A pendulum with an elliptic-type parametric excitation: Stability charts for a damped and undamped system |
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| Institution: | 1. School of Computer & Communication, Lanzhou University of Technology, Lanzhou 730050, China;2. Department of Automation, Tsinghua University, Beijing 100084, China;1. USP – Univ São Paulo, Departamento de Física, Rua do Matão, Cidade Universitária, Travessa R 187, 05508-090 São Paulo, SP, Brazil;2. School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom;3. UNESP – Univ Estadual Paulista, Departamento de Estatística, Matemática Aplicada e Computação, Av. 24A 1515, Bela Vista, 13506-900 Rio Claro, SP, Brazil;4. UNESP – Univ Estadual Paulista, Departamento de Física, Av. 24A 1515, Bela Vista, 13506-900 Rio Claro, SP, Brazil;5. The Abdus Salam – ICTP, Strada Costiera, 11, 34151 Trieste, Italy;1. Centro de Estadística y Software Matemático, Departamento de Cómputo Científico, Universidad Simón Bolívar, Sartenejas, Venezuela;2. Laboratorio de Fenómenos no Lineales, Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, Venezuela;3. Red de Estudios Interdisciplinarios, Academia Nacional de Ciencias Físicas, Matemáticas y Naturales, Venezuela;1. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC – UFABC, Rua Santa Adélia, 166, Bairro Bangu, 09.210-170 Santo André, SP, Brazil;2. Dipartimento di Matematica e Informatica, Università Degli Studi di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy |
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| Abstract: | In this paper, a pendulum parametrically excited by the excitation which has the form of the Jacobi cn elliptic function is considered. Three cases related to the value of the elliptic parameter are distinguished: the case when it is smaller than zero, when it ranges between zero and unity, and when it is higher than unity. First, interpretations of the excitation with such elliptic parameter are given in terms of its period, higher harmonic content and the amplitude. These interpretations enable one to consider the elliptic-type excitation as a type of multi-cosine excitation whose frequency and amplitude are related mutually in a particular way. Stability charts are determined for damped and undamped systems. When the elliptic parameter is equal to zero, the governing equations considered transform to the well-known Mathieu equation. In all other cases, the governing equations considered can be seen as a new generalisation of the Mathieu equation. The influence of an arbitrary real elliptic parameter on the location and shape of the transition curves and instability tongues is investigated, illustrated and discussed in all three cases, which represent new and so far unknown results. |
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| Keywords: | Parametric excitation Jacobi elliptic function Elliptic parameter Stability chart Floquet theory Harmonic balancing |
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