Dealing with Stationary Sinusoidal Responses of Seven Types of Multi-Fractional Vibrators Using Multi-Fractional Phasor

The novelty and main contributions of this paper are reflected in four aspects. First, we introduce multi-fractional phasor in Theorem 1. Second, we propose the motion phasor equations of seven types of multi-fractional vibrators in Theorems 2, 12, 22, 32, 43, 54, and 65, respectively. Third, we present the analytical expressions of response phasors of seven types of multi-fractional vibrators in Theorems 10, 20, 30, 41, 52, 63, and 74, respectively. Fourth, we bring forward the analytical expressions of stationary sinusoidal responses of seven types of multi-fractional vibrators in Theorems 11, 21, 31, 42, 53, 64, and 75, respectively. In addition, by using multi-fractional phasor, we put forward the analytical expressions of vibration parameters (equivalent mass, equivalent damping, equivalent stiffness, equivalent damping ratio, equivalent damping free natural angular frequency, equivalent damped natural angular frequency, equivalent frequency ratio) and frequency transfer functions of seven types of multi-fractional vibrators. Demonstrations exhibit that the effects of multi-fractional orders on stationary sinusoidal responses of those multi-fractional vibrators are considerable.