l7005a report main

Modal Analysis ProjectL7005A – Senior Sound Design Project

Andre Lundkvist

[email protected]

Lule̊a, January 28, 2010

Abstract

This report is about a modal analysis project done in the course L7005A – Senior Sound Design

Project. It is based upon modal measurements from a project in the course L7001A – Experimental

Acoustics and Dynamics, which were previously analysed in I-DEAS Test.

The measurements were imported into Matlab. Functions to find poles (frequency, damping andamplitude) were constructed using the Complex Mode Indicator Function (CMIF) as a base, andmode shapes were extracted from the measurements.

From these parameters, a multiple degrees of freedom (MDOF) modal-parameter model was con-structed by summation of many single degree of freedom (SDOF) models. The modal-parametermodel was analysed by different methods, including mathematical versus measured FRF compar-isons, MAC matrix and visual interpretation of the mathematical mode shapes.

Contents

1 Theory 2

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Analytical mathematical models . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Experimental mathematical models . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Modal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Mode shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Modal frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Modal damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Mathematical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Mobility matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.3 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Parameter estimation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Complex Mode Indicator Function (CMIF) . . . . . . . . . . . . . . . . . . . 8

1.4.2 Least Squares Frequency Domain method (LSFD) . . . . . . . . . . . . . . . 9

1.5 Modal parameter model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.1 Modal Assurance Critereon (MAC) matrix . . . . . . . . . . . . . . . . . . . 10

1.6 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6.1 Measurement estimation and validation . . . . . . . . . . . . . . . . . . . . 11

1.6.2 Measurement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6.3 Excitation and response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6.4 Response transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Measurements 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Measured FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1

3 Method 20

3.1 Complex Mode Indicator Function (CMIF) . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Mode shape estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Modal-parameter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Results 22

4.1 Comparison between I-Deas Test and Matlab poles . . . . . . . . . . . . . . . . . . 22

4.2 Complex Mode Indicator Function (CMIF) . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Mathematical vs. measured FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Mathematical vs. measured mode shapes . . . . . . . . . . . . . . . . . . . . . . . 28

4.5 Modal Assurance Criterion Matrix (MAC) . . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A I-Deas Test stability diagram 34

B Lists 35

B.1 Target modal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

B.2 List of detected frequencies and dampings . . . . . . . . . . . . . . . . . . . . . . . 36

C Matlab code 37

C.1 Import FRFs (impfiles.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

C.2 Process FRFs (process frfs.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

C.3 Complex Mode Indicator Function (cmif.m) . . . . . . . . . . . . . . . . . . . . . . 42

C.4 Find frequencies (find freqs.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

C.5 Find dampings (find damps.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

C.6 SDOF modal model (sdof.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

C.7 MAC matrix (mac.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

C.8 Animate mode (animate mode.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

References 48

2

Chapter 1

Theory

1.1 Introduction

Modal analysis is a method to determine modal parameters of a system. The system can bedefined as a simple structure or a more complex system involving several subsystems [1]. For thepurpose of this project, the system will be a simple structure.

Mathematical dynamic models may be used for a number of reasons.

• Gain understanding how structures behave under dynamic loads

• Curve fitting, to smoothen and reduce data

• Simulation or prediction when external forces are existent

• Physical modifications to the structure, simulate the change of dynamic characteristics

Mathematical models describe generally the dynamic behavior, not the structure itself. Theyare often constrained by assumptions and boundary conditions. The modal parameters thatcompletely describe the dynamics of a system are:

• Modal frequency

• Modal damping

• Mode shape

Any forced dynamic deflection of a structure can be represented as a weighted sum of its modeshapes. Each mode can be described by an Single Degree of Freedom (SDOF) model.

There are mainly two approaches for the study of system vibrations, analytical and experimental.The analytical starts out with knowledge about the structure geometry, boundary conditions andmaterial characteristics (mass, stiffness and damping). The experimental utilize measurements ofdynamic input forces and output responses on the structure or a prototype of the structure.

3

Modal analysis requires some assumptions of the system to work. These are:

Linearity: The system is linear, which means the response is proportional to the force. Also, theexcitation level should not affect the behavior.

Reciprocity: Derives from the Linearity property, and implies that the force and response canswitch place and still produce the same result.

Time Invariant: The properties of the system should not change over time.

Causal: The structure should not be influenced by vibration before excitation.

Stable: The vibrations should die out, and not continue to oscillate and converge to infinity whenthe force is removed.

1.1.1 Analytical mathematical models

Analytical models are often created by using FEM1 based on calculated mass and stiffness underspecific boundary conditions. The model will contain a number of differential equations, whichare coupled [2].

1.1.2 Experimental mathematical models

Experimental modal analysis consists of five different phases. These span from setup, data ac-quisition, system identification, validation and using the obtained information for improving thesystem in a systematic way [2].

Usually measurements are observed in the frequency domain as Frequency Response Functions(FRFs), H(ω). Which is the ratio between output and input as a function of frequency. Seefigure 1.1.

The FRFs can be described by the modal parameters, and the modal parameters can therefore beextracted from the FRFs.

Figure 1.1: Frequency Response Function (FRF) example [3]

1Finite Element Method

4

1.2 Modal parameters

1.2.1 Mode shape

The mode shape ~Ψ is as the name suggests the shape of the mode. It is connected to the modalfrequency and therefore the pole location of the mode, and expresses a deflection pattern in thestructure [1].

From measurements, a mode shape ~Ψ is a vector containing sampled values. The amount ofsamples is directly linked to the degrees of freedom (DOF) used. Often these vectors containcomplex numbers that describe both magnitude and phase, and these describe modes that can beconsidered as propagating waves. Standing waves is considered as a normal mode, meaning ~Ψ isa real number vector.

Unreliable measurements can wrongly indicate complex modes instead of normal modes.

1.2.2 Modal frequency

The resonance frequency of the specific mode is called modal frequency. These can be determinedby observing the maximum magnitudes of the measured or calculated FRF.

1.2.3 Modal damping

Modal damping is the damping to the specific mode, and can be determined from an FRF by findingthe −3 dB bandwidths of the modes. This can be a problem on a lightly dampened structure, asthe resonance peaks are very narrow.

5

1.3 Mathematical theory

1.3.1 Degrees of Freedom

Specifying the Degrees of Freedom (DOF) is based on the purpose of the test, geometry and thefrequency range. A free point in space has 6 DOFs, both rotation and translation on axes x, y & z.There are not many rotational transducers available, but three transitional is usually enough todescribe the the displacement and motion [1].

The number of possible input/output (force/response) combinations will be n2, where n is theselected DOF.

1.3.2 Mobility matrix

The mobility matrix contains all different combinations of the FRFs. For a three-DOF system, themobility matrix would look like equation 1.1. Each row contains FRFs with a common responseDOF, and each column have a common excitation DOF. The diagonal contains the driving pointFRF [1].

H =

H1,1 H1,2 H1,3

H2,1 H2,2 H2,3

H3,1 H3,2 H3,3

(1.1)

1.3.3 Mathematical models

SDOF modal-parameter model

Below is described a Modal-Parameter Model FRF, see equation 1.2, figures 1.2 and 1.3, where λ

(equation 1.4) is the pole location and A (equation 1.3) is the residue. Equation 1.2 describes anSDOF system, but can be expanded to an MDOF system [2].

H(jω) =A1

jω − λ1

+A∗

1

jω − λ∗

1

(1.2)

with

A1 = −j1

2Mω1

A∗

1= j

1

2Mω1

(1.3)

λ1 = σ + jω1

λ∗

1= σ − jω1 (1.4)

In the pole equation, σ is the damping factor and ω1 is the damped natural resonance frequency.The residue describes the mode strength, and is related to the mode shape. Because of thecomplex conjugate part is almost negligible around resonance, the model can be approximated toequation 1.5.

H(jω) ≈A1

jω − λ1

(1.5)

6

Figure 1.2: Example of a magnitude plot of the FRF for a SDOF system [2]

Figure 1.3: Example of a phase plot of the FRF for a SDOF system [2]

The impulse response is described in equation 1.6, and it can look similar to figure 1.4. Theimpulse response is the Fourier transform of the FRF [2].

h(t) = A1eλ1t + A∗

1eλ∗

1t = eσ1t

(

A1ejω1t + A∗

1e−jω1t

)

(1.6)

Figure 1.4: Example of an impulse response for a SDOF system [2]

7

MDOF modal-parameter model

The MDOF model can be seen in equation 1.7 [2].

H(jω) =

N∑

r=1

Ar

jω − λr

+A∗

r

jω − λ∗

r

(1.7)

where the poles can be expressed in matrix form as in equation 1.8.

λr =

σ1 + jω1

. . . 0σN + jωN

σ1 − jω1

0. . .

σN − jωN

(1.8)

If the poles are considered as eigenvalues of a system, its eigenvectors can be interpreted as themodal shapes, where each eigenvector corresponds with a specific eigenvalue. See equation 1.9.

Φ =

[

λ1{Ψ}1 . . . λN{Ψ}N λ∗

1{Ψ}1 . . . λ∗

N{Ψ}∗N{Ψ}1 . . . {Ψ}N {Ψ}∗

1. . . {Ψ}∗N

]

(1.9)

The residues, Ar is described by equation 1.10, where Qr is a scaling factor, and {Ψ} is the modalshape matrix for the specific mode.

Ar = Qr{Ψ}r{Ψ}tr (1.10)

By using the notation above, the FRF can be expanded to the form described in equation 1.11.

H(jω) =N

∑

r=1

(

Qr{Ψ}r{Ψ}tr

(jω − λr)+

Q∗

r{Ψ}∗r{Ψ}∗tr

(jω − λ∗

r)

)

(1.11)

Figure 1.5: Example of a FRF magnitude plot constructed from two SDOF FRFs [2]

8

1.4 Parameter estimation methods

1.4.1 Complex Mode Indicator Function (CMIF)

The CMIF is based upon the singular value decomposition of the FRF matrix, containing all possibleinput – output FRF combinations. The diagonal Singular Matrix ([ Σ ]) is used as the CMIF. Itindicates the existence of real (normal) or complex modes and also gives the relative magnitudeof each mode. It is also capable of yielding the corresponding mode shape and/or participationvector [2].

The Singular Value Decomposition (SVD) of FRF matrix [ H ] is expressed as,

[ H ] = [ U ] · [ Σ ] · [ V ] (1.12)

where,[ H ] is the Frequency Response Function Matrix[ U ] is the Left Singular Matrix (unitary)[ Σ ] is the Singular Matrix (diagonal)[ V ] is the Right Singular Matrix (unitary)

(1.13)

500 1000 1500 2000 2500 3000

20

40

60

80

100

120

Complex Mode Indicator Function (CMIF) of FRF Matrix [H]

Frequency (Hz)

Am

plitu

de (

dB)

Figure 1.6: Example of a CMIF for a FRF matrix

9

1.4.2 Least Squares Frequency Domain method (LSFD)

The non-linear Least Squares Frequency Domain (LSFD) method estimates poles, mode shapes andmodal participation factors of a system. The modal participation factors can only be estimatedif there are multiple inputs. The LSFD method is mostly used for model updating, because of thecomplexity and amount of parameters that needs to be determined [2].

This method is based on the frequency domain modal-parameter model described in section 1.3.3on page 7. If equation 1.7 is expanded, including the modal participation factors, the form will bedescribed as

Hij(jω) =N

∑

r=1

(

ΨirLrj

jω − λr

+Ψ∗

irL∗

rj

jω − λ∗

r

)

+ URij −LRij

ω2(1.14)

where i denotes a response position and j denotes an input station. URij and LRij is the upperand lower residual terms. These terms approximates the modes below and above the frequencyband of interest.

Hij(jω) is the measured FRF at point (i, j). The method is about iteratively determine theunknown parameters λr, Ψir, Lrj , URij and LRij by minimizing the error between the measuredFRF and the model. The amount of parameters is Nu.

The difference between the measured FRF and the model can be expressed as,

eij(jω) = Hij(jω) − Gij(jω, λr,Ψir, Lrj , URij , LRij)|r=1,Nm(1.15)

and the total squared error over the frequency range of interest (0 – Nf ) becomes,

Eij =

Nf∑

f=0

eij(jωf ) · e∗ij(jωf ) (1.16)

The total error between all inputs and outputs can be expressed as,

E =

No∑

i=1

Ni∑

j=1

Eij (1.17)

To minimize the global error, the derivatives of each parameter from the total error E shouldconverge to 0, i.e for parameter rk,

dE

dr1

= 0 (1.18)

dE

dr2

= 0 (1.19)

... (1.20)

dE

drNu

= 0 (1.21)

10

1.5 Modal parameter model validation

1.5.1 Modal Assurance Critereon (MAC) matrix

The Modal Assurance Criterion (MAC) matrix is a mathematical tool to compare two vectors toeach other. It can be used to investigate the validity of estimated modes [2].

The MAC between two mode shape vectors {Ψ}r and {Ψ}s is defined as,

MAC({Ψ}r, {Ψ}s) =

(

{Ψ}∗Tr {Ψ}s

)2

({Ψ}∗Tr {Ψ}r) ({Ψ}∗T

s {Ψ}s)(1.22)

The MAC will approach the value 1 if {Ψ}r and {Ψ}s are the same mode shape. It {Ψ}r and {Ψ}s

are different mode shapes, the MAC value should be low, due to the orthogonality condition of themode shapes.

Figure 1.7: Example of a MAC matrix on 5 randomly generated vectors (Gaussian)

11

1.6 Measurements

1.6.1 Measurement estimation and validation

There are a couple of estimators used to calculate the FRF. For noise at the output, a usefulestimator is H1, see equation 1.23. The function H1 is derived by using least squares method, andis the cross spectrum divided by the autospectrum of the force F. If averaging the FRFs measuredwith H1, the random noise will suppress and H1 will converge towards the true H [1].

H1(ω) ≡GFX(ω)

GFF (ω)(1.23)

For noise at the input H2 is a useful estimator for the FRF. It is derived from the same principle asthe H1 estimator, and is defined as the autospectrum of the response divided by the cross spectrum.The noise at the input is removed more and more from the cross spectrum with increased averages.

H2(ω) ≡GXX(ω)

GXF (ω)(1.24)

A way to validate a measurement is to observe the coherence function, seen in equation 1.26. It isderived from the cross spectrum inequality, equation 1.25 which states that if the autospectrumcontains non-coherent noise, the magnitude of the squared cross spectrum is smaller than theproduct of the autospectrum.

The coherence function have the boundaries described in equation 1.27. For the value 1, themeasurement contains no noise, and for the value 0, there is pure noise in the measurement. Thecoherence function also indicates the linearity between the input and output signal.

When the coherence is 1, estimator H1 and H2 will yield the same result, therefore the estimatorsare overcompensated, and the true FRF will be somewhere in between.

|QXF (ω)|2 ≤ GXX(ω) · GFF (ω) (1.25)

γ(ω)2 ≡|GFX(ω)|2

GXX(ω) · GFF (ω)(1.26)

0 ≤ γ(ω)2 ≤ 1 (1.27)

To calculate the autospectrum, the spectrum is multiplied with its complex conjugate. The com-plex conjugate is the same spectrum but with opposite sign for the imaginary part. The autospec-trum is always real.

The cross spectrum is calculated by multiplying a spectrum with the complex conjugate of adifferent spectrum. For instance the spectrum of the force and the response. Cross spectrum is acomplex entity which describes the phase shift between the different spectrums. The magnitudeof the cross spectrum describes the coherent product of power in the spectrums.[1]

12

1.6.2 Measurement errors

There are some error that can occur when performing mobility measurements. These are listedin table 1.1, and its abbreviations in table 1.2. The different errors can be divided into differentclasses, depending on the nature of the cause. The classes are Random & Bias errors [1].

Random errors are caused by noise, and can occur both at the input and the output of the signalchain. Random noise can be minimized by averaging over many measurements, because the noiseis uncorrelated.

Systematic bias error are the same for every measurement (both magnitude and phase). The onlyway to minimize bias errors is to select another estimator.

Table 1.1: Possible errors for mobility measurementsType of Error H1 H2 γ2

Noise at the output R B +

Noise at the input B R +

Random excitation / non-linearity B/R B/R +

Deterministic excitation B B 0

Scatter of impact (point / direction) R R +

Random excitation B (B) +

Deterministic impact B B 0

Table 1.2: Error table abbreviationsAbbreviation Description

B Systematic Bias error

R Random error

+ Coherence Function can indicate error

0 Coherence Function can not indicate error

13

1.6.3 Excitation and response

Waveform

The choice of excitation waveform depends on which type of measurement is to be performed.There is a list of certain parameters to consider, see table 1.3 [1].

The crest factor is the ratio between the peak and the standard deviation RMS of the signal. Formodal analysis, a high crest factor in the excitation waveform is undesired, as it contributes to acouple of factors. Noise will be introduced because of the sharp peaks, which decreases the signalto noise ratio SNR. The other factor is that nonlinearity might be introduced due to the high peakforces.

Because modal analysis assumes that the system is linear and time invariant, this can be a problem,although there are ways of making linear approximations of a non-linear system.

Table 1.3: Excitation conditional parametersParameter Description

Application What is the targeted system, which type of measurement

Spectrum Control Capability to control the frequency range

Crest Factor Ratio between the peak and RMS in the signal

Linearity Linear and time invariant system

Speed of Test How much time is needed for a measurement

Table 1.4: Excitation waveform comparisonWaveform

Factor Sinusoid Random Pseudo Random Impact Multiple Impact

Analysis Speed Very Slow Slow Fast Fastest Slow

Leakage Error Yes Yes No No Yes

Approximation No Yes No No Some

Crest Factor Good Fair Fair Poor Poor-Fair

Spectrum Control High High High Limited Limited

Zoom Analysis Yes Yes Yes No No

Detect Linearity No Yes No No (Yes)

Detect Leakage No Yes No No Yes

Excitation sources

There are many different exciters to choose from, such as shakers, hammers, pendulums etc. Theseare categorized into two fastening categories, attached and non-attached. The choice of exciterdepends on the excitation conditional parameters, the structure etc [1].

There are advantages and disadvantages for both categories. The attached exciters affects thestructure quite a lot, whereas the non-attached does not. However, the attached exciters give amore controllable excitation than a non-attached.

14

1.6.4 Response transducers

Accelerometers

To be able to measure the structural response, the need for a transducer is imminent. The mostcommonly used transducer is an accelerometer. The standard measurement for an accelerometeris acceleration, but velocity or displacement can be calculated by integration [1].

The piezoelectric accelerometer has good linearity, broad dynamic range, wide frequency rangeand a strong construction. The dynamic range is typically around 160 dB and ranges in frequencybetween 0.2 Hz to at least 10 kHz, with a linearity of 5 %.

Accelerometers can also be very light, about 1 g, and are easy to mount by glue, wax, screwetc. See a list of selected mounting techniques below. Beeswax is a commonly used method, seefigure 1.8. The resonance frequency of the accelerometer is dependent on its weight though, sothis have to be considered, and the mounting also affects the applicable frequency range.

• Steel stud

• Beeswax

• Cement stud

• Thin tape

• Thick tape

• Magnet

For measurements in the driving point, an impedance head may be used if possible by the setup.The impedance head is a transducer with integrated force and response accelerometers. Anothertypical setup is by placing a standard accelerometer transducer on the opposite side of the structureto where the force is applied to.

Figure 1.8: Example of a few accelerometers mounted with beeswax

15

Chapter 2

Measurements

2.1 Introduction

This chapter discusses the measured data of the metal plate showed to the left in figure 2.1. It ishung by springs to reduce external excitation and to avoid influence on the structure.

The measurement grid is showed to the right in figure 2.1, where the point (1,1) would be thelower left hand corner, and it is also the excitation point used. The reference to the measurementgrid is counting from bottom left (row from bottom, column from left), throughout the report.

Figure 2.1: Left : The metal plate used for measurements, mounted with springs. Right : Themeasurement grid used for measuring the metal plate

16

The settings for the measurements were as listed below in table 2.1. An impact hammer was usedfor excitation, and it had a transient window applied to avoid multiple excitations.

Accelerometer 2 was used as a trigger to start the measurement. The hold-off was set to the samelength as the measurement time to avoid the measurement to restart before completion.

Table 2.1: Measurement settingsSetting Parameter Value

Base Resolution 6400 lines

Break Frequency 3200 Hz

Measure Time 2 s

Means 3, linearly weighted

Overload Reject

Accelerometer Window Exponential

— τ 0.3 s

Excitation Window Transient

— Start 0.9 ms

— Stop 1.3 ms

Trigger Signal Accelerometer 2

Level 5 %

Hold-Off 2 s

17

2.2 Measured FRFs

This section presents some chosen FRFs measured on the plate described in figure 2.1.

The driving point FRF is showed in figure 2.2 below. There are clear resonance peaks, and overallthey are quite separated. The phase response has a weird behavior in the low frequency range,probably due to measurement problems.

500 1000 1500 2000 2500 3000

−3

−2

−1

0

1

2

3

Imaginary Response Values (1,1)

Rad

ians

Frequency [Hz]

500 1000 1500 2000 2500 3000−50

0

50

Real Response Values (1,1)

Mag

nitu

de [d

B (

m/s

2 /N)]

Frequency [Hz]

Figure 2.2: Example of a phase and magnitude plot of the FRF associated with the point (1,1)

The measurement in point (3,4), according to the grid in figure 2.1), is showed in figure 2.3.In comparison to the driving point FRF, this response has fewer clear resonance peaks, and theapparent ones has lower amplitude which might suggest higher damping.

Figure 2.4 shows the FRF at point (5,1) which is located at the left edge, just above the middleof the plate. This FRF has clear separated resonances, except at about 680 Hz, where there couldbe a double pole.

18

500 1000 1500 2000 2500 3000

−3

−2

−1

0

1

2

3


Rad

ians

Frequency [Hz]

500 1000 1500 2000 2500 3000−25

−20

−15

−10

−5

0

5

10


Mag

nitu

de [d

B (

m/s

2 /N)]

Frequency [Hz]


500 1000 1500 2000 2500 3000

−3

−2

−1

0

1

2

3


Rad

ians

Frequency [Hz]

500 1000 1500 2000 2500 3000

−40

−20

0

20

40


Mag

nitu

de [d

B (

m/s

2 /N)]

Frequency [Hz]


19

500 1000 1500 2000 2500 3000

−3

−2

−1

0

1

2

3


Rad

ians

Frequency [Hz]

500 1000 1500 2000 2500 3000

−20

−10

0

10

20


Mag

nitu

de [d

B (

m/s

2 /N)]

Frequency [Hz]


500 1000 1500 2000 2500 3000

−3

−2

−1

0

1

2

3


Rad

ians

Frequency [Hz]

500 1000 1500 2000 2500 3000−30

−20

−10

0

10

20

30


Mag

nitu

de [d

B (

m/s

2 /N)]

Frequency [Hz]


20

Chapter 3

Method

3.1 Complex Mode Indicator Function (CMIF)

The complex mode indicator function (CMIF) (see section 1.4.1, page 8) is used to evaluate theresonate frequencies and get an approximation for relative amplitude and damping. The Matlab

code used to calculate the CMIF can be found in appendix C.3 on page 42. The curve is similarto the average FRF used in I-DEAS Test to calculate the stability diagram (see appendix A atpage 34).

The algorithm to search for frequencies is an iterative method. The code for it can be found inappendix C.4 on page 43. It searches for peaks by stepping through the curve, and when the curvehas reached a maximum, it will indicate a new pole by a red line in the CMIF, see figure 3.1.

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plitu

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Figure 3.1: CMIF for all FRFs

The algorithm for finding damping ratios is similar, but it will start out from the detected fre-quencies and iteratively search for the -3 dB decay limits. The code can be found in appendix C.5on page 44. The -3 dB limits will be indicated on the CMIF as green lines, see figure 3.1.

21

It tries to approximate the damping by taking the closest limit for both the left and right limit.Meaning, that if for example the left limit will diverge, but the right will not, it will approximatethe left limit to FP - abs(d2), where FP is the pole location and d2 is the right limit, both insamples.

Figure 3.2 shows a close up display of four detected poles, indicating resonance frequency anddamping for each pole.

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Figure 3.2: Detailed CMIF for all FRFs

3.2 Mode shape estimation

The mode shapes are estimated from the imaginary part of the measured FRF response matrix{H} at each system pole detected by the CMIF [4]. This produces a mode shape matrix for eachpole frequency, with the dimensions of the measurement grid (figure 2.1).

3.3 Modal-parameter model

The modal model was created by using equation 1.7 (page 7). To use the equation in Matlab, itwas rewritten like

Hi,j(jω) ≈

N∑

r=1

Qr{Ψi,j}r{Ψi,j}Tr

jω − σr − jωr

+Q∗

r{Ψi,j}∗

r{Ψi,j}∗Tr

jω − σr + jωr

(3.1)

where Qr is a scaling factor and {Ψi,j}r is the modal shape amplitude and direction informationfor the specific mode and node position. σr is the damping factor and ωr is the damped naturalresonance frequency for the specific mode.

The amplitude of the mode was used as scaling. The Matlab code can be found in appendix C.6on page 45. After creating all SDOF models, these are summed together to create the MDOF model.

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Chapter 4

Results

4.1 Comparison between I-Deas Test and Matlab poles

A comparison between the detected poles in I-DEAS Test and Matlab can be found in table 4.1below. The frequencies detected in Matlab is quite similar to I-DEAS Test, with values bothhigher and lower to I-DEAS Test. The damping is relatively the same for some poles, but candiffer a bit on some. The difference indicates the difference (I-DEAS Test - Matlab).

Poles number 7, 14 and 15 was not detected by I-DEAS Test. Note on pole number 18 and 19,both the frequency and damping is quite close to each other. However, this might be a coincidence.

Table 4.1: Comparison between detected frequencies and damping ratios

Pole Frequency Frequency Frequency Damping Damping Damping

Nr I-DEAS Test Matlab Difference I-DEAS Test Matlab Difference

1 272.544 270.458 2.0860 1.437 0.238 1.199

2 335.348 331.448 3.9000 0.665 0.719 -0.054

3 620.261 621.403 -1.1420 0.202 0.110 0.092

4 644.843 641.900 2.9430 0.090 0.042 0.048

5 783.756 787.377 -3.6210 0.127 0.056 0.071

6 962.906 965.849 -2.9430 0.232 0.158 0.074

7 —— 973.848 —— —— 0.181 ——8 1221.286 1227.31 -6.0240 0.214 0.032 0.182

9 1294.681 1296.80 -2.1190 0.214 0.069 0.145

10 1660.698 1664.74 -4.0420 0.336 0.187 0.149

11 1853.714 1852.21 1.5040 0.280 0.135 0.145

12 1999.744 1998.19 1.5540 0.287 0.113 0.174

13 2089.418 2090.67 -1.2520 0.355 0.219 0.136

14 —— 2110.17 —— —— 0.187 ——15 —— 2373.13 —— —— 0.382 ——16 2401.034 2399.63 1.4040 0.451 0.222 0.229

17 2901.708 2901.55 0.1580 0.268 0.093 0.175

18 3023.607 3023.53 0.0770 0.212 0.273 0.061

19 3093.047 3094.02 -0.9730 0.435 0.452 -0.017

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4.2 Complex Mode Indicator Function (CMIF)

To evaluate the generated mathematical FRFs, first the CMIF function was used. A comparisonbetween the measurement CMIF and the mathematical FRF CMIF can be seen in figure 4.1 below.

Overall, the mathematical poles seem quite close to the measurements. The first pole, locatedat about 270 Hz, which has almost double amplitude. The overall trend seems to be that lowerfrequency poles has higher amplitude than the measured, and higher frequency poles has loweramplitude than measured. This error could probably have something to do with the differencedescribed in the previous section 4.1.

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Figure 4.1: CMIF for all mathematical FRFs

24

4.3 Mathematical vs. measured FRFs

Another evaluation of the generated mathematical FRFs is to compare them to the measured FRFs.The first comparison done is in the driving point (position (1,1)) seen in figure 4.2 below.

The mathematical does not correspond well to the measurement, which indicates that the modelis not accurate at this position of the plate. The mathematical phase is not coherent with themeasurement phase either.

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de [d

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Real

Figure 4.2: Comparison of the mathematical FRF against the measured FRF in point (1,1)

The comparison for position (3,4) can be seen in figure 4.3. It correspond much more with themeasured response at this position, especially between 800 – 2300 Hz. Also the phase is betterthan in the driving point comparison, but still not good.

More comparisons can be seen in figures 4.4 to 4.7. An overall conclusion is that the mathematicalFRFs is mostly accurate for the center positions of the plate. Why this is the case has not beenthoroughly investigated.

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28

4.4 Mathematical vs. measured mode shapes

This section presents the shapes of some of the detected modes. To the left, the mode shapefrom the measurement data is presented, and to the right is the mathematical mode shape. Themodal-parameter model used to create the mathematical FRFs uses an estimate of the mode shapefrom the measurements (section 3.2).

Overall the mode shapes for the lower frequency band looks reasonable, but the “top” and “bot-tom” of the plate indicates possible errors. Intuitively, these rows appear to be 180 degrees out ofphase. The reason for this has not been investigated.

The mode shapes indicates that there are torsional and bending waves present in the modes.

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Figure 4.8: Measured (left) and Mathematical (right) mode shape for 270 Hz

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4.5 Modal Assurance Criterion Matrix (MAC)

The MAC matrix of the mathematical model created with I-DEAS Test can be seen below infigure 4.16. It indicates that the modes are not similar to other modes except for themselves. Thisis the wanted result.

For the Matlab created mathematical modes, the MAC matrix can be found in figure 4.17. Thecode to create this MAC matrix can be found in appendix C.7 on page 46.

Compared to the I-DEAS Test MAC matrix, the model created in Matlab is not optimal. Manymodes are too similar to be trusted. The four most similar modes are listed in table 4.2. Onlymodes 7, 14 and 15 are not detected by I-DEAS Test. If mode 7 is a faulty mode, this couldexplain the similarity between mode 6 and 7.

Figure 4.16: MAC for all mathematical FRFs from I-DEAS Test

Table 4.2: Most similar modesModes Frequency Similarity

Mode 1 Mode 2 Mode 1 Mode 2 MAC Value

4 13 642 2090 0.7367

6 7 966 974 0.3144

9 17 1297 2902 0.2886

5 12 787 1998 0.2226

32

Figure 4.17: MAC for all mathematical FRFs from Matlab

33

4.6 Discussion

By comparing the detected modes (frequency and damping), the Matlab model is fairly close tothe model created with I-DEAS Test. The mathematical FRFs, MAC matrix and the mode shapesall indicate however, that the model is not very good. A reason for this could be that the modeshape estimation process is not very good.

Another reason could be that some of the modes detected are propagating waves through thesurface. In the model used, all modes are handled as standing waves, and might cause problemsif this is the case.

A way to enhance the model is to use the Least Squares Frequency Domain method, describedin section 1.4.2, to iteratively improve the parameters. This process has not been done, or ex-plored further. The method is referred to as “hard” to perform successfully, and requires a lot ofexperience to make the iteration to converge.

There are other approaches available to update the model [5].

34

Appendix A

I-Deas Test stability diagram

Figure A.1: Stability diagram from I-DEAS Test

35

Appendix B

Lists

B.1 Target modal parameters

PARM SHAPE FREQUENCY DAMPING AMPLITUDE PHASE MCF REF,RES

LABEL REC (HERTZ) (%) (RAD)

1 1 272.544 1.437 1.2707E+00 1.571 1.000 1Z-,28Z+

2 2 335.348 0.665 5.1796E+01 1.571 1.000 1Z-,28Z+

3 3 620.261 0.202 2.0356E+02 1.571 1.000 1Z-,28Z+

4 4 644.843 0.090 4.9071E+01 1.571 1.000 1Z-,28Z+

5 5 783.756 0.127 6.0770E+00 1.571 0.999 1Z-,28Z+

6 6 962.906 0.232 1.7722E+01 1.571 1.000 1Z-,28Z+

7 7 1221.286 0.214 5.5194E+01 1.571 1.000 1Z-,28Z+

8 8 1294.681 0.214 5.1247E+02 -1.571 1.000 1Z-,28Z+

9 9 1660.698 0.336 6.8301E-01 1.571 1.000 1Z-,28Z+

10 10 1853.714 0.280 5.8477E+02 -1.571 1.000 1Z-,28Z+

11 11 1999.744 0.287 5.5141E+01 -1.571 1.000 1Z-,28Z+

12 12 2089.418 0.355 4.1940E+01 -1.571 1.000 1Z-,28Z+

13 13 2401.034 0.451 8.8062E+01 -1.571 1.000 1Z-,28Z+

14 14 2901.708 0.268 6.4988E+02 1.571 1.000 1Z-,28Z+

15 15 3023.607 0.212 1.5767E+02 1.571 1.000 1Z-,28Z+

16 16 3093.047 0.435 1.5229E+02 1.571 1.000 1Z-,28Z+

36

B.2 List of detected frequencies and dampings

Pole Frequency Damping Amplitude

(Nr) (Hz) (Rel. dB)

01 270.458 0.2380 37.814

02 331.448 0.7185 22.264

03 621.403 0.1098 81.923

04 641.900 0.0419 119.320

05 787.377 0.0561 89.111

06 965.849 0.1584 75.765

07 973.848 0.1814 77.186

08 1227.31 0.0321 124.489

09 1296.80 0.0687 131.048

10 1664.74 0.1872 74.787

11 1852.21 0.1351 103.584

12 1998.19 0.1134 97.024

13 2090.67 0.2188 68.533

14 2110.17 0.1867 69.634

15 2373.13 0.3822 68.017

16 2399.63 0.2222 89.982

17 2901.55 0.0932 107.287

18 3023.53 0.2727 54.992

19 3094.02 0.4519 57.526

37

Appendix C

Matlab code

C.1 Import FRFs (impfiles.m)

1 % Import all frequency files to workspace2 importfile(’conv/FRF_C1_A1.txt’);importfile(’conv/FRF_C1_A2.txt’);3 importfile(’conv/FRF_C1_A3.txt’);importfile(’conv/FRF_C1_A4.txt’);4 importfile(’conv/FRF_C1_A5.txt’);importfile(’conv/FRF_C2_A1.txt’);5 importfile(’conv/FRF_C2_A2.txt’);importfile(’conv/FRF_C2_A3.txt’);6 importfile(’conv/FRF_C2_A4.txt’);importfile(’conv/FRF_C2_A5.txt’);7 importfile(’conv/FRF_C3_A1.txt’);importfile(’conv/FRF_C3_A2.txt’);8 importfile(’conv/FRF_C3_A3.txt’);importfile(’conv/FRF_C3_A4.txt’);9 importfile(’conv/FRF_C3_A5.txt’);importfile(’conv/FRF_C4_A1.txt’);

10 importfile(’conv/FRF_C4_A2.txt’);importfile(’conv/FRF_C4_A3.txt’);11 importfile(’conv/FRF_C4_A4.txt’);importfile(’conv/FRF_C4_A5.txt’);12 importfile(’conv/FRF_C5_A1.txt’);importfile(’conv/FRF_C5_A2.txt’);13 importfile(’conv/FRF_C5_A3.txt’);importfile(’conv/FRF_C5_A4.txt’);14 importfile(’conv/FRF_C5_A5.txt’);importfile(’conv/FRF_C6_A1.txt’);15 importfile(’conv/FRF_C6_A2.txt’);importfile(’conv/FRF_C6_A3.txt’);16 importfile(’conv/FRF_C6_A4.txt’);importfile(’conv/FRF_C6_A5.txt’);17 importfile(’conv/FRF_C7_A1.txt’);importfile(’conv/FRF_C7_A2.txt’);18 importfile(’conv/FRF_C7_A3.txt’);importfile(’conv/FRF_C7_A4.txt’);19 importfile(’conv/FRF_C7_A5.txt’);importfile(’conv/FRF_C8_A1.txt’);20 importfile(’conv/FRF_C8_A2.txt’);importfile(’conv/FRF_C8_A3.txt’);21 importfile(’conv/FRF_C8_A4.txt’);importfile(’conv/FRF_C8_A5.txt’);22 importfile(’conv/FRF_C9_A1.txt’);importfile(’conv/FRF_C9_A2.txt’);23 importfile(’conv/FRF_C9_A3.txt’);importfile(’conv/FRF_C9_A4.txt’);24 importfile(’conv/FRF_C9_A5.txt’);25

26 % Reconstruct the FRFs by (1, 3, ..., n-1 = REAL), (2, 4, ..., n = IMAG)27 FRF_C1_A1 = 1i*( downsample(FRF_C1_A1 ,2,1))+downsample(FRF_C1_A1 ,2);28 FRF_C1_A2 = 1i*( downsample(FRF_C1_A2 ,2,1))+downsample(FRF_C1_A2 ,2);29 FRF_C1_A3 = 1i*( downsample(FRF_C1_A3 ,2,1))+downsample(FRF_C1_A3 ,2);30 FRF_C1_A4 = 1i*( downsample(FRF_C1_A4 ,2,1))+downsample(FRF_C1_A4 ,2);31 FRF_C1_A5 = 1i*( downsample(FRF_C1_A5 ,2,1))+downsample(FRF_C1_A5 ,2);32 FRF_C2_A1 = 1i*( downsample(FRF_C2_A1 ,2,1))+downsample(FRF_C2_A1 ,2);33 FRF_C2_A2 = 1i*( downsample(FRF_C2_A2 ,2,1))+downsample(FRF_C2_A2 ,2);34 FRF_C2_A3 = 1i*( downsample(FRF_C2_A3 ,2,1))+downsample(FRF_C2_A3 ,2);35 FRF_C2_A4 = 1i*( downsample(FRF_C2_A4 ,2,1))+downsample(FRF_C2_A4 ,2);36 FRF_C2_A5 = 1i*( downsample(FRF_C2_A5 ,2,1))+downsample(FRF_C2_A5 ,2);37 FRF_C3_A1 = 1i*( downsample(FRF_C3_A1 ,2,1))+downsample(FRF_C3_A1 ,2);38 FRF_C3_A2 = 1i*( downsample(FRF_C3_A2 ,2,1))+downsample(FRF_C3_A2 ,2);39 FRF_C3_A3 = 1i*( downsample(FRF_C3_A3 ,2,1))+downsample(FRF_C3_A3 ,2);40 FRF_C3_A4 = 1i*( downsample(FRF_C3_A4 ,2,1))+downsample(FRF_C3_A4 ,2);41 FRF_C3_A5 = 1i*( downsample(FRF_C3_A5 ,2,1))+downsample(FRF_C3_A5 ,2);42 FRF_C4_A1 = 1i*( downsample(FRF_C4_A1 ,2,1))+downsample(FRF_C4_A1 ,2);43 FRF_C4_A2 = 1i*( downsample(FRF_C4_A2 ,2,1))+downsample(FRF_C4_A2 ,2);44 FRF_C4_A3 = 1i*( downsample(FRF_C4_A3 ,2,1))+downsample(FRF_C4_A3 ,2);45 FRF_C4_A4 = 1i*( downsample(FRF_C4_A4 ,2,1))+downsample(FRF_C4_A4 ,2);46 FRF_C4_A5 = 1i*( downsample(FRF_C4_A5 ,2,1))+downsample(FRF_C4_A5 ,2);47 FRF_C5_A1 = 1i*( downsample(FRF_C5_A1 ,2,1))+downsample(FRF_C5_A1 ,2);48 FRF_C5_A2 = 1i*( downsample(FRF_C5_A2 ,2,1))+downsample(FRF_C5_A2 ,2);49 FRF_C5_A3 = 1i*( downsample(FRF_C5_A3 ,2,1))+downsample(FRF_C5_A3 ,2);50 FRF_C5_A4 = 1i*( downsample(FRF_C5_A4 ,2,1))+downsample(FRF_C5_A4 ,2);51 FRF_C5_A5 = 1i*( downsample(FRF_C5_A5 ,2,1))+downsample(FRF_C5_A5 ,2);52 FRF_C6_A1 = 1i*( downsample(FRF_C6_A1 ,2,1))+downsample(FRF_C6_A1 ,2);

38

53 FRF_C6_A2 = 1i*( downsample(FRF_C6_A2 ,2,1))+downsample(FRF_C6_A2 ,2);54 FRF_C6_A3 = 1i*( downsample(FRF_C6_A3 ,2,1))+downsample(FRF_C6_A3 ,2);55 FRF_C6_A4 = 1i*( downsample(FRF_C6_A4 ,2,1))+downsample(FRF_C6_A4 ,2);56 FRF_C6_A5 = 1i*( downsample(FRF_C6_A5 ,2,1))+downsample(FRF_C6_A5 ,2);57 FRF_C7_A1 = 1i*( downsample(FRF_C7_A1 ,2,1))+downsample(FRF_C7_A1 ,2);58 FRF_C7_A2 = 1i*( downsample(FRF_C7_A2 ,2,1))+downsample(FRF_C7_A2 ,2);59 FRF_C7_A3 = 1i*( downsample(FRF_C7_A3 ,2,1))+downsample(FRF_C7_A3 ,2);60 FRF_C7_A4 = 1i*( downsample(FRF_C7_A4 ,2,1))+downsample(FRF_C7_A4 ,2);61 FRF_C7_A5 = 1i*( downsample(FRF_C7_A5 ,2,1))+downsample(FRF_C7_A5 ,2);62 FRF_C8_A1 = 1i*( downsample(FRF_C8_A1 ,2,1))+downsample(FRF_C8_A1 ,2);63 FRF_C8_A2 = 1i*( downsample(FRF_C8_A2 ,2,1))+downsample(FRF_C8_A2 ,2);64 FRF_C8_A3 = 1i*( downsample(FRF_C8_A3 ,2,1))+downsample(FRF_C8_A3 ,2);65 FRF_C8_A4 = 1i*( downsample(FRF_C8_A4 ,2,1))+downsample(FRF_C8_A4 ,2);66 FRF_C8_A5 = 1i*( downsample(FRF_C8_A5 ,2,1))+downsample(FRF_C8_A5 ,2);67 FRF_C9_A1 = 1i*( downsample(FRF_C9_A1 ,2,1))+downsample(FRF_C9_A1 ,2);68 FRF_C9_A2 = 1i*( downsample(FRF_C9_A2 ,2,1))+downsample(FRF_C9_A2 ,2);69 FRF_C9_A3 = 1i*( downsample(FRF_C9_A3 ,2,1))+downsample(FRF_C9_A3 ,2);70 FRF_C9_A4 = 1i*( downsample(FRF_C9_A4 ,2,1))+downsample(FRF_C9_A4 ,2);71 FRF_C9_A5 = 1i*( downsample(FRF_C9_A5 ,2,1))+downsample(FRF_C9_A5 ,2);72

73 % Construct FRF Matrix H74 H = [...75 FRF_C1_A1 ’ FRF_C1_A2 ’ FRF_C1_A3 ’ FRF_C1_A4 ’ FRF_C1_A5 ’...76 FRF_C2_A1 ’ FRF_C2_A2 ’ FRF_C2_A3 ’ FRF_C2_A4 ’ FRF_C2_A5 ’...77 FRF_C3_A1 ’ FRF_C3_A2 ’ FRF_C3_A3 ’ FRF_C3_A4 ’ FRF_C3_A5 ’...78 FRF_C4_A1 ’ FRF_C4_A2 ’ FRF_C4_A3 ’ FRF_C4_A4 ’ FRF_C4_A5 ’...79 FRF_C5_A1 ’ FRF_C5_A2 ’ FRF_C5_A3 ’ FRF_C5_A4 ’ FRF_C5_A5 ’...80 FRF_C6_A1 ’ FRF_C6_A2 ’ FRF_C6_A3 ’ FRF_C6_A4 ’ FRF_C6_A5 ’...81 FRF_C7_A1 ’ FRF_C7_A2 ’ FRF_C7_A3 ’ FRF_C7_A4 ’ FRF_C7_A5 ’...82 FRF_C8_A1 ’ FRF_C8_A2 ’ FRF_C8_A3 ’ FRF_C8_A4 ’ FRF_C8_A5 ’...83 FRF_C9_A1 ’ FRF_C9_A2 ’ FRF_C9_A3 ’ FRF_C9_A4 ’ FRF_C9_A5 ’...84 ];85

86 %H = H(1: round(end/3) ,:); % Use only up to approx 1 kHz87

88 % Set up the frequency scale89 t = 1: length(FRF_C1_A1);90 t = t/t(end)*3200;91 %t = downsample(t,2);92 %t = t(1: round(end/3)); % Use only up to approx 1 kHz

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C.2 Process FRFs (process frfs.m)

1 %===================================================2 % MODAL FREQUENCY AND DAMPING RATIO3 %===================================================4

5 format long6

7 % Start clock8 tic ;9

10 close al l ;11 clear y;12 clear y2;13

14 animation = 0; % Set to ’1’ to animate mode shapes15

16 % Create Complex Mode Indicator Function from all FRFs in H17 h = cmif(H);18

19 % Smoothen the CMIF by a 25 sample long Running Average filter20 hs = smooth(h,25);21

22 % Find Frequencies iteratively23 [freqz ,f_i ,f_l ,f_a] = find_freqs(hs,t);24

25 % Find Damping ratios iteratively26 [d,f_i ,d_i1 ,d_i2] = find_damps(hs,t,f_l);27 d_err = 5; % Damping ’error ’ threshold28

29 % Print out the findings30 fprintf(’Nr Frequency Damping Amplitude\n’);31 lf = length(freqz); % Will be updated in loop32 k = 1; % Reset loop counter33 while k <= lf34 i f d(k) > d_err % If wrong pole (damping factor unrealistic)35 % Remove this pole from list36 d(k) = [];37 freqz(k) = [];38 f_a(k) = [];39 f_l(k) = [];40 lf = length(freqz); % Update length41 else

42 fprintf(’%02g % 7g % 7g % 7g\n’,k,freqz(k),d(k),f_a(k));43 k = k+1;44 end;45 end;46

47 % Plot48 figure;49 plot(t,hs,’b-’); % Plot CMIF50 hold on;51 stem(t,f_i ,’r-.’); % Plot Pole Indicator Lines52 stem(t,d_i1 ,’g--.’); % Plot Left Damping Limit Indicator53 stem(t,d_i2 ,’g--.’); % Plot Right Damping Limit Indicator54 hold off;55 t i t le (’Complex Mode Indicator Function (CMIF) of FRF Matrix [H]’);56 xlabel(’Frequency (Hz)’);57 ylabel(’Amplitude (dB)’);58 axis tight;59

60 outfile = ’../ report/figures/cmif’;61 print(’-depsc’,outfile)62

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78 %===================================================79 % MODE SHAPE VECTORS80 %===================================================81

82 % Step through all detected poles83 for k = 1: length(f_l)84 u = 1;85 % Cycle through all measurement positions86 for j = 1:987 for l = 1:588 % Save the mode shape information to Psi89 Psi(k,j,l) = H(f_l(k),u);90 u = u + 1;91 end;92 end;93 i f ( animation == 1 )94 % Animate the mode (export to .avi -file and save image)95 animate_mode(imag(Psi(k,:,:)),4,freqz(k),sprintf(’anims/real/test%g.avi’,k));96 animate_mode(imag(Psi(k,:,:)) ,3.5,freqz(k));97 outfile = [’../ report/figures/modeshape_ ’ num2str(round(freqz(k)))];98 print(’-depsc’,outfile)99 end;

100 end

101

102 %===================================================103 % MATHEMATICAL FREQUENCY RESPONSE FUNCTIONS104 %===================================================105

106 y = zeros(length(freqz) ,45,length(t));107

108 for k = 1: length(freqz)109 Resid = squeeze(Psi(k,:,:));110 u = 1;111 for m = 1:9112 for n = 1:5113 % Calculate the mode shape scalar for the specific position114 PsiL = Resid(:,n)*Resid(m,:);115 % Create the SDOF modal model for this position and mode116 y(k,u,:) = sdof(t,freqz(k) ,20*d(k),Psi(k,m,n) ,1/f_a(k));117 u = u+1;118 end

119 end

120 end

121

122 ys = squeeze(sum(y,1)); % SDOF summed123

124 % Somehow the real part of the generated frf is the imaginary ..125 % Let ’s flip them..126 for k = 1:45127 ys_temp(k,:) = real(ys(k,:));128 ys(k,:) = imag(ys(k,:));129 ys(k,:) = ys(k,:) - 1i.* ys_temp(k,:);130 end;131

132 %===================================================133 % MATHEMATICAL MODE SHAPE VECTORS134 %===================================================135 for k = 1: length(f_l)136 u = 1;137 for j = 1:9138 for l = 1:5139 Psi_Math(k,j,l) = ys(u,f_l(k));140 PsiG(k,u) = Psi(k,j,l); % In long vector form141 u = u + 1;142 end;143 end;144 i f ( animation == 1 )145 animate_mode(imag(Psi_Math(k,:,:)),4,freqz(k),...146 sprintf(’anims/math/test_math%g.avi’,k));147 animate_mode(imag(Psi_Math(k,:,:)) ,3.5,freqz(k));148 outfile = [’../ report/figures/modeshape_math_ ’ num2str(round(freqz(k)))];149 print(’-depsc’,outfile)150 end;151 end

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158 %===================================================159 % PLOTTING160 %===================================================161

162 % Create the CMIF for the generated FRF matrix ’ys’ (the summed sdofs)163 figure;164 yc = cmif(ys ’);165

166 % Smoothen the CMIF by a 25 sample long Running Average filter167 ycmif = smooth(yc ,25);168

169 % Plot the CMIF170 plot(t,ycmif./max(abs(ycmif)).*max(abs(hs)),t,hs);171 t i t le (’Reconstructed FRF CMIF from SDOF Data’);172 xlabel(’Frequency (Hz)’);173 ylabel(’Amplitude ’);174 axis tight;175 legend(’Reconstructed ’,’CMIF’,’Location ’,’Best’);176 outfile = [’../ report/figures/cmif_generated ’];177 %print(’-depsc ’,outfile)178

179 % Plot all generated FRFs against the measured FRFs , including phase response180 l = 1;181 for m = 1:9182 for n = 1:5183 clear y184 y = squeeze(ys(l,:));185

186 figure (5);187 subplot(2,1,1);188 plot(t,angle(y),’b-’,t,angle(H(:,l)),’r-’); %.*2+pi189 t i t le ([’Comparison Reconstructed FRF and Real FRF (’ num2str(m) ’,’ num2str(n) ’)’]);190 xlabel(’Frequency (Hz)’);191 ylabel(’Phase [rad]’);192 axis tight;193 legend(’Math’,’Real’,’Location ’,’Best’);194

195 subplot(2,1,2);196 plot(t,real(y)./max(abs(real(y))).*max(abs(real(H(:,l)))),’b-’,t,real(H(:,l)),’r-’);197 t i t le ([’Comparison Reconstructed FRF and Real FRF (’ num2str(m) ’,’ num2str(n) ’)’]);198 xlabel(’Frequency (Hz)’);199 ylabel(’Magnitude [dB (m/s^2/N)]’);200 axis tight;201 legend(’Math’,’Real’,’Location ’,’Best’);202 outfile = [’../ report/figures/frf_generated_ ’ num2str(l)];203 %print(’-depsc ’,outfile)204 l = l+1;205 end

206 end

207

208 % Generate MAC Matrix209 MAC = mac(PsiG ,freqz);210

211 % How much time is elapsed?212 toc;

42

C.3 Complex Mode Indicator Function (cmif.m)

1 function h = cmif(H)2 % Function to compute the Complex Mode Indicator Function (CMIF)3 %4 % h = cmif(H);5 %6 % It will compute for the FRF matrix ’H’, with dimensions7 % number_of_linex x number_of_responses x number_of_references8 %9

10 % Extract the resolution of the given FRF matrix11 number_of_lines = size(H,1);12 number_of_responses = size(H,2);13 number_of_references = size(H,3);14

15 % Allocate memory for the result16 h = zeros(number_of_references ,number_of_lines);17

18 % Step through all values19 for i = 1: number_of_lines20 % Calculate the Singular Value Decomposition of the squeezed matrix H21 % Squeeze removes singleton dimensions (if only m x n matrix , k dim rem.)22 h(:,i) = svd(abs(squeeze(H(i,:,:))));23 end

43

C.4 Find frequencies (find freqs.m)

1 function [f,f_i ,f_l ,f_a] = find_freqs(H,F);2 %3 % Function to iteratively find poles of a sampled frf.4 % Syntax:5 % [f] = find_freqs(H,F)6 % where:7 % f : Output vector containing the found freqency poles8 % f_i: Pole indicator lines output9 % f_l: Pole indicator lines sample position output

10 % f_a: Amplitude of the pole11 %12 % H : Input FRF data13 % F : Input Frequency line data14 %15

16 k = 1;17 u = 1;18

19 % Pole indicator lines for plot20 f_i = zeros(length(H) ,1);21

22 for i = 1+u : length(H) - u23 i f H(i+u) < H(i)24 i f H(i-u) < H(i)25 f(k) = F(i);26 f_i(i) = max(H);27 f_l(k) = i;28 f_a(k) = H(i);29 k = k + 1;30 end

31 end

32 end

44

C.5 Find dampings (find damps.m)

1 function [d,f_i ,d_i1 ,d_i2] = find_damps(H,F,FP)2 %3 % Function to iteratively find damping and poles of a sampled frf.4 % Syntax:5 % [d, f_i , d_i1 , d_i2] = find_damps(H,F)6 % where:7 % d : Output vector containing the found dampings8 % f_i : Pole indicator lines output9 % d_i1: Left freq line for pole

10 % d_i2: Right freq line for pole11 %12 % H : Input FRF data13 % F : Input Frequency line data14 % FP : Input Pole Location data15

16 p = 1;17 thr = 10;18

19 % Pole indicator lines for plot20 f_i = zeros(length(H) ,1);21 d_i1 = zeros(length(H) ,1);22 d_i2 = zeros(length(H) ,1);23

24 % Step through all detected frequencies25 for i = 1 : length(FP)26 % Reset parameters27 k = FP(i); % Start sample position for this frequency28 d1 = 0; % Frequency for damping limit left29 d2 = 0; % Frequency for damping limit right30 f_i(k) = max(H);31 while ((H(k) > H(FP(i))-thr) && ((H(k)) > 0) && (k > 1)) % Find limits32 k = k - 1;33 end;34 k1 = k; % Save left limit sample position35 d1 = F(k); % Save frequency of this position36

37 k = FP(i);38 while ((H(k) > H(FP(i))-thr) && ((H(k)) > 0) && (k < length(H) -1))39 k = k + 1;40 end;41 k2 = k;42 d2 = F(k);43

44 % Try to correct if there are many poles close to each other by45 % approximating the correct damping. This is done by using the46 % pole as center location and adjusting the limit that is far away47 % to be the same length from the pole as the ’correct ’ limit.48 i f ( abs(FP(i)-k2) > abs(FP(i)-k1) )49 d2 = F(FP(i) + abs(FP(i)-k1));50 d_i2(FP(i) + abs(FP(i)-k1)) = max(H);51 k2 = FP(i) + abs(FP(i)-k1);52 else

53 d_i2(k2) = max(H);54 end

55 i f ( abs(FP(i)-k1) > abs(FP(i)-k2) )56 d1 = F(FP(i) - abs(k2-FP(i)));57 d_i1(FP(i) - abs(FP(i)-k2)) = max(H);58 k1 = FP(i) - abs(FP(i)-k2);59 else

60 d_i1(k1) = max(H);61 end

62

63 % Calculate damping ratio for the specified pole64 d(p) = (d2 - d1)/H(FP(i));65

66 % Go to next pole67 p = p+1;68

69 end

45

C.6 SDOF modal model (sdof.m)

1 function ret = sdof(w, wd, s, Psi , Q)2 %3 % Function to create a SDOF modal model.4 % Syntax:5 % ret = sdof(w, wd, s, Psi , Q)6 % where:7 % w : Frequency vector8 % wd : Damped natural resonance frequency9 % s : Damping factor

10 % Psi : Mode shape value11 % Q : Scaling factor12

13 % Calculate residuals14 A = Q*Psi;15 Ac = conj(Q)*conj(Psi);16

17 L = s + i*wd;18 Lc = s - i*wd;19

20 for k = 1: length(w)21 ret(k) = (A/(i*w(k) - L)) + (Ac/(i*w(k) - Lc));22 end;

46

C.7 MAC matrix (mac.m)

1 function MAC = mac(Psi ,freqz ,RES)2 %3 % Modal Assurance Critereon Matrix Function4 % Takes a modal shape matrix in form:5 % Psi = [Frequency , Mode Shape Vector]6 % Optional parameter freqz , containing the frequency information for axis.7 % Optional parameter RES , for plotting resolution (if <= 0, no plot is generated)8 %9 % Syntax:

10 % MAC = mac(Psi , freqz , RES);11 %12

13 % Find out how many frequencies14 Freqs = size(Psi ,1);15

16 % If the optional freqz argument is not existant17 i f nargin < 218 freqz = 1:Freqs;19 end;20

21 i f nargin < 322 RES = 100; % Standard resolution for plot23 end;24

25 % Allocate memory26 MAC = zeros(Freqs);27

28 % Compare each frequency29 for u = 1:Freqs30 for l = 1:Freqs31 PsiRed1 = squeeze(Psi(u,:)); % Create temorary vectors32 PsiRed2 = squeeze(Psi(l,:));33 % Calculate MAC value34 MAC(u,l) = (PsiRed1*PsiRed2 ’)^2 / (( PsiRed1*PsiRed1 ’)*( PsiRed2*PsiRed2 ’));35 end

36 end

37

38 MAC = abs(MAC);39

40 i f ( RES > 0 )41 % For a nicer stem 3d plot ...42 MAC_PLOT = zeros(Freqs*RES);43 for u = 0:Freqs -144 for l = 0:Freqs -145 MAC_PLOT(u*RES +1:(u+1)*RES -(RES/2),l*RES +1:(l+1)*RES -(RES/2)) = MAC(u+1,l+1);46 end;47 end;48

49 figure;50 mesh(MAC_PLOT);51 axis square;52 axis tight;53 set(gca,’XTick’,round(RES/2):RES:Freqs*RES+round(RES/2))54 set(gca,’YTick’,round(RES/2):RES:Freqs*RES+round(RES/2))55 set(gca,’XTickLabel ’,freqz)56 set(gca,’YTickLabel ’,freqz)57 colormap(hsv)58 view(57 ,43);59 light(’Position ’ ,[8,-8,20])60 lighting gouraud61

62 t i t le (’Modal Assurance Critereon (MAC) Matrix’);63 xlabel(’Frequency (Hz)’);64 ylabel(’Frequency (Hz)’);65 zlabel(’Value’);66 outfile = [’../ report/figures/mac_generated ’];67 print(’-depsc’,outfile)68

69 end;

47

C.8 Animate mode (animate mode.m)

1 function y = animate_mode(Psi , Maxtime , Freq , Name)2 %3 % Function to animate a mode shape , given it’s matrix form4 % Syntax:5 % y = animate_mode(Psi , Maxtime , Freq , Name);6 % where:7 % Psi : The mode shape matrix in the size of the measurement grid8 % Maxtime: Animation time9 % Freq : Frequency to animate (just for printing the title)

10 % Name : For exporting to .avi -file11 %12

13 i f nargin < 314 Freq = NaN;15 end;16

17 i f nargin < 418 e = 0;19 Name = ’NaN’;20 else

21 aviobj = avifile(Name ,’fps’ ,30);22 e = 1;23 end

24

25 % Remove singleton dimensions26 Psi = squeeze(Psi);27 % Normalize28 Psi = Psi./max(max(abs(abs(Psi))));29

30 la = length(-pi:.1: Maxtime*pi);31

32 k = 0;33 fprintf ([ sprintf(’Exporting Mode Animation %6g Hz to ’,Freq) Name ’ .’]);34 for x=-pi:.1: Maxtime*pi

35 figure (1)36 surf(Psi.*sin(x));37 colormap hot;38 axis equal;39 zlim([-1 1]);40 t i t le (sprintf(’Mode Animation: %6g Hz’,Freq));41 xlabel(’Nodes X’);42 ylabel(’Nodes Y’);43 i f e == 144 frame = getframe(gca);45 aviobj = addframe(aviobj ,frame);46 end;47 k = k+1;48 i f (mod(la,k/la) == 0)49 fprintf(’.’);50 end;51 end

52

53 fprintf(’\n\n’);54

55 i f e == 156 aviobj = close(aviobj);57 end;58

59 % Don ’t return anything useful60 y = 0;

48

Bibliography

[1] O. Dossing and B. & Kjaer, “Structural testing,” April 1988.

[2] P. Sas and W. Heylen, eds., Theory and Practice, International Seminar on Modal Analysis,(Heverlee), September 1993.

[3] “Vibration.” http://en.wikipedia.org/wiki/Vibration, October 2009.

[4] P. Avitabile, “Modal space - in our little world (sem experimental techniques),” August 1999.

[5] R. J. Allemang and D. L. Brown, A Unified Matrix Polynomial Approach to Modal Identi-

fication. Journal of Sound and Vibration 211(3), 301-322: Academic Press Limited, March1998.

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