表面贴有压电片的桁架结构系统静力可靠性分析英文

1、Chinese Journal of Aeronautics 22(2009) 22-31Chinese Journal of AeronauticsStatic Force Reliability Analysis of Truss Structure with Piezoelectric Patches Affixed to Its SurfaceAn Hai, An Weiguang*, Yang Duohe, Wang BinshengDepartment of Aerospace Engineering, Harbin Engineering University, Harbin 1

2、50001, ChinaReceived 18 January 2008; accepted 5 June 2008AbstractFor the truss structure composed of active-elements with piezoelectric patches affixed to its surface, taking the mechani- cal-electric coupling effect under the action of electric loads and mechanical loads into consideration, the fi

3、nite element model for static force analysis is established by using the theory of mechanics. The failure mechanism of piezoelectric elements is discussed and the failure criteria of piezoelectric elements are proposed. The expression of safety margins for the element of piezoelectric truss structur

4、e is proposed with fracture strength, damage electric field strength, the section area of elements, and loads as random variables. On the basis of the research, the stochastic finite element method and structural reliability theory are used to analyze the reliability of the piezoelectric truss struc

5、ture. An example is given to demonstrate the validity of this method.Keywords: piezoelectric truss structure; piezoelectric patch; mechanical-electric coupling; safety margin; stochastic finite element method; structural reliability theory1. IntroductionThe piezoelectric intelligent structure has be

6、en proven to have important application value in the field of aviation and spaceflight. Piezoelectric material has positive and negative piezoelectric effects because of mechanical-electric coupling effect, and has been ap- plied in practical engineering1-4. If there is electric current on the surfa

7、ce of piezoelectric element, the effect of electric field will make the piezoelectric ele- ment distorted. The shape and the mechanical state of the piezoelectric material structure can be changed by the negative piezoelectric effect5-6. However, because piezoelectric materials are brittle, they mig

8、ht be frac- tured due to too large force or puncturability because of over-strong electric current. When we consider the randomness of environmental conditions and structure itself, in order to ensure a certain safety redundancy of structure, it is significant for piezoelectric structure to analyze

9、its reliability of static strength. Because of the idea of intelligent structure has been formed, the de-veloped countries represented by the United States, Japan, and Britain have quickly thrown a great deal of resources into the research and exploration of this area. But on the issue of the reliab

10、ility of intelligent struc- ture, there few published works. The domestic re- search of intelligent structure has been started, but it is virtually zero for the research of the reliability of intelligent structure. In this article, taking load acting on structure, the cross-sectional area and streng

11、th of element as random variables, the system reliability of piezoelectric structure is analyzed using the stochastic finite element method and basic reliability theory for exploring an approach of analyzing the reliability of the piezoelectric intelligent structure.2. Static Finite Element Analysis

12、 of Truss Stru- cture with Piezoelectric Patches Affixed to Its SurfaceBy linear piezoelectricity principle, neglecting the magnetic field effect and thermo-piezoelectric effect, under semi-static electric-field condition, the piezo-7*Corresponding author. Tel.: +86-451-82519209.E-mail address: anwe

13、iguangFoundation items: National Defense Basic Foundation (Z192002A001);electric constitutive equationso = ce - eEare as follows(1)National Defense Monograph Foundation1000-9361/$ - see front matter 2009 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(08)60065-5D = eTe + X E where s and e

14、 are the vectors of stress and strain, re-spectively; E and D are the vectors of electric field intensity and electric displacement; c is the elastic constant matrix; e is the piezoelectricity stress coeffi- cient matrix, and X is the dielectric constant matrix.tric equations of piezoelectric elemen

15、t ares1 = C11e1 - e13 E3D3 = e31e1 + X 33 E3(2)(3)In this article, the truss elements with piezoelectric patches affixed to its surface as shown in Fig.1 and Fig.2 are studied. Piezoelectric patches are affixed toWhen d1 = 0 , the output force from one piezoelec- tric patch is defined asthe surfaces

16、 of elements, and it is assumed that theyy = s= -= -f = -f(4)are identical and the brim-effects of electric field are being neglected.N11 Ae13 AE3whereye13 A he13bN1 is the output force alongX1 axis; A isthe cross-sectional area of the piezoelectric patch, and A = bh ; b is the width of the piezoele

17、ctric patch; and f is the difference of potential at both ends.2.2. Mechanical analysis of element with piezo- electric patches affixed to itFig.1 Element with piezo- Fig.2 Piezoelectric patches electric patches.and the elementarybody.2.1. Piezoelectric equation of piezoelectric patchAssume that the

18、 piezoelectric patch has the charac-The piezoelectric element of the truss structure con- sists of elementary body and piezoelectric patches as shown in Fig.4. Its mechanical analysis is presented below.It is known that A is the cross-sectional area of element; A is the cross-sectional area of the p

19、iezo- electric patch, A = bh ; N is applied load acting on both ends of the element; n is the number of piezo- electric patches at one-side of element; and E and C11 are the elastic constants of the element of the truss and piezoelectric patches, respectively.ters of piezoelectrics shown in Fig.3 (

20、X1 , X 2 , and X 3are local coordinates of Fig.2). Because piezoelectricpatch is only loaded with theX1 axial force of thebody, therefore, only the force and the deformationgenerated along theX1 direction (axis direction of theelement) are considered.Fig.3 Piezoelectric micro-element.For the electri

21、c field intensity, only the effect of E3is considered, and other electric-field components areFig.4 Element structure with piezoelectric driven patches.Assume that the stretch of the element is divided into two parts, wherein one part is the stretch of theset as E2 = E1 = 0. The surface ofX 3 is the

22、 electrodeelementary body with no piezoelectric patches. Thesurface, because the electrode surface is equipotentialoriginal length of the element is L, and the length ofsurfaceE3 = Z3 = 0is held (viz. electrical shortthe element part with piezoelectric patches symmetri-X1X 2cally affixed to it is l.

23、 Then, the total stretch of ele-ment iscircuit boundary conditions), so by selecting E3 and e1as independent variables, and s1 and D3 as responseDL = D(L - nl) + D(nl)(5)variables, and combining with Eq.(1), the piezoelec-where D(L - nl) is the stretch of the element with nopiezoelectric patches; an

24、d D(nl) is the stretch of theelement with piezoelectric patches affixed to it. From the knowledge of material mechanics and structureThe stretch of 0- l2gralcan be obtained through inte-mechanics, the D(L - nl)is given as followsD( l ) =1ll 2 N - (- X )2q dx =(N - N)l(8)D(L - nl) = N ( L - nl )EA(6)

25、2EA 022EAwhere N is the internal force of the element generated by applied loads.Another part of the stretch is combination of a symmetrical piezoelectric patch and the element, and the separations bearing forces are shown in Fig.5.In Fig.5, N is the shearing force. Assume that shearing force N per

26、unit length on the shearing plane obeys uniform distribution, as shown in Fig.6.Fig.7 Schematic diagram of differential unit dx bearing forces.Then, the whole stretch isDl = 2D( l ) = ( N - N )l(9)2EAThe stretch of the element with n piezoelectric patches isD(nl) = ( N - N )nlEA(10)Fig.5 Schematic d

27、iagram of separation bearing forces.For analyzing the deformation of piezoelectric patches, the differential unit dx is also taken, as shown in Fig.8.Fig.8 Schematic diagram of differential unit dx bearing forces.The stretch of 0- l can be obtained through inte-2gral as wellD( l ) = 1 l2 ( l - x)qdx

28、 = N l(11)2C A0 24C AFig.6 Schematic diagram of shearing force distribution.Then, the shearing force per unit length is1111Then, the whole stretch islN lq = N l /2 (7)Dl = 2D( ) =(12)22C11 AActually, the distribution of shearing force per unit length on shearing plane is complex, and the practi- cal

29、 approach is commonly used on the basis of test and experience in engineering8.For analyzing the deformation of 0- l , the differ-2ential unit dx is taken, as shown in Fig.7.The stretch of n piezoelectric patches isD(nl) = N n l 2C11 AFrom Eq.(10) and Eq.(13), it followsN = 2C11 A (N - N )EA(13)(14)

30、Setk = C11 A(15)From Eq.(16), under applied forces, the strain of piezoelectric patches isEAN2kEAThen,N = 2kN (16)e1 =C11 A(2k + 1)C11 ADLL - 2k nl2k + 1(23)2k + 13133Combining Eq.(5), Eq.(6), and Eq.(10), it followsReferring to Eq.(3), the charge balance equation of the piezoelectric patch isN (L -

31、 2knl )Q = ne l2k(EA)DL + 2nXbl fSo,DL = N( L- nl ) + ( N - N )nl =2k +1 EAEAEA(17)(2k +1)C11hL - 2k nl2k +1h(24)N =EA DL L - 2knl 2k + 1(18)As shown in Fig.9, the local coordinates Oxy of the plane truss structure within its global coordinate system are defined as follows, that is, setting i as the

32、It is known from Eq.(4), the force generated by the electric field of piezoelectric patches isorigin, element ij as x axis, the straight line, whichis passing through point i and perpendicular to x axis as y axis, and the angle between O x and Ox is g.e (2 A)F = -ne E (2 A) = -n 13f = -2ne bf(19)D13

33、 3h13According to Eq.(18) and Eq.(19), the sum of in- ternal forces of the element isN =EADL -2k2ne13bf(20)L -nl2k + 1When the displacement is zero, viz. DL = 0 , the output force of the piezoelectric element is defined from Eq.(20)N = -2ne13bf(21)Fig.9 Schematic diagram of local coordinates and glo

34、balwhenN = 0 , the output displacement of the piezo-coordinate system.electric element is defined from Eq.(20)L - 2k nlDL = 2ne bf2k + 1 (22)The balance equation of forces in local coordintes system can be given from Eq.(20) and the balance equation of node forces13EA N u f 1 = K B1 - K3(25)1 2 The

35、displacement derived from Eq.(22) is assumed under an ideal condition that every piezoelectric ele- ment is identical, there is no power consumption,where N2 u2 f4 there is no impact on each other, and the amplitude and phases of their output displacement are all theEA2k0L -nlsame. So, total displac

36、ement is the linear accumula-tive total of displacements of single patches.K = 12k +1EA(26)02k2.3. Static finite element equation of truss structur ewith piezoelectric patches affixed to itL -nl 2k +1 1-1The active element of the piezoelectric truss is shown as Fig.1. The piezoelectric equivalent ax

37、ial force is generated when the voltage is applied alongB = -11 K = 2ne13b0(27)(28)2X3 axis, then, the balance equation of total force is02ne13bshown as Eq.(20).wheref3 andf4 are the potential difference ofupper surface 3 vs lower surface 4 and lower surface 4eeevs upper surface 3, respectively; u1

38、and u2 are the dis-Q = K3 BRz+ K4j(37)placement of end 1 and end 2, respectively; and N1whereeTQ = Q3Q4 .and N2 are the node forces of end 1 and end 2, respec-tively.Transforming local coordinates system into globalGuyan coagulation is used to eliminate potential difference, then,coordinates system

39、for Eq.(25), the load vector P e( ee )eeeand z eof the element can be expressed aseTwhereK + KEz = P+ PQ(38)P = P1x P1 yP2 xP2 y (29)P e = RT 2ne13bV (39)eTQ-2ne bV Q2 4z = z1xz1 yz 2 xz 2 y (30)13Set coordinates transformation matrix asK = RK K -1(40)cos gsin g00 K e = RT K BR(41)1R = (31)E2 43 00c

40、os gsin g K e = RT K K -1 K BR(42)Then, Eq.(25) is transformed intoAccording to electrical knowledge for Qe , each pie-RP = K1BRz- K2j(32)zoelectric patch can be considered as a parallel plateeeeBecause RRT = I , through left multiplying Eq.(32) by RT , the balance equation of force in the globalcap

41、acitor. If there is no current leakage, the charge vector ise 2nX 33 A eQ = V(43)coordinates system can be given aseTeTehP = R K1BRz- R K 2j(33)where V e is applied voltage matrix.According to Eq.(24) and the balance of charge, the balance equation of charge in local coordinates can be given asBy us

42、ing finite element method to assemble units for Eq.(38), the ultimate static finite element equation of the piezoelectric truss structure can be given as=Q3 K B u1 + Kf3 4 (34)( K + KE )z = P + PQ(44)3 Q4 u2 f4 where Q3 and Q4 are the total charge quantities ofwhere z is the nodal displacement vecto

43、r；P is thevector of applied loads to nodes; and PQ is the elec- tric-load vector generated by imposed voltage becausesurfaces 3 and 4 of patches, respectively, and-Q4 = Q0 ;M0 K3 = 0H Q3 =(35)of mechanical-electrical coupling effect.It is known from Eq.(44) that, compared with the ordinary materials

44、, coupling stiffness KE and the electric-load vector PQ are added to the finite ele-and 2nX 33 A0ment equation of the piezoelectric truss structure.3. Reliability Analysis of Truss Structure withhPiezoelectric Patches Affixed to Its SurfaceK 4 = (36)02nX 33 A whereM =2ne13 Akh EA 3.1. Failure mechan

45、ism and safety margins ofpiezoelectric elementA piezoelectric truss structure is often loaded with(2k + 1)C11 A L - 2k nlelectric load and mechanical load simultaneously onH =2ne13 Ak2k + 1 EA use, and the failure of its element is caused by these loads the study on the failure under mechanical and(

46、2k + 1)C11 A L - 2k nl2k +1A is the area of the surface which is perpendicular toE3, A = bl .Transforming local coordinates system into global coordinates system for Eq.(34), it followselectrical coupling loads, then form, a new branch of solid destruction theory, viz. mechatronic reliability9. Defi

47、ciency in terms of several microns might bring fracture in piezoelectric structure when the element only loaded with mechanical and electrical coupling loads. Because the deficiency of piezoelectric structureis stochastic, the critical stress of piezoelectric struc- ture is generated stochastically

48、too.The significant difference between the piezoelectric material failure and ordinary material failure lies in the piezoelectric-induced failure. Apparently, the pie- zoelectric-induced failure is represented by elec- tric-induced fracture and dielectric puncture failure. The electric-induced fract

49、ure is the fracture behavior brought by electric field loading, its failure mecha- nism is that electric field concentration is formed when the electric field is loading, tensile stress is in- duced by electric field, and finally, the failure isbrought by the stress4. Electric load induces highlycon

50、centrated electric field at the crack top, and if the electric field intensity exceeds its puncture intensity, the electric puncture will occur. After the crack is ex- tended, the new electric field concentration and elec- tric puncture will occur in its front. When this occurs recurrently, it will

51、result in the failure of piezoelectric structure.Moreover, when the applied force to the elementary body is too large, it might result in strength failure. The piezoelectric patches are bonded with the body through glue layer, and if the shear strength of glue layer is lower than the applied force,

52、it might lead to the separation of piezoelectric patches and the body, and result in the failure of piezoelectric effect.Assume that the cross-sectional area, yield strength, the cross-sectional area of piezoelectric patch, fracture strength, damage electric field intensity, the shear strength of glue layer, and nodal loads are random variables, viz.where Z1j is the safety margin of the jth piezoelectric elementary body; Z2j is the safety margin of jth piezo- electric patch, and it is assumed that if one piezoelec- tric patch i