This paper contains a survey on one of the mathematical approaches used to solve a fractional differential equation whose solution gives the free dynamic response of viscoelastic single degree of freedom systems (viscosity is actually modelled by a fractional displacement derivative instead of first order one). The paper shall deal with Caputo's fractional derivative since its Laplace Transform (on which the resolution method is based) only depends by lower integer order (and therefore measurable and physically meaningful) derivatives given as initial conditions. The paper provides a deep mathematical analysis of the properties of the solution expressed in terms of the mechanical parameters meaning. Additionally, important physical implications are reported exhibiting a richer dynamic behavior if compared to the standard damping case (velocity linear dependence). Some important consequences in the use of Caputo's fractional derivative are reported, and some limitations to possible viscous parameters values are obtained. Finally, it is shown that free response of fractional derivative equation solution is mathematically equivalent to a suitably forced solution of the integer model.

### A free fractional viscous oscillator as a forced standard damped vibration

#### Abstract

This paper contains a survey on one of the mathematical approaches used to solve a fractional differential equation whose solution gives the free dynamic response of viscoelastic single degree of freedom systems (viscosity is actually modelled by a fractional displacement derivative instead of first order one). The paper shall deal with Caputo's fractional derivative since its Laplace Transform (on which the resolution method is based) only depends by lower integer order (and therefore measurable and physically meaningful) derivatives given as initial conditions. The paper provides a deep mathematical analysis of the properties of the solution expressed in terms of the mechanical parameters meaning. Additionally, important physical implications are reported exhibiting a richer dynamic behavior if compared to the standard damping case (velocity linear dependence). Some important consequences in the use of Caputo's fractional derivative are reported, and some limitations to possible viscous parameters values are obtained. Finally, it is shown that free response of fractional derivative equation solution is mathematically equivalent to a suitably forced solution of the integer model.
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2016
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11589/93240`
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