46701

-0.4743

-0.27966

-178.5

179.52

682.22

1213.8

1005.5

74.105

828.11

1049

276.69

1057.3

-179.48

179.01

178.73

178.61

0.65437

1.6417

4.5746

-179.51

Mode 4 Mode 5


o)

1

2

3

4

5

6

7

8

1609.4

1659.7

368.87

1819.4

656.79

1384.2

1272.6

1383.3

1.4075

2.1882

179.46

-177.66

-177.51

3.9183

4.6923

-176.97

2435.3

519.4

2231.3

108.74

2309.9

90.632

2095.7

1301.2

178.71

177.05

0.27685

-1.8271

-178.23

19.088

3.9724

-178.33

Normalized eigenmodes beam A

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

1

2

3

4

5

6

7

8

-0.040307

-0.11979

-0.23442

-0.36058

-0.50746

-0.66746

-0.84274

-1

0.22768

0.56783

0.84448

0.92029

0.72672

0.29679

-0.32126

-1

0.56127

1

0.82835

0.061046

-0.68112

-0.86058

-0.22367

0.8699

-0.88411

-0.91221

0.20172

1

0.36099

-0.75965

-0.6971

0.76009

-1

-0.21292

0.91486

0.044644

-0.9431

0.028191

0.84607

-0.53146

C-4

Experimental eigenmodes beam B

Mode 1 Mode 2 Mode 3


o) Mag Phase (

o)

1

2

3

4

5

6

7

8

3.8317

12.546

24.672

38.322

57.317

75.542

95.57

112.8

1.6844

-7.6641

2.5776

2.1964

3.9795

2.1099

2.6699

0.76551

126.43

309.13

460.01

503.93

400.23

156.53

177.26

540.88

-176.2

-177.86

-177.54

-177.6

-177.02

-177.65

3.8614

1.5814

710.52

1272.3

1034.3

99.413

864.64

1076

273.26

1120.9

1.2443

-0.4498

-0.71817

-2.7624

-179.75

-179.86

-178.52

-1.6794

Mode 4 Mode 5


o)

1

2

3

4

5

6

7

8

1660.9

1722.5

376.47

1873

659.18

1416.1

1266.5

1506.7

-179.07

179.48

-0.65839

-0.94434

-0.18292

-179.91

-178.02

-2.4715

2529

600.11

2338

228.55

2393.2

108.22

2167.9

1524.7

1.2412

1.0462

179.99

-179

0.75997

-173.9

-178.49

-1.5896

Normalized eigenmodes beam B

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

1

2

3

4

5

6

7

8

0.033951

0.10644

0.21829

0.33931

0.50494

0.66896

0.84538

1

0.23305

0.57142

0.85008

0.93131

0.73908

0.28929

-0.32669

-1

0.55748

1

0.8129

0.077882

-0.67939

-0.84553

-0.21429

0.88019

-0.88486

-0.91955

0.20099

1

0.35181

-0.75557

-0.67267

0.80329

-1

-0.23729

0.92359

0.090368

-0.94617

0.042178

0.85718

-0.59995

C-5

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

dof-nr

Comparison eigenmode no. 1

Experimental model A Experimental model B Analytical

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

0 10 20 30 40 50 60 70 80 90 100-1

0


dof-nr

The normalized eigenmodes are used for visualizing the mode-shapes.

(Some experimental modes are multiplied by a

factor -1 in order to match with the analytical modes)

C-6

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H2,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H2,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H1,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H1,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

C.4. Frequency response functions

C-7

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H4,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H4,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H3,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H3,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

C-8

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H6,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H6,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H5,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H5,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

C-9

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H8,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H8,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H7,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H7,8

|

Frequency [Hz]

Angle

[0]



FE-model A1

FE-modelB1

C-10

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-40

-20

0

20

40

|H1,8

|

Frequency [Hz]

Accele

ration/F

orc

e [

dB

]

Left sensor

Center sensor

Right sensor

0 200 400 600 800 1000 1200 1400 1600 1800 2000-200

-100

0

100

200

∠∠∠∠ |H1,8

|

Frequency [Hz]

Angle

[0]

C.5. Torsional modes

The torsional modes can be analyzed by attaching 2 extra sensors closer to the sides of

the beam. When the beam is excited with the hammer closer to the sides, the torsional

modes are amplified.

2 Torsional modes are present in the frequency domain from 10 to 2000 Hz; the first

torsional mode at 505 Hz, and the second at 1529 Hz. They are recognized because

the left sensor and the right sensor have a resonance peak at that frequency, and the

center sensor does not have this peak, but only a little disturbance.

1st torsional mode

2nd

torsional mode

D-1

D. Flowcharts of data

Flowchart of data:

Matlab file structure:

MAIN.m

transfer.m

Unorm.m

analytical.m

errormatrix.m

MACmatrix.m

plotbar.m

plottransfer.m

reduction.m

label.m

modelA1.m

modelA2.m

modelB1.m

modelB2.m

modelB3.m

importexp.m

main2.m

dampingfactors.m

mnormalize.m

*.vna files

Settings:

• Models to compare

• Output request

• Reduction method

Eigenfrequencies

Eigenmodes

Experiments

Siglab

ME’Scope Matlab Eigenfrequencies

Eigenmodes

Frequency response functions

*.blk *.vna

For practical reasons all the calculations in this project can be done by calling upon

only one M-File, that uses multiple function files.

E-1

E. Indication of the Matlab script

% The MAIN M-file. % Different models can be compared: % Models 'A1', 'A2', 'B1', 'B2' and 'B3' are numerical models. % Models 'Ae' and 'Be' are the experimental models. % Different outputs can be chosen: % 'frf' plots the frequency response functions of the first

% selected model and the second selected model on an arbitrary

% point. 'err' plots the Error matrix of the models. % 'mac' plots the MAC matrix of the models. % 'mod' plots the mode-shapes of the models and the analytical

% mode-shape. 'frq' gives a comparison between the

% eigenfrequencies. clear all; close all; clc; format short g %------- Settings -------% %--- Which models to compare? :: A1/A2/B1/B2/B3/Ae/Be

select(1,:)='Ae'; select(2,:)='Be'; %--- What output? :: frf/err/mac/mod/frq output='frf'; reduc='y'; %Apply a reduction when neccesary? ‘y’/‘n’ method='dynamic'; %Reduction method. 'dynamic'/'static' Enorm='y'; %Normalize error matrices? ‘y’/‘n’ absdK='y'; %visualize error matrix absolute? ‘y’/‘n’ save='n'; %Save figure to harddisk? ‘y’/‘n’

%------- End settings -------%

Nelem = 100; %Number of elements. (Has to be a multiple of 25) n=2*(Nelem+1); %n= number of dofs

%Adding correct labels for plotting the figures [label1,label2]=label(select);

%Geometrical and Physical properties L=0.5; H=0.010; W=0.060; I=1/12*W*H^3; E=2.1e11; rho=7850; A=W*H; l=L/Nelem; x=0:l:L;

%Sensor mass of all the sensors: Ms=(2.3+1.9+1.2)*10^-3;

%8 relevant elements: (corresponding to the 8 deflections) relev=(1:8)*Nelem*6/25-1;

% When the output is 'frf', a point has to be selected. % (Entering multiple points is possible)

if strcmp(output,'frf') pointinput=input('Select point(s) (1:8): '); else pointinput=1; end

COMMENT

Displaying the entire Matlab script requires over 25 pages.

So only the first page is provided here for still giving

an indication of the method used in the script and its possibilities.

When you are interested in the complete script, please contact me:

[email protected]

E-2

for point=pointinput, %Constructing frf, and/or constructing K and M of both models

[freq1,mag1,phase1,K1,M1,relev1]=main2(Nelem,H,W,I,E,rho,A,l,n,Ms,

point,relev,select(1,:));

[freq2,mag2,phase2,K2,M2,relev2]=main2(Nelem,H,W,I,E,rho,A,l,n,Ms,

point,relev,select(2,:)); if strcmp(output,'frf') figure(point); plottransfer(point,freq1,mag1,phase1,'b--') plottransfer(point,freq2,mag2,phase2,'r'); legend(label1,label2,'Location','SouthEast') else if strcmp(select(1,:),'Ae') || strcmp(select(1,:),'Be'), [U1,U1complex,w1]=Unorm(select(1,1)); else %Reducing the K an M matrices from (n x n) to (8 x 8) [Kred1,Mred1]=reduction(K1,M1,relev1,reduc,method); [U1,L1]=eig(Kred1,Mred1); w1=sqrt(diag(L1)); end if strcmp(select(2,:),'Ae') || strcmp(select(2,:),'Be'), [U2,U2complex,w2]=Unorm(select(2,1)); else %Reducing the K an M matrices from (n x n) to (8 x 8) [Kred2,Mred2]=reduction(K2,M2,relev2,reduc,method); [U2,L2]=eig(Kred2,Mred2); w2=sqrt(diag(L2)); end f1=w1/(2*pi); f2=w2/(2*pi); switch output case 'err' % ↑

% Error matrices handling

% ↓ case 'mac' % ↑

% MAC matrices handling

% ↓ case 'mod' % ↑

% Mode-shapes handling

% ↓ case 'frq' % ↑

% Error matrices handling

% ↓ end end

if save=='y' saveas(point,[output '_' select(1,:) '-' select(2,:) '.emf']) end end

Documents

ir. L. Kodde Department Mechanical Engineering Dynamics ...mate.tue.nl/mate/pdfs/8195.pdf · Bachelor final project. Maarten J. Beelen. April 2007.Page 3 of 33 e e+1 ue, F e φe,