IOMC'3 5 th International Operational Modal nalysis Conference 03 May 3-5 Guimarães - Portugal ESTIMTING THE PRMETERS OF MTERILS OF SNDWICH BEM USING DIFERENT METHODS OF OPTIMIZTION Luiz Carlos Winikes, Nilson Barbieri and Renato Barbieri 3 BSTRCT In this work the physical parameters of a 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 are estimated using measured and numeric frequency response functions (FRFs). Two different mathematical models are obtained using the Finite Element Method (FEM) with D elasticity and the Timoshenko beam theories. The computational programs were implemented in FORTRN and/or MTLB in order to compare the Experimental Frequency Response Function (FRFexp) and Numerical Frequency Response Function (FRFnum) to estimate the parameters of the sandwich beam. The FRFexp was obtained by means of modal analysis using a signal analyzer HP and four accelerometers that were placed at points located on the faces of the sandwich beam and FRFnum was obtained through a mathematical model obtained by Finite Element Method (FEM). In the structure of computer programs was performed integration of the Finite Element Method (FEM) and some optimization methods, for example, the Genetic lgorithm (G) and the Particle Swarm Optimizer method (PSO). Through the adjustment of the FRF's numerical and experimental generated by computer simulation of the finite element model coupled to methods G and PSO, it was possible to estimate the parameters, Young s modulus and loss Factor, for each one of the materials that constitute the structure of the sandwich beam. Keywords: Identification of parameters; Optimization; Sandwich Beam, PSO.. INTRODUCTION The proved efficiency of sandwich beams and its current usage in a growing rate demands a higher level of acknowledgment of the mechanical properties, even when the structure is submitted to dynamic loads. For household refrigerators and food freezers, one of the main complaints to the customer care centers is related to noise generation, that is related most of the times with vibration of M.Sc., PUCPR-UTFPR, luiz.winikes@hotmail.com Dr., PUCPR-UTFPR, nilson.barbieri@pucpr.br 3 Dr., PUCPR-UDESC, renato.barbieri@pucpr.br
L. Carlos W., Nilson Barbieri, R. Barbieri the cabinet that produces sound irradiation from internal components like shelves and containers, leaking to the outside of the unit. The efficient numerical models are necessary to simulate (estimate) the dynamical behavior of such systems. When the complex sandwich beams are used the physical parameters are difficult to be estimated. Sandwich structures are extensively used in engineering because of their high specific stiffness and strength. The modeling of sandwich structures has been studied extensively, but less attention has been paid to their material identification []. The work proposes an inverse method for the material identification of sandwich beams by measured flexural resonance frequencies. Various authors [-] have presented various experimental models and analyze to identify the physical parameters of a beam sandwich. 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 assembly of household refrigerators and food freezer are estimated using measured and numeric frequency response functions (FRFs). In previous works, the mathematical models are obtained using the Finite Element Method (FEM) and the Timoshenko beam theory [5-]. The physical parameters were estimated using the amplitude correlation coefficient [7] and Genetic lgorithm (G) []. The experimental data were obtained using the impact hammer and four accelerometers displaced along the sample (cantilevered beam). The parameters estimated were the Young s modulus and the loss factor of the Polyurethane rigid foam and the High Impact Polystyrene. To estimate the initial values of the parameters, separate tests were conduced using cantilevered beams of Polyurethane rigid foam and High Impact Polystyrene. In this work two different mathematical models are obtained using the Finite Element Method (FEM) with D elasticity and the Timoshenko beam theories. The computational programs were implemented in FORTRN and/or MTLB in order to compare the Experimental Frequency Response Function (FRFexp) and Numerical Frequency Response Function (FRFnum) to estimate the parameters of the sandwich beam. The FRFexp was obtained by means of modal analysis using a signal analyzer HP and four accelerometers that were placed at points located on the faces of the sandwich beam and FRFnum was obtained through a mathematical model obtained by Finite Element Method (FEM). In the structure of computer programs was performed integration of the Finite Element Method (FEM) and some optimization methods, for example, the Genetic lgorithm (G), the method of Differential Evolution (DE) and the Particle Swarm Optimizer method (PSO). Through the adjustment of the FRF's numerical and experimental generated by computer simulation of the finite element model coupled to methods G, PSO and DE, it was possible to estimate the parameters: Modulus of Elasticity and Damping Factor for each one of the materials that constitute the structure of the sandwich beam. procedure using wavelet is used for crack identification.. MTHEMTICL MODELS.. Beam theory The equation of motion for the vibration of a sandwich beam according to the Timoshenko beam theory [9] is: w D x * * w w D t S x t w w t x t f (x)e it () where: w (x,t) is the transverse displacement; D is bending stiffness ; * is the mass per unity of surface; S is the shear stiffness ; is the rotational inertia; x is the coordinate along the beam axis;
5 th International Operational Modal nalysis Conference, Guimarães 3-5 May 03 t is the time ; f(x) is the amplitude of the external force applied along the beam span; is the excitation frequency and i. t w w d z e t c t L (a) (b) Figure - (a) Sandwich beam geometric parameters and (b) finite element d.o.f. fter some manipulations obtains the standard finite element equation: ( ) q F K e where: () * * D S (3) S * K ( ) DK M K M e.. D elasticity The displacements, traction components, and distributed body force values are functions of the position indicated by ( x, y). Figure Two dimensional problem [0] 3
L. Carlos W., Nilson Barbieri, R. Barbieri The displacements inside the element are written using the finite element shape functions (isoparametric cubic - nodes) and the nodal values of the unknown displacement field. We have u u v 0 0...... n 0 0 q q n () where q={u,v,u,v,,u n,v n } t and the pair {u i,v i } represent the nodal displacements associated to node i. Using the strain-displacement relations, we get: u / x / x 0... n / x 0 v / y 0 / y... 0 n / y q Bq (5) u / y v / x / y / x... n / y n / x The element elastic strain energy, U e, is given by: U e t ε Dε t d t t t t q B DBq td q DB tdq B () where t is the thickness of the element (variable or not); is the area of the element and D is the constitutive matrix. This expression can be rewritten as U e t t t t ε Dε t d q B DBq td q K q (7) where identifies the element stiffness matrix is t K B DB td () The element kinetic energy, U c, is given by: U c u u t d t q t t q q t t td tdq (9) where is the density of material (kg/m 3 ). This expression can be rewritten as: Uc q t M q (0) where identifies the mass matrix of element as: M t td. ()
FRF amplitude (g/n) 5 th International Operational Modal nalysis Conference, Guimarães 3-5 May 03 3. RESULTS The experimental sample of sandwich beam made with the association of hot-rolled steel, Polyurethane rigid foam and High Impact Polystyrene is shown in Fig. 3. The thickness of the steel is 0. mm; the Polyurethane is 3.5 mm and the Polystyrene is.5 mm and the beam width is 39. mm. The experimental data are obtained using the impact hammer and four accelerometers displaced along the sample (,, 3 and ). Figure shows the typical FRF curve obtained with the four accelerometers. Figure 3 Sandwich beam. 0 0 0 50 00 50 00 50 Frequency (Hz) Figure FRF curve of accelerometer. 5
L. Carlos W., Nilson Barbieri, R. Barbieri The Rational Fraction Polynomial (RFP) method was used to estimate the damping ratio () and the natural frequencies () of the three mode shapes. Table shows the values of these parameters to the accelerometer and the natural frequencies obtained usind the beam theory and D elasticity mathematical models. It can be easily noticed that the errors are smaller for the numerical results obtained with the mathematical model using the D theory of elasticity. Table Experimental damping ratio and natural frequencies. Mode shape Experimental Beam theory D elasticity [Hz] [Hz] [Hz].7 0.05 5.5,50 0.3 0.09 97.7 07,3 3.7 0.05 9.3,50 In previous work [5-], the authors estimated the parameters, Modulus of Elasticity and Damping Factor, for each one of the materials that constitute the structure of the sandwich beam.. Initially the experimental procedure [5] to determine the material s mechanical properties, Young and shear modulus and the density of the components of the sandwich beam are described. The elastic properties were obtained through tension and torsion tests. The shear modulus G c of the Polyurethane rigid foam core was determined using rectangular specimen and the Young s moduli of the steel and High Impact Polystyrene were determined using conventional tension test. To estimate the dynamical values of the parameters in the frequency range varying from 0 to 00 Hz, separated dynamic sweeping test were conduced using cantilevered beams of Polyurethane rigid foam and High Impact Polystyrene [5-]. The experimental data of a three layered sandwich beam were obtained using the impact hammer and four accelerometers displaced along the sample (cantilevered beam). The parameters estimated were the Shear modulus and the loss factor of the Polyurethane rigid foam and the Young s modulus and the loss factor of the High Impact Polystyrene. The physical parameters were estimated using measured and numeric frequency response functions (FRFs). The mathematical models were obtained using the Finite Element Method (FEM) and the Timoshenko beam theory. The physical parameters are estimated using the amplitude correlation coefficient and Genetic lgorithm (G) methods. In this work the mathematical model is obtained using D elasticity theory and the physical parameters are estimated through the adjustment of the FRF's numerical and experimental generated by computer simulation of the finite element model coupled to methods G and PSO. Figures 5-7 show the fitted FRF curves (left) and convergence curves (right) for the first three vibration modes. The objective function is defined as: OF FRFnum FRFexp () n where: Experimental Frequency Response Function (FRFexp); Numerical Frequency Response Function (FRFnum) and n is a number of values in the frequency range around each mode.
FRF mplitude[g/n] Objective function value FRF mplitude[g/n] Objective function value FRF mplitude[g/n] Objective function value 5 th International Operational Modal nalysis Conference, Guimarães 3-5 May 03 3.5 3 G-FEM Experimental 0 9 G-FEM.5.5 7 0.5 0 0 30 3 3 3 Frequency [Hz] 5 0 000 000 3000 000 5000 000 7000 000 9000 0000 Evaluation number of the objective function Figure 5 FRF and convergence curves (first mode) 0 9 G-FEM Experimental 0 G-FEM 7 5 3 0 95 00 05 0 5 0 5 Frequency [Hz] 0 000 000 000 000 0000 000 Evaluation number of the objective function Figure FRF and convergence curves (second mode) 0 G-FEM Experimental 5.5 G-FEM 9 5 7.5 5 3.5 3 5 0 5 30 35 0 Frequency [Hz] 3 0 000 000 3000 000 5000 000 Evaluation number of the objective function Figure 7 FRF and convergence curves (third mode) Table shows the values of the optimal physical parameters for the High Impac Polystyrene and Polyurethane rigid foam. These parameters were optimized using beam theory [5-] for all frequency range and using the G optimization method. The parameters were also optimized using D elasticity model and considering different frequency ranges around the natural frequencies of the first to the third mode of vibration. In this case it was used two optimization methods: G and PSO. The 7
L. Carlos W., Nilson Barbieri, R. Barbieri reference values of the Young s modulus of the High Impac Polystyrene and Polyurethane rigid foam are.05 GPa and 9.70 MPa. Mode shape Optimization Method Table Experimental damping ratio and natural frequencies. High Impact Polystyrene Polyurethane rigid foam Beam theory D elasticity Beam theory D elasticity E (GPa) E (GPa) E (MPa) E (MPa) First G.5 0.0. 0.0 9.5 0.0.79 0.03 PSO. 0.0.7 0.05 Second G.5 0.0. 0.03 9.5 0.0 9. 0.0 PSO.39 0.03 9.3 0.03 Third G.5 0.0.3 0.0 9.5 0.0 9.0 0.05 PSO.3 0.0 9.0 0.05. CONCLUSIONS Two mathematical models using beam theory and elasticity D were used to update the values of physical parameters of mathematical models of sandwich beam made with the association of hotrolled steel, Polyurethane rigid foam and High Impact Polystyrene and used for the assembly of household refrigerators and food freezers. The physical parameters estimated were the Young s modulus and loss factor of the of the Polyurethane rigid foam and the High Impact Polystyrene. Both methods, Genetic lgorithm and Particle Swarm Optimization, presented good results when it compares to the estimated and the experimental FRF curves. It was verified that the parameters are frequency-dependent. The values found with conventional test (static) are good approximations for the initial updated methods starting point. The numerical results obtained through the mathematical model of the D theory of elasticity showed better approximations to the experimental results. REFERENCES [] Y. Shi, H. Sol, H. Hua (00) Material parameter identification of sandwich beams by an inverse method, J. Sound Vib. 90: 3 55. [] R. Caracciolo,. Gasparetto, M. Giovagnoni (00) n experimental technique for complete dynamic characterization of a viscoelastic material, J. Sound Vib. 7:03-03. [3] J. Park (005) Transfer function methods to measure dynamic mechanical properties of complex structures. J. Sound Vib. :57 79. [] S. Kim, K. L. Kreider (00) Parameter identification for nonlinear elastic and viscoelastic plates, pp. Num. Mathematics 5: 53-5. [5] R. Pintelon, P. Guillaume, S. Vanlanduit, K. Belder, Y. Rolain (00) Identification of Young s modulus from broadband modal analysis experiments. Mech. Syst. Signal Proc. : 99 7.
5 th International Operational Modal nalysis Conference, Guimarães 3-5 May 03 [] W-P. Yang, L-W. Chen, C-C. Wang (005) Vibration and dynamic stability of a traveling sandwich beam. J. Sound Vib. 5: 597. [7] R. Singh, P. Davies,. K. Bajaj (003) Estimation of the dynamical properties of polyurethane foam through use of Prony series. J. Sound Vib. : 005 03. [] T. Ohkami, G. Swoboda (999) Parameter identification of viscoelastic materials, Computers and Geotechnics : 79-95. [9] W.-D. Chang (00) n improved real-coded genetic algorithm for parameters estimation of nonlinear systems, Mech. Syst. Signal Proc. 0: 3. [0] D. Backström,. C. Nilsson (007) Modelling the vibration of sandwich beams using frequencydependent parameters, J. Sound Vib. 300: 59-. [] V.L. Tagarielli, V.S. Deshpande (00) N.. Fleck Prediction of the dynamic response of composite sandwich beams under shock loading. International Journal of Impact Engineering 37: 5 [] Wu Zhen, Chen Wanji (00) n assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams. Composite Structures : 337 39. [3] Inés Ivañez, Carlos Santiuste, Sonia Sanchez-Saez (00) FEM analysis of dynamic flexural behaviour of composite sandwich beams with foam core. Composite Structures 9: 5 9 [].R. Damanpack a, S.M.R. Khalili (0) High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method. Composite Structures 9:503 5. [5] N. Barbieri, R. Barbieri, L. C., L. F. Oresten (00) Estimation of parameters of a three-layered sandwich beam. Journal of Mechanics of Materials nd Structures 3: 57-5. [] N. Barbieri, R. Barbieri, L. C. Winikes (00) Parameters estimation o sandwich beam model with rigid polyurethane foam core. Mechanical Systems and Signal Processing : 0 5. [7] H. Grafe (99) Model Updating Structural Dynamics Models Using Measured Response Functions, Ph.D. Thesis, Imperial College of Science, Technology & Medicine, Departament of Mechanical Engineering, London. [] W.-D. Chang (00) n improved real-coded genetic algorithm for parameters estimation of nonlinear systems, Mech. Syst. Signal Proc. 0: 3. [9] D. Zenkert (995), n Introduction to Sandwich Construction, Engineering Materials dvisory Services, Warley, West Midlands, UK. [0] T. R. Chandrupatla,. D.Belegundu (00) Introduction to Finite Elements in Engineering. Ptrentice- Hall, New York. 9