*10.5. Sinusoidal Response to Fractional Oscillators in Class II*

**Theorem 23** (Sinusoidal response II)**.** *Denote by <sup>x</sup>*2*zs*(*t*) *the zero state sinusoidal response to a fractional oscillator in Class II. Then, for t > 0 and 0 < β* ≤ *1, it is expressed by*

*<sup>x</sup>*2*zs*(*t*) = 1 *<sup>m</sup>ωn* <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 \*2 *A* ) *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 −*ω*<sup>2</sup>\*<sup>2</sup>+) <sup>2</sup>*ςωnωβ* sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 \*2 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos *ωt* + <sup>2</sup>*ςωn<sup>ω</sup><sup>β</sup>* sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 sin *ωt* <sup>+</sup>*e*<sup>−</sup> *ςωnωβ*−<sup>1</sup> sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t*⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎡⎢⎢⎢⎣ *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos *<sup>ω</sup>n*"##\$<sup>1</sup>−) *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 \*2*t* <sup>1</sup>− *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎤⎥⎥⎥⎦ − *ςωβ*−<sup>1</sup> sin *βπ* 2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 ) *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 +*ω*2\* "###\$1−⎛⎝ *ςωβ*−<sup>1</sup> sin *βπ* 2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎞⎠2 sin *<sup>ω</sup>n*"##\$<sup>1</sup>−) *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 \*2*t* <sup>1</sup>− *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭. (270)

**Proof.** In the following expression,

$$\begin{aligned} \exp\_{2z}(t) &= \frac{1}{m\_{\text{eq}}\omega\_{\text{eq},2}} \frac{A}{\left(\omega\_{\text{eq},2}^2 - \omega^2\right)^2 + \left(2\zeta\_{\text{eq}}\omega\_{\text{eq},2}\omega\right)^2} \\\\ &\left\{ \begin{aligned} \left(\omega\_{\text{eq},2}^2 - \omega^2\right) &\cos\omega t + 2\zeta\_{\text{eq}}\omega\omega\_{\text{eq},2}\omega\nu\sin\omega t \\ &+ e^{-\zeta\_{\text{eq}}\omega\omega\_{\text{eq},2}t} \left[\left(\omega\_{\text{eq},2}^2 - \omega^2\right)\cos\omega\_{\text{eq},2}t - \frac{\zeta\_{\text{eq}}}{\sqrt{1-\zeta\_{\text{eq}}^2}} \left(\omega\_{\text{eq},2}^2 + \omega^2\right)\sin\omega\omega\_{\text{eq},2}t \right] \end{aligned} \right\}, \end{aligned} \tag{271}$$

replacing *meq*2 with the one in Section 4, *ςeq*2, *<sup>ω</sup>eqd*,<sup>2</sup> and *<sup>ω</sup>eqn*,<sup>2</sup> by those in Section 5, results in (270). The proof completes. 

The stead-state part of *<sup>x</sup>*2*zs*,*<sup>s</sup>*(*t*) is represented by

$$x\_{25,\delta}(t) = \frac{1}{m\omega\_n\sqrt{1-\frac{\epsilon}{\tilde{m}}\omega^{\delta-2}\cos\frac{\delta\pi}{2}}\sqrt{1-\frac{\zeta^2\omega^{2(\delta-1)}\sin^2\frac{\delta\pi}{2}}{1-\frac{\zeta^2\omega^{2(\delta-1)}\sin^2\frac{\delta\pi}{2}}{1-\frac{\zeta^2\omega^{2-2\delta}\cos\frac{\delta\pi}{2}}}}}}} $$

$$\frac{A}{\left(\frac{\omega\_n^2}{1-\frac{\omega}{\tilde{m}}\omega^{\delta-2}\cos\frac{\delta\pi}{2}}-\omega^2\right)^2+\left(\frac{2\omega\omega\omega^{\delta}\sin\frac{\delta\pi}{2}}{1-\frac{\zeta\omega^{\delta-2}}{\tilde{m}}\cos\frac{\delta\pi}{2}}\right)^2}\tag{272}$$

$$\left[\left(\frac{\omega\_n^2}{1-\frac{\zeta}{\tilde{m}}\omega^{\delta-2}\cos\frac{\delta\pi}{2}}-\omega^2\right)\cos\omega t + \frac{2\zeta\omega\omega\_n\omega^{\delta}\sin\frac{\delta\pi}{2}}{1-\frac{\zeta}{\tilde{m}}\omega^{\delta-2}\cos\frac{\delta\pi}{2}}\sin\omega t\right].$$

On the other side, the transient part *<sup>x</sup>*2*zs*,*tr*(*t*) is given by

*<sup>x</sup>*2*zs*,*tr*(*t*) = 1 *<sup>m</sup>ωn* <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 \* *A* ) *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 −*ω*<sup>2</sup>\*<sup>2</sup>+) <sup>2</sup>*ςωnωβ* sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 \*2 ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩*e*− *ςωnωβ*−<sup>1</sup> sin *βπ*2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 *t*⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎡⎢⎢⎢⎣ *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos *<sup>ω</sup>n*"##\$<sup>1</sup>−) *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>1</sup>− *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎤⎥⎥⎥⎦ − *ςωβ*−<sup>1</sup> sin *βπ* 2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 ) *ω*2*n* 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 +*ω*2\* "###\$1−⎛⎝ *ςωβ*−<sup>1</sup> sin *βπ* 2 1− *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎞⎠2 sin *<sup>ω</sup>n*"##\$<sup>1</sup>−) *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>1</sup>− *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭. (273)

Figures 61–63 show the plots of *<sup>x</sup>*2*zs*(*t*), *<sup>x</sup>*2*zs*,*<sup>s</sup>*(*t*), and *<sup>x</sup>*2*zs*,*tr*(*t*).

**Figure 61.** Sinusoidal response *<sup>x</sup>*2*zs*(*t*)to a fractional oscillator in Class II with *β* = 0.9 (solid line) (*ςeq*<sup>2</sup> = 0.14), *β* = 0.6 (dot line) (*ςeq*<sup>2</sup> = 0.07), *β* = 0.3 (dash dot line) (*ςeq*<sup>2</sup> = 0.03) with *m* = *c* = 1, *ωn* = 3 and *ω* = 1.

**Figure 62.** Steady-state sinusoidal part of *<sup>x</sup>*2*zs*(*t*) with *β* = 0.9 (solid line) (*ςeq*<sup>2</sup> = 0.14), *β* = 0.6 (dot line) (*ςeq*<sup>2</sup> = 0.07), *β* = 0.3 (dash dot line) (*ςeq*<sup>2</sup> = 0.03) with *m* = *c* = 1, *ωn* = 3 and *ω* = 1.

**Figure 63.** Transient part of *<sup>x</sup>*2*zs*(*t*) with *β* = 0.9 (solid line) (*ςeq*<sup>2</sup> = 0.14), *β* = 0.6 (dot line) (*ςeq*<sup>2</sup> = 0.07), *β* = 0.3 (dash dot line) (*ςeq*<sup>2</sup> = 0.03) with *m* = *c* = 1, *ωn* = 3 and *ω* = 1.

**Note 10.3:** If *β* = 1, we obtain the zero-state response of the conventional sinusoidal response to a 2-order oscillator in the form

$$\left. \exp\_{2z}(t) \right|\_{\beta=1} = \frac{1}{m\omega\_n\sqrt{1-\varsigma^2}} \frac{A e^{-\varsigma\omega\_n t} \left[ \left(\omega\_n^2 - \omega^2\right) \cos\omega\_n \sqrt{1-\varsigma^2} t - \frac{\varsigma\left(\omega\_n^2 + \omega^2\right) \sin\omega\_n \sqrt{1-\varsigma^2} t}{\sqrt{1-\varsigma^2}} \right]}{\left(\omega\_n^2 - \omega^2\right)^2 + \left(2\varsigma\omega\_n\omega\right)^2}. \tag{274}$$

**Remark 30.** *We discovered that the sinusoidal response to fractional oscillators in Class II for any value of β* ∈ *(0, 1) does have steady-state component <sup>x</sup>*2*zs*,*<sup>s</sup>*(*t*) *described by (272), also see Figure 62.*

*10.6. Sinusoidal Response to Fractional Oscillators in Class III*

**Theorem 24** (Sinusoidal response III)**.** *Let <sup>x</sup>*3*zs*(*t*) *be the zero state sinusoidal response to a fractional oscillator of Class III type. Then, for t > 0, 1 < α* ≤ *2, and 0 < β* ≤ *1, it is written in the form*

*<sup>x</sup>*3*zs*(*t*) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 *<sup>ω</sup>nmeq*3"####\$<sup>1</sup><sup>−</sup> *mωα*−1 sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 2 <sup>4</sup>− *mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 *k* ⎧ *A* ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*22 + *ω<sup>α</sup>* sin *απ*2 <sup>+</sup>2*ςωn<sup>ω</sup>β*−<sup>1</sup> sin *βπ*2 *ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 2 ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos *ωt* + *ω<sup>α</sup>* sin *απ*2 <sup>+</sup>2*ςωn<sup>ω</sup>β*−<sup>1</sup> sin *βπ*2 *ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 sin *ωt* ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ <sup>+</sup>*e*<sup>−</sup> *ωα*−<sup>1</sup> sin *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>(*ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 ) *t*⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos *<sup>ω</sup>eqd*,3*<sup>t</sup>* −⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ωα*−<sup>1</sup> sin *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>*ωn*!− *ωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>2</sup> cos *βπ*2 "#####\$1−⎡⎢⎢⎣ *ωα*−1 sin *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>*ωn*!− *ωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>2</sup> cos *βπ*2 ⎤⎥⎥⎦2 ) *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 + *ω*2\* ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ sin *<sup>ω</sup>eqd*,3*<sup>t</sup>* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭, (275)

*where meq3 and <sup>ω</sup>eqd, 3 are given by (119) and (160), respectively.*

**Proof.** In the following expression,

$$\chi\_{3x}(t) = \frac{1}{m\_{\text{eq}3}\omega\_{\text{eq}3}} \left\{ \frac{\left(\omega\_{\text{eq},3}^2 - \omega^2\right)\cos\omega t + 2\zeta\_{\text{eq}3}\omega\_{\text{eq},3}\omega\sin\omega t}{\left( + e^{-\zeta\_{\text{eq}}\omega\_{\text{eq},3}t} \left[ \begin{array}{c} \left(\omega\_{\text{eq},3}^2 - \omega^2\right)\cos\omega\_{\text{eq},3}t \\\\ -\frac{\zeta\_{\text{eq},3}}{\sqrt{1-\zeta\_{\text{eq},3}^2}} \left(\omega\_{\text{eq},3}^2 + \omega^2\right)\sin\omega\_{\text{eq},3}t \end{array} \right]}{\left(\omega\_{\text{eq},3}^2 - \omega^2\right)^2 + \left(2\zeta\_{\text{eq}}\omega\_{\text{eq},3}\omega\right)^2} \right\},\tag{276}$$

Substituting *ςeq*3, *<sup>ω</sup>eqn*,<sup>3</sup> and *<sup>ω</sup>eqd*,<sup>3</sup> with those in Section 5 yields the Theorem 24. That completes the proof. 

Figure 64 illustrates *<sup>x</sup>*3*zs*(*t*). **Figure 64.** Indicating the sinusoidal response *<sup>x</sup>*3*zs*(*t*) to a fractional oscillator in Class III with (*<sup>α</sup>*, *β*) = (1.8, 0.8) (solid line) (*ςeq*<sup>3</sup> = 0.13), (*<sup>α</sup>*, *β*) = (1.5, 0.8) (dot line) (*ςeq*<sup>3</sup> = 0.22), (*<sup>α</sup>*, *β*) = (1.3, 0.8) (dash dot line) (*ςeq*<sup>3</sup>= 0.40) with *m* = *c* = 1, *k* = 36 (*<sup>ω</sup>n* = 6) and *ω* = 1.

The steady-state part of *<sup>x</sup>*3*zs*(*t*) is in the form

*<sup>x</sup>*3*zs*,*<sup>s</sup>*(*t*) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 *<sup>ω</sup>nmeq*3"####\$<sup>1</sup><sup>−</sup> *mωα*−1 sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 2 <sup>4</sup>− *mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 *k* ⎧ *A* ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*22 + *ω<sup>α</sup>* sin *απ*2 <sup>+</sup>2*ςωn<sup>ω</sup>β*−<sup>1</sup> sin *βπ*2 *ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 2 ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos *ωt* +*ω<sup>α</sup>* sin *απ*2 <sup>+</sup>2*ςωn<sup>ω</sup>β*−<sup>1</sup> sin *βπ*2 sin *ωt ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭. (277)

Its transient part, taking into account (160), is given by

*<sup>x</sup>*3*zs*,*tr*(*t*) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 *<sup>ω</sup>nmeq*3"####\$<sup>1</sup><sup>−</sup> *mωα*−1 sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 2 <sup>4</sup>− *mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 *k* ⎧ *A* ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*22 + *ω<sup>α</sup>* sin *απ*2 <sup>+</sup>2*ςωn<sup>ω</sup>β*−<sup>1</sup> sin *βπ*2 *ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 2 ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩*e*− *ωα*−<sup>1</sup> sin *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>(*ωα*−<sup>2</sup>|cos *απ*2 |−<sup>2</sup>*ςωnωβ*−<sup>2</sup> cos *βπ*2 ) *t*⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 − *ω*2 cos⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ *ωn* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 "##\$1 − *mωα*−<sup>1</sup> sin *απ*2 +*cωβ*−<sup>1</sup> sin *βπ*2 2 <sup>4</sup>−*mωα*−<sup>2</sup> cos *απ*2 +*cωβ*−<sup>2</sup> cos *βπ*2 *k t* ⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭ −⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ωα*−<sup>1</sup> sin *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>*ωn*!− *ωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>2</sup> cos *βπ*2 "#####\$1−⎡⎢⎢⎣ *ωα*−1 sin *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>1</sup> sin *βπ*2 <sup>2</sup>*ωn*!− *ωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>2*ςωnωβ*−<sup>2</sup> cos *βπ*2 ⎤⎥⎥⎦2 ) *ω*2*n* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 + *ω*2\* ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ sin⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩*ωn*"####\$1− *mωα*−1 sin *απ*2 <sup>+</sup>*cωβ*−<sup>1</sup> sin *βπ*2 2 <sup>4</sup>− *mωα*−<sup>2</sup> cos *απ*2 <sup>+</sup>*cωβ*−<sup>2</sup> cos *βπ*2 *k t* −*ωα*−<sup>2</sup> cos *απ*2 + *cm ωβ*−<sup>2</sup> cos *βπ*2 ⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭. (278)

The steady-state component and the transient one of *<sup>x</sup>*3*zs*(*t*) are shown in Figures 65 and 66, respectively.

**Figure 65.** Indicating the steady-state component of *<sup>x</sup>*3*zs*(*t*) with (*<sup>α</sup>*, *β*) = (1.8, 0.8) (solid line) (*ςeq*<sup>3</sup> = 0.13), (*<sup>α</sup>*, *β*) = (1.5, 0.8) (dot line) (*ςeq*<sup>3</sup> = 0.22), (*<sup>α</sup>*, *β*) = (1.3, 0.8) (dash dot line) (*ςeq*<sup>3</sup> = 0.40) with *m* = *c* = 1, *k* = 36 (*<sup>ω</sup>n* = 6) and *ω* = 1.

**Figure 66.** Transient component of *<sup>x</sup>*3*zs*(*t*) with (*<sup>α</sup>*, *β*) = (1.8, 0.8) (solid line) (*ςeq*<sup>3</sup> = 0.13), (*<sup>α</sup>*, *β*) = (1.5, 0.8) (dot line) (*ςeq*<sup>3</sup> = 0.22), (*<sup>α</sup>*, *β*) = (1.3, 0.8) (dash dot line) (*ςeq*<sup>3</sup> = 0.40) with *m* = *c* = 1, *k* = 36 (*<sup>ω</sup>n* = 6) and *ω* = 1.

**Note 10.4:** When (*<sup>α</sup>*, *β*) = (2, 1), *<sup>x</sup>*3*zs*(*t*) reduces to the ordinary zero-state sinusoidal response to a 2-order oscillator in the form

$$\begin{split} \left. \exp\_{3z\varepsilon}(t) \right|\_{\left[ \alpha = 2, \tilde{\rho} = 1 \right]} &= \frac{\frac{A}{m\omega\iota\sqrt{1-\varsigma^{2}}} e^{-\varsigma\omega\_{\mathrm{n}}t} \left[ \left( \omega\_{\mathrm{n}}^{2} - \omega^{2} \right) \cos\omega\_{\mathrm{n}} \sqrt{1-\varsigma^{2}} t - \frac{\varsigma\left( \omega\_{\mathrm{n}}^{2} + \omega^{2} \right) \sin\omega\_{\mathrm{n}} \sqrt{1-\varsigma^{2}} t}{\sqrt{1-\varsigma^{2}}} \right]}{\left( \omega\_{\mathrm{n}}^{2} - \omega^{2} \right)^{2} + \left( 2\zeta\omega\_{\mathrm{n}}\omega \right)^{2}} \\ &= \frac{A e^{-\varsigma\omega\_{\mathrm{n}}t} \left[ \left( \omega\_{\mathrm{n}}^{2} - \omega^{2} \right) \cos\omega\_{\mathrm{d}} t - \frac{\varsigma\left( \omega\_{\mathrm{n}}^{2} + \omega^{2} \right) \sin\omega\_{\mathrm{d}} t}{\sqrt{1-\varsigma^{2}}} \right]}{\left( \omega\_{\mathrm{n}}^{2} - \omega^{2} \right)^{2} + \left( 2\zeta\omega\_{\mathrm{n}}\omega \right)^{2}}. \end{split} \tag{279}$$

**Remark 31.** *We revealed that the sinusoidal response to fractional oscillators in Class III for any value of α* ∈ *(1, 2) and β* ∈ *(0, 1) does have steady-state component <sup>x</sup>*3*zs*,*<sup>s</sup>*(*t*) *described by (277), also see Figure 65.*

**Remark 32.** *The results presented above show that the exact periodic solutions to three classes of fractional oscillators exist.*
