Delay equations with engineering applications Gábor Stépán Department of Applied Mechanics Budapest University of Technology and Economics

Size: px

Start display at page:

Download "Delay equations with engineering applications Gábor Stépán Department of Applied Mechanics Budapest University of Technology and Economics"

Maria Garrison
5 years ago
Views:

1 Delay equatons wth engneerng applcatons Gábor Stépán Department of Appled Mechancs Budapest Unversty of Technology and Economcs Contents Delay equatons arse n mechancal systems by the nformaton system (of control, and by the contact of bodes. - Lnear stablty & subcrtcal Hopf bfurcatons - Force control and balancng human and robotc - Contact problems Shmmyng wheels (of trucs and motorcycles Machne tool vbratons Man references Insperger T, Stepan G, Stablty chart for the delayed Matheu equaton, Proceedngs of the Royal Socety London A 458 ( Insperger T, Stepan G, Sem-dscretzaton for tme-delay systems Engneerng applcatons, to appear, Sprnger, New Yor, 11. Insperger T, Stepan G, Updated sem-dscretzaton method for perodc delay-dfferentlal equatons wth dscrete delay, Internatonal Journal for Numercal Methods n Engneerng 61 ( Stepan G, Insperger T, Szala R, Delay, parametrc exctaton, and the nonlnear dynamcs of cuttng processes, Internatonal Journal of Bfurcaton and Chaos 15 ( Orosz G, Stepan G, Subcrtcal Hopf bfurcatons n a car-followng model wth reacton-tme delay, Proceedngs of the Royal Socety London A 46 ( Non-autonomous lnear RFDEs M&& + d B( ( t + + d K( t + ϑ h Tme-perodc systems: Tral soluton: = p( e + = + = = π π p( t T p( ( A cos( + sn( T t B T t Hll s nfnte dmensonal determnant characterstc functon characterstc roots λ Re λ j <, j=1,, stablty µ j <1, j=1,, λt for characterstc multplers ϑ h B( t + T, = B( K( t + T, = K( λt µ = e of fund. op. at T The delayed Matheu equaton Analytcally constructed stablty chart for testng numercal methods and algorthms & ( + ( δ + ε cos t = b Tme delay and tme perodcty are equal: T = τ = π b Matheu equaton (1868 ε Delayed oscllator (1941 Stablty chart Matheu equaton && + ( δ + ε cos Floquet (1883 Hll (1886 Raylegh(1887 van der Pol & Strutt (198 Swng (BC Strutt Ince dagram (1956 Stephenson (198, Swnney (4, Zele (5 1

2 Stablty chart delayed oscllator & ( + δ = b Pontryagn (194 Nyqust (1949 Bellman & Cooe (1963 n engenerng: Schlppe (1941 Mnorsy (194 Hsu & Bhatt (1966 = = The delayed Matheu equaton & ( + ( δ + ε cos t = b t t λ t t t ( A e + B e e + ( A e + B e = = λ t e ( λ+ t ( λ t = C e + C e Harmonc balance Hll s determnant O O O K ε π ( λ+ ε det K δ + ( λ + be K K O O O Stablty boundares λ = ± ω δ ( ω + b cos(π ω ± b sn(π ω In the man dagonal, there are 4 cases to chec f : 1 ω j /, b then the man dagonal s complex, the Hll s determnant s non-zero b then there are possble stablty boundares 3 b, ω = j / = h δ b ( + j / then D(ω, δ, b, ε = f ( δ b, ε and = e hπ µ =1 (SN 4 b, ω = j / = (h + 1 / δ + b ( + j / (h+ 1 π D (ω, δ, b, ε = f ( δ + b, ε and µ = e = 1(flp Stablty boundares reman lnes n (δ,b Re λ To select stable ones: = b π Γ N j ( π b( 1 Γ N + Ω N b The delayed Matheu stablty charts & ( + ( δ + ε cos t = b Stablty chart of delayed Matheu & ( + ( δ + ε cos t = b b= (Strutt-Incze, 198 ε= (Hsu-Bhat 1966 ε=1 Insperger, Stépán (

3 Sem-dscretzaton method ntroducton & ( + c = c1 τ = π The approxmatng DDE s non-autonomous & x ( + c = c1 (, τ ( = t + ( m nt( t / t t t, t = [ t,( + 1 t t = π /( m + 1/ [ + 1 ( ( m = x m x Introducton to SDM delayed oscllator &&( x + c = c1x m x ( t = x x &( t = = K1 cos( c + K sn( c + c1 x m / c ( = K1 c sn( c + K c cos( c + 1 = a x + a1 + am x m y = x x K x + 1 = a1 x + a11 + a1 m x m y +1 = Ay col( 1 m det( µ I A µ 1,,..., m + < 1 stablty Delayed oscllator stablty chart by SDM & x + c = c (, τ( = t + ( m nt( t / t ( 1 t Full dscretzaton - comparson Dscretzaton also w.r.t. tme dervatves slow convergence Introducton to SDM Matheu equaton & ( + c ( c ( = δ + ε cos t t [ t, t+ 1 && + c x ( t = x x &( t =,1,..., 1 ( c ( t t + sn( c ( t t = x cos c y SDM for Matheu equaton sn t ( ( t c cos t c = A = c ( t c sn( t c cos ( t c π = y (π = A 1 A det( µ I Φ(π µ 1, 1 stablty KA y for = ntervals Messner (1918 for van der Pol, Strut Ince (198 = Φ(π ( 3

4 SDM for Matheu equaton & ( + c ( Messner (1918 c ( δ + ε = δ ε f t < π f π t < π Sem-dscretzaton general case x &( d η( t +, η( t + T, = η( = τ ϑ ε >, M ( ε, m > M ( ε mn U j 1, = = µ ε µ j S ~, j 1, K, mn j µ j < ε, j = mn +1,K Insperger, Stepan: Int. J. of Numercal Methods n Engneerng ( Test on (damped delayed Matheu && x ( + κ ( + ( δ + ε cost = b Examples test on delayed Matheu & ( + ( cδ + cε cos(4πt = c1 t + dϑ 1 π & ( + ( cδ + cε cos(4πt = c1 sn( π t + dϑ 1 & ( + (6 + c cos(πt = 1 + ε 4

Nonlnear RFDEs n Engneerng Unstable lmt cycle ghost vbraton Stablty analyss of steady-states

Supercrtcal case: easy to avod vbratons by nowng the lnear stablty behavor Subcrtcal case:

behavor cannot be predcted by lnear stablty analyss.

5 Nonlnear RFDEs n Engneerng Unstable lmt cycle ghost vbraton Stablty analyss of steady-states s followed by Center Manfold reducton & bfurcaton analyss Hopf bfurcaton self-excted vbratons Supercrtcal case: easy to avod vbratons by nowng the lnear stablty behavor Subcrtcal case: the unstable perodc solutons mean a lmted doman of attracton for the desred steady-state behavor cannot be predcted by lnear stablty analyss. Stc&slp Stc&slp expermental bfurcaton dagram Stc&slp expermental bfurcaton dagram Center Manfold reducton q x 3 s 3 Im s 1 Re s 1 x. ψ ψ x 1 5

6 Shmmy unstable lmt cycle Shmmy quas-perodc oscllatons Machne tool vbratons Machne tool vbraton vdeo The unstable perodc moton Balancng Sh, Tobas (1984 mpact experment 6

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence