NOVEL PROCEDURE TO COMPUTE A CONTACT ZONE MAGNITUDE OF VIBRATIONS OF TWO-LAYERED UNCOUPLED PLATES

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1 NOVEL PROCEDURE TO COMPUTE A CONTACT ZONE MAGNITUDE OF VIBRATION OF TO-LAYERED UNCOUPLED PLATE J. AREJCEICZ, V. A. KRYKO, AND O. OVIANNIKOVA Received February A novel ieraion procedure for dynamical problems, where in each ime sep, a conacing plaes zone is improved, is proposed. Therefore, a zone and magniude of a conac load are also improved. Invesigaions of boundary condiions influence on exernally driven vibraions of uncoupled wo-layer plaes, where for each of he layers, he Kirchhoff hypohesis holds, are carried ou.. Inroducion Uncoupled mulilayer plaes creae a complex dynamical srucure, where depending on inpu parameers and iniial and boundary condiions, various vibraion ypes appear. In spie of undoubable achievemens in saic and dynamical problems of nonlinear heory of plaes, he problems of conacs exhibied by mulilayer plaes subjec o boh longiudinal and ransversal ime-changeable loads are less invesigaed.. Mahemaical models In his work, a model of wo-layer consrucion composed of hin elasic recangular plaes is sudied [7]. The mean surface of an upper plae lies in plane z =, whereas he mean surface of he lower one lies in plane z = (/)(δ + δ )+h,whereh is he disance beween plaes, δ, δ are he hicknesses of upper and lower plaes, correspondingly. The plaes are conacing wih each oher hrough exernal surfaces, projeced ino a corresponding mean surface, wihin a general inkler s hypohesis []. Noe ha each of he plaes is embedded in D space in R wih aached coordinaes. Namely, senses of he axes Ox and Oy go ino direcion of mean surface of upper plae, whereas axis Oz goes ino Earh cenre. In he given coordinaes, he D plaes spaces Ω read { Ω = x, y,z/(x, y) [,a] [,b], δ z δ }, { Ω = x, y,z/(x, y) [,a] [,b], ( ) δ δ + h z } (.) ( ) δ + δ + h, where he space [,a] [,b] defines recangular plae shape; D, D are cylindrical plaes Copyrigh Hindawi Publishing Corporaion Mahemaical Problems in Engineering : () DOI:./MPE..

2 6 Novel procedure o compue a conac zone magniude siffness. The governing equaions are D w (x, y,)+k E h w (x, y,)ψ(x, y,) = q + k E ( ) w + h Ψ(x, y,), h D w (x, y,)+k E h w (x, y,)ψ(x, y,) = q + k E (.) ( ) w + h Ψ(x, y,), h where he funcion Ψ = [ + sign ( w w h )], (.) and Ψ = Ψ(x, y) plays a role of shells conac space Ω indicaor. Noice ha if he iniial plaes locaion (clearance funcion) and he loads do no lead o a conac beween plaes during heir deformaions, hen Ψ and each of he plaes vibraes independenly. Oherwise, he governing equaions are coupled. The sysem (.) is associaed wih one of he following boundary condiions [6] on he boundary Ω, w i = Ωi w i n =, (.) i Ωi w i = Ωi w i n =, (i =,). (.) i Ωi Owing o D Alember principle, qi on he ih plae of he form include boh inerial and damping forces acing q i (x, y,) = q i (x, y,) γ i g w i ε w i i. (.6) The sysem (.) is of high order wih respec o ime and spaial coordinaes (x, y). The compuaional process is as follows. In each ime sep, he following values are accouned from a previous sep w i k = w i (x, y, k ); w i k = w i (x, y, k ), and in order o improve a conac zone, he following ieraion procedure for ime k is applied: D w (m+) + ke Ψw (m+) = q n + ke Ψ ( w (m) ) + h, h h D w (m+) + ke Ψw (m+) = q n + ke Ψ ( w (m) ) (.7) + h. h h Firs PDEs (.) are reduced o he Cauchy problem hrough he second-order mehod of finie difference, hen he problem is solved using fourh-order Runge-Kua mehod. In each ime sep, he Gauss ieraion procedure (.7) is carried ou, and he sysem order is reduced wice, which is imporan owing o compuaion ime (see also []). Finishing he ieraion procedure (.7), he obained values of w i and w i serve as an iniial condiion for a nex sep of he Runge-Kua mehod. The similar like mehods have been also applied in [,, ]. The menioned procedure allows for a conac zone improvemen. Owing o he Runge principle, i has been found ha he opimal sep wih respec o spaial coordinaes is defined hrough a pariion of space Ω j ino pars, whereas

3 J. Awrejcewicz e al. 7 ime sep is equal o =. In wha follows, vibraions of wo-layer plaes wih various boundary condiions along heir conours are sudied. The following four varians of he boundary condiions are accouned: (i) wo plaes are clamped along heir conours (boundary condiions (.)); (ii) wo plaes are ball-ype suppored (boundary condiions (.)); (iii) upper plae is clamped along is conour (boundary condiions (.)), whereas lower plae is ball-ype suppored along is conour (boundary condiions (.)); (iv) upper plae is ball-ype suppored along is conour (boundary condiions (.)), whereas lower one is clamped (boundary condiions (.)).. Two plaes are clamped along heir conours (boundary condiions (.)) Assume ha wo plaes have same hickness (δ = δ = δ), and he frequency of exciaion = P = 9.97, where is he frequency of a linear vibraion of one-layer plae, ε = ε =.6. The clearance beween plaes is h = h /δ =.7 (h is he nondimensional parameer). Boh plaes are subjec o sinusoidal load of he form q i = Q i sinp. Boundary condiions (.) are applied, and he iniial condiions read w = = w = =. (.) Recallhaineachimesep, heieraionprocedure (.7)isapplied.InTable.,pars of ime hisories w i (,;,;), 6, phase porrais w i (w i ), and power specrum s() are presened only for lower plae, since he vibraions of lower plae are mirror reflecions of he upper plae. Furhermore, he resuls are only given for he plae cenre, since vibraions of oher poins are synchronized wih hose of he cenre. During plaes conac, here exis ime insans where vibraions process is an unsaionary one, hen afer a long ime, i achieves a saionary sae. For q = q <., he plaes do no conac wih each oher, and harmonic vibraions occur. This observaion is confirmed hrough he following characerisics. In power specrum, only one frequency is visible, and an ellipse occurs in he phase porrai. However, for q = q = (a conac beween plaes occurred), a picure is changed: vibraions are no longer harmonic, and he phase porrai is of more complexiy. Increasing he ransversal load, a Hopf bifurcaion occurs wih a period ripling (q = q = ) (see [8]). Furher, an inerlace of vibraions on he exciaion frequency and on he frequencies of period ripling and number eigh is observed.. Boundary condiions (.) Consider wo-layer plae ype consrucion wih he same parameers as in he previously analyzed case, bu wih boundary condiions (.). imilarly as in he previous case, for q = q =., harmonic vibraions occur. However, now bifurcaions appear already for q = q =.. Increasing q i, a picure of plaes bending becomes more complicaed. For q = q =, period ripling occurs, hen a ransiion o chaos akes place (q = q = ). This is clearly expressed in boh phase porrai and power specrum. For q = q =, again Hopf bifurcaion appears, hen is collapse akes place (q = q = 6). The scenario is composed of inerlace of Hopf bifurcaions and a ransiion ino harmonic vibraions.

4 8 Novel procedure o compue a conac zone magniude Table. Q ignal Phase porrai Power specrum w(.,.) (a) (b).. (c) w(.,.) (d) 8 (e) 7 9 /.. (f) w(.,.) (g).... (h) /6 /.. / / /6 (i) w(.,.) (j).... (k) (l) w(.,.) (m).... (n) 9 / 7 / 9.. (o)

5 J. Awrejcewicz e al. 9 Table.. Coninued. Q ignal Phase porrai Power specrum 8 w(.,.) (p).... (q) (r) 8 w(.,.) (s).6... () /7 6/7 /7 /7 /7 /7.. (u) 6 w(.,.) (v) 8 (w).. (x) (y) w(.,.) = A for q = q =, 9 conac poins (z) w(.,.) = B for q = q = 8, 6 conac poins. Vibraions associaed wih differen boundary condiions In Table., same characerisics as in Table. are repored, bu now he upper layer is ball-ype suppored along he conour (boundary condiions (.)), whereas he lower layer is clamped (boundary condiions (.)). For q = q =., vibraions of wo plaes are harmonic, and each of he plaes vibraes wih is own frequency. For q = q =., when a conac beween plaes occurs,

6 Novel procedure o compue a conac zone magniude Table. Q ignal Phase porrai Power specrum. w(.,.) (a) (b) (c). w(.,.) (d) (e) / (f) w(.,.) (g) 6 (h) / / (i) 6 w(.,.) (j) 6 (k) 7 / / / (l) 6 w(.,.) (m) 6 (n) / (o)

7 J. Awrejcewicz e al. Table.. Coninued. Q ignal Phase porrai Power specrum 6 6 w(.,.) (p) 6 (q) (r) w(.,.) (s) () 9 / (u) w(.,.) (v).. (w) 9 / (x) (y) w(.,.) = C for q = q =, conac poins (z) w(.,.) = D for q = q = 6, 6 conac poins

8 Novel procedure o compue a conac zone magniude Table. Q ignal Phase porrai Power specrum Q =. =.7 w(.,.) (a) (b).. (c) = w(.,.) (d) (e).. (f) Q =. =.7 8 w(.,.) (g) 8 (h) / / /.. (i) = 9.97 w(.,.) (j) (k) /7 /7 /7 /7 /7 6/7.. (l) in upper-plae power specrum, wo Hopf bifurcaions are remarkable, whereas period seven is associaed wih lower plae (see he power specrum). For q = q =, in he power specrum of lower plae, also wo Hopf bifurcaions appear, and wo plaes begin o vibrae wih one frequency. In Table., he same characerisics as in Table. are repored, bu for differen boundary condiions. Namely, upper plae is clamped hrough is conour (boundary

9 Table.. Coninued. J. Awrejcewicz e al. Q ignal Phase porrai Power specrum Q = = (m) w(.,.) 6 8 (n) / / 9 /.. (o) = w(.,.) (p) (q) / / /.. (r) Q = =.7 w(.,.) (s) () 9 / 7 / /.. (u) 6 = 9.97 w(.,.) (v) 8 (w) 9 / 7 / /.. (x) (y) w(.,.) = E for q = q =, conac poins (z) w(.,.) = F for q = q =, 6 conac poins

10 Novel procedure o compue a conac zone magniude Table. Q ignal Phase porrai Power specrum Q =. = w(.,.) (a) (b).. (c) = (d) w(.,.) (e).. (f) Q = 6 = (g) w(.,.) (h) 9 / / 7 / 9.. (i) =.7 w(.,.) (j).. (k) 9 / 7 / /.. (l) (m) w(.,.) = K for q = q = 6, 6 conac poins (n) w(.,.) = M for q = q = 6, conac poins

11 J. Awrejcewicz e al. condiions (.)), whereas he lower plae is suppored hrough he boundary condiions (.). For q = q =., boh plaes vibraions are harmonic. For q = q = 6, vibraions of boh plaes are synchronized wih one frequency. Then wo Hopf bifurcaions follow. 6. Conclusions The carried ou analysis exhibis complex vibraions of wo-layer sysem of plaes: series of Hopf bifurcaions occurs, where periods hree, five, and seven Hopf bifurcaions are exhibied. The deeced bifurcaions in our complex sysem have been heoreically prediced by harkovskiy while analyzing he logisic curves [8]. I should be emphasized ha conac load value depends essenially on he number of conacing poins. ome novel dynamical phenomena have been deeced. For example, if he upper plae is ball-ype suppored, and he lower one is clamped along is conour, synchronizaion akes place. Namely, boh plaes sar o vibrae wih he same fundamenal frequency =.6 (a ball frequency) earlier han in he case of clamping and ball-ype suppors. Afer he occurring synchronizaion, furher increase of loading has no changed dynamics qualiaively. References [] J.AwrejcewiczandA.V.Krysko,Analysis of complex parameric vibraions of plaes and shells using Bubnov-Galerkin approach, Archive of Appl. Mech. 7 (), 9. [] J. Awrejcewicz and V. A. Krysko, Nonclassical Thermoelasic Problems in Nonlinear Dynamics of hells, cienific Compuaion, pringer-verlag, Berlin,. [] J. Awrejcewicz, V. A. Krysko, and A. V. Krysko, On he economical soluion mehod for a sysem of linear algebraic equaions, Mah. Probl. Eng. (), no., 77. [] J. Awrejcewicz, V. A. Krysko, and G. G. Narkaiis, Bifurcaions of a hin plae-srip excied ransversally and axially, Nonlinear Dynam. (), no., [] B. Ya. Kanor, Conac Problems of Nonlinear hells Theory, Naukova Dumka, Kiev, 99. [6] M.. Kornishin, Nonlinear Problems of Plaes and hallow hells and Mehods of Their oluions, Nauka, Moscow, 96. [7] A. V. Krysko, V. A. Krysko, V. A. Ovsiannikova, and T. B. Babenkova, Complex vibraions of wolayer uncoupled plaes subjec o longiudinal sinusoidal load, Izv. Vuz., rioel svo (), no. 6, (Russian). [8] A. N. harkovskiy, Exisence of cycles exhibied by coninuous mapping of a sraigh line ino iself, Ukrainian Mah. J. 6 (96), no., 6 7 (Russian). J. Awrejcewicz: Deparmen of Auomaics and Biomechanics, Faculy of Mechanical Engineering, Technical Universiy of Lodz, / efanowskiego ree, 9-9 Lodz, Poland address: awrejcew@p.lodz.pl V. A. Krysko: Deparmen of Mahemaics, araov ae Universiy, B. adovaya 96a, araov, Russia address: ak@san.ru O. Ovsiannikova: Deparmen of Mahemaics, araov ae Universiy, B. adovaya 96a, araov, Russia address: ak@san.ru