Geometrical Theory of Nonlinear Modal Analysis Hamid A. Ardeh - PowerPoint PPT Presentation

Geometrical Theory of Nonlinear Modal Analysis Hamid A. Ardeh Department of Mechanical Engineering University of Wisconsin-Madison University of Wisconsin - Madison 1

Acknowledgement (1/1) Acknowledgements • Advi visor • Prof. Matthew S. Allen • Co Committee mem embers • Prof. Dan Negrut • Prof. Daniel C. Kammer • Prof. Melih Eriten • Prof. Gaetan Kerschen • Fundin ing • National Science Foundation (under Grant no. CMMI-0969224) • Air Force Office of Scientific Research (award # FA9550-11-1-0035) • Wisconsin Alumni Research Foundation University of Wisconsin - Madison 2

Acknowledgement Overview (1/1) Overview • Mot • Con otivati tion and back ckgrounds Connecting fu funct ctions • Definition of connecting functions • Non Nonlin inea ear mod odes es of of vib vibrations • Bi-directionally linear connecting functions • • Calculation of connecting functions Three definitions • • Instantaneous Center Manifold (ICM) Linear approximation of connecting functions • Con • Ca Conclu lusions Calc lcula lation of of non onli linear modes • Solving for ICM analytically • Averaging and collocation methods • Averaging ⊕ collocation (MMC) • Stabil ilit ity and bifu ifurcation of of non onli linear modes • Floquet theory • Validation of stability analysis University of Wisconsin - Madison 3

Acknowledgement Overview Motivations (1/2) Motivations and Backgrounds His History • “ In his work on dynamics, Poincare was led to focus attention primarily upon the periodic motions. • He conjectured that any motion of a dynamical system might be approximated by means of those of periodic type, i.e. that the periodic motions to be densely distributed among all possible motions; and it became a task of the first order of importance for him to determine what the actual distribution of the periodic motions was, so as to prove or disprove his conjecture.” [1 ] This conjecture was proved for linear systems by Hilbert (known as spectral theory) and is the • foundation of every technique/method used in modal analysis. Prim rimary ob obje jecti tives es • The primary objective of this work is to provide new insights on how to calculate all periodic • solutions of a class of nonlinear systems efficiently and then use them to arbitrarily accurately approximate any solution of such systems. [1] Birkhoff, George D, "On the periodic motions of dynamical systems", Acta Mathematica 50, 1 (1927), pp. 359--379. University of Wisconsin - Madison 4

Acknowledgement Overview Motivations (2/2) Future Applications Pred edicti ting g th the e lif life e cy cycl cle and gu guid idin ing g des esign ign ch changes es • Engineers prefer to design systems to be linear, many systems are just • intrinsically nonlinear or the linear designs may be suboptimal with respect to the intended purpose. By altering the design the life can change by orders of magnitude. • Pred edicti ting g th the e beh ehavi vior of of non onli linear dynamical l systems • Accurate calculation of periodic solutions and their bifurcations are • required for determining the path of (long-period) comets. University of Wisconsin - Madison 5

Nonlinear modes (1/4) Acknowledgement Overview Motivations Nonlinear modes are periodic solutions. Ros osen enberg defin fined ed a non onli linear mode as a on one-dimen ensional • fu funct ctional l rela elation betw tween coo oordinates es of of a peri eriodic ic olution 𝛉(𝑦 1 ) , solu , i.e i.e. .  Any solution: 𝛉 𝑦 1 (𝑢) = 𝛉 𝑦 1 (𝑢 + 𝑈)  Synchronous: 𝛉 𝑦 1 (0) = 0  Orthogonal to equipotential curves [2]. Vakakis mod odif ified ed Rosenberg’s definition to any peri eriodic ic • solu olution 𝒚(𝑢) i.e i.e.  Any solution: 𝒚(t) = 𝒚 𝑢 + 𝑈 [3]. [2] R.M. Rosenberg. On normal vibrations of a general class of nonlinear dual-mode systems. Journal of Applied Mechanics, 29:714, 1962. [3] A. F. Vakakis. Analysis and identification of linear and nonlinear normal modes in vibrating systems. PhD thesis, California Institute of Technology, 1990. University of Wisconsin - Madison 6

Nonlinear modes (2/4) Acknowledgement Overview Motivations Nonlinear modes are two-dimensional functional relations. Shaw and Pier ierre defin fined a non onlin inear mod ode e as a tw two-dimen ensional tim time in indep epen enden ent t • fu funct ctional l rela elation th that t satis tisfie ies th the e governin ing eq equations of of th the e system i.e. .e. 𝚫 𝑦 1 , 𝑦 1 that 2 2 + ⋯ • is is in invaria iant (tim (time in indep ependent), Γ i = 𝑏 𝑗1 𝑦 1 + 𝑏 𝑗2 𝑦 1 + 𝑏 𝑗3 𝑦 1 𝑦 1 + 𝑏 𝑗4 𝑦 1 𝑦 1 i.e . d 2 𝚫 dt 2 = 𝒈(𝚫, d𝚫 • satis tisfie ies th the e governing g eq equati tions of of mot otion, i.e dt ) [5]. • They ar are tangent to to the vector fi field ld at t its ts fi fixed poi point. • When 𝚫 can be a manifold? • Is 𝚫 invariant? Why is 𝚫 𝑦 1 , 𝑦 1 tangent to the vector field? • Why only fixed points? [4] S.W. Shaw and C. Pierre. Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration, 150(1):170173, 1991. University of Wisconsin - Madison 7

Nonlinear modes (3/4) Acknowledgement Overview Motivations This work presents a new definition for invariance leads to a unified definition for invariant manifolds of both fixed points and periodic solutions. 𝚫 is is an in invariant manif ifold ld under 𝒈 if if and on only ly if if 𝒈 is is alw lways in in th the e tangent • of 𝚫 . bundle le of . We proved th that t a manifold is is in invariant under th the system • , , if if and on only ly if if 𝚫 ’s are especially interesting when calculated around equilibrium , i.e. i.e. fix fixed ed • poin oint and peri eriodic ic solu olutions, of of 𝒈 . . University of Wisconsin - Madison 8

Nonlinear modes (4/4) Acknowledgement Overview Motivations Local invariant manifolds of a nonlinear system can be obtained without an explicit localization of the system. iant manifolds 𝚫 of of 𝒈 can be Ther erefore e all ll in invaria e ob obtained by solv olving th the e same set t of f • PDE’s, weather they are defined around a fixed point z 1 , 𝑨 1 𝑦 1 , 𝑦 1 𝑦 1 , 𝑦 1 or an or an (u (unknown) peri eriodic solu olution • Add dditio ional qu questions tha that ar are no not an answered 𝑦 1 , 𝑦 1 her: Feel her eel fr free ee to o as ask me: e: • Why y is cal alled ICM CM? z 1 , 𝑨 1 • Wha hat t is the the rela elati tionship be between the the cen center r man anifold of of a a system (if f it t exi xists) ) an and its ts ICM CMs? 𝑦 1 , 𝑦 1 • Whe hen do do ICM CMs bec become globally inse nseparable manifolds? • Do we nee Do need hi higher r di dimensional ICM CMs? Do Do the they exis xist? t? University of Wisconsin - Madison 9

Nonlinear modes Acknowledgement Overview Motivations Calculation of NL-modes (1/9) An analytical method was presented to solve the governing PDE’s of each ICM. A com ombin ination of of an averagin ing meth thod (h (harmonic bala lance) and a non onli linea ear (a (alg lgebraic) elim elimination tech echniq ique • was used ed. This is way by id iden enti tify fying (on (only ly) th the e in indep epen enden ent t coo oordin inates, i.e. i.e. • one als on lso id identi tifie ies th the in invaria iant manifold ld (fu (functional rel elation). This is method is is not ot sc scala lable le! University of Wisconsin - Madison 10

Nonlinear modes Acknowledgement Overview Motivations Calculation of NL-modes (2/9) Two classes of methods currently exist. Can a new method, that combines the befits of both averaging and collocation methods without any of their drawbacks, be developed? Averaging meth thods tr try to o make a parametric per eriodic fu function satis tisfy fy th the e governin ing eq equations of of th the e system. • One example: Ha On Harmonic ic ba bala lance. • We e ha have to to integrate the system an analy lytically ly! • They ar are no not t sc scala lable le. • Col Collocation meth thods in integ egrate th the e system numericall lly to o ch chec eck th the e peri eriodic icity of of th the e solu olution. • They ar are sc scalable: we e can integrate nu numericall lly. • They ar are com omputationally exp xpensive: we e of often ha have to to In Integrate the system over an and over. • 𝑈 y 0 = y 𝑢 0 = 𝑧 (𝑢 0 + 𝑜 ) They ar are sen sensit itive to to the init itial conditions. s. • 𝑢 = 𝑢 0 + 𝑈 𝑢 = 𝑢 0 ∆𝑢 = 𝑈 University of Wisconsin - Madison 11