Computational Simulations and Experiments on Vibration Control of a Flexible Two-link Manipulator Using a Piezoelectric Actuator

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1 Egrg Lrs, 3:3, EL_3_3_ Compuaoal Smulaos ad Exprms o Vbrao Corol of a Flxbl Two-lk Mapulaor Usg a Pzolcrc Acuaor Abdul Kadr Muhammad, Shgo Okamoo, Ja Hoo L, Mmbrs, IAENG Absrac Th purposs of hs rsarch ar o formula h quaos of moo of a flxbl wo-lk sysm, o dvlop compuaoal cods by a f-lm mhod ordr o prform dyamcs smulaos wh vbrao corol, o propos a ffcv corol schm ad o cofrm h calculad rsuls by xprms of a flxbl wo-lk mapulaor. Th sysm usd hs papr cosss of wo alumum bams as flxbl lks, wo clamp-pars, wo srvo moors o roa h lks ad a pzolcrc acuaor o corol vbrao. Compuaoal cods o m hsory rsposs, FFT (Fas Fourr Trasform) procssg ad gvalus - gvcors aalyss wr dvlopd o calcula h dyamc bhavor of h lks. Furhrmor, a corol schm usg a pzolcrc acuaor was dsgd o supprss h vbrao. A proporoal-drvav (PD) corol was dsgd ad dmosrad s prformacs. Th sysm ad h proposd corol schm wr cofrmd hrough xprms. Th calculad ad xprmal rsuls rvald ha h vbrao of h flxbl wo-lk mapulaor ca b corolld ffcvly. Idx Trms F-lm mhod, flxbl mapulaor, pzolcrc acuaor, vbrao corol. E I. INTRODUCTION MPLOYMENT of flxbl lk mapulaor s rcommdd h spac ad dusral applcaos ordr o accomplsh hgh prformac rqurms such as hgh-spd bsds saf oprao, crasg of posog accuracy ad lowr rgy cosumpo, amly lss wgh. Howvr, s o usually asy o corol a flxbl mapulaor bcaus of s hrg flxbly. Dformao of h flxbl mapulaor wh s oprad mus b cosdrd by ay corol. Is corollr sysm should b dal wh o oly s moo bu also vbrao du o h flxbly of h lk. I h pas fw dcads, a umbr of modlg mhods ad corol srags usg pzolcrc acuaors o dal wh h vbrao problm hav b vsgad by rsarchrs [ ]. Nshdom ad Kajwara [] vsgad a way o hac prformacs of moo ad vbrao of a flxbl-lk mchasm. Thy usd a modlg Mauscrp rcvd Aprl 3, 5; rvsd May, 5. Evry auhor s wh Mchacal Egrg Cours, Gradua School of Scc ad Egrg, Ehm Uvrsy, 3 Bukyo-cho, Masuyama , Japa. (-mal: y868b@mals.cc.hm-u.ac.jp, kadr_muhammad@yahoo.co.d, okamoo.shgo.mh@hm-u.ac.jp, jhl@hm-u.ac.jp ). Th frs auhor s also wh Cr for Mcharocs ad Corol Sysm, Mchacal Egrg Dparm, Sa Polychc of Ujug Padag, Jl. Prs Kmrdkaa KM Makassar, 9-45, Idosa. mhod basd o modal aalyss usg h f-lm mhod. Th modl was dscrbd as a sa spac form. Thr corol sysm was cosrucd wh a dsgd dyamc compsaor basd o h mxd of H /H. Thy rcommdd sparag h moo ad vbrao corols of h sysm. Yavus Yama al [] ad Krcal al [3] sudd a acv vbrao corol chqu o alumum bam modld calvrd cofgurao. Th suds usd h ANSYS packag program for modlg. Thy vsgad h ffc of lm slco f-lm modlg. Th modl was rducd o sa spac form suabl for applcao of H [] ad spaal H [3] corollrs o supprss vbrao of h bam. Thy showd h ffcvss of hr chqus hrough smulao. Zhag al [4] has sudd a flxbl pzolcrc calvr bam. Th modl of h bam usg f-lms was bul by ANSYS applcao. Basd o h Lar Quadrac Gauss (LQG) corol mhod, hy roducd a procdur o supprss h vbrao of h bam wh h pzolcrc ssors ad acuaors wr symmrcally collocad o boh sds of h bam. Thr smulao rsuls showd h ffcvss of h mhod. Gurss al [5] vsgad vbrao corol of a flxbl sgl-lk mapulaor usg hr pzolcrc acuaors. Th dyamc modlg of h lk had b prsd usg Eulr-Broull bam hory. Compos lar ad agular vlocy fdback corols wr roducd o supprss h vbrao. Thr smulao ad xprmal rsuls showd h ffcvss of h corollrs. Xu ad Koko [6] sudd f-lm aalyss ad dsgd corollr for flxbl srucurs usg pzolcrc maral as acuaors ad ssors. Thy usd a commrcal f-lm cod for modlg ad compld wh a opmal acv vbrao corol sa spac form. Th ffcvss of h corol mhod was cofrmd hrough smulaos. Kusculuoglu al [7] had usd a pzolcrc acuaor for xcao ad corol vbraos of a bam. Th bam ad acuaor wr modld usg Tmoshko bam hory. A opmzd vbrao absorbr usg a lcrcal rssv-ducv shu crcu o h acuaor was usd as a passv corollr. Th ffcvss of rsuls was show by smulaos ad xprm. Furhrmor, Hw al [8] usd h Acv-forc (AF) corol for dformao ad dsurbac auao of a flxbl mapulaor. Th, a PD corol was usd for rajcory rackg of h flxbl mapulaor. Thy usd a moor as a acuaor. Modlg of h mapulaor was do usg vrual lk coorda sysm (VLCS). Thr smulao rsuls had show ha h proposd corol could cacl h dsurbac sasfacorly. Tavakolvour al [9] vsgad h AF corol applcao for a flxbl h pla. Modlg of hr sysm was do usg (Advac ol publcao: July 5)

Egrg Lrs, 3:3, EL_3_3_ f-dffrc mhod. Thr calculad rsuls showd h ffcvss of h proposd corollr o rduc vbrao of h pla. Tavakolvour ad Malah [] sudd h AF corol applcao for a flxbl bam wh a lcromagc acuaor.

2 Egrg Lrs, 3:3, EL_3_3_ f-dffrc mhod. Thr calculad rsuls showd h ffcvss of h proposd corollr o rduc vbrao of h pla. Tavakolvour ad Malah [] sudd h AF corol applcao for a flxbl bam wh a lcromagc acuaor. Modlg of h bam was do usg f-dffrc mhod. Th ffcvss of h proposd corollr was cofrmd hrough smulao ad xprm. I h rc wo yars, Muhammad al [ 5] hav acvly sudd vbrao corol o a flxbl sgl-lk mapulaor wh a pzolcrc acuaor usg f-lm mhod. Modl of h sgl-lk ad h pzolcrc acuaor was bul usg o-dmsoal ad wo-od lms. Thy roducd a smpl ad ffcv corol schm wh h acuaor usg proporoal (P), PD ad AF corols srags. Th ffcvsss of h proposd corol schm ad srags wr show hrough smulaos ad xprms. Th purposs of hs rsarch ar o drv h quaos of moo of a flxbl wo-lk sysm by a f-lm mhod, o dvlop compuaoal cods ordr o prform dyamcs smulaos wh vbrao corol ad o propos a ffcv corol schm of a flxbl wo-lk mapulaor. Th flxbl wo-lk mapulaor usd hs papr cosss of wo alumum bams as flxbl lks, wo alumum clamp-pars, wo srvo moors o roa h lks ad a pzolcrc acuaor o corol vbrao. Compuaoal cods o m hsory rsposs, FFT (Fas Fourr Trasform) procssg ad gvalus - gvcors aalyss wr dvlopd o calcula h dyamc bhavor of h lks. Fally, a PD corollr was dsgd o supprss h vbrao. I was do by addg bdg moms grad by h pzolcrc acuaors o h wo-lk sysm. II. FORMULATION BY FINITE-ELEMENT METHOD Th lks hav b dscrzd by f-lms. Evry f-lm (Elm -h) has wo ods amly Nod ad Nod (+). Evry od (Nod ) has wo dgrs of frdom [ 5], amly h laral dformao v (x,, ad h roaoal agl ψ (x,. Th lgh, h cross-scoal ara ad h ara mom of ra aroud z-axs of vry lm ar dod by l, S ad I z rspcvly. Mchacal proprs of vry lm ar dod as Youg s modulus E ad mass dsy ρ. A. Kmac Fgur shows h poso vcors r p ad r p of arbrary pos P ad P o Lk ad Lk h global ad roag coorda frams. L h lks as flxbl bams hav a moo ha s cofd h horzoal pla as show Fg.. Th O XY fram s h global coorda fram wh Z-axs s fxd. Furhrmor, o x y ad o x y ar h roag coorda frams fxd o h roo of Lk ad Lk, rspcvly (z -axs ad z -axs ar fxd). Th u vcors X, Y, x, y, x ad y axs ar dod by I, J,, j, ad j, rspcvly. Th frs moor s salld o h roo of h Lk. Th scod moor ha rad as a cocrad mass s salld h roo of h Lk. Th roaoal agls of h frs ad scod moor wh h lks roa ar dod by θ ( ad θ (. Lgh of Lk s doad by L. Laral dformao of h arbrary pos P ad P h frs ad h scod lks ar doad by v p ad O-XY : Global coorda fram o -x y : Roag coorda fram fxd o Lk o -x y : Roag coorda fram fxd o Lk r p, r p : Poso vcors of h arbrary pos p ad p h O-XY θ : Roaoal agl of h frs moor θ : Roaoal agl of h scod moor X p, X p : Coordas of h arbrary pos p ad p h X-axs of h O-XY Y p, Y p : Coordas of h arbrary pos p ad p h Y-axs of h O-XY ν p : Laral dformao of h arbrary po p o Lk h o -x y ν p : Laral dformao of h arbrary po p o Lk h o -x y ψ : Roaoal agl of h d-po of Lk v : Laral dformao of h d-po of Lk L : Lgh of Lk Fg.. Poso vcors of arbrary pos P ad P h global ad roag coorda frams v p, rspcvly. Laral dformao ad roaoal agl of h d-po of h frs lk ar doad by v ad ψ, rspcvly. Th poso vcors r p ad r p of h arbrary pos P ad P a m =, masurd h O XY fram show Fg. ar xprssd by r X x,, v, I Y ( x,, v, ) J () p p( p p p r X x,,, v,, v, ) I p p( p Whr X Y Y p( x p,,, v,, v, )J () x cos ( v p ( x, s( ) (3) p x s( v p( x, cos ( ) (4) p X Y L cos ( v ( x, s( ) p x cos ( ( x, ( v x, s ( ( x, ( ) (5) p( L s ( v ( x, cos ( ) p x s ( ( x, ( v x, cos ( ( x, ( ) (6) p ( Th vlocy vcors of h arbrary pos P ad P a m =, show Fg. ar xprssd by r X x,,, v, v, I Y ( x,,, v, v, ) J p p( p p p p p (7) (Advac ol publcao: July 5)

Th four boudary codos ar xprssd as odal vcor as follow v T v δ (9) Th, h hypohszd dformao has four cosas as follows [6] 3 a x a3x a4 x v a () whr x s poso coorda of h arbrary po P h x -axs of h lm

3 Egrg Lrs, 3:3, EL_3_3_ r X x,,,,, v,, v, v,, v, ) I p p( p p Y,,,,, v,, v, v,, v, )J (8) p( x p p B. F-lm Mhod Fgur shows h lm coorda fram of Elm, ad a arbrary po P o Elm. Hr, hr ar four boudary codos oghr a ods ad (+) wh h o-dmsoal ad wo-od lm s usd. Th four boudary codos ar xprssd as odal vcor as follow v T v δ (9) Th, h hypohszd dformao has four cosas as follows [6] 3 a x a3x a4 x v a () whr x s poso coorda of h arbrary po P h x -axs of h lm coorda fram. Th, h rlao bw h laral dformao v ad h roaoal agl ψ of h Nod s gv by v () x Morovr, from mchacs of marals, h sra of Nod ca b dfd by v x y () whr y s poso coorda of h arbrary po P h y -axs of h lm coorda fram. M δ Cδ K f L v v cos g v L v 3 s g M δ (4) whr M, C, ad K, ar h mass marx, dampg marx, sffss marx of Elm o Lk ad Lk. Vcors of f ad g ar h xcao vcors o Lk ad Lk. Th rprsao of h marcs ad h vcor of f ca b foud [] ad [3]. Th vcor of g ca b dfd by Sl g T 6 5l 6 l (5) Fally, h quaos of moo of Lk ad Lk wh lms cosdrg h boudary codos s rspcvly gv by K M δ f M δ Cδ M δ C δ f L v v cos v L K v M 3 s g δ g (6) (7) III. VALIDATION OF FORMULATION AND COMPUTATIONAL CODES A. Exprmal Modl Fgur 3 shows h xprmal modl of h flxbl wo-lk mapulaor. Th flxbl mapulaor cosss of wo flxbl alumum bams, wo clamp-pars, wo srvo moors ad h bas. Lk ad Lk ar aachd o h frs ad scod moors hrough h clamp-pars. Lk ad Lk ar cocd hrough h scod moor. Two sra gags ar bodd o h poso of. [m] ad.38 [m] from h org of h wo-lk sysm. Th frs moor s moud o h bas. I h xprms, h moors wr oprad by a dpd moo corollr. o x y : Elm coorda fram of h Elm Fg.. Elm coorda fram of h Elm C. Equaos of moo Equaos of moo of Elm -h o Lk ad Lk ar rspcvly gv by Fg. 3. Exprmal modl of h flxbl wo-lk mapulaor B. Compuaoal Modls I hs rsarch, w dfd ad usd hr yps of compuaoal modls of h flxbl wo-lk mapulaor. K M δ f Mδ Cδ (3) Modl A A modl of oly a wo-lk mapulaor was usd as Modl A. Fgur 4.a shows Modl A. Th lks ad h clamp-pars wr dscrzd by 35 lms. Two sra gags ar (Advac ol publcao: July 5)

Egrg Lrs, 3:3, EL_3_3_ bodd o h poso of Nod 6 ad Nod of h wo-lk (. [m] ad.38 [m] from h lk s org), rspcvly. Modl B A modl of h flxbl wo-lk mapulaor cludg a pzolcrc acuaor was dfd as Modl B. Fgur 4.

Physcal paramrs of h flxbl wo-lk mapulaor modls ad h pzolcrc acuaor ar show abl.

Th bdg moms proporoal o h volag whch ar xprssd by M 3 6 M d E (8) Hr d s a cosa quay ad M 3 opposs o M 6.

4 Egrg Lrs, 3:3, EL_3_3_ bodd o h poso of Nod 6 ad Nod of h wo-lk (. [m] ad.38 [m] from h lk s org), rspcvly. Modl B A modl of h flxbl wo-lk mapulaor cludg a pzolcrc acuaor was dfd as Modl B. Fgur 4.b shows Modl B. Th pzolcrc acuaor was bodd o h o surfac of Elms 4. Th lks cludg h clamp-pars ad h pzolcrc acuaor wr dscrzd by 36 lms. Schmac rprsaos o modlg of h pzolcrc acuaors ar show Fg. 5. Physcal paramrs of h flxbl wo-lk mapulaor modls ad h pzolcrc acuaor ar show abl. Th pzolcrc acuaor supprsss h vbrao of h wo-lk flxbl mapulaor by addg bdg moms a Nods 3 ad 6 of h wo-lk mapulaor, M 3 ad M 6. Th bdg moms ar grad by applyg volags E o h pzolcrc acuaor. Th bdg moms proporoal o h volag whch ar xprssd by M 3 6 M d E (8) Hr d s a cosa quay ad M 3 opposs o M 6. Furhrmor, h volag o gra h bdg moms s proporoal o h sra masurd by h frs sra gag, ε of h wo-lk du o h vbrao. Th rlao ca b xprssd as follows E (9) d (a) Modl A: Oly wo-lk Fg. 5. Modlg of h pzolcrc acuaor TABLE I PHYSICAL PARAMETERS OF THE FLEXIBLE LINK AND THE PIEZOELECTRIC ACTUATOR [7] l Toal lgh m l Lgh of Lk m.9 - l Lgh of Lk m.5 - l c, l c Lgh of clamp-pars ad m.5 - l a, l a Lgh of Acuaors ad m. - S l, S l Cross sco ara of lks ad m.95-5 S c, S c Cross sco ara of clamp-pars ad Cross sco ara of acuaors ad S a, S a Cross sco ara mom of I zl, I zl ra aroud z-axs of lks ad Cross sco ara mom of I zc, I zc ra aroud z-axs of clamp-pars ad Cross sco ara mom of I za, I za ra aroud z-axs of acuaors ad m m.58-5 m m m E l, E l Youg s Modulus of lks ad GPa 7.3 E c, E c E a, E a Youg s Modulus of clamp-pars ad Youg s Modulus of acuaors ad GPa 7.3 GPa 4.4 ρ l, ρ l Dsy of lks ad kg/m ρ c, ρ c Dsy of clamp-pars ad kg/m ρ a, ρ a Dsy of acuaors ad kg/m (b) Modl B: Two-lk wh a pzolcrc acuaor (b) Modl C: Two-lk wh a pzolcrc acuaor ad a d-ffcor Fg. 4. Compuaoal modls of h flxbl wo-lk mapulaor α, α Dampg facor of lks ad s.5-4 E, E F, F m Maxmum pu volags of acuaors ad Maxmum oupu forcs of acuaors ad Mass of h scod moor ad s clampg sysm V 5. N. g 3.53 Hr d s a cosa quay. Th, d ad d wll b drmd by comparg h calculad rsuls ad xprmal os. Modl C Fgur 4.c shows modl C ha a d-ffcor s cosdrd for a wo-lk mapulaor wh a pzolcrc acuaor. Modl C s usd o show ha h proposd corol schm s also suabl for such sysm. Th d-ffcor s prsd by addg a cocrad mass o Modl B. I hs cas, h quao of moo of h p lm coag h cocrad mass s gv by (Advac ol publcao: July 5)

5 Egrg Lrs, 3:3, EL_3_3_ M K M M f f L v v cos g gcm v L v 3 s g g M cm δ C δ cm cm δ whr h vcors of f cm ad g cm ar rspcvly gv by cm c T cm () f m l l () c T g cm m () ad h cocrad mass marx M cm ca b xprssd as M cm (3) mc whr m c s h mass of h cocrad mass. C. Tm Hsory Rsposs of Fr Vbrao Exprm o fr vbrao was coducd usg a mpuls forc as a xral o. Fgur 6 shows h xprmal m hsory rspos of sras, ε o h fr vbrao a h sam poso h calculao (. [m] from h org of h wo-lk sysm). Furhrmor, h compuaoal cods o m hsory rspos of Modl A wr dvlopd. Fgur 7 shows h calculad sras a Nod 6 of Modl A udr h mpuls forc. Sra, ε ( -6 ) Fg. 6. Exprmal m hsory rspos of sras o fr vbrao of h flxbl wo-lk a. [m] from h org of h wo-lk Sra, ε6 ( -6 ) Fg. 7. Calculad m hsory rspos of sras o fr vbrao a Nod 6 of Modl A D. Fas Fourr Trasform (FFT) Procssg Boh h xprmal ad calculad m hsory rsposs o fr vbrao wr rasfrrd by FFT procssg o fd hr frqucs. Magud of log ε Hz Frqucy, f [Hz] Fg. 8. Exprmal aural frqucy of h flxbl wo-lk Magud of log ε Hz 8.95 Hz Frqucy, f [Hz] Fg. 9. Calculad aural frqucs of Modl A Fgurs 8 ad 9 show h xprmal ad calculad aural frqucs of h flxbl wo-lk mapulaor, rspcvly. Th frs xprmal aural frqucy,.79 [Hz] agrd wh h calculad o,.8 [Hz]. Th scod xprmal aural frqucy could o b masurd. Howvr, could b obad as 8.95 [Hz] h calculao. E. Eg-valus ad Eg-vcors Aalyss Th compuaoal cods o Eg-valus ad Eg-vcors aalyss wr dvlopd for aural frqucs ad vbrao mods. Normalzd dformao Posos of h ods h lk [m] Fg.. Frs vbrao mod ad aural frqucy (f =.79 [Hz]) of Modl A Normalzd dformao Posos of h ods h lk [m] Fg.. Scod vbrao mod ad aural frqucy (f = 8.9 [Hz]) of Modl A (Advac ol publcao: July 5)

6 Egrg Lrs, 3:3, EL_3_3_ Th calculad rsuls for h frs ad scod aural frqucs wr.79 [Hz] ad 8.9 [Hz], rspcvly. Th vbrao mods of aural frqucs ar show Fgurs ad. F. Tm Hsory Rsposs du o Bas Excao Aohr xprm was coducd o vsga h vbrao of h flxbl wo-lk du o h bas xcao grad by roao of h moor. I h xprm, h frs moor wr road by h agl of π/ radas (9 dgrs) wh.5 [s]. Fgurs ad 4 show h xprmal m hsory rsposs of sras of h flxbl wo-lk du o h moor roao a. [m] ad.38 [m] from h org of h lk, rspcvly. Furhrmor, basd o Fgurs ad 4, h m hsory rsposs of sras a Nod 6 ad Nod of Modl A wr calculad as show Fgurs 3 ad 5, rspcvly. Sra, ε ( -6 ) Fg.. Exprmal m hsory rsposs of sras a. [m] from h org of h wo-lk du o h bas xcao Sra, ε6 ( -6 ) Fg. 3. Calculad m hsory rsposs of sras a Nod 6 of Modl A du o h bas xcao Fg. 5. Calculad m hsory rsposs of sras a Nod of Modl A du o h bas xcao IV. CONTROL SCHEME A corol schm o supprss h vbrao of h sgl-lk was dsgd usg h pzolcrc acuaor. I was do by addg bdg moms grad by h pzolcrc acuaor o h sgl-lk. To drv h acuaor, a PDcorollr has b dsgd ad xamd hrough calculaos ad xprms. Th pzolcrc acuaor supprsss h vbrao of h wo-lk flxbl mapulaor by addg bdg moms a ods 3 ad 6 of h wo-lk mapulaor, M 3 ad M 6. Thrfor, h quao of moo of Lk bcom M K M δ f u δ Cδ whr h vcor of u coag M 3 ad M 6 s h corol forc grad by h acuaor o h wo-lk sysm. Furhrmor, subsug Eq. (9) o Eq. (8) gvs M 3 M 6 d (4) d (5) Basd o Eq. (5), h bdg moms ca b dfd s-doma as follows U ( s ) G ( s ) ( s ) 6( s ) (6) Sra, ε ( -6 ) C d whr ε d ad ε 6 do h dsrd ad masurd sras a Nod 6, rspcvly. A block dagram of h PD-corollr for h wo-lk sysm s show Fg. 6. Sra, ε ( -6 ) ε d (s) + - F (s) U + (s) + G c (s) PD-corollr G p (s) Flxbl lk ε (s) Fg. 4. Exprmal m hsory rsposs of sras a.38 [m] from h org of h wo-lk du o h bas xcao Th abov rsuls show h valds of h formulao, compuaoal cods ad modlg h flxbl wo-lk mapulaor. ε d : Dsrd sra F : Bas xcaos ε : Masurd sras a Nod U : Appld bdg moms Fg. 6. Block dagram of proporoal-drvav corol of h flxbl wo-lk mapulaor (Advac ol publcao: July 5)

7 Egrg Lrs, 3:3, EL_3_3_ Morovr, h ga of PD-corollr ca b wr by a vcor s-doma as follows C G ( s) K K s K K s p d V. EXPERIMENT p d (7) A. Exprmal S-up I ordr o vsga h valdy of h proposd corol schm, a xprmal s-up was dsgd. Th s-up s show Fg.7. Th flxbl wo-lk mapulaor cosss of wo flxbl alumum bams, wo clamp-pars, wo srvo moors ad h bas. Lk ad Lk ar aachd o h frs ad scod moors hrough h clamp-pars. Lk ad Lk ar cocd hrough h scod moor. I h xprms, h moors wr oprad by a dpd moo corollr. Two sra gags wr bodd o h posos of. [m] ad.38 [m] from h org of h wo-lk sysm. A d-ffcor was roducd o h sysm ordr o dmosra a compl flxbl wo-lk mapulaor. Th pzolcrc acuaor was aachd o o sd of Lk o provd h blockg forc agas vbraos. A Whaso brdg crcu was dvlopd o masur h chags rssac of h frs sra gag h form of volags as fdback sgals. A amplfr crcu was dsgd o amplfy h small oupu sgal of h Whaso brdg. Aohr Whaso brdg - amplfr crcus wr usd for h scod sra gag. Furhrmor, a daa acquso board ad a compur ha hav fucoaly of A/D (aalog o dgal) covrso, sgal procssg, corol procss ad D/A (dgal o aalog) covrso wr usd. Th daa acquso board cocd o h compur hrough USB por. Fally, h corolld sgals s o a pzo drvr o drv h pzolcrc acuaor s volag rag. T xprm wr rducd by a [µf] capacor aachd o h amplfr. Th oupu volags of h amplfr s o h daa acquso board ad h compur for corol procss. Th PD-corollr was mplmd h compur usg h vsual C++ program. Th aalog oupu volags of h daa acquso board s o h pu chal of h pzo drvr o gra h acuad sgals for h pzolcrc acuaor. VI. CALCULATED AND EXPERIMENTAL RESULTS A. Calculad Rsuls Tm hsory rsposs of sras o h ucorolld ad corolld sysms wr calculad wh h frs ad scod moors road by h agl of π/4 rada (45 dgrs) ad π/ radas (9 dgrs) wh.5 [s], rspcvly. Tm hsory rsposs of sras o h corolld sysm wr calculad for Modls B ad C udr h corol schm show Fg. 6. Th cocrad mass m c usd for Modl C s 4.49 [g]. Examg svral gas of h PD-corollr ladd o K p = [Nm] ad K d =.6 [Nms] as h br os. Fgurs 8 ad show m hsory rsposs of sras a Nod ad Nod for ucorolld Modl B whl fgurs 9 ad show h corolld os. Th maxmum ad mmum sras of ucorolld Modl B a Nod 6 posv ad gav sds wr ad , as show Fg. 8. By usg PD-corollr hy bcam ad , as show Fg. 9. Furhrmor, h maxmum ad mmum sras of ucorolld Modl B a Nod posv ad gav sds wr ad , as show Fg.. By usg PD-corollr hy bcam ad , as show Fg.. Morovr, fgurs ad 4 show m hsory rsposs of sras a Nod ad Nod for ucorolld Modl C whl fgurs 3 ad 5 show h corolld os. Sra, ε6 ( -6 ) Fg. 8. Calculad m hsory rspos of sras a Nod 6 for ucorolld Modl B du o h bas xcaos : Masurm of sras : Vbrao corol : Moo corol Fg. 7. Schmacs of masurm ad corol sysm B. Exprmal Mhod Th roaos of h frs ad scod moors wr s from o π/4 radas (45 dgrs) ad o π/ radas (9 dgrs) wh.5 [s], rspcvly. Oupus of h frs sra gag wr covrd o volags by h Whaso brdg ad magfd by h amplfr. Th oss ha occur h Sra, ε6 ( -6 ) Fg. 9. Calculad m hsory rspos of sras a Nod 6 for corolld Modl B du o h bas xcaos ( K p = [Nm] ad K d =.6 [Nms] ) (Advac ol publcao: July 5)

8 Egrg Lrs, 3:3, EL_3_3_ Th maxmum ad mmum sras of ucorolld Modl C a Nod 6 posv ad gav sds wr ad , as show Fg.. By usg PD-corollr hy bcam ad , as show Fg. 3. Furhrmor, h maxmum ad mmum sras of ucorolld Modl C a Nod posv ad gav sds wr ad , as show Fg. 4. By usg PD-corollr hy bcam ad , as show Fg. 5. Sra, ε ( -6 ) Fg.. Calculad m hsory rspos of sras a Nod for ucorolld Modl B du o h bas xcaos Sra, ε ( -6 ) Fg.. Calculad m hsory rspos of sras a Nod for corolld Modl B du o h bas xcaos (K p = [Nm] ad K d =.6 [Nms]) Sra, ε6 ( -6 ) Fg.. Calculad m hsory rspos of sras a Nod 6 for ucorolld Modl C du o h bas xcaos (m c = 4.49 [g]) Sra, ε6 ( -6 ) Fg. 3. Calculad m hsory rspos of sras a Nod 6 for corolld Modl C du o h bas xcaos (K p = [Nm], K d =.6 [Nms] ad m c = 4.49 [g]) Sra, ε ( -6 ) Fg. 4. Calculad m hsory rspos of sras a Nod for ucorolld Modl C du o h bas xcaos (m c = 4.49 [g]) Sra, ε ( -6 ) Fg. 5. Calculad m hsory rspos of sras a Nod for corolld Modl C du o h bas xcaos (K p = [Nm], K d =.6 [Nms] ad m c = 4.49 [g]) B. Exprmal Rsuls Exprmal m hsory rsposs of sras o h ucorolld ad corolld sysms wr masurd wh h frs ad scod moors road by h agl of π/4 rada (45 dgrs) ad π/ radas (9 dgrs) wh.5 [s], rspcvly. Mass of h d-ffcor usd h xprms s 4.49 [g]. Tm hsory rsposs of sras o h corolld sysm wh ad whou h d-ffcor wr masurd udr h corol schm show Fg. 6. Svral xprmal gas of h PD-corollr, K p (o-dmsoal ga) ad K d wr xamd. Th xamaos of gas ladd o K p = 3 [-] ad K d =.3 [s] as h br os. Fgurs 6 ad 8 show m hsory rsposs of sras a posos of. [m] ad.38 [m] from h lk s org for ucorolld sysm whou a d-ffcor whl fgurs 7 ad 9 show h corolld os. Th maxmum ad mmum sras of ucorolld sysm whou a d-ffcor a posos of. [m] from h lk s org posv ad gav sds wr ad , as show Fg. 6. By usg PD-corollr hy bcam ad , as show Fg. 7. Furhrmor, h maxmum ad mmum sras of ucorolld sysm whou a d-ffcor a poso of.38 [m] from h lk s org posv ad gav sds wr ad , as show Fg. 8. By usg PD-corollr hy bcam ad , as show Fg. 9. Fgurs 3 ad 3 show m hsory rsposs of sras a posos of. [m] ad.38 [m] from h lk s org for ucorolld sysm wh h d-ffcor whl fgurs 3 ad 33 show h corolld os. Th maxmum ad mmum sras of ucorolld sysm wh h d-ffcor a posos of. [m] from h lk s org posv ad gav sds wr ad , as show Fg. 3. By usg PD-corollr (Advac ol publcao: July 5)

9 Egrg Lrs, 3:3, EL_3_3_ hy bcam ad , as show Fg. 3. Furhrmor, h maxmum ad mmum sras of ucorolld sysm wh h d-ffcor a posos of.38 [m] from h lk s org posv ad gav sds wr ad , as show Fg. 3. By usg PD-corollr hy bcam ad , as show Fg. 33. Sra, ε ( -6 ) Fg. 6. Exprmal m hsory rsposs of sras a. [m] from h lk s org for ucorolld sysm whou a d-ffcor du o h bas xcaos Sra, ε ( -6 ) Fg. 7. Exprmal m hsory rsposs of sras a. [m] from h lk s org for corolld sysm whou a d-ffcor du o h bas xcaos (K p = 3 [-] ad K d =.3 [s]) Sra, ε ( -6 ) Fg. 8. Exprmal m hsory rsposs of sras a.38 [m] from h lk s org for ucorolld sysm whou a d-ffcor du o h bas xcaos Sra, ε ( -6 ) Fg. 9. Exprmal m hsory rsposs of sras a.38 [m] from h lk s org for corolld sysm whou a d-ffcor du o h bas xcaos (K p = 3 [-] ad K d =.3 [s]) Sra, ε ( -6 ) Fg. 3. Exprmal m hsory rsposs of sras a. [m] from h lk s org for ucorolld sysm wh h d-ffcor du o h bas xcaos (m c = 4.49 [g]) Sra, ε ( -6 ) Fg. 3. Exprmal m hsory rsposs of sras a. [m] from h lk s org for corolld sysm wh h d-ffcor du o h bas xcaos (K p = 3 [-] ad K d =.3 [s] ad m c = 4.49 [g]) Sra, ε ( -6 ) Fg. 3. Exprmal m hsory rsposs of sras a.38 [m] from h lk s org for ucorolld sysm wh h d-ffcor du o h bas xcaos (m c = 4.49 [g]) Sra, ε ( -6 ) Fg. 33. Exprmal m hsory rsposs of sras a.38 [m] from h lk s org for corolld sysm wh h d-ffcor du o h bas xcaos (K p = 3 [-] ad K d =.3 [s] ad m c = 4.49 [g]) I was vrfd from hs rsuls ha h proposd corol schm ca ffcvly supprss h vbrao of h flxbl wo-lk mapulaor. (Advac ol publcao: July 5)

10 Egrg Lrs, 3:3, EL_3_3_ VII. CONCLUSION Th quaos of moo for h flxbl wo-lk mapulaor had b drvd usg h f-lm mhod. Compuaoal cods had b dvlopd ordr o prform dyamc smulaos of h sysm. Exprmal ad calculad rsuls o m hsory rsposs, aural frqucs ad vbrao mods show h valds of h formulao, compuaoal cods ad modlg of h sysm. Th corol schm usg a proporoal-drvav (PD) corollr was dsgd o supprss h vbrao of h sysm. Th proposd corol schm was xamd hrough h calculaos ad xprms. Th calculad ad xprmal rsuls hav rvald ha h vbrao of h flxbl wo-lk mapulaor ca b supprssd ffcvly. Pzolcrc Acuaor hrough Calculaos ad Exprms, Egrg Lrs, Vol.. No.3, 4, pp [5] A.K. Muhammad al, Acv-forc Corols o Vbrao of a Flxbl Sgl-lk Mapulaor Usg a Pzolcrc Acuaor, Trasacos o Egrg Tchologs: Iraoal MulCofrc of Egrs ad Compur Scss 4, G.-C. Yag al, Ed. Sprgr, 4, pp. 5. [6] M. Lala al, Mchacal Vbrao for Egrs, Joh Wly & Sos Ld, 983, pp [7] Rs Coad Mullayr Pzolcrc Acuaors. REFERENCES [] C. Nshdom, ad I. Kajwara, Moo ad Vbrao Corol of Flxbl-lk Mchasm wh Smar Srucur, JSME Iraoal Joural, Vol.46, No., 3, pp [] Y. Yama al, Acv Vbrao Corol of a Smar Bam, Procdgs of h CANSMART Symposum,, pp [3] O.F. Krcal al, Acv Vbrao Corol of a Smar Bam by Usg a Spaal Approach, Nw Dvlopms Robocs, Auomao ad Corol, 9, pp [4] J. Zhag al, Acv Vbrao Corol of Pzolcrc Illg Srucurs, Joural of Compurs, Vol. 5. No. 3,, pp [5] K. Gurss al, Vbrao corol of a sgl-lk flxbl mapulaor usg a array of fbr opc curvaur ssors ad PZT acuaors, Mcharocs 9, 9, pp [6] S.X. Xu ad T.S. Koko, F Elm Aalyss ad Dsg of Acvly Corolld Pzolcrc Smar Srucurs, F Elms Aalyss ad Dsg 4, 4, pp [7] Z.K. Kusculuoglu al, F Elm Modl of a Bam wh a Pzocramc Pach Acuaor, Joural of Soud ad Vbrao 76, 4, pp [8] J.R. Hw al, Acv Forc Corol of a Flxbl Mapulaor by Dsal Fdback, Mch. Mach. Thory Vol. 3, No. 5, 997, pp [9] A.R. Tavakolpour al, Modlg ad Smulao of a Novl Acv Vbrao Corol Sysm for a Flxbl Srucurs, WSEAS Trasaco o Sysm ad Corol Issu 5, Vol. 6,, pp [] A.R. Tavakolpour ad M. Malah, Corol of Rsoac Phomo Flxbl Srucurs Va Acv Suppor, Joural of Soud ad Vbrao 33,, pp [] A.K. Muhammad al, Compur Smulaos o Vbrao Corol of a Flxbl Sgl-lk Mapulaor Usg F-lm Mhod, Procdg of 9h Iraoal Symposum of Arfcal Lf ad Robocs, 4, pp [] A.K. Muhammad al, Compur Smulaos ad Exprms o Vbrao Corol of a Flxbl Lk Mapulaor Usg a Pzolcrc Acuaor, Lcur Nos Egrg ad Compur Scc: Procdg of Th Iraoal MulCofrc of Egrs ad Compur Scss 4, IMECS 4, 4 March, 4, Hog Kog, pp [3] A.K. Muhammad al, Comparso of Proporoal-drvav ad Acv-forc Corols o Vbrao of a Flxbl Sgl-lk Mapulaor Usg F-lm Mhod, Joural of Arfcal Lf ad Robocs, Vol. 9. No. 4, 4, pp [4] A.K. Muhammad al, Comparso of Proporoal ad Acv-forc Corols o Vbrao of a Flxbl Lk Mapulaor Usg a (Advac ol publcao: July 5)

Lecture 12: Introduction to nonlinear optics II.

Lecture 12: Introduction to nonlinear optics II. Lcur : Iroduco o olar opcs II r Kužl ropagao of srog opc sgals propr olar ffcs Scod ordr ffcs! Thr-wav mxg has machg codo! Scod harmoc grao! Sum frqucy grao! aramrc grao Thrd ordr ffcs! Four-wav mxg! Opcal