*7.3. Impulse Response to Fractional Oscillators in Class II*

**Theorem 14** (Impulse response II)**.** *Denote by h*2(*t*) *the impulse response to a fractional oscillator in Class II. For t* ≥ *0 and 1 < β* ≤ *2, therefore, it is given by*

$$h\_{2}(t) = \frac{e^{-\frac{\xi\omega\_{n}\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{1-\frac{\xi}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}t}\sin\frac{\omega\_{n}\sqrt{1-\frac{\xi^{2}\omega^{2(\beta-1)}\sin\frac{\beta\pi}{2}}}{1-\frac{\xi}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}t}{\omega\_{n}m\sqrt{1-\frac{\xi}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}\sqrt{1-\frac{\xi^{2}\omega^{2(\beta-1)}\sin^{2}\frac{\beta\pi}{2}}{1-\frac{\xi}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}}}.\tag{200}$$

**Proof.** From (196), we have

$$h\_2(t) = e^{-\xi\_{\rm eq2}\omega\_{\rm eq2}t} \frac{1}{m\_{\rm eq2}\omega\_{\rm eq2,2}} \sin\omega\_{\rm eqd,2}t, t \ge 0. \tag{201}$$

By replacing *meq*2 with that in Section 4, *ςeq*2, *<sup>ω</sup>eqn*,2, and *<sup>ω</sup>eqd*,<sup>2</sup> by those in Section 5, we obtain

*h*2(*t*) = *e* <sup>−</sup>*ςeq*2*<sup>ω</sup>eqn*,2*<sup>t</sup>* sin *<sup>ω</sup>eqd*,2*<sup>t</sup> meq*2*ωeqd*,<sup>2</sup> = *e* − *ςωβ*−<sup>1</sup> sin *βπ* 2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *ωn* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t* sin *<sup>ω</sup>eqd*,2*<sup>t</sup><sup>m</sup>*−*<sup>c</sup>ωβ*−<sup>2</sup> cos *βπ*2 *<sup>ω</sup>eqd*,<sup>2</sup> = *e* − *ςωnωβ*−<sup>1</sup> sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t* sin *ωn* "###\$1− *ς*2*ω*<sup>2</sup>(*β*−<sup>1</sup>) sin<sup>2</sup> *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ* 2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t <sup>m</sup>*−*<sup>c</sup>ωβ*−<sup>2</sup> cos *βπ*2 *ωn* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 "##\$1− *ς*2*ω*<sup>2</sup>(*β*−<sup>1</sup>) sin<sup>2</sup> *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 = *e* − *ςωnωβ*−<sup>1</sup> sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t* sin *ωn* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 "##\$1− *ς*2*ω*<sup>2</sup>(*β*−<sup>1</sup>) sin<sup>2</sup> *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t <sup>ω</sup>n<sup>m</sup>* <sup>1</sup><sup>−</sup> *cm ωβ*−<sup>2</sup> cos *βπ*2 "##\$1− *ς*2*ω*<sup>2</sup>(*β*−<sup>1</sup>) sin<sup>2</sup> *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 .

This is (200). Hence, the proof completes. 

Figure 36 illustrates *h*2(*t*) with fixed *ω*. Its plots with variable *ω* are shown in Figure 37. Its pictures in t-ω plane are indicated in Figure 38.

**Figure 36.** Illustrating impulse response *h*2(*t*) for *m* = *c* = *k* = 1. Solid line: *ω* = 30. Dot line: *ω* = 10. (**a**) *β* = 0.3, solid line: *ω* = 30 (*ςeq*<sup>2</sup> = 0.02); dot line: *ω* = 10 (*ςeq*<sup>2</sup> = 0.05). (**b**) *β* = 0.6, solid line: *ω* = 30 (*ςeq*<sup>2</sup> = 0.10); dot line: *ω* = 10 (*ςeq*<sup>2</sup> = 0.16). (**c**) *β* = 0.9, solid line: *ω* = 30 (*ςeq*<sup>2</sup> = 0.35); dot line: *ω* = 10 (*ςeq*<sup>2</sup> = 0.40). (**d**) *β* = 1, solid line: *ω* = 30 (*ςeq*<sup>2</sup> = 0.50); dot line: *ω* = 10 (*ςeq*<sup>2</sup> = 0.50).

**Figure 37.** Plots of impulse response *h*2(*t*) with variable *ω* for *m* = *c* = *k* = 1 in time domain. (**a**) For *β* = 0.63, *ω* = 1, 2, ..., 5 (0.24 ≤ *ςeq*<sup>2</sup> ≤ 0.62). (**b**) For *β* = 0.63, *ω* = 1, 2, ..., 10 (0.18 ≤ *ςeq*<sup>2</sup> ≤ 0.62). (**c**) For *β* = 0.83, *ω* = 1, 2, ..., 5 (0.37 ≤ *ςeq*<sup>2</sup> ≤ 0.56). (**d**) For *β* = 0.83, *ω* = 1, 2, ..., 10 (0.33 ≤ *ςeq*<sup>2</sup> ≤ 0.56).

**Figure 38.** Illustrating impulse response *h*2(*t*) in t-ω plane for *m* = *c* = *k* = 1 with *t* = 0, 1, ..., 50; *ω* = 1, 2, ..., 5. (**a**) *β* = 0.3 (0.08 ≤ *ςeq*<sup>2</sup> ≤ 0.69). (**b**) *β* = 0.6 (0.22 ≤ *ςeq*<sup>2</sup> ≤ 0.63). (**c**) *β* = 0.9 (0.43 ≤ *ςeq*<sup>2</sup> ≤ 0.54). (**d**) *β* = 1 (*ςeq*<sup>2</sup> = 0.50).

**Note 7.2:** The impulse response *h*2(*t*) reduces to the conventional one if *β* = 1. As a matter of fact,

$$\ln\_{2}(t)|\_{\beta=1} = \left[ \frac{e^{\frac{\zeta\omega\_{0}\omega\beta^{-1}}{1-\frac{\beta}{\pi}\omega\beta^{-2}\cos\frac{\beta\pi}{2}}\frac{\beta\pi}{2}t}}{\omega\_{u}m\sqrt{1-\frac{\zeta^{2}\omega^{2}(\beta-1)\sin^{2}\frac{\beta\pi}{2}}{1-\frac{\beta}{\pi}\omega\beta^{-2}\cos\frac{\beta\pi}{2}}}}} \right] \tag{202}$$
 
$$\omega\_{u}m\sqrt{1-\frac{\zeta\omega\_{0}\omega\beta^{-2}\cos\frac{\beta\pi}{2}}{2}}\sqrt{1-\frac{\zeta^{2}\omega^{2}(\beta-1)\sin^{2}\frac{\beta\pi}{2}}{1-\frac{\beta}{\pi}\omega\beta^{-2}\cos\frac{\beta\pi}{2}}}} \right)\_{\beta=1} \tag{202}$$
  $\omega\_{u} = \frac{e^{-\zeta\omega\_{0}t}}{m\omega\_{u}\sqrt{1-\zeta^{2}}}\sin\omega\_{u}\sqrt{1-\zeta^{2}}t$ .
