The driven Rayleigh-van der Pol oscillator

Transcription

1 ENOC 7, Jue 5-, 7, Budapest, Hugary The drive Rayleigh-va der Pol oscillator Reé Bartkowiak Faculty of Mechaical Egieerig ad Marie Techology, Uiversity of Rostock, Geray Suary. Sychroizatio of oscillatory systes is a pheoeo actig fro quatu to celestial scale i ature. It is supposed that the cotrolled sychroizatio also ca be used for applicatios of active vibratio absorptio. For this purpose a self-sustaied oscillator ust be developed that has stable haroic solutios eve whe the oscillator is drive by a exteral haroic force. This cotributio describes how the paraeters for a special oliear oscillator are obtaied, depedig o a give haroic drivig force, i such a way that the stable steady-state respose is haroic. The haroic Rayleigh-va der Pol oscillator This cotributio is a preliiary work for the aalysis if the pheoeo of sychroizatio of oscillators ca be used for applicatios of active vibratio absorptio. For the vibratio absorptio of a syste with haroic vibratio behavior it is iportat that the forced vibratio absorber has haroic steady-state respose. The ost kow self-sustaied oscillators have o-haroic periodic steady-state solutios due to exteral haroic forces. I the followig the paraeters for a special oliear oscillator are derived for a give drivig force i such a aer that the sychroized steady-state oscillatio is haroic. The etraiet of a self-sustaied oscillator by a exteral force ca be see as the siplest case of sychroizatio []. Cosidered is a sprig-ass syste ass, stiffess coefficiet c with a cotrolled force eleet F x, ẋ ad exteral excitatio F t = ˆF siωt + α, accordig to Figure. x c F t = Fˆ si Ωt+ α F x, xɺ Figure : Sprig-ass syste with cotrolled force-eleet ad exteral force. The first step is the developet of a uforced oscillator with a asyptotically stable liit cycle. The equatio of otio of the syste reads ẍ + ω x = F x, ẋ + ˆF c siωt + α, ω =. For asyptotically stable periodic otios of the uforced syste, thus ˆF =, a cotrol schee for the force eleet is obtaied usig speed-gradiet ethod []. By defiig the goal fuctio Q = Hx, ẋ H with the total eergy of the syste Hx, ẋ = ẋ + cx ad the desired costat eergy H > the cotrol schee F x, ẋ γ Q F = γhx, ẋ H ẋ = γ ẋ + cx H ẋ with the gai paraeter γ > ca be obtaied. Itroducig the cotrol force ito the equatio of otio with ˆF = ad rearragig yields the differetial equatio of a oscillator, ẍ + γ ẋ + ω x H ẋ + ω x =. 4 By defiig the diesioless tie τ = ωt with the otatio with l = H ω the stadard for of 4 is give by d dτ = ω d dt ad the diesioless displaceet q = x l q + ε q + q q + q =, ε = γh ω. 5 The differetial equatio 5 with the exact haroic steady state solutio qτ = siτ τ [] is a special case of the geeral o-haroic Rayleigh-va der Pol RvdP equatio, see also [4, 5, 6], q + ε µq + νq q + q =, µ + ν >. 6 Figure shows the phase plot of the haroic oscillator 5 for several iitial coditios, where the steady state solutio becoes a circle with radius ˆq =.

2 ENOC 7, Jue 5-, 7, Budapest, Hugary q Figure : Phase plot of the autooous haroic RvdP oscillator fro 5 for several iitial coditios. The drive Rayleigh-va der Pol oscillator Now the forced sprig-ass syste with a exteral haroic excitatio force F t = F siω t + α with Ω = ηω is cosidered, thus the equatio of otio becoes the equatio of the haroic Rayleigh-va der Pol oscillator 5 with a additioal drivig force, q + ε q + q q + q = F siητ + α, F = F. `ω 7 The fudaetal agular frequecy of the drive oscillator ca becoe a ultiple of the drivig agular frequecy, kp Ω, with the whole ubers k, p called sychroizatio of order k/p. Here the sychroizatio of order / is cosidered oly. I this sectio it will be studied uder which coditios the steady-state respose of the RvdP oscillator due to a haroic excitatio force is haroic as well. I a first step the existece ad stability coditios of the sychroizatio are derived by solvig the questio: which excitatio force is eeded to drive the steady-state solutio of the autooous oscillator 5 with give paraeters to a desired steady-state respose. The secod step is to adjust the paraeters of the oscillator for a give excitatio force i such a way that these existece ad stability coditios are fulfilled. Drivig of the oscillator aplitude I the phase plae the steady-state solutio of the autooous oscillator 5 becoes the ier circle i Figure a with radius q =. The desired steady-state respose of the oscillator where oly the aplitude is drive by a still ukow excitatio force is q = ad η =. The desired otio qτ = si τ, correspodig to the outer circle i Figure a, is as well the steady-state solutio of the autooous oscillator, described by q + ε q + q q + q =. 8 Now the questio is what force is eeded to drive the oscillator fro the ier circle to the outer circle i Figure a. To aswer this questio the equatio of the oscillator i 8 is partitioed ito q + ε q + q q + q = ε q. 9 Itroducig the steady-state solutio qτ = si τ ito the right had side of 9 yields the equatio for the haroically drive haroic RvdP oscillator fro 5, q + ε q + q q + q = ε cos τ. Sice the steady-state solutio qτ = si τ fulfills equatio exactly, the existece coditio of the sychroizatio betwee the excitatio force F τ = ε cos τ ad the oscillator is fulfilled. To deterie if the sychroizatio is stable the equatio is haroically liearized [7] at the diesioless agular frequecy η = ito q + δq + q = ε cos τ, with ε δ = π Zπ! cos τ + si τ cos τ cos τ dτ = ε.

3 ENOC 7, Jue 5-, 7, Budapest, Hugary - autooous oscillator activatio of excitatio drive oscillator - drive oscillator autooous oscillator activatio of excitatio q a q b Figure : Phase plots of steady-state solutios. a Drivig of the oscillator aplitude fro qτ = si τ to qτ = si τ b Drivig of the oscillator phase agle fro qτ = si τ to qτ = si ητ. Thus the sychroizatio of order / is stable for δ = ε >, ad thus for the aplitude >. Now the results are used to deterie the aplitude respose of the oscillator 7 due to a give excitatio force F τ = ˆF siτ + α = ˆF cos α si τ + ˆF si α cos τ. The copariso of the give force with the right had side of, ε cos τ = ˆF cos α si τ + ˆF si α cos τ, ˆF = ˆF lω 4 shows that the aplitude of the oscillator ca oly be drive uder the etioed assuptios for the phase agles α / = ± π. 5 By itroducig the two possible solutios 5 ito 4 the coditios betwee the diesioless force aplitude ˆF ad the aplitude of the oscillator ca be deteried ε = ± ˆF. 6 Equatio 6 cotais two cubic equatios for the ukow aplitude with the six solutios / = σ / + σ /, /4,5/6 = σ/ i 6 σ / σ / ± i 6 σ / 7 with ˆF σ / = 4 ε ˆF 7 ± ε. 8 Itroducig 7 ad 6 ito the stability coditio yields the stability coditios ˆF >, i =,, 5, ε i i = 9 ˆF >, i =, 4, 6. i The two equatios i 6 have for each tie istat oe real solutio i case of ˆF > ε 4 7, aely ad 4, ad each tie three real solutios i case of ˆF < ε 4 7. It ca be show that oly solutios for α = π ad 4 for α = π are real solutios of 6 fulfillig the stability coditios 9 for arbitrary values ˆF >.

4 ENOC 7, Jue 5-, 7, Budapest, Hugary Drivig of the oscillator phase agle Now the questio arises what force is eeded to drive oly the phase agle of the autooous oscillator 5 thus the steady-state respose is give by qτ = si ητ, see also Figure b. This steady-state respose is as well the solutio of the autooous oscillator described by Partitioig of ito q + ε q + η q η q + η q =. q + ε q + q q + q = η q q + η q + η q ad itroducig the steady-state solutio qτ = si ητ ito the right had side yields the equatio for the drive haroic RvdP oscillator fro 5, q + ε q + q q + q = η η si ητ cos ητ + η η cos ητ + η si ητ. Obviously the excitatio force i is periodic but ot haroic for η as desired. As a result the haroic RvdP fro 5 drive by a haroic force ˆF siωt + α has o-haroic steady-state respose i geeral, exeplary see i Figure 4a, except of the case Ω = ω, α = ± π. I the followig it is studied if the paraeters of the o-haroic RvdP oscillator fro 6 ca be adjusted for a give haroic excitatio force i such a way that the steady-state respose is haroic as well. The drive o-haroic Rayleigh-va der Pol oscillator I the followig it is studied which force is eeded to drive the o-haroic RvdP oscillator with give paraeters fro 6 to a desired haroic respose, see also Figure 4b. After solvig the existece ad stability coditios the paraeters of the oscillator are adjusted for a give excitatio force F τ = ˆF siητ + α. The desired steady-state respose qτ = si ητ is as well the steady-state solutio of the autooous oscillator described by q + ε q + η q η q + η q =. Partitioig of yields for arbitrary costat values of a, b, q + ε q + η q η + b εη q + η + a q = a q + b η. 4 Itroducig the haroic solutio qτ = si ητ ito the right had side of 4 the ihoogeeous equatio q + ε q + η q η + b q + η + a q = a si ητ + b cos ητ. 5 εη leads to a drive o-haroic oscillator. Reark: The equatio of a geeral RvdP oscillator q + ε q + νq ξ q + χ q = is equivalet to the stadard for 6 due to a coordiate trasforatio. Due to the excitatio force F τ = a si ητ +b cos ητ the sychroized steady-state respose qτ = si ητ is existig, but the stability has to be studied i the followig. To this ed equatio 5 is haroically liearized at the diesioless agular frequecy η ito q + δq + η + a q = a si ητ + b cos ητ, 6 with δ = ε π π η The sychroizatio of order / is stable for {}}{ η cos ητ + η si ητ η + b cos ητ η cos ητ dτ = b εη η. 7 δ = b >. 8 η Now the paraeters of the oscillator are adjustet to a give haroic excitatio force. If the oharoic RvdP oszillator i 5 will be excited by the exteral force F τ = ˆF siητ + α = ˆF cos α si ητ + ˆF si α cos ητ with the arbitrary diesioless force aplitude ˆF ad the arbitrary phase agle α the sychroized haroic steady-state respose exists if the coditio a si ητ + b cos ητ = ˆF cos α si ητ + ˆF si α cos ητ, 9 ad by this a = ˆF cos α, b = ˆF si α

5 ENOC 7, Jue 5-, 7, Budapest, Hugary drive oscillator drive oscillator autooous oscillator activatio of excitatio activatio of excitatio autooous oscillator a q b q Figure 4: Phase plots. a Haroical drive haroic RvdP oscillator with η. b Haroical drive geeral RvdP oscillator with the steady-state solutio qτ = si ητ. is fulfilled. By itroducig the results ito 5 the equatio of the o-haroic oscillator drive by a give excitatio force ca be obtaied ˆF q + ε q + η q η si α ˆF + q + η cos α + q = ˆF siητ + α. εη Reark: The sae results ca be obtaied by itroducig the asatz qτ = si ητ ito q + ε q + νq ξ q + χ q ˆF siητ +α =. Fro the requireet that all coefficiets of the haroic ters are vaishig a uique solutio for the paraeters ν, ξ, χ ca be foud. Back-trasforatio to the origial coordiates The back-trasforatio of to the origial coordiates of syste with q = x l, = yields ẍ + γ ẋ + Ω x η H ˆF si α ω ẋ + Ω + ˆF cos α γω H H ẋ ωl, = ẍ ω l ad l = H ω x = ˆF siωt + α. where the desired eergy H is a free paraeter. The desired eergy ca be defied reasoably by a desired aplitude respose H ˆx = l = ω H = ˆx ω. Itroducig the desired eergy fro ito yields the equatio for the o-haroic oscillator with adjusted paraeters for the haroic excitatio force, ẍ + γ ẋ + Ω x Ω ˆx ˆF si α ẋ + Ω + ˆF cos α x = ˆF siωt + α. 4 Ω ˆxγ ˆx The back-trasforatio of the stability coditio 8 yields b η = ˆF si α >. 5 ˆxΩ Fro the requireet that the agular eigefrequecy of the haroically liearized syste 6 ust be positive for stable solutios, the additioal coditio Ω + ˆF cos α > 6 ˆx ust be satisfied for stability of the origial syste.

6 ENOC 7, Jue 5-, 7, Budapest, Hugary The searched state depedet cotrol force fro the becoes F x, ẋ = γ ẋ + Ω x Ω ˆx ˆF si α ẋ Ω ω + ˆF cos α x. 7 Ω ˆxγ ˆx If syste is excited by the force ˆF siωt+α, ad cotrol law 7 is used for the state depedet force eleet, the the steady-state respose of the syste reads xt = ˆx si Ωt with a desired aplitude ˆx if the stability coditios 5 ad 6 are fulfilled. Additioal uerical siulatios cofired this result. Now the cotrol law 7 ca be used for applicatios of active vibratio absorptio. Coclusios Based o speed gradiet ethod the haroic Rayleigh-va der Pol oscillator is derived fro a sprig-ass syste with a cotrolled force eleet. By usig the haroic steady-state behavior of the uforced oscillator coditios were defied, that are ecessary for the haroic steady-state respose of the forced o-haroic oscillator. Fro these coditios the paraeters for the forced oscillator, depedig o a give excitatio force, ca be derived to esure stable steady-state haroic solutios. Refereces [] Pikovsky A., Roseblu M., ad Kurths J., Sychroizatio A Uiversal Cocept i Noliear Scieces, Cabridge Uiversity Press,. [] Fradkov, A.L., Miroshik, I.V., Nikiforov, V.O., Noliear ad Adaptive Cotrol of Coplex Systes, Spriger-Sciece+Busiess Media, B.V., 999. [] McLachla, N. W. No-Liear Differetial equatios i Egieerig ad Physical Sciece. Oxford, Claredo Press, 956. [4] Kapla, B. Z. ad Yaffe I. 976 A iproved va-der-pol equatio ad soe of its possible applicatios. Iteratioal Joural of Electroics, 4:, [5] Natha, A. 979 The Rayleigh-va der Pol haroic oscillator. Iteratioal Joural of Electroics, 46:6, [6] Shara, T. N. ad Sigh V. 977 Coets o A "iproved" va-der-pol equatio ad soe of its possible applicatios. Iteratioal Joural of Electroics, 4:6, 6. [7] Magus, K. ad Popp K. Schwiguge. Stuttgart, B. G. Teuber, 997.