Exact Time Domain Solutions of 1-D Transient Dynamic Piezoelectric Problems with Nonlinear Damper Boundary Conditions

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1 Journal of Applid Mathmatics and Physics, 207, 5, ISSN Onlin: ISSN Print: Exact Tim Domain Solutions of -D Transint Dynamic Pizolctric Problms with Nonlinar Dampr Boundary Conditions Naum M. Khutoryansky, Vladimir Gnis Dpartmnt of Enginring Tchnology, Drxl Univrsity, Philadlphia, PA, USA How to cit this papr: Khutoryansky, N.M. and Gnis, V. (207) Exact Tim Domain Solutions of -D Transint Dynamic Pizolctric Problms with Nonlinar Dampr Boundary Conditions. Journal of Applid Mathmatics and Physics, 5, Rcivd: March 4, 207 Accptd: April 27, 207 Publishd: April 30, 207 Copyright 207 by authors and Scintific Rsarch Publishing Inc. This work is licnsd undr th Crativ Commons Attribution Intrnational Licns (CC BY 4.0). Opn Accss Abstract Novl xact solutions of on-dimnsional transint dynamic pizolctric problms for thicknss polarizd layrs and disks, or lngth polarizd rods, ar obtaind. Th solutions ar drivd using a tim-domain Grn s function mthod that lads to an xact analytical rcursiv procdur which is applicabl for a wid varity of boundary conditions including nonlinar cass. A nonlinar dampr boundary condition is considrd in mor dtail. Th corrsponding nonlinar rlationship btwn strsss and vlocitis at a currnt tim momnt is usd in th rcursiv procdur. In addition to th xact rcursiv procdur that is ffctiv for calculations, som nw practically important xplicit xact solutions ar prsntd. Svral xampls of th tim bhavior of th output lctric potntial diffrnc ar givn to illustrat th ffctivnss of th proposd xact approach. Kywords Pizolctric Layr, Transint Dynamic Problms, Tim Domain Solutions, Grn s Function Mthod, Nonlinar Boundary Conditions, Nonlinar Dampr, Output Voltag. Introduction Pizolctric matrials and dvics hav bn widly usd in many tchnical applications. Nowadays, th coupling btwn lctrical and mchanical bhaviors is usd in diffrnt dvics basd both on th so-calld dirct pizolctric ffct or th convrs pizolctric ffct [] [2] [3] [4]. Som nwr rlvant applications includ (among othrs) th high voltag gnration from transint dynamic impact procsss in vhicls [5]. DOI: /jamp April 30, 207

2 Analysis of oprating lctrical and mchanical paramtrs of such procsss can b don by using various analytical and numrical mthods. Although ana- lytical approachs ar limitd to rathr simpl gomtris and othr rstrictions (homognous or picwis-homognous bodis, linar govrning quations, tc.), thy oftn provid xact solutions. Analytical mthods hav bn succssfully usd for many transint dynamic on-dimnsional pizolctric problms [6]-[4]. Among analytical mthods for transint dynamic pizolctric problms, th Laplac transform mthods play a vry significant rol. Thy solv boundary valu problms in th frquncydomain, possibly for complx frquncis, using transformd boundary conditions for a pizolctric body. Aftr obtaining such solutions, th transformation back to th tim-domain mploys spcial mthods for th invrsion of Laplac transforms. Howvr, using th Laplac transform mthods is not instrumntal vn for on-dimnsional problms if nonlinar boundary conditions ar considrd. Tim-domain numrical mthods (.g., finit lmnt or finit diffrnc mthods) that can b usd undr such conditions usually lack prcision associatd with th us of analytical mthods. Thrfor, dvlopmnt of timdomain analytical or smi-analytical mthods combining advantags of analytical and numrical mthods can b of intrst for such problms. In this papr, a tim-domain Grn s function mthod is implmntd for solution of on-dimnsional transint dynamic pizolctric problms for thicknss polarizd disks or lngth polarizd rods. This mthod stms from a timdomain rprsntation formulas approach for transint dynamic pizo-lctric problms dscribd in [5]. For on-dimnsional problms with a varity of boundary conditions including nonlinar ons, this mthod producs xact solutions which ar shown blow. Such solutions can b usd both for analyss of longitudinal mod, pizolctric dvics and as bnchmark solutions for numrical mthods of pizolctricity. 2. Rprsntation Formulas Considr a transvrsly isotropic homognous pizolctric matrial (pizolctric lmnt) with th x3 -axis as th poling dirction and th x x2 plan as th isotropic plan. Lt this pizolctric matrial occupy a disk (or a cylindr) Ω boundd in x 3 -dirction by plans x 3 = 0 and x3 = h whr h > 0 is th thicknss of th disk (or th lngth of th cylindr). Considr a uniaxial strain stat or a strss strss stat in x 3 dirction whn thr is only on nonzro componnt of strain γ 33 or strss σ 33 th othrs bing zro. W assum that th non-zro strss and strain componnts, and also th displacmnt u 3 and lctric displacmnt D 3 in th x3 -dirction, and th lctric potntial φ, dpnd only on x 3 and t which is usually th cas for a longitudinal mod pizolctric lmnt [6]: ( x t) D D ( x t) (, ), φ φ(, ) in σ33 = σ33 3,, 3 = 3 3,, u = u x t = x t Ω () 874

3 Undr conditions (), w can us th following on-dimnsional constitutiv quations (both for th uniaxial strain stat and for th uniaxial strss stat) that rlat th mchanical and lctrical filds in (): σ = Cu + φ ; D = u φ (2) 33 3,3,3 3 3,3,3 whr cofficints ar diffrnt for th uniaxial strain and uniaxial strss cass. Thn th corrsponding quations of motions can b writtn as ρu3 Cu3,33 φ,33 = b3 ; (3) D = φ u = q in Ω 3,3,33 3,33 whr b = b ( x t ) and q q( x t), =, dnot th body forc in x3 -dirction and lctric charg. To simplify furthr notations w will dnot x 3 and drivativs with rspct to x 3 by x and th prim, rspctivly, and will skip subindx 3 for th lastic displacmnt, lctric displacmnt and body forc componnts prsntd in (3). Thn systm (3) bcoms ρu Cu φ = b; D = φ u = q in Ω. Th Grn s functions for vctor { u, φ } can b obtaind using concntratd impulss instad of b or q in (4) whn Ω is substitutd by th infinit mdia. Sinc φ can b xprssd through u du to th scond quation in (4), thn th first quation in (4) can b prsntd as th on-dimnsional wav quation for displacmnt u : whr ρ = D u C u b q C D 2 = C+ is th Young s modulus masurd at constant D. Th wav spd corrsponding to Equation (5) is dnotd blow by c = Th Grn s function for u corrsponding to load { b δ ( x) δ ( t), q 0} = = is th wll-known Grn s function for th on-dimnsional wav Equation (5): U( xt, ) = H( t x c) (6) 2 whr H( t ) is th Havisid stp function (right-continuous), i.. H( t ) = 0 for t < 0 and H( t ) = for t 0. Th corrsponding Grn s function for φ is calculatd using th scond quation in (4): Uφ ( xt, ) = U( xt, ) (7) D C Basd on (6) and (7), th rprsntation formula for th displacmnt vctor ρ (4) (5) 875

4 in 3-D cas dscribd in [5] rducs to th following xprssion for th dis- placmnt componnt u ( xt) u( xt) 3, =, : u( xt, ) = ( 0, ) (, ( ) ) 2 u t xc + u ht h x c t xc / σ ( 0, τ) D ( 0, τ) dτ 2 + t ( h x) c + σ ( h, τ) D( h, τ) dτ 2 + h t ξ x c + b( ξτ, ) q( ξτ, ) dτdξ in Ω, 2 0 whr σ 33 is dnotd by σ and it is takn into account that th outward normals to th lowr and uppr boundaris of th layr 0 x h hav opposit dirctions. In many practical applications, th lctric volum chargs ar absnt. Thr- for, w considr hncforth only th cas whn q = 0. Thn th trms rlatd to D in th abov xprssion can b simplifid sinc, basd on Equation (4) in this cas, D( xt, ) is spatially uniform: D( xt) D( t) (8), =. (9) Du to th proprty (9) th rprsntation formula (8) can b rwrittn as u( xt, ) = ( 0, ) (, ( ) ) 2 u t xc + u ht h x c t xc σ ( 0, τ) D ( τ) dτ 2 + t ( h x) c + σ ( h, τ) D( τ) dτ 2 + h t ξ x c + b( ξτ, ) dτd ξ in Ω. 2 0 To obtain a rprsntation formula for φ ( xt, ) function ψ ( xt, ) φ( xt, ) u( xt, ) (0), lt us considr an auxiliary = () that has th following connction to th lctric displacmnt: D = ψ. According to (3), ψ = D = 0. Thn, using th corrsponding Grn s function x 2 and Equation (), w gt a rprsntation formula for ψ ( xt, ) involving only boundary valu of function ψ ( xt, ) and a spatially uniform lctric displacmnt: h 2x ψ ( xt, ) = ψ ( 0, t) + ψ ( ht, ) + D( t) in Ω. 2 2 Formulas (2) and () lad to th following xprssion for ( xt, ) φ( xt, ) = ( 0, ) (, ) ( 0, ) (, ) 2 φ t + φ ht u t + u ht 2 h 2x + u( xt, ) + D( t) in Ω. 2 φ : (2) (3) 876

5 Aftr u( xt, ) is calculatd, ( xt, ) valu, D( t ) and boundary valus of ( xt, ) φ can b dtrmind using this calculatd φ. Th rprsntation formula (0) allows us to gt rprsntation formulas for th vlocity v( xt, ) = u ( xt, ) and strss σ ( xt, ). Diffrntiating (0) with rspct to tim provids th following rprsntation formula for th vlocity: v( xt, ) = ( 0, ) (, ( ) ) 2 v t xc + v ht h x c σ ( 0, t xc) Dt ( xc) σ 2 + h + b( ξ, t ξ x c) dξ 2 0 in Ω. ( h, t ( h x) c) D( t ( h x) c) To gt a rprsntation formula for th strss w nd to us th first contitutiv quation from (2) (in th nw notations introducd aftr quations (3)) and xprssion (3) which givs th following xprssion for th strss: (4) σ D = C u ( xt, ) D( t). ( xt, ) = Cu ( xt, ) + u ( xt, ) D( t) (5) Aftr diffrntiating (0) with rspct to x and substituting th rsult into (5) w gt σ 2 + ( 0, ) (, ( ) ) 2 σ t xc + σ ht h x c + D( t x c) + D( t ( h x) c) 2D( t) 2 h + b( ξ, t ξ x c) sgn ( x ξ) dξ in Ω. 2 0 ( xt, ) = v( 0, t xc) + v( ht, ( h x) c) (6) A rprsntation formula for D( xt, ) is not ndd undr assumption that q = 0 sinc th lctric displacmnt is uniform in spac in this cas and dtrmind solly by th lctric boundary conditions. 3. Boundary Equations Th vlocity rprsntation formula (4) gnrats two boundary quations whn x tnds to th uppr and lowr boundaris of th pizolctric lmnt, that is, whn x tnds to h or 0: v( ht, ) = v( 0, t θ) + σ ( ht, ) σ ( 0, t θ) h + D( t) D( t θ) + b( ξ, t θ + ξ c) d ξ, 0 (7) 877

6 v( 0, t) = v( ht, θ) + σ ( ht, θ) σ ( 0, t) h + D( t θ) D( t) + b( ξ, t ξ c) dξ 0 (8) whr θ dnots th tim takn by th lastic wav to travl th thicknss of th pizolctric layr: h θ =. c Similarly, th strss rprsntation formula (6) gnrats th following boun- dary quations: ( ht, ) = ( 0, t ) + c v( ht, ) v( 0, t ) σ σ θ ρ θ h + D ( t θ) D ( t ) b ( ξ, t θ + ξ c ) d ξ, 0 ( 0, t) = ( ht, ) + c v( ht, ) v( 0, t) σ σ θ ρ θ h + D ( t θ) D ( t ) + b ( ξ, t ξ c ) d ξ. 0 (9) (20) It is asy to vrify that Equations (7) and (9), though prsntd in diffrnt forms, ar quivalnt. Th sam is tru for th pair of Equations (8) and (20). Thrfor, w shall us th quations in ths pairs intrchangably. W will not work with boundary quations that can b obtaind dirctly from th displacmnt rprsntation formula (0), sinc it is computationally mor ffctiv to dtrmin at first unknown boundary valus of th vlocity v( xt, ), and thn calculat unknown boundary valus of th displacmnt u( xt, ) by intgrating th boundary vlocity ovr tim (using also an initial condition for u( xt, ) ). W also nd to considr boundary valus of th xprssion (3) for th lctric potntial. It is important to mphasiz that two quations obtaind from (3) whn x tnds to h or to 0 ar quivalnt and, thrfor, thy ar prsntd blow as on quation: h φ( ht, ) φ( 0, t) = u ( ht, ) u( 0, t) D( t) (2) Th boundary quations prsntd abov will b usd in th nxt sction to crat an xact tim domain solution procdur in th cas whn nonlinar dampr boundary conditions ar sprcifid. 4. Nonlinar Dampr Boundary Conditions and Exact Solutions Suppos that th lowr nd fac of th pizolctric lmnt is fixd to a nonlinar dampr. Lt F b a damping forc acting on th lowr nd fac which is dfind by th following nonlinar rlationship [7]: ( ) ( ( )) F = k v 0, t sgn v 0, t. (22) whr k > 0 is th damping constant, > 0 is th damping xponnt, and 878

7 sgn (.) is th signum function dfind for all ral numbrs (including 0 whr its valu is also 0). If v( 0, t) 0, th dirction of F is opposit to v( 0, t ). Th xponnt has a valu for a linar dampr, but may vary in practic in th intrval ( 0, 2 ] [7] crating a st of possibl boundary conditions at x = 0. W assum that th forc F is uniformly distributd ovr th lowr nd fac. Thn (22) transforms into th following nonlinar (in gnral) boundary condition at th lowr nd fac: k σ ( 0, t) = v( 0, t) sgn ( v( 0, t) ) (23) A whr A is th lowr nd fac ara. Considr additional assumptions that will b usd to gt xact solutions for th dampr boundary conditions basd on th rsults of th prvious sction. W suppos that th valus of u, σ, b, φ, D ar dfind for < t <. In addition, lt us assum hncforth that u xt, = 0, φ xt, = 0 if 0 < x< ht, < 0 (24) ( ) ( ) which mans, basd on (2) and (3), that σ, D and b ar also zro insid th pizolctric body at ngativ tims. Th nxt additional assumption is that b( xt, ) = 0 (25) insid th pizolctric body at any tim in th sns of gnralizd functions. This also includs th assumption that th initial conditions for th lastic displacmnt u( xt, ) ar zro, as discussd in [5]. Ths assumptions will simplify using boundary Equations (7)-(20) for particular problms considrd blow. Rgarding th dsign of th pizolctric lmnt, w assum that it is a cylindr (or a rod) with two coatd lctrods at x = 0 and x = h. Th lctrods ar considrd to b of ngligibl thicknss (from th mchanical point of viw) and thir dformation is nglctd. Th output voltag, which is dfind as th lctric potntial diffrnc btwn th lowr and uppr lctrods φ = φ( 0, t) φ( ht, ), is of primary intrst blow. Th lctric boundary condition at x = 0 corrsponds to th groundd lctrod: φ 0, t = 0 if t 0 (26) ( ) At th uppr nd fac, th following mchanical boundary condition is usd: σ ht, = p t if t 0 (27) ( ) ( ) whr p( t ) is an applid normal strss load which is assumd to b known and ngativ. Th lctric boundary condition at th uppr nd fac x = h is formulatd as follows: D ht, = 0 if t 0. (28) So, basd on (9), D( t ) = 0. ( ) Using th abov assumptions th rprsntation formulas (4) and (6) for 879

8 th vlocity and strss tak th following simplifid forms: v( xt, ) = ( 0, ) (, ( ) ) 2 v t xc + v ht h x c + pt ( ( h x) c) σ ( 0, t xc) in Ω, 2 σ 2 + σ ( 0, t xc) + pt ( ( h x) c) in Ω 2 ( xt, ) = v( 0, t xc) + v( ht, ( h x) c) whr all th tim dpndnt functions ar qual to zro for ngativ tims. (29) (30) In th rprsntation formulas (29) and (30), thr ar thr unknown boundary functions v( 0, t), σ ( 0, t) and (, ) v ht first two of which ar rlatd by Equation (23). Two additional quations ndd for dtrmination of ths thr functions will b drivd blow basd on (7) and (8). Aftr th th vlocity v( xt, ) is dtrmind for any particular x ovr tim, th corrsponding displacmnt u( xt, ) can b obtaind (du to th zro initial conditions) as t ( ) ( ) u xt, = v xτ, d τ. (3) 0 Boundary valus of th displacmnt provid (according to (2) and (26)) th lctric potntial valu at x = h : φ = ( ht, ) u( ht, ) u( 0, t) 4.. An Exact Rcursiv Procdur Th solution of th abov problm will b obtaind by using an xact rcursiv procdur basd on th following quations obtaind from (7) and (8) undr th boundary conditions (23) (26) (27) (28): v( ht, ) = 2v( 0, t θ) v( ht, 2θ) + p( t) p( t 2 θ), (33) (32) k v( 0, t) + v( 0, t) sgn ( v( 0, t) ) v( h, t θ) p( t θ). A = + (34) Thr ar two unknowns v( ht, ) and v( 0, ) t at ach tim momnt t in ths quations. Th right-hand sids of th quations ar known at ach tim point sinc thy contain ithr p( t ) or tim-dalayd function valus at t θ that had to b dtrmind at a prvious stp of th rcursiv procss. In ordr to simplify driving nxt rsults, w nd to introduc som additional notations: k γ =, ξ v( 0, t), r v( ht, θ) p( t θ). A = = + (35) Thn, Equation (34) rads as ( ) ξ+ γξ sgn ξ = r. (36) 880

9 Lt th lft-hand sid of Equation (36) b dnotd by f ( ξ ). Sinc > 0, f ( ξ ) is a continuous strictly monotonically incrasing function on (, ) ranging from to. Thrfor, for any ral r, thr xists on and only on solution of Equation (36) in (, ). Dnot by Q th oprator that tranforms r into this solution of quation (36). Thus, Q is th lft invrs oprator of th nonlinar oprator acting on ξ in th lft-hand sid of Equation (36). If = 2,, 2 or 3, th corrsponding xprssions of Q r ar vry simpl for computations: Qr 2 = ( rγ ) sgn ( r), 2γ 2 r 2 Qr =, Q2r= ( γ + γ + 4 r) sgn ( r), (37) + γ γ Q3r = γ ( 08r+ 2 2γ + 8 r ) r ( 08r+ 2 2γ + 8r ) Th calculation of Q r for othr valus of can ffctivly b implmntd using a symbolic computation softwar lik Mapl [8]. With hlp of th invrs oprator Q Equation (34) can b rwrittn in th following xplicit form for calculating v( 0, t ) : v( 0, t) = Q v( ht, θ) + p( t θ). Equation (38) combind with (33) crats th rcursiv procdur that can b usd dirctly for calculations or can lad to building xplicit xact solution for vctor { v( 0, t), v( ht, )} stp by stp ovr conscutiv tim intrvals jθ t < ( j+ ) θ ( j = 0,, 2, ). In doing so, it is hlpful to substitut v( 0, t θ ) in (33) by its xprssion obtaind from (38) which provids th v ht : following rcursiv quation for (, ) v( ht, ) = 2 Q v( ht, 2θ) + p( t 2θ) v( h, t 2θ) + p( t 2 θ) p( t), + or, using th idntity oprator I (that lavs unchangd th lmnt on which it oprats), v( ht, ) = ( 2 Q I) v( ht, 2θ) + p( t 2 θ) p( t) Explicit Exact Solutions Now w driv som xplicit xact solutions for v( ht, ) and v( 0, ) (38) (39) t corrs- ponding to thr practically important rangs of th duration t of th strss load at x = h. Our goal is to prsnt th boundary vlocitis dirctly through 88

10 th transint strss load at x pizolctric body Cas : t < 2θ So, p( t ) = 0 if t [ 0,2θ ) condition for conscutiv intrvals θ ( ) θ) obtain th following xplicit xprssion for v( ht, ) : = h that gnrats th dynamic procss in th. Using th rcursiv Equation (39) undr this 2 k,2 k+, k = 0,, 2,, w finally p( t) if 0 t < 2 θ, k 2 v( ht, ) = ( 2Q I) p( t 2 kθ ) if 2kθ t < 2 ( k+ ) θ, k =, 2,. (40) Substituting (40) into (38) w gt th corrsponding xplicit xprssion for v( 0, t ) : Cas 2: t < 4θ 0 if 0 t < θ, k 2 v( 0, t) = Q ( 2Q I) p( t 2kθ ) if ( 2k ) θ t < ( 2k+ ) θ, k =, 2,. In this cas, p( t ) = 0 if t [ 0,4θ ) following xplicit xact solutions: (4). Acting similarly to sction 4 w driv th p( t) if 0 t < 2 θ, k v( ht, ) = ( 2Q I) p( t 2( k ) θ ) (42) 2 + ( 2Q I) p( t 2 kθ) if 2kθ t < 2 ( k + ) θ, k =, 2, ; 0 if 0 t < θ, 2 Q p( t θ) if θ t 3 θ, < k v( 0, t) = Q ( 2Q I) p( t 2kθ + θ) 2 + ( 2Q I) p( t 2kθ θ) if ( 2k ) θ t < ( 2k+ ) θ, k = 2, 3,. (43) Cas 3: t <6θ So, p( t ) = 0 if t [ 0,6θ ) following closd form:, and th corrsponding xact solutions hav th 882

11 p( t) if 0 t < 2 θ, 2 p( t) + ( 2Q I) p( t 2 θ) if 2θ t 4 θ, < 2 v( h, t) = 2Q I p t 2 k + 2Q I p t 2k 2 + ( 2Q I) p( t 2( k + ) θ ) if 2 ( k+ ) θ t < 2 ( k+ 2 ) θ, k =, 2, ; k ( ) ( ( ) θ) ( ) ( θ) (44) 0 if 0 t < θ, 2 Q p( t θ) if θ t 3 θ, < 2 Q p( t θ) + ( 2Q I) p( t 3 θ) if 3θ t < 5 θ, k v( 0, t) = Q ( 2Q I) p( t 2kθ + θ) 2 + ( 2Q I) p( t 2kθ θ) 2 + ( 2Q I) p( t 2kθ 3θ) if ( 2k + 3) θ t < ( 2k + 5 ) θ, k =,2,. Similar xplicit formulas for t 6θ ar xcssivly cumbrsom. In this cas, it is asir to dirctly us th rcursiv procdur basd on (33) and (38) which has th sam simpl form rgardlss of th transint load duration and also provids xact rsults. 5. Exampls and Discussions Considr som xampls of using th rsults of th prvious sction for mathmatical modling of pizolctric cylindrical dvics installd in a car as proposd in [5]. Ths dvics transform th mchanical nrgy of th moving pistons or crank-shafts into lctrical nrgy, which will b stord in th capacitor or th battry chargr. W considr th uniaxial strss stat for a cylindr and assum that th matrial of th cylindr is PZT-5A [4]. In this cas, paramtrs C,, and ρ in Equations (3) hav th following valus: ( ) C = = *0 N m ; 9.89 N V m ; = 76.2*0 farad m; ρ = kg m 0 3 Thn th lastic wav spd c in th pizolctric matrial is qual to m s. Nxt, w tak into account that th total forc instantanously applid to th top of a piston in an intrnal combustion ngin is around 6300 pounds, which corrsponds to approximatly 28,640 N [9]. Suppos that this (45) 883

12 forc F a is applid downward to a pizolctric cylindr with a lngth of h = cm and a diamtr d = cm. So, th ara of th uppr nd fac of th 2 2 cylindr A= πd 4 = cm. Assuming that F a is uniformly distributd ovr th uppr nd fac, w gt th amplitud of th prssur impuls acting on th top of th cylindr: p a = MPA. Lt us assum that th applid normal strss load taks th form of th following rctangular comprssiv load (prssur) impuls: ( ) = ( ) ( ) p t p H t H t t a (46) whr t is th duration of th prssur impuls. 2 2 W assum first that = 0.5 and k = 000 N s m (s,.g., [20]) in th dampr boundary conditions (23). Considr th following thr valus of t : 5, 0, 5 μs. Sinc th transit tim of lastic wavs btwn th uppr and lowr nd facs θ = 2.7 μs, thn ths thr durations corrspond to th thr cass of xplicit xact solutions considrd in 4. Th oprator Q is calculatd according to (37). Th calculatd rsults for th output voltag V = φ (in kv) basd on ths xact solutions ar prsntd for t 50 μs in Figurs -3 blow. Comparing th graphs w can s that th maximum or pak valu dos not dpnd on th prssur impuls duration t. Howvr, th numbr of paks in ach figur dpnds on th t. Th tim distanc btwn two nighboring paks is approximatly qual to 2θ. Aftr th prssur load is rmovd, thr is an attnuation of th output voltag vibrations. Now lt us considr anothr st of th dampr paramtrs: = 2.0 and 2 2 k = 250 N s m. Th paramtrs of th matrial and th impuls durations ar th sam as abov. Th oprator Q is also calculatd according to (37). Figur. Output voltag for 2 2 = 0.5, k = 000 N s m and 5 μs t =. 884

13 Figur 2. Output voltag for 2 2 = 0.5, k = 000 N s m and 0 μs t =. Figur 3. Output voltag for 2 2 = 0.5, k = 000 N s m and 5 μs t =. Th calculatd rsults for th output voltag V = φ (in kv) ar prsntd for t 50 μs in Figurs 4-6. Comparison of ths graphs shows that th maximum valu of th output voltag dos not dpnd on th prssur impuls duration t which similar to th cas whn = 0.5. Th diffrnc is that now thr is only on pak but its width dpnds on th t. Aftr th prssur load is rmovd, th attnuation of th output voltag vibrations is vry pronouncd: th amplitud of vibrations aftr th load rmoval is almost ngligibl. 885

14 Figur 4. Output voltag for 2 2 = 2, k = 250 N s m and 5 μs t =. Figur 5. Output voltag for 2 2 = 2, k = 250 Ns m and 0 μs t =. 6. Conclusion On-dimnsional transint dynamic pizolctric problms for thicknss polarizd layrs and disks, or lngth polarizd rods, ar considrd hr in th framwork of a tim-domain Grn s function mthod. As th rsult, a novl xact analytical rcursiv procdur is drivd which is applicabl for a wid varity of boundary conditions including th nonlinar dampr cas. Som nw practically important xplicit xact solutions ar prsntd. Th ffctivnss of th proposd xact approach is dmonstratd by xampls of th tim bhavior of th output lctric potntial diffrnc btwn two lctrods coatd at th 886

15 Figur 6. Output voltag for 2 2 = 2, k = 250 N s m and 5 μs t =. nd facs of a pizolctric cylindr fixd to a nonlinar dampr at on nd, and subjctd to impulsiv loading at th othr. Acknowldgmnts Th authors would lik to thank th anonymous rviwrs for thir commnts. Rfrncs [] Mason, W.P. (98) Pizolctricity, Its History and Applications. Th Journal of th Acoustical Socity of Amrica, 70, [2] Yang, J. (2006) Analysis of Pizolctric Dvics. World Scintific, Singapor. [3] Fang, D., Wang, J. and Chn, W., Eds. (203) Analysis of Pizolctric Structurs and Dvics. D Gruytr, Brlin, Boston. [4] Vijaya, M.S. (203) Pizolctric Matrials and Dvics: Applications in Enginring and Mdical Scincs. CRC Prss, Boca Raton. [5] Gnis, V. and Soukhomlinoff, A. (200) Unitd Stats Patnt: Pizolctric Powrd Vhicls and Motors, March 200. [6] Cook, E.G. (956) Transint and Stady-Stat Rspons of Ultrasonic Pizolctric Transducrs. 956 IRE National Convntion, Nw York, 9-22 March 956, [7] Filipczynski, L. (960) Transints and th Equivalnt Elctrical Circuit of th Pizolctric Transducr. Acta Acustica unitd with Acustica, 0, [8] Rdwood, M. (96) Transint Prformanc of a Pizolctric Transducr. Th Journal of th Acoustical Socity of Amrica, 33, [9] Stutzr, O.M. (967) Multipl Rflctions in a Fr Pizolctric Plat. Th Journal of th Acoustical Socity of Amrica, 42, [0] Zhang, H.L., Li, M.X. and Ying, C.F. (983) Complt Solutions of th Transint Bhavior of a Transmitting Thicknss-Mod Pizolctric Transducr and Thir 887

16 Physical Intrprtations. Th Journal of th Acoustical Socity of Amrica, 74, [] L, K.C. (999) Vibrations of Shlls and Rods. Springr, Brlin, Hidlbrg. [2] Ma, C.C., Chn, X.H. and Ing, Y.S. (2007) Thortical Transint Analysis and Wav Propagation of Pizolctric Bi-Matrials. Intrnational Journal of Solids and Structurs, 44, [3] Ing, Y.S., Liao, H.F. and Huang, K.S. (203) Th Extndd Durbin Mthod and Its Application for Pizolctric Wav Propagation Problms. Intrnational Journal of Solids and Structurs, 50, [4] Gazonas, G.A., Wildman, R.A., Hopkins, D.A. and Schidlr, M.J. (206) Longitudinal Impact of Pizolctric Mdia. Archiv of Applid Mchanics, 86, [5] Khutoryansky, N.M. and Sosa, H. (995) Dynamic Rprsntation Formulas and Fundamntal Solutions for Pizolctricity. Intrnational Journal of Solids and Structurs, 32, [6] Yuan, F.G. (206) Structural Halth Monitoring (SHM) in Arospac Structurs. Elsvir Scinc, Amstrdam. [7] Dixon, J. (2008) Th Shock Absorbr Handbook. John Wily & Sons, Hobokn. [8] Monagan, M.B., Gdds, K.O., Hal, K.M., Labahn, G., Vorkottr, S.M., McCarron, J. and DMarco, P. (2005) Mapl 0 Programming Guid. Maplsoft, Watrloo, ON, Canada. [9] Pulkrabk, W.W. (997) Enginring Fundamntals of th Intrnal Combustion Engin. Prntic Hall, Uppr Saddl Rivr. [20] Ho, C., Lang, Z.Q., Sapiński, B. and Billings, S.A. (203) Vibration Isolation Using Nonlinar Damping Implmntd by a Fdback-Controlld MR Dampr. Smart Matrials and Structurs, Submit or rcommnd nxt manuscript to SCIRP and w will provid bst srvic for you: Accpting pr-submission inquiris through , Facbook, LinkdIn, Twittr, tc. A wid slction of journals (inclusiv of 9 subjcts, mor than 200 journals) Providing 24-hour high-quality srvic Usr-frindly onlin submission systm Fair and swift pr-rviw systm Efficint typstting and proofrading procdur Display of th rsult of downloads and visits, as wll as th numbr of citd articls Maximum dissmination of your rsarch work Submit your manuscript at: Or contact jamp@scirp.org 888