Sub-Harmonic Solutions of Even Order (, ), To A Weakly Non-Linear Second Order Differential Equation Governed the Motion (MEMS)

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1 Sub-Hrmnic Slutins f Even Order (,, T A Wekly Nn-Liner Secnd Order Differentil Equtin Gverned the Mtin (MEMS A. M. Elnggr, A. F. El-Bssiuny, A. M. Omrn *,, Deprtment f Mthemtics, Fculty f Science, Benh University, Egypt, B. O. 8 * mirmsud.m@gmil.cm Abstrct: Sub-hrmnic peridic slutins f even rder (, t wekly nnliner secnd rder differentil equtins which gverned the mtin f micr- electr mechnicl system (MEMS (Bndpss Filter re investigted nlyticlly. The methd f multiple scles is used t determine the mdultin equtins in the mplitude nd the phse, stedy stte slutins. The frequency-respnse equtin nd stbility nlysis f the stedy stte slutins re btined. Numericl study f the frequency-respnse equtins nd stbility equtins re given fr different vlues f the prmeters. Results re pltted in grup f Figures, n which slid (dshed curves re stble (unstble slutins. Finlly discussin nd cnclusin re given. Key Wrds: MEMS, Wekly nn-liner differentil equtin, Multiple scles methd, Stbility nd Prmetric Excittin. Intrductin The dvent f micr-bifurctin technlgies in the lst cuple f decdes hs led t the birth f n exciting nd revlutinry field clled micr-dynmicl systems. Micr systems re literlly very smll systems r systems mde f very smll cmpnents. Fr micr-electr-mechnicl systems(mems micr-estblishes dimensinl scle electr-suggests either electricity r electrnic (r bth nd mechnicl suggests mving prts f sme kind[]. Lid nd Wng[] studied structurl dynmics f micrsystems, current stte f reserch nd future directins. Mthemticl study f this type f dynmicl systems leds t nn-liner secnd rder differentil equtins r set f cupled nn-liner secnd rder differentil equtins. Peridic slutins f this type f differentil equtins (the respnse f its dynmicl systems is the bject f mny wrks. Elnggr et l.[- ] studied different kinds f peridic slutins (hrmnic, subhrmnic nd super-hrmnic slutins f wekly nn-liner secnd rder differentil equtin. L- Cveticnin et l.[] studied the peridic slutin f the generlized Ryleigh equtin. Yunis nd Nyfeh [8] used the methd f multiple scles t study the respnse f n electrstticlly ctuted resntr t resnnce excittin. Abd El-Rhmn nd Nyfeh [9, ] studied super-hrmnic resnnce excittin f rder ne-hlf. Zhng nd Meng [] nlyzed the nnliner dynmics f the electrstticlly ctuted resnnt sensrs under prmetric excittin. Elnggr et l.[] studied hrmnic nd sub-hrmnic resnnce f micr-electr mechnicl system(mems subjected t wekly nn-liner prmetric nd externl excittin. Zvdney nd Nyfeh [] studied the respnse f single-degree-f-freedm system with qudrtic nd cubic nn-linerities t fundmentl prmetric resnnce. Elnggr nd Alhndwh [] studied prmetric excittin f subhrmnic scilltins. Elnggr nd El-Bssiuny [, ] studied the respnse f self-excited tw-degree nd three-degree-f-freedm systems t multi-frequency excittins. El-Dib [] used the methd f

2 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. multiple scles t determine third-rder slutin fr cubic nn-liner Mthieu equtin. El- Bssiuny nd Eiss [8] investigted nlyticl nd numericl slutins f single-degree-f-freedm with qudrtic nd cubic nnlinerities t hrmnic resnnce. El-Dib [9] nlyzed thereticl nlysis f the prmetric hrmnic respnse f tw resnnt mdes bsed n cubic nn-liner system. El-Bssiuny et l.[] investigted tw-t-ne internl resnnces in nnliner tw-degreef-freedm system with prmetric nd externl excittins. Rhds et l.[] studied the prmetric resnnce f micr-electr mechnicl system, which represented mthemticlly by wekly nnliner secnd rder differentil equtin in which the nn-linerity is mdeled by cubic functin f displcement. El-Bssiuny [] studied n pprch fr implementing n ctive nnliner vibrtin bsrber. The strtegy explits the sturtin phenmenn tht is exhibited by multi-degree-ffreedm systems with cubic nnlinerities pssessing ne-t-ne internl resnnce. The prpsed technique cnsists f intrducing secnd-rder cntrller nd cupling it t the plnt thrugh sensr nd n ctutr where bth the feedbck nd cntrl signls re cubic. El-Bssiuny [] studied the dynmicl stbility nd cmplicted mtins f vessel in regulr se re investigted when the frequncy in the pitch is nerly twice the frequency in the rll. El-Bssiuny[] investigted vibrtin nd chs cntrl f nnliner trsinl vibrting systems. Zhng et l. [] studied the dynmics f nnliner cupled electrsttic micr mechnicl resntrs under tw frequency prmetric nd externl excittins. Elnggr et l.[] used the methd f multiple scles t investigted the sddle nde bifurctin cntrl fr n dd nn-linerity prblem. Elnggr et l.[] nlyzed the perturbtin nlysis f n electrsttic Micr-Electr-Mechnicl System(MEMS subjected t externl nd nn-liner prmetric excittins. Hrmnic, sub-hrmnic nd superhrmnic resnnce f wekly nn-liner dynmicl system subjected t externl excittin, prmetric excittin r bth re investigted by Elnggr et l.[8]. Kcem et l.[9- ] studied respectively nnliner multi-physics mdels including bth mechnicl nd electrsttic nnlinerities nd the fringing field effect. Jeffrey F. Rhds et l.[] studied the tunble micr electr mechnicl filters tht explit prmetric resnnce in which the elstic restring frce is mdeled by cubic functin. A mre cmprehensive review up t but the wrk n the nn-liner dynmics f micr resntrs is presented in literture. In this pper n nlyticl study (perturbtin nlysis fr wekly nn-liner secnd rder differentil equtin, which gverned the mtin f micr electr-mechnicl system(mems(bndpss Filter[] in which the elstic restring frce is mdeled by n dd nnliner functin. The bject f the study is t determine different types f peridic slutins subhrmnic f rder (, nd its stbility. Perturbtin methd (multiple scles methd [-] is used t determine, the mdultin equtins in the mplitude nd the phse. Stedy-stte slutins, the frequency-respnse equtins nd the stbility nlysis re given. Finlly numericl study fr frequency-respnse nd the stbility equtins re crried. The results re pltted in grup f Figures in which slid(dshed lines mens stble(unstble slutins. Discussin nd cnclusins re given. Frmultin f the prblem nd perturbtin nlysis The scilltin mtin the Micr-Electr Mechnicl Systems (MEMS(Bndpss Filter[], is gverned by the fllwing wekly nn-liner secnd rder differentil equtin u u u ( F u F u F u F u ( H u H u cs( = ( Equtin ( represent Mdified Duffing s equtin subjected t wekly nn-liner prmetric excittin, where the dts indicte differentitin with respect t t, is smll prmeter =, is the cefficient f viscus dmping, is the liner nturl frequency, is frequency f the 8 Insn Akdemik Publictins

3 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - externl excittin, F, F, F nd F re the cefficients f the nn-liner terms, H nd H re the cefficients f liner nd nnliner prmetric excittins respectively. T determine first-rder unifrm expnsin f the slutins f Eq.(, ne cn use the methd f multiple scles [-]. Let n u( t; = u ( T, T u ( T, T O(, T = t ( n where T = t is the first scle sscited with chnges ccurring t the frequencies nd, nd T = is slw scle sscited with mdultins in the mplitude. t d d = D D..., = D D D... ( dt dt where Dn =. Substituting Eqs.( nd ( int Eq.( nd equting terms with the sme pwer f Tn n bth sides, we btin system f liner prtil differentil equtins O( : D u u = ( O( : D u u = D D u Fu D u ( H u The slutin f Eq.( cn be expressed by i T i T ( T, T = A( T e A( T e Hu cs( T F u F u F u ( u ( where A is the mplitude f the slutin nd is functin f T nd A is the cmplex cnjugte f A, substitute Eq.(int Eq.(, we get D u u = (i A i A F A F A A F A A i( T H i( ( HA H AA e ( A e T F A A e i T NST. c. c. ( where NST. dentes the terms des nt prduce seculr terms nd c.c. dentes the cmplex cnjugte. Any prticulr slutin f Eq.( cntins seculr terms, which re generted by the first term n the right-hnd side f Eq.(. Mrever, it my cntins smll-divisr terms depending n the resnnce cnditin. Eq.( cntins tw cnditins t btin peridic slutins n ; n =,. i.e their exist sub-hrmnic peridic slutins f even rder (,. ( Sub-hrmnic slutin f rder ( In this sectin, we study subhrmnic slutin f rder. i.e peridic slutins with perid equl tw multiple f the perid f the excittin term i.e (. Intrducing the detuning prmeter in Eq.( t cnvert the smll divisr term int seculr term. i. e = (8 nd write ( T = T T = T T, T = T (9 Using (9, the smll-divisr term rising frm expi ( in Eq.( cn be trnsfrmed int seculr term. Then, eliminting the seculr terms yield 9

4 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. i( T i ( A i A F A F A A F A A F A A HA HAA e = ( One cn tke i ( T A( T = ( T e ( where nd re rel. Inserting Eq.( int Eq.( nd seprte rel nd imginry prts, we btin H = ( H sin ( H = ( F F F F ( H cs ( 8 where Equtins ( nd ( represent the mdultin equtins in the mplitude nd the phse. = T ( It is bvius tht, Eqs.( nd ( hve trivil slutin which crrespnds t the trivil stedy stte slutin. Nn-trivil stedy stte slutin crrespnds t the nn-trivil fixed pints(equilibrium pints f Eqs. ( nd (. Tht is, they stisfy = =, nd re given by H ( H sin = ( H ( H cs = ( F F F F 8 ( where, crrespnd t stedy stte slutins. Eliminting sin nd cs frm Eqs.( nd ( yields the frequency-respnse equtin ( ( F F F F ( ( H = H ( 8 Frm(, we get: F 8 F F F 8 H HH 9 H = (8 The first-rder unifrm expnsin f the slutin (first pprximtin f Eq.( is given by u = cs ( t O( (9 The nlysis f the stbility f the trivil slutins is equivlent t the nlysis f the liner slutins f equtin( by neglecting the nn-liner terms we get i( T i A i A F A HAe = ( T determine the stbility f the trivil stedy stte slutin, it is cnvenient t rewrite A in the frm A = ( B( T ib( T i ( T e ( where B nd b re seprtes rel nd imginry prts, we get b b B = ( B B b = ( Eqs.(nd( dmit slutin f the frm ( B, b ( B, b e, where ( B, b re cnstnts. The eigenvlues f the cefficient mtrix f Eqs.( nd ( re T Insn Akdemik Publictins

5 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - =. ( where = ( F H nd = ( F H. Then, the trivil slutin is stble if the rel prts f bth eigenvlues re less thn r equl zer. T determine the stbility f the nntrivil stedy stte slutins given by Eqs.( nd (. Let = T ( T & = ( ( where nd crrespnd t nn-trivil stedy stte slutins nd nd re perturbtins which re ssumed t be smll cmpred with nd. Substituting ( int equtins ( nd ( nd linerizing the resulting equtins, we btin where = ( ( m m sin (( m m cs ( = 9 [ ( F F [( m m sin ] H m = nd T, ( d, d e F F 8 ( m m cs ] ( H m =. Equtins ( nd ( dmit slutin f the frm ( where d, re cnstnts. Prvided tht, = where ( d H ([( ( zf F H zf F zf F H zf F H H (H H z F F z F z F z F H z F F z F z F z F H z F 8 z F H z z H ] z = 9 H H, z = 8 H 8 HH 8 H z z = H 8 H H 8 H, z = H 8 z = H 8 H, z8 = H 88 HH H, H H = H HH 89 H, z = 8 H H, 8, 9 H = H 8 H, z = 9 H H, 8 = H HH 8 H, z = 9 HH 8 H, = 8H 98 HH 8 Hndz = H 8 H z = H 8 H. z z z z (8 = H 9 H HH, z = 8 H 8 H, nd The slutin is stble if nd nly if the rel prt f ech f the eigenvlues f the cefficient f the mtrix re less thn r equl t zer. H,

6 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. Sub-hrmnic slutin f rder ( In this sectin, we study subhrmnic slutin f rder i.e peridic slutins with perid equl fur multiple f the perid f the excittin term i.e (. Intrducing the detuning prmeter in Eq.( t cnvert the smll divisr term int seculr term. i. e = (9, nd write T = T T = T T, T = T ( ( Eliminting the seculr terms frm the Eq.(9 yields i( ( T i A i A F A F A A HA e = ( i ( T Using the plr frm A( T = ( T e like in the previus sectin int the Eq.( nd seprting rel nd imginry prts, we btin the fllwing mdultin equtins: In Eq.(, where nd re rel, seprte rel nd imginry prts, nd btin = ( H sin ( = ( F F F F ( H cs ( 8 where Eqs.(nd ( represent the mdultins in the mplitude nd the phse, = T fr stedy stte slutin, = =, in Eqs.(nd( we btin = ( H sin ( ( F F F F = H 8 cs ( where nd ( yields the frequency-respnse equtin crrespnd t stedy stte slutins. Eliminting sin nd cs frm Eqs.( nd ( ( F F F F ( ( H = ( 8 Frm (, we get F 8 F F F H = ( The first-rder unifrm expnsin f the slutin (first pprximtin f Eq.( is given by u = cs ( t O( (8 Nw, the nlysis f the stbility f the trivil slutins is equivlent t the nlysis f the liner slutins f equtin ( by neglecting the nn -liner terms we get i A i A FA = (9 i ( T T slve Eq.(9 nd lets A = ( B( T ib( T e, where B nd b re rel nd imginry prts nd get b b ( F B = ( Insn Akdemik Publictins

7 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - B B ( F b = ( Eqs.(nd( dmit slutin f the frm ( B, b ( b, b e, where ( b, b re cnstnts. The eigenvlues f the cefficient mtrix f Eqs.(nd( re = i ( where ( = F Then, the trivil slutin is stble if the rel prts f bth eigenvlues re less thn r equl t zer. T T determine the stbility f the nn-trivil stedy stte slutins given by Eqs.( nd (, we use Eqs.( nd (. Let = T ( T & = ( ( where nd re given by Eqs.( nd (. Inserting Eq.(int Eqs.( nd ( nd using Eqs.( nd ( nd keeping nly the liner terms in nd, we get = ( m sin ( m cs ( 9 = ( ( F F F F m cs ( m sin ( 8 H T where m =. Equtins ( nd ( dmit slutin f the frm (, ( c, c e where c, re cnstnts. Prvided tht, ( c = ([( (8F 8 F F 8 F F F 8 F 8 F F 9 ] F F F F 9 F F 8 8 F ( Cnsequently, slutin is stble if nd nly if the rel prts f bth eigenvlues ( re less thn r equl t zer. Numericl results nd discussins This sectin presents numericl results fr sub-hrmnic slutins f rder (ne-t-tw nd (ne-t-furth by slving the frequency respnse equtins (8 nd ( nd stbility cnditins (, (8, ( nd ( fr different vlues f the prmeter in the equtins. The numericl results re pltted in grup f Figures (-, which represent the vritin f the mplitude with the detuning prmeter ( nd fr given vlues f the ther prmeters in which slid (dshed curves, represent stble (unstble slutins. Figures (- represent the frequency-respnse curves f the subhrmnic slutins f rder. In (Fig., we nte tht, the respnse mplitude hs tw brnches which re bent t the right s tht the upper brnch hs stble fr lrge vlues nd there exist jump phenmen( sddle nde bifurctin nd the lwer brnch hs unstble slutin. Fr incresing the cefficient f the liner term F respectively, we bserve tht the tw brnches re shift

8 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. t the right s tht the regins f definitin, stbility nd multivlued re decresed, (Fig.. When F is decresed with negtive vlues respectively, we nte tht tw brnches re shift t the left nd the sddle nde bifurctins re exist t the pints ( =., =., =.. The znes f definitin, stbility nd multi-vlued re incresed, (Fig.. As the cefficient f the cubic term F tkes the vlues (.9,,, we nte tht tw brnches re shift dwnwrds nd hve decresed mgnitudes respectively nd there exist jump phenmen ( sddle nde bifurctins re exist t the pints ( =., =.9, =.. The regins f definitin, stbility nd multivlued re decresed, Fig.. Fr decresing F respectively, we bserve tht the upper brnch shifts t the left nd mve t the upper nd the lwer brnch shifts upwrds. The tw brnches hve incresed mgnitudes nd there exist sddle nde bifurctins t the pints ( =., =., =.. The regins f definitin, stbility nd multivlued re incresed, (Fig.. Fr the cefficient f the quntic term F incresing nd decresing respectively, we get the sme vritin s in Figures(, nd Figs.(,8. As the cefficient f the seventh nnliner term F tkes the vlues (,,, we nte tht the tw brnches re shift dwnwrd nd hve decresed mgnitudes respectively. The sddle nde bifurctin re exist t the pints( ( =.9, =., =.89,(Fig.9. (Fig.9, represent the vritin f F fr the sme vlues with mgnitudes vlues (-, -, - when F =, we deserve tht the tw brnches re bent t the left. Als, the stbility fr the trivil slutin nd by exmintin f the eigenvlues Eq.(. fr nntrivil slutin fr the secnd cse. Fr further decresing F, f the tw brnches re shift dwnwrds nd hve decresed mgnitudes. When the cefficient f liner prmetric H is incresed respectively, we nte tht the upper brnch shift t the left nd hs incresed mgnitude respectively nd there exist sddle nde bifurctins t the pints ( =., =., =.8. The lwer brnch shifts t the right nd hs decresed mgnitudes respectively, (Fig.. Fr incresing the cefficient f nnliner prmetric excittin H, we get the sme vritin s in (Fig. s tht the sddle nde bifurctin exist t the pints ( =., =., =., (Fig.. Figures (- represent the frequency respnse curves fr sub-hrmnic slutins f rder nefurth. In (Fig., the respnse mplitude hs multivlued curves which bent t the right. The multivlued curve cnsists f tw brnches s tht the upper brnch hs unstble slutin fr smll vlues nd stble slutin fr higher vlues nd there exist sddle nde bifurctin t ( =.. The lwer brnch hs unstble slutin. The minimum pint exist t( =. When F tkes the vlues(., nd, we nte tht the multivlued curve shift t the right nd mve t dwnwrds nd hs decresed mgnitudes s tht the minimum pint shifts t the right respectively. The sddle nde bifurctin re exist t the pints ( =., =., =.. The regins f multivlued, stbility nd definitin re decresed, (Fig.. As F is decresed with negtive vlues respectively, we nte tht the multivlued curve shift t the left mve upwrd respectively nd the minimum pint mve t the left respectively. The sddle nde bifurctin exist t the pints ( =., =., =.. The regins f multivlued, stbility nd definitin re incresed, (Fig.. Fr decresing F respectively, we nte tht the multivlued curve shifts t the left nd mve upwrd respectively s tht the minimum vlue shifts nerest t ( =. The regin f definitin, multivlued nd stbility re incresed, (Fig.8. when F tke vlues (.,,, we bserve tht the multivlued curve cntrcted respectively s tht the upper brnch hs decresed mgnitudes nd the lwer brnch hs incresed mgnitudes. The regin f definitin, multivlued nd stbility re decresed the upper brnch, (Fig.9. Fr incresing nd decresing F, respectively, we get the sme vritin s in Figures(9, s tht the minimum pint des nt ffect nd exist t the sme pint ( =, Figs.(,. Fr incresing F, we get the sme vritin s in (Fig.8 nd (Fig., when F tke negtive vlue ( i. e. F =., we bserve tht the multivlued curves is bent t the left s tht the sddle nde bifurctin exist t the pint ( =.. As F is decresed further, Insn Akdemik Publictins

9 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - the multivlued curve shift dwnwrds s tht it hs the sme mgnitude in the regin nd fter this intervl it hs decresed mgnitudes. The regins f definitin, stbility nd multivlued re decresed, (Fig., H is incresed, we nte tht the multivlued curve is cntrcted nd inclsed inside the min multivlued curves s tht the minimum pint mve upwrds nd tkes incresed vlue respectively, (Fig.. Fr incresing the dmping fctr, we nte the multivlued curve is cntrcting nd given semi-vl s tht the upper brnch hs stble slutins nd hs decresed mgnitudes in smll intervl nd fter this intervl it hs the sme mgnitudes. The lwer brnch hs incresed unstble mgnitudes. As =, the semi-vl is cntrcted s tht the upper brnch hs stble slutins nd the lwer brnch hs unstble incresed mgnitudes. The znes f multivlued, definitin nd stbility re decresed, (Fig.. Fig. Fig. The frequency respnse curves f the sub-hrmnic slutin f rder =, =., F =, F =.9, F =., F =, H =, H = fr the prmeters Fig. Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decreing F

10 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. Fig. Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr fr incresing nd decresing F Fig. Fig. 8 Vritin f the mplitude f the respnse with the detuning prmeter fr fr incresing nd decresing F Fig. 9 Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decresing F Insn Akdemik Publictins

11 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing H Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decresing H... cffe.. cffe Fig. Fig. The frequency respnse curves f the sub-hrmnic slutin f rder, =., F =., F =., F =., F =., = = H fr the prmeters F. F F F F F. Fig. Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decresing F

12 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. F. F F F. F. F. Fig. Fig.8 Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decresing F... F. F F F. F. F..... Fig.9 Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decresing F... F. F F F F. F..... Fig. Fig. Vritin f the mplitude f the respnse with the detuning prmeter fr incresing nd decresing F 8 Insn Akdemik Publictins

13 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - H H H. Fig. Fig. Vritin f the mplitude f the respnse with the Vritin f the mplitude f the respnse with the detuning prmeter fr decresing H detuning prmeter fr incresing Cnclusin Sub-hrmnic slutins f even rder (, t wekly secnd rder differentil equtin gverned the mtin f micr-electr mechnicl system(mems (Bndpss Filter re investigted nlyticlly. Applying the perturbtin methd(multiple scles methd pprximte expressins up t O ( re btined. Fr ech types f peridic slutins the mdultin equtins in the mplitude nd the phses, the stedy-stte slutins, the frequency respnse equtins nd the stbility cnditins re determined. Numericl clcultins nd the results re pltted in grup f Figures, stble(unstble slutins slid(dshed curves nd represent stble(unstble slutins. Generlly their exist tw slutins higher stble nd lwer unstble nd their exist jump phenmen. Bending f the curves in the Right directin f xis. References [] S. D. Senturice, Micr system Design, Kluwer Acdemic Publisher Drdrect, (. [] R. M. Lid nd W. J. Wng, Studied structurl dynmics f micrsystems, current stte f reserch nd future directins, Mechnicl Systems nd Signl Prcessing, ( -. [] A. M. Elngger nd G. M. Hmd-Allh, Anlyticl studies fr even sub-hrmnic syndrniztin f wekly nn-liner cnservetiv physicl system, Kyungpk. Mth. J, ( (98 -. [] A. M. Elngger, Hrmnic nd Sub-hrmnic scilltins s slutins t physicl systems n gverened by qusi-liner differentil equtins x kx k f ( t x =, Acdemic f scienies, (98 -. [] A. M. Elngger, Existence nd determintin f Super-Hrmnic synchrniztin s slutin f qusi-liner physicl system, Indin J.Pwe nd Applied Mthemtics, ( [] A. M. Elngger nd A. F. El-Bssiuny, Hrmnic nd sub-hrmnic slutins f wekly nnliner cnservtive differentil equtin with peridiclly vryiny ceffcient, Prc. Mth. Phys. Sc. Egypt, (98. [] L. Cveticnin, G. M. Abd-El-Ltif, A. M. Elnggr nd G. M. Ismil, Peridic slutin f the generlized Ryleigh equtin, J. f sund nd vibrtin, 8 ( [8] M. I. Yunis nd A. H. Nyfeh, A study f the nn-liner respnse f resnnt micrbem t n electric ctutin, Nn-liner dynmics, (

14 Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - Elngr, et. l. [9] E. M. Abd El-Rhmn nd A. H. Nyfeh, Secndry resnnces f electriclly ctuted resnnt micrsensrs, J.Micrmech, Micreng, ( 9-9. [] A. H. Nyfeh nd M. I. Yunis, Dynmics f MEMS resntrs under super hrmnic nd subhrmnic excittins, J.Micmech, Micreng, ( 8-8. [] Wenming Zhng nd Gung Meng, Nn-liner dynmicl system f micr.cutil ever under cmbined prmetric nd frcing excittins in MEMS sensrs nd ctutrs, A9 ( [] A. M. Elngger, A. F. El-Bssiuny nd G. A. Ms, Hrmnic nd sub-hrmnic resnnce f MEMS subjected t wekly nn-liner prmetric nd externl excittins, Interntinl Jurnl f Applied Mthmticl Reserch (IJAMR Nrth Americn, ( (-. [] L. D. Zvdney, A. H. Nyfeh nd N. E. Snchez, The respnce f single-degree-f-freedm system with qudrtic nd cubic nnlinerities t principl prmetric resnnce, Jurnl f Sund nd Vibrtin, 9 ( [] A. M. Elngger nd A. A. Al-Hndwh, Prmetric excittin f sub-hrmnic scilltins, Int. J, Thereticl Physics, (8 ( [] A. M. Elngger nd A. F. El-Bssiuny, Resnnce f self-excited three-degree-f-freedm systems t multifrequncy excittins, Interntinl Jurnl f thereticl physics, (8 (99-8. [] A. M. Elngger nd A. F. El-Bssiuny, Hrmnic, subhrmnic, superhrmnic, simultneus sub super hrmnic resnnces f self-excited tw cupled secnd rder system t multifrequncy excittins, Acts. Mechnics. Sinic, 8 (99 pp--. [] YO. El-Dib, Nnliner Mthieu equtin nd cupled resnnce mechnism, Chs, Slitins nd Frctls, ( -. [8] A. F. El-Bssiuny nd M. Eiss, Dynmics f single-degree f freedm structure with qudrtic cubic nd qurtic nn-lineritiy t hrmnic resnnce, Applied Mthmticl Cmput., 9 ( -. [9] YO. El-Dib, Instbility f prmetriclly secnd-nd-third-subhrmnic resnnces gverned by nnliner Schrdinger equtins with cmplex cfficients, Chs, Slitins nd Frctls, ( -8. [] A. F. El-Bssiuny, M. M. Kmel nd A. Abdel-Khlik, Tw-t-ne internl resnnces in nnliner tw-degree-f-freedm system with prmetric nd externl excittins, Mth. Cmput. Simult, ( ( -. [] J. F. Rhds, S. W. Shw, K. L. Turner nd R. Bskrn, Tunble micr electrmechnicl filters tht explit prmetric resnnce. Jurnl f Vibrtin nd Acustics, ( ( -. [] A. F. El-Bssiuny, Internl resnnce f nnliner vibrtin bsrber, Physic Script, ( -. [] A. F. El-Bssiuny, Cexistnce f stble slutins in the nnliner ship mtin, Physic Script, ( -. [] A. F. El-Bssiuny, Vibrtin nd chs cntrl f nnliner trsinl vibrting systems, Physic A ( -8. [] W. M. Zhng, G. Meng nd K. X. Wei, Dynmics f nnliner cupled electrsttic micrmechnicl resntrs under tw frequency prmetric nd externl excittins, Shck nd Vibrtin, ( 9-. Insn Akdemik Publictins

15 Elngr, et. l. Interntinl Jurnl f Bsic nd Applied Science, Vl., N., April, pp. - [] A. M. Elnggr, A. F. El-Bssiuny nd K. M. Khlil, Sddle-nde bifurctin cntrl fr n dd nn-linerity prblem, Glbl J. f Pure nd Applied Mthemtics, ( -9. [] A. M. Elnggr, A. F. El-Bssiuny nd G. A. Ms, Perturbtin nlysis f n electrsttic Micr-Electr-Mechnicl System (MEMS subjected t externl nd nn-liner prmetric excittins, Interntinl Jurnl f Bsic nd Applied Sciences, ( ( 9-. [8] A. M. Elnggr nd K. M. Khlil, Cntrl f the nnliner scilltr bifurctin under superhrmnic resnnce. Jurnl f Applied Mechnics nd Technicl Physics, ( ( -. [9] N. Kcem, S. Hentz, D. Pint, B. Reig nd V. Nguyen, Nnliner dynmics f nnmechnicl bem resntrs: imprving the perfrmnce f NEMS- bsed sensr, Nntechnlgy, ( (9. [] N. Kcem, S. Hentz, H. Fntine, V. Nguyen, P. Rbert, B. Legrnd nd L. Buchillt, Frm MEMS t NEMS: mdelling nd chrcteriztin f the nn liner dynmics f resntrs, wy t enhnce the dynmic rnge, NST Nntech, (8 9-. [] N. Kcem, S. Bguet, S. Hentz nd R. Dufur, Cmputtinl nd qusi-nlyticl mdels fr nn-liner vibrtins f resnnt MEMS nd NEMS sensrs, Interntinl Jurnl f Nn- Liner Mechnics, ( ( -. [] J. F. Rhds, S. W. Shw, K. L. Turner nd R. Bskrn, Tunble micr electr mechnicl filters tht explit prmetric resnnce, nd Jurnl f Vibrtin nd Acustics, ( -. [] A. H. Nyfeh nd D. T. Mk, Nn-liner scilltis, Wiley, New-Yrk, (99. [] A. H. Nyfeh. Intrductin t Pertrptin Techniques, Wiley-Interscience, New-Yrk, (98. [] A. H. Nyfeh nd B. Blchndrs, Applied Nnliner Dynmics, Wiley, New-Yrk, (9. [] A. H. Nyfeh nd S. A. Emm, Exct slutin nd stbility f pst buckling cnfigurtins f bems, Nn liner Dynmics, (