Theory of vibrators with variable-order fractional forces

In this paper, we present a theory of six classes of vibrators with variable-order fractional forces of inertia, damping, and restoration. The novelty and contributions of the present theory are reflected in six aspects. 1) Equivalent motion equations of those variable-order fractional vibrators are proposed. 2) The analytical expressions of the effective mass, damping, and stiffness of those variable-order fractional vibrators are presented. 3) The asymptotic properties of the effective mass, damping, and stiffness of a class of variable-order fractional vibrators are given. 4) The restricted effective parameters (damping ratio, damping free natural frequency, damped natural frequency, frequency ratio) of the variable-order fractional vibrators are put forward. 5) We bring forward the analytical representations of the free responses, the impulse responses, and the frequency transfer functions of those variable-order fractional vibrators. 6) We propose a solution to an open problem of how to mathematically explain the Rayleigh damping assumption based on the present theory of variable-order fractional vibrations.