Asymptotic behavior of a laminated beam with nonlinear delay and nonlinear structural damping

Year 2022, , 1517 - 1534, 01.12.2022

Kassimu Mpungu , Tijani Apalara

https://doi.org/10.15672/hujms.947131

Cited By: 4

Abstract

Our concern in the present work is a Timoshenko laminated beam system with nonlinear delay and nonlinear structural damping acting in the equation describing the dynamics of slip. The aim is to establish an explicit and general energy decay rates of the solution under suitable assumptions on the weight of delay and speeds of wave propagation. To achieve our desired stability results, we exploit some properties of convex functions, coupled with the multiplier technique, which involves constructing an appropriate Lyapunov functional equivalent to the energy of the system.

Keywords

Laminated beam, nonlinear structural damping, nonlinear delay, general energy decay

References

• [1] T. Al-Hababi, M. Cao, B. Saleh, N. F. Alkayem, and H. Xu, A critical review of nonlinear damping identification in structural dynamics: Methods, applications, and challenges, Sensors 20(24), 7303, 2020.
• [2] M. S. Alves and R. N. Monteiro, Exponential stability of laminated Timoshenko beams with boundary/internal controls, J. Math. Anal. Appl. 482(1), 123516, 2020.
• [3] T. A. Apalara, Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks, Appl. Anal. 95(1), 187–202, 2016.
• [4] T. A. Apalara, Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys. 68(2), 41, 2017.
• [5] T. A. Apalara, On the stability of a thermoelastic laminated beam, Acta Math. Sci., 39(6), 1517–1524, 2019.
• [6] T. A. Apalara, Exponential stability of laminated beams with interfacial slip, Mech. Solids 56(1), 131–137, 2021.
• [7] T. A. Apalara and S. A. Messaoudi, An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Appl. Math. Optim. 71(3), 449–472, 2015.
• [8] T. A. Apalara, A. M. Nass, and H. Al Sulaimani, On a laminated Timoshenko beam with nonlinear structural damping, Math. Comput. Appl. 25(2), 35, 2020.
• [9] T. A. Apalara, C. A. Raposo, and C. A. Nonato, Exponential stability for laminated beams with a frictional damping, Arch. Math. (Basel), 114(4), 471–480, 2020.
• [10] T. A. Apalara and A. Soufyane, Energy decay for a weakly nonlinear damped porous system with a nonlinear delay, Appl. Anal., pages 1–23, 2021.
• [11] V. I. Arnol’d, Mathematical methods of classical mechanics, volume 60, Springer Science & Business Media, second edition, 1989.
• [12] A. Benaissa and M. Bahlil, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwan. J. Math. 18(5), 1411–1437, 2014.
• [13] Z. Chen, W. Liu, and D. Chen, General decay rates for a laminated beam with memory, Taiwan. J. Math. 23(5), 1227–1252, 2019.
• [14] A. Choucha, D. Ouchenane, and S. Boulaaras, Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term, Math. Methods Appl. Sci. 43(17), 9983–10004, 2020.
• [15] R. Datko, J. Lagnese, and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim. 24(1), 152–156, 1986.
• [16] B. Feng, Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks, Math. Methods Appl. Sci. 41(3), 1162–1174, 2018.
• [17] B. Feng, On a thermoelastic laminated Timoshenko beam: Well posedness and stability, Complexity, 5139419, 13 pages, 2020.
• [18] S. W. Hansen and R. D. Spies, Structural damping in laminated beams due to interfacial slip, J. Sound Vib. 204(2), 183–202, 1997.
• [19] V. Komornik, Exact controllability and stabilization: the multiplier method, volume 36, Elsevier Masson, 1994.
• [20] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations 6(3), 507–533, 1993.
• [21] W. Liu, X. Kong, and G. Li, Asymptotic stability for a laminated beam with structural damping and infinite memory, Math. Mech. Solids 25(10), 1979–2004, 2020.
• [22] W. Liu, Y. Luan, Y. Liu, and G. Li, Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III, Math. Meth. Appl. Sci. 43(6), 3148– 3166, 2020.
• [23] W. Liu and W. Zhao, Stabilization of a thermoelastic laminated beam with past history, Appl. Math. Optim. 80(1), 103–133, 2019.
• [24] A. Lo and N. E. Tatar, Stabilization of laminated beams with interfacial slip, Electron. J. Diff. Equ. 2015(129), 1–14, 2015.
• [25] A. Lo and N. E. Tatar, Uniform stability of a laminated beam with structural memory, Qual. Theory Dyn. Syst. 15(2), 517–540, 2016.
• [26] E. Moyer and M. Miraglia, Peridynamic solutions for Timoshenko beams, Engineering 6(6), 304–317, 2014.
• [27] K. Mpungu, T.A. Apalara and M. Muminov, On the stabilization of laminated beams with delay, Appl. Math. 66(5), 789–812, 2021.
• [28] K. Mpungu and T.A. Apalara, Exponential stability of laminated beam with constant delay feedback, Math. Model. Anal. 26(4), 566–581, 2021.
• [29] K. Mpungu and T.A. Apalara, Stability result of laminated beam with internal distributed delay, J. Math. Inequal. 15(3), 1075–1091, 2021.
• [30] K. Mpungu and T.A. Apalara, Exponential stability of laminated beam with neutral delay, Afr. Mat. 33(2), 30, 2022.
• [31] S. E. Mukiawa, T. A. Apalara, and S. A. Messaoudi, A stability result for a memory-type laminated-thermoelastic system with Maxwell–Cattaneo heat conduction, J. Therm. Stresses 43(11), 1437–1466, 2020.
• [32] S. E. Mukiawa, T. A. Apalara, and S. A. Messaoudi, Stability rate of a thermoelastic laminated beam: case of equal-wave speed and nonequal-wave speed of propagation, AIMS Math. 6(1), 333–361, 2021.
• [33] M. Mustafa, Boundary control of laminated beams with interfacial slip, J. Math. Phys. 59(5), 051508, 2018.
• [34] M. I. Mustafa, Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl. 466(1), 619–641, 2018.
• [35] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45(5), 1561–1585, 2006.
• [36] J. G. Nie and C. S. Cai, Steel–concrete composite beams considering shear slip effects, J. Struct. Eng. 129(4), 495–506, 2003.
• [37] C. Nonato, C. Raposo, and B. Feng, Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay, Asymptot. Anal. 122(1), 1–29, 2021.
• [38] C. A. Raposo, Exponential stability for a structure with interfacial slip and frictional damping, Appl. Math. Lett. 53, 85–91, 2016.
• [39] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Appl. Math. Comput. 217(6), 2857–2869, 2010.
• [40] S. H. Schulze, M. Pander, K. Naumenko, and H. Altenbach, Analysis of laminated glass beams for photovoltaic applications, Int. J. Solids Struct. 49(15-16), 2027–2036, 2012.
• [41] L. Seghour, N. E. Tatar, and A. Berkani, Stability of a thermoelastic laminated system subject to a neutral delay, Math. Methods Appl. Sci. 43(1), 281–304, 2020.
• [42] H. Suh and Z. Bien, Use of time-delay actions in the controller design. IEEE Trans. Automat. Contr. 25(3), 600–603, 1980.
• [43] N. E. Tatar, Stabilization of a laminated beam with interfacial slip by boundary controls, Bound. Value Probl. 2015(1), 169, 2015.
• [44] J. M. Wang, G. Q. Xu, and S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim. 44(5), 1575–1597, 2005.
• [45] P. Wu, D. Zhou, and W. Liu, 2-d elasticity solution of layered composite beams with viscoelastic interlayers, Mech Time Depend Mater 20(1), 65–84, 2016.