Characterization of Constrained Viscoelastic Materials through a FE Model Updating using Genetic Algorithms Giovanni Brattia) Augusto Amador Medeirosb) Murilo Ferreira Santosc) Júlio Apolinário Cordiolid) Arcanjo Lenzie) Laboratory of Noise and Vibration - LVA Department of Mechanical Engineering – EMC Federal University of Santa Catarina – UFSC Florianopolis, SC, Brazil Francisco Keller Klugf) Sideto Futatsugig) Technology Development Team, Embraer Sao Jose dos Campos, SP, Brazil A widely used method to control the internal noise of aircrafts caused by Turbulent Boundary Layer (TBL) is the addition of structural damping - by viscoelastic materials - to fuselage panels. To perform efficient designs, the dynamic properties of fuselage components need to be known. One method widely used for viscoelastic materials characterization is presented by the standard ASTM E-756. However, for multilayer viscoelastic materials characterization, the ASTM E-756 requires the evaluation of each layer individually. Thus, to characterize viscoelastic materials having constrained layer a)
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(CL) glued to its surface, it is necessary to detach the CL. In this process, the glue properties that perform the fixing between the CL and the viscoelastic material are not characterized, and it becomes necessary to characterize the glue in another step. Based on the experimental apparatus recommended by ASTM E-756, this article proposes an alternative method for constrained viscoelastic materials characterization by updating a Finite Element (FE) model using Genetic Algorithms (GA). A viscoelastic material is characterized by the proposed method and the results are compared with those obtained through the ASTM E-756 approach. The advantages and disadvantages of each method are discussed. 1
INTRODUCTION
In order to reduce the noise and vibration generated by the fuselage panels of an aircraft many different methods are used. Sometimes, the change of the system stiffness or mass is sufficient, however, in some cases, the vibration of the panels needs to be isolated or dissipated by isolators and/or damping materials. The passive control by addition of viscoelastic materials is a method that offers a great cost-benefit ratio when compared with others methods. To perform efficient designs, the dynamic properties of the fuselage components need to be well known. In structures with viscoelastic material the dynamic characterization of all components is of fundamental importance to evaluate the materials efficiency and to allow the design of the treatments. One method widely used for viscoelastic materials characterization is presented by the standard ASTM E-7561. This test method determines the properties of a viscoelastic material by indirect measurement in a temperature range. In this work, a viscoelastic material that has a constrained layer glued in your final configuration was characterized through the ASTM-E756 method. This standard suggests that each layer must be tested separately. So the constrained layer of the viscoelastic material was removed and tests were performed in accordance to the ASTM-E756. In order to verify the accuracy of the properties obtained, a numerical validation of a cantilever beam with the viscoelastic material in his final configuration was performed. As will be seen, the results obtained suggest that in the detach process of the CL for the viscoelastic material characterization, the effects that the glue performs on the material are also removed and the glue properties need to be characterized by another process. Based on the experimental apparatus and the procedure proposed by the ASTM E-756, this paper propose an alternative method to the viscoelastic materials characterization by updating FE models using GA. 2
VISCOELASTIC MATERIALS
Viscoelastic material is a material class that has a viscoelastic rheology, i.e., they are materials that in the deformation process suffer elastic and viscous deformation simultaneously2. Most viscoelastic materials exhibit damping behavior which depends strongly upon temperature and frequency3. Two of the main dynamic properties of a viscoelastic material are the Young’s modulus and the loss factor. The Young’s modulus and the damping loss factor may also be represented a complex modulus which is defined as: E* E (1 i ) ,
(1)
where E is the real part of the complex modulus and is the loss factor. The manner in which the modulus ( E * or the shear modulus G * ) and the loss factor vary with temperature and frequency distinguishes one viscoelastic material from another3. A typical data showing the effects of frequency and temperature are illustrated in Fig. 1 for the Young’s modulus and loss factor of a viscoelastic material. Usually both frequency and temperature are varied and a phenomenon allows that a single operation can combine both frequency and temperature into a single variable, which is referred to as the reduced frequency4. So, the data obtained at various frequencies and temperatures can be represented as two curves through the Reduced-Frequency Nomogram, as illustrated in Fig. 2. This nomogram permits the interpolation (in some cases the extrapolation) of data to frequency, or temperature ranges where test data are not available. The variation of E * and with frequency and temperature may be mathematically modeled in various ways. An effective analytical model can be extremely helpful in the process of bridging gaps in the test data, which always occur3. A great simplification in modeling viscoelastic material behavior was reached through the fractional derivative model. This model of viscoelastic behavior employs derivatives of fractional order to relate stress fields and strain fields in viscoelastic materials5 and the complex modulus is defined by: E*
a* b* i 1 c i
,
(2)
where a* , b* , c , and are parameters that must be determined from a suitable curve-fit process applied to test data. Due to the frequency-temperature equivalence principle, can be replaced by the reduced frequency 2 f T , where f is the frequency and the shift factor function T depends upon the temperature3 T . So Eq. (2) takes the form: E *
a* b* i2 f T
1 c i2 f T .
(3)
Of the various forms that have been suggested for the shift factor3 T , a widely used relationship is the Williams-Landel-Ferry (WLF) relationship given by2: E*
C1 T T0 T T ,
(4)
where C1 , T0 , and T are material constants to be determined. 2
CHARACTERIZATION OF VISCOELASTIC MATERIALS
The knowledge of the dynamics properties of viscoelastic materials, as a function of temperature and frequency, is essential to predict their performance, to validate numerical models and to design damping systems.
The widely used method to measure vibration-damping properties of materials is given by the ASTM-E7561. This test method is used to measure the vibration-damping properties of materials: the loss factor ( ) and Young’s modulus ( E * ) or shear modulus ( G * ). However this method presents some limitations to characterize viscoelastic materials having CL glued to its surface and the following alternative method is proposed. 2.1 Measurement Procedure The alternative method proposed in this paper makes use of the procedure and experimental apparatus suggested by the ASTM-E756. The viscoelastic material properties are evaluated in a two-step process. First, a self-supporting, uniform metal beam, called the base beam or bare beam, must be tested to determine its resonant frequencies and corresponding loss factor. Second, the viscoelastic material having CL is applied to the base beam to form a damped composite beam, as shown in Fig. 3, and it needs also to be tested to obtain its resonant frequencies and corresponding composite loss factors over the temperature range of interest. Once the bare beam (and after in the second step the composite beam) has been prepared, it must be clamped in a rigid test fixture to simulate a cantilever boundary condition and placed inside an environmental chamber with control temperature. In order to make the measurements, an noncontact transducer should be positioned at a beam point to apply a force excitation to vibrate the test beam, and another noncontact transducer in another point to measure the response of the test beam to the applied force, using an appropriate instrumentation for generating the excitation signal and measuring the response signal as illustrate in Fig. 4. By measuring several resonances of the base beam, the Young’s modulus and the loss factor of the beam material can be evaluated by the following expressions given in the ASTME7561: E
12 l 4 f n2 H 2Cn ,
(5)
f n fn ,
(6)
where f n is the resonance frequency for mode n , f n half-power bandwidth of mode n , H thickness of beam in vibration direction, l length of beam, the density of beam (all in the SI units) and C1 0.55959 , C2 =3.5069, C3 =9.8194 and Cn =( p / 2)(n - 0.5)2 , for n 3 . Usually for a given temperature range and frequency range this two properties are constants with temperature and frequency and the mean value can be considered. In order to investigate the temperature effects on the viscoelastic material properties, frequency response function (FRF) measurements of the composite beam must be performed at intervals over a wide range of temperatures. 2.2 Calculation of Damping Properties The viscoelastic material characterization procedure makes use of a model update. The Young’s Modulus and the loss factor is determined using the experimental FRF’s of the
composite beam discussed at the before section, FRFs simulated of a FE model and an optimization algorithm. Through the Eq. (3) and Eq. (4), we can write that the complex modulus is a function of the following parameters: E* , T f , a* , b* , c, , C1 , T .
(7)
Often the real part of the complex modulus is called storage modulus and the imaginary part of loss modulus2. The loss factor is related with these two parts as:
, T
Im E * , T
Re E * , T
.
(8)
A way to simulate the experimental FRF’s of the composite beam discussed at the last section is through the Finite Element Method (FEM). For this is necessary to create a composite beam model and set the material properties of each layer. Usually the Young’s modulus and the loss factor of the viscoelastic materials have strong frequency dependence and frequently this variation is considered in the simulations. As all others properties can be obtained through conventional methods and be considered as constants, we can say that the numerical frequency response ( FRFnum ) of the beam with viscoelastic material applied is also function of the parameters: FRFnum , T f , a* , b* , c, , C1 , T ,
(9)
The magnetic transducers (noncontact) can measure correctly the slope of the peaks ( ) and the peaks positions over the frequency ( f n ) but the others parts of the curve are usually distorted. Having this in mind was development an error function ( e ) based on the difference between the experimental and numerical natural frequencies and loss factor as: e( x )
Tmax
mod es
T Tmin
n 1
f n,exp T n,num x, T T f n,num x, T 1 n,exp 2 f n,exp T n,exp T ,
where x a* , b* , c, , C1 , T ;
2
2
(10)
f n,exp T and f n,num x, T is respectively the natural frequency
of mode n obtained experimentally and numerically for the beam with viscoelastic material in the T temperature; n,exp T and n,num x, T is respectively the experimental and numerical loss factor of mode n obtained from Eq. (6) for the beam with constrained viscoelastic material in the T temperature; and 1 and 2 are respectively weighting parameters of squared error of the natural frequencies and loss factors experimental and numerical. The main objective to the development of the Eq. (10) was to be used with the FRF’s obtained from the procedure suggested by the ASTM-E756, which is widely used in viscoelastic material characterization.
In order to compute the x parameters, such that the squared errors sum reach a minimal value, algorithms was developed in the Matlab software to use GA and Nastran solver6. Through the Matlab software, the basic procedure consists in randomly generate parameters x and for each temperature T the Young’s modulus (Eq. (3)) and the loss factor (Eq. (8)) are computed. This properties are imposed in the FE model and a numerical FRF is simulated for obtaining the parameters f n,num x, T and n,num x, T . By doing this for all temperature range, the Eq. (10) can be calculated for each x parameter. Minimizing the Eq. (10), the parameters a* , b* , c, , C1 , T that describe the Young’s modulus and the loss factor of the viscoelastic material are found. 3
RESULTS
A viscoelastic material that has a constrained layer glued in your final configuration was characterized through the ASTM-E756 method and by the proposed method in a temperature range between 258 K and 303 K with step of 5 K. The experimental apparatus used was designed according with the ASTM-E756 recommendations and it is shown in Fig. 5. The base beam with dimensions: 250.00 mm x 10.08 mm x 3.24 mm was made of 1020 steel and the viscoelastic material with 1.24 mm had a thin aluminum layer of 0.13 mm. For the viscoelastic material characterization using the ASTM-E756, frequency response function was measured in all temperature range to the beam with viscoelastic material (without CL) glued on its surface and by the proposed method was measured to the beam with constrained layer viscoelastic material at the same temperature range. Utilizing the experimental FRF’s measured the viscoelastic material properties were obtained by the ASTM-E756 procedure and by the proposed method. For the viscoelastic material characterization by the proposed method was necessary to develop a FE model with three layers like the real composite beam measured. The material properties of the first layer (base beam) was setting with the standard steel properties except the Young’s modulus and the loss factor which was obtained by the Eq. (5) and Eq. (6) from the experimental FRF’s of the base beam and for the last layer (constrained layer) was setting the aluminum standard material properties. The Young’s modulus and loss factor of the middle layer (glue + viscoelastic material + glue) was calculated by the Matlab using respectively the Eq. (3) and Eq. (8) for each set of parameters ( x ). For each temperature step this properties was delivered in table format to Nastran solver simulate the FRF. So in an iterative process using GA the dynamics properties of the viscoelastic material was characterized. The Fig. 6 illustrates the FE model used in the proposed method. The results obtained by the ASTM-E756 method are shown in Fig. 7 and by the proposed method in Fig. 8, both in the reduced-frequency nomogram form. In order to check the properties obtained from both methods, a numerical investigation was made. Utilizing the FE model illustrate in Fig. 6, two FRFs was simulated for the viscoelastic material in 253 K and more two in 303 K using the results obtained by the ASTM-E756 and by the proposed method. The numerical FRFs are compared with experimental FRFs to each temperature in Fig. 9 and Fig. 10. Observing the Fig. 9 and Fig. 10 we can see that natural frequencies simulated using the viscoelastic material properties obtained by the proposed method show to be closest to the experimental results than those obtained by the ASTM-E756 for both temperatures. At high frequency the same agreement was found with respect to the loss factors.
4
CONCLUSIONS
A constrained viscoelastic material was characterized by two methods. Frequency response measurements was performed in a cantilever beam with the constrained viscoelastic material applied inside a environmental chamber with control temperature such as suggested by the ASTM-E756 method in a temperature range. So using the ASTM-E756 equations the Young’s modulus and loss factor of the viscoelastic material could be obtained. To characterizing the constrained viscoelastic material by the proposed method was necessary to develop a FE model. Using the fractional derivative model to describe the viscoelastic material properties, the material was characterized using the experimental end numerical FRFs through of a model update technique using genetic algorithms. The results obtained by both methods in a temperature range were presented in the nomograms form. Finally, the accuracy of the dynamics properties obtained by the both methods was performed by a numerical investigation. The comparison between the numerical FRFs using the properties obtained from each method showed that the properties obtained by the ASTM-E756 method was stiffer than the real comportment and the proposed method showed to be closer. Such as appointed before the viscoelastic material characterization by the ASTM-E756 cannot include the glue effects at the viscoelastic material properties. In the proposed method the viscoelastic material and the glue of the constrained viscoelastic material is considered as only one layer and a higher accuracy can be obtained from this method. 5
ACKNOWLEDGEMENTS
Thanks go to teachers Arcanjo Lenzi and Julio A. Cordioli, who contributed to this work and especially to God for the opportunities that he gives me every day. The first author is also grateful to CAPES (Coordenação de Aperfeioamento de Pessoal de Nível Superior) and EMBRAER, which financed this research. 6
REFERENCES
1. ASTM E756-05:2005, Standard Test Method for Measuring Vibration-Damping Properties of Materials. American Society for Testing and Materials. (2005) 2. Nashif, A. D., Jones, D. I. G., and Henderson, J. P., Vibration Damping. John Wiley & Sons, New York. (1985) 3. Jones, D., Handbook of Viscoelastic Vibration Damping. Wiley, Chichester. (2001) 4. Ferry, J., Viscoelastic properties of polymers. John Wiley & Sons. (1980) 5. Bagley, R. L., “On the Fractional Calculus Model of Viscoelastic Behavior”, Journal of Rheology, 30. (1986) 6. Nastran, MSC, Advanced Dynamic Analysis User’s Guide. (2004)
Fig. 1 – Typical effects of frequency and temperature on the properties of a viscoelastic material.(a) Young’s Modulus. (b) Loss factor.
Fig. 2 – Illustration of the Reduced-Frequency Nomogram.
Fig. 3 – Illustration of the damped composite beam.
Fig. 4 – Typical experimental setup..
Fig. 5 – Picture of the experimental setup. (a)Exterior and (b)Interior.
Fig. 6 – Illustration of the composite FE model.
Fig. 7 – Frequency-reduced nomogram obtained by the ASTM-E756 method.
Fig. 8 – Frequency-reduced nomogram obtained by the proposed method.
Fig. 9 – Comparison between the numerical and experimental FRFs for the composite beam at 253 K.
Fig. 9 – Comparison between the numerical and experimental FRFs for the composite beam at 303 K.
