STABILITY OF THE ACTIVE VIBRATION CONTROL OF CANTILEVER BEAMS

Transcription

1 Iteratioal Coferece o Vibratio Problem September 9-,, Liboa, Portugal STBILITY OF THE CTIVE VIBRTIO COTROL OF CTILEVER BEMS J. Tůma, P. Šuráe, M. Mahdal VSB Techical Uierity of Otraa Czech Republic

2 Outlie. Lumped parameter model of a catileer beam. Modal aalyi 3. Cotroller type for a udamped catileer beam 4. Stable ad potetioaly utable pole of the cotrol ytem 5. Cotroller type for a udamped catileer beam 6. ll-filter 7. Example 8. Cocluio

3 +x Lumped parameter model of a catileer beam catileer beam of the legth L a a cotiuum i diided ito dicrete elemet of the ame legth ΔL that are modelled uig rigid-body dyamic Clampig +y δ y ΔL δ y y - Geeralized coordiate y, y,, y. δ y δ + y + The bedig tiffe K δ relate the applied bedig momet M to the reultig relatie rotatio Δδ of the elemetary beam Lagrage' equatio of motio for a coeratie ytem y - y K δ +z M Cro-ectio b Small agle are aumed i 3EI L x h d T dt y T y Matrix form M y Ky G V y,,,..., ICOVP 3, Portugal, September 9 -, 3 3

4 Ma ad tiffe matrice Parameter B B B B B M 3, 3 4 L h m B L h m L K K.5 mg y y y y y G y Steffe matrix Ma matrix Vector 4 ICOVP 3, Portugal, September 9 -, 3

5 y Modal aalyi Let a beam be parameterized by L =.5 m, b =.4 m, h =.5 m, =. ode Frequecy [Hz] ) x -4 The firt 5 of modal hape of the catileer beam x B) x ICOVP 3, Portugal, September 9 -, 3 5

6 Frequecy trafer fuctio Frequecy trafer fuctio Matrix q H r, q diplacemet of rth elemet force actig at qth elemet H = r H r Excited ibratio, udamped ytem M y Ky F rq Hr, q, r, q,,..., H r j M y Cy Ky F Excited ibratio, damped ytem, icou dampig j r * r j j j Pole r eed elemet y r q actig force f q jω jω -jω Imag jω jω -jω Real Rayleigh dampig α =.59, β =.4 C M K -jω damped -jω udamped ICOVP 3, Portugal, September 9 -, 3 6

7 o-collocated actie ibratio cotrol Localizatio of eor ad actuator ctuator + Seor: - Diplacemet - Velocity - cceleratio Feedbac Seor Cotroller ctuator (ource of force): - Electrodyamic haer - Piezoactuator Reduced arragemet of the VC ytem to a pair of a o-collocated eor ad actuator Set poit y SP y SP y SP + - Cotroller f R Seor Beam H, y = diplacemet = elocity = acceleratio ICOVP 3, Portugal, September 9 -, 3 7

8 Cotroller type for a udamped catileer beam Udamped catileer beam trafer fuctio rq Hr, q, r, q,,..., Pole lie o the imagiary axi (tability margi) SP + - Cotroller f q R Beam H r, q y r Cloed-loop trafer fuctio ~ H r, SP Y Y r SP H R R r, q H r, q R R Structurally table ytem Polyomial with odd power of r q TD, R R Suitable type of cotroller Seor T R T R r r q q r q R r q T D, r q R Polyomial of T - diplacemet (=) - elocity (=). - acceleratio (=).., TD, Cotroller R R R T T T Polyomial of?,,..., ICOVP 3, Portugal, September 9 -, 3 8

9 Stable ad potetioaly utable pole of the cotrol ytem Suppoe that the frequecy pectrum of diturbace affect oly the d-th mode of ibratio drqd Kd Hr, q, r, q,,..., d d K d i a gai factor aociated with the d-th mode of ibratio The trafer fuctio of the cloe loop with the cotroller of the D type Kd T ~ Yr d KdT Hr, SP YSP Kd T KdT d d The dampig ratio ξ ad the decay cotat σ Pole of a real ytem VC O VC OFF Imag jω jω jω -jω -jω Real Gai factor K, T dt Kd d K d d dr qd a a fuctio of table -jω utable K d ICOVP 3, Portugal, September 9 -, 3 9

10 imag imag imag imag Root locu of the catileer beam VC for = 5 Cloed loop trafer fuctio ~ H r, SP Y Y TH T r SP r, q TH r, TH Trafer fuctio pole r q,,, q r, q * 5 Root locu a a fuctio of T T,,, T= 977 T= 53 table table utable real T= T= table table real real T= 775 table real ICOVP 3, Portugal, September 9 -, 3

11 y [m] Simuli model of the catileer beam for = 5 Excited ibratio, damped ytem, icou dampig M y Cy Ky F F I Out Subytem Cotroller Gai P Gai D Cotat5 F Maual Switch Ma Switch F F Py5 Ty 5 Cotat4 Cotat3 Cotat Cotat F F3 F4 F5 F F3 F4 F5 F ctuator y Feedbac PF F K*uec i(m)*k K*uec i(m)*c K*uec i(m) y y3 y4 PF dd y5 Seor Cotroller Itegrator Itegrator Itegrator4 Itegrator6 Itegrator8 dy dy dy3 dy4 dy5 x y dy dy dy3 dy4 dy5 Iitial coditio poitio [m] Itegrator Itegrator3 Itegrator5 Itegrator7 Itegrator9 y y y3 y4 y5 F = -9.8 y y y y3 y4 y5 y Scope ICOVP 3, Portugal, September 9 -, 3

12 Uwraped phae [rad] b d PF PF The ue of the all-pa filter of the ecod order Trafer fuctio GPF, PF PF Magitude of the P filter Phae of the P filter - G j PF, llpa filter frequecy repoe i =.77 i =. Gai ctuator F y y y3 y4 y5 Seor Feedbac Cotroller PF PF Relatie frequecy [-] ICOVP 3, Portugal, September 9 -, 3

13 ll-pa filter Digital P filter Parameter bx cx ax cy by ta f f S c d c b d y d a PF PF PF PF a x() iput Σ y(-) z - z - alog P filter x(-) Σ y(-) C output z - b/a b/a z - iput R C R x(-) Σ c/a c/a Σ - y() output R3 R4 ICOVP 3, Portugal, September 9 -, 3 3

14 y [m] y [m] y [m] Effect of VC o decay of ibratio.5 x VC OFF VC O VC O -3.5 x P filter OFF -3.5 x P filter O T = T = 5 T = Time [] Time [] Time [] ICOVP 3, Portugal, September 9 -, 3 4

15 Mater thei Pael Šuráe: ctie ibratio cotrol ICOVP 3, Portugal, September 9 -, 3 5

16 Cocluio The lumped-parameter model of the catileer beam wa deiged uig the method baed o the modal aalyi. The feedbac of the D type i ufficiet for dampig of the lightly damped ytem. The all pa filter of the ecod order ole the problem with poitie feedbac for ome mode of ibratio ad icreae the efficiecy of dampig. ICOVP 3, Portugal, September 9 -, 3 6