Collective dynamics of disordered two coupled nonlinear pendulums

Collective dynamics of disordered two coupled nonlinear pendulums K. Chikhaoui1,2, D. Bitar 1, N. Bouhaddi1, N. Kacem1, M. Guedri2 1

Univ. Bourgogne Franche-Comté, FEMTO-ST Institute, UMR 6174, CNRS/UFC/ENSMM/UTBM, Department of Applied Mechanics, 25000 Besançon, France {Khaoula.chikhaoui, diala.bitar, noureddine.bouhaddi, najib.kacem}@femto-st.fr 2

National High School of Engineers of Tunis (ENSIT), University of Tunis, 5 Avenue Taha Hussein, BP 56, Bâb Manara, Tunis, Tunisia [email protected]

Abstract. The effect of disorder on the collective dynamics of two coupled nonlinear pendulums is investigated in this paper. Disorder is introduced by slightly perturbing the length of some pendulums in the nearly periodic structure. A generic discrete analytical model combining the multiple scales method and a standingwave decomposition is proposed leading to a set of coupled complex algebraic equations written according to the number and positions of disorder in the structure. The analysis of the disorder impact on the frequency responses and the basins of attraction of two coupled pendulums structure shows that in presence of disorder, the multimode solutions are enhanced and the multistability domain is wider. The disorder introduced by reducing the length of one pendulum favors modal localization on its response.

Keywords: collective dynamics, near-periodicity, coupled nonlinear pendulum, disorder, multi-mode solutions, basins of attraction, modal localization

1

Introduction

Under the hypothesis of perfect periodicity, many works provided interesting insights in the behavior of periodic linear and nonlinear structures (Nayfeh 1983, Lifshitz et al. 2003, Bitar et al. 2015, Manktelow et al. 2011). However, engineering systems are in reality affected by imperfections and the near-periodic term is therefore more adequate to characterize their arranging. In the literature, Kissel (Kissel 1983), for instance, studied the effect of disorder in one-dimensional periodic structure and proved that disorder allows wave attenuation and normal mode

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localization in the frequency range close to the attenuation bands of the associated perfectly periodic structure. A statistical analysis of the disorder effect on the dynamics of coupled one-dimensional periodic structures using the Monte Carlo method was performed by Pierre (Pierre 1990). The effect of the disorder is evaluated through localization factor statistics reflecting the exponential decay of the vibration amplitude. Moreover, Koch (Koch 2003) combined a continuous Timoshenko beam model, the Monte Carlo method and a first order perturbation approach to study the effects of the randomness of flexible joints on the free vibrations of simply-supported periodic trusses. These works has shown that the normal modes of the near-periodic structures are localized in the region where the periodicity is perturbed. Coupled pendulums chain is an example of periodic structures that has been the purpose of several researches in the literature. Recently, energy localization in coupled pendulums array under simultaneous external and parametric excitations has been investigated by Jallouli et al. (Jallouli et al. 2017) using a nonlinear Schrodinger equation. The study proved that adding an external excitation increases the existence region of solitons. Diala et al. (Bitar et al. 2016) investigated the collective nonlinear dynamics of perfectly periodic coupled pendulum structure under primary resonance using multiple scales and standing-wave decomposition. In presence of disorder, Tjavaras et al. (Tjavaras et al. 1996) studied the effect of nonlinearities on the forced response of two disordered pendulums. Results showed the high sensitivity of the structure to small parametric variations and the modal localization generated by the disorder. Moreover, Alexeeva et al. (Alexeeva et al. 2000) proved that the impurity introduced by increasing the length of one pendulum in a coupled parametrically driven damped pendulums chain significantly expands the stability domain. By decreasing the length, solitons are pushed producing effective partition of the chain. Hai-Quing et al. (Hai-Quing et al. 2006) studied the effect of mass impurity on nonlinear modes localization in a damped coupled nonlinear pendulums array under parametric excitation. It was also proven, in (Chen et al. 2002, Zhu et al. 2003) that disorder introduced by increasing or decreasing one pendulum length results in attracting or repelling a breather, respectively. The main purpose of this paper is to investigate the effect of disorder on the collective dynamics of two coupled pendulums chain, when the length of one pendulum is perturbed, through frequency responses computation in generalized coordinates. The analysis of the basins of attractions illustrates the robustness of the multimode solutions against disorder, in terms of attractors and bifurcation topologies, around a chosen frequency in the multistability domain.

Collective dynamics of disordered two coupled nonlinear pendulums

2

3

Model

A generic structure of N coupled nonlinear pendulums is considered, Fig. 1. The pendulums are of identical mass m and viscous damping coefficient c. They are coupled by linear springs of stiffness k and excited by an external force fcos  t  each one. The structure is nearly periodic since some pendulums lengths are perturbed. Two hypotheses are considered here: the linear coupling is very weak and the perturbations are supposed to be very small. Therefore, the eigenfrequencies are supposed to be equal: n  0  g / l , where l is the nominal length.

Fig. 1 Scheme of disordered nonlinear coupled-pendulums

The equation of motion of the nth pendulum is expressed as

n 

 m

n  n2n 

k g 3 f 2n  n 1  n 1   n  cos  Ωt  2  6ln mln mln 2

(1)

If the nth pendulum is perturbed, n is replaced by n and ln  l 1   l  , where  l is the introduced perturbation. To solve Eq. (1), the multiple scales method is applied. The single time variable t is thus replaced by an infinite sequence of independent time scales ( Ti   i t ), where  is a small dimensionless parameter. The scaled equation of motion the nth pendulum is thus expressed as follows

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n   

 m

0  n2n  

k  2n  n 1  n 1  mln 2

(2)

g 3 f n   cos  Ωt  6ln mln 2

where the excitation frequency Ω=0   , with  the detuning parameter. The solution of Eq. (2) can generally be given by a formal power series expansion n   iin . Up to the first order, one obtains an equation of the form i

 21n  20 n  0 n k 2      2  20 n  0 n 1  0 n 1 0 1n 2 m t t T mln 2 t





g 3 f  0 n  exp i 0t   T   6ln 2mln 2

(3)

Projecting the displacement of the nth pendulum on standing wave modes with slowly varying amplitudes ( sin  nqm  , qm  m  N  1 , m  1 N ) (Lifshitz et al. 2003, Bitar et al. 2015, Bitar et al. 2016), for boundary conditions 0  N 1  0 , leads to: N

n  Am sin  nqm  exp  i0t   c.c.   1n

(4)

m 1

0 n

Substituting Eq. (4) into Eq. (3) leads to N equations of the form N  21n 2      mth secular terms  ei0t  other terms  0 1n t 2 m 1

(5)

The multiple scales method requires vanishing the secular terms. Projecting the response on the standing-wave modes leads to the generic complex equation of the mth amplitude Am


2i0 S 

Am  k  i0 Am   2 Am  Gm cos  qm   T m mln 2 

2 k N  1 mln 2

g 8ln

5

N

N

n 1

x 1

 A A A Δ  j

k



 sin  nqm   cos  nqx  sin  qx  Ax  Ax * l

1 jkl , m



j , k ,l



(6)

N 1 f e xp i  T sin  nqm      N  1 mln 2 n 1

1 where Δjkl , m is the delta function defined in terms of the Kronecker deltas

(Lifshitz 2003, Bitar et al. 2015). Gm and S depend on the number and the positions of the perturbed pendulums in the structure. Under perfect periodicity, Gm  2 Am and S  0 . If the concerned pendulum is perfect but the adjacent is perturbed, Gm  Am  Am , where Am refers to the perturbed amplitude. In this case, S  1 or S  1 if the perturbed pendulum is, respectively, the previous or the next. When the concerned pendulum is perturbed and his neighbors are perfect, Am replaces Am in Eq. (6), Gm  2 Am and S  0 . To solve the obtained complex equations, a Cartesian form of the amplitude is introduced: Am   am  ibm  exp  i T  . Accordingly, to each complex equation (6) correspond two algebraic equations obtained for the real and imaginary parts of the amplitude Am . Consequently, 2N  p  q  d  algebraic equations are obtained, where p is the number of perturbed pendulums, q the number of perfect pendulums neighbors of the perturbed ones and d  1 if the structure contains perfect pendulums having perfect neighbors and 0 otherwise.

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Numerical example

A nearly periodic structure of two coupled nonlinear pendulums is considered here. The length of the first pendulum is perturbed by  l  2% , such as

l1  l  l   l  . The design parameters of the structure are listed in Table 1. Table 1. Design parameters of the two coupled pendulums structure. Parameters m (kg)

l (m)

k (N.m)

α (kg.s-1)

f (N.m)

ω0 (rad.s-1)

Values

0.062

9.10-4

0.16

0.01

12.58

0.25

6

2N  p  q  d   8

K. Chikhaoui et al.

algebraic

equations

are

generated

in

this

case

( p  1, q  1, d  0 ). However, if perfectly periodic structure is considered, 4 algebraic equations are obtained. Figure 2 illustrates the amplitudes of the responses of the associated perfectly periodic structure in generalized coordinates.

Fig. 2 Amplitudes of the pendulums responses in generalized coordinates, under perfect periodicity.

The collective dynamics generates modal interactions between the pendulums responses. Up to three stable solutions exist at several frequencies in the multistability domain and are classified as Single (SM) and Double mode (DM) solutions. The null trivial solution of the second amplitude generates the only SM branch presented by red curve, which corresponds to the two SM branches in the first amplitude: a resonant branch (SM-RB) and a non-resonant branch (SM-NRB). The coupling between the pendulums generates the DM branches, presented by blue curves. The bifurcation topology transfer results in the correspondence between the amplitudes in term of bifurcation points and branch types. This phenomena is more finely illustrated through the basins of attraction, which are plotted in the Nyquist plane

 a  , b   for a fixed detuning parameter   1 and 1 0

1 0

initial conditions  a2 0   b2 0  0.25 , Fig. 3. Two and three attractors correspond to the generated stable solutions, for the second and the first amplitudes, respectively.


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Fig. 3 Basins of attraction of the amplitudes of the pendulums responses, under perfect periodicity, in the Nyquist plane

 a  , b   1 0

SM-RB,

1 0

for   1 and  a2 0   b2 0  0.25 ,

SM-NRB,

DM,

SM.

The analysis of the basins of attraction shows the correspondence between the DM branches for the two responses amplitudes. However, when the amplitude of the first pendulum response jumps between the SM-RB and SM-NRB, the amplitude of the second pendulum response is stabilized on the SM. The DM attractors are the most robust, since they dominate the basins of attraction with 60.9% of the total area. However, the contributions of the SM-RB and the DM attractors are quantified by 15.7% and 23.4%, respectively.

Fig. 4 Amplitudes of the first pendulum responses in generalized coordinates, for l1  l  l   l  .

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Fig. 5 Amplitudes of the second pendulum responses in generalized coordinates, for l1  l  l   l  .

As illustrated in figures 4 and 5, introducing disorder results in adding other stable solutions. The Multistability domain is thus wider. More stable solutions are, in fact, added to the amplitude of the first pendulum response than to the amplitude of the second pendulum response. DM solutions are enhanced in both cases. Modal localization is more important on the SM-RB for the first pendulum response. However, few additional stable solutions are detected on the branches DM and SM-NRB of the amplitude of the second pendulum response. The robustness of the basins of attractions against perturbation is illustrated through figures 6 and 7.

Fig. 6 Basins of attraction of the amplitudes of the first pendulum responses in the Nyquist plane

 a  , b   for   1 ,  a  1 0

1 0

2 0

  b2 0  0.25 and l1  l  l   l  .


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Fig. 7 Basins of attraction of the amplitudes of the second pendulum responses in the Nyquist plane

 a  , b   for   1 ,  a  1 0

1 0

2 0

  b2 0  0.25 and l1  l  l   l  .

The attractors of the additional stable solutions, in the basins of attraction showed in figures 6 and 7, are distinguishable through colors which slightly differ from the initially defined colors for the same type of attractor. The attractors of the DM remain the most robust since they dominate the basins with 59.6% of their size. The size of the attractors of the SM-RB of the first pendulum response increases, when introducing the perturbation, from 15.7% to 23.4%. The SM-RB attractors become more robust than those corresponding to the SM-NRB. It can be concluded that introducing disorder results in enhancing the stability of the system. The perturbation introduced by reducing the length of one pendulum favors modal localization on its response.

4

Conclusion

Modal localization in disordered two coupled nonlinear pendulums was investigated through the robustness analysis of the collective dynamics against slight parametric perturbation. The obtained results prove that in presence of disorder, the stability of the nearly periodic structure is enhanced and a modal localization is generated. More multimode solutions introduced by the collective dynamics and interactions between pendulums are, in fact, generated and the multistatbility domain is wider.

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