Periodic behavior of a nonlinear third order vibrating system |
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| Institution: | 1. Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA;2. Newcomb&Boyd, Consulting Engineering Group, Atlanta, GA 30303-1277, USA;1. Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal;2. Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK;3. Faculty of Mathematics, University of Vienna, Oskar-Morgnstern Platz, 1, 1090 Wien, Austria;1. Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria;2. Department of Physics, The University of Burdwan, Golapbag 713104, West Bengal, India;3. Department of Physics, School of Physical, Chemical and Applied Sciences, Pondicherry University, Pondicherry 605014, India;4. Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303, CNRS, Univ. de Bourgogne Franche-Comte, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France;1. Computer Engineering Department, KFUPM, Dhahran, Saudi Arabia;2. Center for Communications and IT Research, Research Institute, KFUPM, Dhahran, Saudi Arabia;1. Department of Land Surveying an Geo-Informatics, HK PolyU, Hong Kong;2. Department of Geography, University of Sussex, UK;1. CSIRO Land and Water, Floreat, Western Australia, 6014, Australia;2. Department of Civil Engineering, Curtin University, Bentley, Western Australia, 6102, Australia;3. School of Engineering and Information Technology, Murdoch University, Murdoch, Western Australia, 6150, Australia;4. School of Pathology and Laboratory Medicine, and Oceans Institute, University of Western Australia, Nedlands, Western Australia, 6009, Australia;1. Department of Electrical Engineering, Capital University of Science and Technology, Islamabad, Pakistan;2. Department of Electrical and Computer Engineering, COMSATS University Islamabad, Islamabad, Pakistan |
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| Abstract: | A theorem is proved to show that the third order differential equation x‴+f(t,x,x′,x″)=0 has nontrivial solutions characterized by x′(0)=x′(τ)=0 when x,x′,x″ and f(t,x,x′,x″) are bounded. A second condition is introduced to prove the existence of periodic solution for this equation. It is shown that the equation has a τ-periodic solution if f(t,x,x′,x″) is an even function with respect to x′. The existence and periodicity conditions would be applied to third order systems such as viscoelastic mechanical vibration isolator system. The concepts of Green’s function and the Schauder’s fixed-point theorem have been used for proving the third-order-existence theorem. |
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