Received in revised form

18 December 2008

Accepted 20 August 2009

assembly of household refrigerators and food freezers are estimated using measured

and numeric frequency response functions (FRFs). The mathematical models are

ich b

terialured

l, bydetermining Poissons ratio and complex dynamic Youngs modulus of a small beam-like specimen subject to seismic

shaker. The longitudinal and the transversal deformations are measured by strain gauges to get Poissons ratio, whereas the

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

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Mechanical Systems and Signal Processing 24 (2010) 4064150888-3270/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2009.08.005 Corresponding author at: Pontifcia Universidade Catolica do Parana PUCPR, Programa de Pos-Graduac- ~ao em Engenharia Mecanica, Rua ImaculadaConceic- ~ao, 1155, CEP: 80215-901, Curitiba, Parana, Brazil.

E-mail addresses: nilson.barbieri@pucpr.br (N. Barbieri), renato.barbieri@pucpr.br (R. Barbieri), luiz.winikes@ig.com.br (L.C. Winikes).excitation, together with the theoretical background. The same experimental device is used basically for both kinds of tests:the specimen is instrumented, placed into a temperature-controlled chamber and excited by means of an electrodynamicmodeling of sandwich structures has been studied extensively, but less attention has been paid to their maidentication [1]. The work proposes an inverse method for the material identication of sandwich beams by measexural resonance frequencies.

Caracciolo et al. [2] presented an experimental technique for completely characterizing a viscoelastic materiaacknowledgment of the mechanical properties, even when the structure is submitted to dynamic loads. For householdrefrigerators and food freezers, one of the main complaints to the customer care centers is related to noise generation, thatis related most of the times with vibration of the cabinet that produces sound irradiation from internal components likeshelves and containers, leaking to the outside of the unit.

The efcient numerical models are necessary to simulate (estimate) the dynamical behavior of such systems. When thecomplex sandwich beams are used the physical parameters are difcult to be estimated.

Sandwich structures are extensively used in engineering because of their high specic stiffness and strength. TheKeywords:

Sandwich beam

Genetic Algorithm

Amplitude correlation coefcient

Parameter updating

1. Introduction

The proved efciency of sandwphysical parameters are estimated using the amplitude correlation coefcient and

genetic algorithm (GA). The experimental data are obtained using the impact hammer

and four accelerometers displaced along the sample (cantilevered beam). The

parameters estimated are Youngs modulus and the loss factor of the Polyurethane

rigid foam and the High Impact Polystyrene.

& 2009 Elsevier Ltd. All rights reserved.

eams and its current usage in a growing rate demands a higher level ofAvailable online 22 August 2009obtained using the nite element method (FEM) and the Timoshenko beam theory. TheParameters estimation of sandwich beam model with rigidpolyurethane foam core

Nilson Barbieri a,b,, Renato Barbieri a, Luiz Carlos Winikes b

a Pontifcia Universidade Catolica do Paran a PUCPR, Programa de Pos-Graduac- ~ao em Engenharia Mecanica, Rua Imaculada Conceic- ~ao, 1155,

CEP: 80215-901, Curitiba, Paran a, Brazilb Universidade Tecnologica Federal do Paran a UTFPR, Brazil

a r t i c l e i n f o

Article history:

Received 26 February 2007

a b s t r a c t

In this work, the physical parameters of sandwich beams made with the association of

hot-rolled steel, Polyurethane rigid foam and High Impact Polystyrene, used for the

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

small number of vibration measurements. Sensitivity analysis to investigate the effects of experimental variables on the

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N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415 407measured properties and to study optimal sensor locations of the vibration measurements is performed. Using thedeveloped methods, the complex bending and shear stiffnesses of sandwich beams of different core materials and apolymer beam are measured. Continuous variations of the measured bending and shear stiffnesses and their loss factorswith frequency were obtained. To further illustrate the measurements of frequency-dependent variation of dynamicproperties of complex structures, the damping of structural vibration using porous and granular materials is investigated.

Kim and Kreider [4] studied the parameter identication in nonlinear elastic and viscoelastic plates by solving aninverse problem numerically. The material properties of the plate, which appear in the constitutive relations, are recoveredby optimizing an objective function constructed from reference strain data. The resulting inverse algorithm consists of anoptimization algorithm coupled with a corresponding direct algorithm that computes the strain elds given a set ofmaterial properties. Numerical results are presented for a variety of constitutive models; they indicate that themethodology works well even with noisy data.

Pintelon et al. [5] analyzed the stressstrain relationship of linear viscoelastic materials characterized by a complex-valued, frequency-dependent elastic modulus (Youngs modulus). Using system identication techniques it is shown theelastic modulus can be measured accurately in a broad frequency band from forced exural (transverse) and longitudinalvibration experiments on a beam under freefree boundary conditions. The approach is illustrated on brass, copper,plexiglass and PVC beams.

Yang et al. [6] analyzed the vibration and dynamic stability of a traveling sandwich beam using the nite elementmethod. The damping layer is assumed to be linear viscoelastic and almost incompressible. The extensional and shearmoduli of the viscoelastic material are characterized by complex quantities. Complex-eigenvalue problems are solved bythe state-space method, and the natural frequencies and modal loss factors of the composite beam are extracted. Theeffects of stiffness and thickness ratio of the viscoelastic and constrained layers on natural frequencies and modal lossfactors are reported. Tension uctuations are the dominant source of excitation in a traveling sandwich material, and theregions of dynamic instability are determined by modied Bolotins method. Numerical results show that the constraineddamping layer stabilizes the traveling sandwich beam.

Singh et al. [7] formulated a system identication procedure for estimation of parameters associated with adynamic model of a single-degree-of-freedom foam-mass system. Ohkami and Swoboda [8] presented two parameteridentication procedures for linear viscoelastic materials. Chang [9] uses the Genetic Algorithm for parameters estimationof nonlinear systems.

Backstrom and Nilsson [10] indicate the need for simple methods describing the dynamics of these complex structures.By implementing frequency-dependent parameters, the vibration of sandwich composite beams can be approximatedusing simple fourth-order beam theory. A higher-order sandwich beam model is utilized in order to obtain estimates ofthe frequency-dependent bending stiffness and shear modulus of the equivalent BernoulliEuler and Timoshenko models.The resulting predicted eigenfrequencies and transfer accellerance functions are compared to the data obtained from thehigher-order model and from measurements. It can be noticed that for lower order wavenumber the ordinary Timoshenkotheory and the higher-order theory show satisfactory agreement.

In this work, the physical parameters of sandwich beams made with the association of hot-rolled steel, Polyurethanerigid foam and High Impact Polystyrene, used for the assembly of household refrigerators and food freezer are estimatedusing measured and numeric frequency response functions (FRFs). The mathematical models are obtained using the niteelement method (FEM) and the Timoshenko beam theory. The physical parameters are estimated using the amplitudecorrelation coefcient [11] and Genetic Algorithm (GA) [9]. The experimental data are obtained using the impact hammerand four accelerometers displaced along the sample (cantilevered beam). The parameters estimated are Youngs modulusand the loss factor of the Polyurethane rigid foam and the High Impact Polystyrene. To estimate the initial values of theparameters, separate tests were conduced using cantilevered beams of Polyurethane rigid foam and High ImpactPolystyrene.

2. Mathematical model

A lot of research has been done on nite element models of cantilever beams based on EulerBernoulli beam theory. InEulerBernoulli beam theory the assumption made is, plane cross section before bending remains plane and normal to theneutral axis after bending. This assumption is valid if length to thickness ratio is large and for small deection of beam.However, if length to thickness ratio is small, plane deection before bending will not remain normal to the neutral axisafter bending. In practical situations a large number of modes of vibrations contribute to the structures performance.vertical displacement of the specimen and the acceleration of the support are measured to get Youngs modulus of thetested material. The experimental curves of Poissons ratio and of Youngs modulus, obtained at different temperatures, arethen gathered into a unique master curve by using the reduced variables method. The two master curves, respectively,represent Poissons ratio and Youngs modulus for the tested material in a very broad frequency range.

Park [3] used experimental methods to measure frequency-dependent dynamic properties of complex structures.Flexural wave propagations are analyzed using the Timoshenko beam, the classical beam, and the shear beam theories.Wave speeds, bending and shear stiffnesses of the structures are measured through the transfer function method requiring

EulerBernoulli beam theory gives inaccurate results for higher modes of vibration. Timoshenko beam theory corrects thesimplifying assumptions made in EulerBernoulli beam theory. In this theory cross sections remain plane and rotate aboutthe same neutral axis as the EulerBernoulli model, but do not remain normal to the deformed longitudinal axis. Thedeviation from normality is produced by a transverse shear that is assumed to be constant over the cross section. Thus theTimoshenko beam model is superior to the EulerBernoulli model in precisely predicting the beam response [10] for lowernumber of vibration modes.

The equation of motion for the vibration of a beam according to the Timoshenko beam theory [12] is

D@4w

@x4 r @

2w

@t2 r

SD

@4w

@x2@t2R @

4w

@t4

!R @

4w

@x2@t2 f x eiot 1

wherew(x,y) is the transverse displacement of the beam from an equilibrium state, D is bending stiffness, r* is the mass perunity of surface, S is the shear stiffness, R is the rotational inertia, x is the coordinate along the beam axis, t is the time, f(x)p

ARTICLE IN PRESS

N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415408is the amplitude of the external generalized force along the beam span, o is the excitation frequency and i 1.The dimensions and parameters of sandwich beam showed in Fig. 1a are: E1 is Youngs modulus, r1 is the density and t1

is the thickness (steel); E2 is Youngs modulus, r2 is the density and t2 is the thickness (High Impact Polystyrene); Ec isYoungs modulus, Gc is the shear modulus, rc is the density and tc is the thickness (Polyurethane rigid foam), e is theposition of the neutral line, d is the distance between centerline of the steel and High Impact Polystyrene beam and z is theposition of the reference axis. According to Fig. 1a and theory of sandwich beam [12]:

d tc t12 t2

22

e E1t1t12 tc

t22

Ectc

tc2 t2

2

=E1t1 Ectc E2t2 3

D E1t31

12 E2t

32

12 Ect

3c

12 E1t1d e2 E2t2e2 Ectc

tc t22

e 2

4

r r1t1 r2t2 rctc 5

R r1t31

12 r2t

32

12 rct

3c

12 r1t1d e2 r2t2e2 rctc

tc t22

e 2

6

S Gcd2

tc7

Making w=w(x,t) an harmonic function, it is possible to admit that:

wx; t Wx eiot 8

Replacing Eq. (8) into (1) we obtain

D@4W

@x4 ro2Wx r

SDo2 @

2W

@x2Ro4Wx

!Ro2 @

2W

@x2 f x 9

e z

d

t 1

t 2

t c 1 2

L

w1 w2

Fig. 1. (a) Sandwich beam geometric parameters and (b) nite element d.o.f.

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N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415 409The exact solution W(x) needs to satisfy (9) at every point x and in general is unknown. To overcome this problem theapproximate solution ~W(x) is used. This approximate solution is interpolated over a nite element (Fig. 1b), with 2 nodesaccording to the expression:

~W x ffqg 10

where [/(x)] is the shape function matrix (14) and the four fj(x) are the well-known Hermitian interpolation functions[13] for C1 continuity. The vector {q} is the generalized displacement vector, {q}={w1, y1, w2, y2}t where wi denote the nodaldisplacement and yi is the rotation at element node i.

Replacing the approximate solution into (9) it introduces an residual error, e(x,o), which is minimized usingthe Galerkin weighted residual method. In mathematical terms, the residual error is made orthogonal to the weightfunctions:

Z L0ft D @

4 ~W

@x4 ro2 ~W x r

SDo2 @

2 ~W

@x2Ro4 ~W x

!Ro2 @

2 ~W

@x2 f x

!dx 0 11

where L is the element length.After some manipulations obtains the standard nite element equation:

Keofqg fFg 12

where

Keo DK o2 rM rDS

R

Ks

o4 rRS

M 13

K 1L3

12 6L 12 6L6L 4L2 6L 2L212 6L 12 6L6L 2L2 6L 4L2

26664

37775 14

Ks 1

30L

36 3L 36 3L3L 4L2 3L L236 3L 36 3L3L L2 3L 4L2

26664

37775 15

M l420

156 22L 54 13L22L 4L2 13L 3L54 13L 156 22L

13L 3L 22L 4L2

26664

37775 16

and

fFg Z L

0f xft dx 17

2.1. Numerical estimation methods

To approximate the experimental and numeric FRF data, the predictor-corrector updating technique [11] based on twocorrelation coefcients (shape and amplitude) and their sensitivities can be used. In this work, only the amplitudecorrelation coefcient is used. This coefcient is dened as

waok 2jfHXokgT fHAokgj

fHXokgTfHXokg fHAokgTfHAokg18

where HX(ok) and HA(ok) are the measured and predicted response vectors at matching excitation/response locations.

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N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415410In cases where the measurements are use directly and/or a damped nite element solution is available, the responseswill be complex and the corresponding sensitivity is [11]

@waok@j

2jfHXgTfHAgjfHXgTfHXg fHAgTfHAg

RefHXgT HAf g RefHXgT@RefHAg

@j IfHXgT @IfHAg

@j

IfHXgT HAf g RefHXgT

@IfHAg@j IfHXg

T @RefHAg@j

2jfHXgTfHAgj2

fHXgT fHXg fHAgT fHAgIfHAgT

@IfHAg@j

RefHAgT@RefHAg

@j

19

where Re real and I imaginary parts of responses.It is therefore proposed to make use of wa(ok) and its sensitivity in a combined manner to improve the overall level of

correlation. Based on a truncated Taylor series expansion, one can write therefore one equation for frequency point ok:

1 waok @waok

@j1@waok

@j2 @waok

@jNj

" #1Nj

Dj 20

where Nj is the number of updating parameters and Eq. (20) is recognized to be in the standard form of sensitivity-basedmodel updating formulations:

feg SfDjg 21

An extended weighted least-square approach is proposed which minimizes:

Jfjg fegT Wf feg fDjgT WjfDjg 22

where [Wf] and [Wj] are diagonal weighting matrices for the frequency points and updating parameters, respectively.Some frequency points should be excluded. Eq. (20) use implicitly functions of the predicted FRFs and their sensitivities.These are not dened at resonances and care should be taken to avoid them. As the updating computations progress,updating parameter changes may lead to changes in natural frequencies and therefore, the frequency points to be excludedgenerally vary from one iteration to another [11]. The elimination of derivatives, either analytically or numericallyevaluated not only reduces either formulation and programming time or function evaluation computer time, but alsomakes the strategy less sensitive to irregular or discontinuous functions. The principal problem is the choice of Dji. Itshould be as small as possible to keep the inherent approximation error small. However, it is made too small, then roundofferror may creep in, particularly when the derivatives are small in value. It is not surprising then that the direct searchmethods have consistently done well, competing successfully with the more sophisticated rst-order methods that requirederivatives [14].

Another update method uses Genetic Algorithm (GA). This method is vastly used and it is based on evolutionarybiological process [9] and the GAs parameters used in this application are: mutation rate=0.02; population size=50 andnumber of generations=5000. The objective function is dened by

f Xnpi1

jFRFexp FRFFEMj 23

where FRFexp is the FRF experimental obtained with laser transducer or accelerometer; FRFFEM is the numeric FRF estimatedusing GA or amplitude correlation coefcient; np is the number of points (normally np=1600). The data were collectedusing frequency range varying from 0 to 400Hz and the frequency increment Do=0.25Hz. A norm-1 cost function has beenchosen, although this may not be the optimal choice. A norm-2 is often seen in mathematical literature, i.e. the least-squaresmethod, but the analytical advantage of continuous derivatives is assumingly irrelevant when utilizing computationalmethods. Also, with norm-2, erroneous data could have a disproportionately large inuence on the nal result [15]. The GAwas used to minimize cost function (23) due to faster convergence than the amplitude correlation coefcient method. It isshown in the literature that GA always converges to a maximum or minimum global. This fact is not easily achieved byoptimization methods that use gradients. This is one of the major advantages of GA approach. While the authors do notknow any popular modal estimator using the GA approach, in this work the viability of this application is showed.

FRF numeric is obtained using frequency sweeping in the range of interest with the same increment of themeasurements. The nal nite elements system of equations is solved for each frequency and the unitary external forceapplied in the node of the excitation. Adopting this procedure, the FRFs numeric values correspond to the acceleration ofthe nodes of xation of accelerometers because the amplitude of force is unitary.

3. Results

To validate the mathematical model was considered the three layered cantilever sandwich beam model and the resultsare compared with Helgesson [16], Mead and Markus [17], Ahmed [18], and Sakiyama et al. [19]. The beam specicproperties such as width, height, modulus of elasticity, etc., are chosen to be the same as the former investigators in orderto be able to compare the results.

The specic dimensions and material properties of the sandwich beam are:Length of beam L=0.7112m; modulus of elasticity E=6.891010N/m2; core shear modulus G=82.68106N/m2;

thickness of faces h1=h3=0.45720mm; thickness of core h2=12.7mm; core density r2=32.8 kg/m3; face densityr1=r3=2680 kg/m3; and width of beam=1m.

The out come of the program and analysis are natural modes, frequencies. The numerical results of the natural modefrequencies or eigenfrequencies are presented and compared with earlier investigators in Table 1.

The resulting natural frequencies in Table 1 compare quite well with the results obtained by earlier investigators,principally with Helgesson. Worth noting is that the obtained frequencies are always belowMead and always above Ahmedand Sakiyama. The fact that frequencies of Mead are always above the results from this work is due to the fact that Meadsimplies the harmonic forces by using complex damped modes. Ahmed has not considered axial forces and the theorypresented is therefore softer than the present; hence the frequencies are below the present. Sakiyama has derived thegoverning differential equations by using the Green function and complex shear module G=G0(1+iv) [16]. Fig. 2 shows theconvergence rate for the rst vibration mode. The fast convergence is obtained using the mathematical model with 300nite elements. Higher modes in Table 1 presented the same convergence behavior.

The experimental sample of sandwich beam made with the association of hot-rolled steel, Polyurethane rigid foam andHigh Impact Polystyrene is shown in Fig. 3. The thickness of the steel is 0.6mm; the polyurethane is 38.25mm and thepolystyrene is 1.25mm and the beam width is 39.18mm.

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Table 1Natural frequency (Hz) and comparison with earlier authors.

Mode Present Helgesson Mead Ahmed Sakiyama

1 33.75 33.75 34.24 32.79 33.15

2 198.81 198.77 201.85 193.50 195.96

3 511.45 511.27 520.85 499 503.43

4 905.34 904.87 925.40 886 893.28

5 1346.58 1345.60 1381.30 1320 1328.50

6 1811.94 1810.30 1867 1779 1790.70

7 2288.17 2285.80 2374 2249 2260.20

8 2767.82 2764.70 2905 2723 2738.90

9 3247.08 3243.30 3212.80

N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415 4110 100 200 300 400 500 600 70033.74

33.75

33.76

33.77

33.78

33.79

33.8

33.81

33.82

Number of finite elements

Freq

uenc

y (H

z)

Fig. 2. Convergence rate.

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N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415412150

150A 4

A 3

polyurethane

steel

polystyrene The experimental data are obtained using the impact hammer and four accelerometers displaced along thesample (A1A4). Fig. 4 shows the typical FRF curve obtained with the accelerometer placed at the middle of sample (A2).

The rational fraction polynomial (RFP) [20] method was used to estimate the damping ratio (x) and the naturalfrequencies (o) of the three mode shapes. Table 2 shows the values of these parameters to the four accelerometers.

The position of the accelerometer A3 is near to the nodal point of the second mode shape. This justies the resultssuppressed in Table 2.

According to the results shown in Table 2, it tried to estimate some physical parameters of the system: Youngs modulusand the loss factor of the Polyurethane rigid foam and the High Impact Polystyrene.

The loss factor Z was estimated considering complex Youngs modulus:

E E1 jZ

150

150A2

A1

Rigid base

Dimensions (mm)

Fig. 3. Sandwich beam.

0 50 100 150 200 2500

2

4

6

8

10

12

14

Freuency (Hz)

FRF

ampl

itude

(ms2 /

N)

Fig. 4. Experimental FRF.

To obtain initial values of these parameters separated studies were conduced to the Polyurethane rigid foam and HighImpact Polystyrene. The rst approximation to the loss factor was Z=x0.05.

Fig. 5 shows the experimental specimen for the High Impact Polystyrene and Polyurethane rigid foam cantilever beam.The High Impact Polystyrene beam dimensions and material property are: length L=0.145m; width=0.02m;thickness=0.0018m; density r=1040kg/m3. The mini shaker was used to the beam random excitation with thefrequency range varying from 0 to 400Hz. One accelerometer (PCB model 353B18) and one laser velocity transducer (B&Kmodel 3544) were used to collect the vibration data. The accelerometer was placed at the base excitation point (shaker)and the laser sensor at the position d=0.135m. Fig. 6 shows the experimental and estimated curves of the acceleration ratio(non-dimensional parameter) Paa (acceleration of laser sensor/acceleration of the accelerometer). The parameters wereestimated using the Genetic Algorithm and the objective function as being the difference between the values of theexperimental and numeric Paa. The acceleration ratio had been used in the same way that in [2] and also due to the lowweight of the material that could cause errors in the measurements of the low variation of dynamic loads.

ARTICLE IN PRESS

L

d

laser sensor

shaker clamped edge

free edge

accelerometer

Fig. 5. Experimental specimen of cantilever beam with position sensor.

0

101

102

n ra

tio P

aa

Estimated Experimental

Table 2Experimental damping ratio and natural frequencies.

Mode shape Accelerometers

A1 A2 A3 A4

o (Hz) x o (Hz) x o (Hz) x o (Hz) x

1 25.42 0.050 25.39 0.050 25.33 0.052 25.32 0.053

2 109.41 0.0171 109.45 0.0165 109.39 0.0167

3 223.97 0.0154 224.00 0.0152 224.05 0.0159 224.18 0.0157

N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415 4130 50 100 150 200 250 300 350 400102

101

10

Frequency (Hz)

Acc

eler

atio

Fig. 6. Acceleration ratio curves of High Impact Polystyrene cantilever beam.

The parameters updated are Youngs modulus of the High Impact Polystyrene and the loss factor. The optimal valuesfound for these parameters are E=1.425GPa and Z=0.045.

The Polyurethane rigid foam cantilever beam dimensions and material property are: length L=0.225m, width=0.03m,thickness=0.03m and density r=29kg/m3. The laser sensor position is d=0.1125m. Fig. 7 shows the experimental andestimated curves of the acceleration ratio (non-dimensional parameter) Paa (acceleration laser sensor/acceleration of theaccelerometer). The parameters updated are Youngs modulus of the Polyurethane rigid foam and the loss factor. Theoptimal values found for these parameters are E=8.394MPa and Z=0.045.

Figs. 6 and 7 show good agreement between the estimated and experimental curves, even near to the resonances.The physical parameters of the cantilever beam shown in Fig. 3 were estimated using the amplitude correlation

coefcient (ACC) and Genetic Algorithm (GA). The frequency range varying from 0 to 250Hz was chosen due to the goodrelation signal/noise. Fig. 8 shows the experimental and numeric (estimated) FRF curves for the sandwich beam (Fig. 3).The curves obtained using ACC and GA are practically superposed.

The optimal parameter values found with the two numeric methods are shown in Table 3.

ARTICLE IN PRESS

e (g/

N)

Estimated GA Estimated ACCExperimental

101

100

0 50 100 150 200 250 300 350 400

103

102

104

105

106

101

Frequency (Hz)

Acc

eler

atio

n ra

tio P

aa

Estimated Experimental

Fig. 7. Acceleration ratio curves of Polyurethane rigid foam cantilever beam.

N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 4064154140 50 100 150 200 250Frequency (Hz)

FRF

ampl

itud

104

103

102

Fig. 8. Experimental and estimated FRF curves.

ARTICLE IN PRESS

N. Barbieri et al. / Mechanical Systems and Signal Processing 24 (2010) 406415 4154. Conclusions

Amplitude correlation coefcient method and an optimization algorithm (Genetic Algorithm) were used to update thevalues of physical parameters of mathematical models of sandwich beam made with the association of hot-rolled steel,Polyurethane rigid foam and High Impact Polystyrene, used for the assembly of household refrigerators and food freezers.

The physical parameters estimated were: Youngs modulus and the loss factor of the Polyurethane rigid foam and theHigh Impact Polystyrene.

Both techniques using Genetic Algorithm and Amplitude Correlation Coefcient, presented good results when itcompares the estimated and experimental FRF curves. Similar results were obtained by Backstrom [15] using the built-inMatlab function fminsearch. The particular function is based on the NelderMead method.

Genetic Algorithm does not use derivatives, thus is a good estimated method even for resonance region. GAs are notguaranteed to nd the global optimum solution to a problem, but they are generally good at nding acceptably goodsolutions to problems acceptably quickly. Where specialized techniques exist for solving particular problems, they arelikely to outperform GAs in both speed and accuracy of the nal result. A number of different methods for optimizing well-behaved continuous functions have been developed which rely on using information about the gradient of the function toguide the direction of search. If the derivative of the function cannot be computed, because it is discontinuous, for example,these methods often fail. Such methods are generally referred to as hillclimbing. They can perform well on functions withonly one peak (unimodal functions). But on functions with many peaks, (multimodal functions), they suffer from theproblem that the rst peak found will be climbed, and this may not be the highest peak. Having reached the top of a localmaximum, no further progress can be made.

Amplitude correlation coefcient method use derivatives and in this way it is recommended that some frequency pointsshould be excluded. As the method use implicitly functions and their sensitivities that are not dened at resonances andcare should be taken to avoid them. For computational efciency, it is recommended to concentrate only on thosefrequency regions where the level correlation needs to be improving while ignoring the others [11].

The conventional Timoshenko beam theory was applied to model the sandwich beam and the numeric responsepresented good results when compared with results of literature and experimental data (only three vibration modes).

References

[1] Y. Shi, H. Sol, H. Hua, Material parameter identication of sandwich beams by an inverse method, J. Sound Vib. 290 (2006) 12341255.[2] R. Caracciolo, A. Gasparetto, M. Giovagnoni, An experimental technique for complete dynamic characterization of a viscoelastic material, J. Sound Vib.

272 (2004) 10131032.[3] J. Park, Transfer function methods to measure dynamic mechanical properties of complex structures, J. Sound Vib. 288 (2005) 5779.[4] S. Kim, K.L. Kreider, Parameter identication for nonlinear elastic and viscoelastic plates, Appl. Numer. Math. 56 (2006) 15381544.[5] R. Pintelon, P. Guillaume, S. Vanlanduit, K. Belder, Y. Rolain, Identication of Youngs modulus from broadband modal analysis experiments, Mech.

Syst. Signal Process. 18 (2004) 699726.[6] W.-P. Yang, L.-W. Chen, C.-C. Wang, Vibration and dynamic stability of a traveling sandwich beam, J. Sound Vib. 285 (2005) 597614.[7] R. Singh, P. Davies, A.K. Bajaj, Estimation of the dynamical properties of polyurethane foam through use of Prony series, J. Sound Vib. 264 (2003)

10051043.[8] T. Ohkami, G. Swoboda, Parameter identication of viscoelastic materials, Comput. Geotech. 24 (1999) 279295.[9] W.-D. Chang, An improved real-coded genetic algorithm for parameters estimation of nonlinear systems, Mech. Syst. Signal Process. 20 (2006)

236246.[10] D. Backstrom, A.C. Nilsson, Modelling the vibration of sandwich beams using frequency-dependent parameters, J. Sound Vib. 300 (2007) 589611.[11] H. Grafe, Model updating structural dynamics models using measured response functions, Ph.D. Thesis, Imperial College of Science, Technology &

Medicine, Department of Mechanical Engineering, London, 1998.[12] D. Zenkert, An Introduction to Sandwich Construction, Engineering Materials Advisory Services, Warley, West Midlands, UK, 1995.

Table 3Optimal parameters.

High impact polystyrene Polyurethane rigid foam

E (GPa) Z E (MPa) Z

GA 1.5830 0.0446 9.5159 0.0645

ACC 1.5800 0.0471 9.5284 0.0635[13] R.D. Cook, D.S. Malkus, M.E. Plesha, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York, USA, 1989.[14] J.N. Siddall, Optimal Engineering DesignPrinciples and Applications, Marcel Dekker, Inc., New York, USA, 1982.[15] D. Backstrom, Estimation of the material parameters of a sandwich beam from measured eigenfrequencies, Doctoral Thesis, Kungliga Tecniska

Hogskolan, Stockholm, Sweden, 2006.[16] J. Helgesson, Response matrix formulation and free vibration analysis of a three-layered sandwich beam, Master Thesis, School of Mechanical

Engineering, Lund University, Sweden, 2003.[17] D.J. Mead, S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vib. 10 (1969)

163175.[18] K.H. Ahmed, Dynamic analysis of sandwich beam, J. Sound Vib. 21 (1972) 263276.[19] T. Sakiyama, H. Matsuda, C. Morita, Free vibration analysis of sandwich beamwith elastic or viscoelastic core by applying the discreet green function,

J. Sound Vib. 191 (1996) 189206.[20] N.M.M. Maia, J.M. Silva (Eds.), Theoretical and Experimental Modal Analysis, John Wiley & Sons Inc., New York, 1997.

Parameters estimation of sandwich beam model with rigid polyurethane foam coreIntroductionMathematical modelNumerical estimation methods

ResultsConclusionsReferences