Рассмотрена модель вынужденных колебаний осциллятора, основанная на применении математического аппарата дробного интегро-дифференцирования. Показано, что данная модель вполне сопоставима с классической моделью вынужденных колебаний осциллятора с вязким демпфированием. Вычислены параметры частотной характеристики установившихся колебаний дробного осциллятора и определена связь между порядком дробной производной в дифференциальном уравнении колебаний и добротностью системы.

We propose a new method of describing for vibration damping of a beam built-in at one end. The method takes into account dynamic hysteresis, which determines the dissipation of mechanical energy due to viscoelasticity. As a mathematical tool, we use fractional integro-differentiation. Experimental studies of rapidly damped vibrations of a foam polypropylene beam have been carried out. It is shown that the theoretical model successfully describes the experimental results.

A new model of damped vibrations of an oscillator based on the use of the mathematical apparatus of fractional integro-differentiation with an exponential-power function of dynamic memory is considered. Using the Wright function, the exact solution of the equation of motion of the oscillator is obtained. Theoretical analysis shows that the model can be effectively used to describe vibrational processes in low-Q dynamic systems (in particular, mechanical or electrical systems).

The spectral energy density of the oscillator radiation is calculated in the dipole approximation. The oscillator motion is described by an equation with fractional integro-differentiation. The fractional oscillator model can describe various types of radiation, including those with a nonexponential relaxation law. Found the shape of the spectral line of radiation. The obtained result is compared with the classical Lorentzian spectrum and experimental emission spectra of monochromatic and phosphor LEDs. The order of fractional integro-differentiation in the model sets the magnitude of the broadening of the radiation spectrum.