Table Of ContentLocalization in finite asymmetric vibro-impact chains
Itay Grinberg1,∗ and Oleg V. Gendelman1,†
1Faculty of Mechanical Engineering
Technion - Israel Institute of Technology
(Dated: January 12, 2017)
We explore the dynamics of strongly localized periodic solutions (discrete solitons, or discrete
breathers) in a finite one-dimensional chain of asymmetric vibro-impact oscillators. The model
involves a parabolic on-site potential with asymmetric rigid constraints (the displacement domain
of each particle is finite), and a linear nearest-neighbor coupling. When the particle approaches
the constraint, it undergoes an impact (not necessarily elastic), that satisfies Newton impact law.
Nonlinearity of the system stems from the impacts; their possible non-elasticity is the sole source
ofdampinginthesystem. Wedemonstratethatthisvibro-impactmodelallowsderivationofexact
analytic solutions for the asymmetric discrete breathers, both in conservative and forced-damped
7 settings. The asymmetry makes two types of breathers possible: breathers that impact both or
1 only one constraint. Transition between these two types of the breathers corresponds to a grazing
0 bifurcation. Specialcharacterofthenonlinearitypermitsexplicitderivationofamonodromymatrix.
2 Therefore,thestabilityoftheobtainedbreathersolutionscanbeexactlystudiedintheframeworkof
simplemethodsoflinearalgebra,andwithrathermoderatecomputationalefforts. Allthreegeneric
n
scenarios of the loss of stability (pitchfork, Neimark-Sacker and period doubling bifurcations) are
a
J observed.
1 PACS numbers: 05.45.Yv, 63.20.Pw, 63.20.Ry
1
] INTRODUCTION chains with homogeneous interaction[26], and vibro-
S impact chains[19]. Recently the latter approach was ex-
P
Localization is an important and widely studied phe- tendedtotheforced-dampedvibro-impactchains[21,23],
.
n nomenon in discrete dynamical systems [1–9]. Contrary chains with self-excitation[27], and, most recently, to
li tothelocalizationinlinearsystems,innonlinearsystems Multi-Breather(MB)solutions[22],namelytheDBswith
n it is possible without any disorder, i.e. localization may more than a single localization site.
[
occur even in a purely homogeneous nonlinear lattice. Theaforementionedvibro-impactchainsareessentially
1 Interesting examples of such localized responses are Dis- linear, except for the possibility of collisions, i.e. all the
v creteBreathers(DBs), sometimesreferredtoasIntrinsic on-site and coupling interactions, that are not impacts,
5 Localized Modes (ILMs) or Discrete Solitons. The DB is are linear. This feature not only allows the derivation
5 a periodic strongly localized response of the lattice sys- oftheexactsolution, butalsoconsiderablysimplifiesthe
0
tem. Itcanbesimplyimaginedasanoscillatingenvelope stability analysis. The stability of the periodic DB solu-
3
localized in the vicinity of a single or several sites of the tion is determined by location of the eigenvalues of the
0
. lattice. The DB’s localization is typically exponential; Monodromy matrix [28]. In most cases, the monodromy
1
however, in the systems with strong nonlinearity, it may matrix can only by obtained numerically by integration
0
be hyper-exponential[2]. The DBs are known in various of the equations of motion over the period of the solu-
7
1 branches in physics. They were experimentally observed tion. This task can be extremely difficult when treat-
: andtheoreticallydiscussedinavarietyofmodelsystems, ing systems with a large number of particles, to the ex-
v
such as superconducting Josephson junctions [10], non- tentthatsuper-computersmaybenecessary. Theconsid-
i
X linear magnetic metamaterials[11], electrical lattices[12], ered vibro-impact models allow explicit derivation of the
r micro-mechanicalcantileverarrays[13–17],Bose-Einstein monodromy matrix [21–23]. Thus the stability analysis
a condensates[18], and chains of mechanical oscillators[19– is reduced to evaluation of the spectrum of easily com-
23]. puted matrices. Consequently, even simple PCs suffice
The exact DB solutions in specific nonlinear chain forchainswiththousandsofparticles. Furthermore,even
models remain scarce due to the nonlinearity and dis- forthefinitechains,onecanaccomplishthesederivations
creteness of the systems that encumbers the derivation without further approximations.
ofexactsolutions. Currenttheoreticalresearchprimarily This work is based on the approach used in refs.
concentratesonnumericalexplorationsandapproximate [19, 21–23], but introduces important novel feature into
analytic approaches [1, 2, 9, 24]. Few known exceptions themodel. Insymmetricmodels,abreakdownoftheDB
are the completely integrable Ablowitz-Ladic model[25], symmetry is one of the instability scenarios [21]. In cur-
rent model, the asymmetry is imbedded into the lattice
itself. To be more specific, the model comprises a finite
number of linearly coupled oscillators; each of the latter
∗ [email protected] includes on-site coupling with two asymmetric rigid bar-
† [email protected] riers bounding the movement. Thus one can obtain the
A Discrete Breathers in the Conservative ModelII EXACT SOLUTION FOR THE ASYMMETRIC BREATHER
asymmetricDBsandexploretheirzonesofexistenceand the dissipation. Equations of motion for the chain can
stability properties in the space of parameters. Further- be easily obtained from the Hamiltonian given in (3).
more,theconsideredasymmetryallowsanewtypeofthe Furthermore, the periodicity of the DB allows introduc-
DB–thesingle-sidedDB–wheretheimpactonlyoccurs ing the impact into the equations of motion as exter-
at one of the constraints. In sec. I we present a detailed nal forcing in the the form of a sum of delta functions
description of the system. For this new type of DBs we with advanced-delayed arguments. Without restricting
observeforthefirsttimetheperioddoublingbifurcations the generality, we adopt that the DB is localized at the
andobtainananalyticsolutionfortheemergingsolution particle with n=0. One obtains the following equations
in a similar manner. Section II contains the derivations of motion:
of an exact analytic solution for both conservative and
u¨ +γ u +γ (2u −u −u )=
forced/damped DBs, as well as for the single-sided DB. 0 1 0 2 0 1 N
(cid:18) 2πj(cid:19)
The method of the stability analysis is explained in sec. =2p (cid:80)∞ δ t−φ− −
III. Numerical validation of the results is presented in 1 j=−∞ ω (5)
sec. IV, followed by concluding remarks in sec. V. −2p (cid:80)∞ δ(cid:18)t− 2πj(cid:19)
2 j=−∞ ω
I. DESCRIPTION OF THE MODEL
u¨ +γ u +γ (2u −u −u )=0 (6)
n 1 n 2 n n+1 n−1
We consider a chain of (N +1) identical unit masses,
coupled to their neighbors via linear springs, and with u¨ +γ u +γ (2u −u −u )=0 (7)
N 1 N 2 N 0 N−1
periodic boundary conditions. In addition, all masses
are subject to identical on-site potentials. The on-site where 2p1 and 2p2 correspond to the amounts of mo-
interactionisvialinearspring,butthemotionisbounded mentum transferred in the course ofeach ofthe impacts,
by a set of asymmetric impact barriers. The on-site and δ is the Dirac delta and φ is the phase instance of the
coupling potentials can be described as follows: secondary impact in the period of the DB.
Theexpressionfortheimpactforcingcanbere-written
(cid:40)
γ u2 |u−a|<1 in the form of generalized Fourier series:
V(u)= 1 (1)
Impact |x−a|=1
u¨ +γ u +γ (2u −u −u )=
= ω (cid:80)∞0 1(p0cos(2jω(t0−φ)1)−pNcos(jωt)) (8)
π j=−∞ 1 2
W(u)=γ u2 (2)
2
u¨ +γ u +γ (2u −u −u )=0 (9)
n 1 n 2 n n+1 n−1
where a ≥ 0 is the parameter of asymmetry and γ and
1
γ are the on-site and coupling stiffnesses, respectively.
2
ThisyieldsthefollowingHamiltonianforthefullfinite u¨ +γ u +γ (2u −u −u )=0 (10)
N 1 N 2 N 0 N−1
chain:
Thus, one further obtains:
N (cid:18)1 (cid:19)
N(cid:80)−H1 =n(cid:80)=0 2p2n+V(un) + (3) u¨0++2πωγ1(cid:80)u0∞j=+1γ(p21(2cous0(−jωu(1t−−uφN)))−=pωπ2c(ops1(−jωpt2)))+ (11)
+ W(u −u )+W(u −u )
n n+1 N 0
n=0
u¨ +γ u +γ (2u −u −u )=0 (12)
The impact, which could be either elastic or non- n 1 n 2 n n+1 n−1
elastic, obeys the following traditional Newton impact
law: u¨ +γ u +γ (2u −u −u )=0 (13)
N 1 N 2 N 0 N−1
u˙(ti+)=−eu˙(ti−) (4) To obtain the exact solution, the displacement of each
particleisalsosearchedintheformofFourierseries,and
wheret isthetimeinstanceoftheimpactand0<e≤1
i the following anzats is used:
is the coefficient of restitution.
∞
(cid:88)
u =u + (u cos(jω(t−φ))+u cos(jωt))
n n,0 n,j,1 n,j,2
II. EXACT SOLUTION FOR THE j=1
ASYMMETRIC BREATHER (14)
where,
A. Discrete Breathers in the Conservative Model
u =A fn+B f−n (15)
n,0 0 0 0 0
u =A fn+B f−n (16)
If the external forcing is absent, one should set the n,j,1 j j j j
coefficient of restitution to e = 1 in order to preclude u =C fn+D f−n (17)
n,j,2 j j j j
2
A Discrete Breathers in the Conservative ModelII EXACT SOLUTION FOR THE ASYMMETRIC BREATHER
Linearityoftheequationsofmotionbetweentheimpacts u (φ)= ω(cid:0)f0−N−1+10(cid:1)(p2−p1) −
yields: 0 πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1)
2 0 0 0
γ1+2γ2−j2ω2±(cid:113)(j2ω2−γ1−2γ2)2−4γ22 −(cid:80)∞j=1 πγ (cid:0)2fω−(cid:0)ffj−−N1−(cid:1)1(cid:0)f+−1N(cid:1)−p11−1(cid:1)+ (29)
f = = 2 j j j
=j γ1+2γ2−j2ω2±(cid:112)(j22ωγ22−γ1−4γ2)(j2ω2−γ1) +(cid:80)∞j=1 πγ (cid:0)2fω−(cid:0)ffj−−N1−(cid:1)1(cid:0)f+−1N(cid:1)−p12−1(cid:1)cos(jωφ)=
2γ 2 j j j
2 =−1+a
(18)
Account of the periodic boundary conditions in eq.
Reordering these equations, one can write them down
(13) yields the following relations:
in a somewhat simplified form:
A =B f−N−1 (19)
j j j
−p χ (φ)+p χ =1+a (30)
C =D f−N−1 (20) 1 1 2 2
j j j −p χ +p χ (φ)=−1+a (31)
1 2 2 1
Finally, substituting these derivations into the first
equation of system (11), that describes the dynamics of where
impacting mass, one obtains for j >0:
(f−N−1+1)
0 +
Bj =−πγ2(cid:0)fj −fj−21ω(cid:1)p(cid:0)1fj−N−1−1(cid:1) (21) χ1(φ)≡ πωγ2+(cid:80)∞ (f0−2f(0−f1j−)N(f−0−1N+−1)1−1) cos(jωφ)
2ωp j=1 (fj−fj−1)(fj−N−1−1)
D = 2 (22) (32)
j πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1)
2 j j j
(f−N−1+1)
and for j =0: χ ≡ ω (f0−f0−01)(f0−N−1−1)+ (33)
ω(p −p ) 2 πγ2+(cid:80)∞ 2(fj−N−1+1)
B0 = πγ (cid:0)f −f−21(cid:1)(cid:0)f1−N−1−1(cid:1) (23) j=1 (fj−fj−1)(fj−N−1−1)
2 0 0 0
So far there are 3 unknowns (p , p and φ) and only 2
Summarizing, one obtains the following exact solution 1 2
equations. Additionalequationisderivedfromthecondi-
for the DB:
tion of energy conservation in the course of each impact:
∞
(cid:88)
u =u + (u cos(jω(t−φ))+u cos(jωt)) ∞
n n,0 n,j,1 n,j,2 (cid:88)
u˙ (0)=−ω u jsin(jωφ)=0 (34)
j=1 0 0,j,1
(24) j=1
where
Duetoorthogonalityofsin(jωφ),theseconditionscan
ω(p −p )(cid:0)fn−N−1+f−n(cid:1)
u = 2 1 0 0 (25) hold only for φ = π/ω. In other terms, the DB solution
n,0 πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1) turns out to be symmetric with respect to the time in-
2 0 0 0
2ωp (cid:0)fn−N−1+f−n(cid:1) version. Fortunately, this conclusion crucially simplifies
un,j,1 =−πγ2(cid:0)fj1−fjj−1(cid:1)(cid:0)fj−N−j1−1(cid:1) (26) tfohremp:roblem as the expression for χ1 takes the following
2ωp (cid:0)fn−N−1+f−n(cid:1)
u = 2 j j (27) (f−N−1+1)
n,j,2 πγ2(cid:0)fj −fj−1(cid:1)(cid:0)fj−N−1−1(cid:1) χ = ω (f0−f0−01)(f0−N−1−1)+ (35)
1 πγ2+(cid:80)∞ 2(−1)j(fj−N−1+1)
To obtain the values of the unknown parameters, we j=1 (fj−fj−1)(fj−N−1−1)
explicitly take into account the conditions of impacts,
that should be enforced when the particle achieves the
With φ no longer an unknown, eq. (30)-(31) can now
barriers:
easily be solved:
ω(cid:0)f−N−1+1(cid:1)(p −p )
u (0)= 0 2 1 − 1 a
0 π2ωγ2(cid:0)(cid:0)ff−0N−−1f0−+11(cid:1)(cid:1)(cid:0)fp0−N−1−1(cid:1) p1 = χ2−χ1− χ1+χ2 (36)
−(cid:80)∞ j 1 cos(jωφ)+ (28) 1 a
j=1 πγ2(cid:0)fj −fj−1(cid:1)(cid:0)fj−N−1−1(cid:1) p2 = χ −χ + χ +χ (37)
2ω(cid:0)f−N−1+1(cid:1)p 2 1 1 2
+(cid:80)∞ j 2 =1+a
j=1 πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1) Summarizing, we obtain the following exact solution
2 j j j
3
B Discrete Breathers in the Forced-Damped SetItIingEXACT SOLUTION FOR THE ASYMMETRIC BREATHER
for the conservative DB: inthesameformasabove. Thesolutionshouldobeythe
(cid:16) (cid:17) following set of equations:
4ω fn−N−1+f−n a
0 0
+(cid:80)∞un(t)=(cid:16) πγ28ω(cid:16)af0(cid:16)−f(cid:17)2njf(cid:16)−0−N1−(cid:17)1(cid:16)+f0−fN2−j−n(cid:17)1(cid:17)−1(cid:17)(χ1+χco2s)+(jωt)+ v¨0+γ+1v20p+1(cid:80)γ2∞j(=2−v∞0−δ(cid:18)vt1−−φvN−)2=ωπjF(cid:19)(t−+ψ)+ (44)
+(cid:80)∞j=1 πγ2(cid:16) f2j−8ωf2−(cid:16)j1f2nj−−(cid:17)fN12−(cid:16)−jN1−+1f−2−j1n−1(cid:17)(χ(cid:17)1+χ2) cos(jωt) −2p2(cid:80)∞j=−∞δ(cid:18)t− 2ωπj(cid:19)
j=1 πγ2 f2j−1−f2−j1−1 f2−jN−1−1−1 (χ2−χ1)
(38) v¨n+γ1vn+γ2(2vn−vn+1−vn−1)=F(t+ψ) (45)
v¨ +γ v +γ (2v −v −v )=F(t+ψ) (46)
N 1 N 2 N 0 N−1
1. Single-Sided Discrete Breathers
where ψ is the phase of the external force with respect
to DB’s impacts.
The asymmetric barriers allow a new type of DB so-
TheexternalforceF(t)canberemovedfromtheequa-
lution – the single-sided DB, i.e. the regime, in which
tions with the help of a simple transformation. Let
the impacting mass does not reach more distant barrier
v (t) = u (t)+G(t+ψ) where G¨(t)+γ G(t) = F(t).
and impacts only one of the barriers. Consequently, this n n 1
Substitution into the above equations yields:
regime can be described by the following equations of
motion:
u¨ +γ u +γ (2u −u −u )=
0 1 0 2 0 1 N
(cid:88)∞ (cid:18) 2πj(cid:19) =2p (cid:80)∞ δ(cid:18)t−φ− 2πj(cid:19)−
u¨0+γ1u0+γ2(2u0−u1−uN)=2pj=−∞δ t− ω −12p j(cid:80)=−∞∞ δ(cid:18)t− 2πjω(cid:19) (47)
(39) 2 j=−∞ ω
u¨ +γ u +γ (2u −u −u )=0 (40)
n 1 n 2 n n+1 n−1
u¨ +γ u +γ (2u −u −u )=0 (48)
n 1 n 2 n n+1 n−1
u¨ +γ u +γ (2u −u −u )=0 (41)
N 1 N 2 N 0 N−1 u¨ +γ u +γ (2u −u −u )=0 (49)
N 1 N 2 N 0 N−1
As previously, the system is closed with the help of Similarly, the impact forcing terms are re-written in
equations, that fix the impact at the desired location: the form of generalized Fourier series:
u (0)=−1+a (42)
0 u¨ +γ u +γ (2u −u −u )=
= ω (cid:80)∞0 1(p0cos(2jω(t0−φ)1)−pNcos(jωt)) (50)
Finally, the solution is written as follows: π j=−∞ 1 2
u¨ +γ u +γ (2u −u −u )=0 (51)
ω(1−a)(cid:0)fn−N−1+f−n(cid:1) n 1 n 2 n n+1 n−1
u =− 0 0 −
n πγ χ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1)
2ω(12−2a)0(cid:0)fn−0N−1+0f−n(cid:1) (43) u¨N +γ1uN +γ2(2uN −u0−uN−1)=0 (52)
−(cid:80)∞j=1 πγ χ (cid:0)f −f−j1(cid:1)(cid:0)f−N−1j−1(cid:1)cos(jωt) Theequationsareidenticaltothoseoftheconservative
2 2 j j j model. Hence, the solution is similar:
The complete derivation of the solution is available in
Appendix A. ∞
(cid:88)
u =u + (u cos(jω(t−φ))+u cos(jωt))
n n,0 n,j,1 n,j,2
j=1
(53)
B. Discrete Breathers in the Forced-Damped
where
Setting
ω(p −p )(cid:0)fn−N−1+f−n(cid:1)
u = 2 1 0 0 (54)
Now let us adopt that all masses are subjected to an n,0 πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1)
2 0 0 0
external force F(t). We examine the case of a sym-
2ωp (cid:0)fn−N−1+f−n(cid:1)
metric force F(t) which satisfies F(t) = F(t+2π/ω) u =− 1 j j (55)
and F(t) = −F(t+π/ω). Additionally, the damping n,j,1 πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1)
2 j j j
isintroducedthroughthe non-unitrestitution coefficient 2ωp (cid:0)fn−N−1+f−n(cid:1)
0<e<1. Similarlytotheconservativecase,welookfor u = 2 j j (56)
theperiodicsolution,thustheimpactscanbeintroduced n,j,2 πγ (cid:0)f −f−1(cid:1)(cid:0)f−N−1−1(cid:1)
2 j j j
4
B Discrete Breathers in the Forced-Damped SetItIingEXACT SOLUTION FOR THE ASYMMETRIC BREATHER
Asintheconservativesetting,thesolutionmustsatisfy Similarly, it is possible to perform the same procedure
the impact location equations: for the second impact:
v (0)=−p χ (φ)+p χ +G(ψ)=1+a (57)
0 1 1 2 2
u˙ (cid:0)φ+(cid:1)+eu˙(cid:0)φ−(cid:1)=−G˙(φ+ψ)(1+e) (67)
0
v0(φ)=−p1χ2+p2χ1(φ)+G(ψ+φ)=−1+a (58) ThegeneralizedFourierseriesconvergestotheaverage
of the velocities on both sides of the discontinuity:
Also, the impact law must be satisfied:
v˙0(0+)=u˙0(0+)+G˙(ψ)= u˙(φ+)+u˙(φ−)
(cid:16) (cid:17) (59) =−p χ (68)
=−e u˙0(0−)+G˙(ψ) =−ev˙0(0−) 2 2 3
Conservation of momentum during the impact yields:
u˙0(cid:0)0+(cid:1)+eu˙0(cid:0)0−(cid:1)=−G˙(ψ)(1+e) (60) u˙(φ+)−u˙(φ−)
=p (69)
2 1
Forthesymmetriccaseitisclearthattheenergymust
be conserved during each impact in terms of u , namely, From these equations we extract the terms for the ve-
0
in terms of the reduced un-forced system. Hence, for locities:
the un-forced system and the symmetric DB u˙ (0+) =
−u˙0(0−) and similarly for the second impact. H0owever, u˙(cid:0)φ+(cid:1)=p1−p2χ3 (70)
thisisnottruefortheasymmetricDB–theenergymust
be conserved for the reduced un-forced system (other-
u˙(cid:0)φ−(cid:1)=−p −p χ (71)
wise, the DB solution cannot exist), but it holds for the 1 2 3
complete period of oscillations, and not necessarily in
Note that the energy loss in this impact is ∆E =
each single impact. So, more refined treatment is re- 2
−2p p χ ;hencetheenergyofthereducedun-forcedsys-
quired in this case. 1 2 3
tem is conserved throughout the period as expected.
ThegeneralizedFourierseriesconvergestotheaverage
Plugging into eq. (67), one obtains:
of the velocities on both sides of the discontinuity:
G˙(φ+ψ)=p χ −qp (72)
2 3 1
u˙ (0+)+u˙ (0−)
0 0
=−p χ (61)
2 1 3
1. Harmonic Excitation
where
χ ≡(cid:88)∞ 2jω2(cid:0)fj−N−1+1(cid:1) sin(jωφ) (62) forIcninogrdfuerncttoiosno,lvtehatthesaetqisufiaetsiotnhsewsyemnmeeedtrtyoccohnodoisteiotnhse.
3 j=1 πγ2(cid:0)fj −fj−1(cid:1)(cid:0)fj−N−1−1(cid:1) For simplicity, let us choose F(t) = Acos(ωt). Solving
the ODE, we obtain:
Conservation of momentum during the impact yields:
G(t)=A˜cos(ωt) (73)
u˙ (0+)−u˙ (0−)
0 0
=−p (63)
2 2 where,
From these equations we extract terms for the veloci- A
A˜= (74)
ties:
γ −ω2
1
u˙0(cid:0)0+(cid:1)=−p2−p1χ3 (64) Pluggingthesolutionintoeq. (57),(58),(66)and(72),
one obtains the following expressions:
u˙0(cid:0)0−(cid:1)=p2−p1χ3 (65) −p χ (φ)+p χ +A˜cos(ωψ)=1+a (75)
1 1 2 2
Note that the energy gain for the reduced un-forced
system during the impact is ∆E1 = 2p1p2χ3. Energy −p χ +p χ (φ)+A˜cos(ω(ψ+φ))=−1+a (76)
1 2 2 1
gain is possible since the reduced un-forced system does
not represent a physical system.
Plugging into eq. (60), one obtains:
−A˜ωsin(ω(ψ+φ))=p χ −qp (77)
2 3 1
G˙(ψ)=p χ +qp (66)
1 3 2
where q =(1−e)/(1+e). −A˜ωsin(ωψ)=p χ +qp (78)
1 3 2
5
B Discrete Breathers in the Forced-Damped SetItIingEXACT SOLUTION FOR THE ASYMMETRIC BREATHER
To find the exact solution explicitly, we assume that φ A˜ω(qsin(ω(ψ+φ))−χ sin(ωψ))
3
p = (80)
is known and the barrier asymmetry a is the unknown. 1 q2+χ2
3
Solutionoftheabovesetofequationsunderthisassump-
tion yields:
A˜ω(χ sin(ω(ψ+φ))+qsin(ωψ))
3
p2=− (81)
q2+χ2
3
(cid:32)2(cid:0)q2+χ2(cid:1)(cid:33)
±arccos σA˜ 3 +α a=−p1χ1(φ)+p2χ2+A˜cos(ωψ)−1 (82)
ψ = (79)
ω
where,
(cid:118)
(cid:117)2ω2χ2(χ −χ )2(1+cos(ωφ))+4ωχ (cid:0)q2+χ2(cid:1)(χ −χ )sin(ωφ)+
(cid:117) 3 1 2 3 3 1 2
σ =(cid:116) (cid:16) (cid:16) (cid:17)(cid:17) (83)
+2 q4+χ4+q2 ω2(χ −χ )2+2χ3 (1−cos(ωφ))
3 1 2
(cid:32)(cid:0)q2+χ2(cid:1)(1−cos(ωφ))−(q−χ )(χ −χ )ωsin(ωφ)(cid:33)
α=±arccos 3 3 1 2 (84)
σ
2. Single-Sided Forced-Damped Discrete Breathers Harmonic Excitation Let F(t)=Acos(ωt) and,
The single-sided DB is also possible in the forced- G(t)=A˜cos(ωt) (91)
damped model. The equations of motion can be written
as follows: where,
=v¨F0(+t+γ1ψv)0++2γp2(cid:80)(2v∞0−v1δ(cid:18)−tv−N)2π=j(cid:19) (85) A˜= γ1−Aω2 (92)
j=−∞ ω
In a similar manner to the regular DB, we obtain the
following equations:
v¨ +γ v +γ (2v −v −v )=F(t+ψ) (86)
n 1 n 2 n n+1 n−1
−pχ +A˜cos(ωψ)=−1+a (93)
2
v¨ +γ v +γ (2v −v −v )=F(t+ψ) (87)
N 1 N 2 N 0 N−1
where ψ is the phase of the external force with respect
to the DB’s impacts. −ωA˜sin(ωψ)=−qp (94)
TheexternalforceF(t)canberemovedfromtheequa-
tions in the same manner as the previous case. Solving the equation yields:
Asintheconservativemodel,thesolutionmustsatisfy
the impact location equations: (cid:32) q(a−1) (cid:33)
±arccos +α
A˜(cid:112)q2+ω2χ2
v0(0)=−pχ2+G(ψ)=−1+a (88) ψ = 2 (95)
ω
Also, the impact law must be satisfied:
v˙(0+)=u˙(0+)+G˙(ψ)=p+G˙(ψ)= ωA˜
(cid:16) (cid:17) (cid:16) (cid:17) p= sin(ωψ) (96)
=−e −p+G˙(ψ) =−e u˙(0−)+G˙(ψ) =−ev˙(0−) q
(89)
where,
By further simplification, one obtains:
(cid:32) (cid:33)
G˙(ψ)=−qp (90) q
α=±arccos (97)
(cid:112)
q2+ω2χ2
The full derivations are given in AppendixB. 2
6
B Discrete Breathers in the Forced-Damped Setting III STABILITY
3. Single-Sided Forced-Damped Discrete Breathers with is given in detail in Appendix C.
Period Doubling
The stability analysis discussed in detail in the follow-
III. STABILITY
ing Section shows that one of the mechanisms for the
loss of stability of the single-sided forced DB is the pe-
rioddoublingbifurcation. Numericalinvestigationshows The stability of the derived DB solutions will be in-
that the period doubling is reflected by a consecutive set vestigated by Floquet theory[28]. The Floquet multipli-
of collisions with different exchange of momentum in the ers are often evaluated numerically, but, as mentioned
new doubled period of the DB. This type of solution can above, the special nature of the system allows explicit
also be obtained analytically in a similar manner to that construction of the monodromy matrix. Then, computa-
oftheforcedDBwithminormodifications. Thelocation tionofitseigenvaluesisarelativelysimplecomputational
ofthesecondimpactintheperiodissettothesamebar- task, and comprehensive study of the stability patterns
rier as the first collision, i.e. the closer barrier, and the in the space of parameters becomes possible [21]. More-
period of the DB is doubled. However, obtaining a solu- over, eigenvectors corresponding to the unstable Floquet
tioninthismannerisonlypossibleifthefrequencyofthe multiplierscanbeeasilycomputedandexaminedtogain
doubled period solution is in the attenuation zone of the some qualitative insight into the mechanism of the loss
chain for a given set of parameters. The full derivation of stability.
The governing equations of motion can also be written in the following equivalent form:
(cid:126)u˙ =A(cid:126)u (98)
(cid:2) (cid:3)T
where (cid:126)u= u ··· u u˙ ··· u˙ and:
0 N 0 N
(cid:20) (cid:21)
0 I
A= (N+1)×(N+1) (N+1)×(N+1) (99)
A˜ 0
(N+1)×(N+1) (N+1)×(N+1)
γ +2γ −γ 0 ··· 0 −γ
1 2 2 2
−γ2 γ1+2γ2 −γ2 0 ··· 0
A˜= 0... −..γ.2 ...... γ1+...2γ2 −..γ.2 0... (100)
0 ··· 0 −γ2 γ1+2γ2 −γ2
−γ 0 ··· 0 −γ γ +2γ
2 2 1 2
or for the forced-damped model:
(cid:126)v˙ =A(cid:126)v+F(cid:126) (101)
where F(cid:126) =F(t)(cid:2)1 ··· 1(cid:3)T.
From the above equation we can derive the evolution must be constructed to take into account the linear per-
of the perturbed phase trajectory between two impacts: turbations of the mapping and of the flight time to the
discontinuity[29]. The saltation matrix for the adopted
L =exp(φA) (102)
1 impact law obtains the following form:
(cid:20)S˜ 0 (cid:21)
S = (N+1)×(N+1) (N+1)×(N+1) (105)
(cid:18)(cid:18)2π (cid:19) (cid:19) 1,2 Sˆ(N+1)×(N+1) S˜(N+1)×(N+1)
L =exp −φ A (103)
2 ω where,
or for the single-sided impact: −e 0 ··· ··· 0
.
.
(cid:18)2π (cid:19) 0 1 0 .
L=exp ωA (104) S˜ = ... 0 ... ... ... (106)
The impacts mapping cannot simply be based on the ... ... 1 0
impactlawforthestabilityanalysis,butsaltationmatrix 0 ··· ··· 0 1
7
A Conservative Model IV NUMERICAL VALIDATION AND STABILITY PATTERNS
(1+e)∆
1,2
0 ··· ··· 0
Γ
1,2
..
0 0 0 .
Sˆ1,2 = ... 0 ... ... ... (107)
... ... 0 0
0 ··· ··· 0 0
where ∆ = u¨ (φ−),∆ = u¨ (0−), , Γ = −p ,Γ = p
1 0 2 0 1 1 2 2
for the conservative model; ∆ = u¨ (φ−),∆ = u¨ (0−),
1 0 2 0
, Γ =−p −p χ +G˙(φ+ψ),Γ =p −p χ +G˙(ψ)for
1 1 2 3 2 2 1 3
the forced-damped model. Similarly for the Single-sided
DB – ∆ = u¨ (0−) and Γ = −p for the conservative
1 0 1
model; ∆ = u¨ (0−), , Γ = −p+G˙(ψ) for the forced-
1 0 1
dampedmodel. Notethatfortheconservativemodelthe
coefficient of restitution e is set to unity. FIG. 1. The displacements of the masses for a DB solution
at the instances of the two impacts for N = 10 (Black) and
The Monodromy matrix can be written compactly as
N =100 (Dashed gray).
follows:
M=L S L S (108)
1 1 2 2
or for the single-sided DB:
M=LS (109)
1
Figure 2 shows the strong effect of the asymmetry pa-
rameter a on the DB shape. However, when examining
Then, the stability of the DB solution is assessed just
the displacement of the impacting mass throughout the
byeasycomputationofthisMonodromymatrixandeval-
period of the DB , the difference is only a small change
uation of its spectrum.
in the curvature as demonstrated in fig. 3. It appears
thatfortheconservativeDB,theasymmetrymainlycon-
tributes to the value to which the DB converges apart
IV. NUMERICAL VALIDATION AND from the localization site.
STABILITY PATTERNS
A. Conservative Model
In order to qualitatively examine the properties of the
asymmetric DBs, and to validate the accuracy of our re-
sults, we turn to numerical methods. The simulations
in this section are performed using MatLab. The vibro-
impact response was modeled according to the impact
law using event-driven algorithm. The numerical results
were in agreement with the analytical results, as one
should expect for the exact solutions. Thus, one can see
theseresultsasillustrations. Unlessotherwisestated,the
simulationsweredoneforthefollowingsetofparameters:
γ =0.2 γ =0.1 ω =1.5
1 2 (110)
N =20 a=0.4
As was the case in the symmetric DB, the oscillatory
profile is qualitatively the same when the length of the FIG. 2. The displacements of the masses for a DB solution
chainismodified,asshowninfig. 1. Thisresultconforms attheinstancesofthetwoimpactsfora=0(Black),a=0.2
(Gray) and a=0.5 (Dashed Black).
to the strong localization of the DB solution.
8
B Forced-Damped Model IV NUMERICAL VALIDATION AND STABILITY PATTERNS
unless stated otherwise, the parameters are as follows:
γ =0.2 γ =0.1 ω =1.5 N =20
1 2 (111)
A=0.1 e=0.9 a=0.4
FIG.3. ThedisplacementsofthefirstmassforaDBsolution
fora=0(Black),a=0.2(Gray)anda=0.5(DashedBlack).
Another type of DB enabled by the asymmetry of the
system is the single-sided DB. Figure 4 presents the ex-
ample of the single-sided DB; note that the impacting
mass does not reach the more distant barrier at (1+a). FIG.5. ThedisplacementsofthemassesfortheforcedDBso-
lutionattheinstancesofthetwoimpactsforN =10(Black)
and N =100 (Dashed gray).
FIG. 4. The displacements of the masses for a single-sided
DB solution at the instances of the two impacts.
FIG. 6. The displacements of the masses for the forced DB
solution at the instances of the two impacts for A=0.9 and
B. Forced-Damped Model
a=0 (Black), a=0.2 (Gray) and a=0.4 (Dashed Black).
This model is a bit more complicated to examine. As
mentioned in sec. IIB, we are unable to find φ without In general, the effect of the asymmetry in the forced-
approximations with unknown error. Therefore, φ is re- damped model is similar to one observed in the conser-
garded as a known parameter and instead we obtain the vative model. In fig. 5 we see that there is no notable
asymmetry parameter a. Fortunately, numerical investi- change in the DB profile as a result of adding masses to
gation revealsthat therelation between a and φ behaves the chain. Figure 6 shows that, while the shape is gener-
in a manner that allows finding the wanted value of a ally different since the oscillating term converges to G(t)
bymeansofiterativeextrapolationwiththemaximaler- and not to zero, as the mass is farther away from the
ror of our choice. For time consumption purposes, the localization site, the most profound consequence of the
allowed error in a was taken to be 10−12. Furthermore, asymmetry is still the shift of the center of oscillations.
9
B Forced-Damped Model IV NUMERICAL VALIDATION AND STABILITY PATTERNS
sided DBs exist as well. An example is presented in fig.
8.
FIG. 7. The displacements of the first mass for A=0.9 and
a=0 (Black), a=0.2 (Gray) and a=0.5 (Dashed Black).
FIG. 9. The displacements of the first mass for some of the
solutions for γ = 0.1, γ = 0.05, ω = 1.33, A = 1.5 and
Figure 7 demonstrates a main difference from the con- 1 2
a=0.
servative DB; it clearly shows that there is a shift of the
second impact, i.e. φ is diverted from π/ω. Numerical
investigation shows this difference is typically very small An interesting phenomenon appearing for stronger ex-
until the appearance of multiple solutions mentioned be- ternal forcing, i.e. larger values of A, is a multitude of
low. solutions. For certain sets of parameters the analytic so-
lution yields more than a single solution. Figure 9 shows
that these solutions can even be very different from each
other. It is interesting to note that it is possible that
more than one of these solutions are stable.
Another interesting fact is that even for a symmetric
model, namely a = 0, an asymmetric solution could ex-
ist as predicted by Grinberg and Gendelman [22]. one
ormore ofthesesolutions appearto become stable when
thesymmetricDBlossesstabilityviathepitchforkbifur-
cation.
C. Stability
The procedure for the stability analysis is described
in detail in sec. III. Additionally, the following stability
FIG.8. Thedisplacementsofthemassesfortheforcedsingle- maps also refer to existence of the solution, i.e. solu-
sided DB solution at the instances of the two impacts for tions that are not physical, e.g. some masses exceed the
ω=0.92. boundaries, are marked as non-existent. The set of pa-
rameters is similar to that in the previous sub-sections,
Just like in the conservative model, the forced single- unless stated otherwise.
We begin with investigation of the forced-damped model; since the stable solutions of the forced-damped model
generically are hyperbolic attractors, the numeric validation of the stability analysis is easier than in the conservative
model.
10
