Non-horizontally Suspended Cable Dynamics with Flexible Tower Modulations
DOI:
https://doi.org/10.5890/JVTSD.2018.03.003Abstract
Based upon an asymptotically reduced coupled model, nonlinear forced vibrations of a non-horizontally suspended cable is investigated in this paper, which is coupled with a flexible oscillating tower. The cable’s nonlinear coupled dynamics is a modulated version of cable’s uncoupled dynamics, i.e., the cable dynamics with fixed rigid towers. Nonlinear frequency responses of the cable-tower coupled system are found, with saddle-node bifurcations, Hopf bifurcations, and quasi-periodic behaviors detected. Special attentions are paid to the dynamic effects caused by cable-tower coupling, boundary damping, and the inclinations.References
[1] Hilfer, R. (1999), Application of fractional Calculus in Physics, World Scientific, Singapore.
[2] Podlubny, I. (1999), Fractional differential equations, Academic Press, San Diego.
[3] Katugampola, U.N. (2011), New approach to generalized fractional derivatives, J. Math. Anal. Appl., 218(3), 860-865.
[4] Katugampola, U.N. (2011), New approach to a generalized fractional integral, Appl. Math. Comput., 218, 860-865.
[5] Katugampola, U.N. (2014), Existence and uniqueness results for a class of generalized fractional differential equations, arXiv:1411.5229, pp. 1-9.
[6] Vivek, D., Kanagarajan, K., and Harikrishnan, S., (2017), Existence and uniqueness results for pantograph equations with generalized fractional derivative, J. Nonlinear Anal. Appl., 2, 105-112.
[7] Vivek, D., Kanagarajan, K., and Harikrishnan, S. (2017), Existence and uniqueness results for implicit differential equations with generalized fractional derivative, J. Nonlinear Anal. Appl., (Accepted articleID:2017/jnaa-00370).
[8] Vivek, D., Kanagarajan, K., and Sivasundaram, S. (2017), Theory and analysis of nonlinear neutral pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 24(3), 699-712.
[9] Rahimkhani, P., Ordokhani, Y., and Babolian, E. (2017), Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309, 493-510.
[10] Trif, D. (2012), Direct operatorial tau method for pantograph-type equations, Appl. Math. Comput., 219(4), 2194-2203.
[11] Balachandran, K., Kiruthika, S., and Trujillo, J.J. (2013), Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci., 33B, 1-9.
[12] Vivek, D., Kanagarajan, K., and Sivasundaram, S. (2016), Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4), 685-698.
[13] Guan, K.Z., Wang, Q.S., Wang, Q.S., and He, X.B. (2012), Oscillation of a pantograph differential equation with impulsive perturbations, Appl. Math. Comput., 219, 3147-3153.
[14] Benchohra, M., Henderson, J., and Ntouyas, S.K. (2006), Impulsive differential equations and inclusions, Hindawi Publishing Corporation, 2, New York.
[15] Graef, J.R., Henderson, J., and Ouahab, A. (2013), Impulsive differential inclusions, A Fixed Point Approch, De Gruyter, Berlin/Boston.
[16] Ndiyo, E.E. (2017), Existence result for solution of second order impulsive differential inclusion to dynamic evolutionary processes, Am. J. Appl. Math., 7(2), 89-92.
[17] Zhang, G.L., Song, M.H., and Liu, M.Z., Asymptotic stability of a class of impulsive delay differential equations, Journal of Applied Mathematics, 2012, Article ID 723893, 9 pages, 2012. DOI:10.1155/2012/723893.
[18] Deepak, D., Ashok, K., and Ganga R.G. (2017), Existence of solution to fractional order delay differential equations with impulses, Advanced Math. Models and Applications, 2, 155-165.
[19] Wang, J.R., Zhou, Y., and Michal, F. (2012), Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64, 3389-3405.
[20] Wang, J.R., Michal, F., and Zhou, Y. (2016), A survey on impulsive fractional differential equations, Frac. Cal. Appl. Anal., 19, 825-1052.
[21] Nemat, N. (2017), Existence and multiplicity of solutions for impulsive fractional differential equations, Mediterr. J. Math.,14(85), 1-17.
[22] Benchohra, M. and Bouriah, S. (2015), Existence and stability results for nonlinear boundary valur problem for implicit differential equations of fractional order, Moroccan J. Pure and Appl. Anal., 1(1), 22-37.
[23] Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328, 1075-1081.
[24] Bainov, D.D. and Hristova, S.G. (1997), Integral inequalities of Gronwall type for piecewise continuous functions, J. Appl. Math. Stoc. Anal., 10, 89-94.
Article Metrics
Usage tracking begins September 1, 2026.