Identification of model parameters of a sandwich beam incorporating magnetorheological fluid

Transcription

1 XXIV Symposium Vibrations in Physical Systems, Poznan Bedlewo, May 12-15, 2010 Identification of model parameters of a sandwich beam incorporating magnetorheological fluid Bogdan SAPIŃSKI AGH University of Science and Technology, al. Mickiewicza 30, Kraków, deep@agh.edu.pl Jacek SNAMINA AGH University of Science and Technology, al. Mickiewicza 30, Kraków, snamina@agh.edu.pl Mateusz ROMASZKO AGH University of Science and Technology, al. Mickiewicza 30, Kraków, matek@agh.edu.pl Abstract The study covers the identification of model parameters of a sandwich (three-layered) cantilever beam incorporating magnetorheological (MR) fluid. The beam comprises two outer layers made of aluminium and a MR fluid layer in between, sealed with silicone rubber. The beam finite element (FE) model is created using the ANSYS software. Interactions of the magnetic field are taken into account by varying the FE model parameters. Data required for identification are collected from results of measurement of the beam s free vibration. The identification procedure assumes the good agreement between the frequencies of the beam s free vibrations and dimensionless damping factors obtained from research and computation data. Keywords: MR fluid, beam, vibrations, damping, identification 1. Introduction Control of vibration of flexible structures using smart fluids has received a great deal of attention since the early 1990s. The literature on the subjects abounds in reports on adaptive features of beam and plate structures incorporating electrorheological (ER) fluid. Fewer reports are available that explore similar applications of MR fluids [3, 4, 5, 6, 7]. The distinctive feature of such structure is that their structural behaviour can be controlled by the magnetic/electric field which activates the smart fluids present in them. Interactions of the magnetic/electric field cause the stiffens and damping characteristics to change, hence enabling vibration reduction. At this stage of the research programs, the authors focus on a three-layered cantilever beam, incorporating MR fluid layer of 140CG type of Lord Corporation [8] in between two flexible layers. The purpose of the research program is to reduce the free transverse vibration of the beam under the applied magnetic field. The paper is concerned with identification of model parameters of a beam incorporating MR fluid, assuming the fluid is represented by finite elements in the form of a rheological structure of the Voigt-Kelvin type and of the modified Bingham structure. The model is intended to be used for testing the control algorithms of beam vibration.

2 2. Model The structure of the beam with a MR fluid layer, is shown in Figure 1. Development of the beam model involves three stages. The beam considered in the first stage is made of a single layer of aluminium 400 mm in length, 30 mm in width and 2 mm in thickness (Beam1). The beam modelled in the second stage comprises two aluminium layers and a silicone rubber sealing 2 mm in thickness (Beam2). Finally, the beam modelled in the third stage has a MR fluid layer in between the aluminium layers (Beam3). The beam is modelled using the finite elements readily available in the ANSYS library: solid45 (aluminium layer), solid185 (silicone rubber sealing), combin40 and mass21 (MR fluid). The schematic diagram of the finite element combin40 is shown in Figure 2a. The MR fluid layer is modelled by the rheological Voigt-Kelvin structure (Figure 2b) and the modified Bingham structure (Figure 2c) assuming shear mode of MR fluid operation. In order to assure the shearing mode operation it is required that vertical displacements of relevant cross-sections of the upper and lower aluminium layers should be the same [2]. Thus the relative displacement of modes of the element combin40 is confined to direction coinciding with the beam s axis. b) a) c) Figure 1. Beam structure Figure 2. combin40 element and its modifications 3. Identification experiment Experiments were performed in the set-up shown schematically in Figure 3. Data for identification were collected by testing the beam s free vibration and recording the displacement z of the point P. Experiments were performed on the beams: Beam1, Beam2 and Beam3. Experiments were performed in the absence of magnetic field and under the applied field. The free end of the beam was deflected from the equilibrium state and displacement z of the point P was registered with the laser vibrometer. The height of the slit between the poles of an electromagnet was 20 mm. Of particular interest is the relationship between the electromagnet s position and dimensionless damping coefficient. The electromagnet position was measured by the distance y m between the beam attachment point and the slit centre. The following values of y m were assumed: 43, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140 mm.

3 Vibrations in Physical Systems Vol.24 (2010) Figure 3. Experimental set-up: 1 beam; 2 electromagnet; 3 laser vibrometer In each position of the electromagnet, it was supplied with current I=5 A. The current induced the magnetic field with flux density 0.16 T. Figure 4 shows time histories of displacement of the point P for the Beam1 and Beam3 with no magnetic field and under the applied field, for y m =80 mm. Figure 4. Displacement of the point P Thus obtained time histories yield natural frequencies f and dimensionless damping coefficient ζ of the beam s vibration, depending on the position of the electromagnet for the current I=5 A. Computation data, obtained for selected positions of the electromagnet, are compiled in Table 1. Frequency and the dimensionless damping coefficient for Beam3 with no activated magnetic field become 8.63 Hz and , respectively.

4 Table 1. Frequency and dimensionless damping coefficient y m [mm] f [Hz] ζ [ ] Identification procedure The main purpose of identification was to find the values of major model parameters: the internal damping factor for aluminium and the rubber sealing, Young modulus for aluminium and the rubber sealing and, most importantly, of model parameters of the MR fluid layer. Identification of parameters of the MR fluid layers involved two procedures. In accordance with the procedure 1, parameter values of the Voigt-Kelvin structure c p, k p (Figure 2b) are determined basing on the equality between experimental and calculated values of natural frequencies and dimensionless damping coefficients obtained for the electromagnet position y m =80 mm. In the procedure 2 we sought such values of parameters c p, k p that the error defined by the formula (1) takes minimum value: min ( ζ j ( c p, k p ) ζ j ) + W ( f j ( c p, k p ) f j ) (1) cp, k p j= 1 j= 1 where: f j, f j natural frequency, experimental and theoretical; ζ j, ζ j dimensionless damping coefficient, experimental and theoretical; j=1, 2,...,11 index corresponding to successive positions of the electromagnet; W=0.001 weighting coefficient. Values of c p and k p obtained in the first procedure are equal to c p =19.2 [N s/m], k p =2544 [N/m] and in the second procedure are equal to c p =21.1 [N s/m], k p =2162 [N/m]. These values yield the relationship between frequency and the dimensionless damping coefficient and the electromagnet s position, shown in Figures 5 and 6. The values of c p and k p calculated for various current levels are compiled in Tables 2 and 3. Figure 5. Frequency as the function of electromagnet s position

5 Vibrations in Physical Systems Vol.24 (2010) Figure 6. Dimensionless damping coefficient as the function of electromagnet s position Table 2. Values of c p and k p in accordance with the procedure 1 Table 3. Values of c p and k p in accordance with the procedure 2 I =3 A I =5 A I =9 A I =3 A I =5 A I =9 A c p [N s/m] c p [N s/m] k p [N/m] k p [N/m] The proposed model correctly predicts the changes of parameters ζ and f for successive position of the electromagnet along the beam axis. A good approximation of the dimensionless damping coefficient is achieved for all electromagnet s positions. Frequencies are well predicted for the electromagnet in the position y m <100 mm, which might be associated with enhanced interactions of a non-homogenous magnetic field at the edges of the space between the poles of an electromagnet [1]. The influence of nonhomogeneity of the field on the beam s motion enhances with amplitude increase. It was observed at points of the beam at a larger distance from the attachment point. At higher amplitudes of vibration, points of the beam are temporarily placed in the space where concentration of magnetic field lines is so high that attraction of ferromagnetic particles causes the natural frequency to be vastly reduced. 5. Summary A model of a sandwich beam incorporating MR fluid is proposed and its parameters are identified. The model enables us to accurately predict the vibration damping capability over the entire range of electromagnet s positions. Natural frequencies of beam vibration are established with sufficient accuracy for the electromagnet in the position given as y m <100 mm. Phenomena due to non-homogeneity of the magnetic field in the slit of the electromagnet can be neglected for the electromagnet positions y m < 100 mm. In the light of the assumptions made, the predictions of the beam s motion are sufficiently accurate, enabling us to the test the control algorithms of beam s vibration.

6 Acknowledgments This study is a part of the research program no N References 1. B. Sapiński, J. Snamina: Vibration of a beam with magnetorheological fluid in non-homogenous magnetic field. Engineering Modeling, 6, , B. Sapiński, J. Snamina: Modeling of an adaptive beam with MR fluid. Solid State Phenomena , , M. Yalcinitas, H. Dai: Vibration suppression capabilities of magneto-rheological materials based adaptive structures. Smart Materials and Structures, 13, 1 11, Q. Sun, J. X. Zhou, L. Zhang: An adaptive beam model and dynamic characteristics of magnetorheological materials. Journal of Sound and Vibration, 261, , Z. F. Yeh, Y. S. Shih: Dynamic characteristics and dynamic instability of magnetorheological based adaptive beams. Journal of Composite Materials, 40, , V. Lara-Prieto, R. Parkin, M. Jackson, V. Siberschmidt, Z. Kęsy: Vibration characteristics of MR cantilever sandwich beams experimental study. Smart Materials and Structures, 19, 1 9, US Patent Magnetorheological Fluid Composite Structure 8. Identyfikacja parametrów modelu trójwarstwowej belki z cieczą magnetoreologiczną W pracy dokonano identyfikacji parametrów modelu trójwarstwowej belki wspornikowej z cieczą magnetoreologiczną (MR). Belka składa się z dwóch zewnętrznych warstw aluminiowych, pomiędzy którymi znajduje się warstwa cieczy MR uszczelniona gumą silikonową. Model belki zbudowano przy wykorzystaniu elementów skończonych dostępnych w programie ANSYS. Oddziaływanie pola magnetycznego na belkę uwzględniono przez zmianę wartości parametrów uŝytych w modelu elementów skończonych. Dane do identyfikacji pozyskano z badań drgań własnych belki. W procedurze identyfikacji kierowano się zgodnością wartości częstotliwości drgań własnych belki oraz bezwymiarowych współczynników tłumienia uzyskanych z badań i obliczeń.