*4.4. Summary*

We have proposed three equivalent oscillation equations with order 2 to equivalently characterize three classes of fractional oscillators, opening a novel way of studying fractional oscillators. The analytic expressions of equivalent mass *meqj* and damping *ceqj* (*j* = 1, 2, 3) for each equivalent oscillator have been presented. One general thing regarding *meqj* and damping *ceqj* is that they follow power laws. Another thing in common is that they are dependent on oscillation frequency *ω* and fractional order.

### **5. Equivalent Natural Frequencies and Damping Ratio of Three Classes of Fractional Oscillators**

We have presented three equivalent oscillation equations corresponding to three classes of fractional oscillators in the last section. Functionally, they are abstracted in a unified form

$$k\_{\rm eqj} \frac{d^2 \mathbf{x}\_j(t)}{dt^2} + c\_{\rm eqj} \frac{d \mathbf{x}\_j(t)}{dt} + k \mathbf{x}\_j(t) = f(t), j = 1, 2, 3. \tag{128}$$

In each equivalent oscillator, either *meqj* or *ceqj* is not a constant in general. Instead, either is a function of the oscillation frequency *ω* and the fractional order *α* for *meq*1 and *ceq*1, *β* for *meq*2 and *ceq*2, (*<sup>α</sup>*, *β*) for *meq*3 and *ceq*3. Consequently, natural frequencies and damping ratios of fractional oscillators should rely on *ω* and fractional order. We shall propose their analytic expressions in this section.
