Affiliations: Institute of Science, PLA University of Science and
Technology, 211101 Nanjing, China | Institute of Vibration Engineering Research, Nanjing
University of Aeronautics and Astronautics, 210016 Nanjing, China
Abstract: Fractional-order derivative has been shown an adequate tool to the
study of so-called "anomalous" social and physical behaviors, in reflecting
their non-local, frequency- and history-dependent properties, and it has been
used to model practical systems in engineering successfully, including the
famous Bagley-Torvik equation modeling forced motion of a rigid plate immersed
in Newtonian fluid. The solutions of the initial value problems of linear
fractional differential equations are usually expressed in terms of
Mittag-Leffler functions or some other kind of power series. Such forms of
solutions are not good for engineers not only in understanding the solutions
but also in investigation. This paper proves that for the linear SDOF
oscillator with a damping described by fractional-order derivative whose order
is between 1 and 2, the solution of its initial value problem free of external
excitation consists of two parts, the first one is the 'eigenfunction
expansion' that is similar to the case without fractional-order derivative, and
the second one is a definite integral that is independent of the eigenvalues
(or characteristic roots). The integral disappears in the classical linear
oscillator and it can be neglected from the solution when stationary solution
is addressed. Moreover, the response of the fractionally damped oscillator
under harmonic excitation is calculated in a similar way, and it is found that
the fractional damping with order between 1 and 2 can be used to produce
oscillation with large amplitude as well as to suppress oscillation, depending
on the ratio of the excitation frequency and the natural frequency.
Keywords: Linear oscillator, fractional-order damping, characteristic root, eigenfunction expansion
