Modeling of a Circular Plate with Piezoelectric Actuators of Arbitrary Shape

Transcription

1 Vol ACTA PHYSICA POLONICA A No. 6 Acoustic Biomedical Engineering Modeling of a Circular Plate with Piezoelectric Actuators of Arbitrary Shape M. Wiciak a, R. Trojanowski b a Cracow University of Technology, Institute of Mathematics, Warszawska St. 4, Cracow, Pol b AGH University of Science Technology, Faculty of Mechanical Engineering Robotics Department of Mechanics Vibroacoustics, al. A. Mickiewicza 30, Krakow, Pol This paper is concerned with mathematical aspects numerical modeling of vibration of a circular plate with piezoelectric actuators. Particularly, a thin Kirchho-Love plate with arbitrary shaped actuators e.g. pie-shaped, trapezoid,, rectangular is considered. In the theoretical model, the moments that act upon a structure are induced by piezoelectric actuators are described by the generalized tensor product of a distribution distribution-valued function. Numerical computations utilize the FEM approach supported by Ansys software. DOI: /APhysPolA PACS: At, Vn 1.Introduction The possibility of active vibration control for minimization of acoustic energy radiated by vibrating surface elements was explored for over 50 years [1. The advantage of using dierently shaped, distributed actuators for active control was demonstrated by a number of researchers, e.g. [5. Many works deal with dierent problems like optimal placement of actuators [6, 7, using specic congurations [8. There are also works that try to use elements typically used in active methods in passive systems [9, 10. Among plates of various shapes, circular plates seem to have a particular importance [5 due to their axial symmetry. Circular geometries are used in a wide variety of applications are often easily manufactured. In [11, a thin rectangular plate with arbitrary shaped actuators e.g. triangles, parallelograms, s is considered. The theoretical model for a structure is given in a language of distribution-valued function. Also, the formula for the solution of the Cauchy problem in the class of absolutely continuous tempered distribution-valued functions is derived. This paper presents an analytical approach to modeling circular plates with piezoelectric actuators of arbitrary shape. In particular, pie-, trapezoid-, -, rectangular-shaped actuators are considered. The natural tool to describe a structure that consists of a plate a piezoelectric actuator is the theory of distributions. The moments that act upon a structure are induced by piezoelectric actuators are expressed as the generalized tensor product of a distribution distributionvalued function. For two actuator shapes rectangular, numerical models of a circular plate with two piezo elements attached are created. The analysis uses the FEM method for structural vibrations. One of the piezo elements acts as an exciter the other as an actuator for vibration reduction.. Mathematical formulation of the problem.1 Equation of motion Let us consider a thin circular plate of thickness h 0 radius R, under the action of external forces moments. The plate is also assumed to be made of linearly elastic, homogeneous isotropic material of mass density ρ. The equation of motion for this plate is [5: D 4 w + Dµ w t 4 w + ρh 0 = F, 1 t where w = wt, r, θis transverse displacements, 4 =, = r + 1 r r + 1 r θ is the Laplacian operator in polar coordinates r, θ. Moreover, F = F t, r, θ is the external excitation, µ is internal damping loss factor, D = Eh 3 0/11 ν is the exural rigidity of the plate, ν E are the Poisson ratio Young's modulus, respectively. Additionally, consider a piezoelectric actuator adhered to the plate. The actuator consists of two identical pie-shaped piezoelectric elements of thickness h, bonded symmetrically on two opposite surfaces of the plate. The two elements are excited by opposite voltages the actuator generates external excitation of the distributed moment type in the plate. It is assumed that the reaction moments m r m θ in polar coordinates are equal uniformly distributed. Let F pe = F pe t, r, θ be an additional external stress caused by activating the piezoelectric element. Then [1, 5: corresponding author; mwiciak@pk.edu.pl F pe = m r r + r m r r + 1 r m θ θ 1 r m θ r

2 Modeling of a Circular Plate with Piezoelectric Actuators m r = m θ = m = C 0 ɛ pe χ, 4 where C 0 denotes the piezoelectric strain-plate moment coupling term, ɛ pe = d 31 V/h is the magnitude of the induced strain, d 31 piezoelectric-strain constant, V is the applied voltage, χ is a characteristic function dened on the area of the piezoelectric actuator Γ, { 1 if r, θ Γ χr, θ = 0 if r, θ/ Γ. 5 The function χ is not dierentiable even not continuous but the derivatives of m r = m θ = m can be understood in the sense of the theory of distribution. Namely, any locally summable function χ : R R denes regular distribution by means the formula [χϕ = χr, θϕr, θdr dθ 6 R for any test function ϕ DR. Since χ is assumed to be a characteristic function on Γ [χϕ = ϕr, θdr dθ 7 Γ for ϕ DR. Let us also recall that for any multi-index α N, the derivative D α T of distribution T D R is given by α D α T ϕ = 1 α T ϕ 8 r α1 α θ for ϕ DR, where α = α 1, α α = α 1 + α. Thus one can see that the formulae for distributions m m r, m θ m r in 3 depend on the shape of Γ... Pie-shaped actuator r, Consider at rst a pie-shaped actuator bonded to the plate as shown in Fig.1. In this case Γ = {r, θ : R 1 r R, θ 1 θ θ }. 9 It is convenient [1,,4,11 to treat the distribution [χ as a tensor product, [χ = [H θ1 H θ [H R1 H R, 10 where H a is a Heaviside step function, H a x = 1 for x a H a x = 0 for x < a. Fig. 1. Pie-shaped piezo actuator. Let us recall [1, 11 that tensor product of distributions T, S D R is a distribution S T D R dened as S T ϕ = Sθ T ϕ, θ. 11 In the above, the distribution T D R operates on test functions of variable r, while S D R is a distribution on test functions of variable θ. It is well known that for any ϕ DR, the mappings R r ϕr, θ R for xed θ, R θ T ϕ, θ R are test functions of variables r θ, respectively. It is also known [1 that the following formulae for derivatives are true: k dk k S T = S T, rk drk θ k S T = dk S T. 1 dθk According to 8 one can easy compute d dx [H a ϕ = [H a ϕ = ϕ xdx = a ϕa = δ a ϕ 13 for ϕ DR,where δ a is the Dirac distribution for any ϕ DR, δ a ϕ = ϕa. Thus d dx [H a = δ a. 14 Thereby on account of 4, 10, 1, 14 one can obtain m r = C 0ɛ pe [H θ1 H θ δ R1 δ R, 15 m r = C 0ɛ pe [H θ1 H θ δ R 1 δ R, 16 m θ = C 0ɛ pe δ θ1 δ θ [HR1 H R, 17 where, for short, δ R i, δ θ d i denote dr δ R i d dθ δ θ i, derivatives of distributions δ Ri, δ θi DR, respectively i = 1,. Finally, the external loads due to the actuator are F pe = C 0 ɛ pe [H θ1 H θ δ R1 δ R + 1r δ R1 1r δ R + 1 δ r θ1 δ θ [HR1 H R Trapezoid or disk-shaped actuator Now, let us turn to the case of a bit more complicated shapes of piezoelectric actuators. Consider trapezoid or circular actuator which is bonded to the plate as shown in Fig. Fig. 3, respectively. It is assumed that parallel sides of a trapezoid-shaped actuator are also parallel to a tangent to the plate edge, are given by the equations y = ax + b 1, y = ax + b, 0 < b 1 < b < R. As far as a disk-shaped actuator is considered it is claimed that the actuator is enclosed by a circle with the centre at the point a, b a radius r 0. Analogously to the case of pie-shaped actuator the internal moments across the piezoelectric can be expressed as 4 5, where Γ = {r, θ : r 1 θ r r θ, θ 1 θ θ }, 19

3 1050 M. Wiciak, R. Trojanowski Fig.. Trapezoid shaped piezo actuator. k θ k S T = k i=0 k d i i dθ i S T k i, 4 where the symbols k d, k, k d, i r k dr k θ k dθ denote derivatives i of distributions in accordance with 8, while T j means the j-th derivative of the map T. Consequently, the internal moments across the trapezoid- disk-shaped piezoelectric element can be expressed as the generalized tensor product m r = m θ = m = C 0 ɛ pe [H θ1 H θ [ H r1θ H rθ. 5 Since for any θ R ϕ DR, [ Hriθ ϕ = ϕrdr, i = 1,, 6 r iθ the derivative of the map R θ [ H riθ D R is given by [ Hriθ = r i θ δ riθ for i = 1,, 7 [Hriθ = r 1 θ δ r iθ r i θ δ riθ, Fig. 3. Disc-shaped piezo actuator. r i θ = for i = 1,, 0 sin θ a cos θ in the case of trapezoid-shaped actuator, r 1, θ = a cos θ + b sin θ r0 a sin θ b cos θ, 1 in the case of disk-shaped actuator, where θ 1 θ are the slopes of the tangent to the circle from the center of the plate [5. Unfortunately, in both above cases the distribution [χ cannot be treated as a tensor product of distributions since r i is a function of θ [H ri is not longer a distribution but a distribution-valued function, R θ [ H riθ D R. A simple example showing that even when a distribution S depends only on θ T θ D R is a distribution depending on r, the tensor product is not a distribution, can be constructed [11. For this reason, a generalized tensor product will be used. For the convenience of the reader, we recall that if S D R a distribution-valued function T : R θ T θ D R is of class C, then the generalized tensor product of S T, S T, is a distribution, is dened by S T ϕ = Sθ T θϕ, θ for any ϕ DR. The following formulae for derivatives of S T are also true [11: k dk S T = S T, 3 rk drk for i = 1,. 8 Therefore from 5, 4 the second derivative of m with respect to θ is obtained as equaling m θ = C 0ɛ pe [H θ1 H θ [ r 1θ δ r 1θ r θ δ r + θ r 1 θ δ r1θ r θ δ rθ δ θ1 δ θ r 1θ δ r1θ r θ δ rθ + δ θ 1 δ θ [ Hr1θ H rθ. 9 Similarly, in accordance with 5, 3, the derivatives of m with respect to r are m r = C 0ɛ pe m [H θ1 H θ d dr [ Hr1θ H rθ = C 0 ɛ pe [Hθ1 H θ δ r1θ δ rθ, 30 R = C 0ɛ pe Consequently, F pe = [H θ1 H θ [H θ1 H θ 1 + 1r rθ δ r θ δ r 1θ δ r θ 1 + 1r r1θ δ r 1θ + 1 r r+r 1θ δ r1θ r+r θ δ rθ + 1 δ r θ1 δ θ [ Hr1θ H rθ r δ θ 1 δ θ r 1θ δ r1θ r θ δ rθ. 3 In particular, in the case of trapezoid-shaped actuator,

4 Modeling of a Circular Plate with Piezoelectric Actuators r iθ = cos θ + a sin θ, 33 sin θ a cos θ r i θ = sin θ a cos θ 3 + a 1 a sin θ + a sinθ, 34 whereas, in the case of disk-shaped actuator, derivatives of r i are given by r 1,θ = a sin θ + b cos θ a b sin θ ab cos θ ± so r, 35 r0 a sin θ b cos θ 1,θ = a cos θ b sin θ a b cos θ + ab sin θ ± r0 a sin θ b cos θ a b sin θ ab cos θ 1 8 r 0 a sin θ b cos θ Rectangular actuators Consider now a rectangular actuator which is bonded to the plate as shown in Fig. 4. Let us assume that two parallel sides of the rectangle are parallel to a tangent to the plate edge, are represented by the equations y = ax + b 1, y = ax + b, 0 < b 1 < b < R. Let y = 1/ax +, i = 3, 4, b 3 < b 4, denote the equations of two other parallel sides of rectangle. r i θ = sin θ + 1 for i = 3, a cos θ Thus the distribution [χ can be expressed as a sum [χ = [H θ H θ1 [ H r1θ H r4θ + [H θ4 H θ [ H r1θ H rθ + [H θ3 H θ4 [ H r1θ H r3θ. 38 With 3 taken into account an easy computation shows that F pe = C 0 ɛ pe [H θ3 H θ1 1r [ r + r 1θ δ r 1θ r + r 1 θ δ r1θ + [H θ H θ4 1 [ r r + r θ δ r θ r + r θ δ rθ + [H θ4 H θ3 1 r [ r + r 3θ δ r 3θ r + r 3 θ δ r3θ + [H θ1 H θ 1 [ r r + r 4θ δ r 4θ r + r 4 θ δ r4θ 1 [ r δ θ 1 [H r4 H r1 + δ θ [H r H r4 +δ θ 3 [H r1 H r3 + δ θ 4 [H r3 H r + r δ θ 1 [ r 1 δ r1 r 4δ r4 + δθ r 4δ r4 r δ r +δ θ3 r 3δ r3 r 1δ r1 + δθ4 r δ r r 3δ r3. Fig. 4. Rectangular shaped piezo actuator. Denote also by x i, y i the vertexes of the rectangle for i = 1,..., 4 θ i = tan 1 y i /x i, as shown in Fig. 4. Analogously to previous sections, the reaction moments across the rectangular actuator can be described by m r = m θ = m = C 0 ɛ pe [χ, where χ is a characteristic function of the region Γ covered by the rectangular actuator [3. To provide the parameterization of Γ in polar coordinates we need to decompose it into three subregions, Γ = Γ 1 Γ Γ 3, determined by Γ 1 = {r, θ : r 1 θ r r 4 θ, θ θ θ 1 }, Γ = {r, θ : r 1 θ r r θ, θ 4 θ θ }, Γ 3 = {r, θ : r 1 θ r r 3 θ, θ 3 θ θ 4 }, where r i θ = for i = 1,, sin θ a cos θ 39 Note that derivatives of curves r i = r i θ for i = 1, are given by formulae 33, 34, while for i = 3, 4 one can obtain r i = cos sin θ + 1 θ 1a a cos sin θ, 40 θ r i = sin θ + 1 a cos θ 3 + 1a 1 1a sin θ 1a sin θ. 41 Finally it should be noted that it is possible to describe external loads due to another shapes of actuators in analogous way. Also one can build a model for transverse displacement when a set of actuators is considered. In this case the term of external loads is a sum of distributions that describe single actuator. Let us note that the solution of 1 exists in the sense of the theory of distributions. The theorem on existence

5 105 M. Wiciak, R. Trojanowski of the solution analytic formulae for the solution can be found in [13. Equation 1 can be also solved using classical methods based on the separation of variables [, 3. An approximate solution can be obtained by the nite element method. 3. Numerical calculations From the above analytical considerations two actuator shapes were chosen for numerical simulations. Those were where the added external is describe by equations 3, 35, 36 rectangle where the added external is described by equations 3941, 33, 34. Four models consisting of a steel plate with a diameter equal to 0.15 m thickness equal to m clamped on the edge two piezo elements attached were created. The rst element marked by white color is used for plate excitation but the other one marked by violet color for vibration reduction. In the rst two models the primary excitation was obtained with the use of actuator for reduction a one a -shaped one were used Fig. 5a b. For the next models the dierence was that the piezo element used for plates excitation were -shaped Fig. 5c d. Fig. 6. Modes shapes analyzed. Model parameters. TABLE I Object Element type Other info plate SOLSH190 ρ = 7500 kg/m 3, E = Pa, ν = 0.3 piezo element SOLID6 equivalent to PZT4, A = 600 mm Fig. 5. Created models. All of the modeled piezo actuators had a base area of 1600 m 1 mm thickness. The element used for modeling were SOLSH190 for the plate SOLID6 for actuators. Material constants can be found in Table I. After performing modal analyses, 6 rst modes were chosen for harmonic analyses although mode 3 occur for almost the same frequency, so in harmonic analyses they were treated as one. Fig. 6 shows those mode shapes. For frequencies corresponding to the chosen mode shapes, harmonic analyses were performed with optimization procedure for the voltage applied to the element used for vibration reduction. The goal function for optimization was n J = min u zi i=1, 4 n where u zi are vibrations of node i of the plate in the plane perpendicular to its base, n number of nodes creating plates base. Optimization parameters are shown in Table II, where V 0 is the voltage amplitude applied to the element used for plates excitation while V 1 ϕ 1 are design variables. V 1 is the voltage amplitude applied to the actuator ϕ 1 is a phase angle of applied voltage as described by the equation V = V 1 e jωt+ϕ1, 43 where j = 1. The maximum number of iterations was set to 00. TABLE II Optimization parameters. Quantity V 0 V 1 ϕ 1 Range 100 V V 0360

6 Modeling of a Circular Plate with Piezoelectric Actuators Results conclusions Obtained results are presented in Table III. Results of the analyses. TABLE III mode shape V 1 φ 1 reduction excitation reduction [V [deg [db or Examining the results one can say that the dierences in vibration reduction obtained with the use of dierent shapes of piezo actuators are rather small with one exception, depend on the relation of shapes between the element used for excitation the one used for reduction. Another thing is that the largest reduction occurs for modes 1, 0, 1,, 0, 0, that occurs not only when our conguration is symmetrical but also when there is a proper placement of actuators. The dierence in results for - conguration for modes or 3 is probably the eect of the second mode shape from this pair being observed. It also appears that for most cases the shaped elements need lower voltage to achieve similar eect although the dierence is rather small it does not exceed 6%. In the paper an analytical model of circular plates with piezoelectric actuators of arbitrary shape was presented. The internal moments across the actuator were described in terms of generalized tensor product of a distribution a distribution-valued function. The solution of the problem exists in the sense of the theory of distributions. It can be derived by analytic formulae or using classical methods based on separation of variables. On the other h an approximate solution can be obtained by the - nite element method. Numerical analyses results showed large vibration reduction more than 30 db for the rst, fourth sixth mode, almost no reduction for the rest of the analyzed modes. If the actuator was placed symmetrically to the exciter, all 6 modes could easily be reduced. Acknowledgments This study is a part of the research project N N supported by Polish Ministry of Science Higher Education. References [1 E.K. Dimitriadis, C.R. Fuller, C.A. Rogers, AIAA Journal 9, [ E.K. Dimitriadis, C.R. Fuller, C.A. Rogers, J. Vib. Acoust. 113, [3 E.M. Sekouri, Y. Hu, A.D. Ngo, Mechatronics 14, [4 M. Sullivan, J.E. Hubbard, Jr., S.E. Burke, J. Acoust. Soc. Amer. 99, [5 J. Wiciak, Archives of Acoustics 3, [6 A. Bra«ski, M. Borkowski, S. Szela, Acta Phys. Pol. A 118, [7 A. Bra«ski, G. Lipi«ski, Acta Phys. Pol. A 119, [8 M. S. Kozie«, J. Wiciak, Acta Phys. Pol. A 116, [9 R. Filipek, J. Wiciak, European Physical Journal - Special Topics 154, [10 M.S. Kozie«, B. Koltowski, Acta Phys. Pol. A 119, [11 M. Wiciak, Acta Phys. Polon. A 11, A [1 W. Rudin, Functional Analysis, McGraw-Hill, New York, [13 M. Wiciak, Univ. Iagel. Acta Math. 4,