Free damped vibrations of two hereditarily elastic oscillators, the hereditary properties of which are described by the Boltzmann-Volterra relationships with the weakly singular Rzhanitsyn kernel taken as the creep kernel (the first model) or as the relaxation kernel (the second model), are considered. These integral relationships are equivalent to the differential relationships involving infinite sums of various order time derivatives of excitation (force) or response (displacement) depending on whether the Rzhanitsyn kernel is used for the creep kernel or relaxation kernel. The problem is solved by the Laplace transform method. When passing from image to pre-image one is led to find the roots of an algebraic (characteristic) equation with fractional exponents. A method for solving such equations is proposed which allows one to investigate the roots' behaviour in a wide range of single-mass system parameters. It is shown that, if the Rzhanitsyn kernel is the creep kernel in the Volterra equations, then the characteristic equation does not possess real roots, but has two complex conjugate roots; i.e., the test single-mass system subjected to the impulse excitation does not pass into an aperiodic regime. On the contrary, the oscillator model with the Rzhanitsyn relaxation kernel may be both in vibrating motion and in the aperiodic regime, depending on the intervals over which the relaxation times for the given model vary, as well as on the order of fractional power and the ratio of the relaxed modulus (rubbery modulus) to the non-relaxed modulus (glassy modulus). However, contrary to the standard linear solid model with ordinary time derivatives for which the dimensions of the domain of aperiodicity as well as its existence are governed only by the magnitude of the ratio of the relaxed modulus to the non-relaxed modulus, for the model with the Rzhanitsyn relaxation kernel all of the above-listed factors essentially depend in addition on an order of a fractional operator parameter. The main characteristics of the vibratory and aperiodic motions of the single-mass system as functions of the relaxation time or creep time, which are equivalent to the temperature dependences, are constructed and analyzed for both models.