Dynamic Stiffness Approach to Frequency Analysis of FGM Beam Bonded with a Piezoelectric Layer

Nội dung text: Dynamic Stiffness Approach to Frequency Analysis of FGM Beam Bonded with a Piezoelectric Layer

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 357-361, DOI 10.15625/vap.2019000302 Dynamic Stiffness Approach to Frequency Analysis of FGM Beam Bonded with a Piezoelectric Layer Nguyen Tien Khiem1*), Tran Thanh Hai1), Nguyen Ngoc Huyen2) and Luu Quynh Huong2) 1) Institute of Mechanics, Vietnam Academy of Science and Technology 2) Thuy Loi University, Chua Boc, Dong Da, Hanoi, Vietnam E-mail: ntkhiem@imech.vast.vn; khiemvch@gmail.com Abstract Frequency analysis of a functionally graded beam bonded by a piezoelectric layer is carried out by using the dynamic stiffness (2.1) method. First, governing equations are conducted for FGM beam element with a piezoelectric layer based on the Timoshenko beam theory and power law of material grading. The established equations are solved to get frequency dependent shape functions that are employed then for constructing dynamic stiffness matrix of the beam element. Natural frequencies of an FGM beam with a piezoelectric layer representing a distributed sensor are examined in dependence Fig.1. FGM beam with piezoelectric layer upon the material properties and thickness of the piezoelectric layer where E, G, ρ denote Young’s, shear modulus and mass Keywords: FGM Beam, Piezoelectric Material, Dynamic density of the material respectively, the subscripts b and t Stiffness Method. indicate bottom and top material components and z is the ordinate measured from the central axis of the beam. 1. Introduction Following Timoshenko beam theory, the constitutive equations are Piezoelectric material has got more and more used for control of engineering structures [1-3] due to the property ; that couples its elastic evolution with electric field. ; (2.2) Particularly, it is a specially prevailing device for structural health monitoring as both sensor and actuators [4-6]. (2.3) The authors of References [7,8] have shown that piezoelectric patch can be efficiently used also for where the index b at components in the deformable field repairing a notched or cracked beam. Since a typical now denote those of the base beam, the index 0 implies property of the piezoelectric material is its undergoing at that is measured at the mid-plan of the beam. Using the high frequency band, the frequency domain methods are equations strain energy of the base beam can be most appropriate in modelling piezoelectric structures. calculated as Lee and his co-workers [9,10] and Park et al [11] have extended the spectral element method for composite beam with piezoelectric layers. Li and Shi studied free vibration of a functionally graded piezoelectric beam via state-space based differential quadrature [12]. In this report, the dynamic stiffness method is developed for frequency analysis of functionally graded (2.4) material (FGM) beam with piezoelectric layer. This is where first effort of the authors to apply the piezoelectric material for health monitoring FGM structures. (2.5) 2. Governing equations 2.1. Constitutive equations for base FGM beam 2.2. Constitutive equations for piezo-electric layer Suppose that FGM properties of the base beam are Let’s consider the piezoelectric layer as a Timoshenko varying accordingly to the power law beam element, so that constitutive equations can be
2. Nguyen Tien Khiem , Tran Thanh Hai , Nguyen Ngoc Huyen and Luu Quynh Huong expressed as the constants , , , , , , become those of the host FGM beam , , (without asterisk superscript). The piezoelectric material is described by 3. Dynamic stiffness model 3.1. General vibration mode shape (2.6) Seeking solution of Eq. (2.11) in the form where , are elastic modulus, piezoelectric and dielectric constants respectively. and are electric field and displacement of the piezoelectric layer. (3.1) Perfect bonding between the base beam and piezo-electric leads that equation to layer is represented by the conditions (2.7) that yield (3.2) So that one obtains characteristic equation for seeking the (2.8) wave number as follows Therefore (2.9) and or (3.3) Let roots of Eq. (3.3) with respect to be found in the form (3.4) (2.10) where are roots of the cubic equation +c . Hence, so-called vibration mode 2.3. Equations of motion for FGM beam element with shape a piezoelectric layer can be expressed as Using Hamilton’s principle, equations of motion of the electromechanical system are established as + + . (2.11) (3.5) where the following notations have been used where + ; ; (2.12) . (3.6) + + ; 3.2. Dynamic stiffness matrix Using the shape functions (3.5), vector of nodal . (2.14) displacements , defined in Fig. 2, can be calculated as The resultant forces are determined for the beam as ; (3.7) (2.15) Where and Obviously, in case of beam without piezoelectric layer all
3. Dynamic Stiffness Approach to Frequency Analyssis of FGM Beam Bonded with a Piezoelectric Layer Eliminating constant vector C from Eqs. (3.7) and (3.11) gives (3.14) with matrix that is acknowledged as the dynamic (3.9) stiffness matrix of the beam element. For a structure composed off a number of such the beam (3.8) elements, the total dynamic stiffness matrix is assembled from those of the component elements by (3.15) 3.3. Modal analysis of FGM beam bonded by piezoelectric layer Free vibration of a structure formulated by using the dynamic stiffness model is governed by solving the equation (3.16) Fig. 2. Nodal displacements and forces of a beam element where total vector of nodal displacements is Similarly, the internal forces assembled also from the local vector . Solution of Eq. (3.16) gives natural frequencies determined as positive roots of equation = (3.10) can be calculated at the nodes as (3.17) (3.11) where and and normalized solution of equation (3.18) (3.12) The matrix in Eq. (3.11) has the elements Therefore, mode shape corresponding to natural frequency would be determined by ; ; ; 4. Examples ; Let’s consider an FGM beam of length L bonded by a ; piezoelectric layer along all the beam length as shown in ; ; Fig. 3. Note that if the beam is clamped at the ends, for which boundary conditions are 0 then Eq. (3.7) leads to ; (4.1) ; ; The latter equation gives rise immediately the frequency equation for the beam (4.2) ; Numerical computation is carried out for FGM with ; following parameters 3 ; E1=390e9Pa; ρ1=3960kg/m ; µ1=0.25; 3 E2=210e9Pa; ρ2=7800 kg/m ; µ2=0.31; L=1.0m; b=0.1m; h=L/10 with varying volume distribution index n. The parameters ; of piezoelectric material are ; H13=-7.70394e8; ρ=7750; C11=69.0084e9; . (3.13) B33=7.38857e7; C55=21.0526e9.
4. Nguyen Tien Khiem , Tran Thanh Hai , Nguyen Ngoc Huyen and Luu Quynh Huong Table 1. Variation of natural frequencies versus power law index n of FGM and thickness h of piezoelectric layer for clamped-clamped end beam n Freq. hp=0.05 hp=0.08 hp=0.1 1 9.6862 10.0951 10.5041 Fig. 3. FGM beam with a piezoelectric patch 2 24.5118 25.0041 25.5356 Thickness of the piezoelectric layer varies from 0 0.1 3 42.4221 38.1190 36.0326 (without piezoelectric layer) to 0.1 (the beam thickness) 4 43.8183 43.9338 44.2979 and boundary conditions for the beam are clamped ends. 5 65.9328 65.1768 65.0498 Results of the computation are depicted in Table 1 and 1 9.3367 9.7948 10.2236 Fig. 4. Table 1. Variation of natural frequencies versus power 2 23.6387 24.2734 24.8692 law index n of FGM and thickness h of piezoelectric 0.2 3 41.0480 37.0744 35.1281 layer for clamped-clamped end beam 4 42.2748 42.6621 43.1463 5 63.6249 63.2950 63.3530 n Freq. hp = 0 hp = 0.01 hp = 0.02 1 8.6362 9.1808 9.6429 1 10.8205 10.2595 9.9140 2 21.8717 22.7578 23.4656 2 27.7924 26.3334 25.3985 0.5 3 38.1419 34.8241 33.1659 0.1 3 50.3343 47.6680 45.8992 4 39.1248 39.9972 40.6970 4 56.3731 52.3676 49.1608 5 58.8860 59.3243 59.7221 5 76.5667 72.4797 69.6949 1 8.0408 8.6423 9.1229 1 10.2319 9.7532 9.4652 2 20.3397 21.3910 22.1673 2 26.2854 25.0404 24.2569 1.0 3 35.3747 32.6293 31.2342 0.2 3 47.6118 45.3378 43.8498 4 36.3465 37.5453 38.3877 4 53.5455 50.0200 47.1606 5 54.6515 55.6204 56.2574 5 72.4358 68.9492 66.5975 1 7.5627 8.1945 8.6783 1 9.1182 8.7782 8.5895 2 19.0653 20.2000 20.9988 2 23.4254 22.5401 22.0174 2.0 3 32.7219 30.4779 29.3231 0.5 3 42.4315 40.8149 39.8092 4 33.9690 35.3399 36.2473 4 48.0050 45.3109 43.0747 5 50.9504 52.2180 52.9881 5 64.5588 62.0764 60.4676 1 7.1817 7.8352 8.3128 1 8.2292 7.9864 7.8688 2 17.9958 19.1716 19.9596 2 21.1256 20.4903 20.1522 5.0 3 30.1578 28.3557 27.4216 1.0 3 38.2389 37.0750 36.4073 4 31.9003 33.3536 34.2684 4 43.1884 41.1057 39.3475 5 47.6508 49.0759 49.8996 5 58.1469 56.3521 55.2607 1 7.0083 7.6850 8.1632 1 7.5376 7.3663 7.3015 2 17.5053 18.7226 19.5086 2 19.3119 18.8570 18.6522 10 3 29.0126 27.3954 26.5562 2.0 3 34.8940 34.0499 33.6202 4 30.9497 32.4704 33.3916 4 38.9091 37.2893 35.9051 5 46.1361 47.6676 48.5185 5 52.9791 51.6631 50.9295 Observing the results given in Table 1 and graphs shown 1 6.9493 6.8451 6.8320 in Fig. 4 one can make the following notices: 2 17.7560 17.4632 17.3820 Natural frequencies of FGM beam bonded with a 5.0 3 32.0050 31.4374 31.2166 piezoelectric layer decrease with growing volume 4 35.0282 33.7687 32.6832 fraction index n independently upon thickness of the 5 48.4886 47.5736 47.1418 piezoelectric layer; 1 6.6339 6.5709 6.5922 Under growing thickness of piezoelectric layer from zero to thickness of the host beam, flexural vibration 2 16.9432 16.7477 16.7462 frequencies of the electro-mechanical system at first 10 3 30.5295 30.1234 30.0336 reduce to a minimum, then they get monotonically 4 33.3620 32.2413 31.2729 increasing. Curvature of the graphs at minimum value 5 46.2376 45.5513 45.3036 is increasing with reducing volume fraction index n.
5. Dynamic Stiffness Approach to Frequency Analyssis of FGM Beam Bonded with a Piezoelectric Layer Longitudinal frequencies of the beam are all decreasing layer increasing from zero to a level makes flexural with growing thickness of piezoelectric layer regardless frequencies decreased before they get monotonical the index n. increase, while the axial vibration frequencies are unaffected by increasing of the thickness. Next study of the authors is to investigate influence of piezoelectric layer on natural frequencies of cracked FGM beam. References [1]. Saravanos, D.A., Heyliger, P.A. (1999) Mechanics and computational models for laminated piezoelectric beams, plates and shells. ASME Appl Mech Rev 52(10) 305-320. [2]. Maurini, C., Porfiri, M., Pouget, J. (2006) Numerical method for modal analysis of stepped piezoelectric beams. J. of Sound and Vib. Vol. 298: 918-933. [3]. Yang, S.M., Lee, Y.J. (1994) Modal analysis of stepped beams with piezoelectric materials. J. of Sound and Vib. Fig. 4. Variation of fundamental frequency versus 176(3) 289-300. thickness of piezoelectric layer for clamped end FGM beam with n = 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10. [4]. Bhalla, S., Soh, C.-K. (2006) Progress in Structural Health Monitoring and Non-destructive Evaluation Using The decrease and increase of natural frequency with Piezo-impedance Transducers. Smart Material and growing thickness of piezoelectric layer can be explained Structures: New Research, Editor P.L. Reece, Chapter 6, as following: as well-known in the theory of vibration, pp. 177-228. natural frequency of a system is defined as ratio of its [5]. Lee, C.-K. and Moon, F. C. (1990) Modal stiffness to mass. Therefore, frequency could be increasing Sensors/Actuators. Journal of Applied Mechanics, or decreasing in dependence on whether the stiffness grow Transactions of ASME, Vol. 57: 434-441. is more or less than that of the mass. Actually, as shown in [6]. Mateescu, D., Han, Y., Misra, A. (2007) Dynamics of Fig. 4, frequency of the beam bonded with piezoelectric Structures with Piezoelectric Sensors and Actuators for layer is always less than that of beam without the bonded Structural Health Monitoring. Key Engineering Materials layer (when n = 0.1). This implies that contribution of the Vol. 347: 493-498. layer in stiffness is less than its contribution in mass. In the [7]. Wang, Q., Duan, W.H., Quek, S.T. (2004) Repair of case when n > 0.2 the contribution in stiffness is less only notched beam under dynamic load using piezoelectric for small thickness of bonded layer and it gets more than patch. International Journal of Mechanical Sciences 46, mass contribution from a value of the thickness that results 1517–1533. in increasing natural frequency. [8]. Ariaei, A., Ziaei-Rad, S., Ghayour, M. (2010) Repair of 5. Conclusion cracked Timoshenko beam subjected to a moving mass using piezoelectric patches. International Journal of In the present report, governing equations of FGM Mechanical Sciences 52, 1074–1091. beam bonded by a piezoelectric layer have been derived on the base of Timoshenko beam theory, power law of [9]. Lee, U., Kim, J. (2000) Dynamics of elastic-piezo-electric material grading. The established equations have the form two-layer beams using spectral element method. Intern. J. of single FGM beam with those coefficients added by the of Solids and Struct. Vol. 37: 4403-4417. terms representing contribution of the piezoelectric layer. [10]. Lee, U., Kim, D., Park, I. (2013) Dynamic modeling and These equations are then solved to get analysis of the PZT-bonded composite Timoshenko beams: frequency-dependent shape functions needed to develop Spectral element method. J. Sound and Vib. Vol. 332: the dynamic stiffness method for modal analysis of the 1585-1609. double beam. [11]. Park, H.-W., Kim E.J., Lim, K.L., Sohn, H. (2010) The developed method has been employed to frequency analysis of an FGM beam with piezoelectric Spectral element formulation for dynamic analysis of a layer. The analysis shows that the piezoelectric layer does coupled piezo-electric wafer and bean system. Computers not modify the well-known properties of FGM beam even and Structures Vol. 88: 567-580. thickness of the layer reaches to thickness of the base [12]. Li, Y., Shi, Z.F. (2009) Free vibration of a functionally beam. However, thickness of the piezoelectric layer graded piezoelectric beam via state-space based makes a remarkable influence on natural frequencies of differential quadrature. Composite Structures 87, 257-264. the bonded beam. Namely, thickness of piezoelectric