**Problem 4.** *What is the expression of <sup>ω</sup>eqn*,1?

If we find the solutions to the above four, a consequent problem is as follows.

**Problem 5.** *How to represent response (free, or impulse, or step, or sinusoidal) with meq*1, *ceq*1, *<sup>ω</sup>eqn*,1, *and <sup>ω</sup>eqd*,<sup>1</sup> *to a fractional oscillator in Class I?*

If we solve the above problems, the solution to the following problem is ready.

### **Problem 6.** *What is the physical mechanism of a fractional oscillator in Class I?*

Note that the intrinsic damping for a Class II fractional oscillator must differ from its primary damping *c* owing to the fractional friction *c <sup>d</sup>βy*2(*t*) *dtβ* for *β* = 1. We call it the equivalent damping denoted by *ceq*2. Because *ceq*2= *c* if *β* = 1, the equivalent mass of a fractional oscillator in Class II, denoted by *meq*2, is not equal to the primary *m* for *β* = 1. Thus, the six stated above are also unsolved problems for fractional oscillators in Class II. They are, consequently, the problems unsolved for Class III fractional oscillators.

Note that there are other problems regarding with three classes of fractional oscillators. For example, the explicit expression of the sinusoidal response (37) in closed form needs investigation because of the difficulty in finding the solution to ∞ 0 *e* <sup>−</sup>*stKα*(*s*)*ds*. We shall deal with them in separate sections. Thesolutionstotheproblemsdescribedaboveconstitutemainhighlightsofthisresearch.

We note that the damping nature of a fractional oscillator in Class I was also observed by other researchers, not explicitly stated though, as can be seen from, e.g., Zurigat ([26], Figure 1), Blaszczyk et al. ([28], Figure 2), Al-rabtah et al. ([29], Figure 2), Ryabov and Puzenko ([36], Equation (5)), Uchaikin ([38], Chapter 7), Duan et al. ([39], Equation (4.3), Figure 2), Gomez-Aguilar et al. ([53], Equation (15), Figures 2 and 3), Chung and Jung ([77], Figure 1). One thing remarkable is by Tofighi, who explored the intrinsic damping of an oscillator in Class I, see ([35], pp. 32–33). That was an advance regarding with the damping implied in (31) but it may be unsatisfactory if one desires its closed form of analytic expression.
