Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 4, pp. 1197-1204, Warsaw 2016
DOI: 10.15632/jtam-pl.54.4.1197

METAMORPHOSES OF RESONANCE CURVES IN SYSTEMS

OF COUPLED OSCILLATORS

Jan Kyzioł, Andrzej Okniński

Kielce University of Technology, Kielce, Poland

e-mail: kyziol@tu.kielce.pl; fizao@tu.kielce.pl

We study dynamics of two coupled periodically driven oscillators in a general case and com-
pare itwith two simplifiedmodels. Periodic steady-state solutions to these system equations
are determinedwithin theKrylov-Bogoliubov-Mitropolskyapproach.Amplitude profiles are
computed. These two equations, each describing a surface, define a 3D curve – intersection
of these surfaces. In the present paper, we analyse metamorphoses of amplitude profiles
induced by changes of control parameters in three dynamical systems studied. It is shown
that changes of the dynamics occur in the vicinity of singular points of these 3D curves.

Keywords: coupled oscillators, amplitude profiles, singular points

1. Introduction

Westudydynamics of two couplednonlinear oscillators, one ofwhichbeingdrivenbyan external
periodic force. Equations of motion are

mẍ−V (ẋ)−R(x)+Ve(ẏ)+Re(y)= F(t)

me(ẍ+ ÿ)−Ve(ẏ)−Re(y)= 0
(1.1)

wherex is the position of primarymassm, y is the relative position of anothermassme attached
to m and R, V and Re, Ve are nonlinear elastic restoring and nonlinear forces of internal friction
formassesm,me, respectively (weuse convention ẋ ≡ dx/dt, etc.).Adynamicvibrationabsorber
is a typical mechanical model described by (1.1) (in this case m is usuallymuch larger than me)
(Den Hartog, 1985; Oueini et al., 1999).

Dynamics of coupled, externally and/or parametrically driven oscillators, is very complex.
Indeed, there are many interesting nonlinear phenomena present in this class of dynamical sys-
tems. There exists a large body of analytical and numerical studies documentingmultistability,
symmetry breaking, attractors merging, synchronisation, existence of exotic attractors and va-
rious transitions to chaos (Bi, 2004; Brezetskyi et al., 2015; Chen and Xu, 2010; Danzl and
Moehlis, 2010; Dudkowski et al., 2014; Kuznetsov et al., 2009; Laxalde et al., 2006; McFarland
et al., 2005; Pikovsky et al., 2003; Sabarathinam et al., 2013; Warmiński, 2010).

In our earlier papers,wehave designed amethodbased on the theory of singular points of 2D
curves, permitting computation of parameter values at which qualitative changes (metamorpho-
ses) of 2D amplitude curves occur, see Kyzioł and Okniński (2013) and references therein. We
have also shown that metamorphoses of amplitude profiles are visible in bifurcation diagrams
as qualitative changes of dynamics (bifurcations). Recently, our approach has been generalized
to the case of 3D resonance curves and applied to compute bifurcations in dynamical system
(1.1) with small nonlinearities in the main mass frame (Kyzioł, 2015). It is thus possible to
treat system (1.1) as a small perturbation of model with linear functions R(x), V (ẋ) analyzed
in Kyzioł and Okniński (2013) (let us recall that in this case internal motion can be separated



1198 J. Kyzioł, A. Okniński

off, leading to a simpler equation for the corresponding amplitude profile) and use the results
obtained byKyzioł andOkniński (2013).We show that themethod is a powerful tool to predict
bifurcations of nonlinear resonances present in such dynamical systems.
In the paper byAwrejcewicz (1995), the author outlined a programme, based on the Implicit

Function Theorem, to “define and find different branches intersecting at singular points” of
amplitude equations. In the present paper, we are working in a more general context of theory
of singular points of algebraic curves (Wall, 2004; Hartmann, 2003).
We investigate the following hierarchy of dynamical systems of form (1.1): we consider func-

tions R, V , Re, Ve forwhich (a) systemof equations can be reduced to one second-order effective
equation of relativemotion, (b) fourth-order equation for variable y can be separated off, (c) it is
impossible to separate variables.We analyse approximate analytic solutions (amplitude profiles)
obtainedwithin theKrylov-Bogoliubov-Mitropolsky (KBM)method (Nayfeh, 1981;Awrejcewicz
and Krysko, 2006), using theory of algebraic curves. More exactly, singular points of amplitu-
de profiles are computed. We demonstrate that qualitative changes of dynamics, referred to as
metamorphoses, induced by changes of control parameters, occur in neighbourhoods of singular
points of amplitude profiles, see also Kyzioł and Okniński (2011, 2013) and references therein.
The paper is organized as follows. In the next Section, equations (1.1), (2.1) are transformed

into non-dimensional form. In Section 4, implicit equations for resonance surfaces A(ω), B(ω)
are derived within the Krylov-Bogoliubov-Mitropolsky approach, where the amplitudes A, B
correspond to small and large masses, respectively. The problem is more difficult than before
because these two equations are coupled. In Section 5, we review necessary facts from theory
of algebraic curves which are used to compute singular points on three-dimensional resonance
curve (intersection of resonance surfaces A(ω), B(ω)). In Section 6, computational results are
presented. Our results are summarized in the last Section.

2. Equations of motion

In what follows the function F(t) is assumed in form F(t) = f cos(ωt). When all the functions
R, V , Re, Ve are nonlinear, namely

R(x)=−αx−γx3 Re(y)=−αey−γey
3

V (ẋ)=−νẋ−βẋ3 Ve(ẏ)=−νeẏ−βeẏ
3 (2.1)

then we deal with the general case of Eq. (1.1). For linear functions R, V

R(x)=−αx Re(y)=−αey−γey
3

V (ẋ)=−νẋ Ve(ẏ)=−νeẏ+βeẏ
3 (2.2)

it is possible to separate off the variable y to obtain the following equation for relative motion
(Kyzioł and Okniński, 2013)

L̂(µÿ−Ve(ẏ)−Re(y))+ ǫmeK̂y = F cos(ωt) (2.3)

where

L̂ = M
d2

dt2
+ν

d

dt
+α K̂ =

(
ν
d

dt
+α
) d2

dt2
F = meω

2f

ǫ =
me
M

µ =
mme
M

M = m+me

Finally, assuming me ≪ m, i.e. ǫ ≪ 1, we can reject the term proportional to ǫ to obtain
an approximate equation which can be integrated partly to yield the effective equation (Kyzioł
and Okniński, 2011)



Metamorphoses of resonance curves in systems of coupled oscillators 1199

µÿ+νeẏ−βeẏ
3+αey+γey

3 = F(t)

F(t)=
−meω

2f
√
M2
(
ω2− α

M

)2
+ν2ω2

cos(ωt+ δ) (2.4)

where transient states are neglected.

3. Equations in non-dimensional form

Equations (1.1), (2.1) are transformed into non-dimensional form (Kyzioł, 2015). We introduce
non-dimensional time τ and frequency Ω and rescale variables x, y

t =

√
µ

αe
τ ω =

√
αe
µ
Ω x =

√
αe
γe
u y =

√
αe
γe
z (3.1)

to get

ü+ Ĥu̇+ cu̇3+ âu+du3− κ̂(hż + bż3+z +z3)= λcos(Ωτ)

z̈ +hż + bż3+z +z3− Ĥu̇− cu̇3− âu−du3 =−λcos(Ωτ)
(3.2)

and new parameters read

a =
µα

Mαe
b =

βe
γe

(αe
µ

)3
2

c =
β(αe)

3

2

√
µmγe

d =
µγ

mγe

h =
νe
√
µαe

H =
ν

M

√
µ

αe
G =

1

αe

√
γe
αe
f κ =

me
m

λ =
κ

κ+1
G Ĥ = H(1+κ) â = a(1+κ) κ̂ =

κ

κ+1

(3.3)

where

M = m+me µ =
mme
M

u̇ ≡
du

dτ
ż ≡

dz

dτ

Note that ü is eliminated from the second of Eqs. (1.1).

4. Nonlinear resonances

System of equations (3.2) is written in form

d2u

dτ2
+Ω2u+ε(σu+g(u̇,u, ż,z,τ)) = 0

d2z

dτ2
+Ω2z+ε(σz +k(u̇,u, ż,z,τ)) = 0

(4.1)

where

εσ = Θ2−Ω2 Ĥ = εĤ0 â = εâ0 b = εb0 c = εc0
Θ2 = εΘ20 d = εd0 h = εh0 εδ0 =1 λ = ελ0

(4.2)

and functions g(u̇,u, ż,z,τ), k(u̇,u, ż,z,τ) are defined in (Kyzioł, 2015). Equations (4.1) ha-
ve been prepared in such a way that for ε = 0 the solutions are u(τ) = Bcos(Ωτ + ψ),
z(τ)= Acos(Ωτ +ϕ).



1200 J. Kyzioł, A. Okniński

We shall now look for 1 : 1 resonance using the Krylov-Bogoliubov-Mitropolsky (KBM)
perturbation approach (Nayfeh, 1981; Awrejcewicz and Krysko, 2006). For a small nonzero ε,
the solutions to Eqs. (4.1) are assumed in form

u(τ) =Bcos(Ωτ +ψ)+εu1(B,ψ,τ)+ . . .

z(τ) = Acos(Ωτ +ϕ)+εz1(A,ϕ,τ)+ . . .
(4.3)

with slowly varying amplitudes and phases

dA

dτ
= εM1(A,ϕ)+ . . .

dB

dτ
= εP1(B,ψ)+ . . .

dϕ

dτ
= εN1(A,ϕ)+ . . .

dψ

dτ
= εQ1(B,ψ)+ . . .

(4.4)

Proceeding as described in (Kyzioł, 2015), we obtain finally equations for the amplitudes A, B

L1(A,B,Ω;Λ)= 0 L2(A,B,Ω;Λ) = 0 (4.5)

where Λ denotes parameters and

L1 = Z((η
2
4 +u

2
2)(u

2
1+η

2
2)+ κ̂X

2u3)− (κ̂−1)
2(u21+η

2
2)λ
2

u1 = κ̂X +η1 u2 = κ̂X +η3 u3 = κ̂X
2+2η2η4−2u1u2

(4.6)

L2 =
Y

(κ̂−1)2
[u24+(κ̂Xu5+η1η3−η2η4)

2]−λ2X2

u4 = κ̂X(η2+η4)+η1η4+η2η3 u5 =(κ̂−1)X +η1+η3

(4.7)

η1 =
(3
4
Y +1−X

)
(κ̂−1) η2 = Ω

(3
4
bXY +h

)
(κ̂−1)

η3 =
(
â+
3

4
dZ −X

)
(κ̂−1) η4 = Ω

(
3
4
cXZ + Ĥ

)
(κ̂−1)

(4.8)

X = Ω2 Y = A2 Z = B2 (4.9)

If we put c = d = 0 in Eqs.(4.6), (4.7) and (4.8) (or β = γ = 0 in Eqs. (2.1)) then the
function L2 becomes independent on B. In this case, it is possible to separate variables in Eqs.
(1.1), (2.2) obtaining the fourth-order effective equation for the smallmass (Kyzioł andOkniński,
2013). The function L2, defined above, for c = d =0 is equal to the function L(X,Y ) defined in
Eq. (4.1) in (Kyzioł andOkniński, 2013).

5. Metamorphoses of the amplitude profiles

In the preceding Section, we have obtained two implicit equations (4.5) for amplitude profiles.
Each of these equations describes a surface in a three dimensional space (A,B,Ω). Intersection
of the surfaces L1 =0, L2 =0 is a 3D curve, and in singular points of this curve all threeminors
of the rectangular matrix

M=

[
L′1,A L

′

1,B L
′

1,Ω

L′2,A L
′

2,B L
′

2,Ω

]
(5.1)

are zero (Hartmann, 2003), where L′1,A = ∂L1/∂A, etc. Equations (4.5) and these conditions are
used to compute singular points.We have shown in our previous papers that qualitative changes
of dynamics, induced by changes of control parameters, occur in neighbourhoods of singular
points of amplitude profiles (Kyzioł and Okniński, 2011, 2013; Kyzioł, 2015).



Metamorphoses of resonance curves in systems of coupled oscillators 1201

6. Amplitude profiles and bifurcation diagrams

Applying the KBM method to effective equation (2.4) we obtain approximate formula
y(t)= Acos(ωt+ϕ)wheredependenceofAonω is givenbyan implicit equationF1(A,ω;Λ) = 0.
The formof the functionF1 can be found inKyzioł andOkniński (2011). InFig. 1a, this implicit
function is shown just after an isolated point (A,ω) = (1.124,1.784) has been born.

Fig. 1. Amplitude profile F1(A,ω;Λ)= 0 (a) and F2(A,ω;Λ)=0 (b) with an isolated point

Then, applying theKBMmethod to fourth-order equation (2.3)we obtain the corresponding
implicit amplitude equation F2(A,ω;Λ) = 0. The form of F2 has been described in Kyzioł and
Okniński (2013). In Fig. 1b, we see that an isolated point (A,ω) = (1.274,1.899) has been
just born. Similarity of the amplitude profiles shows that effective equation (2.4) is a good
approximation to fourth-order equation (2.3). Bifurcation diagrams show indeed the birth of
new branches of solutions in both models (Kyzioł and Okniński, 2011, 2013).
Nowweconsider the general casewith small nonlinearities in themainmass frame, c =0.001,

d = 0.02, so that the system of equations (1.1), (2.1) is a small perturbation of model (1.1),
(2.2), with other parameters being equal a = 6, b = 0.001, h = 0.5, H = 0.7, κ = 0.05,

γ =2.011615
df
= γcr.

Fig. 2. Resonance surfaces before the singular point is formed, left figure. The conical structure does not
intersect the lower surface, right figure

Resonance surfaces (4.5) are shown inFigs. 2 before the singular point is formed, the singular
point being (A,B,Ω)= (1.276,0.620,1.902) and γ =2.015 > γcr, where the surfaces in the right



1202 J. Kyzioł, A. Okniński

figure have been rotated to show that the additional conical surface does not pierce the other
surface yet, and, after formation of the singular point, γ =1.995 < γcr, in Fig. 3.

Fig. 3. Resonance surfaces L1(A,B,Ω;Λ)= 0 and L2(A,B,Ω;Λ)=0 with an additional tubular
structure intersecting the lower surface

The corresponding bifurcation diagrams, one with a new branch near Ω =1.9, are shown in
Figs. 4.

Fig. 4. Bifurcation diagram before formation of the singular point (left figure) and after (right figure)

7. Discussion

In thepresentwork,we continue studyof thegeneral case of dynamics of two coupledperiodically
driven oscillators, cf. Eq. (1.1), initiated in Kyzioł (2015). More exactly, we have investigated:
(a) model (1.1), (2.1) with small parameters α, γ; (b) model (1.1), (2.2) with α = 0, γ = 0
(in this case, dynamics of small mass can be separated off, see Eq. (2.3)); (c) and approximate
effective equation (2.4).



Metamorphoses of resonance curves in systems of coupled oscillators 1203

We have studied the amplitude (resonance) equations for steady states, obtained via the
KBMapproach, within the theory of singular points of 2D and 3D algebraic curves (Wall, 2004;
Hartmann, 2003). Analysis of the resonance curves in cases (b), (c) has been relatively simple
since we have been dealing with one implicit equation of form F(A,Ω;Λ) = 0, describing a 2D
curve only (Kyzioł andOkniński, 2011, 2013). The general case is more difficult since there are
two amplitude equations, L1(A,B,Ω;Λ) = 0, L2(A,B,Ω;Λ) = 0, which describe two surfaces.
Conditions for singular points of the 3D curve – intersection of these surfaces – are also more
complex: they are given by two equations (4.5) and three equations det(M)= 0withmatrixM
given by (5.1).

Bifurcations diagrams shown inFigs. 4 confirm that a qualitative change of dynamics – birth
of a new branch of a nonlinear resonance – occurs in the neighbourhood of the singular point of
the 3D curve defined by amplitude equations (4.5).

Acknowledgement

The present paper is an extended version of the article published in the Proceedings of the confe-

rence: PCM-CMM-2015 – 3rd Polish Congress of Mechanics and 21st Computer Methods inMechanics,

September 8th-11th 2015, Gdańsk, Poland.

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Manuscript received October 1, 2015; accepted for print February 18, 2016