Nonlinear Saturation Controller for Suppressing Inclined Beam Vibrations. Usama H. Hegazy *, Noura A Salem

Transcription

1 Nonlinar Saturation Controllr for Suppring Inlind Bam Vibration Uama H. Hgazy * Noura A Salm Abtrat In thi papr prnt th numrial prturbation olution of an inlind bam to xtrnal paramtri for ith to diffrnt ontrollr poitiv poition fdbak (PPF) nonlinar aturation (NS) ontrollr found that th (NS) on i an fftiv ontrollr. Th frquny rpon funtion th pha plan mthod ar ud to invtigat th ytm bhavior it tability. All poibl ronan a ill b xtratd fft of diffrnt paramtr on ytm bhavior at ronan ar tudid. Kyord: Bam Vibration ontrol Prturbation u INTRODUCTION availabl ronan a. Thy rportd th ourrn of aturation phnomna at diffrnt paramtr valu. Kak ibration ar th au of diomfort diturban Ho [8] prntd fftivn of th PPF algorithm Vdamag omtim dtrution of mahin applid for a modl of a olar panl hr th firt four trutur. It mut b rdud or ontrolld or mod of vibration hav bn onidrd. Si Hgazy liminatd. On of th mot ommon mthod of vibration [9] applid diffrnt ativ ontrollr to uppr th ontrol i th dynami aborbr. It ha th advantag of lo vibration of a miromhanial ronator ytm. Morovr a ot impl opration at on modl frquny. In th tim-varying tiffn i introdud to ontrol th haoti domain of many mhanial vibration ytm th oupld nonlinar vibration of uh motion of th onidrd ytm. Eia Amr [] ytm an b rdud to Yaman Sn [] tudid th vibration ontrol of a nonlinar ond ordr diffrntial quation hih ar olvd antilvr bam ubjt to both xtrnal paramtri analytially numrially. xitation but ith diffrnt ontrollr. Golnaraghi [] Elhfnay Baiouny [] tudid th nonlinar indiatd that hn th ytm i xitd at a frquny nar intability problm of to uprpod diltri fluid by th high natural frquny th trutur rpond at th uing th mthod of multipl al. Frquny rpon frquny of th xitation th amplitud of th rpon urv ar prntd graphially. Th tability of th propod inra ith th xitation amplitud. Ouini t al. [3] olution i dtrmind. Numrial olution ar prntd propod a nonlinar ontrol la hih i bad on ubi graphially for th fft of th diffrnt paramtr on th vloity fdbak to uppr th vibration of th firt mod ytm tability rpon hao. El Bhady El-Zahar of a antilvr bam hn ubjtd to a prinipal paramtri [] tudid th fft of th nonlinar ontrollr on th xitation. Th mthod of multipl al i ud to driv to vibrating ytm. Th approximat olution up to th ond firt-ordr diffrntial quation govrning th tim volution ordr ar drivd uing th mthod of multipl al of th amplitud pha of th rpon. Thn a prturbation thniqu nar th primary prinipal paramtri bifuration analyi i ondutd to xamin th tability of intrnal ronan a. Morovr thy invtigatd th th lod-loop ytm to invtigat th prforman of tability of th olution uing both pha plan mthod th ontrol la. Th thortial xprimntal finding frquny rpon quation th fft of diffrnt indiat that th ontrol la lad to fftiv vibration paramtr on th vibration of th ytm. Warminki t al. [3] upprion bifuration ontrol. El-Srafi t al. [45] tudid ativ upprion of nonlinar ompoit bam hod ho fftiv i th ativ ontrol on vibration vibration by ltd ontrol algorithm. un t al. [45] rdution of diffrnt mod of motion at ronan. Thy xtnivly tudid thortial xprimntal rarh on dmontratd th advantag of ativ ontrol ovr th th aturation phnomnon. Eia t al. [67] invtigatd a paiv on. Hgazy [6] tudid th nonlinar dynami ingl-dgr-of-frdom nonlinar oillating ytm ubjt vibration ontrol of an ltromhanial imograph ytm to multi-paramtri /or xtrnal xitation. Th multipl ith tim-varying tiffn. An ativ ontrol mthod i tim al prturbation thniqu i applid to obtain olution applid to th ytm bad on ubi vloity fdbak. In up to th third ordr approximation to xtrat tudy th [7] Hgazy invtigatd Th problm of uppring th vibration of a hingd hingd flxibl bam that i ubjtd to primary prinipal paramtri xitation. Diffrnt Uama H. Hgazy i an Aoiat Profor of Applid Mathmati in Dpartmnt of Mathmati Faulty of Sin Al-Azhar Univrity- ontrol la ar propod aturation phnomnon i Gaza Paltin. u.hjazy@alazhar.du.p uhijazy@yahoo.om invtigatd to uppr th vibration of th ytm. El- Noura A. Salm i urrntly puruing matr dgr program in Ganaini t al. [8] applid poitiv poition fdbak ativ Dpartmnt of Mathmati Faulty of Sin in Al-Azhar Univrity- Gaza Paltin ontrollr to uppr th vibration of a nonlinar ytm hn ubjtd to xtrnal primary ronan xitation. Th multipl al prturbation mthod i applid to obtain a firt

2 965 ordr approximat olution. Th quilibrium urv for variou ontrollr paramtr ar plottd. Th tability of th tady tat olution i invtigatd uing frquny rpon quation. Th approximat olution i numrially vrifid. Thy found that all prdition from analytial olution ar in good agrmnt ith th numrial imulation. SYSTEM MODEL Th modifid ond-ordr nonlinar ordinary diffrntial quation that drib th motion of th inlind bam i givn by [] u mu u bu b u - d uu uu 3 5 = f o( W t) o( a) uf o( W t)in( a) tf ( t). Whr uu u rprnt diplamnt vloity ontrol ignal ontrol F F f = v fdbak ignal Ff F = v f () = u for (PPF) = uv for (NS) ontrol. So th lod loop ytm quation to th both ontrollr ar: (i) Poitiv Poition Fdbak (PPF) ontrol u mu u bu b u - d uu uu = 3 5 f o( W t)o( a) uf o( W t)in( a) tv (3) v x v v = ruv 3 NUMERICAL INTEGRATION Th numrial tudy of th rpon th tability of to nonlinar ytm ar ondutd. Eah ytm i rprntd by to (th plant th aborbr) oupld ond ordr nonlinar diffrntial quation. Th plant (orintd bam) ha quadrati ubi quinti nonlinariti i ubjtd to xtrnal paramtri xitation. Th oupling trm ar ithr produ th poitiv poition aborbr or nonlinar ink aborbr. All poibl ronan a ar xtratd fft of diffrnt paramtr ontrollr on th plant ar diud rportd. 3. TIME-RESPONSE SOLUTION Th tim rpon of th nonlinar ytm (3) (4) (5) (6) alration of th vibrating bam rptivly i th ha bn invtigatd applying fourth ordr Rung-Kutta natural frquny m i th damping offiint b b numrial mthod th rult ar hon in Fig. () () rptivly. Th pha plan mthod i ud to giv an d ar nonlinar offiint f f ar th xtrnal indiation about th tability of th ytm. Fig. (a) (b) paramtri foring amplitud rptivly W i th ho th non-ronant bhavior of th main ytm th xitation frquny a i th orintation angl t i th gain PPF aborbr rptivly ith fin limit yl for th plant. Whra a haoti bhavior i illutratd in Fig.() (d) F () t i th ontrol ignal. for both th plant th aborbr at th imultanou primary ronan a. Fig. () ho th rpon of th W introdu a to ond-ordr nonlinar ontrollr hih plant th NS aborbr at non-ronan Fig. (a) ar oupld to th main ytm through a ontrol la. Thn (b) rptivly at to ronan a Fig. () th quation govrning th dynami of th ontrollr i (d). It i lar that th rpon of th plant ith th NS uggtd a aborbr i muh bttr than of PPF aborbr. Th NS might b mor fftiv in ontrolling th bhavior of th main v xv v = rff ( t) () ytm at ronan hih rultd in a light haoti ronant rpon Fig. () or a modulatd amplitud Fig. hr vv v rprnt diplamnt vloity (). Thrfor th NS aborbr ill b onidrd alration of th ontrollr i th natural frquny z i oupld ith th main ytm for furthr invtigation in th th damping offiint r i th gain. W hoo th folloing tion. 4 MULTIPLE-TIME SCALES ANALYSIS Th nonlinar diffrntial quation (5) ith NS ontrollr (6) i ald uing th prturbation paramtr a follo u m u u b u b u - d uu uu = 3 5 f t a uf t a tv o( W )o o( W )in (6) (5a) v x v v = ru (ii) Nonlinar Saturation (NS) ontrol 3 5 ( u mu u bu bu d uu uu ) - = f t uf t v o( W )o( a) o( W )in( a) t (4) (5) v v v = uv (6a) x r. Applying th multipl al mthod obtain firt ordr approximat olution for quation (3) (4) by king th olution in th form 6

3 u( T T ) = u ( T T ) u ( T T ) v( T T ) = v ( T T ) v ( T T ). hr i a mall dimnionl book kping prturbation paramtr T = t T = T ar th fat lo tim al rptivly. Th tim drivativ tranform i rat in trm of th n tim al a d dt hr D d = D D = D DD (8) dt = D T Subtituting (8) gt u = u u (7) =. (9) T u tim drivativ from quation (7) u = Du Du Du Du u = D u D u D Du D Du () 5 5i T iwt = - ba f o( a) () i( ) W T i( -W) T f A in ( a) f A. it tb tbb m it ( D ) v = (-ib -ixb ) u u u u Du () i ( ) T i ( ) T rba rab. fo( t)o uf o( t)in v - v v v v = Dv Dv Dv Dv v = D v D v D Dv D Dv Subtituting quation () () into quation (5a) (6a) gt Du Du DDu Du b b - d - W a - W a - t = Dv Dv DDv x Dv v v - ruv =. (3) Equating th offiint of am por of in quation () (3) giv O : ( D ) = (4) u ( D ) v. O( ): = (5) ( ) D u =-D Du -md u -bu -b u 3 5 dd u f o( Wt)o( a) 3 u f o( W t)in( a) tv (6) ( ) D v =-DDv - x Dv ruv. (7) Th gnral olution of quation (4) (5) i givn by u = A( T ) A( T ) (8) it -it v = BT BT. (9) it -it hr th quantiti AT ( ) BT ( ) ar unknon funtion in T. No to olv quation (6) (7) ubtitut quation (8) (9) into thm thn uing th form iwt -W i T o( W T) = in( W T) = obtain D u= -ia -mia-3baa ( ) ( 3 it -baa -6d AA ) (-b -5b -8d ) A A A A 3i T hr dnot th omplx onjugat trm. - i iwt -W i T in ( a) () () Th partiular olution of quation () () an b rittn in th folloing form u ( T T ) = 6 A - - A - A A - A ( b 5b 8d ) it 3iT 8 b A f o( a) 4 5 5iT iwt ( -W)( W) - f A W W ( ) f A W -W ( ) ( W) ( -W) (- )( ) i i T T in in ( a) ( a) t t it B BB 966 ()

4 967 (b) Non-ronant tim ri of th ontrollr v ( T T ) = B - rba it i ( ) T ( ) i( - ) T - r ( - ) AB. (3) () Ronant tim ri of th plant hn W= = (a) Non-ronant tim ri of th plant (d) Ronant tim ri of th ontrollr hn = W= 6 Fig. Non-ronant ronant tim hitory olution of th plant th (PPF) ontrollr hn:

5 968 =. b = 5. d =.3 m =.5 W=.7 b = 5. f =.4 f =. a= 3 t =. x =. r =. = 6.5. () Ronant tim ri of th plant hn (a) Non-ronant tim ri of th plant W= = (b) Non-ronant tim ri of th ontrollr (d) Ronant tim ri of th ontrollr hn = W= 6

6 969 =. b = 5. d =.3 m =.5 W=.7 b = 5. f =.4 f =. a= 3 t =. x =. r =. = STABILITY ANALYSIS W hall invtigat th tability of th ytm at th imultanou ronan ondition W= Subtituting quation (4) into quation () () liminating th trm that produ ular trm prforming om algbrai manipulation obtain 3 -ia -mia-3baa -baa -6d AA () Ronant tim ri of th plant hn W= (5) i = T i T f o ( a) tb = -i T (-ib - ixb ) rab =. (6) i Subtituting A = a q i B = a q obtain th folloing quation that drib th modulation of amplitud pha of th motion =. In thi a introdu th dtuning paramtr uh that W= = (4) a =- ma f in - q T o a tain( - q q T) aq - ba - ba - da f o - ( q T ) o ( a) tao (- q q T) =. (7) (8) (f) Ronant tim ri of th ontrollr hn = W= Fig. Non-ronant ronant tim hitory olution of th plant th (NS) ontrollr ytm hn: a =-x a - raa in - q q T (9) 4 a q r aa o - q q T =. (3) 4 6

7 97 L = f L = ta L = raa 3 4 Lt = (- ) g = (- q q T ) g q T Thn quation (7) - (3) bom a =- ma L in g o a L in g (3) ag = a - ba - ba - da ( g ) ( a) ( g ) L o o L o. ( g ) (3) a =-xa-l 3in (33) æ ift ift -i ç ( p 3 - ip 4) i f( p3 -ip4) a( g - g ) = a( - ) L 3o ( g). (34) è (44) ift -ix ( p3 - ip4) =. Th tady tat olution orrpond to ontant a a i T Dividing both id of quation (43) by g g that i a = a = g = g =. Thu gt f both of ma =L in ( g) o ( a) i T id of quation (44) by L in ( g) (35) f giv ip - p fp-i fp - ipm- pm =. (45) - a ba ba d a 4 8 (36) =L o( g) o ( a) Lo ( g). -ip 3 - p 4 fp3 -ifp4 -ip3x - p4x =. (46) x ( g ) = -L in (37) a 3 - ( ) 3o ( ). a - =L g (38) From quation (35) - (38) hav æ ( ma) ç - a ba ba d a è 4 8 ( a) ( a) =L o L LL o a æ a - =L3 x ç ( ). è (39) (4) Equation (39) (4) ar alld frquny rpon quation of th plant th NS ontrollr rptivly. (A) TRIVIAL SOLUTION To dtrmin th tability of th trivial olution invtigat th olution of th linarizd form quation (5) (6) that i -i A - mi A = (4) - - = (4) ib ixb. W xpr A B in th folloing Cartian form A p ip f p p p p ar ral. W obtain i T i T = ( - ) B = ( p -ip ) f hr æ ift ift -iç ( p - ip ) if( p-ip) è ift - im ( p- ip) =. (43) Sparating ral imaginary part in quation (45) (46) to gt æ p = ç - m p (-f) p è (47) æ p = ( f) p ç - m p è (48) p = - x p - f p (49) ( f ) ( x) p = p - p (5) Stting =- m =-f 33 =-z =-f. 34 Th tability of th trivial olution i invtigatd by valuating th ignvalu of th aobian matrix of quation (47) - (5) 6 -l - -l -l l =

8 97 hih giv 4 3 l hl hl h3l h4 =. (5) hr h h h3 h 4 ar funtion in th ytm paramtr. Aording to th Routh-Huritz ritrion th nary uffiint ondition for all th root of quation (5) to hav ngativ ral part hn a tabl olution ar h > hh - h3 > h ( hh -h )- hh > h 4>. (B)NON-TRIVIAL SOLUTION To dtrmin th tability of th non-trivial olution lt a = b b( T) a = ( T) j j j = ( T) y y y T = (5) Subtituting quation (5) into quation (3) - (34) imilarly a in abov hav ç - rb oy y è 4 b b =-mb- mb f( inj joj) o ( a) t ( iny yoy) t ( iny yo y) æ (53) y = ç - bb b d b- roy b bj bj bj bj = b b è b æ b( b 3 bb...) - b( b 5 bb...) ç - ( -) - toy- rboy 4 8 è b æ æ 3 3 (54) rbiny tin y y finjo ( a) j. - d ( b 3 bb... ) f( oj-jin j) o( a ç ç ) è 4b èb t ( o y - y iny ) t ( o y -y in y ) =-x -x - rb y y y ( in o ) 4 - rb ( iny y oy )- rb ( in y y o y ) 4 4 ( j j j j -y -y -y -y ) = - - rb o y-yiny 4 (55) rb ( o y- yiny) rb ( o y-yin y). 4 4 Sin b j y ar olution of quation (3) - (34) b j y ar a vry mall trm j j = j = y y = y = thn thy an b liminatd hav (56) æ æ b = ç - m bç f ojo( a) j è è æ æ ç tiny ç t o y y è4 è4 æ j = ç - b - b - d b b èb æ æ ç - f in j o( a) j ç t oy è b è b æ ç - tin y y è 4b æ æ = ç - r iny b ç -x - rb iny è 4 è 4 Lt æ =- m = f ojo ( a) 3 = tin y 4 = t o y = b b b - b b d =- f injo ( a) b 3 = o b t y 4 =- in b t y =- r in y (57) (58) (59) (6) =-x - rb in y 34 =- rb o y 4 6

9 =- b b db - ro y 3 4 b =- - - t o y - r b o y 43 b = rb iny t in y. 44 4b Th tability of th non-trivial olution i invtigatd by valuating th ignvalu of th aobian matrix of quation (57) - (6) -l 3 4 -l 3 4 -l l hih giv = l h l h l h l h =. (6) Th non-trivial olution i tabl if h > hh - h3 > h3( hh -h3)- hh 4 > h 4 >. 6 THEORETICAL FREQUENCY RESPONSE SOLUTION Th ronant frquny rpon quation of th main ytm (39) ith NS ontrollr (4) ar olvd numrially. Th rult ar hon in Fig. (3) (4) hih rprnt th variation of th tady tat amplitud a againt th dtuning paramtr σ rptivly for diffrnt valu of th othr paramtr. Fig. (3) ho th thortial frquny rpon urv of th main ytm to primary ronan a. It an b notd from Fig. (3b-3d) (3g) that tady tat amplitud inra a ah of th natural frquny ω th linar damping offiint µ th nonlinar offiint β δ dra. Th inra in th quinti nonlinar paramtr β bnd th frquny rpon urv to th right ith trivial fft on th tady tat amplitud a hon in Fig. (3). Fig. (3f) indiat that a th xitation for amplitud f inra th branh of th rpon urv divrg aay th amplitud inra. Th fft of th gain i hon in Fig. (3h). Fig. (4) illutrat th ronant frquny rpon urv of th NS ontrollr to ubharmoni intrnal ronan for variou paramtr. Eah figur onit of to urv that ithr divrg aay hn th gain ρ th tady tat amplitud of th plant inra Fig.(4b 4f). Or thy onvrg to ah othr a th natural frquny ω th linar damping ζ ar drad a hon in Fig.(4 4d). Th urv in Fig. (4) ar hiftd to th right a th dtuning paramtr σ inra. (a) Bai a (b) Th natural frquny () Th damping offiint (d) Th ubi nonlinar offiint 6

10 973 () Th quinti nonlinar offiint (a) Bai a (f) Th foring amplitud (b) Th gain (g) Nonlinar offiint () Th damping offiint (h) Th gain Fig. 3 Thortial ronant frquny rpon urv of th plant hn: =.7 b = 5. d =.3 m =.5 b = 5. f =.4 a= 3 t =.. (d) Th natural frquny 6

11 974 [4] L. un H. Hongxing S. Rongying Saturation-bad ativ aborbr for a non-linar plant to prinipal xtrnal xitation Mhanial Sytm Signal Proing (3) (7). [5] L. un L. Xiaobin H. Hongxing Ativ nonlinar aturation-bad ontrol for uppring th fr vibration of a lf-xitd plant Communiation in Nonlinar Sin Numrial Simulation (). [6] M. Eia W.A.A. El-Ganaini Y. S. Hamd On th aturation phnomna ronan of non-linar diffrntial quation Mnufiya ournal of Eltroni Enginring Rarh (MEER) (5). [7] M. Eia W.A.A. El-Ganaini Y. S. Hamd Saturation tability ronan of non-linar ytm Phyia A (5). () Th dtuning paramtr [8] M.K. Kak S. Ho Ativ vibration ontrol of mart grid trutur by multi-input multi-output poitiv poition fdbak ontrollr ournal of Sound Vibration (7). [9] M. SiSi U. H. Hgazy Homolini bifuration hao ontrol in MEMS Ronator Applid Mathmatial Modlling 35() (). [] M. Eia Y.A. Amr Vibration ontrol of a antilvr bam ubjt to both xtrnal paramtri xitation Applid mathmati omputation (4). [] M. Yaman S. Sn Vibration ontrol of a antilvr bam of varying orintation Intrnational ournal of Solid Strutur 44 -(7). [] M.F. Golnaraghi Rgulation of flxibl trutur via non-linar oupling" Dynami Control (4)45-48(99). (f) Th tady tat amplitud of th plant [3] S.S. Ouni C. M. Chin A.H. Nayfh Dynami of a ubi nonlinar vibration aborbr Nonlinar Dynami 83-95(999). [4] S. El-Srafi M. Eia H. El-Shrbiny T.H. El-Gharb On paiv Fig. 4 Thortial ronant frquny rpon urv of th (NS) Ativ ontrol of vibrating ytm" Intrnational ournal of ontrollr hn: =.7 x =. =. a =. Applid Mathmati (5). [5] S.A. El-Srafi M.H. Eia H. M. El-Shrbiny T.H. El-Gharb r =.. Comparion btn paiv ativ ontrol of a non-linar dynamial ytm apan ournal of Indutrial Applid Mathmati (6). 7 CONCLUSIONS [6] U. H. Hgazy Dynami Control of a Slf-utaind Eltromhanial Simograph With Tim-Varying Stiffn Th ontrol tability of a nonlinar diffrntial quation Mania (9). [7] U. H. Hgazy Singl-Mod Rpon Control of a Hingdrprnting th on-dgr-of-frdom nonlinar inlind Hingd Flxibl Bam Arhiv of Applid Mhani bam ar tudid. Th inlind bam ha ubi quinti 345(9). nonlinariti ubjtd to xtrnal paramtri xitation [8] W. A.A. El-Ganaini N. A. Sad M. Eia Poitiv poition for. To ontrollr thniqu hav bn applid to th fdbak (PPF) ontrollr for upprion of nonlinar ytm inlind bam ytm undr diffrnt ronan ondition vibration Nonlinar Dynami 7(3) (3). th rult of numrial Rung-Kutta intgration ho that NS ontrollr i th mot fftiv. Th analytial olution of th plant ith NS ontrollr ar obtaind applying th multipl al prturbation thniqu. Th tability of th oupld ytm i invtigatd applying th frquny rpon quation for variou valu of th paramtr of th plant NS ontrollr. In addition a good ritrion of both tability hao i th pha plan. REFERENCES [] A. R. F. Elhfnay A. F. El-Baiouny Nonlinar tability hao in ltrodynami Chao Soliton & Fratal (5). [] E. E. El Bhady E. R. El-Zahar Vibration rdution tability tudy of a dynamial ytm undr multi-xitation for via ativ aborbr Intrnational ournal of Phyial Sin (). [3]. Warminki M. Bohnki W. arzyna P. Filipk M. Augutyniak Ativ upprion of nonlinar ompoit bam vibration by ltd ontrol algorithm Communiation in Nonlinar Sin Numrial Simulation 6(5)37-48 (). 6