AMPLITUDE CONTROL OF A SELF-VIBRATION MACHINE

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1 ENOC 28, San Peersburg, Russa, June, 3 July, 4 28 AMPLITUDE CONTROL OF A SELF-VIBRATION MACHINE Yuusuke Uchyama Grauae School of Sysems an Informaon Engneerng Unversy of Tsukuba Japan uchyama@aosuna.esys.sukuba.ac.jp Hrosh Yabuno Deparmen of Mechanacal Engneerng Keo Unversy Japan ybuno@keo.mech.ac.jp Absrac In hs paper, a sprng-mass-amper sysem wh an elecrcal crcu (3r-orer sysem) s nvesgae o esablsh a self-vbraon machne. The esgn of feeback gan makes he sysem a self-exce oscllaor. The mechancal par (sprng-mass-amper sysem) n he 3r-orer sysem becomes self-vbraon machne whch s robus agans varaon of he mass. x m K A - kx -c x Key wors Self-vbraon machne, Conrol of oscllaon, Cener manfol, Normal form. Inroucon Self-vbraon machne [], [2] s a machne vbrang wh he naural frequency whch s vare epenng on he varaon of he sysem parameers of mass an sffness. Because he energy effcency of he self-exce machne s kep nepenen of he varaon, here are many applcaons for realzng hghperformance machnes, for example, ulrasonc ransucer[3] an vbraory rllng[4]. Also, he characersc ha he resonance frequency s race o he vare naural frequency epenng on he varaon of he parameers s applcable o he measuremen of he naural frequency an such a characersc s recenly ulze o make AFM (aomc force mcroscope) much hgher resoluon []. In hs paper, we conser a sprng-mass-amper sysem whch s acuae by a lnear moor an esablsh a self-vbraon machne uner feeback conrol. A meho of he amplue conrol s also scusse by nroucng he cener manfol heory[6]. 2 Realzaon of self-vbraon machne 2. Analycal moel an equaon of moon We conser an analycal moel of sprng-massamper sysem as shown n Fg. an nvesgae he meho o make he sysem a self-vbraon machne Fgure. Analycal moel. an o conrol he magnue of he amplue. The hrus force of a lnear moor s proporonal o he curren. The equaon of moon of he sprng-massamper sysem s expresse as follows: m 2 x 2 + cx + kx = K, () where K s he hrus consan of he lnear moor. To realze a self-vbraon machne, we acuae he npu volage of he lnear moor. The ynamcs of he lnear moor s escrbe by an equvalen L-R crcu. We npu he volage o he crcu whch s proporonal o he velocy of he mass. Uner hs feeback npu, he crcu equaon s expresse as follows: L x + R + P = V, (2) where, L R an P are reacance, ressance, an back elecromove force consan of he lnear moor. The npu volage s se as V = K l x, (3) where K l s he lnear feeback gan. Inroucng represenave lengh values of X = KI/k I (rang

2 curren of he lnear moor) T = m/k an usng mensonless splacemen x = x/x mensonless curren = /I an mensonless me = /T yel he followng equaons of he sprng-massamper sysem an he crcu for Eqs. () an(2): 2 x x + γ 2 + x =, (4) + R + P x = V, () Imagnary par.. -. where he mensonless parameers are γ = c/ mk R = RT/L P = KP/kL. The mensonless npu volage s where K l = K l K/LI. V = K l x, (6) 2.2 Analyss of equaon of moon an equaon of crcu Inroucng he sae varables of x = x = x / = yels he sae equaon: or x = x = Ax, x = γ K l P R x x (7),A= γ K l P R (8) In he above equaons an hereafer, he symbol o enoe mensonless value s roppe for smplcy. We clarfy ha he mechancal par of he sprngmass-amper sysem n he above 3r orer sysem nclung he ynamcs of he crcu can accomplsh he behavor of he self-vbraon machne. In such a sysem, he 3 egenvalues of he 3r-orer sysem have o be one negave real an a par of complex wh posve real par. The real egenvalue has o be negave for rf-free oscllaon of he mechancal par. Fgure 2 shows he roo locus corresponng o he parameer values of an expermenal sysem. Uner he approprae choce of he feeback gan K l, he above egenvalues for he self-vbraon machne are obane. Fgure 3 shows me hsores of he splacemen of he mechancal sysem an he curren n he lnear moor. I can be seen ha he rf-free self-exce oscllaon occurs an he amplue grows wh me. 3 Amplue conrol of self-vbraon machne In he preceeng secon, a self-vbraon machne s realze, bu he amplue grows wh me. In hs secon, we propose a nonlnear feeback conrol meho Real par Fgure 2. Roo locus of A wh γ=.272, R=3.4, an P =.36 uner changng K l from o (: K l =2, : K l =). o keep he resonance amplue consan. We esgn he approprae nonlnear feeback gan on he cener manfol. The esgn s much easer han ha for he orgnal sysem because he sysem orer s reuce o secon orer. 3. Feeback conrol o make cener subspace We seek he lnear feeback conrol so ha he egensubspace of he ynamcs of he 3r-orer sysem consss of he cener an sable subspace. Then, he lnear operaor A has a par of pure magnary egenvalues an a negave real egenvalue, an by x = Qy he lnear operaor A can be ransforme no ω Q AQ = ω. (9) η where ω an η are posve. The characersc equaon of A s Φ A (λ) = e(λi A) s expresse as follows: Φ A (λ) =λ 3 +(γ + R)λ 2 whch s equal ha of Eq. (9): + {γr K l + P +}λ + R, () Φ A (λ) =λ 3 + ηλ 2 + ω 2 λ + ηω 2. () From Eqs. () an () he followng equaons for he above reques for he egenvalues: η = γ + R>, (2)

3 x x -3 x (a) Dmensonless splacemen 2 2 (b) Dmensonless curren Fgure 3. Tme hsores of mensonless splacemen an curren wh γ=.272, R=3.4, an P =.36, uner K l =.. ω 2 = γ(r K l )+P +>, (3) K l = γr + P + R R + γ, (4) Equaon (2) can always be sasfe. We nee o choose he feeback gan accorng o Eq. (4) uner he conon Eq. (3) an he feeback gan s expresse K cr. 3.2 Desgn of nonlnear feeback gan on cener manfol To perform he amplue conrol, we apply nonlnear feeback conrol an esgn he nonlnear feeback gan on he cener manfol. We se he mensonless npu volage as follows: Then, he sae equaon s expresse as follows: x = γ x K cr P R +(K cr ɛ + µ 2) (6) The ransformaon by Q yels y y 2 = ω ω y y 2 y 3 η y 3 +{K cr (q 2 y + q 22 y 2 + q 23 y 3 ) +µ(q 2 y + q 22 y 2 + q 23 y 3 ) 3 } q 3 q 23 q 33 (7) where q j an q j are j componens for Q an Q, respecvely. By suspenson-rck we oban he cener manfol as follows: [6] y 3 = χ 3 y 3 + χ 2 yy χ 2 y y2 2 + χ 3 y2 3 + χ ɛy + χ ɛy 2. (8) The ynamcs reuce on he cener manfol s governe wh [ ] [ ][ ] y K = cr q 3 q 3 ɛ ω + K cr q 3 q 32 ɛ y y 2 ω + K cr q 23 q 3 ɛ K cr q 23 q 32 ɛ y 2 [ ] α y +µ 3 + α 2 yy α 3 y y2 2 + α 4 y2 3 α y 3 + α 6 yy α 7 y y2 2 + α 8 y2 3,(9) where α = q 3 q2,α 3 2 =3 q 3 q2q 2 22,α 3 =3 q 3 q 2 q22, 2 α 4 = q 23 q2,α 3 = q 23 q2,α 3 6 =3 q 23 q2q 2 22, α 7 =3 q 23 q 2 q22,α 2 8 = q 23 q22 3 (2) Furhermore, he normal form [7] of (9) uner he nonlnear coornae ransformaon: y = z + h(z) (2) s [ ] [ ][ ] z K = cr q 3 q 3 ɛ ω + K cr q 3 q 32 ɛ z z 2 ω + K cr q 23 q 3 ɛ K cr q 23 q 32 ɛ z 2 [ ] (az + bz +µ 2 )(z 2 + z2) 2 (az 2 bz )(z 2 + z2) 2, (22) where ɛ<< V = K cr ( + ɛ) x ( ) 3 x + µ, () where a = 8 (3α + α 3 + α 6 +3α 8 ), b = 8 (α 2 +3α 4 3α α 7 ) (23)

4 x -6 z 2 z x x = = z 2 z x -6 Fgure 4. Phase space wh γ=.272, R=3.4 P =.36, K l =.387 an ɛ=.. : µ=, : µ=. Fgure 4 shows he phase of wo kns of nonlnear feeback gan. From comparson beween hem, by changng he nonlnear feeback gan, we carry ou he amplue conrol. Fgure are corresponng me hsores. The amplues of he splacemen of he mass an he curren n he lnear moor are consan n he seay sae. Therefore, can be seen ha he seng of he nonlnear feeback gan realzes esre response amplue of he self-vbraon machne. 4 Expermenal apparaus an proceure Fgure6 shows he expermenal apparaus. The prmary col of he lnear moor s fxe o he base. The seconary permanen magne of he lnear moor (Syowa-Densen-Denran Corporaon; VCM26-2R; maxmum hrus force of 8 N an hrus consan of 2 N/A) s suppore by a sle bearng (IKO Corporaon, BSU66-A) an s relave poson can be move wh respec o he base. The mass of he seconary permanen magne correspons o he mass of he sysem n Fg.. To prove he lnear resorng force, we aach a sprng (Samn Corporaon) o lnear-moor. The splacemen s measure by usng a laser sensor (KEYENCE Corporaon, LB-/LB-6) an he splacemen sgnal s npu o a PC hrough an AD-boar (Inerface Corporaon, PCI-323A), n whch he conrol volage V s calculae n real me. The sgnal corresponng o V of Eq. s fe from he PC o he power amplfer (KIKUSUI Corporaon, PBX4- ) hrough a DA-boar (Inerface Corporaon, PCI- 323A). The amplfe sgnal s npu o he lnear moor. For he amplue conrol, he applcaon of cubc velocy feeback s propose an he feeback gan s esgn on he cener manfol. References Babsky, V. I. (99). Auoresonan Mecharonc Sysems. Mechroncs,, pp Kura, Y. an Muraguch, Y. (996). Self-Exce Drvng of Resonance-Type Vbraon Machne by Vbraon Quany Feeback. Trans. JSME., 62, pp Babsky, V., Asashev, V. K., an Kalashnkov, A. N. (24). Auoresonan Conrol of Nonlnear Moe n Ulrasonc Transucer for Machnng Applcaons. Ulrasonc, 42, pp Pegne, G., Kanmev, E., Brssaou, D., an Gouskov, A. (2). Self-Exce Vbraon Drllng. Proc. of he Insuon of Mechancal Engneers Par B: J. Eng. Manuf., 29, pp Okajma, T., Sekguch, H., Arakawa, H. an Ika, A. (23), Self-Oscllaon Technque for AFM n Lqus. Apple Surface Scence, 2, pp Carr, J. (98). Applcaon of Cener Manfol Theory. Sprnger. New York. Nayfeh, A. H. (993). Meho of Normal Forms. Wley-Inerscences. New York. Concluson A meho s presene o make he mass-sprngamper sysem a self-vbraon sysem. The sysem s regare as he 3r-orer sysem. By egenvalue analyss, s shown ha he mass-sprng-amper sysem s self-exce an a self-vbraon sysem s esablshe.

5 x x (a) Dmensonless splacemen x x x x (b) Dmensonless curren Fgure. Tme hsores of mensonless splacemen an curren wh γ=.272, R=3.4 P =.36, K l =.387 an ɛ=.. : µ=, : µ=. Sensor Amplfer Seconary Permanen Magne Dsplacemen Sensor DC/CC Power Supply Prmary Col Lnear Moor FFT Analyzer Sle Bearng Base PC A/D D/A Fgure 6. Expermenal apparaus.