Summary.
Existence, uniqueness and dissipativity is established for a class of nonlinear dynamical systems including systems with fractional damping. The problem is reduced to a system of fractional-order differential equations for numerical integration. The method is applied to a non-linear pendulum with fractional damping as well as to a nonlinear pendulum suspended on an extensible string. An example of such a fractional damping is a pendulum with the bob swinging in a viscous fluid and subject to the Stokes force (proportional to the velocity of the bob) and the Basset-Boussinesq force (proportional to the Caputo derivative of order 1/2 of the angular velocity). An existence and uniqueness theorem is proved and dissipativity is studied for a class of discrete mechanical systems subject to fractional-type damping. Some particularities of fractional damping are exhibited, including non-monotonic decay of elastic energy. The 2:1 resonance is compared with nonresonant behavior.
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Seredyńska, M., Hanyga, A. Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum. Acta Mechanica 176, 169–183 (2005). https://doi.org/10.1007/s00707-005-0220-8
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DOI: https://doi.org/10.1007/s00707-005-0220-8