Journal of Applied Nonlinear Dynamics 11(4) (2022) 833--844 | DOI:10.5890/JAND.2022.12.005

St\`{e}ve Cloriant Mba Feulefack, Blaise Rom\'{e}o Nana Nbendjo

Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Faculty of Science, University of Yaound'{e} I, PO Box 812 Yaound'{e}, Cameroon

Abstract

In this paper, the effect of delay on a network of indirectly coupled Euler's beams is analyzed. The system consists of a number of hinged-hinged beams indirectly interconnected to piezoelectric patches in a parallel conformation. A stability analysis is provided in order to give the threshold values of the parameter space for the onset of unstable motion. It is found that the stable region is reduced as the time delay increases. Disturbance-induced by time-delay on the synchronization state and the strong amplitude reduction (SAR) state is also presented. It is conventionally known that delay induces instability in coupled systems, but we find here that this delay can also contribute to stabilize these systems by synchronizing them.

References

1. [1]	Pikovsky, A., Rosenblum M., and Kurths J. (2001), Synchronization: a universal concept in nonlinear sciences, The Cambridge nonlinear science series, Cambridge University Press.

2. [2]	Resmi, V., Ambika, G., and Amritkar, R.E. (2011), General mechanism for amplitude death in coupled systems, Physical Review E, 84(4), 046212.

3. [3]	Djanan, A.A.N., Nbendjo, B.R.N., and Woafo, P. (2014), Effect of self-synchronization of dc motors on the amplitude of vibration of a rectangular plate, European Physical Journal Special Topics, 223(4), 813-825.

4. [4]	Atay, F.M. (2010), Complex time-delay systems: theory and applications, Springer.

5. [5]	Jeevarathinam, C., Rajasekar, S., and Sanju{a}n, M.A.F. (2013), Effect of multiple time-delay on vibrational resonance, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(1), 013136.

6. [6]	Rusinek, R., Weremczuk, A., Kecik, K., and Warminski, J. (2014), Dynamics of a time delayed Duffing oscillator, International Journal of Non-Linear Mechanics, 65, 98-106.

7. [7]	Tchakui, M.V. and Woafo, P. (2016), Dynamics of three unidirectionally coupled autonomous duffing oscillators and application to inchworm piezoelectric motors: Effects of the coupling coefficient and delay, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(11), 113108.

8. [8]	Cao, Y. (2019), Bifurcations in an Internet congestion control system with distributed delay, Applied Mathematics and Computation, 347,54-63.

9. [9]	Morgan, R.A. and Wang, A.W. (2002), An active-passive piezoelectric absorber for structural vibration control under harmonic excitations with time-varying frequency, part 1: algorithm development and analysis, Journal of Sound and Vibration, 124(1),77-83.

10. [10]	Nbendjo, B.R.N., Salissou, Y., and Woafo, P. (2005), Active control with delay of catastrophic motion and horseshoes chaos in a single well duffing oscillator, Chaos, Solitons and Fractals, 23(3), 809-816.

11. [11]	Zhang, L., Yang, C.Y., Chajes, M.J., and Cheng, A.H.D. (1993), Stability of active tendon structural control with time delay, Journal of Engineering Mechanics, 119(5), 1017-1024.

12. [12]	Quintero-Quiroz, C. and Cosenza, M.G. (2015), Collective behavior of chaotic oscillators with environmental coupling, Chaos, Solitons and Fractals, 71, 41-45.

13. [13]	Sethia, G.C., Sen, A., and Atay, F.M. (2008), Clustered chimera states in delay-coupled oscillator systems, Physical Review Letters, 100(14), 144102.

14. [14]	Verma, U.K., Kamal, N.K., and Shrimali, M.D. (2018), Co-existence of in-phase oscillations and oscillation death in environmentally coupled limit cycle oscillators, Chaos, Solitons and Fractals, 110, 55-63.

15. [15]	Feulefack, S.C.M., Nbendjo, B.R.N., Vincent, U.E., and Woafo, P. (2017), Dynamical clustering, synchronization and strong amplitude reduction in a network of Euler's beams coupled via a dynamic environment, Nonlinear Dynamics, 88(1), 455-464.

16. [16]	Tuwa P.R.N. and Woafo, P. (2016), Analysis of an electrostatically actuated micro-plate subject to proportional-derivative controllers, Journal of Vibration and Control, 24(10), 2020-2029.

17. [17]	Kolmanovskii, V.B. and Nosov, V.R. (1986), Stability of functional differential equations, 180, Elsevier.

18. [18]	Cao, Y. (2013), A new hybrid chaotic map and its application on image encryption and hiding, Mathematical Problems in Engineering, 2013, Article ID 728375, 13 pages.

19. [19]	Palazzi, M.J. and Cosenza, M.G. (2014), Amplitude death in coupled robust-chaos oscillators, European Physical Journal Special Topics, 223(13), 2831-2836.