Vibration Control of an Electromechanical Model with Time-Dependent Magnetic Field

Transcription

1 Journal of Mchanical Dsign and Vibration, 06, Vol. 4, No., -9 Availabl onlin at Scinc and Education Publishing DOI:0.69/jmdv-4-- Vibration Control of an Elctromchanical Modl with Tim-Dpndnt Magntic Fild Usama H. Hgazy *, Jihad Y. Abu Ful Dpartmnt of Mathmatics, Faculty of Scinc, Al-Azhar Univrsity, Gaza, Palstin *Corrsponding author: Abstract This papr prsnts a study of th nonlinar rspons of th lctromchanical (sismograph systm undr paramtric xcitations in th mchanical and lctrical parts with priodically tim-varying magntic fild. Th cas of subharmonic (paramtric rsonanc is considrd and xamind. Approximatd solutions ar sought applying th mthod of multipl scals. Numrical simulations ar carrid out to illustrat th stady-stat rspons and th stability of th solutions using th frquncy rspons function and tim sris solution. Kywords: simographs, vibration control, subharmonic rsonanc Cit This Articl: Usama H. Hgazy, and Jihad Y. Abu Ful, Vibration Control of an Elctromchanical Modl with Tim-Dpndnt Magntic Fild. Journal of Mchanical Dsign and Vibration, vol. 4, no. (06: -9. doi: 0.69/jmdv Introduction Considrabl ffort has bn dvotd to th study of nonlinar oscillations and chaotic motion of som coupld oscillators in svral scintific filds [,]. Th nonlinar oscillations and chaotic motions in a controlld lctromchanical sismograph systm with tim-varying stiffnss is invstigatd []. Th stiffnss in th sismograph is considrd as th tim varying in a priodic form. Th mthod of multipl tim scal prturbation tchniqu is usd to solv th nonlinar diffrntial quations dscribing th controlld systm up to scond ordr of accuracy, whr diffrnt activ controllrs ar applid. Th dynamics and synchronization of two coupld lctromchanical dvics and ffcts of highr nonlinarity wr invstigatd [4], whr harmonic balanc is utilizd to driv th amplitud quations in th gnral cas. Th avraging mthod is usd to study th oscillatory stat of a slf-sustaind lctromchanical systm [5]. Th ffcts of th nonlinar coupling, dtuning paramtr and chaotic bhavior wr invstigatd. Chaos control, bifurcation diagrams of th lctromchanical sismograph systm wr studid [6]. Th amplituds of th fifth subharmonic and suprharmonic vibrations stats of th systm wr found using th multipl scals mthod. Th appropriat coupling paramtr of an lctromchanical damping dvic with magntic coupling was found analytically and numrically [7]. Th problm of supprssing th vibrations of a hingd-hingd flxibl bam whn subjctd to xtrnal harmonic and paramtric xcitations is considrd and studid. Th multipl scal prturbation mthod is applid to obtain a first-ordr approximat solution. Th quilibrium curvs for various controllr paramtrs ar plottd. Th stability of th stady stat solution is invstigatd using frquncy-rspons quations. Th approximat solution was numrically vrifid. It is found that all prdictions from analytical solutions wr in good agrmnt with th numrical simulations [8]. Th bifurcation structur of th modl was analyzd and th ffcts of th coupling paramtr on th bifurcation structur was studid. An xtndd Duffing-Van dr Pol oscillator was considrd and th mthod of multipl scals is utilizd to obtain th approximat solution both in th cas of non-rsonant and rsonant stats [9]. Morovr, th Mlinkov thorm is usd to invstigat th chaotic bhavior of th systm. A particular cas of th micro-lctro-mchanical systm (MEMS modl is introducd by including a tim-varying stiffnss, and th motion of th modl is studid using th phas portrait, tim sris, and Poincar map [0]. It is shown from th phas portrait, and Poincar map that thr xist diffrnt motions in th MEMS rsonator systm undr crtain conditions. Mlnikov s mthod has bn mployd to dfin th rgions of paramtr spac whr homoclinic and htroclinic chaos can occur. Numrical simulations ar prformd to study th systm s bhavior for various paramtr sts and to vrify th rsult from Mlnikov s mthod. in th slf-sustaind lctromchanical systm with multipl functions is considrd, and h avraging and th harmonic balanc mthods wr usd to analyz th amplituds of th oscillatory stats in th autonomous and nonautonomous cass, rspctivly []. Diffrnt bifurcation structurs, stability chart and th variation of th Lyapunov xponnt wr obtaind. Th dynamics and chaos control of th slf-sustaind lctromchanical dvic with and without discontinuity wr studid []. Th ffcts of th amplitud of th paramtric modulation and som cofficints along with th transition to chaotic bhavior wr studid. In this papr, th stady-stat rspons of th sismograph modl with tim-dpndnt magntic fild is invstigatd for various systm paramtrs undr subharmonic rsonanc condition. Th stability of th numrical solution is studid using th frquncy rspons

2 Journal of Mchanical Dsign and Vibration function mthod. Thr activ control laws hav bn applid and thir prformanc is invstigatd. Numrical intgration using Rung-Kutta fourth ordr is prformd to vrify analytical rsults obtaind by th mthod of multipl scals prturbation tchniqu.. Approximat Prturbation Solution Th considrd sismograph systm will b subjctd to paramtric xcitations with priodically tim-varying magntic fild. Th modifid govrning quations of motion ar givn by [], x+ ω x+ εµ x + εγx εα( + f cos ωtq = ε xf cos Ω t + Cr, 5 q + ωq εµ q + εµ q + εγ q + εγ q + εα( + f cos ωt x = ε qe cos ωt + Cr, ( ( whr, th cofficints ar th sam as thos dfind in []. Th mthod of multipl scals is usd to obtain approximat solutions of th nonlinar quations ( and (. Assuming x and q in th form x( t, ε = x ( T, T + εx ( T, T +..., ( q( t, ε = q ( T, T + εq ( T, T +..., ( whr T 0 = t is th fast tim scal associatd with changs occurring at th frquncis ω, ω, and Ω, and T = εt is th slow tim scal associatd with modulations in th amplituds and phass causd by th nonlinarity, damping, and rsonancs. In trms of T 0 and T, th tim drivativs bcom: d 0 ε 0 ε 0 d = D + D+..., = D + DD+..., dt dt whr Dj =, j = 0,. T j Thn substituting quations ( and (4 and th tim drivativs into quations ( and ( and comparing th cofficints of th sam powrs of ε, w obtain 0 O( ε : O( ε : ( D + ω 0 x ( D ω ( D q 0 + x0 = 0, ( = 0, (6 = DDx 0 0µ D0x0 γx0+ α + f cos ωt Dq x0fcos ΩtGD 0x0, ( D0 + q = DDq µ Dq 0 0 µ ( Dq 0 0 ( cos 5 γq0 γq0 α + f ωt Dx qe 0 cos ωtgdq 0 0. (7 (8 Th solutions of (5 and (6 can b xprssd as: iω0 T iω0 T 0 0 = it0 it0 0 0, = 0 + 0, x T, T A A, q T T B B whr A 0, B 0 ar complx functions in T. Substituting for x 0 and q 0 into quations (7 and (8, thn th gnral solutions of quations ( and ( can b xprssd in th following simplifid form iα B it x T T A iω0 T 0 0 ( 0, = + ( ω+ ( ω iα fb + ( ω+ ω+ ( ωω iω0 T + γa 0 8ω iα fb + ( ω ω+ ( ω+ ω FA0 it 0( ω+ω Ω ( Ω+ ω FA0 it 0( ωω + cc, Ω Ω ( ω 0 it0 0 it0 ( + ω ( ω it 4 (, = ( µ γ 5γ q T0 T B i B0 B0 B0B iT iαω A 0 0 iω0 T + γb 0 4 ( + ω( ω iαω fa 0 it0( ω+ ω ( + ω+ ω( ωω iαω fa 0 it0( ω ω ( + ωω( ω+ ω EB0 it0( + ω EB0 it0( ω + cc, ω ω+ ω ω 0 it0. Subharmonic Rsonanc Analysis (0 (9 In this sction, th principal paramtric rsonanc (subharmonic which occurs whn th frquncy of th xcitation is clos to twic that of th natural frquncis of th systm, that is ( Ω ω and ( ω ω, in th prsnc of on-to-on intrnal rsonanc ( ω ω =. Th prvious rsonant rlations can b xprssd as follows Ω= ω + εσ, ω = + εσ, ω = + εσ. ( Eliminating th scular trms from quations (9 and (0, th solvability condition yilds ( ω 0 ( µ + ω 0 γ 0 0 iω0 T i A i G A A A it0 it0( Ωω + iα B0 + FA0 = 0, ib + i µ G B iµ B B γbb 0 0 0γBB 0 0 iω0 T it0 ( ω iαω A0 + EB0 = it0 ( (

3 Journal of Mchanical Dsign and Vibration Substituting quation ( into quations ( and ( i thn dividing by 0 T ω it and 0, rspctivly, w gt iω A i µ + G ω A γ A A i T i T FA0 σ + + iαb0 σ = 0, ( µ ( µ γ ib + i G B i + B B iσ T iσ T 0γB0B0 iαω A0 + EB0 = 0. (4 (5 i Substituting th polar forms A 0 a θ i =, B 0 = b θ, whr ab, and θ, θ ar th stady-stat amplituds and th phass of motions, rspctivly, and simplifying w obtain iθ iθ iθ iω a + ωaθ i µ + G ωa (6 iθ i( σ T θ i i( σ T + θ γ a + Fa + αb = 0, 8 4 iθ iθ iθ ib + bθ + i µ G b iθ 5 5 iθ iµ + γ b γb 8 6 i( σ T + θ i( σ T θ iαω a + Eb = 0. 4 i Dividing (6 by θ i ω and (7 by θ yild ia + aθ i µ + G a γ a 4ω cosυ i cosυ4 + Fa + αb = 0, ω + isinυ ω + isinυ4 ib + bθ + i ( µ G b ( iµ + γ b γb iαω a( cosυ + isinυ 8 + Eb( cosυ + i sinυ = 0, whr υ= σ T θ, υ= σt θ, υ= σ T + θ θ, υ4= σ T + θ θ= υ iυ = cosυ+ isinυ., (7 (8 (9 Sparating ral and imaginary parts, givs th govrning quations of th amplituds a, b and phass υ i α b a = µ + G a + Fa sinυ + cos υ, (0 4 ω ω a a a ( υ υ = ( σ σ γ 4ω α b + cos sin, ω Fa υ υ4 ω ( b = ( µ G b µ b 4 αω a cosυ + Ebsin υ, ( b( υ + υ = b( σ+ σ γb 4 ( 5 5 γb + αω a sinυ + Eb cos υ, 8 Th stady-stat solutions corrspond to constant a, b, υ i, (i =,,4; that is a = b = υ = υ = υ = 0. Thn quations (0 (, which dscrib th modulation of th amplituds and th phass for th paramtric rsonanc rspons of th systm ( and (, bcom α b µ sinυ cos υ, ( G a Fa + = 4 ω + ω (4 a ( σ σ γ a 4ω α b = cos + sin, ω Fa υ υ4 ω (5 G b b = a cos Ebsin, (6 4 µ µ αω υ υ 5 b + + b + b 4 8 = αω a sinυ + Eb cos υ, 5 ( σ σ γ γ (7 Squaring (4-7, adding (4 and (5, thn (6 and (7 yilds th following frquncy rspons quations 6 4 a a a 4 0, Γ +Γ +Γ +Γ = ( b 6b 7b 8b 9b 0 0. Γ +Γ +Γ +Γ +Γ +Γ = (9 Th cofficints Γ i, for i=,,,0 ar dfind in Appndix C. 4. Numrical Rsults In this sction, th stady-stat rspons of th systm is invstigatd for various systm paramtrs undr th subharmonic rsonanc condition. Th stability of th numrical solution is studid using th frquncy rspons function mthod. Th frquncy rspons Equations (8 and (9 which ar nonlinar algbraic quations in th amplitud a and b rspctivly ar solvd. Th rsults ar shown in Figur as th amplitud a of th mchanical part against th dtuning paramtr σ, and in Figur as th amplitud b of th lctrical part against th dtuning paramtr σ for diffrnt valus of th systm paramtrs. Figurs (a, (c, (d and (h show that th stady-stat amplitud a dcrass as ach of th natural frquncy ω, th damping cofficint μ, th nonlinar cofficint γ and th gain G incras. It can also b sn that th branchs of th rspons curvs convrg to ach othr and th rgion of unstabl solutions gt smallr. But in Figurs ( and (g th stady- stat amplitud a incrass as ach of

4 4 Journal of Mchanical Dsign and Vibration th scond mod amplitud b and th mchanical xcitation amplitud F incras. Figur (f indicats that th curvs ar shiftd as th dtuning paramtr σ varis. Th ffct of varying th coupling cofficint α on th frquncy rspons is trivial, as shown in Figurs (b. Figur. Frquncy rspons curvs of th mchanical part at subharmonic rsonanc cas γ=.0, ω=.0, σ=0.0, μ=0., α=.5, b=0.0, G=0. It is shown from Figur (g that th branchs of frquncy rspons curv divrg and th unstabl rgion incrass as th first mod amplitud a incras. This would xplain that thr might b a transfr of nrgy from th first mod to th scond mod through th coupling paramtrs and th intrnal rsonanc as wll. Whras Figur (i, indicats that th stady-stat amplitud b incrass as th lctrical amplitud E incras. Figurs (a and (b show that th ffcts of Ω= ω = ω with linar vlocity fdback whn: F=.0, varying th natural frquncy ω and th coupling cofficint α ar trivial. But from Figurs (c, (d, (, (f and (j w can s that th stady-stat amplitud b incrass as ach of th linar damping cofficint μ, th nonlinar damping cofficint μ, th nonlinar cofficint γ, th nonlinar cofficint γ and th gain G dcras. Figur (h indicats a shifting ffct to th lft and an incras in th amplitud as th dtuning paramtr σ incrass. Figur. Frquncy rspons curvs of th lctrical part at rsonanc cas with linar vlocity fdback, whn: E=.0, γ =0.07, γ =0.7, ω =.0, σ =0.0, μ =0.0, μ =0., α =0.0, a=0.0, G =0. Figur and Figur 4 show th forc rspons curvs of th mchanical part and lctrical part, rspctivly, at rsonanc cas with linar vlocity fdback. W can s in Figurs (a, (c, (d, (f and (h that th amplitud of th mchanical part a incrass as ach of th natural frquncy ω, th linar cofficint µ, th nonlinar cofficint γ, th dtuning paramtrs σ, and th gain G dcras. Figur (b shows that as th mchanical forc

5 Journal of Mchanical Dsign and Vibration 5 lis in th rang 0 < F < 0.4, th amplitud a incrass as th coupling cofficint α is incrasd. But if F > 0.4, thn any furthr incras in th coupling cofficint α has insignificant ffct on th amplitud and may lad to saturation phnomnon. From Figur ( w can s that th first mod amplitud a incrass as scond mod amplitud b incrass. Figur. Forc rspons curvs of th mchanical part at rsonanc cas with linar vlocity fdback, whn: σ =0.05, γ =.0, ω =.0, σ =0.0, μ =0., α =.5, b=0.0, G =0. Figur 4. Forc rspons curvs of th lctrical part at rsonanc cas with linar vlocity fdback whn: σ =0.05, γ =0.07, γ =0.7, ω =.0, σ =0.0, μ =0.0, μ =0., α =0.6, a=0.0, G =0. It can b shown from Figurs 4(c, (d and (j that th lctrical xcitation amplitud E incrass as ach of th linar cofficint µ, th nonlinar cofficint µ, and th gain G dcras. But it can b sn from Figurs 4 (a, (b and (g that th lctrical xcitation amplitud E incrass as ach of th natural frquncy ω, th coupling cofficint α and th first mod amplitud a incras. Morovr, a saturation phnomnon is noticd for th systm as ths paramtrs ar incrasd whn E > 0.. Figurs 4(, (f, (h and (i show th ffcts of ach of th nonlinar cofficints γ, and th dtuning paramtrs σ and σ. To vrify th analytic prdictions, Eqs. ( and ( ar numrically intgratd using a fourth ordr Rung-Kutta algorithm. A non-rsonanc systm bhavior is shown in Figur 5. Diffrnt rsonanc cass ar invstigatd and shown in Figur 6, whr th stady-stat amplitud x has maximum pak at th subharmonic rsonanc cas ( Ω ω ω in Figur 6 (d. It is noticd that compard to th nonrsonant cas, th stady-stat amplitud x is incrasd by 900%, whil th stady-stat amplitud q has not changd. Thus, it is considrd as th worst rsonanc cas of th systm bhavior.

6 6 Journal of Mchanical Dsign and Vibration Figur 5. Non-rsonant tim history without control whn: Ω=.0, ω =.7, ω=.5, F=0.05, E=0.05, γ =0.04, γ =0.07, γ =0.7, f =0., μ =0., μ =0., μ =0., α =.5, α =0.0 Figur 6. Tim history solution at diffrnt rsonanc cass without control whn: Ω=.0, ω =.7, ω=.5, F=0.05, E=0.05, γ =0.04, γ =0.07, γ =0.7, f =0., μ =0., μ =0., μ =0., α =.5, α =0.0 Figur 7. Subharmonic rsonanc tim history with various control laws, whn: Ω=.0, ω =.0, ω=.0, F=0.05, E=0.05, γ =0.04, γ =0.07, γ =0.7, f =0., μ =0., μ =0., μ =0., α =.5, α =0.0, G =G =0.

7 Journal of Mchanical Dsign and Vibration 7 From Figur 7(b w can s that th bst control mthod is th ngativ linar vlocity fdback. Th ffct of varying th gains G and G on th x and q amplituds ar shown in Figurs 8(a (. Figur 8. Effcts of th linar vlocity fdback gains G and G on th subharmonic rsonanc tim solution, whn : Ω=.0, ω =.0, ω=.0, F=0.05, E=0.05, γ =0.04, γ =0.07, γ =0.7, f =0., μ =0., μ =0., μ =0., α =.5, α =0.0 Figur 9. Effct of th paramtric xcitation amplitud f at rsonanc cas without control

8 8 Journal of Mchanical Dsign and Vibration Th ffct of paramtric xcitation f, th amplitud of tim varying magntic fild, is studid at subharmonic rsonanc cas and shown in Figur 9, which rprsnts th tim-sris solutions (t, x for th mchanical part, and (t, q for th lctrical part. Considring Figur 9(a as basic cas for comparison, it can b sn from Figur 9 that as th paramtric xcitation incrass, a chaotic motion occurs for th x amplitud and th q amplitud. Figurs 9(b ( indicat that a continuous incras in f changs th shap of th chaotic motion. 5. Conclusions Th rspons and stability of th systm of coupld non-linar diffrntial quations rprsnting th nonlinar dynamical two-dgr-of-frdom lctromchanical systm including cubic and quintic nonlinaritis subjct to paramtric xcitation forcs with tim-varying magntic fild ar solvd and studid. Th invstigation includs th solutions applying both th prturbation tchniqu and Rung-Kutta numrical mthod. Th stability of th systm is invstigatd applying both th frquncy rspons quation and tim sris solution. From th study it is concludd that th activ controllrs ar vry ffctiv tool in vibration rduction at many diffrnt rsonanc cass. Thy ar vry hlpful in supprssing th undsird vibration or somtims liminat thm. Th nd of th work givs som rcommndations rgarding th dsign of such systm. Also a comparison with similar publishd work is rportd. In this sction, th main conclusions of th systm ar rportd brifly.. It is found that th frquncy rspons curvs consist of two branchs. Ths curvs ar bnt to right or lft for som curvs du to th nonlinarity ffct (hardning or softning. This lads to multi-valud solutions and hnc to a jump phnomnon occurrnc, which ar typical charactristics of th bhavior of nonlinar dynamical systm and ar an important challng in th controllrs dsign. Ths rsults ar in agrmnt with th rsults in [].. Th stady-stat amplituds x and q of th mchanical and lctrical parts, rspctivly, of th systm ar monotonic incrasing functions in th mchanical and lctrical forcs F and E, rspctivly.. Th stady-stat amplitud x of th mchanical part is i a monotonic incrasing function in th stady-stat amplitud q and th coupling cofficint α. ii a monotonic dcrasing function in th linar damping cofficint μ, th nonlinar cofficint γ and th gain G. 4. Th stady-stat amplitud q of th lctrical part is i a monotonic incrasing function in th stady-stat amplitud x, th natural frquncy of th mchanical part ω and th coupling cofficint α. ii a monotonic dcrasing function in th linar and nonlinar damping cofficints μ and μ, th nonlinar cofficints γ and γ and th gain G. 5. Th forc rspons curvs could dtct th bhavior of som paramtrs that show insignificant (trivial ffct in th frquncy rspons curvs. It also givs an indication to th valus of som paramtrs at which saturation may occur. 6. Th applid paramtric forcs produc various rsonanc cass. Th rsonanc cas with maximum pak of amplituds is considrd as th worst rsonanc cas. Numrical rsults show that th worst rsonanc cas undr paramtric forcs occurs at th subharmonic rsonanc cas at which th frquncy of th xcitation is twic that of th natural frquncy of th systm. 7. As for th paramtric xcitations f and f, which ar th amplituds of th tim-varying magntic fild. Whn th magnitud of any xcitation is incrasd a chaotic bhavior occurs in both mods of th systm. Th chaotic motion changs as th paramtric xcitation changs. Morovr, th shaps of chaotic motions for both mods ar diffrnt. Whras in [], th paramtric xcitation, which rsults from varying th stiffnss of th mchanical oscillator priodically, has an ffct on th x-amplitud only. Thus, stability and controllability in lctromchanical sismograph modls with tim-varying magntic fild ar bttr than sismograph modls with constant or tim-varying stiffnss. 8. Diffrnt control laws wr applid to th systm undr paramtric xcitation forcs. It is noticd that th bst control mthod is th ngativ vlocity fdback. Morovr, th highr th gain (G or G is, th fastr is th approach to th quilibrium solution, which is in agrmnt with []. This control forc has also an ffct in rducing or liminating nonlinarity ffct in th frquncy rspons curvs. Rfrncs [] Xiujing, H. and Qinshng, B, Bursting oscillations in Duffing s quation with slowly damping xtrnal forcing, Commun Nonlinar Sci Numr Simul, [] Kitio Kwuimy, C.A. and Woafo P, Exprimntal ralization and simulation a slf-sustaind macro lctromchanical systm, Mch Rs Commun, 7 ( [] Hgazy, U.H., Dynamics and control of a slf-sustaind lctromchanical sismographs with tim varying stiffnss, Mccanica, 44, [4] Mbouna Nguutu, G.S., Yamapi, R. and Woafo, P, Effcts of highr nonlinarity on th dynamics and synchronization of two coupld lctromchanical dvics, Commun Nonlinar Sci Numr Simul, [5] Kitio Kwuimy, C.A. and Woafo, P, Dynamics of a slf-sustaind lctromchanical systm with flxibl arm and cubic coupling, Commun Nonlinar Sci Numr Simul,, [6] Siw Siw, M., Moukam Kakmni, F.M., Bowong S. and Tchawoua C, Non-linar rspons of a slf sustaind lctromchanical sismographs to fifth rsonanc xcitations and chaos control, Chaos Solitons Fractals, [7] Yamapi, R, Dynamics of an lctromchanical damping dvic with magntic coupling, Commun Nonlinar Sci Numr Simul, [8] Hgazy, U.H, Singl-Mod rspons and control of a hingdhingd flxibl bam, Arch Appl Mch,

9 Journal of Mchanical Dsign and Vibration 9 [9] Siw Siw, M., Moukam Kakmni, F.M., Tchawoua C, Rsonant oscillation and homoclinic bifurcation in a Ф 6 -Van dr Pol oscillator, Chaos Solitons Fractals, [0] Siw Siw, M., Hgazy U.H, Homoclinic bifurcation and chaos control in MEMS rsonators, App Math Modll, [] Yamapi, R., Aziz-Alaoui, M.A, Vibration analysis and bifurcations in th slf-sustaind lctromchanical systm with multipl functions, Commun Nonlinar Sci Numr Simul, [] Yamapi, R., Bowong, S, Dynamics and chaos control of th slfsustaind lctromchanical dvic with and without discontinuity, Commun Nonlinar Sci Numr Simul,