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

SLIDE 1

Geometrical Theory of Nonlinear Modal Analysis

Hamid A. Ardeh

Department of Mechanical Engineering University of Wisconsin-Madison

University of Wisconsin - Madison 1

SLIDE 2

Acknowledgements

University of Wisconsin - Madison 2 Acknowledgement (1/1)

• 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

SLIDE 3

Overview

University of Wisconsin - Madison 3 Acknowledgement

• Mot
• tivati

tion and back ckgrounds

• Non

Nonlin inea ear mod

• des

es of

• f vib

vibrations

• Three definitions
• Instantaneous Center Manifold (ICM)
• Ca

Calc lcula lation of

• f non
• nli

linear modes

• Solving for ICM analytically
• Averaging and collocation methods
• Averaging ⊕ collocation (MMC)
• Stabil

ilit ity and bifu ifurcation of

• f non
• nli

linear modes

• Floquet theory
• Validation of stability analysis
• Con

Connecting fu funct ctions

• Definition of connecting functions
• Bi-directionally linear connecting functions
• Calculation of connecting functions
• Linear approximation of connecting functions
• Con

Conclu lusions

Overview (1/1)

SLIDE 4

Motivations and Backgrounds

University of Wisconsin - Madison 4 Acknowledgement Overview

• 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

• f 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

• bje

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.

Motivations (1/2)

SLIDE 5

Future Applications

University of Wisconsin - Madison 5 Acknowledgement Overview

• 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

• f non
• nli

linear dynamical l systems

• Accurate calculation of periodic solutions and their bifurcations are

required for determining the path of (long-period) comets.

Motivations (2/2)

SLIDE 6

Nonlinear modes are periodic solutions.

University of Wisconsin - Madison 6 Acknowledgement Overview

• Ros
• sen

enberg defin fined ed a non

• nli

linear mode as a on

• ne-dimen

ensional fu funct ctional l rela elation betw tween coo

• ordinates

es of

• f a peri

eriodic ic solu

• lution 𝛉(𝑦1),

, i.e i.e. .

 Any solution: 𝛉 𝑦1(𝑢) = 𝛉 𝑦1(𝑢 + 𝑈)  Synchronous: 𝛉 𝑦1(0) = 0  Orthogonal to equipotential curves [2].

• Vakakis mod
• dif

ified ed Rosenberg’s definition to any peri eriodic ic solu

• lution 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.

Motivations Nonlinear modes (1/4)

SLIDE 7

Nonlinear modes are two-dimensional functional relations.

University of Wisconsin - Madison 7 Acknowledgement Overview

• Shaw and Pier

ierre defin fined a non

• nlin

inear mod

• de

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

• f th

the e system i.e.

.e.

𝚫 𝑦1, 𝑦1 that

• is

is in invaria iant (tim (time in indep ependent), Γi = 𝑏𝑗1𝑦1 + 𝑏𝑗2

𝑦1 + 𝑏𝑗3𝑦1

2

𝑦1 + 𝑏𝑗4𝑦1 𝑦1

2 + ⋯

• satis

tisfie ies th the e governing g eq equati tions of

• f mot
• tion, i.e

i.e. d2𝚫

dt2 = 𝒈(𝚫, d𝚫 dt) [5].

• They ar

are tangent to to the vector fi field ld at t its ts fi fixed poi point.

[4] S.W. Shaw and C. Pierre. Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration, 150(1):170173, 1991.

Motivations Nonlinear modes (2/4)

• When 𝚫 can be a manifold?
• Is 𝚫 invariant? Why is 𝚫 𝑦1,

𝑦1 tangent to the vector field?

• Why only fixed points?

SLIDE 8

This work presents a new definition for invariance leads to a unified definition for invariant manifolds of both fixed points and periodic solutions.

University of Wisconsin - Madison 8 Acknowledgement Overview

• 𝚫 is

is an in invariant manif ifold ld under 𝒈 if if and on

• nly

ly if if 𝒈 is is alw lways in in th the e tangent bundle le of

• f 𝚫.

.

• We proved th

that t a manifold is is in invariant under th the system , , if if and on

• nly

ly if if

• 𝚫’s are especially interesting when calculated around equilibrium, i.e.

i.e. fix fixed ed poin

• int and peri

eriodic ic solu

• lutions, of
• f 𝒈 .

.

Motivations Nonlinear modes (3/4)

SLIDE 9

Local invariant manifolds of a nonlinear system can be obtained without an explicit localization of the system.

University of Wisconsin - Madison 9 Acknowledgement Overview

• Ther

erefore e all ll in invaria iant manifolds 𝚫 of

• f 𝒈 can be

e ob

• btained by solv
• lving th

the e same set t of f PDE’s, weather they are defined around a fixed point

• r
• r an

an (u (unknown) peri eriodic solu

• lution

Motivations Nonlinear modes (4/4)

z1, 𝑨1 𝑦1, 𝑦1 𝑦1, 𝑦1 z1, 𝑨1 𝑦1, 𝑦1 𝑦1, 𝑦1

Add dditio ional qu questions tha that ar are no not an answered her her: Feel eel fr free ee to

• as

ask me: e:

• Why

y is cal alled ICM CM?

• Wha

hat t is the the rela elati tionship be between the the cen center r man anifold of

• f

a a system (if f it t exi xists) ) an and its ts ICM CMs?

• Whe

hen do do ICM CMs bec become globally inse nseparable manifolds?

• Do

Do we nee need hi higher r di dimensional ICM CMs? Do Do the they exis xist? t?

SLIDE 10

An analytical method was presented to solve the governing PDE’s of each ICM.

University of Wisconsin - Madison 10 Acknowledgement Overview

• A com
• mbin

ination of

• f an averagin

ing meth thod (h (harmonic bala lance) and a non

• nli

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

• ordin

inates, i.e. i.e.

Motivations Nonlinear modes Calculation of NL-modes (1/9)

• n
• ne als

lso id identi tifie ies th the in invaria iant manifold ld (fu (functional rel elation).

This is method is is not

• t sc

scala lable le!

SLIDE 11

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?

University of Wisconsin - Madison 11 Acknowledgement Overview

• Averaging meth

thods tr try to

• make a parametric per

eriodic fu function satis tisfy fy th the e governin ing eq equations of

• f th

the e system.

• On

One example: Ha 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

• ch

chec eck th the e peri eriodic icity of

• f th

the e solu

• lution.
• They ar

are sc scalable: we e can integrate nu numericall lly.

• They ar

are com

• mputationally exp

xpensive: we e of

• ften ha

have to to In Integrate the system over an and over.

• They ar

are sen sensit itive to to the init itial conditions. s.

Motivations Nonlinear modes Calculation of NL-modes (2/9)

𝑢 = 𝑢0 𝑢 = 𝑢0 + 𝑈 ∆𝑢 = 𝑈 y0 = y 𝑢0 = 𝑧 (𝑢0 +

𝑈 𝑜)

SLIDE 12

Averaging methods try to make a parametric periodic function satisfy the governing equations of the system!

University of Wisconsin - Madison 12 Acknowledgement Overview

• Assume

e a sum of

• f (or

(orthogonal) peri eriodic fu functi tions

𝒕(𝑢) = 𝒍=𝟐

𝒐

𝛽𝑙𝒒𝑙 𝑢

• Approximates a solu
• lution of
• f th

the e system, i.e i.e.

𝒕 𝑢 − 𝒈 𝒕 𝑢 = 𝟏

• To
• fin

find th the e unknown coe

• efficie

ients

𝟏 𝑼

𝒕. 𝒒𝑙 𝜐 𝑒𝜐 =

𝟏 𝑼 𝒈 𝒕 . 𝒒𝑙(𝜐) 𝑒𝜐

• Res

esult lts in n an an al algebraic system of

• f eq

equatio ions 𝒉(𝛽1, … , 𝛽1) = 𝟏

• The ri

right-hand-sid ide is s a a weig eighted average of

• f the system

𝟏 𝑼 𝒈 𝒕 . 𝒒𝑙(𝜐) 𝑒𝜐 = 𝑈 𝑛=0

𝑁

𝑔 𝒕 𝑛𝑈

𝐿

𝒒𝑙(𝑛𝑈

𝐿 )

𝑁

, 𝑁 → ∞

Motivations Nonlinear modes Calculation of NL-modes (3/9)

SLIDE 13

In effect, averaging methods minimize a weighted average of the difference in acceleration over infinitely many points.

University of Wisconsin - Madison 13 Acknowledgement Overview

• In

Integ egration is is just t a weig eighed aver eragin ing

𝟏 𝑼

𝒈 𝒕 . 𝒒𝑙(𝜐) 𝑒𝜐 = 𝑈 𝑛=0

𝑁

𝑔 𝒕 𝑛𝑈 𝑁 𝒒𝑙(𝑛𝑈 𝑁 ) 𝑁 , 𝑁 → ∞

𝟏 𝑼

𝒕. 𝒒𝑙(𝜐) 𝑒𝜐 =

𝑈 𝑛=0

𝑁

𝒕 𝑛𝑈

𝑁 𝒒𝑙(𝑛𝑈 𝑁 )

𝑁

, 𝑁 → ∞

• Averaging meth

thods jus ust t min inimize a a weig eighted average of

• f the dif

difference in n ac acceleratio ions, i.e .e.

𝒋

𝑈 𝑁

𝒕 𝑢𝑗 − 𝒈 𝒕 𝑢𝑗

𝒒𝑙(𝑢𝑗) = 𝟏 𝑢𝑗 =

𝑗𝑈 𝑁 , 𝑗 = 0, … 𝑁 → ∞

• The

e peri eriodic sum approximates es a solu

• lution of
• f th

the e system bec ecause, in in aver erage, it it matches with ith it it at t in infin finitely many poin

• ints in

in on

• ne

e peri eriod

• Ind

ndefin finite ans answer

• We

e ha have to

• do

do it t ana analy lytic icall lly!

Motivations Nonlinear modes Calculation of NL-modes (4/9)

SLIDE 14

Collocation methods integrate the system numerically to check the periodicity of the solution.

University of Wisconsin - Madison 14 Acknowledgement Overview

• The

e solu

• lution of
• f th

the e system is is peri eriodic bec ecause it it cr crosses itse itself lf aft fter ONE peri eriod.

• De

Defi finit ition of

• f peri

eriodic fu functions + + Uniq iquen eness of

• f solu
• lutions

𝒛 𝒖 = 𝒛𝟏 +

𝟏 𝒖 𝒉 𝒛 𝝊

𝒆𝝊, 𝒛 = 𝒚 𝒚 , 𝒉 = 𝒚 𝒈(𝒚)

• Find T, 𝒛𝟏 th

that 𝒛 𝑼 = 𝒛𝟏

• We

e can integrate nu numerically ly

• On

One constr traint  de definite an answer r

• In

Integrate the system over an and over er

Motivations Nonlinear modes Calculation of NL-modes (5/9)

𝑢 = 𝑢0 𝑢 = 𝑢0 + 𝑈 ∆𝑢 = 𝑈 y0 = y 𝑢0 = 𝑧 (𝑢0 +

𝑈 𝑜)

SLIDE 15

Averaging Collocation: Can we replace the infinitely many integration points with only a few collocation points?

University of Wisconsin - Madison 15 Acknowledgement Overview

Averaging: Col Collocation: One e tim time Analyti tical l in integ egration Many tim times Numerical l in integ egration In Indefinite answer De Defi finit ite answer Rec ecall l ou

• ur con
• ndit

itions

𝒋

𝑈 𝑁

𝒕 𝑢𝑗 − 𝒈 𝒕 𝑢𝑗 𝒒𝑙(𝑢𝑗) = 𝟏 𝑢𝑗 =

𝑗𝑈 𝑁 , 𝑗 = 0, … 𝑁 → ∞

𝒛 𝑈 − 𝒛0 =

𝟏 𝑈

𝒉 𝒛 𝝊 𝒆𝝊 = 𝟏, 𝒛 𝟏 = 𝒛0

• Ca

Can we e rep eplace th the e aver erage sum with ith a set t of

• f m

much str tronger coll

• llocation con
• ndit

itions?

• Ca

Can we e do

• it

it for on

• nly

ly a few 𝑛 pairs of

• f coll
• llocati

tion poin

• ints

ts 𝑛 ≪ 𝑁, , i.e. i.e. with ithout any In Integ egration?

Motivations Nonlinear modes Calculation of NL-modes (6/9)

SLIDE 16

Multi-harmonic Multiple-point Collocation (MMC) provides the solution!

University of Wisconsin - Madison 16 Acknowledgement Overview

• MMC uses

es a Fou

• urier serie

eries as th the e parametric peri eriodic sum

𝒕 𝑢 = 𝑫 +

𝒍=𝟐 𝒐

𝑩𝑙𝑑𝑝𝑡(2𝜌𝑙𝑢 𝑈 ) + 𝑪𝑙sin(2𝜌𝑙𝑢 𝑈 )

to

• min

inimize e th the e coll

• llocation con
• ndit

ition

min

{𝑩,𝑪,𝑫,𝑈} 𝑛

𝒈 𝒕

𝑛𝑈 𝐿

− 𝒕

𝑛𝑈 𝐿

+ 𝒈 𝒕

𝑛𝑈 𝐿 + 𝑈

− 𝒕

𝑛𝑈 𝐿 + 𝑈

= 0

• I

I proved ed th that t MMC con

• nverges if

iff

𝒈 is

is monotonic in in th the neigh ighborhood of

• f coll
• llocati

tion poin

• ints

ts.

• Co

Comes in in tw two fla flavors:

• Ne

Newton

• nia

ian: Very ry fas ast but but wi with th a a very ry small l rad adiu ius of

• f con
• nvergence
• St

Steepest desc descent (wi (with Wolf

• lfe

e con

• ndit

itio ion) : : Slo Slower but but has has a a very ry lar arge rad adiu ius of

• f con
• nvergence

Motivations Nonlinear modes Calculation of NL-modes (7/9)

𝒆𝑛(𝑛𝑈 𝐿 ) 𝒆𝑛(𝑛𝑈 𝐿 + 𝑈)

SLIDE 17

Homogenous convergence refers to the cases where all collocation points converge to the same solution.

University of Wisconsin - Madison 17 Acknowledgement Overview

• All

ll poin

• ints con
• nverge on
• n th

the e same solu

• lution.
• It

It has a ver ery la large radius of

• f con
• nvergence

e (in (in th the e stee eepest des esce cent fla flavor)!

Motivations Nonlinear modes Calculation of NL-modes (8/9)

SLIDE 18

An added benefit: a heterogeneous convergence is possible!

University of Wisconsin - Madison 18 Acknowledgement Overview

• With

ithout in integ egration, th ther ere is is no

• con
• nstrain

int to

• force th

the e poin

• ints to
• be

e on

• n on
• ne

e solu

• luti

tion, i.e. i.e. coll

• llocation poin
• ints can

con

• nverge

e on

• n dif

ifferent solu

• luti

tions.

Motivations Nonlinear modes Calculation of NL-modes (9/9)

Add ddit itio ional l que questio ions:

• What if

f 𝒈 is s no not t mon

• notonic

ic ar around a a collo llocation po poin int?

• Is

Is MMC the on

• nly

ly com

• mbinatio

ion of

• f aver

eragin ing an and collo llocation meth thod pos possib ible?

• Can

Can one

• ne use

use continuation with th MMC?

SLIDE 19

Which solutions are stable? When will we jump from one to another? The main tool for stability analysis of periodic solutions is Floquet theory.

University of Wisconsin - Madison 19 Acknowledgement Overview

• Stabil

ilit ity analy lysis is is is im important in in

• De

Desig ign, i.e. ide dentify ifyin ing g the the ran ange of

• f safe op
• per

eration, of

• f no

nonl nlin inear systems.

• In

n exp xperimental ide dentification

• n of
• f no

nonl nlinear systems, , to

• pr

predict or

• r expla

xplain whi which set of

• f no

nonl nlinear mo mode des can or

• r cannot
• t be

be experim imentally ly excit ited.

• Bi

Bifu furcatio ion an analy lysis is of

• f per

perio iodic ic solut

• lutio

ions of

• f no

nonl nlin inear system.

• Ca

Can be performed usin ing g Lyapunov or

• r Poin
• incare or
• r Floq

loquet methods

• Al

All pr provid ide interchangeable le resu esult

• lts. Fl

Floq

• quet me

methods can be be imp mple lemented mu much ea easie ier!

• Floq

loquet stabil ilit ity

• Li

Line nearize the the system abo about the the per periodic solu

• lution
• n.
• De

Determin ine the the gr growt wth of

• f a

a sma mall ll per perturbatio ion ar arou

• und the

the per perio iodic ic solu

• lutio

ion.

Motivations Nonlinear modes Calculation of NL-modes Stability analysis (1/2)

SLIDE 20

New algorithm by Ardeh-Allen validates near-zero Floquet exponents and returns accurate stability analysis results!

University of Wisconsin - Madison 20 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis (2/2)

Add ddit itio ional l que questio ions:

• Why ar

are e nea near-zero Fl Floquet exponents important?

• Ho

How efficient is s this al algorithm?

• Can

Can this is al algorit ithm improve the efficiency of

• f cu

current al algorithms of

• f fin

finding per periodic ic solu solutions?

SLIDE 21

Overview

University of Wisconsin - Madison 21

• Mot
• tivati

tion and back ckgrounds

• Non

Nonlin inea ear mod

• des

es of

• f vib

vibrations

• Three definitions
• Instantaneous Center Manifold (ICM)
• Ca

Calc lcula lation of

• f non
• nli

linear modes

• Solving for ICM analytically
• Averaging and collocation methods
• Averaging ⊕ collocation (MMC)
• Stabil

ilit ity and bifu ifurcation of

• f non
• nli

linear modes

• Floquet theory
• Validation of stability analysis
• Con
• nnectin

ing fu functions

• Definition of connecting functions
• Bi-directionally linear connecting functions
• Calculation of connecting functions
• Linear approximation of connecting functions
• Co

Conclu lusions

Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (0/15)

SLIDE 22

Superposition describes a set of two properties. General solution is a function with a specific domain and codomain!

University of Wisconsin - Madison 22 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (1/15)

• Linear Systems: linear System of Autonomous, Homogenous Second Order ODE’s

𝒚 = 𝒈 𝒚 = 𝐵𝒚

• Add

Addit itiv ivit ity

If If 𝒓1 an and 𝒓2 ar are solu solutions, s, so so is s 𝒓1 + 𝒓2. .

• Ho

Homo mogeneit ity

If If 𝒓1 is a a solu solution, so so is s 𝛽𝒓1, 𝛽 ∈ ℝ. .

 Addit itivit ity a and Ho Homogeneit ity  Superpositi tion

If f 𝒓𝑗 𝑗 = 1, … , 𝑂 ∈ ℕ ar are solut

• lutio

ions, so

• is every 𝑮 𝒓1, … , 𝒓𝑂 = 𝒋 𝛽𝑗 𝒓𝑗

 The general l solu

• lution of
• f a lin

linear systems represents any solu

• lution of
• f th

the system as as a superposit ition of

• f a fin

finit ite number of

• f solu
• lutions.

SLIDE 23

General solution of a linear systems exists because linear systems accept superposition.

University of Wisconsin - Madison 23 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (2/15)

 Gen eneral l solu

• lution is

is a fu funct ction th that t can rep epresen ents ts all ll solu

• lutions of
• f th

the e system as super erposit ition of

• f a fin

finit ite number

• f
• f solu
• lutions, i.e.

i.e. a fu funct ction 𝐆 𝐫1, … , 𝐫𝑂 such ch th that t

 𝐺 spa spans the the en entir ire se set t of

• f sol

solutio ions of

• f the

the system.

 For linear systems, if 𝐫i are linearly independent,  𝐆 𝐫1, … , 𝐫𝑜 = 𝑗=1

𝑜

𝒓𝑗 spans the entire set of solutions of the system.  𝐫i can be linear modes of the system.

 Th The e se set t {𝐫1, … , 𝐫𝑂}, , mus ust t be be a a fi finit ite se set. t.

 There are infinitely many linear modes.  But the system is Homogeneous: we can normalize them! The finite set can be the set of “Linear Normal Modes”.

SLIDE 24

An example: superposition dictates the form of the general solution for a 2DOF linear oscillatory system.

University of Wisconsin - Madison 24 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (3/15)

• Co

Consider th the 2DO DOF lin linear system

𝑦1 𝑦2 = −2 1 1 −2 𝑦1 𝑦2

 The e system has tw two

• in

infin finite sets ts of

• f lin

linea ear mod

• des.

 The e cr cross secti ection of

• f th

thes ese tw two sets ts with ith th the e pla lane e 𝑢 = 0 gen enerates tw two

• lin

lines es.

 The (e (evolu lutio ion of

• f)

) un unit it vec ectors s al along eac each br branch ar are the lin inear nor normal mod

• des.

 With ithout th the hom

• mogeneity property (i.e

(i.e. nor

• rmalization), no
• gen

eneral solu

• lution cou
• uld exis

xist, bec ecause th the e set t of

• f b

basis is vect ector wou

• uld be

e in infin finit ite.

• 𝒗0(𝑢) =

𝒓1

𝑀𝑁(𝑢) +

𝒓2

𝑀𝑁(𝑢)

• 𝒗0 0 = 𝛽1

𝒓1

𝑀𝑂𝑁 𝑢 + 𝛽2

𝒓2

𝑀𝑂𝑁 𝑢

• 𝛽𝑗 =

𝒓𝑗

𝑀𝑁(0)

SLIDE 25

Connecting functions are like the general solution, i.e. they’re functions, but without the constraints on their domain and codomain.

University of Wisconsin - Madison 25 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (4/15)

 Con Connecting Funct ction  Any fu funct ction of

• f solu
• luti

tions of

• f th

the system th that t is is als lso

• a solu
• lution of
• f th

the system.

De Defined as as an any 𝑮 𝒓1, … , 𝒓𝑂 , 𝑂 ∈ ℕ that sa sati tisfy 𝑒2𝑮

𝑒𝑢2 = 𝒈 𝑮 ar

are connectin ing fu functions. s.

 Con Connecting Funct ction Vs. Superposition  Superposition defin fines a famil ily of

• f lin

linear, hom

• mogenous con
• nnecti

ting g fu funct ctions.  Con Connecting Funct ction Vs. Gen eneral l Solu

• lution

 If If th there exis xists a famil ily, 𝑮 , , of

• f a fin

finit ite {𝒓1, … , 𝒓𝑂} th that t spans th the entir tire set t of

• f solu
• lutions of
• f th

the system, th then it it is is a gen eneral solu

• lution.

SLIDE 26

Lie proved that no general solution exists in the absence of superposition [5], but it does not mean that a nonlinear connecting function does not exist.

University of Wisconsin - Madison 26 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (5/15)

 With ithout super erposition, gen eneral l solu

• lution Doe

Does No Not t Exis xist (E (Except for

• r Ricc

iccati Equations)!  Lie Lie Theorem: For any fin finit ite {𝒓1, … , 𝒓𝑂} th there exis xist no

• fu

funct ction, 𝑮, th that t can span th the entir tire set t of

• f solu
• lutions.

 What t Abou

• ut Con

Connecting Funct ctions?  The set t of

• f non
• nli

linear modes of

• f (e

(eig igensolutions of

• f)

) non

• nlin

inear systems, i.e i.e. {𝒓1, … , 𝒓𝑂}, , is is in infin finit ite!  The e set t of

• f non
• nli

linea ear modes cannot be e nor

• rmali

lized (n (non

• nli

linear systems are not

• t hom
• mogenous)!

 If If we e rel elax th the e fin finit iteness con

• ndit

ition and glob global l coverage of

• f 𝑮:

 Is Is th there a non

• nlin

inear loc local l con

• nnecting fu

funct ction?

[5] S. Lie and G. Scheers. Vorlesungen uuber continuierliche gruppen mit geometrischen und anderen anwendungen, edited and revised by g. Scheers, Teubner, Leipzig, 1893. 5.1.3

SLIDE 27

Nonlinear connecting functions exist and can be considered as local general solutions!

University of Wisconsin - Madison 27 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (6/15)

 I I proved ed th that t if if a non

• nlin

inea ear loc local con

• nnectin

ing fu function exis xists, it it must satis tisfy fy  For a con

• nserv

rvative system 𝒈 wit ith solu

• lutions {𝒓1, … , 𝒓𝑂}

 𝚾 = 𝚾 𝒓1, … , 𝒓𝑂, 𝒓1, … , 𝒓𝑂  𝕂𝒓𝒋 𝛂𝒓𝒌𝐺𝑙 = 𝟏 and 𝕂

𝒓𝒋 𝛂 𝒓𝒌𝐺𝑙

= 𝟏  𝚾 𝒓1, … , 𝒓𝑂, 𝒓1, … , 𝒓𝑂 = 𝒆 + 𝑗 𝐵𝑗 𝒓𝑗 + 𝑗 𝐶𝑗 𝒓𝑗 + 𝑙 𝑗 𝑘 𝒓𝑗

𝑈𝐷𝑗𝑘𝑙

𝒓𝑘 𝒇𝑙  A non

• nli

linear loc

• cal con
• nnecti

ting g funct ction must als lso

• satis

tisfy  𝚾 = 𝒚 

𝑒𝚾 𝑒𝑢 =

𝒚 

𝑒2𝚾 𝑒𝑢2 = 𝒈(𝚾)

Add ddit itio ional l que questio ions:

• Ho

How did did you

• u pr

prove this?

• What abo

about no non-conserv rvativ ive systems?

• What ass

assumptions did did you

• u

mak ake abo about 𝒈 an and 𝑮?

• Did

Did you

• u assu

assume a a form

• rm for
• r 𝑮?

?

• What is

s N? Ho How do do you

• u

de determin ine this nu number?

• Is

Is this s the only

• nly adm

admiss ssible le form

• rm for
• r 𝑮 or
• r one
• ne of
• f man

any pos possib ible ch choices?

SLIDE 28

The first step to identify nonlinear connecting functions is to assemble a system of algebraic equations for connecting functions.

University of Wisconsin - Madison 28 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (7/15)

 Rem emember: a non

• nli

linea ear con

• nnectin

ing function must satis tisfy fy

 𝚾 = 𝒚,

𝑒𝚾 𝑒𝑢 =

𝒚, an and

𝑒2𝚾 𝑒𝑢2 = 𝒈(𝚾)

 We e pick ick non

• nlin

linear mod

• des as

as 𝒓’s, i.e.

𝒓𝑗 𝑢 = 𝜹𝑗 + 𝑘 𝜷𝑘 𝑈𝑗 𝑑𝑝𝑡 𝑘 2𝜌

𝑈𝑗 𝑢 + 𝜸𝑘 𝑈𝑗 sin 𝑘 2𝜌 𝑈𝑗 𝑢

 Update𝚾with 𝒓’s, i.e.

𝚾 = 𝒆(𝝊) + 𝑗 𝐵𝑗 (𝝊)𝒓𝑗 + 𝑗 𝐶𝑗 (𝝊) 𝒓𝑗 + 𝑙 𝑗 𝑘 𝒓𝑗

𝑈𝐷𝑗𝑘𝑙(𝝊)

𝒓𝑘 𝒇𝑙

 Substi titute 𝚾 in in th the e nece ecessary con

• nditi

tion, i.e i.e. .

𝐡 𝚾, 𝒚, 𝒚 = 𝟏

 Giv iven an arb rbitrary ry set t of

• f in

initia itial l con

• ndition 𝒚 0 = 𝒗,

𝒚 0 = 𝒘 ,

, th then we dis iscretize tim time, i.e i.e. 𝑢 = 0, 𝑢1, … , 𝑢𝑙 and 𝒚 𝑢𝑙 = 𝒗 + 𝒘𝑢𝑙 +

1 2 𝒈 𝒗 𝑢𝑙 2 + 1 3! 𝕂𝒚 𝒈 𝒗 𝒘𝑢𝑙 3+O{𝑢𝑙 4},

 This is result lts in in a set t of

• f non
• nli

linear alg lgebraic equati tions 𝐡 𝒆, 𝐵𝑗, 𝐶𝑗, 𝑑𝑗𝑘𝑙, 𝝊, 𝒗, 𝒘 = 𝟏

SLIDE 29

One approach to solve the nonlinear system of algebraic equations is Homotopy Analysis.

University of Wisconsin - Madison 29 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (8/15)

 Remember 𝐡 𝒆, 𝐵𝑗, 𝐶𝑗, 𝑑𝑗𝑘𝑙, 𝝊, 𝒗, 𝒘

 is s a a no nonlin inear al algebraic system of

• f eq

equations: it t req equires an an initia ial l gue guess.  A solu solutio ion is s a a connecting fu functio ion identified by y a a  a a se set t of

• f coe
• efficients,

s, an and  a vec ector of

• f per

periods of

• f se

set t of

• f per

periodic ic solu solutions s 𝒓𝑗(𝑢)  Also so no note, a a solu solutio ion is s al also

• a

a solu solution of

• f th

the orig

• riginal system de

defined by y 𝒈. .

 On e e meth thod, calle lled Ho Homotopy analy lysis, , defin fines a Ho Homotopy rel elation (p (path th) betw tween th the e known and unknown solu

• lutions of
• f th

the e system.

An An example of

• f Hom
• motop
• py Rela

elation

• n (pa

(path)

SLIDE 30

Homotopy paths must be defined! Here’s an example for a 2DOF system.

University of Wisconsin - Madison 30 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (9/15)

10

• 3

10

• 2

10

• 1

10 10

1

0.15 0.2 0.25 0.3 0.35 0.4 0.45

Energy Frequency (Hz)

s1 s2

 The Ho Homotopy path th  The e known solu

• lutions is

is lin linea ear con

• nnecti

ting g fu funct ction of

• f lin

linea ear modes of

• f th

the e system at t a poin

• int very clos

close to

• th

the fix fixed poin

• int of
• f th

the system.

𝑡 1

SLIDE 31

𝒗1 𝒗2

University of Wisconsin - Madison 31 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (10/15)

 Two Ho Homotopy path ths wer ere e defin fined as str traig ight t lin lines es, which ich cr cross eq equip ipotentia ial con

• ntours tr

transversall lly, starting fr from tw two

• poin
• ints 𝒗0

(1) and 𝒗0 (2) very

ery clo close to

• th

the e fix fixed poin

• int

t of

• f th

the e system.

Two examples of Homotopy paths.

SLIDE 32

University of Wisconsin - Madison 32 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (11/15)

Two examples of Homotopy paths.

 Ho Homotopy path ths can be e used ed to

• fin

find con

• nnecting fu

funct ctions at t any arbi rbitrary ry poin

• int in

in th the e state e space ce.  Dr Drawback: on

• ne must be able

le to

• defin

fine a Ho Homotopy path th and a mon

• notonic Ho

Homotopy parameter 𝑡: 0 → 1.  It’s computationally con

• nvenient. One can t

to

• fin

find a solu

• luti

tion (c (con

• nnecti

tion fu funct ction) at t any poin

• int usin

ing g peri eriodic solu

• lutions with

ith th the e same en ener ergy.  The e res esult lts show con

• nsis

istent accu ccuracy even en in in sign ignif ificantl tly non

• nli

linear reg

• egion. The

e fr freq equen encies es of

• f th

the e fir first and th the e sec econd can rise rise up to

• 36%

% and 145 145% % compared to

• th

their eir lin linea ear natu tural l fr frequencie ies.

SLIDE 33

A second approach to solve the system of algebraic equations is to use continuation.

University of Wisconsin - Madison 33 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (12/15)

𝒗1 𝒗2

 Two con

• ntin

inuation path ths wer ere ob

• btain

ined, starti ting g fr from th the e same tw two

• poin
• ints 𝒗0

(1) and 𝒗0 (2) very

ry clo close to

• th

the fix fixed poin

• int

t of

• f th

the system.

SLIDE 34

University of Wisconsin - Madison 34 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions (13/15)

Two examples of continuation paths.

 Con Continuation path ths are deter ermined by th the e dynamics of

• f 𝑲𝝊

𝒉,

, 𝑲𝒗

𝒉 and 𝑲𝒘 𝒉 and th

the e con

• nti

tinuati tion parameter (s (step ep-size) e), i.e i.e. th the e des estin tination of

• f con
• ntin

inuation path ths cannot be e set t in in advance.

 Co Contin inuatio ion pa paths foll

• llow

w the the Ho Homo motop

• py pa

paths very ry clos

• sely

ly in n the the line near regi egion

• n!

SLIDE 35

University of Wisconsin - Madison 35 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis

A third approach is also provided which does not require starting from a known solution in the linear region!

Connecting functions (14/15)

 This is approach calcu lcula lates a lin linea ear approximate e con

• nnecting fu

funct ction (of

• f a set

t of

• f alm

lmost ort

• rthogonal

l peri eriodic solu

• lutions) and th

then en uses es it it as th the in initia itial l gu guess for solv

• lvin

ing th the non

• nli

linear alg lgebraic equations.  In In or

• rder to
• avoid

id solv

• lvin

ing th the e lin linea ear system, th this is approach uses es alm lmost ort

• rthogonal

l peri riodic ic solu

• lutions.

𝚾 = 𝐯, 𝑒𝚾 𝑒𝑢 = 𝒘, 𝑒2𝚾 𝑒𝑢2 = 𝒈(𝚾) 𝚾L = 𝐯 𝑒𝚾𝑀 𝑒𝑢 = 𝒘 𝑒2𝚾L 𝑒𝑢2 = 𝒈𝑀(𝚾𝑀) Linear Algebraic System

Two fu functi tions are alm lmost ort

• rthogonal if

if th their ir projecti tions on

• n a fin

finit ite ort

• rthogonal

l basis is is is ort

• rthogonal!

𝑇 = {cos 𝜕0𝑢 , … cos(5𝜕0𝑢)} 𝐵11 … . 𝐵15 , 𝐵21 … 𝐵25 𝜕1 𝜕2 = 𝑞1𝜕0 𝑞2𝜕0 = 𝑞1 𝑞2 𝐵11 … . 𝐵15 𝐵21 … 𝐵25 T = 0 𝑞1, 𝑞2 > 5 →

SLIDE 36

University of Wisconsin - Madison 36 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis

The linear approximation to the connecting function provides a good initial guess for solving the nonlinear algebraic system even in the nonlinear region.

Connecting functions (15/15)

 The e fir first and th the e sec econd mod

• des

es show 19% and 35% in incr crease in in th thei eir fr frequencie ies com

• mpared to
• th

the e fir first and sec econd lin linea ear natu tural fr freq equencies es.

Add ddit itio ional l que questio ions:

• Ho

How do do you

• u fin

find the se set t of

• f alm

almost t

• rth
• rthogonal per

periodic solu solutions?

SLIDE 37

University of Wisconsin - Madison 37 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis

This work has provided a new definition and new methods of calculation and accurate stability analysis of nonlinear modes.

Connecting functions Conclusions (1/2)

 A new defin finiti tion for

• r a non
• nli

linear mode of

• f, speci

cifically, , In Instantaneous Ce Center Manifold (IC (ICM) was proposed.

I. I. Th The e pr prop

• pos
• sed de

defin init itio ion en encompasses al all the the pr previo ious no nonl nlin inear mo mode de defin finit itions. II. It t al also

• lead

eads to

• ne

new me method

• ds of
• f calc

alcula latio ion of

• f no

nonl nlin inear mo modes th that do do no not t req equir ire an any pr previously ly kno known solut

• lutio

ion as as an an ini nitia ial l gu guess ess.

 An extremely sim imple yet t effecti tive meth thod, i.e i.e. . Multi lti-harmonic ic Multi ltiple le-point Co Collocation (M (MMC), , for

• r fin

findin ing peri riodic ic solu

• lutions of
• f con
• nserv

rvative non

• nlin

linear systems was pres esen ented.

I. I. MM MMC uses uses a a simil ilar con

• ndit

itio ion to

• the

the on

• ne

e used used in n mul multip iple le-poi

• int sho

hoot

• tin

ing g me methods, ho however, it t do does es no not t req equir ire integration of

• f the

the vector

• r

fi field eld over er an any per perio iod of

• f ti

time. II. MM MMC is capable le of

• f fi

find ndin ing mo more tha than on

• ne

e per perio iodic ic solut

• lutio

ions of

• f the

the no nonl nlin inear system in n eac each solut

• lutio

ion. III. Al Alth though no not t pr presented in n this this do document, the the al algor

• rit

ithm has has be been imp mple lemented in n a a con

• ntin

inuatio ion fr framewor

• rk and

and see eems to

• be

be mo more com

• mputatio

ionall lly effic ficie ient tha than sho hoot

• tin

ing g al algor

• rit

ithms.

 A set t of

• f lim

limits for

• r tw

two

• sou
• urces

es of

• f er

error in in th the e process of

• f calcu

lcula latin ing Floq loquet exp xponen ents ts is is pres esented.

I. I. Th These ese limit its wer ere used used to

• pr

prop

• pose a

a crit riteria ia for

• r valid

idatio ion of

• f ne

near-zero

• Fl

Floq

• quet expon
• nents.

. II. An An al algor

• rit

ithm was as al also

• pr

prop

• pos
• sed whi

which adju adjusted the the integr gratio ion toler

• lerance and

and app approxim imation level l for

• r the

the per periodic ic solu

• lutio

ions usin using g the the crit riteria ia, to

• as

assure tha that valid id Fl Floq

• quet expo

ponents wer ere calc lcula lated al alon

• ng

g an an en entir ire br branch. III. Th The e pr prop

• pos
• sed al

algor

• rit

ithm al also

• red

educes the the com

• mputatio

ional l cos

• st of
• f fi

find ndin ing per periodic ic solu

• lutio

ions by y pr provid idin ing g an an ada adaptiv ive toler

• lerance for
• r the

the per perio iodic ic orbi

• rbit sol
• lver.

SLIDE 38

University of Wisconsin - Madison 38 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis Connecting functions Conclusions (1/2)

This work has presented the general form and three methods of calculation of nonlinear connecting functions.

 The e gen eneral l for

• rm of
• f non
• nli

linear con

• nnecti

ting g fu funct ctions is is provid ided ed.  The e con

• ncept of
• f alm

lmost or

• rth

thogonalit ity and or

• rder of
• f alm

lmost ort

• rthogonali

lity is is defin fined for

• r peri

eriodic ic solu

• lutions.

 A A nu numerical sch cheme for

• r fi

find nding a a set of

• f al

almos

• st orth
• rthog
• gon
• nal per

periodic solu

• lution
• n (of

(of an any or

• rde

der) is pr provided.

 Three ee numerical approaches of

• f calcu

lcula lation of

• f con
• nnecti

ting fu funct ctions are provided.

I. I. The fi first t two ap approaches s use use Ho Homotopy an analy lysis is an and contin inuation res espectiv ively to to identif ify continuous s br branches s of

• f

connecting fu functions.

I. I. Ho However bo both th me methods dem demand startin ing fr from

• m a

a kno known con

• nnectin

ing fun function.

II. II. The thir ird meth thod, ho however, overcomes this is constr traint by y fin finding a a lin inear ap approximatio ion of

• f a

a connecting fu functio ion at t an any ar arbitrary ry po poin int in the state spa space an and usin using it t as as the starting poi point in the sea search for

• r the no

nonlin inear r connectin ing fu function.

I. I. Th The e con

• ncept of
• f alm

almos

• st orth
• rthog
• gonali

lity mak makes it t po possib ible le for

• r the

the thir third app approa

• ach to
• effic

ficie iently ly fi find nd the the a a line near ap approxim imatio ion of

• f

con

• nnectin

ing g fun functio ions at t an any ar arbit itrary ry po point reg egardle less of

• f its

ts pr proxim imit ity to

• the

the equ equili libriu ium of

• f the

the system  .

SLIDE 39

University of Wisconsin - Madison 39 Acknowledgement Overview Motivations Nonlinear modes Calculation of NL-modes Stability analysis

Thanks!

Connecting functions Conclusions

Questions?